Math 3
Washington State K-12
Mathematics Learning Standards
Prepared by
Charles A. Dana Center for Science and Mathematics Education
The University of Texas at Austin
Mathematics Teaching and Learning
Office of Superintendent of Public Instruction
Dr. Terry Bergeson
Superintendent of Public Instruction
Catherine Davidson, Ed. D.
Chief of Staff
Lexie Domaradzki
Assistant Superintendent, Teaching and Learning
July 2008
Math 3
SUPERINTENDENT OF PUBLIC INSTRUCTION
DR. TERRY BERGESON OLD CAPITOL BUILDING • PO BOX 47200 • OLYMPIA WA 98504-7200 • http://www.k12.wa.us
It is with great pride that I, Dr. Terry Bergeson, State Superintendent of Public Instruction officially adopt the
revised K-12 Mathematics Standards as the new essential academic learning requirements for the state of
Washington.
Teams of national experts and talented Washington state mathematics educators, curriculum directors, and
mathematicians have worked tirelessly since October 2007 to develop the best set of K-12 mathematics
standards for our state. Since the announcement of the first draft in December 2007, these standards have
received input from thousands of educators and stakeholders throughout the state, including in-depth input
from the State Board of Education’s Math Panel.
As per 2008 Senate Bill 6534, the State Board of Education (SBE) contracted with an independent contractor,
Strategic Teaching to conduct a final review and analysis of the K-12 standards. On April 28, 2008, the SBE
voted to approve the K-8 standards for adoption by the Office of Superintendent of Public Instruction (OSPI).
The 9-12 standards were approved for OSPI adoption on July 30, 2008.
These standards set more challenging and rigorous expectations at each grade level. In addition, they provide
more clarity to support all students in developing and sharpening their mathematical skills, deepening their
understanding of concepts and processes, and utilizing their problem-solving, reasoning and communication
abilities. For students to develop this deeper level of understanding, their knowledge must be connected
not only to a variety of ideas and skills across topic areas and grade levels in mathematics, but also to other
subjects taught in school and to situations outside the classroom.
The revised K-12 Mathematics Standards are the first step in improving the mathematics learning of all
students in Washington and are now at the vanguard of the nation’s mathematics education improvement
movement. The standards will strongly support teachers as they prepare the state’s young people for
graduation, college and the workforce.
K-12 Mathematics Standards adopted on this 1st day of August, 2008 by
Sincerely,
Dr. Terry Bergeson, Superintendent
Office of Superintendent of Public Instruction
Math 3
Table of Contents
Introduction..................................................................................................................................i
Kindergarten...............................................................................................................................1
K.1. Core Content: Whole numbers.............................................................................................................3
K.2. Core Content: Patterns and operations................................................................................................5
K.3. Core Content: Objects and their locations . .........................................................................................6
K.4. Additional Key Content ........................................................................................................................7
K.5. Core Processes: Reasoning, problem solving, and communication....................................................8
Grade 1........................................................................................................................................9
1.1. Core Content: Whole number relationships ....................................................................................... 11
1.2. Core Content: Addition and subtraction..............................................................................................14
1.3. Core Content: Geometric attributes....................................................................................................17
1.4. Core Content: Concepts of measurement..........................................................................................18
1.5. Additional Key Content .......................................................................................................................19
1.6. Core Processes: Reasoning, problem solving, and communication..................................................20
Grade 2......................................................................................................................................21
2.1. Core Content: Place value and the base ten system..........................................................................23
2.2. Core Content: Addition and subtraction..............................................................................................24
2.3. Core Content: Measurement...............................................................................................................26
2.4. Additional Key Content .......................................................................................................................27
2.5. Core Processes: Reasoning, problem solving, and communication..................................................29
Grade 3......................................................................................................................................31
3.1. Core Content: Addition, subtraction, and place value.........................................................................33
3.2. Core Content: Concepts of multiplication and division ......................................................................34
3.3. Core Content: Fraction concepts........................................................................................................38
3.4. Core Content: Geometry ....................................................................................................................40
3.5. Additional Key Content .......................................................................................................................41
3.6. Core Processes: Reasoning, problem solving, and communication..................................................42
Grade 4......................................................................................................................................43
4.1. Core Content: Multi-digit multiplication................................................................................................45
4.2. Core Content: Fractions, decimals, and mixed numbers ..................................................................48
4.3. Core Content: Concept of area ..........................................................................................................51
4.4. Additional Key Content........................................................................................................................53
4.5. Core Processes: Reasoning, problem solving, and communication..................................................55
Grade 5......................................................................................................................................57
5.1. Core Content: Multi-digit division.........................................................................................................59
5.2. Core Content: Addition and subtraction of fractions and decimals.....................................................61
5.3. Core Content: Triangles and quadrilaterals........................................................................................63
5.4. Core Content: Representations of algebraic relationships.................................................................65
5.5. Additional Key Content .......................................................................................................................67
5.6. Core Processes: Reasoning, problem solving, and communication..................................................68
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Washington State K–12 Mathematics Standards
Grade 6......................................................................................................................................69
6.1. Core Content: Multiplication and division of fractions and decimals...................................................71
6.2. Core Content: Mathematical expressions and equations ..................................................................74
6.3. Core Content: Ratios, rates, and percents..........................................................................................76
6.4. Core Content: Two- and three-dimensional figures............................................................................78
6.5. Additional Key Content........................................................................................................................80
6.6. Core Processes: Reasoning, problem solving, and communication..................................................81
Grade 7......................................................................................................................................83
7.1. Core Content: Rational numbers and linear equations.......................................................................85
7.2. Core Content: Proportionality and similarity........................................................................................88
7.3. Core Content: Surface area and volume............................................................................................92
7.4. Core Content: Probability and data.....................................................................................................93
7.5. Additional Key Content .......................................................................................................................95
7.6. Core Processes: Reasoning, problem solving, and communication..................................................96
Grade 8......................................................................................................................................97
8.1. Core Content: Linear functions and equations ..................................................................................99
8.2. Core Content: Properties of geometric figures ................................................................................101
8.3. Core Content: Summary and analysis of data sets .........................................................................103
8.4. Additional Key Content......................................................................................................................107
8.5. Core Processes: Reasoning, problem solving, and communication................................................109
Algebra 1................................................................................................................................. 111
A1.1. Core Content: Solving problems . .................................................................................................. 113
A1.2. Core Content: Numbers, expressions, and operations ................................................................. 116
A1.3. Core Content: Characteristics and behaviors of functions.............................................................120
A1.4. Core Content: Linear functions, equations, and inequalities..........................................................123
A1.5. Core Content: Quadratic functions and equations.........................................................................126
A1.6. Core Content: Data and distributions.............................................................................................128
A1.7. Additional Key Content ..................................................................................................................131
A1.8. Core Processes: Reasoning, problem solving, and communication..............................................132
Geometry.................................................................................................................................133
G.1. Core Content: Logical arguments and proofs .................................................................................135
G.2. Core Content: Lines and angles . ....................................................................................................137
G.3. Core Content: Two- and three-dimensional figures.........................................................................138
G.4. Core Content: Geometry in the coordinate plane............................................................................143
G.5. Core Content: Geometric transformations ......................................................................................145
G.6. Additional Key Content ....................................................................................................................146
G.7. Core Processes: Reasoning, problem solving, and communication...............................................149
Algebra 2.................................................................................................................................151
A2.1. Core Content: Solving problems.....................................................................................................153
A2.2. Core Content: Numbers, expressions, and operations..................................................................157
A2.3. Core Content: Quadratic functions and equations.........................................................................159
A2.4. Core Content: Exponential and logarithmic functions and equations............................................161
A2.5. Core Content: Additional functions and equations.........................................................................163
A2.6. Core Content: Probability, data, and distributions......................................................................... 165
A2.7. Additional Key Content....................................................................................................................167
A2.8. Core Processes: Reasoning, problem solving, and communication..............................................168
July 2008
Washington State K–12 Mathematics Standards
Mathematics 1.........................................................................................................................171
M1.1. Core Content: Solving problems ...................................................................................................173
M1.2. Core Content: Characteristics and behaviors of functions............................................................176
M1.3. Core Content: Linear functions, equations, and relationships...................................................... 179
M1.4. Core Content: Proportionality, similarity, and geometric reasoning...............................................183
M1.5. Core Content: Data and distributions.............................................................................................185
M1.6. Core Content: Numbers, expressions, and operations..................................................................187
M1.7. Additional Key Content...................................................................................................................190
M1.8. Core Processes: Reasoning, problem solving, and communication.............................................192
Mathematics 2.........................................................................................................................193
M2.1. Core Content: Modeling situations and solving problems.............................................................195
M2.2. Core Content: Quadratic functions, equations, and relationships.................................................198
M2.3. Core Content: Conjectures and proofs .........................................................................................202
M2.4. Core Content: Probability ............................................................................................................. 208
M2.5. Additional Key Content...................................................................................................................209
M2.6. Core Processes: Reasoning, problem solving, and communication............................................. 211
Mathematics 3.........................................................................................................................213
M3.1. Core Content: Solving problems....................................................................................................215
M3.2. Core Content: Transformations and functions...............................................................................218
M3.3. Core Content: Functions and modeling.........................................................................................220
M3.4. Core Content: Quantifying variability..............................................................................................223
M3.5. Core Content: Three-dimensional geometry . ...............................................................................225
M3.6. Core Content: Algebraic properties................................................................................................227
M3.7. Additional Key Content...................................................................................................................229
M3.8. Core Processes: Reasoning, problem solving, and communication.............................................231
July 2008
Washington State K–12 Mathematics Standards
Introduction
Overview
The Washington State K–12 Mathematics Standards outline the mathematics learning expectations for
all students in Washington. These standards describe the mathematics content, procedures, applications,
and processes that students are expected to learn. The topics and mathematical strands represented
across grades K–12 constitute a mathematically complete program that includes the study of numbers,
operations, geometry, measurement, algebra, data analysis, and important mathematical processes.
Organization of the standards
The Washington State K–12 Mathematics Standards are organized by grade level for grades K–8 and by
course for grades 9–12, with each grade/course consisting of three elements: Core Content, Additional
Key Content, and Core Processes. Each of these elements contains Performance Expectations and
Explanatory Comments and Examples.
Core Content areas describe the major mathematical focuses of each grade level or course. A limited
number of priorities for each grade level in grades K–8 and for each high school course are identified, so
teachers know which topics call for the most time and emphasis. Each priority area includes a descriptive
paragraph that highlights the mathematics addressed and its role in a student’s overall mathematics
learning.
Additional Key Content contains important expectations that do not warrant the same amount of
instructional time as the Core Content areas. These are expectations that might extend a previously
learned skill, plant a seed for future development, or address a focused topic, such as scientific notation.
Although they need less classroom time, these expectations are important, are expected to be taught, and
may be assessed as part of Washington State’s assessment system. The content in this section allows
students to build a coherent knowledge of mathematics from year to year.
Core Processes include expectations that address reasoning, problem solving, and communication.
While these processes are incorporated throughout other content expectations, they are presented in
this section to clearly describe the breadth and scope of what is expected in each grade or course. In
Core Processes, at least two rich problems that cut across Core or Key Content areas are included as
examples for each grade or course. These problems illustrate the types and breadth of problems that
could be used in the classroom.
Performance Expectations, in keeping with the accepted definition of standards, describe what students
should know and be able to do at each grade level. These statements are the core of the document.
They are designed to provide clear guidance to teachers about the mathematics that is to be taught and
learned.
Explanatory Comments and Examples accompany most of the expectations. These are not technically
performance expectations. However, taken together with the Performance Expectations, they provide a
full context and clear understanding of the expectation.
The comments expand upon the meaning of the expectations. Explanatory text might clarify the
parameters regarding the type or size of numbers, provide more information about student expectations
regarding mathematical understanding, or give expanded detail to mathematical definitions, laws,
principles, and forms included in the expectation.
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Washington State K–12 Mathematics Standards
i
The example problems include those that are typical of the problems students should do, those that
illustrate various types of problems associated with a particular performance expectation, and those that
illustrate the expected limits of difficulty for problems related to a performance expectation. Teachers are
not expected to teach these particular examples or to limit what they teach to these examples. Teachers
and quality instructional materials will incorporate many different types of examples that support the
teaching of the content described in any expectation.
In some instances, comments related to pedagogy are included in the standards as familiar illustrations to
the teacher. Teachers are not expected to use these particular teaching methods or to limit the methods
they use to the methods included in the document. These, too, are illustrative, showing one way an
expectation might be taught.
Although, technically, the performance expectations set the requirements for Washington students, people
will consider the entire document as the Washington mathematics standards. Thus, the term standards,
as used here, refers to the complete set of Performance Expectations, Explanatory Comments and
Examples, Core Content, Additional Key Content, and Core Processes. Making sense of the standards
from any grade level or course calls for understanding the interplay of Core Content, Additional Key
Content, and Core Processes for that grade or course.
What standards are not
Performance expectations do not describe how the mathematics will be taught. Decisions about
instructional methods and materials are left to professional teachers who are knowledgeable about the
mathematics being taught and about the needs of their students.
The standards are not comprehensive. They do not describe everything that could be taught in a
classroom. Teachers may choose to go beyond what is included in this document to provide related
or supporting content. They should teach beyond the standards to those students ready for additional
challenges. Standards related to number skills, in particular, should be viewed as a floor—minimum
expectations—and not a ceiling. A student who can order and compare numbers to 120 should be given
every opportunity to apply these concepts to larger numbers.
The standards are not test specifications. Excessive detail, such as the size of numbers that can be tested
and the conditions for assessment, clouds the clarity and usability of a standards document, generally,
and a performance expectation, specifically. For example, it is sufficient to say “Identify, describe, and
classify triangles by angle measure and number of congruent sides,” without specifying that acute, right,
and obtuse are types of triangles classified by their angle size and that scalene, isosceles, and equilateral
are types of triangles classified by their side length. Sometimes this type of information is included in the
comments section, but generally this level of detail is left to other documents.
What about strands?
Many states’ standards are organized around mathematical content strands—generally some combination
of numbers, operations, geometry, measurement, algebra, and data/statistics. However, the Washington
State K–12 Mathematics Standards are organized according to the priorities described as Core Content
rather than being organized in strands. Nevertheless, it is still useful to know what content strands are
addressed in particular Core Content and Additional Key Content areas. Thus, mathematics content
strands are identified in parentheses at the beginning of each Core Content or Additional Key Content
area. Five content strands have been identified for this purpose: Numbers, Operations, Geometry/
Measurement, Algebra, and Data/Statistics/Probability. For each of these strands, a separate K–12 strand
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Washington State K–12 Mathematics Standards
document allows teachers and other readers to track the development of knowledge and skills across
grades and courses. An additional strand document on the Core Processes tracks the development of
reasoning, problem solving, and communication across grades K–12.
A well-balanced mathematics program for all students
An effective mathematics program balances three important components of mathematics—conceptual
understanding (making sense of mathematics), procedural proficiency (skills, facts, and procedures),
and problem solving and mathematical processes (using mathematics to reason, think, and apply
mathematical knowledge). These standards make clear the importance of all three of these components,
purposefully interwoven to support students’ development as increasingly sophisticated mathematical
thinkers. The standards are written to support the development of students so that they know and
understand mathematics.
Conceptual understanding (making sense of mathematics)
Students who understand a concept are able to identify examples as well as non-examples, describe
the concept (for example, with words, symbols, drawings, tables, or models), provide a definition of
the concept, and use the concept in different ways. Conceptual understanding is woven throughout
these standards. Expectations with verbs like demonstrate, describe, represent, connect, and justify,
for example, ask students to show their understanding. Furthermore, expectations addressing both
procedures and applications often ask students to connect their conceptual understanding to the
procedures being learned or problems being solved.
Procedural proficiency (skills, facts, and procedures)
Learning basic facts is important for developing mathematical understanding. In these standards, clear
expectations address students’ knowledge of basic facts. The use of the term basic facts typically
encompasses addition and multiplication facts up to and including 10 + 10 and 10 x 10 and their related
subtraction and division facts. In these standards, students are expected to “quickly recall” basic facts.
“Quickly recall” means that the student has ready and effective access to facts without having to go
through a development process or strategy, such as counting up or drawing a picture, every time he or
she needs to know a fact. Simply put, students need to know their basic facts.
Building on a sound conceptual understanding of addition, subtraction, multiplication, and division,
Washington’s standards include a specific discussion of students’ need to understand and use the
standard algorithms generally seen in the United States to add, subtract, multiply, and divide whole
numbers. There are other possible algorithms students might also use to perform these operations and
some teachers may find value in students learning multiple algorithms to enhance understanding.
Algorithms are step-by-step mathematical procedures that, if followed correctly, always produce a correct
solution or answer. Generalized procedures are used throughout mathematics, such as in drawing
geometric constructions or going through the steps involved in solving an algebraic equation. Students
should come to understand that mathematical procedures are a useful and important part of mathematics.
The term fluency is used in these standards to describe the expected level and depth of a student’s
knowledge of a computational procedure. For the purposes of these standards, a student is considered
fluent when the procedure can be performed immediately and accurately. Also, when fluent, the student
knows when it is appropriate to use a particular procedure in a problem or situation. A student who is
fluent in a procedure has a tool that can be applied reflexively and doesn’t distract from the task of solving
the problem at hand. The procedure is stored in long-term memory, leaving working memory available to
focus on the problem.
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Washington State K–12 Mathematics Standards
iii
Problem solving and mathematical processes (reasoning and thinking to apply mathematical
content)
Mathematical processes, including reasoning, problem solving, and communication, are essential
in a well-balanced mathematics program. Students must be able to reason, solve problems, and
communicate their understanding in effective ways. While it is impossible to completely separate
processes and content, the standards’ explicit description of processes at each grade level calls attention
to their importance within a well-balanced mathematics program. Some common language is used to
describe the Core Processes across the grades and within grade bands (K–2, 3–5, 6–8, and 9–12). The
problems students will address, as well as the language and symbolism they will use to communicate
their mathematical understanding, become more sophisticated from grade to grade. These shifts across
the grades reflect the increasing complexity of content and the increasing rigor as students deal with
more challenging problems, much in the same way that reading skills develop from grade to grade with
increasingly complex reading material.
Technology
The role of technology in learning mathematics is a complex issue, because of the ever-changing
capabilities of technological tools, differing beliefs in the contributions of technology to a student’s
education, and equitable student access to tools. However, one principle remains constant: The focus
of mathematics instruction should always be on the mathematics to be learned and on helping students
learn that mathematics.
Technology should be used when it supports the mathematics to be learned, and technology should not
be used when it might interfere with learning.
Calculators and other technological tools, such as computer algebra systems, dynamic geometry
software, applets, spreadsheets, and interactive presentation devices are an important part of today’s
classroom. But the use of technology cannot replace conceptual understanding, computational fluency, or
problem-solving skills.
Washington’s standards make clear that some performance expectations are to be done without the aid of
technology. Elementary students are expected to know facts and basic computational procedures without
using a calculator. At the secondary level, students should compute with polynomials, solve equations,
sketch simple graphs, and perform some constructions without the use of technology. Students should
continue to use previously learned facts and skills in subsequent grade levels to maintain their fluency
without the assistance of a calculator.
At the elementary level, calculators are less useful than they will be in later grades. The core of
elementary school—number sense and computational fluency—does not require a calculator. However,
this is not to say that students couldn’t use calculators to investigate mathematical situations and to solve
problems involving complicated numbers, lots of numbers, or data sets.
As middle school students deal with increasingly complex statistical data and represent proportional
relationships with graphs and tables, a calculator or technological tool with these functions can be useful
for representing relationships in multiple ways. At the high school level, graphing calculators become
valuable tools as all students tackle the challenges of algebra and geometry to prepare for a range of
postsecondary options in a technological world. Graphing calculators and spreadsheets allow students to
explore and solve problems with classes of functions in ways that were previously impossible.
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Washington State K–12 Mathematics Standards
While the majority of performance expectations describe skills and knowledge that a student could
demonstrate without technology, learning when it is helpful to use these tools and when it is cumbersome
is part of becoming mathematically literate. When students become dependent upon technology to solve
basic math problems, the focus of mathematics instruction to help students learn mathematics has failed.
Connecting to the Washington Essential Academic Learning Requirements
(EALRs) and Grade Level Expectations (GLEs)
The new Washington State K–12 Mathematics Standards continue Washington’s longstanding
commitment to teaching mathematics content and mathematical thinking. The new standards replace
the former Essential Academic Learning Requirements (EALRs) and Grade Level Expectations (GLEs).
The former mathematics EALRs, listed below, represent threads in the mathematical content, reasoning,
problem solving, and communication that are reflected in these new standards.
EALR 1: The student understands and applies the concepts and procedures of
mathematics.
EALR 2: The student uses mathematics to define and solve problems.
EALR 3: The student uses mathematical reasoning.
EALR 4: The student communicates knowledge and understanding in both everyday
and mathematical language.
EALR 5: The student understands how mathematical ideas connect within
mathematics, to other subjects.
System-wide standards implementation activities
These mathematics standards represent an important step in ramping up mathematics teaching and
learning in the state. The standards provide a critical foundation, but are only the first step. Their success
will depend on the implementation efforts that match many of the activities outlined in Washington’s Joint
Mathematics Action Plan. This includes attention to:
•
Aligning the Washington Assessment for Student Learning to these standards;
•
Identifying mathematics curriculum and instructional support materials;
•
Providing systematic professional development so that instruction aligns with the standards;
•
Developing online availability of the standards in various forms and formats, with additional
example problems, classroom activities, and possible lessons embedded.
As with any comprehensive initiative, fully implementing these standards will not occur overnight. This
implementation process will take time, as teachers become more familiar with the standards and as
students enter each grade having learned more of the standards from previous grades. There is always
a tension of balancing the need to raise the bar with the reality of how much change is possible, and how
quickly this change can be implemented in real schools with real teachers and real students.
Change is hard. These standards expect more of students and more of their teachers. Still, if
Washington’s students are to be prepared to be competitive and to reach their highest potential,
implementing these standards will pay off for years to come.
July 2008
Washington State K–12 Mathematics Standards
v
Kindergarten
Kindergarten
Kindergarten
K.1. Core Content: Whole numbers
(Numbers, Operations)
S
tudents begin to develop basic notions of numbers and use numbers to think about objects and
the world around them. They practice counting objects in sets, and they think about how numbers
are ordered by showing the numbers on the number line. As they put together and take apart simple
numbers, students lay the groundwork for learning how to add and subtract. Understanding numbers is
perhaps the most central idea in all of mathematics, and if students build and maintain a strong foundation
of number sense and number skills, they will be able to succeed with increasingly sophisticated numerical
knowledge and skills from year to year.
Performance Expectations
Explanatory Comments and Examples
Students are expected to:
K.1.A
Rote count by ones forward from 1 to 100 and
backward from any number in the range of 10
to 1.
K.1.B
Read aloud numerals from 0 to 31.
Shown numeral cards in random order from 0 to
31, students respond with the correct name of the
numerals. Students also demonstrate that they can
distinguish 12 from 21 and 13 from 31—a common
challenge for kindergartners.
The choice of 31 corresponds to the common use of
calendar activities in kindergarten.
K.1.C
Fluently compose and decompose
numbers to 5.
Students should be able to state that 5 is made up
of 4 and 1, 3 and 2, 2 and 3, or 1 and 4. They should
understand that if I have 3, I need 2 more to make
5, or that if I have 4, I need only 1 more to make 5.
Students should also be able to recognize the number
of missing objects without counting.
The words compose and decompose are used to
describe actions that young students learn as they
acquire knowledge of small numbers by putting them
together and taking them apart. This understanding
is a bridge between counting and knowing number
combinations. It is how instant recognition of small
numbers develops and leads naturally to later
understanding of fact families.
Example:
•
K.1.D
Order numerals from 1 to 10.
July 2008
Washington State K–12 Mathematics Standards
Here are 5 counters. I will hide some. If you see 2,
how many am I hiding?
The student takes numeral cards (1 to 10) that have
been shuffled and puts them in correct ascending order.
3
Kindergarten
Performance Expectations
Explanatory Comments and Examples
Students are expected to:
4
K.1.E
Count objects in a set of up to 20, and count
out a specific number of up to 20 objects from
a larger set.
K.1.F
Compare two sets of up to 10 objects each
and say whether the number of objects in one
set is equal to, greater than, or less than the
number of objects in the other set.
K.1.G Locate numbers from 1 to 31 on the number line.
Students should be able to do this without having to
start counting at 1.
K.1.H Describe a number from 1 to 9 using 5 as a
benchmark number.
Students should make observations such as
“7 is 2 more than 5” or “4 is 1 less than 5.” This is
helpful for mental math and lays the groundwork for
using 10 as a benchmark number in later work with
base-ten numbers and operations.
July 2008
Washington State K–12 Mathematics Standards
Kindergarten
Kindergarten
K.2. Core Content: Patterns and operations
(Operations, Algebra)
S
tudents learn what it means to add and subtract by joining and separating sets of objects. Working
with patterns helps them strengthen this understanding of addition and subtraction and moves them
toward the important development of algebraic thinking. Students study simple repetitive patterns in
preparation for increasingly sophisticated patterns that can be represented with algebraic expressions in
later grades.
Performance Expectations
Explanatory Comments and Examples
Students are expected to:
K.2.A
Copy, extend, describe, and create simple
repetitive patterns.
Students can complete these activities with specified
patterns of the type AB, AAB, AABB, ABC, etc.
Examples:
•
Make a type AB pattern of squares and circles with
one square, one circle, one square, one circle, etc.
•
Here is a type AAB pattern using colored cubes: red,
red, blue, red, red, blue, red, red. What comes next?
•
A shape is missing in the type AB pattern below.
What is it?
K.2.B
Translate a pattern among sounds, symbols,
movements, and physical objects.
Red, red, yellow, red, red, yellow could translate to
clap, clap, snap, clap, clap, snap.
Students should be able to translate patterns among
all of these representations. However, when they have
demonstrated they can do this, they need not use all
representations every time.
K.2.C Model addition by joining sets of objects that
have 10 or fewer total objects when joined and
model subtraction by separating a set of 10 or
fewer objects.
Seeing two sets of counters or other objects, the
student determines the correct combined total. The
student may count the total number of objects in the set
or use some other strategy in order to arrive at the sum.
The student establishes the total number of counters or
objects in a set; then, after some have been removed,
the student figures out how many are left.
Examples:
K.2.D
Describe a situation that involves the actions
of joining (addition) or separating (subtraction)
using words, pictures, objects, or numbers.
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Washington State K–12 Mathematics Standards
•
Get 4 counting chips. Now get 3 counting chips.
How many counting chips are there altogether?
•
Get 8 counting chips. Take 2 away. How many
are left?
Students can be asked to tell an addition story or a
subtraction story.
5
Kindergarten
Kindergarten
K.3. Core Content: Objects and their locations (Geometry/Measurement)
S
tudents develop basic ideas related to geometry as they name simple two- and three-dimensional
figures and find these shapes around them. They expand their understanding of space and location
by describing where people and objects are. Students sort and match shapes as they begin to develop
classification skills that serve them well in both mathematics and reading—matching numbers to sets,
shapes to names, patterns to rules, letters to sounds, and so on.
Performance Expectations
Explanatory Comments and Examples
Students are expected to:
6
K.3.A
Identify, name, and describe circles, triangles,
rectangles, squares (as special rectangles),
cubes, and spheres.
Students should be encouraged to talk about the
characteristics (e.g., round, four-cornered) of the
various shapes and to identify these shapes in a
variety of contexts regardless of their location, size,
and orientation. Having students identify these shapes
on the playground, in the classroom, and on clothing
develops their ability to generalize the characteristics
of each shape.
K.3.B
Sort shapes using a sorting rule and explain
the sorting rule.
Students could sort shapes according to attributes
such as the shape, size, or the number of sides, and
explain the sorting rule. Given a selection of shapes,
students may be asked to sort them into two piles and
then describe the sorting rule. After sorting, a student
could say, “I put all the round ones here and all the
others there.”
K.3.C
Describe the location of one object relative to
another object using words such as in, out,
over, under, above, below, between, next to,
behind, and in front of.
Examples:
•
Put this pencil under the paper.
•
I am between José and Katy.
July 2008
Washington State K–12 Mathematics Standards
Kindergarten
Kindergarten
K.4. Additional Key Content S
(Geometry/Measurement)
tudents informally develop early measurement concepts. This is an important precursor to Core
Content on measurement in later grades, when students measure objects with tools. Solving
measurement problems connects directly to the student’s world and is a basic component of learning
mathematics.
Performance Expectations
Explanatory Comments and Examples
Students are expected to:
K.4.A Make direct comparisons using measurable
attributes such as length, weight, and capacity.
July 2008
Washington State K–12 Mathematics Standards
Students should use language such as longer than,
shorter than, taller than, heavier than, lighter than,
holds more than, or holds less than.
7
Kindergarten
Kindergarten
K.5. Core Processes: Reasoning, problem solving, and communication
S
tudents begin to build the understanding that doing mathematics involves solving problems and
discussing how they solved them. Problems at this level emphasize counting and activities that
lead to emerging ideas about addition and subtraction. Students begin to develop their mathematical
communication skills as they participate in mathematical discussions involving questions like “How did
you get that?” and “Why is that true?”
Performance Expectations
Explanatory Comments and Examples
Students are expected to:
K.5.A
Identify the question(s) asked in a problem.
K.5.B
Identify the given information that can be used
to solve a problem.
K.5.C
Recognize when additional information is
required to solve a problem.
K.5.D
Select from a variety of problem-solving
strategies and use one or more strategies to
solve a problem.
K.5.E
Answer the question(s) asked in a problem.
K.5.F
Describe how a problem was solved.
K.5.G Determine whether a solution to a problem
is reasonable.
8
Descriptions of solution processes and explanations
can include numbers, words (including mathematical
language), pictures, or physical objects. Students
should be able to use all of these representations as
needed. For a particular solution, students should be
able to explain or show their work using at least one of
these representations and verify that their answer
is reasonable.
Examples:
•
Grandma went to visit her two grandchildren and
discovered that the gloves they were each wearing
had holes in every finger, even the thumbs. She
will fix their gloves. How many glove fingers
(including thumbs) need to be fixed?
•
Students are given drinking straws or coffee
stirrers cut to a variety of different lengths: 6″, 5″,
4″, 3″, and 2″. They are to figure out which sets of
three lengths, when joined at their ends, will form
triangles and which sets of three will not.
July 2008
Washington State K–12 Mathematics Standards
Grade 1
July 2008
Washington State K–12 Mathematics Standards
9
Grade 1
Grade 1
1.1. Core Content: Whole number relationships (Numbers, Operations)
S
tudents continue to work with whole numbers to quantify objects. They consider how numbers
relate to one another. As they expand the set of numbers they work with, students start to develop
critical concepts of ones and tens that introduce them to place value in our base ten number system.
An understanding of how ones and tens relate to each other allows students to begin adding and
subtracting two-digit numbers, where thinking of ten ones as one ten and vice versa is routine. Some
students will be ready to work with numbers larger than those identified in the Expectations and should
be given every opportunity to do so.
Performance Expectations
Explanatory Comments and Examples
Students are expected to:
1.1.A Count by ones forward and backward from 1
to 120, starting at any number, and count by
twos, fives, and tens to 100.
Research suggests that when students count past 100,
they often make errors such as “99, 100, 200” and
“109, 110, 120.” However, once a student counts to
120 consistently, it is highly improbable that additional
counting errors will be made.
Example:
•
Start at 113. Count backward. I’ll tell you when
to stop. [Stop when the student has counted
backward ten numbers.]
1.1.B
Name the number that is one less or one more
than any number given verbally up to 120.
1.1.C
Read aloud numerals from 0 to 1,000.
The patterns in the base ten number system become
clearer to students when they count in the hundreds.
Therefore, learning the names of three-digit numbers
supports the learning of more difficult two-digit
numbers (such as numbers in the teens and numbers
ending in 0 or 1).
1.1.D
Order objects or events using ordinal numbers.
Students use ordinal numbers to describe positions
through the twentieth.
Example:
•
1.1.E
Write, compare, and order numbers to 120.
John is fourth in line.
Students arrange numbers in lists or talk about the
relationships among numbers using the words equal to,
greater than, less than, greatest, and least.
Example:
•
Write the numbers 27, 2, 111, and 35 from least
to greatest.
Students might also describe which of two numbers is
closer to a given number. This is part of developing an
understanding of the relative value of numbers.
July 2008
Washington State K–12 Mathematics Standards
11
Grade 1
Performance Expectations
Explanatory Comments and Examples
Students are expected to:
1.1.F
Fluently compose and decompose numbers
to 10.
Students put together and take apart whole numbers
as a precursor to addition and subtraction.
Examples:
1.1.G
Group numbers into tens and ones in more
than one way.
•
Ten is 2 + 5 + 1 + 1 + 1.
•
Eight is five and three.
•
Here are twelve coins. I will hide some. If you
see three, how many am I hiding? [This example
demonstrates how students might be encouraged
to go beyond the expectation.]
Students demonstrate that the value of a number
remains the same regardless of how it is grouped.
Grouping of numbers lays a foundation for future work
with addition and subtraction of two-digit numbers,
where renaming may be necessary.
For example, twenty-seven objects can be grouped
as 2 tens and 7 ones, regrouped as 1 ten and 17
ones, and regrouped again as 27 ones. The total (27)
remains constant.
be
27 can
shown as
27 = 10 + 10 + 7
be
27 can
shown as
27 = 10 + 17
be
27 can
shown as
1.1.H
12
Group and count objects by tens, fives,
and twos.
Given 23 objects, the student will count them by tens
as 10, 20, 21, 22, 23; by fives as 5, 10, 15, 20, 21, 22,
23; and by twos as 2, 4, 6, 8, 10, 12, 14, 16, 18, 20,
22, 23.
July 2008
Washington State K–12 Mathematics Standards
Grade 1
Performance Expectations
Explanatory Comments and Examples
Students are expected to:
1.1.I
Classify a number as odd or even and
demonstrate that it is odd or even.
Students use words, objects, or pictures to
demonstrate that a given number is odd or even.
Examples:
July 2008
Washington State K–12 Mathematics Standards
•
13 is odd because 13 counters cannot be
regrouped into two equal piles.
•
20 is even because every counter in this set of
20 counters can be paired with another counter
in the set.
13
Grade 1
Grade 1
1.2. Core Content: Addition and subtraction
(Operations, Algebra)
S
tudents learn how to add and subtract, when to add and subtract, and how addition and
subtraction relate to each other. Understanding that addition and subtraction undo each other is an
important part of learning to use these operations efficiently and accurately. Students notice patterns
involving addition and subtraction, and they work with other types of patterns as they learn to make
generalizations about what they observe.
Performance Expectations
Explanatory Comments and Examples
Students are expected to:
1.2.A
Connect physical and pictorial representations
to addition and subtraction equations.
The intention of the standard is for students to
understand that mathematical equations represent
situations. Simple student responses are adequate.
Combining a set of 3 objects and a set of 5 objects
to get a set of 8 objects can be represented by the
equation 3 + 5 = 8. The equation 2 + 6 = 8 could be
represented by drawing a set of 2 cats and a set of
6 cats making a set of 8 cats. The equation 9 – 5 = 4
could be represented by taking 5 objects away from a
set of 9 objects.
1.2.B
Use the equal sign (=) and the word equals to
indicate that two expressions are equivalent.
Students need to understand that equality means is
the same as. This idea is critical if students are to
avoid common pitfalls in later work with numbers and
operations, where they may otherwise fall into habits of
thinking that the answer always follows the equal sign.
Examples:
1.2.C
Represent addition and subtraction on the
number line.
•
7=8–1
•
5 + 3 equals 10 – 2
Examples:
•
4+3=7
0
•
2
3
4
5
6
7
8
2
3
4
5
6
7
8
7–4=3
0
14
1
1
July 2008
Washington State K–12 Mathematics Standards
Grade 1
Performance Expectations
Explanatory Comments and Examples
Students are expected to:
1.2.D
Demonstrate the inverse relationship between
addition and subtraction by undoing an addition
problem with subtraction and vice versa.
The relationship between addition and subtraction
is an important part of developing algebraic thinking.
Students can demonstrate this relationship using
physical models, diagrams, numbers, or actingout situations.
Examples:
1.2.E
Add three or more one-digit numbers using
the commutative and associative properties
of addition.
•
3 + 5 = 8, so 8 – 3 = 5
•
Annie had ten marbles, but she lost three. How
many marbles does she have? Joe found her
marbles and gave them back to her. Now how
many does she have?
Examples:
•
3 + 5 + 5 = 3 + 10
(Associativity allows us to add the last two
addends first.)
•
(5 + 3) + 5 = 5 + (5 + 3) = (5 +5) + 3 = 13
(Commutativity and associativity allow us to
reorder addends.)
This concept can be extended to address a problem like
3 +  + 2 = 9, which can be rewritten as 5 +  = 9.
1.2.F
Apply and explain strategies to compute
addition facts and related subtraction facts for
sums to 18.
Strategies for addition include counting on, but
students should be able to move beyond counting
on to use other strategies, such as making 10, using
doubles or near doubles, etc.
Subtraction strategies include counting back, relating
the problem to addition, etc.
1.2.G
Quickly recall addition facts and related
subtraction facts for sums equal to 10.
Adding and subtracting zero are included.
1.2.H
Solve and create word problems that match
addition or subtraction equations.
Students should be able to represent addition and
subtraction sentences with an appropriate situation,
using objects, pictures, or words. This standard is about
helping students connect symbolic representations
to situations. While some students may create word
problems that are detailed or lengthy, this is not
necessary to meet the expectation. Just as we want
students to be able to translate 5 boys and 3 girls sitting
at a table into 5 + 3 = 8, we want students to look at an
expression like 7 – 4 = 3 and connect it to a situation or
problem using objects, pictures, or words.
July 2008
Washington State K–12 Mathematics Standards
15
Grade 1
Performance Expectations
Explanatory Comments and Examples
Students are expected to:
1.2.H Cont.
Example:
•
For the equation 7 + ? = 10, a possible story
might be:
Jeff had 7 marbles in his pocket and some
marbles in his drawer. He had 10 marbles
altogether. How many marbles did he have in his
drawer? Use pictures, words, or objects to show
your answer.
1.2.I
Recognize, extend, and create number patterns.
Example:
•
Extend the simple addition patterns below and tell
how you decided what numbers come next:
1, 3, 5, 7, . . .
2, 4, 6, 8, 10, . . .
50, 45, 40, 35, 30, . . .
16
July 2008
Washington State K–12 Mathematics Standards
Grade 1
Grade 1
1.3. Core Content: Geometric attributes
(Geometry/Measurement)
S
tudents expand their knowledge of two- and three-dimensional geometric figures by sorting,
comparing, and contrasting them according to their characteristics. They learn important
mathematical vocabulary used to name the figures. Students work with composite shapes made out of
basic two-dimensional figures as they continue to develop their spatial sense of shapes, objects, and the
world around them.
Performance Expectations
Explanatory Comments and Examples
Students are expected to:
1.3.A
Compare and sort a variety of two- and
three-dimensional figures according to their
geometric attributes.
1.3.B Identify and name two-dimensional figures,
including those in real-world contexts,
regardless of size or orientation.
The student may sort a collection of two-dimensional
figures into those that have a particular attribute (e.g.,
those that have straight sides) and those that do not.
Figures should include circles, triangles, rectangles,
squares (as special rectangles), rhombi, hexagons,
and trapezoids.
Contextual examples could include classroom clocks,
flags, desktops, wall or ceiling tiles, etc. Triangles
should appear in many positions and orientations and
should not all be equilateral or isosceles.
1.3.C
Combine known shapes to create shapes and
divide known shapes into other shapes.
Students could be asked to trace objects or use
a drawing program to show different ways that a
rectangle can be divided into three triangles. They
can also use pattern blocks or plastic shapes to make
new shapes. The teacher can give students cutouts of
shapes and ask students to combine them to make a
particular shape.
Example:
•
July 2008
Washington State K–12 Mathematics Standards
What shapes can be made from a rectangle and a
triangle? Draw a picture to show your answers.
17
Grade 1
Grade 1
1.4. Core Content: Concepts of measurement
(Geometry/Measurement)
S
tudents start to learn about measurement by measuring length. They begin to understand what it
means to measure something, and they develop their measuring skills using everyday objects. As
they focus on length, they come to understand that units of measure must be equal in size and learn
that standard-sized units exist. They develop a sense of the approximate size of those standard units
(like inches or centimeters) and begin using them to measure different objects. Students learn that when
a unit is small, it takes more of the unit to measure an item than it does when the units are larger, and
they relate and compare measurements of objects using units of different sizes. Over time they apply
these same concepts of linear measurement to other attributes such as weight and capacity. As students
practice using measurement tools to measure objects, they reinforce their numerical skills and continue to
develop their sense of space and shapes.
Performance Expectations
Explanatory Comments and Examples
Students are expected to:
1.4.A
Recognize that objects used to measure an
attribute (length, weight, capacity) must be
consistent in size.
Marbles can be suitable objects for young children to
use to measure weight, provided that all the marbles
are the same weight. Paper clips are appropriate for
measuring length as long as the paper clips are all the
same length.
1.4.B
Use a variety of non-standard units to
measure length.
Use craft sticks, toothpicks, coffee stirrers, etc., to
measure length.
1.4.C Compare lengths using the transitive property.
Example:
•
1.4.D
Use non-standard units to compare objects
according to their capacities or weights.
Examples can include using filled paper cups to
measure capacity or a balance with marbles or cubes
to measure weight.
1.4.E
Describe the connection between the size of
the measurement unit and the number of units
needed to measure something.
Examples:
1.4.F
18
If Jon is taller than Jacob, and Jacob is taller than
Luisa, then Jon is taller than Luisa.
Name the days of the week and the months of
the year, and use a calendar to determine a
day or month.
•
It takes more toothpicks than craft sticks to
measure the width of my desk. The longer the unit,
the fewer I need.
•
It takes fewer marbles than cubes to balance my
object. The lighter the unit, the more I need.
•
It takes more little medicine cups filled with water
than larger paper cups filled with water to fill my
jar. The less my unit holds, the more I need.
Examples:
•
Name the days of the week in order.
•
Name the months of the year in order.
•
How many days until your birthday?
•
What month comes next?
•
What day was it yesterday?
July 2008
Washington State K–12 Mathematics Standards
Grade 1
Grade 1
1.5. Additional Key Content (Data/Statistics/Probability)
S
tudents are introduced to early ideas of statistics by collecting and visually representing data. These
ideas reinforce their understanding of the Core Content areas related to whole numbers and addition
and subtraction as students ask and answer questions about the data. As they move through the grades,
students will continue to apply what they learn about data, making mathematics relevant and connecting
numbers to applied situations.
Performance Expectations
Explanatory Comments and Examples
Students are expected to:
1.5.A
Represent data using tallies, tables, picture
graphs, and bar-type graphs.
In a picture graph, a single picture represents a single
object. Pictographs, where a symbol represents more
than one unit, are introduced in grade three when
multiplication is developed.
Students are expected to be familiar with all
representations, but they need not use them all in
every situation.
1.5.B Ask and answer comparison questions
about data.
Students develop questions that can be answered
using information from their graphs. For example,
students could look at tallies showing the number of
pockets on pants for each student today.
Andy
Sara
Chris
They might ask questions such as:
— Who has the most pockets?
— Who has the fewest pockets?
— How many more pockets does Andy have
than Chris?
July 2008
Washington State K–12 Mathematics Standards
19
Grade 1
Grade 1
1.6. Core Processes: Reasoning, problem solving, and communication
S
tudents further develop the concept that doing mathematics involves solving problems and discussing
what they did to solve them. Problems in first grade emphasize addition, subtraction, and solidifying
number concepts, and sometimes include precursors to multiplication. Students continue to develop their
mathematical communication skills as they participate in mathematical discussions involving questions
like “How did you get that?”; “Why did you do that?”; and “How do you know that?” Students begin to build
their mathematical vocabulary as they use correct mathematical language appropriate to first grade.
Performance Expectations
Explanatory Comments and Examples
Students are expected to:
1.6.A
Identify the question(s) asked in a problem.
1.6.B
Identify the given information that can be used
to solve a problem.
1.6.C
Recognize when additional information is
required to solve a problem.
1.6.D
Select from a variety of problem-solving
strategies and use one or more strategies to
solve a problem.
1.6.E
Answer the question(s) asked in a problem.
1.6.F
Identify the answer(s) to the question(s)
in a problem.
1.6.G
Describe how a problem was solved.
1.6.H
Determine whether a solution to a problem
is reasonable.
Descriptions of solution processes and explanations
can include numbers, words (including mathematical
language), pictures, or physical objects. Students
should be able to use all of these representations as
needed. For a particular solution, students should be
able to explain or show their work using at least one of
these representations and verify that their answer
is reasonable.
Examples:
•
Think about how many feet a person has. How
many feet does a cat have? How many feet does a
snail have? How about a fish or a snake?
There are ten feet living in my house. Who could
be living in my house?
Come up with a variety of ways you can have a total
of ten feet living in your house. Use pictures, words,
or numbers to show how you got your answer.
•
You are in charge of setting up a dining room with
exactly twenty places for people to sit. You can use
any number and combination of different-shaped
tables. A hexagon-shaped table seats six people.
A triangle-shaped table seats three people. A
square-shaped table seats four people.
Draw a picture showing which tables and how
many of each you could set up so that twenty
people have a place to sit. Is there more than one
way to do this? How many ways can you find?
20
July 2008
Washington State K–12 Mathematics Standards
Grade 2
July 2008
Washington State K–12 Mathematics Standards
21
Grade 2
Grade 2
2.1. Core Content: Place value and the base ten system
(Numbers)
S
tudents refine their understanding of the base ten number system and use place value concepts of
ones, tens, and hundreds to understand number relationships. They become fluent in writing and
renaming numbers in a variety of ways. This fluency, combined with the understanding of place value, is a
strong foundation for learning how to add and subtract two-digit numbers.
Performance Expectations
Explanatory Comments and Examples
Students are expected to:
2.1.A Count by tens or hundreds forward and backward from 1 to 1,000, starting at any number.
Example:
2.1.B Connect place value models with their
numerical equivalents to 1,000.
Understanding the relative value of numbers using place
value is an important element of our base ten number
system. Students should be familiar with representing
numbers using words, pictures (including those involving
grid paper), or physical objects such as base ten blocks.
Money can also be an appropriate model.
2.1.C Identify the ones, tens, and hundreds place in
a number and the digits occupying them.
Examples:
2.1.D Write three-digit numbers in expanded form.
2.1.E Group three-digit numbers into hundreds, tens,
and ones in more than one way.
•
Count forward by tens out loud starting at 32.
•
4 is located in what place in the number 834?
•
What digit is in the hundreds place in 245?
Examples:
•
573 = 500 + 70 + 3
•
600 + 30 + 7 = 637
Students should become fluent in naming and
renaming numbers based on number sense and their
understanding of place value.
Examples:
2.1.F Compare and order numbers from 0 to 1,000.
July 2008
Washington State K–12 Mathematics Standards
•
In the number 647, there are 6 hundreds, there are
64 tens, and there are 647 ones.
•
There are 20 tens in 200 and 10 hundreds in 1,000.
•
There are 23 tens in 230.
•
3 hundreds + 19 tens + 3 ones describes the same
number as 4 hundreds + 8 tens + 13 ones.
Students use the words equal to, greater than, less
than, greatest, or least and the symbols =, <, and >.
23
Grade 2
Grade 2
2.2. Core Content: Addition and subtraction
(Operations, Geometry/Measurement, Algebra)
S
tudents focus on what it means to add and subtract as they become fluent with single-digit addition
and subtraction facts and develop addition and subtraction procedures for two-digit numbers.
Students make sense of these procedures by building on what they know about place value and number
relationships and by putting together or taking apart sets of objects. This is students’ first time to deal
formally with step-by-step procedures (algorithms)—an important component of mathematics where
a generalizable technique can be used in many similar situations. Students begin to use estimation to
determine if their answers are reasonable.
Performance Expectations
Explanatory Comments and Examples
Students are expected to:
2.2.A
Quickly recall basic addition facts and related
subtraction facts for sums through 20.
2.2.B
Solve addition and subtraction word problems
that involve joining, separating, and comparing
and verify the solution.
Problems should include those involving take-away
situations, missing addends, and comparisons.
The intent of this expectation is for students to show
their work, explain their thinking, and verify that the
answer to the problem is reasonable in terms of the
original context and the mathematics used to solve the
problem. Verifications can include the use of numbers,
words, pictures, or physical objects.
Example:
•
Hazel and Kimmy each have stamp collections.
Kimmy’s collection has 7 more stamps than Hazel’s.
Kimmy has a total of 20 stamps. How many stamps
are in Hazel’s collection? Explain your answer.
[Students may verify their work orally, with
pictures, or in writing. For instance, students might
give the equation below or might use the picture.]
20 – 7 = 13
and
USA
USA
USA
USA
USA
USA
USA
are 20
Hazel’s
2.2.C
Add and subtract two-digit numbers efficiently
and accurately using a procedure that works
with all two-digit numbers and explain why the
procedure works.
Kimmy’s
Students should be able to connect the numerical
procedures with other representations, such as words,
pictures, or physical objects.
This is students’ first exposure to mathematical
algorithms. It sets the stage for all future work with
computational procedures.
The standard algorithms for addition and subtraction
are formalized in grade three.
24
July 2008
Washington State K–12 Mathematics Standards
Grade 2
Performance Expectations
Explanatory Comments and Examples
Students are expected to:
2.2.D
2.2.E
Add and subtract two-digit numbers mentally
and explain the strategies used.
Estimate sums and differences.
Examples of strategies include
•
Combining tens and ones:
68 + 37 = 90 + 15 = 105
•
Compensating: 68 + 37 = 65 + 40 = 105
•
Incremental: 68 + 37 = 68 + 30 + 7 = 105
Example:
•
2.2.F
Create and state a rule for patterns that can be
generated by addition and extend the pattern.
Examples:
•
2, 5, 8, 11, 14, 17, . . .
•
Look at the pattern of squares below. Draw a
picture that shows what the next set of squares
might look like and explain why your answer
makes sense.
2.2.G
Solve equations in which the unknown number
appears in a variety of positions.
Students might estimate that 198 + 29 is a little
less than 230.
A
B
C
Students need this kind of experience with equivalence
to accompany their first work with addition and
subtraction. Flexible use of equivalence and missing
numbers sets the stage for later work when solving
equations in which the variable is in different positions.
Examples:
2.2.H
Name each standard U.S. coin, write its value
using the $ sign and the ¢ sign, and name
combinations of other coins with the same
total value.
2.2.I Determine the value of a collection of coins
totaling less than $1.00.
July 2008
Washington State K–12 Mathematics Standards
•
8+3=+5
•
10 – 7 = 2 + 
•
=9+4+2
Students should be expected to express, for example,
the value of a quarter as twenty-five cents, $0.25,
and 25¢, and they should be able to give other
combinations of coins whose value is 25¢. This is a
precursor to decimal notation.
25
Grade 2
Grade 2
2.3. Core Content: Measurement
S
(Geometry/Measurement)
tudents understand the process of measuring length and progress from measuring length with
objects such as toothpicks or craft sticks to the more practical skill of measuring length with standard
units and tools such as rulers, tape measures, or meter sticks. As students are well acquainted with twodigit numbers by this point, they tell time on different types of clocks.
Performance Expectations
Explanatory Comments and Examples
Students are expected to:
26
2.3.A
Identify objects that represent or approximate
standard units and use them to measure length.
At this level, students no longer rely on non-standard
units. Students find and use approximations for
standard length units, like a paper clip whose length
is about an inch, or the width of a particular student’s
thumbnail that might be about a centimeter. They might
also use commonly available classroom objects like
inch tiles or centimeter cubes.
2.3.B
Estimate length using metric and U.S.
customary units.
Students could make observations such as, “The
ceiling of the classroom is about 8 feet high.”
2.3.C
Measure length to the nearest whole unit in
both metric and U.S. customary units.
Standard tools may include rulers, yardsticks, meter
sticks, or centimeter/inch measuring tapes. Students
should measure some objects that are longer than the
measurement tool being used.
2.3.D
Describe the relative size among minutes,
hours, days, weeks, months, and years.
Students should be able to describe relative sizes
using statements like, “Since a minute is less than an
hour, there are more minutes than hours in one day.”
2.3.E
Use both analog and digital clocks to tell time
to the minute.
July 2008
Washington State K–12 Mathematics Standards
Grade 2
Grade 2
2.4. Additional Key Content S
(Numbers, Operations, Geometry/Measurement,
Data/Statistics/Probability)
tudents make predictions and answer questions about data as they apply their growing understanding
of numbers and the operations of addition and subtraction. They extend their spatial understanding
of Core Content in geometry developed in kindergarten and grade one by solving problems involving
two- and three-dimensional geometric figures. Students are introduced to a few critical concepts that will
become Core Content in grade three. Specifically, they begin to work with multiplication and division and
learn what a fraction is.
Performance Expectations
Explanatory Comments and Examples
Students are expected to:
2.4.A Solve problems involving properties of twoand three-dimensional figures.
A critical component in the development of students’
spatial and geometric understanding is the ability to
solve problems involving the properties of figures. At
the primary level, students must move from judging
plane and space shapes by their appearance as whole
shapes to focusing on the relationship of the sides,
angles, or faces. At the same time, students must
learn the language important for describing shapes
according to their essential characteristics. Later, they
will describe properties of shapes in more formal ways
as they progress in geometry.
Examples:
•
How many different ways can you fill the outline of
the figure with pattern blocks? What is the greatest
number of blocks you can use? The least number?
Can you fill the outline with every whole number of
blocks between the least number of blocks and the
greatest number of blocks?
•
Build a figure or design out of five blocks. Describe
it clearly enough so that someone else could build
it without seeing it. Blocks may represent twodimensional figures (i.e., pattern blocks) or threedimensional figures (i.e., wooden geometric solids).
2.4.B Collect, organize, represent, and interpret data
in bar graphs and picture graphs.
In a picture graph, a single picture represents a single
object. Pictographs, where a symbol represents more
than one unit, are introduced in grade three when
multiplication skills are developed.
2.4.C Model and describe multiplication situations in
which sets of equal size are joined.
Multiplication is introduced in grade two only at a
conceptual level. This is a foundation for the more
systematic study of multiplication in grade three. Small
numbers should be used in multiplication problems that
are posed for students in grade two.
Example:
•
July 2008
Washington State K–12 Mathematics Standards
You have 4 boxes with 3 apples in each box. How
many apples do you have?
27
Grade 2
Performance Expectations
Explanatory Comments and Examples
Students are expected to:
2.4.D Model and describe division situations in which
sets are separated into equal parts.
Division is introduced in grade two only at a conceptual
level. This is a foundation for the more systematic
study of division in grade three. Small numbers
should be used in division problems that are posed for
students in grade two.
Example:
•
2.4.E Interpret a fraction as a number of equal parts
of a whole or a set.
28
You have 15 apples to share equally among
5 classmates. How many apples will each
classmate get?
Examples:
•
Juan, Chan, and Hortense are going to share
a large cookie in the shape of a circle. Draw a
picture that shows how you can cut up the cookie
in three fair shares, and tell how big each piece is
as a fraction of the whole cookie.
•
Ray has two blue crayons, one red crayon, and
one yellow crayon. What fraction of Ray’s crayons
is red? What fraction of the crayons is blue?
July 2008
Washington State K–12 Mathematics Standards
Grade 2
Grade 2
2.5. Core Processes: Reasoning, problem solving, and communication
S
tudents further develop the concept that doing mathematics involves solving problems and talking
about what they did to solve those problems. Second-grade problems emphasize addition and
subtraction with increasingly large numbers, measurement, and early concepts of multiplication and
division. Students communicate their mathematical thinking and make increasingly more convincing
mathematical arguments. Students participate in mathematical discussions involving questions like “How
did you get that?”; “Why did you use that strategy?”; and “Why is that true?” Students continue to build
their mathematical vocabulary as they use correct mathematical language appropriate to grade two when
discussing and refining solutions to problems. Performance Expectations
Explanatory Comments and Examples
Students are expected to:
2.5.A
Identify the question(s) asked in a problem and
any other questions that need to be answered
in order to solve the problem.
2.5.B
Identify the given information that can be used
to solve a problem.
2.5.C
Recognize when additional information is
required to solve a problem.
2.5.D
Select from a variety of problem-solving
strategies and use one or more strategies to
solve a problem.
2.5.E
Identify the answer(s) to the question(s)
in a problem.
2.5.F
Describe how a problem was solved.
2.5.G
Determine whether a solution to a problem
is reasonable.
July 2008
Washington State K–12 Mathematics Standards
Descriptions of solution processes and explanations
can include numbers, words (including mathematical
language), pictures, or physical objects. Students
should be able to use all of these representations as
needed. For a particular solution, students should be
able to explain or show their work using at least one of
these representations and verify that their answer
is reasonable.
Examples:
•
A bag full of jellybeans is on the table. There are
10 black jellybeans in the bag. There are twice as
many red jellybeans as black jellybeans. There
are 2 fewer red jellybeans than yellow jellybeans.
There are half as many pink jellybeans as yellow
jellybeans. How many jellybeans are in the bag?
Explain your answer.
•
Suzy, Ben, and Pedro have found 1 quarter, 1
dime, and 4 pennies under the sofa. Their mother
has lots of change in her purse, so they could
trade any of these coins for other coins adding up
to the same value. She says they can keep the
money if they can tell her what coins they need so
the money can be shared equally among them.
How can they do this?
29
Grade 2
Grade 3
July 2008
Washington State K–12 Mathematics Standards
31
Grade 3
Grade 3
3.1. Core Content: Addition, subtraction, and place value
(Numbers, Operations)
S
tudents solidify and formalize important concepts and skills related to addition and subtraction. In
particular, students extend critical concepts of the base ten number system to include large numbers,
they formalize procedures for adding and subtracting large numbers, and they apply these procedures in
new contexts.
Performance Expectations
Explanatory Comments and Examples
Students are expected to:
3.1.A
Read, write, compare, order, and represent
numbers to 10,000 using numbers, words,
and symbols.
This expectation reinforces and extends place
value concepts.
Symbols used to describe comparisons include <, >, =.
Examples:
3.1.B
•
Fill in the box with <, >, or = to make a true
sentence: 3,546  4,356.
•
Is 5,683 closer to 5,600 or 5,700?
Round whole numbers through 10,000 to the
nearest ten, hundred, and thousand.
Example:
3.1.C
Fluently and accurately add and subtract
whole numbers using the standard
regrouping algorithms.
Teachers should be aware that in some countries the
algorithms might be recorded differently.
3.1.D
Estimate sums and differences to approximate
solutions to problems and determine
reasonableness of answers.
Example:
Solve single- and multi-step word problems
involving addition and subtraction of whole
numbers and verify the solutions.
The intent of this expectation is for students to show
their work, explain their thinking, and verify that the
answer to the problem is reasonable in terms of the
original context and the mathematics used to solve the
problem. Verifications can include the use of numbers,
words, pictures, or equations.
3.1.E
July 2008
Washington State K–12 Mathematics Standards
•
•
Round 3,465 to the nearest ten and then to the
nearest hundred.
Marla has $10 and plans to spend it on items
priced at $3.72 and $6.54. Use estimation to
decide whether Marla’s plan is a reasonable one,
and justify your answer.
33
Grade 3
Grade 3
3.2. Core Content: Concepts of multiplication and division (Operations, Algebra)
S
tudents learn the meaning of multiplication and division and how these operations relate to each
other. They begin to learn multiplication and division facts and how to multiply larger numbers.
Students use what they are learning about multiplication and division to solve a variety of problems. With
a solid understanding of these two key operations, students are prepared to formalize the procedures for
multiplication and division in grades four and five.
Performance Expectations
Explanatory Comments and Examples
Students are expected to:
3.2.A Represent multiplication as repeated addition,
arrays, counting by multiples, and equal
jumps on the number line, and connect each
representation to the related equation.
Students should be familiar with using words,
pictures, physical objects, and equations to represent
multiplication. They should be able to connect
various representations of multiplication to the related
multiplication equation. Representing multiplication
with arrays is a precursor to more formalized area
models for multiplication developed in later grades
beginning with grade four.
The equation 3 × 4 = 12 could be represented in the
following ways:
— Equal sets:
Equal groups:
Equal sharing:
— An array:
— Repeated addition: 4 + 4 + 4
— Three equal jumps forward from 0 on the
number line to 12:
0
34
1
2
3
4
5
6
7
8
9 10 11 12
July 2008
Washington State K–12 Mathematics Standards
Grade 3
Performance Expectations
Explanatory Comments and Examples
Students are expected to:
3.2.B
Represent division as equal sharing, repeated
subtraction, equal jumps on the number line,
and formation of equal groups of objects,
and connect each representation to the
related equation.
Students should be familiar with using words,
pictures, physical objects, and equations to represent
division. They should be able to connect various
representations of division to the related equation.
Division can model both equal sharing (how many in
each group) and equal groups (how many groups).
The equation 12 ÷ 4 = 3 could be represented in the
following ways:
— Equal
groups:
Equal groups:
Equal
Equal sharing:
sharing:
— An array:
— Repeated subtraction: The expression
12 – 4 – 4 – 4 involves 3 subtractions of 4.
— Three equal jumps backward from 12 to 0
on the number line:
0
3.2.C
Determine products, quotients, and missing
factors using the inverse relationship between
multiplication and division.
1
2
3
4
5
6
7
8
9 10 11 12
Example:
•
To find the value of N in 3 x N = 18, think
18 ÷ 3 = 6.
Students can use multiplication and division fact
families to understand the inverse relationship between
multiplication and division.
Examples:
•
3 × 5 = 15 15 ÷ 3 = 5 5 × 3 = 15
15 ÷ 5 = 3
15
x or ÷
3
July 2008
Washington State K–12 Mathematics Standards
5
35
Grade 3
Performance Expectations
Explanatory Comments and Examples
Students are expected to:
3.2.D
Apply and explain strategies to compute
multiplication facts to 10 X 10 and the related
division facts.
Strategies for multiplication include skip counting
(repeated addition), fact families, double-doubles
(when 4 is a factor), “think ten” (when 9 is a factor,
think of it as 10 – 1), and decomposition of arrays into
smaller known parts.
Number properties can be used to help remember
basic facts.
5 × 3 = 3 × 5 (Commutative Property)
1 × 5 = 5 × 1 = 5 (Identity Property)
0 × 5 = 5 × 0 = 0 (Zero Property)
5 × 6 = 5 × (2 × 3) = (5 × 2) × 3 = 10 × 3 = 30
(Associative Property)
4 × 6 = 4 (5 + 1) = (4 × 5) + (4 × 1) = 20 + 4 = 24
(Distributive Property)
4 groups of 1
4 groups of 5
4x6
Division strategies include using fact families and
thinking of missing factors.
36
3.2.E
Quickly recall those multiplication facts for
which one factor is 1, 2, 5, or 10 and the
related division facts.
Many students will learn all of the multiplication facts
to 10 X 10 by the end of third grade, and all students
should be given the opportunity to do so.
3.2.F
Solve and create word problems that match
multiplication or division equations.
The goal is for students to be able to represent
multiplication and division sentences with an
appropriate situation, using objects, pictures, or written
or spoken words. This standard is about helping
students connect symbolic representations to the
situations they model. While some students may create
word problems that are detailed or lengthy, this is not
necessary to meet the expectation. Just as we want
students to be able to translate 5 groups of 3 cats into
5 x 3 = 15; we want students to look at an equation like
12 ÷ 4 = 3 and connect it to a situation using objects,
pictures, or words.
July 2008
Washington State K–12 Mathematics Standards
Grade 3
Performance Expectations
Explanatory Comments and Examples
Students are expected to:
3.2.G Multiply any number from 11 through 19 by
a single-digit number using the distributive
property and place value concepts.
Example:
•
Equation: 3 × 9 = ?
[Problem situation:
There are 3 trays of cookies with 9 cookies on
each tray. How many cookies are there in all?]
Example:
•
6 × 12 can be thought of as 6 tens and 6 twos,
which equal 60 and 12, totaling 72.
10
6
6 x 10 = 60
6 groups of 10
3.2.H Solve single- and multi-step word problems
involving multiplication and division and verify
the solutions.
2
6 x 2 = 12
3.2.F cont.
6 groups of 2
Problems include using multiplication to determine
the number of possible combinations or outcomes
for a situation, and division contexts that require
interpretations of the remainder.
The intent of this expectation is for students to show
their work, explain their thinking, and verify that the
answer to the problem is reasonable in terms of the
original context and the mathematics used to solve the
problem. Verifications can include the use of numbers,
words, pictures, physical objects, or equations.
Examples:
July 2008
Washington State K–12 Mathematics Standards
•
Determine the number of different outfits that can be
made with four shirts and three pairs of pants.
•
There are 14 soccer players on the boys’ team and
13 on the girls’ team. How many vans are needed
to take all players to the soccer tournament if each
van can take 5 players?
37
Grade 3
Grade 3
3.3. Core Content: Fraction concepts
(Numbers, Algebra)
S
tudents learn about fractions and how they are used. Students deepen their understanding of
fractions by comparing and ordering fractions and by representing them in different ways. With a
solid knowledge of fractions as numbers, students are prepared to be successful when they add, subtract,
multiply, and divide fractions to solve problems in later grades.
Performance Expectations
Explanatory Comments and Examples
Students are expected to:
3.3.A
Represent fractions that have denominators
of 2, 3, 4, 5, 6, 8, 9, 10, and 12 as parts of
a whole, parts of a set, and points on the
number line.
The focus is on numbers less than or equal to 1.
Students should be familiar with using words, pictures,
physical objects, and equations to represent fractions.
3.3.B
Compare and order fractions that have
denominators of 2, 3, 4, 5, 6, 8, 9, 10, and 12.
Fractions can be compared using benchmarks
1
(such as 2 or 1), common numerators, or common
denominators. Symbols used to describe comparisons
include <, >, =.
Fractions with common denominators may be compared
and ordered using the numerators as a guide.
2
3
5
<
<
6
6
6
Fractions with common numerators may be compared
and ordered using the denominators as a guide.
3
3
3
<
<
10
8
4
Fractions may be compared using
3.3.C
38
Represent and identify equivalent fractions with
denominators of 2, 3, 4, 5, 6, 8, 9, 10, and 12.
1
8
1
2
1
as a benchmark.
2
5
6
Students could represent fractions using the number
line, physical objects, pictures, or numbers.
July 2008
Washington State K–12 Mathematics Standards
Grade 3
Performance Expectations
Explanatory Comments and Examples
Students are expected to:
3.3.D
Solve single- and multi-step word problems
involving comparison of fractions and verify
the solutions.
The intent of this expectation is for students to show
their work, explain their thinking, and verify that the
answer to the problem is reasonable in terms of the
original context and the mathematics used to solve the
problem. Verifications can include the use of numbers,
words, pictures, physical objects, or equations.
Examples:
•
Emile and Jordan ordered a medium pizza. Emile
ate
1
1
of it and Jordan ate of it. Who ate more
3
4
pizza? Explain how you know.
•
Janie and Li bought a dozen balloons. Half of them
were blue,
1
1
were white, and were red. Were
3
6
there more blue, red, or white balloons? Justify
your answer.
July 2008
Washington State K–12 Mathematics Standards
39
Grade 3
Grade 3
3.4. Core Content: Geometry (Geometry/Measurement)
S
tudents learn about lines and use lines, line segments, and right angles as they work with
quadrilaterals. Students connect this geometric work to numbers, operations, and measurement as
they determine simple perimeters in ways they will use when calculating perimeters of more complex
figures in later grades.
Performance Expectations
Explanatory Comments and Examples
Students are expected to:
3.4.A
Identify and sketch parallel, intersecting, and
perpendicular lines and line segments.
3.4.B
Identify and sketch right angles.
3.4.C
Identify and describe special types
of quadrilaterals.
Special types of quadrilaterals include squares,
rectangles, parallelograms, rhombi, trapezoids and kites.
3.4.D
Measure and calculate perimeters
of quadrilaterals.
Example:
Solve single- and multi-step word problems
involving perimeters of quadrilaterals and
verify the solutions.
Example:
3.4.E
40
•
•
Sketch a parallelogram with two sides 9 cm long
and two sides 6 cm long. What is the perimeter of
the parallelogram?
Julie and Jacob have recently created two
rectangular vegetable gardens in their backyard.
One garden measures 6 ft by 8 ft, and the other
garden measures 10 ft by 5 ft. They decide to
place a small fence around the outside of each
garden to prevent their dog from getting into their
new vegetables. How many feet of fencing should
Julie and Jacob buy to fence both gardens?
July 2008
Washington State K–12 Mathematics Standards
Grade 3
Grade 3
3.5. Additional Key Content S
(Algebra, Geometry/Measurement, Data/Statistics/Probability)
tudents solidify and formalize a number of important concepts and skills related to Core Content
studied in previous grades. In particular, students demonstrate their understanding of equivalence as
an important foundation for later work in algebra. Students also reinforce their knowledge of measurement
as they use standard units for temperature, weight, and capacity. They continue to develop data
organization skills as they reinforce multiplication and division concepts with a variety of types of graphs.
Performance Expectations
Explanatory Comments and Examples
Students are expected to:
3.5.A
Determine whether two expressions are equal
and use “=” to denote equality.
Examples:
•
Is 5 × 3 = 3 × 5 a true statement?
•
Is 24 ÷ 3 = 2 × 4 a true statement?
A common error students make is using the
mathematical equivalent of a run-on sentence to solve
some problems—students carry an equivalence from
a previous expression into a new expression with an
additional operation. For example, when adding 3 + 6 +
7, students sometimes incorrectly write:
3 + 6 = 9 + 7 = 16
Correct sentences:
3.5.B
Measure temperature in degrees Fahrenheit
and degrees Celsius using a thermometer.
3+6=9
9 + 7 = 16
The scale on a thermometer is essentially a vertical
number line. Students may informally deal with
negative numbers in this context, although negative
numbers are not formally introduced until grade six.
Measure temperature to the nearest degree.
3.5.C
Estimate, measure, and compare weight and
mass using appropriate-sized U.S. customary
and metric units.
3.5.D
Estimate, measure, and compare capacity
using appropriate-sized U.S. customary and
metric units.
3.5.E
Construct and analyze pictographs, frequency
tables, line plots, and bar graphs.
Students can write questions to be answered with
information from a graph. Graphs and tables can be
used to compare sets of data.
Using pictographs in which a symbol stands for
multiple objects can reinforce the development of
both multiplication and division skills. Determining
appropriate scale and units for the axes of various
types of graphs can also reinforce multiplication and
division skills.
July 2008
Washington State K–12 Mathematics Standards
41
Grade 3
Grade 3
3.6. Core Processes: Reasoning, problem solving, and communication
S
tudents in grade three solve problems that extend their understanding of core mathematical
concepts—such as geometric figures, fraction concepts, and multiplication and division of whole
numbers—as they make strategic decisions that bring them to reasonable solutions. Students use
pictures, symbols, or mathematical language to explain the reasoning behind their decisions and
solutions. They further develop their problem-solving skills by making generalizations about the processes
used and applying these generalizations to similar problem situations. These critical reasoning, problemsolving, and communication skills represent the kind of mathematical thinking that equips students to
use the mathematics they know to solve a growing range of useful and important problems and to make
decisions based on quantitative information.
Performance Expectations
Explanatory Comments and Examples
Students are expected to:
42
3.6.A
Determine the question(s) to be answered
given a problem situation.
3.6.B
Identify information that is given in a problem
and decide whether it is necessary or
unnecessary to the solution of the problem.
3.6.C
Identify missing information that is needed to
solve a problem.
3.6.D
Determine whether a problem to be solved
is similar to previously solved problems, and
identify possible strategies for solving
the problem.
3.6.E
Select and use one or more appropriate
strategies to solve a problem.
3.6.F
Represent a problem situation using
words, numbers, pictures, physical
objects, or symbols.
3.6.G
Explain why a specific problem-solving
strategy or procedure was used to
determine a solution.
3.6.H
Analyze and evaluate whether a solution is
reasonable, is mathematically correct, and
answers the question.
3.6.I
Summarize mathematical information, draw
conclusions, and explain reasoning.
3.6.J
Make and test conjectures based on data
(or information) collected from explorations
and experiments.
Descriptions of solution processes and explanations
can include numbers, words (including mathematical
language), pictures, physical objects, or equations.
Students should be able to use all of these
representations as needed. For a particular solution,
students should be able to explain or show their work
using at least one of these representations and verify
that their answer is reasonable.
Examples:
•
Whitney wants to put a fence around the perimeter
of her square garden. She plans to include a gate
that is 3 ft wide. The length of one side of the
garden is 19 ft. The fencing comes in two sizes:
rolls that are 18 ft long and 24 ft long. Which rolls
and how many of each should Whitney buy in
order to have the least amount of leftover fencing?
Justify your answer.
•
A soccer team is selling water bottles with soccer
balls painted on them to raise money for new
equipment. The team bought 10 boxes of water
bottles. Each box cost $27 and had 9 bottles. At
what price should the team sell each bottle in order
to make $180 profit to pay for new soccer balls?
Justify your answer.
July 2008
Washington State K–12 Mathematics Standards
Grade 4
July 2008
Washington State K–12 Mathematics Standards
43
Grade 4
Grade 4
4.1. Core Content: Multi-digit multiplication
(Numbers, Operations, Algebra)
S
tudents learn basic multiplication facts and efficient procedures for multiplying two- and three-digit
numbers. They explore the relationship between multiplication and division as they learn related
division and multiplication facts in the same fact family. These skills, along with mental math and
estimation, allow students to solve problems that call for multiplication. Building on an understanding of
how multiplication and division relate to each other, students prepare to learn efficient procedures for
division, which will be developed in fifth grade. Multiplication of whole numbers is not only a basic skill,
it is also closely connected to Core Content of area in this grade level, and this connection reinforces
understanding of both concepts. Multiplication is also central to students’ study of many other topics in
mathematics across the grades, including fractions, volume, and algebra.
Performance Expectations
Explanatory Comments and Examples
Students are expected to:
4.1.A
Quickly recall multiplication facts through
10 X 10 and the related division facts.
4.1.B
Identify factors and multiples of a number.
4.1.C
Represent multiplication of a two-digit number
by a two-digit number with place value models.
Examples:
•
The factors of 12 are 1, 2, 3, 4, 6, 12.
•
The multiples of 12 are 12, 24, 36, 48, . . .
Representations can include pictures or physical
objects, or students can describe the process in words
(14 times 16 is the same as 14 times 10 added to
14 times 6).
The algorithm for multiplication is addressed in
expectation 4.1.F.
Example:
•
14 × 16 = 224
4 tens
100
16
24
ones
6 tens
14
100 + 40 + 60 + 24 = 224
July 2008
Washington State K–12 Mathematics Standards
45
Grade 4
Performance Expectations
Explanatory Comments and Examples
Students are expected to:
4.1.D
Multiply by 10, 100, and 1,000.
Multiplying by 10, 100, and 1,000 extends place
value concepts to large numbers through the millions.
Students can use place value and properties of
operations to determine these products.
Examples:
4.1.E
4.1.F
•
10 × 5,000 = 50,000
100 × 5,000 = 500,000
1,000 × 5,000 = 5,000,000
•
40 × 300
= (4 × 10) × (3 × 100)
= (4 × 3) × (10 × 100)
= 12 × 1,000
= 12,000
Compare the values represented by digits in
whole numbers using place value.
Example:
Fluently and accurately multiply up to a threedigit number by one- and two-digit numbers
using the standard multiplication algorithm.
Example:
•
•
Compare the values represented by the digit 4 in
4,000,000 and 40,000. [The value represented by
the 4 in the millions place is 100 times as much
as the value represented by the 4 in the tenthousands place.]
245
7
x
3
3
245
7
x
1 7 15
Teachers should be aware that in some countries the
algorithm might be recorded differently.
4.1.G
4.1.H
4.1.I
Mentally multiply two-digit numbers by
numbers through 10 and by multiples of 10.
Examples:
•
4 × 32 = (4 × 30) + (4 × 2)
•
4 × 99 = 400 – 4
•
25 × 30 = 75 × 10
Estimate products to approximate solutions
to problems and determine reasonableness
of answers.
Example:
Solve single- and multi-step word problems
involving multi-digit multiplication and verify
the solutions.
The intent of this expectation is for students to show
their work, explain their thinking, and verify that the
answer to the problem is reasonable in terms of the
original context and the mathematics used to solve the
problem. Verifications can include the use of numbers,
words, pictures, or equations.
•
28 × 120 is approximately 30 times 100, so the
product should be around 3,000.
Problems could include multi-step problems that use
operations other than multiplication.
46
July 2008
Washington State K–12 Mathematics Standards
Grade 4
Performance Expectations
Explanatory Comments and Examples
Students are expected to:
4.1.J
Solve single- and multi-step word problems
involving division and verify the solutions.
The intent of this expectation is for students to show
their work, explain their thinking, and verify that the
answer to the problem is reasonable in terms of the
original context and the mathematics used to solve the
problem. Verifications can include the use of numbers,
words, pictures, or equations.
Division problems should reinforce connections
between multiplication and division. The example
below can be solved using multiplication along with
some addition and subtraction.
Example:
•
A group of 8 students shares a box containing 187
animal crackers. What is each student’s equal
share? How many crackers are left over?
Division algorithms, including long division, are
developed in fifth grade.
July 2008
Washington State K–12 Mathematics Standards
47
Grade 4
Grade 4
4.2. Core Content: Fractions, decimals, and mixed numbers S
(Numbers, Algebra)
tudents solidify and extend their understanding of fractions (including mixed numbers) to include
decimals and the relationships between fractions and decimals. Students work with common factors
and common multiples as preparation for learning procedures for fraction operations in grades five and six.
When they are comfortable with and knowledgeable about fractions, students are likely to be successful
with the challenging skills of learning how to add, subtract, multiply, and divide fractions.
Performance Expectations
Explanatory Comments and Examples
Students are expected to:
4.2.A
Represent decimals through hundredths with
place value models, fraction equivalents, and
the number line.
4.2.B Read, write, compare, and order decimals
through hundredths.
Students should know how to write decimals and
show them on the number line and should understand
their mathematical connections to place value models
and fraction equivalents. Students should be able to
represent decimals with words, pictures, or physical
objects, and connect these representations to the
corresponding decimal.
Decimals may be compared using benchmarks, such
as 0, 0.5, 1, or 1.5. Decimals may also be compared
using place value.
Examples:
•
List in increasing order: 0.7, 0.2, 1.4.
•
Write an inequality that compares 0.05 and 0.50.
4.2.C
Convert a mixed number to a fraction and vice
versa, and visually represent the number.
Students should be able to use either the fraction or
mixed-number form of a number as appropriate to
a given situation, and they should be familiar with
representing these numbers with words, pictures, and
physical objects.
4.2.D
Convert a decimal to a fraction and vice versa,
and visually represent the number.
Students should be familiar with using pictures and
physical objects to visually represent decimals and
fractions. For this skill at this grade, fractions should
be limited to those that are equivalent to fractions with
denominators of 10 or 100.
Examples:
•
48
33
= 0.3
1010
July 2008
Washington State K–12 Mathematics Standards
Grade 4
Performance Expectations
Explanatory Comments and Examples
Students are expected to:
4.2.D cont.
•
•
4.2.E
Compare and order decimals and fractions
(including mixed numbers) on the number line,
in lists, and with the symbols <, >, or =.
•
•
•
Write a fraction equivalent to a given fraction.
5
20
5
20
42
100
= 0.25
Examples:
4.2.F
0.42 =
Compare each pair of numbers using <, >, or =:
6
10
10
0.8
11
11
22
3
2
0.75
0.75
11
22
3
1
, 0.35, 3 on the number line.
2
35
1
3
5 numbers
2 from least to greatest:
Order the following
Correctly show
7 , 6.2, 1 , 0.88.
6
12
7
6
Example:
•
Write at least two fractions equivalent to each
fraction given below:
1 5 2
2, 6 , 3
4.2.G
Simplify fractions using common factors.
4.2.H
Round fractions and decimals to the nearest
whole number.
July 2008
Washington State K–12 Mathematics Standards
49
Grade 4
Performance Expectations
Explanatory Comments and Examples
Students are expected to:
4.2.I
Solve single- and multi-step word problems
involving comparison of decimals and fractions
(including mixed numbers), and verify
the solutions.
The intent of this expectation is for students to show
their work, explain their thinking, and verify that the
answer to the problem is reasonable in terms of the
original context and the mathematics used to solve the
problem. Verifications can include the use of numbers,
words, pictures, or equations.
Example:
•
Ms. Ortiz needs 11
11
pounds of sliced turkey. She
22
picked up a package labeled “1.12 lbs.” Would she
have enough turkey with this package? Explain
why or why not.
50
July 2008
Washington State K–12 Mathematics Standards
Grade 4
Grade 4
4.3. Core Content: Concept of area (Geometry/Measurement, Algebra)
S
tudents learn how to find the area of a rectangle as a basis for later work with areas of other
geometric figures. They select appropriate units, tools, and strategies, including formulas, and use
them to solve problems involving perimeter and area. Solving such problems helps students develop
spatial skills, which are critical for dealing with a wide range of geometric concepts. The study of area is
closely connected to Core Content on multiplication, and connections between these concepts should be
emphasized whenever possible.
Performance Expectations
Explanatory Comments and Examples
Students are expected to:
4.3.A
Determine congruence of two-dimensional
figures.
At this grade level, students determine congruence
primarily by making direct comparisons (e.g., tracing
or cutting). They may also use informal notions of
transformations described as flips, turns, and slides.
Both the language and the concepts of transformations
are more formally developed in grade eight.
4.3.B
Determine the approximate area of a figure
using square units.
Examples:
4.3.C Determine the perimeter and area of a
rectangle using formulas, and explain why the
formulas work.
•
Draw a rectangle 3.5 cm by 6 cm on centimeter
grid paper. About how many squares fit inside
the rectangle?
•
Cover a footprint with square tiles or outline it on
grid paper. About how many squares fit inside
the footprint?
This is an opportunity to connect area to the concept
of multiplication, a useful model for multiplication that
extends into algebra. Students should also work with
squares as special rectangles.
Example:
•
4.3.D
Determine the areas of figures that can be
broken down into rectangles.
Outline on grid paper a rectangle that is 4 units
long and 3 units wide. Without counting the
squares, how can you determine the area? Other
than measuring, how could you use a shortcut to
find the perimeter of the rectangle?
Example:
•
Find the area of each figure:
1
3
1
July 2008
Washington State K–12 Mathematics Standards
3
7
3
6
6
51
Grade 4
Performance Expectations
Explanatory Comments and Examples
Students are expected to:
4.3.E
4.3.F
Demonstrate that rectangles with the same
area can have different perimeters, and that
rectangles with the same perimeter can have
different areas.
Example:
Solve single- and multi-step word problems
involving perimeters and areas of rectangles
and verify the solutions.
The intent of this expectation is for students to show
their work, explain their thinking, and verify that the
answer to the problem is reasonable in terms of the
original context and the mathematics used to solve the
problem. Verifications can include the use of numbers,
words, pictures, or equations.
•
Draw different rectangles, each with an area of
24 square units, and compare their perimeters.
What patterns do you notice in the data? Record
your observations.
Problems include those involving U.S. customary and
metric units, including square units.
52
July 2008
Washington State K–12 Mathematics Standards
Grade 4
Grade 4
4.4. Additional Key Content
S
(Geometry/Measurement, Algebra, Data/Statistics/Probability)
tudents use coordinate grids to connect numbers to basic ideas in algebra and geometry. This
connection between algebra and geometry runs throughout advanced mathematics and allows
students to use tools from one branch of mathematics to solve problems related to another branch.
Students also extend and reinforce their work with whole numbers and fractions to describe sets of data
and find simple probabilities. Students combine measurement work with their developing ideas about
multiplication and division as they do basic measurement conversions. They begin to use algebraic
notation while solving problems in preparation for formalizing algebraic thinking in later grades.
Performance Expectations
Explanatory Comments and Examples
Students are expected to:
4.4.A
4.4.B
Represent an unknown quantity in simple
expressions, equations, and inequalities using
letters, boxes, and other symbols.
Solve single- and multi-step problems involving
familiar unit conversions, including time, within
either the U.S. customary or metric system.
4.4.C
Estimate and determine elapsed time using a
calendar, a digital clock, and an analog clock.
4.4.D
Graph and identify points in the first quadrant
of the coordinate plane using ordered pairs.
Example:
•
There are 5 jars. Lupe put the same number
of marbles in each jar. Write an equation or
expression that shows how many marbles are in
each jar if there are 40 marbles total.
[5 × = 40 or 5 × M = 40;
M represents the number of marbles]
Examples:
•
Jill bought 3 meters of ribbon and cut it into pieces
25 centimeters long. How many 25-centimeter
pieces of ribbon did she have?
•
How many quarts of lemonade are needed to
make 25 one-cup servings?
Example:
y
(5, 5)
5
(1, 4)
4
(5, 3)
3
(2, 2)
2
(3, 1)
1
(0, 0)
July 2008
Washington State K–12 Mathematics Standards
0
1
2
3
4
5
x
53
Grade 4
Performance Expectations
Explanatory Comments and Examples
Students are expected to:
4.4.E
Determine the median, mode, and range of a
set of data and describe what each measure
indicates about the data.
Example:
•
What is the median number of siblings that
students in this class have? What is the mode
of the data? What is the range of the number of
siblings? What does each of these values tell you
about the students in the class?
Siblings of Class 4A
Frequency
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
0
1
2
3
4
X
5
6
Number of Siblings of Class 4A
4.4.F
Describe and compare the likelihood of events.
For this introduction to probability, an event can be
described as certain, impossible, likely, or unlikely.
Two events can be compared as equally likely, not
equally likely, or as one being more likely or less
likely than the other.
4.4.G
Determine a simple probability from a context
that includes a picture.
Probability is expressed as a number from 0 to 1.
Example:
•
4.4.H
54
Display the results of probability experiments
and interpret the results.
What is the probability of a blindfolded person
choosing a black marble from the bowl?
Displays include tallies, frequency tables, graphs,
pictures, and fractions.
July 2008
Washington State K–12 Mathematics Standards
Grade 4
Grade 4
4.5. Core Processes: Reasoning, problem solving, and communication
S
tudents in grade four solve problems that extend their understanding of core mathematical
concepts—such as multiplication of multi-digit numbers, area, probability, and the relationships
between fractions and decimals—as they make strategic decisions that bring them to reasonable
solutions. Students use pictures, symbols, or mathematical language to explain the reasoning behind
their decisions and solutions. They further develop their problem-solving skills by making generalizations
about the processes used and applying these generalizations to similar problem situations. These critical
reasoning, problem-solving, and communication skills represent the kind of mathematical thinking that
equips students to use the mathematics they know to solve a growing range of useful and important
problems and to make decisions based on quantitative information.
Performance Expectations
Explanatory Comments and Examples
Students are expected to:
4.5.A
Determine the question(s) to be answered
given a problem situation.
4.5.B
Identify information that is given in a
problem and decide whether it is essential or
extraneous to the solution of the problem.
4.5.C
Identify missing information that is needed to
solve a problem.
4.5.D
Determine whether a problem to be solved
is similar to previously solved problems, and
identify possible strategies for solving the
problem.
4.5.E
Select and use one or more appropriate
strategies to solve a problem and explain why
that strategy was chosen.
Descriptions of solution processes and explanations
can include numbers, words (including mathematical
language), pictures, physical objects, or equations.
Students should be able to use all of these
representations as needed. For a particular solution,
students should be able to explain or show their work
using at least one of these representations and verify
that their answer is reasonable.
Examples:
•
Jake’s family adopted a small dog, Toto. They
have a rectangular dog pen that is 10 feet by 20
feet. Toto needs only half that area, so Jake plans
to make the pen smaller by cutting each dimension
in half. Jake’s mother asked him to rethink his plan
or Toto won’t have the right amount of space.
— Whose reasoning is correct—Jake’s or his
mother’s? Why?
4.5.F
Represent a problem situation using
words, numbers, pictures, physical objects,
or symbols.
— According to Jake’s plan, what fractional
part of the old pen will be the area of the
new pen? Give the answer in simplest form.
4.5.G
Explain why a specific problem-solving
strategy or procedure was used to determine
a solution.
— Make a new plan so that the area of the
new pen is half the area of the old pen.
4.5.H
Analyze and evaluate whether a solution is
reasonable, is mathematically correct, and
answers the question.
4.5.I
Summarize mathematical information, draw
conclusions, and explain reasoning.
4.5.J
Make and test conjectures based on data
(or information) collected from explorations
and experiments.
July 2008
Washington State K–12 Mathematics Standards
•
The city is paying for a new deck around the
community pool. The rectangular pool measures
50 meters by 25 meters. The deck, which will
measure 5 meters wide, will surround the pool like
a picture frame. If the cost of the deck is $25 for
each square meter, what will be the total cost for
the new deck? Explain your solution.
55
Grade 4
Grade 5
July 2008
Washington State K–12 Mathematics Standards
57
Grade 5
Grade 5
5.1. Core Content: Multi-digit division
(Operations, Algebra)
S
tudents learn efficient ways to divide whole numbers. They apply what they know about division
to solve problems, using estimation and mental math skills to decide whether their results are
reasonable. This emphasis on division gives students a complete set of tools for adding, subtracting,
multiplying, and dividing whole numbers—basic skills for everyday life and further study of mathematics.
Performance Expectations
Explanatory Comments and Examples
Students are expected to:
5.1.A
Represent multi-digit division using
place value models and connect the
representation to the related equation.
Students use pictures or grid paper to represent
division and describe how that representation connects
to the related equation. They could also use physical
objects such as base ten blocks to support the visual
representation. Note that the algorithm for long division
is addressed in expectation 5.1.C.
5.1.B Determine quotients for multiples of 10 and
100 by applying knowledge of place value and
properties of operations.
Example:
5.1.C
The use of ‘R’ or ‘r’ to indicate a remainder may
be appropriate in most of the examples students
encounter in grade five. However, students should
also be aware that in subsequent grades, they will
learn additional ways to represent remainders, such as
fractional or decimal parts.
Fluently and accurately divide up to a four-digit
number by one- or two-digit divisors using the
standard long-division algorithm.
•
Using the fact that 16 ÷ 4 = 4, students can
generate the related quotients 160 ÷ 4 = 40 and
160 ÷ 40 = 4.
Example:
132 r1
6 793
-6
19
-18
13
-1 2
1
Teachers should be aware that in some countries the
algorithm might be recorded differently.
5.1.D
Estimate quotients to approximate solutions
and determine reasonableness of answers in
problems involving up to two-digit divisors.
Example:
•
The team has saved $45 to buy soccer balls. If the
balls cost $15.95 each, is it reasonable to think
there is enough money for more than two balls?
Problems like 54,596 ÷ 798, which can be estimated by
56,000 ÷ 800, while technically beyond the standards,
could be included when appropriate. The numbers
are easily manipulated and the problems support the
ongoing development of place value.
July 2008
Washington State K–12 Mathematics Standards
59
Grade 5
Performance Expectations
Explanatory Comments and Examples
Students are expected to:
5.1.E
Mentally divide two-digit numbers by one-digit
divisors and explain the strategies used.
5.1.F
Solve single- and multi-step word problems
involving multi-digit division and verify
the solutions.
The intent of this expectation is for students to show
their work, explain their thinking, and verify that the
answer to the problem is reasonable in terms of the
original context and the mathematics used to solve the
problem. Verifications can include the use of numbers,
words, pictures, or equations.
Problems include those with and without remainders.
60
July 2008
Washington State K–12 Mathematics Standards
Grade 5
Grade 5
5.2. Core Content: Addition and subtraction of fractions and decimals
(Numbers, Operations, Algebra)
S
tudents extend their knowledge about adding and subtracting whole numbers to learning procedures
for adding and subtracting fractions and decimals. Students apply these procedures, along with
mental math and estimation, to solve a wide range of problems that involve more of the types of numbers
students see in other school subjects and in their lives.
Performance Expectations
Explanatory Comments and Examples
Students are expected to:
5.2.A
Represent addition and subtraction of fractions
and mixed numbers using visual and numerical
models, and connect the representation to the
related equation.
This expectation includes numbers with like and unlike
denominators. Students should be able to show these
operations on a number line and should be familiar
with the use of pictures and physical materials (like
fraction pieces or fraction bars) to represent addition
and subtraction of mixed numbers. They should be
able to describe how a visual representation connects
to the related equation.
Example:
3 3
–
=
2 4
0
1
4
1
2
3
4
1
5
4
3
2
7
4
2
5.2.B
Represent addition and subtraction of decimals
using place value models and connect the
representation to the related equation.
Students should be familiar with using pictures
and physical objects to represent addition and
subtraction of decimals and be able to describe how
those representations connect to related equations.
Representations may include base ten blocks, number
lines, and grid paper.
5.2.C
Given two fractions with unlike denominators,
rewrite the fractions with a common
denominator.
Fraction pairs include denominators with and without
common factors.
When students are fluent in writing equivalent
fractions, it helps them compare fractions and helps
prepare them to add and subtract fractions.
Examples:
•
Write equivalent fractions with a common
2
3
and .
3
4
3
4 with a common
Write equivalent fractions
1
3
denominator for
and .
6
38
denominator for
•
8
July 2008
Washington State K–12 Mathematics Standards
61
Grade 5
Performance Expectations
Explanatory Comments and Examples
Students are expected to:
5.2.D
Determine the greatest common factor and
the least common multiple of two or more
whole numbers.
Least common multiple (LCM) can be used to
determine common denominators when adding
and subtracting fractions.
Greatest common factor (GCF) can be used to
simplify fractions.
5.2.E
Fluently and accurately add and subtract
fractions, including mixed numbers.
Fractions can be in either proper or improper form.
Students should also be able to work with whole
numbers as part of this expectation.
5.2.F
Fluently and accurately add and subtract
decimals.
Students should work with decimals less than 1 and
greater than 1, as well as whole numbers, as part of
this expectation.
5.2.G
Estimate sums and differences of fractions,
mixed numbers, and decimals to approximate
solutions to problems and determine
reasonableness of answers.
Example:
•
10
5.2.H
Solve single- and multi-step word problems
involving addition and subtraction of whole
numbers, fractions (including mixed numbers),
and decimals, and verify the solutions.
Jared is making a frame for a picture that is
33
1
inches wide and 15 inches tall.
44
8
He has a 4-ft length of metal framing material.
Estimate whether he will have enough framing
material to frame the picture.
The intent of this expectation is for students to show
their work, explain their thinking, and verify that the
answer to the problem is reasonable in terms of the
original context and the mathematics used to solve the
problem. Verifications can include the use of numbers,
words, pictures, or equations.
Multi-step problems may also include previously
learned computational skills like multiplication and
division of whole numbers.
62
July 2008
Washington State K–12 Mathematics Standards
Grade 5
Grade 5
5.3. Core Content: Triangles and quadrilaterals
(Geometry/Measurement, Algebra)
S
tudents focus on triangles and quadrilaterals to formalize and extend their understanding of these
geometric shapes. They classify different types of triangles and quadrilaterals and develop formulas
for their areas. In working with these formulas, students reinforce an important connection between
algebra and geometry. They explore symmetry of these figures and use what they learn about triangles
and quadrilaterals to solve a variety of problems in geometric contexts.
Performance Expectations
Explanatory Comments and Examples
Students are expected to:
5.3.A
Classify quadrilaterals.
Students sort a set of quadrilaterals into their various
types, including parallelograms, kites, squares, rhombi,
trapezoids, and rectangles, noting that a square can
also be classified as a rectangle, parallelogram,
and rhombus.
5.3.B
Identify, sketch, and measure acute, right, and
obtuse angles.
Example:
Identify, describe, and classify triangles by
angle measure and number of congruent sides.
Students classify triangles by their angle size using the
terms acute, right, or obtuse.
5.3.C
•
Use a protractor to measure the following angles
and label each as acute, right, or obtuse.
Students classify triangles by the length of their sides
using the terms scalene, isosceles, or equilateral.
5.3.D
Determine the formula for the area of a parallelogram by relating it to the area of a rectangle.
Students relate the area of a parallelogram to the area
of a rectangle, as shown below.
5.3.E
Determine the formula for the area of a triangle
by relating it to the area of a parallelogram.
Students relate the area of a triangle to the area of a
parallelogram, as shown below.
5.3.F
Determine the perimeters and areas of
triangles and parallelograms.
July 2008
Washington State K–12 Mathematics Standards
Students may be given figures showing some side
measures or may be expected to measure sides of
figures. If students are not given side measures, but
instead are asked to make their own measurements,
it is important to discuss the approximate nature of
any measurement.
63
Grade 5
Performance Expectations
Explanatory Comments and Examples
Students are expected to:
5.3.G
5.3.H
Draw quadrilaterals and triangles from given
information about sides and angles.
Determine the number and location of lines of
symmetry in triangles and quadrilaterals.
Examples:
•
Draw a triangle with one right angle and no
congruent sides.
•
Draw a rhombus that is not a square.
•
Draw a right scalene triangle.
Example:
•
Draw and count all the lines of symmetry in the
square and isosceles triangle below. (Lines of
symmetry are shown as dotted lines.)
5.3.I
64
Solve single- and multi-step word problems
about the perimeters and areas of
quadrilaterals and triangles and verify
the solutions.
The intent of this expectation is for students to show
their work, explain their thinking, and verify that the
answer to the problem is reasonable in terms of the
original context and the mathematics used to solve the
problem. Verifications can include the use of numbers,
words, pictures, or equations.
July 2008
Washington State K–12 Mathematics Standards
Grade 5
Grade 5
5.4. Core Content: Representations of algebraic relationships
(Operations, Algebra)
S
tudents continue their development of algebraic thinking as they move toward more in-depth study
of algebra in middle school. They use variables to write simple algebraic expressions describing
patterns or solutions to problems. They use what they have learned about numbers and operations
to evaluate simple algebraic expressions and to solve simple equations. Students make tables and
graphs from linear equations to strengthen their understanding of algebraic relationships and to see the
mathematical connections between algebra and geometry. These foundational algebraic skills allow
students to see where mathematics, including algebra, can be used in real situations, and these skills
prepare students for success in future grades.
Performance Expectations
Explanatory Comments and Examples
Students are expected to:
5.4.A
Describe and create a rule for numerical and
geometric patterns and extend the patterns.
Example:
•
5.4.B
Write a rule to describe the relationship between
two sets of data that are linearly related.
The picture shows a sequence of towers
constructed from cubes. The number of cubes
needed to build each tower forms a numeric
pattern. Determine a rule for the number of cubes in
each tower and use the rule to extend this pattern.
Tower 1
Tower 2
Tower 3
Rules can be written using words or algebraic expressions.
Example:
•
The table below shows numerators (top row) and
denominators (bottom row) of fractions equivalent to
1
3
a given fraction ( ). Write a rule that could be used
to describe how the two rows could be related.
July 2008
Washington State K–12 Mathematics Standards
1
2
3
4
3
6
9
?
65
Grade 5
Performance Expectations
Explanatory Comments and Examples
Students are expected to:
5.4.C
Write algebraic expressions that represent
simple situations and evaluate the expressions,
using substitution when variables are involved.
Students should evaluate expressions with and without
parentheses. Evaluating expressions with parentheses is
an initial step in learning the proper order of operations.
Examples:
•
Evaluate (4 × n) + 5 when n = 2.
•
If 4 people can sit at 1 table, 8 people can sit at
2 tables, and 12 people can sit at 3 tables, and
this relationship continues, write an expression
to describe the number of people who can sit at
n tables and tell how many people can sit at 67
tables.
•
Compare the answers to A and B below.
A: (3 x 10) + 2
B: 3 x (10 + 2)
5.4.D
Graph ordered pairs in the coordinate plane
for two sets of data related by a linear rule and
draw the line they determine.
Example:
•
The table shows the total cost of purchasing
different quantities of equally priced DVDs.
number
purchased
total cost
0
2
5
$0
$10
$25
Graph the ordered pairs (0, 0), (2, 10), and (5, 25)
and the line connecting the ordered pairs. Use
the line to determine the total cost when 3 DVDs
are purchased.
Cost of DVDs Purchased
Cost and Number of DVDs Purchased
25
20
15
10
5
0
66
1
2
3
4
5
Number of DVDs Purchased
July 2008
Washington State K–12 Mathematics Standards
Grade 5
Grade 5
5.5. Additional Key Content (Numbers, Data/Statistics/Probability)
S
tudents extend their work with common factors and common multiples as they deal with prime
numbers. Students extend and reinforce their use of numbers, operations, and graphing to describe
and compare data sets for increasingly complex situations they may encounter in other school subjects
and in their lives.
Performance Expectations
Explanatory Comments and Examples
Students are expected to:
5.5.A
Classify numbers as prime or composite.
Divisibility rules can help determine whether a number
has particular factors.
5.5.B
Determine and interpret the mean of a small
data set of whole numbers.
At this grade level, numbers for problems are selected
so that the mean will be a whole number.
Examples:
•
Seven families report the following number of pets.
Determine the mean number of pets per family.
0, 3, 3, 3, 5, 6, and 8
[One way to interpret the mean for this data set is
to say that if the pets are redistributed evenly, each
family will have 4 pets.]
•
The heights of five trees in front of the school are
given below. What is the average height of these
trees? Does this average seem to represent the
‘typical’ size of these trees? Explain your answer.
3 ft, 4 ft, 4 ft, 4 ft, 20 ft
5.5.C
Construct and interpret line graphs.
Line graphs are used to display changes in data over time.
Example:
•
Below is a line graph that shows the temperature
of a can of juice after the can has been placed in
ice and salt over a period of time. Describe any
conclusions you can make about the data.
Temperature of Apple Juice After Cooling in Salt and Ice
60
Temperature in
Degrees Fahrenheit
55
50
45
40
35
30
1
2
3
4
5
Time in Minutes
July 2008
Washington State K–12 Mathematics Standards
67
Grade 5
Grade 5
5.6. Core Processes: Reasoning, problem solving, and communication
S
tudents in grade five solve problems that extend their understanding of core mathematical
concepts—such as division of multi-digit numbers, perimeter, area, addition and subtraction of
fractions and decimals, and use of variables in expressions and equations—as they make strategic
decisions leading to reasonable solutions. Students use pictures, symbols, or mathematical language to
explain the reasoning behind their decisions and solutions. They further develop their problem-solving
skills by making generalizations about the processes used and applying these generalizations to similar
problem situations. These critical reasoning, problem-solving, and communication skills represent the kind
of mathematical thinking that equips students to use the mathematics they know to solve a growing range
of useful and important problems and to make decisions based on quantitative information.
Performance Expectations
Explanatory Comments and Examples
Students are expected to:
5.6.A
Determine the question(s) to be answered
given a problem situation.
5.6.B
Identify information that is given in a
problem and decide whether it is essential or
extraneous to the solution of the problem.
5.6.C
Determine whether additional information is
needed to solve the problem.
5.6.D
Determine whether a problem to be solved is
similar to previously solved problems,
and identify possible strategies for solving
the problem.
5.6.E
Select and use one or more appropriate
strategies to solve a problem, and explain the
choice of strategy.
5.6.F
Represent a problem situation using words,
numbers, pictures, physical objects, or
symbols.
5.6.G
Explain why a specific problem-solving
strategy or procedure was used to determine
a solution.
5.6.H
Analyze and evaluate whether a solution is
reasonable, is mathematically correct, and
answers the question.
5.6.I
Summarize mathematical information, draw
conclusions, and explain reasoning.
5.6.J
Make and test conjectures based on data
(or information) collected from explorations
and experiments.
Descriptions of solution processes and explanations
can include numbers, words (including mathematical
language), pictures, physical objects, or equations.
Students should be able to use all of these
representations as needed. For a particular solution,
students should be able to explain or show their work
using at least one of these representations and verify
that their answer is reasonable.
Examples:
•
La Casa Restaurant uses rectangular tables.
One table seats 6 people, with 1 person at each
end and 2 people on each long side. However, 2
tables pushed together, short end to short end,
seat only 10 people. Three tables pushed together
end-to-end seat only 14 people. Write a rule that
describes how many can sit at n tables pushed
together end-to-end. The restaurant’s long banquet
hall has tables pushed together in a long row to
seat 70. How many tables were pushed together to
seat this many people? How do you know?
•
The small square in the tangram figure below is
8
the area of the large square.
For each of the 7 tangram pieces that make up the
large square, tell what fractional part of the large
square that piece represents. How do you know?
1
68
July 2008
Washington State K–12 Mathematics Standards
Grade 6
July 2008
Washington State K–12 Mathematics Standards
69
Grade 6
Grade 6
6.1. Core Content: Multiplication and division of fractions and decimals
S
(Numbers, Operations, Algebra)
tudents have done extensive work with fractions and decimals in previous grades and are now
prepared to learn how to multiply and divide fractions and decimals with understanding. They can
solve a wide variety of problems that involve the numbers they see every day—whole numbers, fractions,
and decimals. By using approximations of fractions and decimals, students estimate computations and
verify that their answers make sense.
Performance Expectations
Explanatory Comments and Examples
Students are expected to:
6.1.A
Compare and order non-negative fractions,
decimals, and integers using the number line,
lists, and the symbols <, >, or =.
Examples:
•
List the numbers 2
1
3
,
4
5
, 0.94,
5
4
,
1.1, and
43
in
50
1 then
4 graph
5 the numbers
43 on
increasing order, and
2 , , 0.94, , 1.1, and
the number line. 3 5
50
4
•
6.1.B
Represent multiplication and division of nonnegative fractions and decimals using area
models and the number line, and connect each
representation to the related equation.
Compare each pair of numbers using <, >, or =.
4
5
5
1.2
7
4
4
11
22
7
7
8
8
33
44
2.5
2
.5
This expectation addresses the conceptual meaning
of multiplication and division of fractions and decimals.
Students should be familiar with the use of visual
representations like pictures (e.g., sketching the
problem, grid paper) and physical objects (e.g.,
tangrams, cuisenaire rods). They should connect the
visual representation to the corresponding equation.
The procedures for multiplying fractions and decimals
are addressed in 6.1.D and 6.1.E.
6.1.C
Estimate products and quotients of fractions
and decimals.
Example:
•
0.28 ÷ 0.96 ≈ 0.3 ÷ 1; 0.3 ÷ 1 = 0.3
0.24 x 12.4
July 2008
Washington State K–12 Mathematics Standards
20
3
x
41
13
1
1
x 12.4 ;
x 12.4 = 3.1
4
4
1 ; 1
1 = 1
1
x
x
2
2
8
4
4
71
Grade 6
Performance Expectations
Explanatory Comments and Examples
Students are expected to:
6.1.D
Fluently and accurately multiply and divide
non-negative fractions and explain the inverse
relationship between multiplication and division
with fractions.
Students should understand the inverse relationship
between multiplication and division, developed in grade
three and now extended to fractions. Students should
work with different types of rational numbers, including
whole numbers and mixed numbers, as they continue
to expand their understanding of the set of rational
numbers.
Example:
•
Multiply or divide.
5
4
2
6.1.E
Multiply and divide whole numbers and
decimals by 1000, 100, 10, 1, 0.1, 0.01,
and 0.001.
×
1
4
2
6÷
3
×3
1
2
4
1
5
3
8
÷1
2
3
This expectation extends what students know about
the place value system and about multiplication and
division and expands their set of mental math tools.
As students work with multiplication by these powers
of 10, they can gain an understanding of how numbers
relate to each other based on their relative sizes.
Example:
•
6.1.F
Fluently and accurately multiply and divide
non-negative decimals.
Mentally compute 0.01 x 435.
Students should understand the inverse relationship
between multiplication and division, developed in
grade three and now extended to decimals. Students
should work with different types of decimals, including
decimals greater than 1, decimals less than 1, and
whole numbers, as they continue to expand their
understanding of the set of rational numbers.
Example:
•
6.1.G
72
Describe the effect of multiplying or dividing
a number by one, by zero, by a number
between zero and one, and by a number
greater than one.
Multiply or divide.
0.84 × 1.5
7.85 ÷ 0.32
2.04 × 32
17.28 ÷ 1.2
Examples:
•
Without doing any computation, list
74, 0.43 × 74, and 74 ÷ 0.85
in increasing order and explain your reasoning.
•
Explain why
4
is undefined.
0
July 2008
Washington State K–12 Mathematics Standards
Grade 6
Performance Expectations
Explanatory Comments and Examples
Students are expected to:
6.1.H
Solve single- and multi-step word problems
involving operations with fractions and
decimals and verify the solutions.
The intent of this expectation is for students to show
their work, explain their thinking, and verify that the
answer to the problem is reasonable in terms of the
original context and the mathematics used to solve the
problem. Verifications can include the use of numbers,
words, pictures, or equations.
Example:
•
July 2008
Washington State K–12 Mathematics Standards
Every day has 24 hours. Ali sleeps 3/8 of the day.
Dawson sleeps 1/3 of the day. Maddie sleeps
7.2 hours in a day. Who sleeps the longest? By
how much?
73
Grade 6
Grade 6
6.2. Core Content: Mathematical expressions and equations (Operations, Algebra)
S
tudents continue to develop their understanding of how letters are used to represent numbers in
mathematics—an important foundation for algebraic thinking. Students use tables, words, numbers,
graphs, and equations to describe simple linear relationships. They write and evaluate expressions and
write and solve equations. By developing these algebraic skills at the middle school level, students will be
able to make a smooth transition to high school mathematics.
Performance Expectations
Explanatory Comments and Examples
Students are expected to:
6.2.A
Write a mathematical expression or equation
with variables to represent information in a
table or given situation.
Examples:
•
•
6.2.B
Draw a first-quadrant graph in the coordinate
plane to represent information in a table or
given situation.
6.2.C Evaluate mathematical expressions when the
value for each variable is given.
6.2.D
74
Apply the commutative, associative, and
distributive properties, and use the order of
operations to evaluate mathematical expressions.
What expression can be substituted for the
question mark?
x
1
2
3
4
...
x
y
2.5
5
7.5
10
...
?
A t-shirt printing company charges $7 for each
t-shirt it prints. Write an equation that represents
the total cost, c, for ordering a specific quantity, t,
of these t-shirts.
Example:
•
Mikayla and her sister are making beaded
bracelets to sell at a school craft fair. They can
make two bracelets every 30 minutes. Draw a
graph that represents the number of bracelets the
girls will have made at any point during the 6 hours
they work.
Examples:
•
Evaluate 2s + 5t when s = 3.4 and t = 1.8.
•
Evaluate 3 x – 14 when x = 60.
2
Examples:
•
 1 1
Simplify 6  +  , with and without the use of
 2 3
the distributive property.
•
Evaluate b – 3(2a – 7) when a = 5.4 and b = 31.7.
July 2008
Washington State K–12 Mathematics Standards
Grade 6
Performance Expectations
Explanatory Comments and Examples
Students are expected to:
6.2.E
Solve one-step equations and verify solutions.
Students solve equations using number sense,
physical objects (e.g., balance scales), pictures, or
properties of equality.
Example:
•
Solve for the variable in each equation below.
112 = 7a
1.4y = 42
6.2.F
Solve word problems using mathematical
expressions and equations and verify solutions.
2
1
1
= b+
2
3
y
7
=
45 15
The intent of this expectation is for students to show
their work, explain their thinking, and verify that the
answer to the problem is reasonable in terms of the
original context and the mathematics used to solve the
problem. Verifications can include the use of numbers,
words, pictures, or equations.
Example:
•
July 2008
Washington State K–12 Mathematics Standards
Zane and his friends drove across the United
States at an average speed of 55 mph. Write
expressions to show how far they traveled in 12
hours, in 18 hours, and in n hours. How long did it
take them to drive 1,430 miles? Verify your solution.
75
Grade 6
Grade 6
6.3. Core Content: Ratios, rates, and percents
S
(Numbers, Operations, Geometry/Measurement,
Algebra, Data/Statistics/Probability)
tudents extend their knowledge of fractions to develop an understanding of what a ratio is and how
it relates to a rate and a percent. Fractions, ratios, rates, and percents appear daily in the media
and in everyday calculations like determining the sale price at a retail store or figuring out gas mileage.
Students solve a variety of problems related to such situations. A solid understanding of ratios and rates
is important for work involving proportional relationships in grade seven.
Performance Expectations
Explanatory Comments and Examples
Students are expected to:
6.3.A
6.3.B
Identify and write ratios as comparisons of
part-to-part and part-to-whole relationships.
Example:
Write ratios to represent a variety of rates.
Example:
•
•
6.3.C
Represent percents visually and numerically,
and convert between the fractional, decimal,
and percent representations of a number.
If there are 10 boys and 12 girls in a class, what
is the ratio of boys to girls? What is the ratio of the
number of boys to the total number of students in
the class?
Julio drove his car 579 miles and used 15 gallons
of gasoline. How many miles per gallon did his car
get during the trip? Explain your answer.
In addition to general translations among these
representations, this expectation includes the quick
recall of equivalent forms of common fractions (with
denominators like 2, 3, 4, 5, 8, and 10), decimals, and
percents. It also includes the understanding that a
fraction represents division, an important conceptual
background for writing fractions as decimals.
Examples:
•
Represent
75
as a percent using numbers, a
100
picture, and a circle graph.
76
•
Represent 40% as a fraction and as a decimal.
•
Write
13
as a decimal and as a percent.
16
July 2008
Washington State K–12 Mathematics Standards
Grade 6
Performance Expectations
Explanatory Comments and Examples
Students are expected to:
6.3.D
Solve single- and multi-step word problems
involving ratios, rates, and percents, and verify
the solutions.
The intent of this expectation is for students to show
their work, explain their thinking, and verify that the
answer to the problem is reasonable in terms of the
original context and the mathematics used to solve the
problem. Verifications can include the use of numbers,
words, pictures, or equations.
Examples:
6.3.E
Identify the ratio of the circumference to
the diameter of a circle as the constant
π, and recognize
22
and 3.14 as
7
•
An item is advertised as being 25% off the regular
price. If the sale price is $42, what was the original
regular price? Verify your solution.
•
Sally had a business meeting in a city 100 miles
away. In the morning, she drove an average speed
of 60 miles per hour, but in the evening when she
returned, she averaged only 40 miles per hour.
How much longer did the evening trip take than the
morning trip? Explain your reasoning.
Example:
•
common approximations of π.
6.3.F
Determine the experimental probability
of a simple event using data collected in
an experiment.
The term experimental probability refers here to the
relative frequency that was observed in an experiment.
Example:
•
6.3.G
Determine the theoretical probability of an
event and its complement and represent the
probability as a fraction or decimal from 0 to 1
or as a percent from 0 to 100.
July 2008
Washington State K–12 Mathematics Standards
Measure the diameter and circumference of several
circular objects. Divide each circumference by its
diameter. What do you notice about the results?
Tim is checking the apples in his orchard for
worms. Selecting apples at random, he finds 9
apples with worms and 63 apples without worms.
What is the experimental probability that a given
apple from his orchard has a worm in it?
Example:
•
A bag contains 4 green marbles, 6 red marbles,
and 10 blue marbles. If one marble is drawn
randomly from the bag, what is the probability it
will be red? What is the probability that it will not
be red?
77
Grade 6
Grade 6
6.4. Core Content: Two- and three-dimensional figures
(Geometry/Measurement, Algebra)
S
tudents extend what they know about area and perimeter to more complex two-dimensional figures,
including circles. They find the surface area and volume of simple three-dimensional figures. As they
learn about these important concepts, students can solve problems involving more complex figures than
in earlier grades and use geometry to deal with a wider range of situations. These fundamental skills of
geometry and measurement are increasingly called for in the workplace and they lead to a more formal
study of geometry in high school.
Performance Expectations
Explanatory Comments and Examples
Students are expected to:
6.4.A
6.4.B
Determine the circumference and area
of circles.
Determine the perimeter and area of a
composite figure that can be divided into
triangles, rectangles, and parts of circles.
Examples:
•
Determine the area of a circle with a diameter of
12 inches.
•
Determine the circumference of a circle with a
radius of 32 centimeters.
Although students have worked with various
quadrilaterals in the past, this expectation includes
other quadrilaterals such as trapezoids or irregular
quadrilaterals, as well as any other composite figure
that can be divided into figures for which students have
calculated areas before.
Example:
•
Determine the area and perimeter of each of the
following figures, assuming that the dimensions on
the figures are in feet. The curved portion of the
second figure is a semi-circle.
3
6
7
2
8
6.4.C
Solve single- and multi-step word problems
involving the relationships among radius,
diameter, circumference, and area of circles,
and verify the solutions.
8
The intent of this expectation is for students to show their
work, explain their thinking, and verify that the answer
to the problem is reasonable in terms of the original
context and the mathematics used to solve the problem.
Verifications can include the use of numbers, words,
pictures, or equations.
Example:
•
78
Captain Jenkins determined that the distance
around a circular island is 44 miles. What is the
distance from the shore to the buried treasure in the
center of the island? What is the area of the island?
July 2008
Washington State K–12 Mathematics Standards
Grade 6
Performance Expectations
Explanatory Comments and Examples
Students are expected to:
6.4.D
Recognize and draw two-dimensional
representations of three-dimensional figures.
The net of a rectangular prism consists of six rectangles
that can then be folded to make the prism. The net of a
cylinder consists of two circles and a rectangle.
Example:
6.4.E
Determine the surface area and volume of
rectangular prisms using appropriate formulas
and explain why the formulas work.
6.4.F
Determine the surface area of a pyramid.
6.4.G
Describe and sort polyhedra by their attributes:
parallel faces, types of faces, number of faces,
edges, and vertices.
Students may determine surface area by calculating
the area of the faces and adding the results.
Prisms and pyramids are the focus at this level.
Examples:
•
How many pairs of parallel faces does each
polyhedron have? Explain your answer.
•
July 2008
Washington State K–12 Mathematics Standards
What type of polyhedron has two parallel triangular
faces and three non-parallel rectangular faces?
79
Grade 6
Grade 6
6.5. Additional Key Content
(Numbers, Operations)
S
tudents extend their mental math skills now that they have learned all of the operations—addition,
subtraction, multiplication, and division—with whole numbers, fractions, and decimals. Students
continue to expand their understanding of our number system as they are introduced to negative numbers
for describing positions or quantities below zero. These numbers are a critical foundation for algebra, and
students will learn how to add, subtract, multiply, and divide positive and negative numbers in seventh
grade as further preparation for algebraic study.
Performance Expectations
Explanatory Comments and Examples
Students are expected to:
6.5.A
Use strategies for mental computations
with non-negative whole numbers, fractions,
and decimals.
Examples:
•
John wants to find the total number of hours he
worked this week. Use his time card below to find
the total.
Days
•
80
Monday Tuesday Wednesday Thursday
Days
41
4
3
61
2
71
2
Friday
11
2
What is the total cost for items priced at $25.99
and $32.95? (A student may think of something
like 25.99 + 32.95 = (26 + 33) – 0.06 = 58.94.)
6.5.B
Locate positive and negative integers on the
number line and use integers to represent
quantities in various contexts.
Contexts could include elevation, temperature, or debt,
among others.
6.5.C
Compare and order positive and negative
integers using the number line, lists, and the
symbols <, >, or =.
Examples:
•
Compare each pair of numbers using <, >, or =.
-11  -14
-7  4
-101  -94
July 2008
Washington State K–12 Mathematics Standards
Grade 6
Grade 6
6.6. Core Processes: Reasoning, problem solving, and communication
S
tudents refine their reasoning and problem-solving skills as they move more fully into the symbolic
world of algebra and higher-level mathematics. They move easily among representations—
numbers, words, pictures, or symbols—to understand and communicate mathematical ideas, to make
generalizations, to draw logical conclusions, and to verify the reasonableness of solutions to problems.
In grade six, students solve problems that involve fractions and decimals as well as rates and ratios in
preparation for studying proportional relationships and algebraic reasoning in grade seven.
Performance Expectations
Explanatory Comments and Examples
Students are expected to:
6.6.A
Analyze a problem situation to determine the
question(s) to be answered.
6.6.B
Identify relevant, missing, and extraneous
information related to the solution to a problem.
6.6.C
Analyze and compare mathematical strategies
for solving problems, and select and use one
or more strategies to solve a problem.
6.6.D
Represent a problem situation, describe the
process used to solve the problem, and verify
the reasonableness of the solution.
6.6.E
Descriptions of solution processes and explanations
can include numbers, words (including mathematical
language), pictures, physical objects, or equations.
Students should be able to use all of these
representations as needed. For a particular solution,
students should be able to explain or show their work
using at least one of these representations and verify
that their answer is reasonable.
Examples:
•
5
mile on each side. On Monday, she
8
2
started at one corner of the park and jogged
3
Communicate the answer(s) to the
question(s) in a problem using appropriate
representations, including symbols and
informal and formal mathematical language.
6.6.F
Apply a previously used problem-solving
strategy in a new context.
6.6.G
Extract and organize mathematical information
from symbols, diagrams, and graphs to make
inferences, draw conclusions, and justify
reasoning.
6.6.H
Make and test conjectures based on data (or
information) collected from explorations and
experiments.
July 2008
Washington State K–12 Mathematics Standards
As part of her exercise routine, Carmen jogs
twice around the perimeter of a square park that
measures
of the way around in 17 minutes before stopping
at a small pond in the park to feed some ducks.
How far had Carmen run when she reached the
pond? What percent of her planned total distance
had Carmen completed when she stopped to feed
the ducks? If it took Carmen 17 minutes to jog to
the point where she stopped, assuming that she
continued running in the same direction at the
same pace and did not stop again, how long would
it have taken her to get back to her starting point?
Explain your answers.
•
At Springhill Elementary School’s annual fair,
Vanessa is playing a game called “Find the Key.”
A key is randomly placed somewhere in one of the
rooms shown on the map below. (The key cannot
be placed in the hallway.)
81
Grade 6
Performance Expectations
Explanatory Comments and Examples
Students are expected to:
To win the game, Vanessa must correctly guess
the room where the key is placed. Use what you
know about the sizes of the rooms to determine
the probability that the key is placed in the gym,
the office, the café, the book closet, or the library.
Write each probability as a simplified fraction,
a decimal, and a percent. Which room should
Vanessa select in order to have the best chance of
winning? Justify the solution.
Gym
82
Office
Hallway
6.6 cont.
Book
Closet
Library
Cafe
July 2008
Washington State K–12 Mathematics Standards
Grade 7
July 2008
Washington State K–12 Mathematics Standards
83
Grade 7
Grade 7
7.1. Core Content: Rational numbers and linear equations
(Numbers, Operations, Algebra)
S
tudents add, subtract, multiply, and divide rational numbers—fractions, decimals, and integers—including
both positive and negative numbers. With the inclusion of negative numbers, students can move more
deeply into algebraic content that involves the full set of rational numbers. They also approach problems that
deal with a wider range of contexts than before. Using generalized algebraic skills and approaches, students
can approach a wide range of problems involving any type of rational number, adapting strategies for solving
one problem to different problems in different settings with underlying similarities.
Performance Expectations
Explanatory Comments and Examples
Students are expected to:
7.1.A
Compare and order rational numbers using the
number line, lists, and the symbols <, >, or =.
Examples:
•
List the numbers
2 2
7
4 4
, - , 1.2, , - , -1.2, and 3 3
4
3 3
in increasing order, and graph the numbers on the
number line.
•
7.1.B
Represent addition, subtraction, multiplication,
and division of positive and negative integers
visually and numerically.
Compare each pair of numbers using <, >, or =.
- 11
20
- 13
21
-7
5
-1.35
3
4
-2.75
-2
Students should be familiar with the use of the number
line and physical materials, such as colored chips,
to represent computation with integers. They should
connect numerical and physical representations to the
computation. The procedures are addressed in 7.1.C.
Examples:
July 2008
Washington State K–12 Mathematics Standards
•
Use a picture, words, or physical objects to
illustrate 3 – 7; -3 – 7; -3 – (-7); (-3)(-7);
21 ÷ (-3).
•
At noon on a certain day, the temperature was
13°; at 10 p.m. the same day, the temperature was
-8°. How many degrees did the temperature drop
between noon and 10 p.m.?
85
Grade 7
Performance Expectations
Explanatory Comments and Examples
Students are expected to:
7.1.C
Fluently and accurately add, subtract, multiply,
and divide rational numbers.
This expectation brings together what students know
about the four operations with positive and negative
numbers of all kinds—integers, fractions, and decimals.
Some of these skills will have been recently learned and
may need careful development and reinforcement.
This is an opportunity to demonstrate connections
among the operations and to show similarities and
differences in the performance of these operations
with different types of numbers. Visual representations
may be helpful as students begin this work, and they
may become less necessary as students become
increasingly fluent with the operations.
Examples:
•
•
•
7.1.D
Define and determine the absolute value
of a number.
-
4 3
− =
3 4
272
=
8
(3.5)(-6.4) =
Students define absolute value as the distance of the
number from zero.
Examples:
7.1.E
Solve two-step linear equations.
•
Explain why 5 and -5 have the same absolute value.
•
Evaluate |7.8 – 10.3|.
Example:
•
7.1.F
Write an equation that corresponds to a given
problem situation, and describe a problem
situation that corresponds to a given equation.
Solve 3.5x – 12 = 408 and show each step in
the process.
Students have represented various types of problems
with expressions and particular types of equations in
previous grades. Many students at this grade level will
also be able to deal with inequalities.
Examples:
86
•
Meagan spent $56.50 on 3 blouses and a pair of
jeans. If each blouse cost the same amount and
the jeans cost $25, write an algebraic equation that
represents this situation and helps you determine
how much one blouse cost.
•
Describe a problem situation that could be solved
using the equation 15 = 2x – 7.
July 2008
Washington State K–12 Mathematics Standards
Grade 7
Performance Expectations
Explanatory Comments and Examples
Students are expected to:
7.1.G
Solve single- and multi-step word problems
involving rational numbers and verify
the solutions.
The intent of this expectation is for students to show
their work, explain their thinking, and verify that the
answer to the problem is reasonable in terms of the
original context and the mathematics used to solve the
problem. Verifications can include the use of numbers,
words, pictures, or equations.
Example:
•
Tom wants to buy some candy bars and
magazines for a trip. He has decided to buy three
times as many candy bars as magazines. Each
candy bar costs $0.70 and each magazine costs
$2.50. The sales tax rate on both types of items is
1
6 %. How many of each item can he buy if he has
2
$20.00 to spend?
July 2008
Washington State K–12 Mathematics Standards
87
Grade 7
Grade 7
7.2. Core Content: Proportionality and similarity
(Operations, Geometry/Measurement, Algebra)
S
tudents extend their work with ratios to solve problems involving a variety of proportional
relationships, such as making conversions between measurement units or finding the percent
increase or decrease of an amount. They also solve problems involving the proportional relationships
found in similar figures, and in so doing reinforce an important connection between numerical
operations and geometric relationships. Students graph proportional relationships and identify the rate
of change as the slope of the related line. The skills and concepts related to proportionality represent
some of the most important connecting ideas across K–12 mathematics. With a good understanding
of how things grow proportionally, students can understand the linear relationships that are the basis
for much of high school mathematics. If learned well, proportionality can open the door for success in
much of secondary mathematics.
Performance Expectations
Explanatory Comments and Examples
Students are expected to:
7.2.A
7.2.B
Mentally add, subtract, multiply, and divide
simple fractions, decimals, and percents.
Example:
Solve single- and multi-step problems involving
proportional relationships and verify the solutions.
The intent of this expectation is for students to show
their work, explain their thinking, and verify that the
answer to the problem is reasonable in terms of the
original context and the mathematics used to solve the
problem. Verifications can include the use of numbers,
words, pictures, or equations.
•
A shirt is on sale for 20% off the original price of
$15. Use mental math strategies to calculate the
sale price of the shirt.
Problems include those that involve rate, percent
increase or decrease, discount, markup, profit, interest,
tax, or the conversion of money or measurement
(including multiplying or dividing amounts in recipes).
More complex problems, such as dividing 100
into more than two proportional parts (e.g., 4:3:3),
allow students to generalize what they know about
proportional relationships to a range of situations.
Examples:
88
•
At a certain store, 48 television sets were sold
in April. The manager at the store wants to
encourage the sales team to sell more TVs and is
going to give all the sales team members a bonus
if the number of TVs sold increases by 30% in
May. How many TVs must the sales team sell in
May to receive the bonus? Explain your answer.
•
After eating at a restaurant, you know that the bill
before tax is $52.60 and that the sales tax rate is
8%. You decide to leave a 20% tip for the waiter
based on the pre-tax amount. How much should
you leave for the waiter? How much will the total
bill be, including tax and tip? Show work to support
your answers.
July 2008
Washington State K–12 Mathematics Standards
Grade 7
Performance Expectations
Explanatory Comments and Examples
Students are expected to:
7.2.B cont.
•
7.2.C
Students should recognize the constant ratios in
similar figures and be able to describe the role of a
scale factor in situations involving similar figures. They
should be able to connect this work with more general
notions of proportionality.
Describe proportional relationships in
similar figures and solve problems involving
similar figures.
Joe, Sam, and Jim completed different amounts
of yard work around the school. They agree to
split the $200 they earned in a ratio of 5:3:2,
respectively. How much did each boy receive?
Example:
•
7.2.D
Make scale drawings and solve problems
related to scale.
The length of the shadow of a tree is 68 feet at the
same time that the length of the shadow of a 6-foot
vertical pole is 8 feet. What is the height
of the tree?
Example:
•
On an 80:1 scale drawing of the floor plan of
a house, the dimensions of the living room are
1
7′′
1′′
× 2 . What is the actual area of the living
8
2
room in square feet?
7.2.E
Represent proportional relationships using
graphs, tables, and equations, and make
connections among the representations.
Proportional relationships are linear relationships
whose graphs pass through the origin and can be
written in the form y = kx.
Example:
•
7.2.F Determine the slope of a line corresponding
to the graph of a proportional relationship and
relate slope to similar triangles.
July 2008
Washington State K–12 Mathematics Standards
The relationship between the width and length
of similar rectangles is shown in the table below.
Write an equation that expresses the length, l, in
terms of the width, w, and graph the relationship
between the two variables.
width
4
12
18
...
w
length
10
30
45
...
?
This expectation connects the constant rate of change
in a proportional relationship to the concept of slope
of a line. Students should know that the slope of a line
is the same everywhere on the line and realize that
similar triangles can be used to demonstrate this fact.
They should recognize how proportionality is reflected
in slope as it is with similar triangles. A more complete
discussion of slope is developed in high school.
89
Grade 7
Performance Expectations
Explanatory Comments and Examples
Students are expected to:
7.2.G
Determine the unit rate in a proportional
relationship and relate it to the slope of the
associated line.
The associated unit rate, constant rate of change
of the function, and slope of the graph all represent
the constant of proportionality in a proportional
relationship.
Example:
•
7.2.H
Determine whether or not a relationship is
proportional and explain your reasoning.
Coffee costs $18.96 for 3 pounds. What is the
cost per pound of coffee? Draw a graph of the
proportional relationship between the number of
pounds of coffee and the total cost, and describe
how the unit rate is represented on the graph.
A proportional relationship is one in which two
quantities are related by a constant scale factor, k.
It can be written in the form y = kx. A proportional
relationship has a constant rate of change and its
graph passes through the origin.
Example:
•
Determine whether each situation represents a
proportional relationship and explain your reasoning.
—
x
1
2
3
4
y
4.5
9
13.5
18
— y = 3x + 2
— One way to calculate a person’s maximum
target heart rate during exercise in beats per
minute is to subtract the person’s age from
200. Is the relationship between the maximum
target heart rate and age proportional? Explain
your reasoning.
90
July 2008
Washington State K–12 Mathematics Standards
Grade 7
Performance Expectations
Explanatory Comments and Examples
Students are expected to:
7.2.I
Solve single- and multi-step problems involving
conversions within or between measurement
systems and verify the solutions.
The intent of this expectation is for students to show
their work, explain their thinking, and verify that the
answer to the problem is reasonable in terms of the
original context and the mathematics used to solve the
problem. Verifications can include the use of numbers,
words, pictures, or equations.
Students should be given the conversion factor when
converting between measurement systems.
Examples:
July 2008
Washington State K–12 Mathematics Standards
•
The lot that Dana is buying for her new one-story
house is 35 yards by 50 yards. Dana’s house
plans show that her house will cover 1,600 square
feet of land. What percent of Dana’s lot will not be
covered by the house? Explain your work.
•
Joe was planning a business trip to Canada, so he
went to the bank to exchange $200 U.S. dollars for
Canadian dollars (at a rate of $1.02 CDN per $1
US). On the way home from the bank, Joe’s boss
called to say that the destination of the trip had
changed to Mexico City. Joe went back to the bank
to exchange his Canadian dollars for Mexican
pesos (at a rate of 10.8 pesos per $1 CDN). How
many Mexican pesos did Joe get?
91
Grade 7
Grade 7
7.3. Core Content: Surface area and volume
(Algebra, Geometry/Measurement)
S
tudents extend their understanding of surface area and volume to include finding surface area and
volume of cylinders and volume of cones and pyramids. They apply formulas and solve a range of
problems involving three-dimensional objects, including problems people encounter in everyday life, in
certain types of work, and in other school subjects. With a strong understanding of how to work with both
two-dimensional and three-dimensional figures, students build an important foundation for the geometry
they will study in high school.
Performance Expectations
Explanatory Comments and Examples
Students are expected to:
7.3.A
Determine the surface area and volume of
cylinders using the appropriate formulas and
explain why the formulas work.
Explanations might include the use of models such as
physical objects or drawings.
A net can be used to illustrate the formula for finding
the surface area of a cylinder.
7.3.B
Determine the volume of pyramids and cones
using formulas.
7.3.C
Describe the effect that a change in scale
factor on one attribute of a two- or threedimensional figure has on other attributes of
the figure, such as the side or edge length,
perimeter, area, surface area, or volume of a
geometric figure.
Examples:
Solve single- and multi-step word problems
involving surface area or volume and verify
the solutions.
The intent of this expectation is for students to show
their work, explain their thinking, and verify that the
answer to the problem is reasonable in terms of the
original context and the mathematics used to solve the
problem. Verifications can include the use of numbers,
words, pictures, or equations.
7.3.D
•
A cube has a side length of 2 cm. If each side
length is tripled, what happens to the surface
area? What happens to the volume?
•
What happens to the area of a circle if the
diameter is decreased by a factor of 3?
Examples:
92
•
Alexis needs to paint the four exterior walls of a
large rectangular barn. The length of the barn is 80
feet, the width is 50 feet, and the height is 30 feet.
The paint costs $28 per gallon, and each gallon
covers 420 square feet. How much will it cost
Alexis to paint the barn? Explain your work.
•
Tyesha has decided to build a solid concrete
pyramid on her empty lot. The base will be a
square that is forty feet by forty feet and the height
will be thirty feet. The concrete that she will use to
construct the pyramid costs $70 per cubic yard.
How much will the concrete for the pyramid cost
Tyesha? Justify your answer.
July 2008
Washington State K–12 Mathematics Standards
Grade 7
Grade 7
7.4. Core Content: Probability and data
(Data/Statistics/Probability)
S
tudents apply their understanding of rational numbers and proportionality to concepts of probability.
They begin to understand how probability is determined, and they make related predictions. Students
revisit how to interpret data, now using more sophisticated types of data graphs and thinking about the
meaning of certain statistical measures. Statistics, including probability, is considered one of the most
important and practical fields of study for making sense of quantitative information, and it plays an
important part in secondary mathematics in the 21st century.
Performance Expectations
Explanatory Comments and Examples
Students are expected to:
7.4.A
Represent the sample space of probability
experiments in multiple ways, including tree
diagrams and organized lists.
The sample space is the set of all possible outcomes.
Example:
•
7.4.B
7.4.C
Determine the theoretical probability of a
particular event and use theoretical probability
to predict experimental outcomes.
Describe a data set using measures of center
(median, mean, and mode) and variability
(maximum, minimum, and range) and evaluate
the suitability and limitations of using each
measure for different situations.
July 2008
Washington State K–12 Mathematics Standards
José flips a penny, Jane flips a nickel, and Janice
flips a dime, all at the same time. List the possible
outcomes of the three simultaneous coin flips
using a tree diagram or organized list.
Example:
•
A triangle with a base of 8 units and a height of
7 units is drawn inside a rectangle with an area
of 90 square units. What is the probability that a
randomly selected point inside the rectangle will
also be inside the triangle?
•
There are 5 blue, 4 green, 8 red, and 3 yellow
marbles in a paper bag. Rosa runs an experiment
in which she draws a marble from the bag, notes
the color on a sheet of paper, and puts the marble
back in the bag, repeating the process 200 times.
About how many times would you expect Rosa to
draw a red marble?
As a way to understand these ideas, students could
construct data sets for a given mean, median, mode,
or range.
Examples:
•
Kiley keeps track of the money she spends each
week for two months and records the following
amounts: $6.30, $2.25, $43.00, $2.25, $11.75,
$5.25, $4.00, and $5.20. Which measure of center
is most representative of Kiley’s weekly spending?
Support your answer.
•
Construct a data set with five data points, a mean
of 24, a range of 10, and without a mode.
•
A group of seven adults have an average age of
36. If the ages of three of the adults are 45, 30,
and 42, determine possible ages for the remaining
four adults.
93
Grade 7
Performance Expectations
Explanatory Comments and Examples
Students are expected to:
Construct and interpret histograms, stem-andleaf plots, and circle graphs.
7.4.E
Evaluate different displays of the same data for
effectiveness and bias, and explain reasoning.
Example:
•
The following two bar graphs of the same data
show the number of five different types of sodas
that were sold at Blake High School. Compare and
contrast the two graphs. Describe a reason why
you might choose to use one graph over the other.
Number of Cans of Soda Sold
7.4.D
1,545
1,544
1,543
1,542
1,541
Fizzy Mountain Baken Dr. Salt Snapcrackle
Soda
Pop
Cola Don’t
Figure 1
Number of Cans of Soda Sold
1,555
1,550
1,545
1,540
1,535
Fizzy Mountain Baken Dr. Salt Snapcrackle
Soda
Pop
Cola Don’t
94
Figure 2
July 2008
Washington State K–12 Mathematics Standards
Grade 7
Grade 7
7.5. Additional Key Content (Numbers, Algebra)
S
tudents extend their coordinate graphing skills to plotting points with both positive and negative
coordinates on the coordinate plane. Using pairs of numbers to locate points is a necessary skill for
reading maps and tables and a critical foundation for high school mathematics. Students further prepare
for algebra by learning how to use exponents to write numbers in terms of their most basic (prime) factors.
Performance Expectations
Explanatory Comments and Examples
Students are expected to:
7.5.A
7.5.B
Graph ordered pairs of rational numbers and
determine the coordinates of a given point in
the coordinate plane.
Write the prime factorization of whole numbers
greater than 1, using exponents when
appropriate.
Example:
•
Graph and label the points A(1, 2), B(-1, 5),
C(-3, 2), and D(-1, -5). Connect the points in
the order listed and identify the figure formed by
the four points.
•
Graph and label the points A(1, -2), B(-4, -2), and
C(-4, 3). Determine the coordinates of the fourth
point (D) that will complete the figure to form a
square. Graph and label point D on the coordinate
plane and draw the resulting square.
Writing numbers in prime factorization is a useful tool
for determining the greatest common factor and least
common multiple of two or more numbers.
Example:
•
July 2008
Washington State K–12 Mathematics Standards
Write the prime factorization of 360 using exponents.
95
Grade 7
Grade 7
7.6. Core Processes: Reasoning, problem solving, and communication
S
tudents refine their reasoning and problem-solving skills as they move more fully into the symbolic
world of algebra and higher-level mathematics. They move easily among representations—
numbers, words, pictures, or symbols—to understand and communicate mathematical ideas, to make
generalizations, to draw logical conclusions, and to verify the reasonableness of solutions to problems.
In grade seven, students solve problems that involve positive and negative numbers and often involve
proportional relationships. As students solve these types of problems, they build a strong foundation for
the study of linear functions that will come in grade eight.
Performance Expectations
Explanatory Comments and Examples
Students are expected to:
7.6.A
Analyze a problem situation to determine the
question(s) to be answered.
7.6.B
Identify relevant, missing, and extraneous
information related to the solution to a problem.
7.6.C
Analyze and compare mathematical strategies
for solving problems, and select and use one
or more strategies to solve a problem.
7.6.D
Represent a problem situation, describe the
process used to solve the problem, and verify
the reasonableness of the solution.
7.6.E
Communicate the answer(s) to the
question(s) in a problem using appropriate
representations, including symbols and
informal and formal mathematical language.
7.6.F
Apply a previously used problem-solving
strategy in a new context.
7.6.G
Extract and organize mathematical information
from symbols, diagrams, and graphs to
make inferences, draw conclusions, and
justify reasoning.
7.6.H
96
Make and test conjectures based on data
(or information) collected from explorations
and experiments.
Descriptions of solution processes and explanations
can include numbers, words (including mathematical
language), pictures, physical objects, or equations.
Students should be able to use all of these
representations as needed. For a particular solution,
students should be able to explain or show their work
using at least one of these representations and verify
that their answer is reasonable.
Examples:
•
When working on a report for class, Catrina
read that a person over the age of 30 can lose
approximately 0.06 centimeters of height per
year. Catrina’s 80-year-old grandfather is 5 feet
7 inches tall. Assuming her grandfather’s height
has decreased at this rate, about how tall was he
at age 30? Catrina’s cousin, Richard, is 30 years
old and is 6 feet 3 inches tall. Assuming his height
also decreases approximately 0.06 centimeters
per year after the age of 30, about how tall will you
expect him to be at age 55? (Remember that
1 inch ≈ 2.54 centimeters.) Justify your solution.
•
If one man takes 1.5 hours to dig a
5-ft × 5-ft × 3-ft hole, how long will it take three
men working at the same pace to dig a
10-ft × 12-ft × 3-ft hole? Explain your solution.
July 2008
Washington State K–12 Mathematics Standards
Grade 8
July 2008
Washington State K–12 Mathematics Standards
97
Grade 8
Grade 8
8.1. Core Content: Linear functions and equations (Algebra)
S
tudents solve a variety of linear equations and inequalities. They build on their familiarity with
proportional relationships and simple linear equations to work with a broader set of linear
relationships, and they learn what functions are. They model applied problems with mathematical
functions represented by graphs and other algebraic techniques. This Core Content area includes topics
typically addressed in a high school algebra or a first-year integrated math course, but here this content
is expected of all middle school students in preparation for a rich high school mathematics program that
goes well beyond these basic algebraic ideas.
Performance Expectations
Explanatory Comments and Examples
Students are expected to:
8.1.A
Solve one-variable linear equations.
Examples:
Solve each equation for x.
8.1.B
Solve one- and two-step linear inequalities and
graph the solutions on the number line.
•
91 – 2.5x = 26
•
7
( x − 2) = 119
8
•
-3x + 34 = 5x
•
114 = -2x – 8 + 5x
•
3(x – 2) – 4x = 2(x + 22) – 5
The emphasis at this grade level is on gaining
experience with inequalities, rather than on becoming
proficient at solving inequalities in which multiplying or
dividing by a negative is necessary.
Example:
•
8.1.C
Represent a linear function with a verbal
description, table, graph, or symbolic
expression, and make connections among
these representations.
8.1.D Determine the slope and y-intercept of a linear
function described by a symbolic expression,
table, or graph.
Translating among these various representations
of functions is an important way to demonstrate a
conceptual understanding of functions.
Examples:
•
•
July 2008
Washington State K–12 Mathematics Standards
Graph the solution of 4x – 21 > 57 on the number line.
Determine the slope and y-intercept for the
function described by
y=
2
x−5
3
The following table represents a linear function.
Determine the slope and y-intercept.
x
2
3
5
8
12
y
5
8
14
23
35
99
Grade 8
Performance Expectations
Explanatory Comments and Examples
Students are expected to:
8.1.E
8.1.F
Interpret the slope and y-intercept of the
graph of a linear function representing a
contextual situation.
Example:
Solve single- and multi-step word problems
involving linear functions and verify the solutions.
The intent of this expectation is for students to show
their work, explain their thinking, and verify that the
answer to the problem is reasonable in terms of the
original context and the mathematics used to solve the
problem. Verifications can include the use of numbers,
words, pictures, or equations.
•
A car is traveling down a long, steep hill. The
elevation, E, above sea level (in feet) of the car
when it is d miles from the top of the hill is given
by E = 7500 – 250d, where d can be any number
from 0 to 6. Find the slope and y-intercept of the
graph of this function and explain what they mean
in the context of the moving car.
Example:
•
8.1.G
Determine and justify whether a given
verbal description, table, graph, or symbolic
expression represents a linear relationship.
Mike and Tim leave their houses at the same time
to walk to school. Mike’s walk can be represented
by d1 = 4000 – 400t, and Tim’s walk can be
represented by d2 = 3400 – 250t, where d is
the distance from the school in feet and t is the
walking time in minutes. Who arrives at school
first? By how many minutes? Is there a time when
Mike and Tim are the same distance away from
the school? Explain your reasoning.
Examples:
•
•
Could the data presented in the table represent a
linear function? Explain your reasoning.
x
y
-1
0
0
-1
1
0
2
3
3
8
4
15
5
24
1
4
Does y = x − 5 represent a linear function?
Explain your reasoning.
100
July 2008
Washington State K–12 Mathematics Standards
Grade 8
Grade 8
8.2. Core Content: Properties of geometric figures (Numbers, Geometry/Measurement)
S
tudents work with lines and angles, especially as they solve problems involving triangles. They use
known relationships involving sides and angles of triangles to find unknown measures, connecting
geometry and measurement in practical ways that will be useful well after high school. Since squares
of numbers arise when using the Pythagorean Theorem, students work with squares and square roots,
especially in problems with two- and three-dimensional figures. Using basic geometric theorems such as
the Pythagorean Theorem, students get a preview of how geometric theorems are developed and applied
in more formal settings, which they will further study in high school.
Performance Expectations
Explanatory Comments and Examples
Students are expected to:
8.2.A
Identify pairs of angles as complementary,
supplementary, adjacent, or vertical, and
use these relationships to determine missing
angle measures.
Example:
•
Determine the measures of ∠BOA, ∠EOD, ∠FOB,
and ∠FOE and explain how you found each
measure. As part of your explanation, identify
pairs of angles as complementary, supplementary,
or vertical.
A
B
F
C
36
O
E
D
8.2.B
Determine missing angle measures using the
relationships among the angles formed by
parallel lines and transversals.
Example:
•
Determine the measures of the indicated angles.
∠1: _____ ∠2: _____ ∠3: _____ ∠4: _____
m n
3
1
40º
4
25º
2
m
n
July 2008
Washington State K–12 Mathematics Standards
101
Grade 8
Performance Expectations
Explanatory Comments and Examples
Students are expected to:
8.2.C
8.2.D
8.2.E
Demonstrate that the sum of the angle
measures in a triangle is 180 degrees, and
apply this fact to determine the sum of the
angle measures of polygons and to determine
unknown angle measures.
Examples:
Represent and explain the effect of one or
more translations, rotations, reflections, or
dilations (centered at the origin) of a geometric
figure on the coordinate plane.
Example:
Quickly recall the square roots of the perfect
squares from 1 through 225 and estimate the
square roots of other positive numbers.
Students can use perfect squares of integers to
determine squares and square roots of related
numbers, such as 1.96 and 0.0049.
•
Determine the measure of each interior angle in a
regular pentagon.
•
In a certain triangle, the measure of one angle is
four times the measure of the smallest angle, and
the measure of the remaining angle is the sum of
the measures of the other two angles. Determine
the measure of each angle.
•
Consider a trapezoid with vertices (1, 2), (1, 6),
(6, 4), and (6, 2). The trapezoid is reflected across
the x-axis and then translated four units to the left.
Graph the image of the trapezoid after these two
transformations and give the coordinates of the
new vertices.
Examples:
8.2.F
Demonstrate the Pythagorean Theorem and its
converse and apply them to solve problems.
•
Determine:
•
Between which two consecutive integers does the
square root of 74 lie?
36 ,
0.25 , 144 , and
196 .
One possible demonstration is to start with a right
triangle, use each of the three triangle sides to form the
side of a square, and draw the remaining three sides
of each of the three squares. The areas of the three
squares represent the Pythagorean relationship.
Examples:
8.2.G Apply the Pythagorean Theorem to determine
the distance between two points on the
coordinate plane.
102
•
Is a triangle with side lengths 5 cm, 12 cm, and
13 cm a right triangle? Why or why not?
•
Determine the length of the diagonal of a rectangle
that is 7 ft by 10 ft.
Example:
•
Determine the distance between the points (-2, 3)
and (4, 7).
July 2008
Washington State K–12 Mathematics Standards
Grade 8
Grade 8
8.3. Core Content: Summary and analysis of data sets (Algebra, Data/Statistics/Probability)
S
tudents build on their extensive experience organizing and interpreting data and apply statistical
principles to analyze statistical studies or short statistical statements, such as those they might
encounter in newspapers, on television, or on the Internet. They use mean, median, and mode to
summarize and describe information, even when these measures may not be whole numbers. Students
use their knowledge of linear functions to analyze trends in displays of data. They create displays for
two sets of data in order to compare the two sets and draw conclusions. They expand their work with
probability to deal with more complex situations than they have previously seen. These concepts of
statistics and probability are important not only in students’ lives, but also throughout the high school
mathematics program.
Performance Expectations
Explanatory Comments and Examples
Students are expected to:
8.3.A
Summarize and compare data sets in terms of
variability and measures of center.
Students use mean, median, mode, range, and interquartile range to summarize and compare data sets,
and explain the influence of outliers on each measure.
Example:
8.3.B
Select, construct, and analyze data displays,
including box-and-whisker plots, to compare
two sets of data.
July 2008
Washington State K–12 Mathematics Standards
•
Captain Bob owns two charter boats, the SockEye-To-Me and Old Gus, which take tourists on
fishing trips. On Saturday, the Sock-Eye-To-Me
took four people fishing and returned with eight
fish weighing 18, 23, 20, 6, 20, 22, 18, and 20
pounds. On the same day, Old Gus took five
people fishing and returned with ten fish weighing
38, 18, 12, 14, 17, 42, 12, 16, 12, and 14 pounds.
Using measures of center and variability, compare
the weights of the fish caught by the people in the
two boats.
Make a summary statement telling which boat
you would charter for fishing based on these data
and why.
What influence, if any, do outliers have on the
particular statistics for these data?
Previously studied displays include stem-and-leaf
plots, histograms, circle graphs, and line plots. Here
these displays are used to compare data sets. Boxand-whisker plots are introduced here for the first time
as a powerful tool for comparing two or more data sets.
103
Grade 8
Performance Expectations
Explanatory Comments and Examples
Students are expected to:
8.3.B cont.
Example:
•
As part of their band class, Tayla and Alyssa are
required to keep practice records that show the
number of minutes they practice their instruments
each day. Below are their practice records for the
past fourteen days:
Tayla:
55, 45, 60, 45, 30, 30, 90, 50, 40, 75, 25,
90, 105, 60
Alyssa: 20, 120, 25, 20, 0, 15, 30, 15, 90, 0, 30,
30, 10, 30
8.3.C
Create a scatterplot for a two-variable data set,
and, when appropriate, sketch and use a trend
line to make predictions.
Of stem-and-leaf plot, circle graph, or line plot,
select the data display that you think will best
compare the two girls’ practice records. Construct
a display to show the data. Compare the amount
of time the two girls practice by analyzing the data
presented in the display.
Example:
•
Kera randomly selected seventeen students from
her middle school for a study comparing arm span
to standing height. The students’ measurements
are shown in the table below.
Comparison of Arm Span and Standing Height
(in cm) at Icicle River Middle School
104
Height
(cm)
Arm Span
(cm)
Height
(cm)
Arm Span
(cm)
138
145
155
150
135
135
175
177
142
147
162
160
158
145
150
152
177
174
142
143
150
152
149
149
158
162
160
165
160
160
173
170
160
158
Create a scatterplot for the data shown.
If appropriate, sketch a trendline.
Use these data to estimate the arm span of a
student with a height of 180 cm, and the height of
a student with an arm span of 130 cm. Explain any
limitations of using this process to make estimates.
July 2008
Washington State K–12 Mathematics Standards
Grade 8
Performance Expectations
Explanatory Comments and Examples
Students are expected to:
8.3.D
Describe different methods of selecting
statistical samples and analyze the strengths
and weaknesses of each method.
Students should work with a variety of sampling
techniques and should be able to identify strengths
and weaknesses of random, census, convenience, and
representative sampling.
Example:
•
8.3.E
Determine whether conclusions of statistical
studies reported in the media are reasonable.
8.3.F
Determine probabilities for mutually exclusive,
dependent, and independent events for small
sample spaces.
July 2008
Washington State K–12 Mathematics Standards
Carli, Jamar, and Amberly are conducting a
survey to determine their school’s favorite Seattle
professional sports team. Carli selects her sample
using a convenience method—she surveys
students on her bus during the ride to school.
Jamar uses a computer to randomly select 30
numbers from 1 through 600, and then surveys
the corresponding students from a numbered,
alphabetical listing of the student body. Amberly
waits at the front entrance before school and
surveys every twentieth student entering. Analyze
the strengths and weaknesses of each method.
Examples:
•
Given a standard deck of 52 playing cards, what is
the probability of drawing a king or queen? [Some
students may be unfamiliar with playing cards, so
alternate examples may be desirable.]
•
Leyanne is playing a game at a birthday party.
Beneath ten paper cups, a total of five pieces of
candy are hidden, with one piece hidden beneath
each of five cups. Given only three guesses,
Leyanne must uncover three pieces of candy to
win all the hidden candy. What is the probability
she will win all the candy?
•
A bag contains 7 red marbles, 5 blue marbles, and
8 green marbles. If one marble is drawn at random
and put back in the bag, and then a second marble
is drawn at random, what is the probability of
drawing first a red marble, then a blue marble?
105
Grade 8
Performance Expectations
Explanatory Comments and Examples
Students are expected to:
8.3.G
Solve single- and multi-step problems using
counting techniques and Venn diagrams and
verify the solutions.
The intent of this expectation is for students to show
their work, explain their thinking, and verify that the
answer to the problem is reasonable in terms of the
original context and the mathematics used to solve the
problem. Verifications can include the use of numbers,
words, pictures, or equations.
Counting techniques include the fundamental counting
principle, lists, tables, tree diagrams, etc.
Examples:
106
•
Jack’s Deli makes sandwiches that include a
choice of one type of bread, one type of cheese,
and one type of meat. How many different
sandwiches could be made given 4 different bread
types, 3 different cheeses, and 5 different meats?
Explain your reasoning.
•
A small high school has 57 tenth-graders. Of
these students, 28 are taking geometry, 34 are
taking biology, and 10 are taking neither geometry
nor biology. How many students are taking both
geometry and biology? How many students are
taking geometry but not biology? How many
students are taking biology but not geometry?
Draw a Venn diagram to illustrate this situation.
July 2008
Washington State K–12 Mathematics Standards
Grade 8
Grade 8
8.4. Additional Key Content
(Numbers, Operations)
S
tudents deal with a few key topics about numbers as they prepare to shift to higher level mathematics
in high school. First, they use scientific notation to represent very large and very small numbers,
especially as these numbers are used in technological fields and in everyday tools like calculators or
personal computers. Scientific notation has become especially important as “extreme units” continue to
be identified to represent increasingly tiny or immense measures arising in technological fields. A second
important numerical skill involves using exponents in expressions containing both numbers and variables.
Developing this skill extends students’ work with order of operations to include more complicated
expressions they might encounter in high school mathematics. Finally, to help students understand the full
breadth of the real-number system, students are introduced to simple irrational numbers, thus preparing
them to study higher level mathematics in which properties and procedures are generalized for the entire
set of real numbers.
Performance Expectations
Explanatory Comments and Examples
Students are expected to:
8.4.A
8.4.B
Represent numbers in scientific notation, and
translate numbers written in scientific notation
into standard form.
Solve problems involving operations with
numbers in scientific notation and verify solutions.
Examples:
•
Represent 4.27 x 10-3 in standard form.
•
Represent 18,300,000 in scientific notation.
•
Throughout the year 2004, people in the city of
Cantonville sent an average of 400 million text
messages a day. Using this information, about how
many total text messages did Cantonville residents
send in 2004? (2004 was a leap year.) Express your
answer in scientific notation.
Units include those associated with technology, such
as nanoseconds, gigahertz, kilobytes, teraflops, etc.
Examples:
July 2008
Washington State K–12 Mathematics Standards
•
A supercomputer used by a government agency
will be upgraded to perform 256 teraflops (that
is, 256 trillion calculations per second). Before
the upgrade, the supercomputer performs
1.6 x 1013 calculations per second. How many
more calculations per second will the upgraded
supercomputer be able to perform? Express the
answer in scientific notation.
•
A nanosecond is one billionth of a second. How
many nanoseconds are there in five minutes?
Express the answer in scientific notation.
107
Grade 8
Performance Expectations
Explanatory Comments and Examples
Students are expected to:
8.4.C
Evaluate numerical expressions involving nonnegative integer exponents using the laws of
exponents and the order of operations.
Example:
•
Simplify and write the answer in exponential form:
(7 4 ) 2
7
3
Some students will be ready to solve problems
involving simple negative exponents and should be
given the opportunity to do so.
Example:
8.4.D Identify rational and irrational numbers.
•
Simplify and write the answer in exponential form:
(54)25-3
Students should know that rational numbers are
numbers that can be represented as the ratio of two
integers; that the decimal expansions of rational
numbers have repeating patterns, or terminate; and
that there are numbers that are not rational.
Example:
•
108
Identify whether each number is rational or
irrational and explain your choice.
1
3.14, 4.6,
, 2 , 25 , π
11
July 2008
Washington State K–12 Mathematics Standards
Grade 8
Grade 8
8.5. Core Processes: Reasoning, problem solving, and communication
S
tudents refine their reasoning and problem-solving skills as they move more fully into the symbolic
world of algebra and higher level mathematics. They move easily among representations—
numbers, words, pictures, or symbols—to understand and communicate mathematical ideas, to make
generalizations, to draw logical conclusions, and to verify the reasonableness of solutions to problems.
In grade eight, students solve problems that involve proportional relationships and linear relationships,
including applications found in many contexts outside of school. These problems dealing with
proportionality continue to be important in many applied contexts, and they lead directly to the study of
algebra. Students also begin to deal with informal proofs for theorems that will be proven more formally in
high school.
Performance Expectations
Explanatory Comments and Examples
Students are expected to:
8.5.A
Analyze a problem situation to determine the
question(s) to be answered.
8.5.B
Identify relevant, missing, and extraneous
information related to the solution to a problem.
8.5.C
Analyze and compare mathematical strategies
for solving problems, and select and use one
or more strategies to solve a problem.
8.5.D
Represent a problem situation, describe the
process used to solve the problem, and verify
the reasonableness of the solution.
•
8.5.E
Communicate the answer(s) to the
question(s) in a problem using appropriate
representations, including symbols and
informal and formal mathematical language.
The dimensions of a room are
12 feet by 15 feet by 10 feet. What is the furthest
distance between any two points in the room?
Explain your solution.
•
Miranda’s phone service contract ends this
month. She is looking for ways to save money
and is considering changing phone companies.
Her current cell phone carrier, X-Cell,
calculates the monthly bill using the equation
c = $15.00 + $0.07m, where c represents the total
monthly cost and m represents the number of
minutes of talk time during a monthly billing cycle.
Another company, Prism Cell, offers 300 free
minutes of talk time each month for a base fee of
$30.00 with an additional $0.15 for every minute
over 300 minutes. Miranda’s last five phone
bills were $34.95, $36.70, $37.82, $62.18, and
$36.28. Using the data from the last five months,
help Miranda decide whether she should switch
companies. Justify your answer.
8.5.F
Apply a previously used problem-solving
strategy in a new context.
8.5.G
Extract and organize mathematical information
from symbols, diagrams, and graphs to
make inferences, draw conclusions, and
justify reasoning.
8.5.H
Make and test conjectures based on data
(or information) collected from explorations
and experiments.
July 2008
Washington State K–12 Mathematics Standards
Descriptions of solution processes and explanations
can include numbers, words (including mathematical
language), pictures, or equations. Students should
be able to use all of these representations as needed.
For a particular solution, students should be able to
explain or show their work using at least one of these
representations and verify that their answer
is reasonable.
Examples:
109
Grade 8
Algebra 1
July 2008
Washington State K–12 Mathematics Standards
111
Algebra I
Algebra 1
A1.1. Core Content: Solving problems (Algebra)
S
tudents learn to solve many new types of problems in Algebra 1, and this first core content area
highlights the types of problems students will be able to solve after they master the concepts and skills
in this course. Students are introduced to several types of functions, including exponential and functions
defined piecewise, and they spend considerable time with linear and quadratic functions. Each type of
function included in Algebra 1 provides students a tool to solve yet another class of problems. They learn
that specific functions model situations described in word problems, and so functions are used to solve
various types of problems. The ability to determine functions and write equations that represent problems
is an important mathematical skill in itself. Many problems that initially appear to be very different from
each other can actually be represented by identical equations. Students encounter this important and
unifying principle of algebra—that the same algebraic techniques can be applied to a wide variety of
different situations.
Performance Expectations
Explanatory Comments and Examples
Students are expected to:
A1.1.A Select and justify functions and equations to
model and solve problems.
Students can analyze the rate of change of a function
represented with a table or graph to determine if the
function is linear. Students also analyze common ratios
to determine if the function is exponential.
After selecting a function to model a situation, students
describe appropriate domain restrictions. They use the
function to solve the problem and interpret the solution
in the context of the original situation.
Examples:
•
A cup is 6 cm tall, including a 1.1 cm lip. Find a
function that represents the height of a stack of
cups in terms of the number of cups in the stack.
Find a function that represents the number of cups
in a stack of a given height.
•
For the month of July, Michelle will be dog-sitting
for her very wealthy, but eccentric, neighbor, Mrs.
Buffett. Mrs. Buffett offers Michelle two different
salary plans:
— Plan 1: $100 per day for the 31 days of
the month.
— Plan 2: $1 for July 1, $2 for July 2, $4
for July 3, and so on, with the daily rate
doubling each day.
a. Write functions that model the amount of
money Michelle will earn each day on Plan 1
and Plan 2. Justify the functions you wrote.
b. State an appropriate domain for each of the
models based on the context.
c. Which plan should Michelle choose to
maximize her earnings? Justify your
recommendation mathematically.
d. Extension: Write an algebraic function for the
cumulative pay for each plan based on the
number of days worked.
July 2008
Washington State K–12 Mathematics Standards
113
Algebra I
Performance Expectations
Explanatory Comments and Examples
Students are expected to:
A1.1.B Solve problems that can be represented by
linear functions, equations, and inequalities.
It is mathematically important to represent a word
problem as an equation. Students must analyze the
situation and find a way to represent it mathematically.
After solving the equation, students think about the
solution in terms of the original problem.
Examples:
•
The assistant pizza maker makes 6 pizzas an
hour. The master pizza maker makes 10 pizzas
an hour but starts baking two hours later than his
assistant. Together, they must make 92 pizzas.
How many hours from when the assistant starts
baking will it take?
What is a general equation, in function form, that
could be used to determine the number of pizzas
that can be made in two or more hours?
•
A1.1.C Solve problems that can be represented by a
system of two linear equations or inequalities.
A swimming pool holds 375,000 liters of water.
Two large hoses are used to fill the pool. The
first hose fills at the rate of 1,500 liters per hour
and the second hose fills at the rate of 2,000 liters
per hour. How many hours does it take to fill the
pool completely?
Examples:
•
An airplane flies from Baltimore to Seattle (assume
a distance of 2,400 miles) in 7 hours, but the return
1
flight takes only 4 2 hours. The air speed of the
plane is the same in both directions. How many
miles per hour does the plane fly with respect to the
wind? What is the wind speed in miles per hour?
•
A coffee shop employee has one cup of 85% milk
(the rest is chocolate) and another cup of 60% milk
(the rest is chocolate). He wants to make one cup
of 70% milk. How much of the 85% milk and 60%
milk should he mix together to make the 70% milk?
•
Two plumbing companies charge different rates
for their service. Clyde’s Plumbing Company
charges a $75-per-visit fee that includes one hour
of labor plus $45 dollars per hour after the first
hour. We-Unclog-It Plumbers charges a $100-pervisit fee that includes one hour of labor plus $40
per hour after the first hour. For how many hours
of plumbing work would Clyde’s be less expensive
than We-Unclog-It?
Note: Although this context is discrete, students
can model it with continuous linear functions.
114
July 2008
Washington State K–12 Mathematics Standards
Algebra I
Performance Expectations
Explanatory Comments and Examples
Students are expected to:
A1.1.D Solve problems that can be represented by
quadratic functions and equations.
A1.1.E Solve problems that can be represented by
exponential functions and equations.
Examples:
•
Find the solutions to the simultaneous equations
y = x + 2 and y = x2.
•
If you throw a ball straight up (with initial height of
4 feet) at 10 feet per second, how long will it take
to fall back to the starting point? The function
h(t) = -16t2 + v0t + h0 describes the height, h in feet,
of an object after t seconds, with initial velocity v0
and initial height h0.
•
Joe owns a small plot of land 20 feet by 30 feet.
He wants to double the area by increasing both
the length and the width, keeping the dimensions
in the same proportion as the original. What will be
the new length and width?
•
What two consecutive numbers, when multiplied
together, give the first number plus 16? Write the
equation that represents the situation.
Students approximate solutions with graphs or tables,
check solutions numerically, and when possible, solve
problems exactly.
Examples:
•
E. coli bacteria reproduce by a simple process
called binary fission—each cell increases in size
and divides into two cells. In the laboratory, E. coli
bacteria divide approximately every 15 minutes. A
new E. coli culture is started with 1 cell.
a. Find a function that models the E. coli
population size at the end of each 15-minute
interval. Justify the function you found.
b. State an appropriate domain for the model
based on the context.
c. After what 15-minute interval will you have at
least 500 bacteria?
•
July 2008
Washington State K–12 Mathematics Standards
Estimate the solution to 2x = 16,384.
115
Algebra I
Algebra 1
A1.2. Core Content: Numbers, expressions, and operations (Numbers, Operations, Algebra)
S
tudents see the number system extended to the real numbers represented by the number line. They
work with integer exponents, scientific notation, and radicals, and use variables and expressions to
solve problems from purely mathematical as well as applied contexts. They build on their understanding
of computation using arithmetic operations and properties and expand this understanding to include the
symbolic language of algebra. Students demonstrate this ability to write and manipulate a wide variety of
algebraic expressions throughout high school mathematics as they apply algebraic procedures to solve
problems.
Performance Expectations
Explanatory Comments and Examples
Students are expected to:
A1.2.A Know the relationship between real numbers
and the number line, and compare and order
real numbers with and without the number line.
Although a formal definition of real numbers is beyond
the scope of Algebra 1, students learn that every point
on the number line represents a real number, either
rational or irrational, and that every real number has its
unique point on the number line. They locate, compare,
and order real numbers on the number line.
Real numbers include those written in scientific
notation or expressed as fractions, decimals,
exponentials, or roots.
Examples:
•
Without using a calculator, order the following on
the number line:
82 , 3π, 8.9, 9,
•
37
, 9.3 × 100
4
A star’s color gives an indication of its temperature
and age. The chart shows four types of stars and
the lowest temperature of each type.
Type
Lowest Temperature
(in ºF)
A
1.35 x 104
Blue-White
B
2.08 x 104
Blue
G
9.0 x 103
Yellow
P
4.5 x 104
Blue
Color
List the temperatures in order from lowest to highest.
116
July 2008
Washington State K–12 Mathematics Standards
Algebra I
Performance Expectations
Explanatory Comments and Examples
Students are expected to:
A1.2.B Recognize the multiple uses of variables,
determine all possible values of variables that
satisfy prescribed conditions, and evaluate
algebraic expressions that involve variables.
Students learn to use letters as variables and in other
ways that increase in sophistication throughout high
school. For example, students learn that letters can
be used:
•
To represent fixed and temporarily unknown values
in equations, such as 3x + 2 = 5;
•
To express identities, such as x + x = 2x for all x;
•
As attributes in formulas, such as A = lw;
•
As constants such as a, b, and c in the equation
y = ax2 + bx + c;
•
As parameters in equations, such as the m and b
for the family of functions defined by y = mx + b;
•
To represent varying quantities, such as x in f(x) = 5x;
•
To represent functions, such as f in f(x) = 5x; and
•
To represent specific numbers, such as π.
Expressions include those involving polynomials,
radicals, absolute values, and integer exponents.
Examples:
A1.2.C Interpret and use integer exponents and
square and cube roots, and apply the laws
and properties of exponents to simplify and
evaluate exponential expressions.
•
For what values of a and n, where n is an integer
greater than 0, is an always negative?
•
For what values of a is
•
For what values of a is 5 − a defined?
•
For what values of a is -a always positive?
Examples:
•
1
2-3 =
3
2
2
5
•
-2
3
2 3 5
= 4
2
-3 2
2 5
2 3 5
•
a-2 b2 c
b5
=
a 2 b -3 c 2 a 4 c
•
•
July 2008
Washington State K–12 Mathematics Standards
1
an integer?
a
8 = 2• 2• 2 = 2 2
3
a•b = 3 a • 3 b
117
Algebra I
Performance Expectations
Explanatory Comments and Examples
Students are expected to:
A1.2.D Determine whether approximations or exact
values of real numbers are appropriate,
depending on the context, and justify
the selection.
Decimal approximations of numbers are sometimes
used in applications, such as carpentry or engineering;
while at other times, these applications may require
exact values. Students should understand the
difference and know that the appropriate approximation
depends upon the necessary degree of precision
needed in given situations.
For example, 1.414 is an approximation and not an
exact solution to the equation x2 – 2 = 0, but 2 is an
exact solution to this equation.
Example:
•
Using a common engineering formula, an
engineering student represented the maximum
safe load of a bridge to be 1000(99 – 70 2) tons.
He used 1.41 as the approximation for 2 in his
calculations. When the bridge was built and tested
in a computer simulation to verify its maximum
weight-bearing load, it collapsed! The student had
estimated the bridge would hold ten times the
weight that was applied to it when it collapsed.
— Calculate the weight that the student
thought the bridge could bear using 1.41
as the estimate for 2.
— Calculate other weight values using
estimates of 2 that have more decimal
places. What might be a reasonable
degree of precision required to know
how much weight the bridge can handle
safely? Justify your answer.
A1.2.E Use algebraic properties to factor and combine
like terms in polynomials.
Algebraic properties include the commutative,
associative, and distributive properties.
Factoring includes:
118
•
Factoring a monomial from a polynomial, such as
4x2 + 6x = 2x(2x + 3);
•
Factoring the difference of two squares, such as
36x2 – 25y2 = (6x + 5y)(6x – 5y) and
x4 – y4 = (x + y)(x – y)(x2 + y2);
•
Factoring perfect square trinomials, such as
x2 + 6xy + 9y2 = (x + 3y)2;
•
Factoring quadratic trinomials, such as
x2 + 5x + 4 = (x + 4)(x + 1); and
•
Factoring trinomials that can be expressed as the
product of a constant and a trinomial, such as
0.5x2 – 2.5x – 7 = 0.5(x + 2)(x – 7).
July 2008
Washington State K–12 Mathematics Standards
Algebra I
Performance Expectations
Explanatory Comments and Examples
Students are expected to:
A1.2.F Add, subtract, multiply, and divide polynomials.
Write algebraic expressions in equivalent forms using
algebraic properties to perform the four arithmetic
operations with polynomials.
Students should recognize that expressions are
essentially sums, products, differences, or quotients.
For example, the sum 2x2 + 4x can be written as a
product, 2x(x + 2).
Examples:
July 2008
Washington State K–12 Mathematics Standards
•
(3x2 – 4x + 5) + (-x2 + x – 4) + (2x2 + 2x + 1)
•
(2x2 – 4) – (x2 + 3x – 3)
•
2x2
6
• 4
9 2x
•
x 2 – 2x – 3
x +1
119
Algebra I
Algebra 1
A1.3. Core Content: Characteristics and behaviors of functions
(Algebra)
S
tudents formalize and deepen their understanding of functions, the defining characteristics and uses
of functions, and the mathematical language used to describe functions. They learn that functions are
often specified by an equation of the form y = f(x), where any allowable x-value yields a unique y-value.
While Algebra 1 has a particular focus on linear and quadratic equations and systems of equations,
students also learn about exponential functions and those that can be defined piecewise, particularly
step functions and functions that contain the absolute value of an expression. Students learn about
the representations and basic transformations of these functions and the practical and mathematical
limitations that must be considered when working with functions and when using functions to model
situations.
Performance Expectations
Explanatory Comments and Examples
Students are expected to:
A1.3.A Determine whether a relationship is a function
and identify the domain, range, roots, and
independent and dependent variables.
Functions studied in Algebra 1 include linear, quadratic,
exponential, and those defined piecewise (including
step functions and those that contain the absolute
value of an expression).
Given a problem situation, students should describe
further restrictions on the domain of a function that are
appropriate for the problem context.
Examples:
•
Which of the following are functions? Explain why
or why not.
— The age in years of each student in your
math class and each student’s shoe size.
— The number of degrees a person rotates a
spigot and the volume of water that comes
out of the spigot.
•
•
120
A function f(n) = 60n is used to model the distance
in miles traveled by a car traveling 60 miles per
hour in n hours. Identify the domain and range of
this function. What restrictions on the domain of
this function should be considered for the model to
correctly reflect the situation?
What is the domain of f(x) = 5 − x ?
July 2008
Washington State K–12 Mathematics Standards
Algebra I
Performance Expectations
Explanatory Comments and Examples
Students are expected to:
A1.3.A cont.
•
Which of the following equations, inequalities, or
graphs determine y as a function of x?
— y = 2
— x = 3
— y = |x|
 x + 3, x ≤ 1
 x − 2, x > 1
— y = 
— x2 + y2 = 1
y
1
-1
1
x
-1
A1.3.B Represent a function with a symbolic
expression, as a graph, in a table, and using
words, and make connections among these
representations.
This expectation applies each time a new class (family)
of functions is encountered. In Algebra 1, students
should be introduced to a variety of additional functions
that include expressions such as x3,
1
x , , and
x
absolute values. They will study these functions in
depth in subsequent courses.
Students should know that f(x) =
a
represents an
x
inverse variation. Students begin to describe the graph
of a function from its symbolic expression, and use
key characteristics of the graph of a function to infer
properties of the related symbolic expression.
Translating among these various representations
of functions is an important way to demonstrate
conceptual understanding of functions.
Students learn that each representation has particular
advantages and limitations. For example, a graph
shows the shape of a function, but not exact values.
They also learn that a table of values may not uniquely
determine a single function without some specification
of the nature of that function (e.g., it is quadratic).
July 2008
Washington State K–12 Mathematics Standards
121
Algebra I
Performance Expectations
Explanatory Comments and Examples
Students are expected to:
A1.3.C Evaluate f(x) at a (i.e., f(a)) and solve for x in
the equation f(x) = b.
Functions may be described and evaluated with symbolic
expressions, tables, graphs, or verbal descriptions.
Students should distinguish between solving for f(x)
and evaluating a function at x.
Example:
•
Roses-R-Red sells its roses for $0.75 per stem
and charges a $20 delivery fee per order.
— What is the cost of having 10 roses delivered?
— How many roses can you have delivered
for $65?
122
July 2008
Washington State K–12 Mathematics Standards
Algebra I
Algebra 1
A1.4. Core Content: Linear functions, equations, and inequalities
(Algebra)
S
tudents understand that linear functions can be used to model situations involving a constant rate
of change. They build on the work done in middle school to solve sets of linear equations and
inequalities in two variables, learning to interpret the intersection of the lines as the solution. While the
focus is on solving equations, students also learn graphical and numerical methods for approximating
solutions to equations. They use linear functions to analyze relationships, represent and model problems,
and answer questions. These algebraic skills are applied in other Core Content areas across high school
courses.
Performance Expectations
Explanatory Comments and Examples
Students are expected to:
A1.4.A Write and solve linear equations and
inequalities in one variable.
This expectation includes the use of absolute values in
the equations and inequalities.
Examples:
•
Write an absolute value equation or inequality for
— all the numbers 2 units from 7, and
— all the numbers that are more than b units
from a.
•
Solve |x – 6| ≤ 4 and locate the solution on the
number line.
•
Write an equation or inequality that has
— no real solutions;
— infinite numbers of real solutions; and
— exactly one real solution.
A1.4.B Write and graph an equation for a line given
the slope and the y-intercept, the slope and a
point on the line, or two points on the line, and
translate between forms of linear equations.
•
Solve for x in 2(x – 3) + 4x = 15 + 2x.
•
Solve 8.5 < 3x + 2 ≤ 9.7 and locate the solution on
the number line.
Linear equations may be written in slope-intercept,
point-slope, and standard form.
Examples:
•
Find an equation for a line with y-intercept equal to
2 and slope equal to 3.
•
Find an equation for a line with a slope of 2 that
goes through the point (1, 1).
•
Find an equation for a line that goes through the
points (-3, 5) and (6, -2).
•
For each of the following, use only the equation
(without sketching the graph) to describe the graph.
— y = 2x + 3
— y – 7 = 2(x – 2)
July 2008
Washington State K–12 Mathematics Standards
•
Write the equation 3x + 2y = 5 in slope intercept form.
•
Write the equation y – 1 = 2(x – 2) in standard form.
123
Algebra I
Performance Expectations
Explanatory Comments and Examples
Students are expected to:
A1.4.C Identify and interpret the slope and intercepts
of a linear function, including equations for
parallel and perpendicular lines.
Examples:
•
The graph shows the relationship between time and
distance from a gas station for a motorcycle and a
scooter. What can be said about the relative speed
of the motorcycle and scooter that matches the
information in the graph? What can be said about
the intersection of the graphs of the scooter and
the motorcycle? Is it possible to tell which vehicle
is further from the gas station at the initial starting
point represented in the graph? At the end of the
time represented in the graph? Why or why not?
Distance
scooter
motorcycle
Time
•
A 1,500-gallon tank contains 200 gallons of water.
Water begins to run into the tank at the rate of
75 gallons per hour. When will the tank be full?
Find a linear function that models this situation,
draw a graph, and create a table of data points.
Once you have answered the question and
completed the tasks, explain your reasoning.
Interpret the slope and y-intercept of the function
in the context of the situation.
•
Given that the figure below is a square, find the
slope of the perpendicular sides AB and BC.
Describe the relationship between the two slopes.
A
B
D
q
p
q
C
p
124
July 2008
Washington State K–12 Mathematics Standards
Algebra I
Performance Expectations
Explanatory Comments and Examples
Students are expected to:
A1.4.D Write and solve systems of two linear
equations and inequalities in two variables.
Students solve both symbolic and word problems,
understanding that the solution to a problem is given by
the coordinates of the intersection of the two lines when
the lines are graphed in the same coordinate plane.
Examples:
•
Solve the following simultaneous linear equations
algebraically:
— -2x + y = 2
— x + y = -1
A1.4.E Describe how changes in the parameters
of linear functions and functions containing
an absolute value of a linear expression
affect their graphs and the relationships they
represent.
•
Graph the above two linear equations on the same
coordinate plane and use the graph to verify the
algebraic solution.
•
An academic team is going to a state mathematics
competition. There are 30 people going on the trip.
There are 5 people who can drive and 2 types of
vehicles, vans and cars. A van seats 8 people, and
a car seats 4 people, including drivers. How many
vans and cars does the team need for the trip?
Explain your reasoning.
Let v = number of vans and c = number of cars.
v+c≤5
8v + 4c > 30
In the case of a linear function y = f(x), expressed
in slope-intercept form (y = mx + b), m and b are
parameters. Students should know that f(x) = kx
represents a direct variation (proportional relationship).
Examples:
•
Graph a function of the form f(x) = kx, describe
the effect that changes on k have on the graph
and on f(x), and answer questions that arise in
proportional situations.
•
A gas station’s 10,000-gallon underground storage
tank contains 1,000 gallons of gasoline. Tanker
trucks pump gasoline into the tank at a rate of 400
gallons per minute. How long will it take to fill the
tank? Find a function that represents this situation
and then graph the function. If the flow rate increases
from 400 to 500 gallons per minute, how will the
graph of the function change? If the initial amount
of gasoline in the tank changes from 1,000 to 2,000
gallons, how will the graph of the function change?
•
Compare and contrast the functions y = 3|x| and
y= -
July 2008
Washington State K–12 Mathematics Standards
1
x .
3
125
Algebra I
Algebra 1
A1.5. Core Content: Quadratic functions and equations
(Algebra)
S
tudents study quadratic functions and their graphs, and solve quadratic equations with real roots in
Algebra 1. They use quadratic functions to represent and model problems and answer questions
in situations that are modeled by these functions. Students solve quadratic equations by factoring
and computing with polynomials. The important mathematical technique of completing the square is
developed enough so that the quadratic formula can be derived.
Performance Expectations
Explanatory Comments and Examples
Students are expected to:
A1.5.A Represent a quadratic function with a symbolic
expression, as a graph, in a table, and with
a description, and make connections among
the representations.
Example:
•
Kendre and Tyra built a tennis ball cannon that
launches tennis balls straight up in the air at an
initial velocity of 50 feet per second. The mouth of
the cannon is 2 feet off the ground. The function
h(t) = -16t2 + 50t + 2 describes the height, h, in
feet, of the ball t seconds after the launch.
Make a table from the function. Then use the table
to sketch a graph of the height of the tennis ball
as a function of time into the launch. Give a verbal
description of the graph. How high was the ball after
1 second? When does it reach this height again?
A1.5.B Sketch the graph of a quadratic function,
describe the effects that changes in the
parameters have on the graph, and interpret the
x-intercepts as solutions to a quadratic equation.
Note that in Algebra 1, the parameter b in the term bx
in the quadratic form ax2 + bx + c is not often used to
provide useful information about the characteristics of
the graph.
Parameters considered most useful are:
•
a and c in f(x) = ax2 + c
•
a, h, and k in f(x) = a(x – h)2 + k, and
•
a, r, and s in f(x) = a(x – r)(x – s)
Example:
•
A particular quadratic function can be expressed in
the following two ways:
f(x) = -(x – 3)2 + 1
f(x) = -(x – 2)(x – 4)
— What information about the graph can be
directly inferred from each of these forms?
Explain your reasoning.
— Sketch the graph of this function, showing
the roots.
126
July 2008
Washington State K–12 Mathematics Standards
Algebra I
Performance Expectations
Explanatory Comments and Examples
Students are expected to:
A1.5.C Solve quadratic equations that can be factored
as (ax + b)(cx + d) where a, b, c, and d
are integers.
Students learn to efficiently solve quadratic equations
by recognizing and using the simplest factoring
methods, including recognizing special quadratics as
squares and differences of squares.
Examples:
A1.5.D Solve quadratic equations that have real roots
by completing the square and by using the
quadratic formula.
•
2x2 + x – 3 = 0; (x – 1)(2x + 3) = 0; x = 1, −-
•
4x2 + 6x = 0; 2x(2x + 3) = 0; x = 0, −-
•
36x2 – 25 = 0; (6x + 5)(6x – 5) = 0; x = ±
•
x2 + 6x + 9 = 0; (x + 3)2 = 0; x = -3
3
2
3
2
5
6
Students solve those equations that are not easily
factored by completing the square and by using the
quadratic formula. Completing the square should also
be used to derive the quadratic formula.
Students learn how to determine if there are two, one,
or no real solutions.
Examples:
•
Complete the square to solve x2 + 4x = 13.
x2 + 4x – 13 = 0
x2 + 4x + 4 = 17
(x + 2)2 = 17
x + 2 = ± 17
x = -2 ± 17
x ≈ 2.12, -6.12
•
Use the quadratic formula to solve 4x2 – 2x = 5.
x=
x=
- b ± b 2 – 4 ac
2a
- (- 2 )
x=
2 ± 84
8
x=
2 ± 2 21
8
x=
1 ± 21
4
( - 2)2 – 4(4 -5)
2(4)
x ≈ 1.40, -0.90
July 2008
Washington State K–12 Mathematics Standards
127
Algebra I
Algebra 1
A1.6. Core Content: Data and distributions
(Data/Statistics/Probability)
S
tudents select mathematical models for data sets and use those models to represent, describe, and
compare data sets. They analyze data to determine the relationship between two variables and make
and defend appropriate predictions, conjectures, and generalizations. Students understand limitations
of conclusions based on results of a study or experiment and recognize common misconceptions and
misrepresentations in interpreting conclusions.
Performance Expectations
Explanatory Comments and Examples
Students are expected to:
A1.6.A Use and evaluate the accuracy of summary
statistics to describe and compare data sets.
A univariate set of data identifies data on a single
variable, such as shoe size.
This expectation extends what students have learned
in earlier grades to include evaluation and justification.
They both compute and evaluate the appropriateness
of measure of center and spread (range and
interquartile range) and use these measures to
accurately compare data sets. Students will draw
appropriate conclusions through the use of statistical
measures of center, frequency, and spread, combined
with graphical displays.
Examples:
•
The local minor league baseball team has a salary
dispute. Players claim they are being underpaid,
but managers disagree.
— Bearing in mind that a few top players
earn salaries that are quite high, would it
be in the managers’ best interest to use
the mean or median when quoting the
“average” salary of the team? Why?
— What would be in the players’ best interest?
•
Each box-and-whisker plot shows the prices of
used cars (in thousands of dollars) advertised for
sale at three different car dealers. If you want to go
to the dealer whose prices seem least expensive,
which dealer would you go to? Use statistics from
the displays to justify your answer.
Cars are US
Better-than-New
Yours Now
128
0
5
10
July 2008
Washington State K–12 Mathematics Standards
Algebra I
Performance Expectations
Explanatory Comments and Examples
Students are expected to:
A1.6.B Make valid inferences and draw conclusions
based on data.
Determine whether arguments based on data
confuse association with causation. Evaluate the
reasonableness of and make judgments about
statistical claims, reports, studies, and conclusions.
Example:
•
A1.6.C Describe how linear transformations affect the
center and spread of univariate data.
A1.6.D Find the equation of a linear function that best
fits bivariate data that are linearly related,
interpret the slope and y-intercept of the line,
and use the equation to make predictions.
Mr. Shapiro found that the amount of time his
students spent doing mathematics homework is
positively correlated with test grades in his class. He
concluded that doing homework makes students’
test scores higher. Is this conclusion justified?
Explain any flaws in Mr. Shapiro’s reasoning.
Examples:
•
A company decides to give every one of its
employees a $5,000 raise. What happens to
the mean and standard deviation of the salaries
as a result?
•
A company decides to double each of its
employee’s salaries. What happens to the mean
and standard deviation of the salaries as a result?
A bivariate set of data presents data on two variables,
such as shoe size and height.
In high school, the emphasis is on using a line of best
fit to interpret data and on students making judgments
about whether a bivariate data set can be modeled with
a linear function. Students can use various methods,
including technology, to obtain a line of best fit.
Making predictions involves both interpolating and
extrapolating from the original data set.
Students need to be able to evaluate the quality of their
predictions, recognizing that extrapolation is based
on the assumption that the trend indicated continues
beyond the unknown data.
July 2008
Washington State K–12 Mathematics Standards
129
Algebra I
Performance Expectations
Explanatory Comments and Examples
Students are expected to:
A1.6.E Describe the correlation of data in scatterplots in
terms of strong or weak and positive or negative.
Example:
•
Which words—strong or weak, positive or
negative—could be used to describe the
correlation shown in the sample scatterplot below?
Scatterplot
-1
-1.5
x
-2
Y
x
x x
x x
x x
x
xx
-2.5
x
-3
-3.5
-4
130
100
200
300
x
x xx
xx x
x
400
x
500
X
July 2008
Washington State K–12 Mathematics Standards
Algebra I
Algebra 1
A1.7. Additional Key Content (Algebra)
S
tudents develop a basic understanding of arithmetic and geometric sequences and of exponential
functions, including their graphs and other representations. They use exponential functions to analyze
relationships, represent and model problems, and answer questions in situations that are modeled by
these nonlinear functions. Students learn graphical and numerical methods for approximating solutions to
exponential equations. Students interpret the meaning of problem solutions and explain limitations related
to solutions.
Performance Expectations
Explanatory Comments and Examples
Students are expected to:
A1.7.A Sketch the graph for an exponential function
of the form y = abn where n is an integer,
describe the effects that changes in the
parameters a and b have on the graph, and
answer questions that arise in situations
modeled by exponential functions.
Examples:
•
Sketch the graph of y = 2n by hand.
•
You have won a door prize and are given a choice
between two options:
$150 invested for 10 years at 4% compounded
annually.
$200 invested for 10 years at 3% compounded
annually.
— How much is each worth at the end of
each year of the investment periods?
— Are the two investments ever equal in
value? Which will you choose?
A1.7.B Find and approximate solutions to exponential
equations.
Students can approximate solutions using graphs or
tables with and without technology.
A1.7.C Express arithmetic and geometric sequences
in both explicit and recursive forms, translate
between the two forms, explain how rate of
change is represented in each form, and use the
forms to find specific terms in the sequence.
Examples:
•
Write a recursive formula for the arithmetic
sequence 5, 9, 13, 17, . . . What is the slope of the
line that contains the points associated with these
values and their position in the sequence? How is
the slope of the line related to the sequence?
•
Given that u(0) = 3 and u(n + 1) = u(n) + 7 when n
is a positive integer,
a. find u(5);
b. find n so that u(n) = 361; and
c. find a formula for u(n).
•
Write a recursive formula for the geometric sequence
5, 10, 20, 40, . . . and determine the 100th term.
•
Given that u(0) = 2 and u(n + 1) = 3u(n),
a. find u(4), and
b. find a formula for u(n).
A1.7.D Solve an equation involving several variables by
expressing one variable in terms of the others.
July 2008
Washington State K–12 Mathematics Standards
Examples:
•
Solve A = p + prt for p.
•
Solve V = πr 2h for h or for r.
131
Algebra I
Algebra 1
A1.8. Core Processes: Reasoning, problem solving, and communication
S
tudents formalize the development of reasoning in Algebra 1 as they use algebra and the properties
of number systems to develop valid mathematical arguments, make and prove conjectures, and find
counterexamples to refute false statements, using correct mathematical language, terms, and symbols in
all situations. They extend the problem-solving practices developed in earlier grades and apply them to
more challenging problems, including problems related to mathematical and applied situations. Students
formalize a coherent problem-solving process in which they analyze the situation to determine the
question(s) to be answered, synthesize given information, and identify implicit and explicit assumptions
that have been made. They examine their solution(s) to determine reasonableness, accuracy, and
meaning in the context of the original problem. The mathematical thinking, reasoning, and problemsolving processes students learn in high school mathematics can be used throughout their lives as they
deal with a world in which an increasing amount of information is presented in quantitative ways and more
and more occupations and fields of study rely on mathematics.
Performance Expectations
Explanatory Comments and Examples
Students are expected to:
A1.8.A Analyze a problem situation and represent it
mathematically.
Examples:
•
A1.8.B Select and apply strategies to solve problems.
A1.8.C Evaluate a solution for reasonableness, verify
its accuracy, and interpret the solution in the
context of the original problem.
Three teams of students independently conducted
experiments to relate the rebound height of a
ball to the rebound number. The table gives the
average of the teams’ results.
Construct a scatterplot of the data, and describe
the function that relates the height of the ball to the
rebound number. Predict the rebound height of the
ball on the tenth rebound. Justify your answer.
A1.8.D Generalize a solution strategy for a single
problem to a class of related problems, and
apply a strategy for a class of related problems
to solve specific problems.
A1.8.E Read and interpret diagrams, graphs, and
text containing the symbols, language, and
conventions of mathematics.
A1.8.F Summarize mathematical ideas with precision
and efficiency for a given audience and purpose.
A1.8.G Synthesize information to draw conclusions,
and evaluate the arguments and conclusions
of others.
A1.8.H Use inductive reasoning about algebra and the
properties of numbers to make conjectures,
and use deductive reasoning to prove or
disprove conjectures.
132
Rebound
Number
Rebound
Height (cm)
0
200
1
155
2
116
3
88
4
66
5
50
6
44
•
Prove (a + b)2 = a2 + 2ab + b2.
•
A student writes (x + 3)2 = x2 + 9. Explain why this
is incorrect.
•
Prove formally that the sum of two odd numbers is
always even.
July 2008
Washington State K–12 Mathematics Standards
Geometry
July 2008
Washington State K–12 Mathematics Standards
133
Geometry
Geometry
G.1. Core Content: Logical arguments and proofs (Logic)
S
tudents formalize the reasoning skills they have developed in previous grades and solidify their
understanding of what it means to prove a geometric statement mathematically. In Geometry,
students encounter the concept of formal proof built on definitions, axioms, and theorems. They use
inductive reasoning to test conjectures about geometric relationships and use deductive reasoning
to prove or disprove their conclusions. Students defend their reasoning using precise mathematical
language and symbols.
Performance Expectations
Explanatory Comments and Examples
Students are expected to:
G.1.A
Distinguish between inductive and
deductive reasoning.
Students generate and test conjectures inductively and
then prove (or disprove) their conclusions deductively.
Example:
•
G.1.B Use inductive reasoning to make conjectures,
to test the plausibility of a geometric statement,
and to help find a counterexample.
G.1.C Use deductive reasoning to prove that a valid
geometric statement is true.
A student first hypothesizes that the number of
degrees in a polygon = 180 • (s – 2), where s
represents the number of sides, and then proves
this is true. When was the student using inductive
reasoning? When was s/he using deductive
reasoning? Justify your answers.
Examples:
•
Investigate the relationship among the medians of
a triangle using paper folding. Make a conjecture
about this relationship.
•
Using dynamic geometry software, decide if the
following is a plausible conjecture: If segment AM
is a median in triangle ABC, then ray AM bisects
angle BAC.
Valid proofs may be presented in paragraph, twocolumn, or flow-chart formats. Proof by contradiction is
a form of deductive reasoning.
Example:
•
G.1.D Write the converse, inverse, and contrapositive
of a valid proposition and determine their validity.
G.1.E Identify errors or gaps in a mathematical
argument and develop counterexamples
to refute invalid statements about geometric
relationships.
July 2008
Washington State K–12 Mathematics Standards
Prove that the diagonals of a rhombus are
perpendicular bisectors of each other.
Examples:
•
If m and n are odd integers, then the sum of m
and n is an even integer. State the converse and
determine whether it is valid.
•
If a quadrilateral is a rectangle, the diagonals have
the same length. State the contrapositive and
determine whether it is valid.
Example:
•
Identify errors in reasoning in the following proof:
Given ∠ABC ≅ ∠PRQ, AB ≅ PQ, and BC ≅ QR,
then ∆ABC ≅ ∆PQR by SAS.
135
Geometry
Performance Expectations
Explanatory Comments and Examples
Students are expected to:
G.1.F
Distinguish between definitions and undefined
geometric terms and explain the role of
definitions, undefined terms, postulates
(axioms), and theorems.
Students sketch points and lines (undefined terms) and
define and sketch representations of other common
terms. They use definitions and postulates as they
prove theorems throughout geometry. In their work
with theorems, they identify the hypothesis and the
conclusion and explain the role of each.
Students describe the consequences of changing
assumptions or using different definitions for
subsequent theorems and logical arguments.
Example:
•
There are two definitions of trapezoid that can be
found in books or on the web. A trapezoid is either
— a quadrilateral with exactly one pair of
parallel sides or
— a quadrilateral with at least one pair of
parallel sides.
Write some theorems that are true when applied
to one definition but not the other, and explain
your answer.
136
July 2008
Washington State K–12 Mathematics Standards
Geometry
Geometry
G.2. Core Content: Lines and angles (Geometry/Measurement)
S
tudents study basic properties of parallel and perpendicular lines, their respective slopes, and the
properties of the angles formed when parallel lines are intersected by a transversal. They prove
related theorems and apply them to solve both mathematical and practical problems.
Performance Expectations
Explanatory Comments and Examples
Students are expected to:
G.2.A
Know, prove, and apply theorems about
parallel and perpendicular lines.
Students should be able to summarize and explain
basic theorems. They are not expected to recite lists
of theorems, but they should know the conclusion of a
theorem when given its hypothesis.
Examples:
G.2.B Know, prove, and apply theorems about
angles, including angles that arise from
parallel lines intersected by a transversal.
•
Prove that a point on the perpendicular bisector of
a line segment is equidistant from the ends of the
line segment.
•
If each of two lines is perpendicular to a given line,
what is the relationship between the two lines?
How do you know?
Examples:
•
Prove that if two parallel lines are cut by a
transversal, then alternate-interior angles are equal.
•
Take two parallel lines l and m, with (distinct)
points A and B on l and C and D on m.
If AC intersects BD at point E, prove that
∆ABE ≅ ∆CDE.
G.2.C Explain and perform basic compass and
straightedge constructions related to parallel
and perpendicular lines.
Constructions using circles and lines with dynamic
geometry software (i.e., virtual compass and straightedge)
are equivalent to paper and pencil constructions.
Example:
•
Construct and mathematically justify the steps to:
— Bisect a line segment.
— Drop a perpendicular from a point to a line.
— Construct a line through a point that is
parallel to another line.
G.2.D Describe the intersections of lines in the plane
and in space, of lines and planes, and of
planes in space.
July 2008
Washington State K–12 Mathematics Standards
Example:
•
Describe all the ways that three planes can
intersect in space.
137
Geometry
Geometry
G.3. Core Content: Two- and three-dimensional figures
(Geometry/Measurement)
S
tudents know and can prove theorems about two- and three-dimensional geometric figures, both
formally and informally. They identify necessary and sufficient conditions for proving congruence,
similarity, and properties of figures. Triangles are a primary focus, beginning with general properties
of triangles, working with right triangles and special triangles, proving and applying the Pythagorean
Theorem and its converse, and applying the basic trigonometric ratios of sine, cosine, and tangent.
Students extend their learning to other polygons and the circle, and do some work with threedimensional figures.
Performance Expectations
Explanatory Comments and Examples
Students are expected to:
G.3.A
Know, explain, and apply basic postulates
and theorems about triangles and the special
lines, line segments, and rays associated with
a triangle.
G.3.B Determine and prove triangle congruence,
triangle similarity, and other properties
of triangles.
Examples:
•
Prove that the sum of the angles of a triangle is 180°.
•
Prove and explain theorems about the incenter,
circumcenter, orthocenter, and centroid.
•
The rural towns of Atwood, Bridgeville, and
Carnegie are building a communications tower to
serve the needs of all three towns. They want to
position the tower so that the distance from each
town to the tower is equal. Where should they
locate the tower? How far will it be from each town?
Students should identify necessary and sufficient
conditions for congruence and similarity in triangles,
and use these conditions in proofs.
Examples:
G.3.C Use the properties of special right triangles (30°–60°–90° and 45°–45°–90°) to
solve problems.
138
•
Prove that congruent triangles are similar.
•
For a given ∆RST, prove that ∆XYZ, formed by
joining the midpoints of the sides of ∆RST, is
similar to ∆RST.
•
Show that not all SSA triangles are congruent.
Examples:
•
Determine the length of the altitude of an equilateral
triangle whose side lengths measure 5 units.
•
If one leg of a right triangle has length 5 and the
adjacent angle is 30°, what is the length of the
other leg and the hypotenuse?
•
If one leg of a 45°–45°–90° triangle has length 5,
what is the length of the hypotenuse?
July 2008
Washington State K–12 Mathematics Standards
Geometry
Performance Expectations
Explanatory Comments and Examples
Students are expected to:
G.3.C cont.
•
The pitch of a symmetrical roof on a house 40
feet wide is 30º. What is the length of the rafter, r,
exactly and approximately?
r
40
‘
30º
G.3.D Know, prove, and apply the Pythagorean
Theorem and its converse.
Examples:
•
Consider any right triangle with legs a and b
and hypotenuse c. The right triangle is used to
create Figures 1 and 2. Explain how these figures
constitute a visual representation of a proof of the
Pythagorean Theorem.
P
c
b
Q
a
Figure 1
•
Figure 2
A juice box is 6 cm by 8 cm by 12 cm. A straw is
inserted into a hole in the center of the top of the box.
The straw must stick out 2 cm so you can drink from
it. If the straw must be long enough to touch each
bottom corner of the box, what is the minimum length
the straw must be? (Assume the diameter of the
straw is 0 for the mathematical model.)
12 cm
July 2008
Washington State K–12 Mathematics Standards
8 cm
6 cm
•
In ∆ABC, with right angle at C, draw the altitude
CD from C to AB. Name all similar triangles in the
diagram. Use these similar triangles to prove the
Pythagorean Theorem.
•
Apply the Pythagorean Theorem to derive the
distance formula in the (x, y) plane.
139
Geometry
Performance Expectations
Explanatory Comments and Examples
Students are expected to:
G.3.E Solve problems involving the basic
trigonometric ratios of sine, cosine,
and tangent.
G.3.F
Know, prove, and apply basic theorems
about parallelograms.
Examples:
•
A 12-foot ladder leans against a wall to form a 63°
angle with the ground. How many feet above the
ground is the point on the wall at which the ladder
is resting?
•
Use the Pythagorean Theorem to establish that
sin2ø + cos2ø = 1 for ø between 0° and 90°.
Properties may include those that address symmetry
and properties of angles, diagonals, and angle sums.
Students may use inductive and deductive reasoning
and counterexamples.
Examples:
G.3.G Know, prove, and apply theorems about
properties of quadrilaterals and other polygons.
•
Are opposite sides of a parallelogram always
congruent? Why or why not?
•
Are opposite angles of a parallelogram always
congruent? Why or why not?
•
Prove that the diagonals of a parallelogram bisect
each other.
•
Explain why if the diagonals of a quadrilateral
bisect each other, then the quadrilateral is a
parallelogram.
•
Prove that the diagonals of a rectangle are
congruent. Is this true for any parallelogram?
Justify your reasoning.
Examples:
•
What is the length of the apothem of a regular
hexagon with side length 8? What is the area of
the hexagon?
•
If the shaded pentagon were removed, it could be
replaced by a regular n-sided polygon that would
exactly fill the remaining space. Find the number of
sides, n, of a replacement polygon that makes the
three polygons fit perfectly.
P
140
July 2008
Washington State K–12 Mathematics Standards
Geometry
Performance Expectations
Explanatory Comments and Examples
Students are expected to:
G.3.H Know, prove, and apply basic theorems
relating circles to tangents, chords, radii,
secants, and inscribed angles.
G.3.I
Explain and perform constructions related to
the circle.
Examples:
•
Given a line tangent to a circle, know and explain
that the line is perpendicular to the radius drawn to
the point of tangency.
•
Prove that two chords equally distant from the
center of a circle are congruent.
•
Prove that if one side of a triangle inscribed in a circle
is a diameter, then the triangle is a right triangle.
•
Prove that if a radius of a circle is perpendicular to
a chord of a circle, then the radius bisects
the chord.
Students perform constructions using straightedge
and compass, paper folding, and dynamic geometry
software. What is important is that students understand
the mathematics and are able to justify each step in a
construction.
Example:
•
In each case, explain why the constructions work:
a. Construct the center of a circle from two chords.
b. Construct a circumscribed circle for a triangle.
c. Inscribe a circle in a triangle.
G.3.J
Describe prisms, pyramids, parallelepipeds,
tetrahedra, and regular polyhedra in terms of
their faces, edges, vertices, and properties.
July 2008
Washington State K–12 Mathematics Standards
Examples:
•
Given the number of faces of a regular polyhedron,
derive a formula for the number of vertices.
•
Describe symmetries of three-dimensional
polyhedra and their two-dimensional faces.
•
Describe the lateral faces that are required for a
pyramid to be a right pyramid with a regular base.
Describe the lateral faces required for an oblique
pyramid that has a regular base.
141
Geometry
Performance Expectations
Explanatory Comments and Examples
Students are expected to:
G.3.K Analyze cross-sections of cubes, prisms,
pyramids, and spheres and identify the
resulting shapes.
142
Examples:
•
Start with a regular tetrahedron with edges of unit
length 1. Find the plane that divides it into two
congruent pieces and whose intersection with
the tetrahedron is a square. Find the area of the
square. (Requires no pencil or paper.)
•
Start with a cube with edges of unit length 1. Find
the plane that divides it into two congruent pieces
and whose intersection with the cube is a regular
hexagon. Find the area of the hexagon.
•
Start with a cube with edges of unit length 1.
Find the plane that divides it into two congruent
pieces and whose intersection with the cube is a
rectangle that is not a face and contains four of the
vertices. Find the area of the rectangle.
•
Which has the larger area, the above rectangle or
the above hexagon?
July 2008
Washington State K–12 Mathematics Standards
Geometry
Geometry
G.4. Core Content: Geometry in the coordinate plane
(Geometry/Measurement, Algebra)
S
tudents make connections between geometry and algebra by studying geometric properties and
attributes that can be represented on the coordinate plane. They use the coordinate plane to
represent situations that are both purely mathematical and that arise in applied contexts. In this way, they
use the power of algebra to solve problems about shapes and space.
Performance Expectations
Explanatory Comments and Examples
Students are expected to:
G.4.A
Determine the equation of a line in
the coordinate plane that is described
geometrically, including a line through two
given points, a line through a given point
parallel to a given line, and a line through a
given point perpendicular to a given line.
G.4.B Determine the coordinates of a point that is
described geometrically.
G.4.C Verify and apply properties of triangles and
quadrilaterals in the coordinate plane.
July 2008
Washington State K–12 Mathematics Standards
Examples:
•
Write an equation for the perpendicular bisector of
a given line segment.
•
Determine the equation of a line through the points
(5, 3) and (5, -2).
•
Prove that the slopes of perpendicular lines are
negative inverses of each other.
Examples:
•
Determine the coordinates for the midpoint of a
given line segment.
•
Given the coordinates of three vertices of a
parallelogram, determine all possible coordinates
for the fourth vertex.
•
Given the coordinates for the vertices of a
triangle, find the coordinates for the center of the
circumscribed circle and the length of its radius.
Examples:
•
Given four points in a coordinate plane that are the
vertices of a quadrilateral, determine whether the
quadrilateral is a rhombus, a square, a rectangle,
a parallelogram, or none of these. Name all the
classifications that apply.
•
Given a parallelogram on a coordinate plane,
verify that the diagonals bisect each other.
•
Given the line with y-intercept 4 and x-intercept 3,
find the area of a square that has one corner on the
origin and the opposite corner on the line described.
143
Geometry
Performance Expectations
Explanatory Comments and Examples
Students are expected to:
G.4.C cont.
•
Below is a diagram of a miniature golf hole as
drawn on a coordinate grid. The dimensions of
the golf hole are 4 feet by 12 feet. Players must
start their ball from one of the three tee positions,
located at (1, 1), (1, 2), or (1, 3). The hole is
located at (10, 3). A wall separates the tees from
the hole. At which tee should the ball be placed to
create the shortest “hole-in-one” path? Sketch the
intended path of the ball, find the distance the ball
will travel, and describe your reasoning. (Assume
the diameters of the golf ball and the hole are 0 for
the mathematical model.)
..
.
Wall
Tees
.
Hole
G.4.D Determine the equation of a circle that is
described geometrically in the coordinate plane
and, given equations for a circle and a line,
determine the coordinates of their intersection(s).
144
Examples:
•
Write an equation for a circle with a radius of 2
units and center at (1, 3).
•
Given the circle x2 + y2 = 4 and the line y = x, find
the points of intersection.
•
Write an equation for a circle given a line segment
as a diameter.
•
Write an equation for a circle determined by a
given center and tangent line.
July 2008
Washington State K–12 Mathematics Standards
Geometry
Geometry
G.5. Core Content: Geometric transformations (Geometry/Measurement)
S
tudents continue their study of geometric transformations, focusing on the effect of such
transformations and the composition of transformations on the attributes of geometric figures.
They study techniques for establishing congruence and similarity by means of transformations.
Performance Expectations
Explanatory Comments and Examples
Students are expected to:
G.5.A
Sketch results of transformations and
compositions of transformations for a given
two-dimensional figure on the coordinate
plane, and describe the rule(s) for performing
translations or for performing reflections about
the coordinate axes or the line y = x.
G.5.B Determine and apply properties of
transformations.
Transformations include translations, rotations,
reflections, and dilations.
Example:
•
Line m is described by the equation y = 2x + 3.
Graph line m and reflect line m across the line
y = x. Determine the equation of the image of the
reflection. Describe the relationship between the
line and its image.
Students make and test conjectures about
compositions of transformations and inverses of
transformations, the commutativity and associativity of
transformations, and the congruence and similarity of
two-dimensional figures under various transformations.
Examples:
G.5.C Given two congruent or similar figures in a
coordinate plane, describe a composition of
translations, reflections, rotations, and dilations
that superimposes one figure on the other.
G.5.D Describe the symmetries of two-dimensional
figures and describe transformations, including
reflections across a line and rotations about
a point.
July 2008
Washington State K–12 Mathematics Standards
•
Identify transformations (alone or in composition)
that preserve congruence.
•
Determine whether the composition of two
reflections of a line is commutative.
•
Determine whether the composition of two rotations
about the same point of rotation is commutative.
•
Find a rotation that is equivalent to the composition
of two reflections over intersecting lines.
•
Find the inverse of a given transformation.
Examples:
•
Find a sequence of transformations that
superimposes the segment with endpoints (0, 0)
and (2, 1) on the segment with endpoints (4, 2)
and (3, 0).
•
Find a sequence of transformations that
superimposes the triangle with vertices (0, 0),
(1, 1), and (2, 0) on the triangle with vertices (0, 1),
(2, -1), and (0, -3).
Although the expectation only addresses twodimensional figures, classroom activities can easily
extend to three-dimensional figures. Students can also
describe the symmetries, reflections across a plane,
and rotations about a line for three-dimensional figures.
145
Geometry
Geometry
G.6. Additional Key Content (Measurement)
S
tudents extend and formalize their work with geometric formulas for perimeter, area, surface area, and
volume of two- and three-dimensional figures, focusing on mathematical derivations of these formulas
and their applications in complex problems. They use properties of geometry and measurement to solve
problems in purely mathematical as well as applied contexts. Students understand the role of units in
measurement and apply what they know to solve problems involving derived measures like speed or density.
They understand that all measurement is approximate and specify precision in measurement problems.
Performance Expectations
Explanatory Comments and Examples
Students are expected to:
G.6.A
Derive and apply formulas for arc length and
area of a sector of a circle.
Example:
•
Find the area and perimeter of the Reuleaux
triangle below.
The Reuleaux triangle is constructed with
three arcs. The center of each arc is located at
the vertex of an equilateral triangle. Each arc
extends between the two opposite vertices of the
equilateral triangle.
The figure below is a Reuleaux triangle that
circumscribes equilateral triangle ABC. ∆ABC has
side length of 5 inches. AB has center C, BC has
center A, and CA has center B, and all three arcs
have the same radius equal to the length of the
sides of the triangle.
A
C
G.6.B Analyze distance and angle measures on a
sphere and apply these measurements to the
geometry of the earth.
146
B
Examples:
•
Use a piece of string to measure the distance
between two points on a ball or globe; verify that
the string lies on an arc of a great circle.
•
On a globe, show with examples why airlines use
polar routes instead of flying due east from Seattle
to Paris.
•
Show that the sum of the angles of a triangle on a
sphere is greater than 180 degrees.
July 2008
Washington State K–12 Mathematics Standards
Geometry
Performance Expectations
Explanatory Comments and Examples
Students are expected to:
G.6.C Apply formulas for surface area and volume of
three-dimensional figures to solve problems.
Problems include those that are purely mathematical
as well as those that arise in applied contexts.
Three-dimensional figures include right and oblique
prisms, pyramids, cylinders, cones, spheres, and
composite three-dimensional figures.
Examples:
•
As Pam scooped ice cream into a cone, she began
to formulate a geometry problem in her mind. If the
ice cream was perfectly spherical with diameter
2.25'' and sat on a geometric cone that also had
diameter 2.25'' and was 4.5'' tall, would the cone
hold all the ice cream as it melted (without her eating
any of it)? She figured the melted ice cream would
have the same volume as the unmelted ice cream.
Find the solution to Pam’s problem and justify
your reasoning.
•
G.6.D Predict and verify the effect that changing
one, two, or three linear dimensions has on
perimeter, area, volume, or surface area of
two- and three-dimensional figures.
G.6.E Use different degrees of precision in
measurement, explain the reason for using
a certain degree of precision, and apply
estimation strategies to obtain reasonable
measurements with appropriate precision for
a given purpose.
July 2008
Washington State K–12 Mathematics Standards
A rectangle is 5 inches by 10 inches. Find the
volume of a cylinder that is generated by rotating
the rectangle about the 10-inch side.
The emphasis in high school should be on verifying the
relationships between length, area, and volume and on
making predictions using algebraic methods.
Examples:
•
What happens to the volume of a rectangular
prism if four parallel edges are doubled in length?
•
The ratio of a pair of corresponding sides in two
similar triangles is 5:3. The area of the smaller triangle
is 108 in2. What is the area of the larger triangle?
Example:
•
The U.S. Census Bureau reported a national
population of 299,894,924 on its Population Clock
in mid-October of 2006. One can say that the U.S.
population is 3 hundred million (3 × 108) and be
precise to one digit. Although the population had
surpassed 3 hundred million by the end of that month,
explain why 3 × 108 remained precise to one digit.
147
Geometry
Performance Expectations
Explanatory Comments and Examples
Students are expected to:
G.6.F
Solve problems involving measurement
conversions within and between systems,
including those involving derived units, and
analyze solutions in terms of reasonableness
of solutions and appropriate units.
This performance expectation is intended to build on
students’ knowledge of proportional relationships.
Students should understand the relationship between
scale factors and their inverses as they relate to
choices about when to multiply and when to divide in
converting measurements.
Derived units include those that measure speed,
density, flow rates, population density, etc.
Example:
•
148
A digital camera takes pictures that are 3.2
megabytes in size. If the pictures are stored on a
1-gigabyte card, how many pictures can be taken
before the card is full?
July 2008
Washington State K–12 Mathematics Standards
Geometry
Geometry
G.7. Core Processes: Reasoning, problem solving, and communication
S
tudents formalize the development of reasoning in Geometry as they become more sophisticated
in their ability to reason inductively and begin to use deductive reasoning in formal proofs. They
extend the problem-solving practices developed in earlier grades and apply them to more challenging
problems, including problems related to mathematical and applied situations. Students use a coherent
problem-solving process in which they analyze the situation to determine the question(s) to be answered,
synthesize given information, and identify implicit and explicit assumptions that have been made.
They examine their solution(s) to determine reasonableness, accuracy, and meaning in the context of
the original problem. They use correct mathematical language, terms, symbols, and conventions as
they address problems in Geometry and provide descriptions and justifications of solution processes.
The mathematical thinking, reasoning, and problem-solving processes students learn in high school
mathematics can be used throughout their lives as they deal with a world in which an increasing amount
of information is presented in quantitative ways, and more and more occupations and fields of study rely
on mathematics.
Performance Expectations
Explanatory Comments and Examples
Students are expected to:
G.7.A
Analyze a problem situation and represent
it mathematically.
G.7.B Select and apply strategies to solve problems.
G.7.C Evaluate a solution for reasonableness, verify
its accuracy, and interpret the solution in the
context of the original problem.
Examples:
•
AB is the diameter of the semicircle and the radius
of the quarter circle shown in the figure below. BC is
the perpendicular bisector of AB.
G.7.D Generalize a solution strategy for a single
problem to a class of related problems, and
apply a strategy for a class of related problems
to solve specific problems.
G.7.E Read and interpret diagrams, graphs, and
text containing the symbols, language, and
conventions of mathematics.
G.7.F Summarize mathematical ideas with precision
and efficiency for a given audience and purpose.
G.7.G Synthesize information to draw conclusions
and evaluate the arguments and conclusions
of others.
E
D
F
A
C
B
Imagine all of the triangles formed by AB and any
arbitrary point lying in the region bounded by AC,
CD, and AD , seen in bold below.
E
D
F
G.7.H Use inductive reasoning to make conjectures,
and use deductive reasoning to prove or
disprove conjectures.
A
C
B
Use inductive reasoning to make conjectures about
what types of triangles are formed based upon
the region where the third vertex is located. Use
deductive reasoning to verify your conjectures.
July 2008
Washington State K–12 Mathematics Standards
149
Geometry
Performance Expectations
Explanatory Comments and Examples
Students are expected to:
G.7 cont.
•
Rectangular cartons that are 5 feet long need to
be placed in a storeroom that is located at the end
of a hallway. The walls of the hallway are parallel.
The door into the hallway is 3 feet wide and the
width of the hallway is 4 feet. The cartons must be
carried face up. They may not be tilted. Investigate
the width and carton top area that will fit through
the doorway.
C
S
3
4
T
5
A
R
Generalize your results for a hallway opening of x
feet and a hallway width of y feet if the maximum
carton dimensions are c feet long and x2 + y2 = c2.
150
July 2008
Washington State K–12 Mathematics Standards
Algebra 2
July 2008
Washington State K–12 Mathematics Standards
151
Algebra 2
Algebra 2
A2.1. Core Content: Solving problems
T
he first core content area highlights the type of problems students will be able to solve by the end
of Algebra 2, as they extend their ability to solve problems with additional functions and equations.
When presented with a word problem, students are able to determine which function or equation
models the problem and use that information to solve the problem. They build on what they learned
in Algebra 1 about linear and quadratic functions and are able to solve more complex problems.
Additionally, students learn to solve problems modeled by exponential and logarithmic functions,
systems of equations and inequalities, inverse variations, and combinations and permutations. Turning
word problems into equations that can be solved is a skill students hone throughout Algebra 2 and
subsequent mathematics courses.
Performance Expectations
Explanatory Comments and Examples
Students are expected to:
A2.1.A Select and justify functions and equations to
model and solve problems.
Examples:
•
A manufacturer wants to design a cylindrical
soda can that will hold 500 milliliters (ml) of soda.
The manufacturer’s research has determined
that an optimal can height is between 10 and 15
centimeters. Find a function for the radius in terms
of the height, and use it to find the possible range
of radius measurements in centimeters. Explain
your reasoning. •
Dawson wants to make a horse corral by creating
a rectangle that is divided into 2 parts, similar to
the following diagram. He has a 1200-foot roll of
fencing to do the job.
— What are the dimensions of the enclosure
with the largest total area?
— What function or equation best models
this situation?
l
w
July 2008
Washington State K–12 Mathematics Standards
153
Algebra 2
Performance Expectations
Explanatory Comments and Examples
Students are expected to:
A2.1.B Solve problems that can be represented by
systems of equations and inequalities.
Examples:
•
Mr. Smith uses the following formula to calculate
students’ final grades in his Algebra 2 class:
0.4E + 0.6T = C, where E represents the score
on the final exam, and T represents the average
score of all tests given during the grading period.
All tests and the final exam are worth a maximum
of 100 points. The minimum passing score on
tests, the final exam, and the course is 60.
Determine the inequalities that describe the
following situation and sketch a system of graphs
to illustrate it. When necessary, round scores to
the nearest tenth.
— Is it possible for a student to have a failing
test score average (i.e., T < 60 points)
and still pass the course?
— If you answered “yes,” what is the
minimum test score average a student can
have and still pass the course? What final
exam score is needed to pass the course
with a minimum test score average?
— A student has a particular test score
average. How can (s)he figure out the
minimum final exam score needed to pass
the course?
•
154
Data derived from an experiment seems to be
parabolic when plotted on a coordinate grid. Three
observed data points are (2, 10), (3, 8), and (4, 4).
Write a quadratic equation that passes through
the points.
July 2008
Washington State K–12 Mathematics Standards
Algebra 2
Performance Expectations
Explanatory Comments and Examples
Students are expected to:
A2.1.C Solve problems that can be represented by
quadratic functions, equations, and inequalities.
In addition to solving area and velocity problems by
factoring and applying the quadratic formula to the
quadratic equation, students use the vertex form of
the equation to solve problems about maximums,
minimums, and symmetry.
Examples:
•
The Gateway Arch in St. Louis has a special
shape called a catenary, which looks a lot like a
parabola. It has a base width of 600 feet and is
630 feet high. Which is taller, this catenary arch
or a parabolic arch that has the same base width
but has a height of 450 feet at a point 150 feet
from one of the pillars? What is the height of the
parabolic arch?
•
Fireworks are launched upward from the ground
with an initial velocity of 160 feet per second. The
formula for vertical motion is h(t) = 0.5at2 + vt + s,
where the gravitational constant, a, is -32 feet per
square second, v represents the initial velocity, and
s represents the initial height. Time t is measured
in seconds, and height h is measured in feet.
For the ultimate effect, the fireworks must explode
after they reach the maximum height. For the
safety of the crowd, they must explode at least 256
ft above the ground. The fuses must be set for the
appropriate time interval that allows the fireworks
to reach this height. What range of times, starting
from initial launch and ending with fireworks
explosion, meets these conditions?
A2.1.D Solve problems that can be represented
by exponential and logarithmic functions
and equations.
July 2008
Washington State K–12 Mathematics Standards
Examples:
•
If you need $15,000 in 4 years to start college,
how much money would you need to invest
now? Assume an annual interest rate of 4%
compounded monthly for 48 months.
•
The half-life of a certain radioactive substance is
65 days. If there are 4.7 grams initially present,
how long will it take for there to be less than 1
gram of the substance remaining?
155
Algebra 2
Performance Expectations
Explanatory Comments and Examples
Students are expected to:
A2.1.E Solve problems that can be represented by
a
inverse variations of the forms f(x) = + b,
x
a
a
f(x) =
x
2
+ b, and f(x) =
Examples:
•
.
(bx + c)
At the You’re Toast, Dude! toaster company, the
weekly cost to run the factory is $1400, and the
cost of producing each toaster is an additional
$4 per toaster.
— Find a function to represent the weekly
cost in dollars, C(x), of producing x
toasters. Assume either unlimited
production is possible or set a maximum
per week. — Find a function to represent the total
production cost per toaster for a week.
— How many toasters must be produced
within a week to have a total production
cost per toaster of $8? •
A person’s weight varies inversely as the square of
his distance from the center of the earth. Assume
the radius of the earth is 4000 miles. How much
would a 200-pound man weigh
— 1000 miles above the surface of the earth?
— 2000 miles above the surface of the earth?
A2.1.F Solve problems involving combinations and
permutations.
156
Examples:
•
The company Ali works for allows her to invest
in her choice of 10 different mutual funds, 6 of
which grew by at least 5% over the last year. Ali
randomly selected 4 of the 10 funds in which to
invest. What is the probability that 3 of Ali’s funds
grew by 5%?
•
Four points (A, B, C, and D) lie on one straight
line, n, and five points (E, F, G, H, and J) lie on
another straight line, m, that is parallel to n. What
is the probability that three points, selected at
random, will form a triangle?
July 2008
Washington State K–12 Mathematics Standards
Algebra 2
Algebra 2
A2.2. Core Content: Numbers, expressions, and operations
(Numbers, Operations, Algebra)
S
tudents extend their understanding of number systems to include complex numbers, which they
will see as solutions for quadratic equations. They grow more proficient in their use of algebraic
techniques as they continue to use variables and expressions to solve problems. As problems become
more sophisticated and the level of mathematics increases, so does the complexity of the symbolic
manipulations and computations necessary to solve the problems. Students refine the foundational
algebraic skills they need to be successful in subsequent mathematics courses.
Performance Expectations
Explanatory Comments and Examples
Students are expected to:
A2.2.A Explain how whole, integer, rational, real, and
complex numbers are related, and identify
the number system(s) within which a given
algebraic equation can be solved.
Example:
•
Within which number system(s) can each of the
following be solved? Explain how you know.
— 3x + 2 = 5
— x2 = 1
— x2 =
1
4
— x2 = 2
— x2 = -2
—
A2.2.B Use the laws of exponents to simplify and
evaluate numeric and algebraic expressions
that contain rational exponents.
x
=π
7
Examples:
•
Convert the following from a radical to exponential
form or vice versa.
— 24
—
—
—
•
July 2008
Washington State K–12 Mathematics Standards
5
1
3
16
x2 + 1
x2
x
Evaluate x-2/3 for x = 27.
157
Algebra 2
Performance Expectations
Explanatory Comments and Examples
Students are expected to:
A2.2.C Add, subtract, multiply, divide, and simplify
rational and more general algebraic expressions.
In the same way that integers were extended to
fractions, polynomials are extended to rational
expressions. Students must be able to perform the
four basic arithmetic operations on more general
expressions that involve exponentials.
The binomial theorem is useful when raising
expressions to powers, such as (x + 3)5.
Examples:
•
x + 1 3x − 3
−
( x + 1)2 x 2 − 1
•
Divide
3/2
158
( x + 2)
x +1
by
x+ 2
x 2− 1
July 2008
Washington State K–12 Mathematics Standards
Algebra 2
Algebra 2
A2.3. Core Content: Quadratic functions and equations
(Algebra)
A
s students continue to solve quadratic equations and inequalities in Algebra 2, they encounter
complex roots for the first time. They learn to translate between forms of quadratic equations,
applying the vertex form to evaluate maximum and minimum values and find symmetry of the graph, and
they learn to identify which form should be used in a particular situation. This opens up a whole range of
new problems students can solve using quadratics. These algebraic skills are applied in subsequent high
school mathematics and statistics courses.
Performance Expectations
Explanatory Comments and Examples
Students are expected to:
A2.3.A Translate between the standard form of a
quadratic function, the vertex form, and the
factored form; graph and interpret the meaning
of each form.
Students translate among forms of a quadratic function
to convert to one that is appropriate—e.g., vertex
form—to solve specific problems.
Students learn about the advantages of the standard
form (f(x) = ax2 + bx + c), the vertex form
(f(x) = a(x – h)2 + d), and the factored form
(f(x) = a(x – r)(x – s)). They produce the vertex form
by completing the square on the function in standard
form, which allows them to see the symmetry of the
graph of a quadratic function as well as the maximum
or minimum. This opens up a whole range of new
problems students can solve using quadratics.
Students continue to find the solutions of the equation,
which in Algebra 2 can be either real or complex.
Example:
•
Find the minimum, the line of symmetry, and
the roots for the graphs of each of the following
functions:
f(x) = x2 – 4x + 3
f(x) = x2 – 4x + 4
f(x) = x2 – 4x + 5
A2.3.B Determine the number and nature of the roots
of a quadratic function.
Students should be able to recognize and interpret the
discriminant.
Students should also be familiar with the Fundamental
Theorem of Algebra, i.e., that all polynomials, not just
quadratics, have roots over the complex numbers. This
concept becomes increasingly important as students
progress through mathematics.
Example:
•
July 2008
Washington State K–12 Mathematics Standards
For what values of a does f(x) = x2 – 6x + a have 2
real roots, 1 real root, and no real roots?
159
Algebra 2
Performance Expectations
Explanatory Comments and Examples
Students are expected to:
A2.3.C Solve quadratic equations and inequalities,
including equations with complex roots.
Students solve equations that are not easily factored by
completing the square and by using the quadratic formula.
Examples:
•
x2 – 10x + 34 = 0
•
3x2 + 10 = 4x
•
Wile E. Coyote launches an anvil from 180
feet above the ground at time t = 0. The
equation that models this situation is given by
h = -16t2 + 96t + 180, where t is time measured
in seconds and h is height above the ground
measured in feet.
a. What is a reasonable domain restriction for t in
this context?
b. Determine the height of the anvil two seconds
after it was launched.
c. Determine the maximum height obtained by
the anvil.
d. Determine the time when the anvil is more
than 100 feet above ground.
•
160
Farmer Helen wants to build a pigpen. With 100
feet of fence, she wants a rectangular pen with
one side being a side of her existing barn. What
dimensions should she use for her pigpen in order
to have the maximum number of square feet?
July 2008
Washington State K–12 Mathematics Standards
Algebra 2
Algebra 2
A2.4. Core Content: Exponential and logarithmic functions and equations
(Algebra)
S
tudents extend their understanding of exponential functions from Algebra 1 with an emphasis on
inverse functions. This leads to a natural introduction of logarithms and logarithmic functions. They
learn to use the basic properties of exponential and logarithmic functions, graphing both types of function
to analyze relationships, represent and model problems, and answer questions. Students employ these
functions in many practical situations, such as applying exponential functions to determine compound
interest and applying logarithmic functions to determine the pH of a liquid.
Performance Expectations
Explanatory Comments and Examples
Students are expected to:
A2.4.A Know and use basic properties of exponential
and logarithmic functions and the inverse
relationship between them.
Examples:
•
Given f(x) = 4x, write an equation for the inverse
of this function. Graph the functions on the same
coordinate grid.
— Find f(-3).
— Evaluate the inverse function at 7.
•
Derive the formulas:
— logba ⋅ logab = 1
— logaN = logbN ⋅ logab
•
Find the exact value of x in:
— logx16 =
4
3
— log381 = x
•
Solve for y in terms of x:
— loga
y
=x
x
— 100 = x ⋅ 10y
A2.4.B Graph an exponential function of the form
f(x) = abx and its inverse logarithmic function.
Students expand on the work they did in Algebra 1 to
functions of the form y = abx. Although the concept of
inverses is not fully developed until Precalculus, there
is an emphasis in Algebra 2 on students recognizing
the inverse relationship between exponential and
logarithmic functions and how this is reflected in the
shapes of the graphs.
Example:
•
July 2008
Washington State K–12 Mathematics Standards
Find the equation for the inverse function of y = 3x.
Graph both functions. What characteristics of each
of the graphs indicate they are inverse functions?
161
Algebra 2
Performance Expectations
Explanatory Comments and Examples
Students are expected to:
A2.4.C Solve exponential and logarithmic equations.
Examples:
•
A recommended adult dosage of the cold
medication NoMoreFlu is 16 ml. NoMoreFlu
causes drowsiness when there are more than 4 ml
in one’s system, making it unsafe to drive, operate
machinery, etc. The manufacturer wants to print a
warning label telling people how long they should
wait after taking NoMoreFlu for the drowsiness
to pass. If the typical metabolic rate is such that
one quarter of the NoMoreFlu is lost every four
hours, and a person takes the full dosage, how
long should adults wait after taking NoMoreFlu to
ensure that there will be
— Less than 4 ml of NoMoreFlu in their
system?
— Less than 1 ml in their system?
— Less than 0.1 ml in their system?
162
2
•
Solve for x in 256 = 2 x
•
Solve for x in log5(x – 4) = 3.
−1
.
July 2008
Washington State K–12 Mathematics Standards
Algebra 2
Algebra 2
A2.5. Core Content: Additional functions and equations
(Algebra)
S
tudents learn about additional classes of functions including square root, cubic, logarithmic, and those
involving inverse variation. Students plot points and sketch graphs to represent these functions and
use algebraic techniques to solve related equations. In addition to studying the defining characteristics of
each of these classes of functions, students gain the ability to construct new functions algebraically and
using transformations. These extended skills and techniques serve as the foundation for further study and
analysis of functions in subsequent mathematics courses.
Performance Expectations
Explanatory Comments and Examples
Students are expected to:
A2.5.A Construct new functions using the
transformations f(x – h), f(x) + k, cf(x), and by
adding and subtracting functions, and describe
the effect on the original graph(s).
Students perform simple transformations on functions,
including those that contain the absolute value of
expressions, quadratic expressions, square root
expressions, and exponential expressions, to make
new functions.
Examples:
•
What sequence of transformations changes
f(x) = x2 to g(x) = -5(x – 3)2 + 2?
•
Carly decides to earn extra money by making
glass bead bracelets. She purchases tools for
$40.00. Elastic bead cord for each bracelet costs
$0.10. Glass beads come in packs of 10 beads,
and one pack has enough beads to make one
bracelet. Base price for the beads is $2.00 per
pack. For each of the first 100 packs she buys, she
gets $0.01 off each of the packs. (For example, if
she purchases three packs, each pack costs $1.97
instead of $2.00.) Carly plans to sell each bracelet
for $4.00. Assume Carly will make a maximum of
100 bracelets.
— Find a function C(b) that describes
Carly’s costs.
— Find a function R(b) that describes
Carly’s revenue.
Carly’s profit is described by P(b) = R(b) – C(b).
— Find P(b).
— What is the minimum number of bracelets
that Carly must sell in order to make a profit?
— To make a profit of $100?
July 2008
Washington State K–12 Mathematics Standards
163
Algebra 2
Performance Expectations
Explanatory Comments and Examples
Students are expected to:
A2.5.B Plot points, sketch, and describe the graphs
of functions of the form f (x) = a x − c + d , and
solve related equations.
Students solve algebraic equations that involve
the square root of a linear expression over the
real numbers. Students should be able to identify
extraneous solutions and explain how they arose.
Students should view the function g(x) = x as the
inverse function of f(x) = x2, recognizing that the
functions have different domains for x greater than or
equal to 0.
Example:
•
Analyze the following equations and tell what
you know about the solutions. Then solve the
equations.
—
A2.5.C Plot points, sketch, and describe the graphs
a
of functions of the form f (x) = + b,
x
a
a
f ( x) = 2 + b , and f (x) =
, and solve
( bx + c )
x
related equations.
—
5 x − 6 = −2
—
2 x + 15 = x
—
2x − 5 = x + 7
Examples:
•
Sketch the graphs of the four functions
f ( x) =
•
164
a
+ b when a = 4 and 8 and b = 0 and 1.
x2
Sketch the graphs of the four functions
f (x) =
A2.5.D Plot points, sketch, and describe the graphs
of cubic polynomial functions of the form
f(x) = ax3 + d as an example of higher order
polynomials and solve related equations.
2 x+5 =7
4
when b = 1 and 4 and c = 2 and 3.
( bx + c )
Example:
•
Solve for x in 60 = -2x3 + 6.
July 2008
Washington State K–12 Mathematics Standards
Algebra 2
Algebra 2
A2.6. Core Content: Probability, data, and distributions
(Data/Statistics/Probability)
S
tudents formalize their study of probability, computing both combinations and permutations to
calculate the likelihood of an outcome in uncertain circumstances and applying the binominal theorem
to solve problems. They extend their use of statistics to graph bivariate data and analyze its shape to
make predictions. They calculate and interpret measures of variability, confidence intervals, and margins
of error for population proportions. Dual goals underlie the content in the section: students prepare for the
further study of statistics and become thoughtful consumers of data.
Performance Expectations
Explanatory Comments and Examples
Students are expected to:
A2.6.A Apply the fundamental counting principle
and the ideas of order and replacement to
calculate probabilities in situations arising from
two-stage experiments (compound events).
A2.6.B Given a finite sample space consisting of
equally likely outcomes and containing
events A and B, determine whether A and B
are independent or dependent, and find the
conditional probability of A given B.
Example:
A2.6.C Compute permutations and combinations, and
use the results to calculate probabilities.
Example:
A2.6.D Apply the binomial theorem to solve problems
involving probability.
The binominal theorem is also applied when computing
with polynomials.
•
•
What is the probability of drawing a heart from
a standard deck of cards on a second draw,
given that a heart was drawn on the first draw
and not replaced?
Two friends, Abby and Ben, are among five students
being considered for three student council positions.
If each of the five students has an equal likelihood of
being selected, what is the probability that Abby and
Ben will both be selected?
Examples:
•
Use Pascal’s triangle and the binomial theorem
to find the number of ways six objects can be
selected four at a time.
•
In a survey, 33% of adults reported that they
preferred to get the news from newspapers rather
than television. If you survey 5 people, what is the
probability of getting exactly 2 people who say they
prefer news from the newspaper?
— Write an equation that can be used to
solve the problem.
— Create a histogram of the binomial
distribution of the probability of getting 0
through 5 responders saying they prefer
the newspaper.
A2.6.E Determine if a bivariate data set can be
better modeled with an exponential or a
quadratic function and use the model to
make predictions.
July 2008
Washington State K–12 Mathematics Standards
In high school, determining a formula for a curve
of best fit requires a graphing calculator or similar
technological tool.
165
Algebra 2
Performance Expectations
Explanatory Comments and Examples
Students are expected to:
A2.6.F Calculate and interpret measures of variability
and standard deviation and use these
measures and the characteristics of the normal
distribution to describe and compare data sets.
Students should be able to identify unimodality,
symmetry, standard deviation, spread, and the shape
of a data curve to determine whether the curve could
reasonably be approximated by a normal distribution.
Given formulas, students should be able to calculate
the standard deviation for a small data set, but
calculators ought to be used if there are very many
points in the data set. It is important that students
be able to describe the characteristics of the normal
distribution and identify common examples of data that
are and are not reasonably modeled by it. Common
examples of distributions that are approximately
normal include physical performance measurements
(e.g., weightlifting, timed runs), heights, and weights.
Apply the Empirical Rule (68–95–99.7 Rule) to
approximate the percentage of the population meeting
certain criteria in a normal distribution.
Example:
•
A2.6.G Calculate and interpret margin of error and
confidence intervals for population proportions.
Which is more likely to be affected by an outlier
in a set of data, the interquartile range or the
standard deviation?
Students will use technology based on the complexity
of the situation.
Students use confidence intervals to critique various
methods of statistical experimental design, data
collection, and data presentation used to investigate
important problems, including those reported in
public studies.
Example:
•
In 2007, 400 of the 500 10th graders in Local High
School passed the WASL. In 2008, 375 of the 480
10th graders passed the test. The Local Gazette
headline read “10th Grade WASL Scores Decline
in 2008!” In response, the Superintendent of Local
School District wrote a letter to the editor claiming
that, in fact, WASL performance was not significantly
lower in 2008 than it was in 2007. Who is correct,
the Local Gazette or the Superintendent?
Use mathematics to find the margin of error to justify
your conclusion. (Formula for the margin of error
(E): E = zc
p(1−p)
; z95 = 1.96, where n is the sample
n
size, p is the proportion of the sample with the trait
of interest, c is the confidence level, and zc is the
multiplier for the specified confidence interval.)
166
July 2008
Washington State K–12 Mathematics Standards
Algebra 2
Algebra 2
A2.7. Additional Key Content
(Algebra)
S
tudents study two important topics here. First, they extend their ability to solve systems of two
equations in two variables to solving systems of three equations in three variables, which leads
to the full development of matrices in Precalculus. Second, they formalize their work with series as
they learn to find the terms and partial sums of arithmetic series and the terms and partial and infinite
sums of geometric series. This conceptual understanding of series lays an important foundation for
understanding calculus.
Performance Expectations
Explanatory Comments and Examples
Students are expected to:
A2.7.A Solve systems of three equations with
three variables.
Students solve systems of equations using algebraic
and numeric methods.
Examples:
•
Jill, Ann, and Stan are to inherit $20,000. Stan is
to get twice as much as Jill, and Ann is to get twice
as much as Stan. How much does each get?
•
Solve the following system of equations.
2x – y – z = 7
3x + 5y + z = -10
4x – 3y + 2z = 4
A2.7.B Find the terms and partial sums of arithmetic
and geometric series and the infinite sum for
geometric series.
Students build on the knowledge gained in Algebra 1
to find specific terms in a sequence and to express
arithmetic and geometric sequences in both explicit and
recursive forms.
Examples:
•
A ball is dropped from a height of 10 meters. Each
time it hits the ground, it rebounds
3
of the distance
4
it has fallen. What is the total sum of the distances it
falls and rebounds before coming to rest?
•
Show that the sum of the first 10 terms of the
geometric series 1 +
1 1
1
+ +
+ ... is twice the
3 9 27
sum of the first 10 terms of the geometric series
1
3
1
9
1 – + – July 2008
Washington State K–12 Mathematics Standards
1
+ ...
27
167
Algebra 2
Algebra 2
A2.8. Core Processes: Reasoning, problem solving, and communication
S
tudents formalize the development of reasoning at high school as they use algebra and the properties
of number systems to develop valid mathematical arguments, make and prove conjectures, and find
counterexamples to refute false statements using correct mathematical language, terms, and symbols in
all situations. They extend the problem-solving practices developed in earlier grades and apply them to
more challenging problems, including problems related to mathematical and applied situations. Students
formalize a coherent problem-solving process in which they analyze the situation to determine the
question(s) to be answered, synthesize given information, and identify implicit and explicit assumptions
that have been made. They examine their solution(s) to determine reasonableness, accuracy, and
meaning in the context of the original problem. The mathematical thinking, reasoning, and problemsolving processes students learn in high school mathematics can be used throughout their lives as they
deal with a world in which an increasing amount of information is presented in quantitative ways and more
and more occupations and fields of study rely on mathematics.
Performance Expectations
Explanatory Comments and Examples
Students are expected to:
A2.8.A Analyze a problem situation and represent
it mathematically.
A2.8.B Select and apply strategies to solve problems.
A2.8.C Evaluate a solution for reasonableness, verify
its accuracy, and interpret the solution in the
context of the original problem.
A2.8.D Generalize a solution strategy for a single
problem to a class of related problems and
apply a strategy for a class of related problems
to solve specific problems.
Examples:
•
Show that a + b ≠ a + b , for all positive real
values of a and b.
•
Show that the product of two odd numbers is
always odd.
•
Leo is painting a picture on a canvas that
measures 32 inches by 20 inches. He has divided
the canvas into four different rectangles, as shown
in the diagram.
A2.8.E Read and interpret diagrams, graphs, and
text containing the symbols, language, and
conventions of mathematics.
A2.8.F Summarize mathematical ideas with precision
and efficiency for a given audience and purpose.
He would like the upper right corner to be a
rectangle that has a length 1.6 times its width. Leo
wants the area of the larger rectangle in the lower
left to be at least half the total area of the canvas.
168
July 2008
Washington State K–12 Mathematics Standards
Algebra 2
Performance Expectations
Explanatory Comments and Examples
Students are expected to:
A2.8.G Use inductive reasoning and the properties
of numbers to make conjectures, and use
deductive reasoning to prove or disprove
conjectures.
Describe all the possibilities for the dimensions of
the upper right rectangle to the nearest hundredth,
and explain why the possibilities are valid.
If Leo uses the largest possible dimensions for
the smaller rectangle:
A2.8.H Synthesize information to draw conclusions
and evaluate the arguments and conclusions
of others.
— What will the dimensions of the larger
rectangle be?
— Will the larger rectangle be similar to the
rectangle in the upper right corner? Why
or why not?
— Is the original canvas similar to the
rectangle in the upper right corner?
(A rectangle whose length and width are in the ratio
1+ 5
(approximately equal to 1.6) is called a “golden
2
rectangle” and is often used in art and architecture.)
•
A relationship between variables can be represented
with a table, a graph, an equation, or a description
in words.
— How can you decide from a table
whether a relationship is linear, quadratic,
or exponential?
— How can you decide from a graph
whether a relationship is linear, quadratic,
or exponential?
— How can you decide from an equation
whether a relationship is linear, quadratic,
or exponential?
July 2008
Washington State K–12 Mathematics Standards
169
Algebra 2
Mathematics 1
July 2008
Washington State K–12 Mathematics Standards
171
Mathematics 1
In Mathematics 1, students begin to formalize mathematics by exploring function concepts with emphasis
on the family of linear functions and their applications. Students extend their work with graphical and
numerical data analysis to include bivariate data involving linear relationships. Students identify and
prove relationships about lines in the plane and similar triangles. Proportionality is a common thread in
Mathematics 1 that connects linear functions, data analysis, and coordinate geometry. Throughout this
course, students develop their reasoning skills by making conjectures and predictions or creating simple
proofs related to algebraic, geometric, and statistical relationships.
172
July 2008
Washington State K–12 Mathematics Standards
Math 1
Mathematics 1
M1.1. Core Content: Solving problems (Algebra)
S
tudents learn to solve many new types of problems in Mathematics 1, and this first core content area
highlights the types of problems students will be able to solve after they master the concepts and skills
in this course. Throughout Mathematics 1, students spend considerable time with linear functions and are
introduced to other types of functions, including exponential functions and functions defined piecewise.
They learn that specific functions model situations described in word problems, and thus they learn the
broader notion that functions are used to solve various types of problems. The ability to write an equation
that represents a problem is an important mathematical skill in itself, and each new function provides
students the tool to solve yet another class of problems. Many problems that initially appear to be very
different from each other can actually be represented by identical equations. This is an important and
unifying principle of algebra—that the same algebraic techniques can be applied to a wide variety of
different situations.
Performance Expectations
Explanatory Comments and Examples
Students are expected to:
M1.1.A Select and justify functions and equations to
model and solve problems.
Students can analyze the rate of change of a function
represented with a table or graph to determine if the
function is linear. Students also analyze common
ratios to determine if the function is exponential. After
selecting a function to model a situation, students
describe appropriate domain restrictions. They use the
function to solve the problem and interpret the solution
in the context of the original situation.
Examples:
•
A cup is 6 cm tall, including a 1.1 cm lip. Find a
function that represents the height of a stack of
cups in terms of the number of cups in the stack.
Find a function that represents the number of cups
in a stack of a given height.
•
For the month of July, Michelle will be dog-sitting
for her very wealthy, but eccentric, neighbor, Mrs.
Buffett. Mrs. Buffett offers Michelle two different
salary plans:
Plan 1: $100 per day for the 31 days of the month.
Plan 2: $1 for July 1, $2 for July 2, $4 for July 3,
and so on, with the daily rate doubling each day.
a. Write functions that model the amount of
money Michelle will earn each day on Plan 1
and Plan 2. Justify the functions you wrote.
b. State an appropriate domain for each of the
models based on the context.
c. Which plan should Michelle choose to
maximize her earnings? Justify your
recommendation mathematically.
d. Extension: Write an algebraic function for the
cumulative pay for each plan based on the
number of days worked.
July 2008
Washington State K–12 Mathematics Standards
173
Math 1
Performance Expectations
Explanatory Comments and Examples
Students are expected to:
M1.1.B Solve problems that can be represented by
linear functions, equations, and inequalities.
It is mathematically important to represent a word
problem as an equation. Students must analyze the
situation and find a way to represent it mathematically.
After solving the equation, students think about the
solution in terms of the original problem.
Examples:
•
The assistant pizza maker makes 6 pizzas an
hour. The master pizza maker makes 10 pizzas
an hour but starts baking two hours later than his
assistant. Together, they must make 92 pizzas.
How many hours from when the assistant starts
baking will it take?
What is a general equation, in function form, that
could be used to determine the number of pizzas
that can be made in two or more hours?
•
M1.1.C Solve problems that can be represented by a
system of two linear equations or inequalities.
A swimming pool holds 375,000 liters of water.
Two large hoses are used to fill the pool. The
first hose fills at the rate of 1,500 liters per hour
and the second hose fills at the rate of 2,000 liters
per hour. How many hours does it take to fill the
pool completely?
Examples:
•
An airplane flies from Baltimore to Seattle (assume
a distance of 2,400 miles) in 7 hours, but the return
1
4
flight takes only 4 hours. The air speed of the
plane is the same in both directions. How many
miles per hour does the plane fly with respect to the
wind? What is the wind speed in miles per hour?
174
•
A coffee shop employee has one cup of 85% milk
(the rest is chocolate) and another cup of 60% milk
(the rest is chocolate). He wants to make one cup
of 70% milk. How much of the 85% milk and 60%
milk should he mix together to make the 70% milk?
•
Two plumbing companies charge different rates
for their service. Clyde’s Plumbing Company
charges a $75-per-visit fee that includes one hour
of labor plus $45 dollars per hour after the first
hour. We-Unclog-It Plumbers charges a $100-pervisit fee that includes one hour of labor plus $40
per hour after the first hour. For how many hours
of plumbing work would Clyde’s be less expensive
than We-Unclog-It?
Note: Although this context is discrete, students
can model it with continuous linear functions.
July 2008
Washington State K–12 Mathematics Standards
Math 1
Performance Expectations
Explanatory Comments and Examples
Students are expected to:
M1.1.D Solve problems that can be represented by
exponential functions and equations.
Students recognize common examples of exponential
growth or decay, such as applying exponential
functions to determine compound interest, population
growth, and radioactivity. They approximate solutions
with graphs or tables, check solutions numerically, and
when possible, solve problems exactly.
Example:
•
July 2008
Washington State K–12 Mathematics Standards
Mr. Tsu invests $1000 in a 5-year CD that pays 4%
interest compounded yearly. Present to Mr. Tsu his
expected balance at the end of years 1, 3, and 5
and the process you used to arrive at each value.
175
Math 1
Mathematics 1
M1.2. Core Content: Characteristics and behaviors of functions
(Algebra)
S
tudents formalize and deepen their understanding of functions, the defining characteristics and uses
of functions, and the mathematical language used to describe functions. They learn that functions
are often specified by an equation of the form y = f(x), where any allowable x-value yields a unique
y-value. Mathematics 1 has a particular focus on linear functions, equations, and systems of equations
and on functions that can be defined piecewise, particularly step functions and functions that contain the
absolute value of an expression. Students compare and contrast non-linear functions, such as quadratic
and exponential, with linear functions. They learn about the representations and basic transformations of
these functions and the practical and mathematical limitations that must be considered when working with
functions and when using functions to model situations.
Performance Expectations
Explanatory Comments and Examples
Students are expected to:
M1.2.A Determine whether a relationship is a function
and identify the domain, range, roots, and
independent and dependent variables.
Functions studied in Mathematics 1 include linear and
those defined piecewise (including step functions and
those that contain the absolute value of an expression).
They compare and contrast non-linear functions, such as
quadratic and exponential, to linear functions.
Given a problem situation, students should describe
further restrictions on the domain of a function that are
appropriate for the problem context.
Examples:
•
Which of the following are functions? Explain why
or why not.
— The age in years of each student in your
math class and each student’s shoe size.
— The number of degrees a person rotates a
spigot and the volume of water that comes
out of the spigot.
176
•
A function f(n) = 60n is used to model the distance
in miles traveled by a car traveling 60 miles per
hour in n hours. Identify the domain and range of
this function. What restrictions on the domain of
this function should be considered for the model to
correctly reflect the situation?
•
What is the domain of f(x) = 5 − x ?
July 2008
Washington State K–12 Mathematics Standards
Math 1
Performance Expectations
Explanatory Comments and Examples
Students are expected to:
M1.2.A cont.
•
Which of the following equations, inequalities, or
graphs determine y as a function of x?
— y = 2
— x = 3
— y = |x|
 x + 3, x ≤ 1
 x − 2, x > 1
— y = 
— x2 + y2 = 1
y
1
-1
1
x
-1
M1.2.B Represent a function with a symbolic
expression, as a graph, in a table, and using
words, and make connections among these
representations.
This expectation applies each time a new class
(family) of functions is encountered. In Mathematics 1,
students should be introduced to a variety of additional
functions that include expressions such as x3,
1
x, ,
x
and absolute values. They will study these functions in
depth in subsequent courses.
a
Students should know that f(x) = x represents an
inverse variation. Students begin to describe the graph
of a function from its symbolic expression, and use
key characteristics of the graph of a function to infer
properties of the related symbolic expression.
Translating among these various representations
of functions is an important way to demonstrate
conceptual understanding of functions.
Students learn that each representation has particular
advantages and limitations. For example, a graph
shows the shape of a function, but not exact values.
They also learn that a table of values may not uniquely
determine a single function without some specification
of the nature of that function (e.g., it is quadratic).
July 2008
Washington State K–12 Mathematics Standards
177
Math 1
Performance Expectations
Explanatory Comments and Examples
Students are expected to:
M1.2.C Evaluate f(x) at a (i.e., f(a)) and solve for x in
the equation f(x) = b.
Functions may be described and evaluated with symbolic
expressions, tables, graphs, or verbal descriptions.
Students should distinguish between solving for f(x)
and evaluating a function at x.
Example:
•
Roses-R-Red sells its roses for $0.75 per stem
and charges a $20 delivery fee per order.
— What is the cost of having 10 roses delivered?
— How many roses can you have delivered
for $65?
M1.2.D Plot points, sketch, and describe the graphs of
a
functions of the form f(x) = + b.
x
Mathematics 1 addresses only rational functions
a
+ b. Rational functions of the
x
a
form f(x) = 2 + b and f(x) = a are addressed in
x
(bx + c)
of the form f(x) =
Mathematics 3.
Example:
•
Sketch the graphs of the four functions f(x) =
when a = 4 and 8 and b = 0 and 1.
178
a
+b
x
July 2008
Washington State K–12 Mathematics Standards
Math 1
Mathematics 1
M1.3. Core Content: Linear functions, equations, and relationships
(Algebra, Geometry/Measurement,
Data/Statistics/Probability)
S
tudents understand that linear functions can be used to model situations involving a constant rate
of change. They build on the work done in middle school to solve systems of linear equations and
inequalities in two variables, learning to interpret the intersection of lines as the solution. While the focus is
on solving equations, students also learn graphical and numerical methods for approximating solutions to
equations. They use linear functions to analyze relationships, represent and model problems, and answer
questions. These algebraic skills are applied in other Core Content areas across high school courses.
Performance Expectations
Explanatory Comments and Examples
Students are expected to:
M1.3.A Write and solve linear equations and
inequalities in one variable.
This expectation includes the use of absolute values in
the equations and inequalities.
Examples:
M1.3.B Describe how changes in the parameters of
linear functions and functions containing an
absolute value of a linear expression affect their
graphs and the relationships they represent.
•
Write an absolute value equation or inequality for
all the numbers 2 units from 7, and all the numbers
that are more than b units from a.
•
Solve |x – 6| ≤ 4 and locate the solution on the
number line.
•
Write an equation or inequality that has no real
solutions; infinite numbers of real solutions; and
exactly one real solution.
•
Solve for x in 2(x – 3) + 4x = 15 + 2x.
•
Solve 8.5 < 3x + 2 ≤ 9.7 and locate the solution on
the number line.
In the case of a linear function y = f(x), expressed
in slope-intercept form (y = mx + b), m and b are
parameters. Students should know that f(x) = kx
represents a direct variation (proportional relationship).
Examples:
•
Graph a function of the form f(x) = kx, describe
the effect that changes on k have on the graph
and on f(x), and answer questions that arise in
proportional situations.
•
A gas station’s 10,000-gallon underground storage
tank contains 1,000 gallons of gasoline. Tanker
trucks pump gasoline into the tank at a rate of 400
gallons per minute. How long will it take to fill the
tank? Find a function that represents this situation
and then graph the function.
If the flow rate increases from 400 to 500 gallons
per minute, how will the graph of the function
change? If the initial amount of gasoline in the tank
changes from 1,000 to 2,000 gallons, how will the
graph of the function change?
•
Compare and contrast the functions y = 3|x| and
1
3
y = - |x|.
July 2008
Washington State K–12 Mathematics Standards
179
Math 1
Performance Expectations
Explanatory Comments and Examples
Students are expected to:
M1.3.C Identify and interpret the slope and intercepts
of a linear function, including equations for
parallel and perpendicular lines.
Examples:
•
The graph shows the relationship between time and
distance from a gas station for a motorcycle and a
scooter. What can be said about the relative speed
of the motorcycle and scooter that matches the
information in the graph? What can be said about
the intersection of the graphs of the scooter and
the motorcycle? Is it possible to tell which vehicle
is further from the gas station at the initial starting
point represented in the graph? At the end of the
time represented in the graph? Why or why not?
Distance
scooter
motorcycle
Time
•
A 1,500-gallon tank contains 200 gallons of water.
Water begins to run into the tank at the rate of
75 gallons per hour. When will the tank be full?
Find a linear function that models this situation,
draw a graph, and create a table of data points.
Once you have answered the question and
completed the tasks, explain your reasoning.
Interpret the slope and y-intercept of the function in
the context of the situation.
•
Given that the figure below is a square, find the
slope of the perpendicular sides AB and BC.
Describe the relationship between the two slopes.
A
B
D
q
p
q
180
C
p
July 2008
Washington State K–12 Mathematics Standards
Math 1
Performance Expectations
Explanatory Comments and Examples
Students are expected to:
M1.3.D Write and graph an equation for a line given
the slope and the y-intercept, the slope and a
point on the line, or two points on the line, and
translate between forms of linear equations.
Linear equations may be written in slope-intercept,
point-slope, and standard form.
Examples:
•
Find an equation for a line with y-intercept equal to
2 and slope equal to 3.
•
Find an equation for a line with a slope of 2 that
goes through the point (1, 1).
•
Find an equation for a line that goes through the
points (-3, 5) and (6, -2).
•
For each of the following, use only the equation
(without sketching the graph) to describe the graph.
— y = 2x + 3
— y – 7 = 2(x – 2)
M1.3.E Write and solve systems of two linear
equations and inequalities in two variables.
•
Write the equation 3x + 2y = 5 in slope intercept form.
•
Write the equation y – 1 = 2(x – 2) in standard form.
Students solve both symbolic and word problems,
understanding that the solution to a problem is given by
the coordinates of the intersection of the two lines when
the lines are graphed in the same coordinate plane.
Examples:
•
Solve the following simultaneous linear equations
algebraically:
-2x + y = 2
x + y = -1
•
Graph the above two linear equations on the same
coordinate plane and use the graph to verify the
algebraic solution.
•
An academic team is going to a state mathematics
competition. There are 30 people going on the trip.
There are 5 people who can drive and 2 types of
vehicles, vans and cars. A van seats 8 people, and
a car seats 4 people, including drivers. How many
vans and cars does the team need for the trip?
Explain your reasoning.
Let v = number of vans and c = number of cars.
v+c≤5
8v + 4c > 30
July 2008
Washington State K–12 Mathematics Standards
181
Math 1
Performance Expectations
Explanatory Comments and Examples
Students are expected to:
M1.3.F Find the equation of a linear function that best
fits bivariate data that are linearly related,
interpret the slope and y-intercept of the line,
and use the equation to make predictions.
A bivariate set of data presents data on two variables,
such as shoe size and height.
In high school, the emphasis is on using a line of best
fit to interpret data and on students making judgments
about whether a bivariate data set can be modeled with
a linear function. Students can use various methods,
including technology, to obtain a line of best fit.
Making predictions involves both interpolating and
extrapolating from the original data set.
Students need to be able to evaluate the quality of their
predictions, recognizing that extrapolation is based
on the assumption that the trend indicated continues
beyond the unknown data.
M1.3.G Describe the correlation of data in scatterplots in
terms of strong or weak and positive or negative.
Example:
•
Which words—strong or weak, positive or
negative—could be used to describe the
correlation shown in the sample scatterplot below?
Scatterplot
-1
-1.5
x
-2
Y
x
x x
x x
x x
x
xx
-2.5
x
-3
-3.5
-4
100
M1.3.H Determine the equation of a line in
the coordinate plane that is described
geometrically, including a line through two
given points, a line through a given point
parallel to a given line, and a line through a
given point perpendicular to a given line.
182
200
300
x
x xx
xx x
x
400
x
500
X
Examples:
•
Write an equation for the perpendicular bisector of
a given line segment.
•
Determine the equation of a line through the points
(5, 3) and (5, -2).
•
Prove that the slopes of perpendicular lines are
negative inverses of each other.
July 2008
Washington State K–12 Mathematics Standards
Math 1
Mathematics 1
M1.4. Core Content: Proportionality, similarity, and geometric reasoning
(Geometry/Measurement)
S
tudents extend and formalize their knowledge of two-dimensional geometric figures and their
properties, with a focus on properties of lines, angles, and triangles. They explain their reasoning
using precise mathematical language and symbols. Students study basic properties of parallel and
perpendicular lines, their respective slopes in the coordinate plane, and the properties of the angles
formed when parallel lines are intersected by a transversal. They prove related theorems and apply them
to solve problems that are purely mathematical and that arise in applied contexts. Students formalize their
prior work with similarity and proportionality by making and proving conjectures about triangle similarity.
Performance Expectations
Explanatory Comments and Examples
Students are expected to:
M1.4.A Distinguish between inductive and deductive
reasoning.
Students generate and test conjectures inductively and
then prove (or disprove) their conclusions deductively.
Example:
•
A student first hypothesizes that the sum of the
angles of a triangle is 180 degrees and then proves
this is true. When was the student using inductive
reasoning? When was s/he using deductive
reasoning? Justify your answers.
M1.4.B Use inductive reasoning to make conjectures,
to test the plausibility of a geometric statement,
and to help find a counterexample.
Example:
M1.4.C Use deductive reasoning to prove that a valid
geometric statement is true.
Valid proofs may be presented in paragraph, twocolumn, or flow-chart formats. Proof by contradiction is
a form of deductive reasoning.
•
Using dynamic geometry software, decide if the
following is a plausible conjecture: If two parallel
lines are cut by a transversal, then alternate
interior angles are equal.
Example:
•
M1.4.D Determine and prove triangle similarity.
Prove that if two parallel lines are cut by a
transversal, then alternate interior angles are equal.
Similarity in Mathematics 1 builds on proportionality
concepts from middle school mathematics. Determining
and proving triangle congruence and other properties of
triangles are included in Mathematics 2.
Students should identify necessary and sufficient
conditions for similarity in triangles, and use these
conditions in proofs.
Example:
•
July 2008
Washington State K–12 Mathematics Standards
For a given ∆RST, prove that ∆XYZ, formed by
joining the midpoints of the sides of ∆RST, is
similar to ∆RST.
183
Math 1
Performance Expectations
Explanatory Comments and Examples
Students are expected to:
M1.4.E Know, prove, and apply theorems about
parallel and perpendicular lines.
Students should be able to summarize and explain
basic theorems. They are not expected to recite lists
of theorems, but they should know the conclusion of a
theorem when given its hypothesis.
Examples:
M1.4.F Know, prove, and apply theorems about
angles, including angles that arise from
parallel lines intersected by a transversal.
•
Prove that a point on the perpendicular bisector of
a line segment is equidistant from the ends of the
line segment.
•
If each of two lines is perpendicular to a given line,
what is the relationship between the two lines?
How do you know?
Example:
•
Take two parallel lines l and m, with (distinct)
points A and B on l and C and D on m.
If AC intersects BD at point E, prove that
∆ABE ≅ ∆CDE.
M1.4.G Explain and perform basic compass and
straightedge constructions related to parallel
and perpendicular lines.
Constructions using circles and lines with dynamic
geometry software (i.e., virtual compass and straightedge) are equivalent to paper and pencil constructions.
Example:
•
Construct and mathematically justify the steps to:
— Bisect a line segment.
— Drop a perpendicular from a point to a line.
— Construct a line through a point that is
parallel to another line.
184
July 2008
Washington State K–12 Mathematics Standards
Math 1
Mathematics 1
M1.5. Core Content: Data and distributions
(Data/Statistics/Probability)
S
tudents select mathematical models for data sets and use those models to represent, describe, and
compare data sets. They analyze the linear relationship between two statistical variables and make
and defend appropriate predictions, conjectures, and generalizations based on data. Students understand
limitations of conclusions drawn from the results of a study or an experiment and recognize common
misconceptions and misrepresentations.
Performance Expectations
Explanatory Comments and Examples
Students are expected to:
M1.5.A Use and evaluate the accuracy of summary
statistics to describe and compare data sets.
A univariate set of data identifies data on a single
variable, such as shoe size.
This expectation extends what students have learned
in earlier grades to include evaluation and justification.
They both compute and evaluate the appropriateness of
measure of center and spread (range and interquartile
range) and use these measures to accurately compare
data sets. Students will draw appropriate conclusions
through the use of statistical measures of center,
frequency, and spread, combined with graphical displays.
Examples:
•
The local minor league baseball team has a salary
dispute. Players claim they are being underpaid,
but managers disagree.
— Bearing in mind that a few top players
earn salaries that are quite high, would it
be in the managers’ best interest to use
the mean or median when quoting the
“average” salary of the team? Why?
— What would be in the players’ best interest?
•
Each box-and-whisker plot shows the prices of
used cars (in thousands of dollars) advertised for
sale at three different car dealers. If you want to go
to the dealer whose prices seem least expensive,
which dealer would you go to? Use statistics from
the displays to justify your answer.
Cars are US
Better-than-New
Yours Now
0
July 2008
Washington State K–12 Mathematics Standards
5
10
185
Math 1
Performance Expectations
Explanatory Comments and Examples
Students are expected to:
M1.5.B Describe how linear transformations affect the
center and spread of univariate data.
M1.5.C Make valid inferences and draw conclusions
based on data.
Examples:
•
A company decides to give every one of its
employees a $5,000 raise. What happens to
the mean and standard deviation of the salaries
as a result?
•
A company decides to double each of its
employee’s salaries. What happens to the mean
and standard deviation of the salaries as a result?
Determine whether arguments based on data
confuse association with causation. Evaluate the
reasonableness of and make judgments about
statistical claims, reports, studies, and conclusions.
Example:
•
186
Mr. Shapiro found that the amount of time his
students spent doing mathematics homework is
positively correlated with test grades in his class. He
concluded that doing homework makes students’
test scores higher. Is this conclusion justified?
Explain any flaws in Mr. Shapiro’s reasoning.
July 2008
Washington State K–12 Mathematics Standards
Math 1
Mathematics 1
M1.6. Core Content: Numbers, expressions, and operations
(Numbers, Operations, Algebra)
S
tudents see the number system extended to the real numbers represented by the number line.
They use variables and expressions to solve problems from purely mathematical as well as applied
contexts. They build on their understanding of and ability to compute with arithmetic operations and
properties and expand this understanding to include the symbolic language of algebra. Students
demonstrate this ability to write and manipulate a wide variety of algebraic expressions throughout high
school mathematics as they apply algebraic procedures to solve problems.
Performance Expectations
Explanatory Comments and Examples
Students are expected to:
M1.6.A Know the relationship between real numbers
and the number line, and compare and order
real numbers with and without the number line.
Although a formal definition of real numbers is beyond
the scope of Mathematics 1, students learn that every
point on the number line represents a real number,
either rational or irrational, and that every real number
has its unique point on the number line. They locate,
compare, and order real numbers on the number line.
Real numbers include those written in scientific
notation or expressed as fractions, decimals,
exponentials, or roots.
Examples:
•
Without using a calculator, order the following on
the number line:
82 , 3π, 8.9, 9,
•
37
, 9.3 × 100
4
A star’s color gives an indication of its temperature
and age. The chart shows four types of stars and
the lowest temperature of each type.
Type
Lowest Temperature
(in ºF)
A
1.35 x 104
Blue-White
B
2.08 x 104
Blue
G
9.0 x 103
Yellow
P
4.5 x 104
Blue
Color
List the temperatures in order from lowest to highest.
July 2008
Washington State K–12 Mathematics Standards
187
Math 1
Performance Expectations
Explanatory Comments and Examples
Students are expected to:
M1.6.B Determine whether approximations or exact
values of real numbers are appropriate,
depending on the context, and justify
the selection.
Decimal approximations of numbers are sometimes
used in applications, such as carpentry or engineering;
while at other times, these applications may require
exact values. Students should understand the
difference and know that the appropriate approximation
depends upon the necessary degree of precision
needed in given situations.
For example, 1.414 is an approximation and not an
exact solution to the equation x2 – 2 = 0, but 2 is an
exact solution to this equation.
Example:
•
Using a common engineering formula, an
engineering student represented the maximum
safe load of a bridge to be 1000(99 – 70 2) tons.
He used 1.41 as the approximation for 2 in his
calculations. When the bridge was built and tested
in a computer simulation to verify its maximum
weight-bearing load, it collapsed! The student had
estimated the bridge would hold ten times the
weight that was applied to it when it collapsed.
— Calculate the weight that the student
thought the bridge could bear using 1.41
as the estimate for 2.
— Calculate other weight values using
estimates of 2 that have more decimal
places. What might be a reasonable
degree of precision required to know
how much weight the bridge can handle
safely? Justify your answer.
M1.6.C Recognize the multiple uses of variables,
determine all possible values of variables that
satisfy prescribed conditions, and evaluate
algebraic expressions that involve variables.
188
Students learn to use letters as variables and in other
ways that increase in sophistication throughout high
school. For example, students learn that letters can
be used:
•
To represent fixed and temporarily unknown values
in equations, such as 3x + 2 = 5;
•
To express identities, such as x + x = 2x for all x;
•
As attributes in formulas, such as A = lw;
•
As constants such as a, b, and c in the equation
y = ax2 + bx + c;
•
As parameters in equations, such as the m and b
for the family of functions defined by y = mx + b;
•
To represent varying quantities, such as x in
f(x) = 5x;
•
To represent functions, such as f in f(x) = 5x; and
•
To represent specific numbers, such as π.
July 2008
Washington State K–12 Mathematics Standards
Math 1
Performance Expectations
Explanatory Comments and Examples
Students are expected to:
M1.6.C cont.
Expressions include those involving polynomials,
radicals, absolute values, and integer exponents.
Examples:
M1.6.D Solve an equation involving several variables by
expressing one variable in terms of the others.
July 2008
Washington State K–12 Mathematics Standards
•
For what values of a and n, where n is an integer
greater than 0, is an always negative?
•
For what values of a is
•
For what values of a is 5 − a defined?
•
For what values of a is -a always positive?
1
an integer?
a
Examples:
•
Solve A = p + prt for p.
•
Solve V = πr 2h for h or for r.
189
Math 1
Mathematics 1
M1.7. Additional Key Content
(Numbers, Algebra)
S
tudents develop a basic understanding of arithmetic and geometric sequences and of exponential
functions, including their graphs and other representations. They use exponential functions to analyze
relationships, represent and model problems, and answer questions in situations that are modeled by
these nonlinear functions. Students learn graphical and numerical methods for approximating solutions to
exponential equations. Students interpret the meaning of problem solutions and explain limitations related
to solutions.
Performance Expectations
Explanatory Comments and Examples
Students are expected to:
M1.7.A Sketch the graph for an exponential function
of the form y = abn where n is an integer,
describe the effects that changes in the
parameters a and b have on the graph, and
answer questions that arise in situations
modeled by exponential functions.
Examples:
•
Sketch the graph of y = 2n by hand.
•
You have won a door prize and are given a choice
between two options:
$150 invested for 10 years at 4% compounded
annually.
$200 invested for 10 years at 3% compounded
annually.
How much is each worth at the end of each year of
the investment periods?
Are the two investments ever equal in value?
Which will you choose?
M1.7.B Find and approximate solutions to exponential
equations.
Students can approximate solutions using graphs or
tables with and without technology.
M1.7.C Interpret and use integer exponents and
square and cube roots, and apply the laws
and properties of exponents to simplify and
evaluate exponential expressions.
Examples:
1
•
2-3 =
•
-2
3
2 3 5
= 4
2
-3 2
2 5
2 3 5
•
a-2 b2 c
b5
= 4
2 -3 2
ab c
ac
3
2
2
•
•
190
5
8 = 2• 2• 2 = 2 2
3
a•b = 3 a • 3 b
July 2008
Washington State K–12 Mathematics Standards
Math 1
Performance Expectations
Explanatory Comments and Examples
Students are expected to:
M1.7.D Express arithmetic and geometric sequences
in both explicit and recursive forms, translate
between the two forms, explain how rate of
change is represented in each form, and use
the forms to find specific terms in the sequence.
Examples:
•
Write a recursive formula for the arithmetic
sequence 5, 9, 13, 17, . . . What is the slope of the
line that contains the points associated with these
values and their position in the sequence? How is
the slope of the line related to the sequence?
•
Given that u(0) = 3 and u(n + 1) = u(n) + 7 when n
is a positive integer,
a. find u(5);
b. find n so that u(n) = 361; and
c. find a formula for u(n).
•
Write a recursive formula for the geometric
sequence 5, 10, 20, 40, . . . and determine the
100th term.
•
Given that u(0) = 2 and u(n + 1) = 3u(n),
a. find u(4), and
b. find a formula for u(n).
July 2008
Washington State K–12 Mathematics Standards
191
Math 1
Mathematics 1
M1.8. Core Processes: Reasoning, problem solving, and communication
S
tudents formalize the development of reasoning in Mathematics 1 as they use algebra, geometry, and
statistics to make and defend generalizations. They justify their reasoning with accepted standards
of mathematical evidence and proof, using correct mathematical language, terms, and symbols in all
situations. They extend the problem-solving practices developed in earlier grades and apply them to
more challenging problems, including problems related to mathematical and applied situations. Students
formalize a coherent problem-solving process in which they analyze the situation to determine the
question(s) to be answered, synthesize given information, and identify implicit and explicit assumptions
that have been made. They examine their solution(s) to determine reasonableness, accuracy, and
meaning in the context of the original problem. The mathematical thinking, reasoning, and problemsolving processes students learn in high school mathematics can be used throughout their lives as they
deal with a world in which an increasing amount of information is presented in quantitative ways and more
and more occupations and fields of study rely on mathematics.
Performance Expectations
Explanatory Comments and Examples
Students are expected to:
M1.8.A Analyze a problem situation and represent it
mathematically.
Examples:
•
M1.8.B Select and apply strategies to solve problems.
M1.8.C Evaluate a solution for reasonableness, verify
its accuracy, and interpret the solution in the
context of the original problem.
Three teams of students independently conducted
experiments to relate the rebound height of a
ball to the rebound number. The table gives the
average of the teams’ results.
M1.8.D Generalize a solution strategy for a single
problem to a class of related problems, and
apply a strategy for a class of related problems
to solve specific problems.
M1.8.E Read and interpret diagrams, graphs, and
text containing the symbols, language, and
conventions of mathematics.
M1.8.F Summarize mathematical ideas with precision
and efficiency for a given audience and
purpose.
M1.8.G Synthesize information to draw conclusions,
and evaluate the arguments and conclusions
of others.
M1.8.H Use inductive reasoning to make conjectures,
and use deductive reasoning to prove or
disprove conjectures.
192
Rebound
Number
Rebound
Height (cm)
0
200
1
155
2
116
3
88
4
66
5
50
6
44
Construct a scatterplot of the data, and describe
the function that relates the height of the ball to the
rebound number. Predict the rebound height of the
ball on the tenth rebound. Justify your answer.
•
Prove formally that the sum of two odd numbers is
always even.
July 2008
Washington State K–12 Mathematics Standards
Mathematics 2
July 2008
Washington State K–12 Mathematics Standards
193
Mathematics 2
Mathematics 2 extends the study of functions to include quadratic functions, providing tools for modeling
a greater variety of real-world situations. Students develop computational and algebraic skills that support
analysis of these functions and their multiple representations. Students extend their ability to reason
mathematically. They distinguish between inductive and deductive thinking, make conjectures, and
prove theorems. Students become skilled in writing more involved proofs through their study of triangles,
lines, and quadrilaterals. Finally, the study of probability extends students’ understanding of proportional
reasoning and relationships with the inclusion of counting methods and lays the groundwork for the study
of data and variability in the next course.
194
July 2008
Washington State K–12 Mathematics Standards
Math 2
Mathematics 2
M2.1. Core Content: Modeling situations and solving problems
(Algebra)
T
his first core content area highlights the types of problems students will be able to solve by the end of
Mathematics 2. Students extend their ability to model situations and solve problems with additional
functions and equations in this course. Additionally, they deepen their understanding and proficiency with
functions they encountered in Mathematics 1 and use these functions to solve more complex problems.
When presented with a word problem, students determine which function or equation models the problem
and then use that information to write an equation to solve the problem. Turning word problems into
equations that can be solved is a skill students hone throughout the course.
Performance Expectations
Explanatory Comments and Examples
Students are expected to:
M2.1.A Select and justify functions and equations to
model and solve problems.
Example:
•
Dawson wants to make a horse corral by creating
a rectangle that is divided into 2 parts, similar to
the following diagram. He has a 1200-foot roll of
fencing to do the job.
What are the dimensions of the enclosure with the
largest total area?
What function or equation best models this situation?
l
w
M2.1.B Solve problems that can be represented by
systems of equations and inequalities.
Example:
M2.1.C Solve problems that can be represented by
quadratic functions, equations, and inequalities.
Students solve problems by factoring and applying the
quadratic formula to the quadratic equation, and use
the vertex form of the equation to solve problems about
maximums, minimums, and symmetry.
•
Data derived from an experiment seems to be
parabolic when plotted on a coordinate grid. Three
observed data points are (2, 10), (3, 8), and (4, 4).
Write a quadratic equation that passes through
the points.
Examples:
•
July 2008
Washington State K–12 Mathematics Standards
Find the solutions to the simultaneous equations
y = x + 2 and y = x2.
195
Math 2
Performance Expectations
Explanatory Comments and Examples
Students are expected to:
M2.1.C cont.
M2.1.D Solve problems that can be represented by
exponential functions and equations.
•
If you throw a ball straight up (with initial height of
4 feet) at 10 feet per second, how long will it take
to fall back to the starting point? The function
h(t) = -16t2 + v0t + h0 describes the height, h in feet,
of an object after t seconds, with initial velocity v0
and initial height h0.
•
Joe owns a small plot of land 20 feet by 30 feet.
He wants to double the area by increasing both
the length and the width, keeping the dimensions
in the same proportion as the original. What will be
the new length and width?
•
What two consecutive numbers, when multiplied
together, give the first number plus 16? Write the
equation that represents the situation.
•
The Gateway Arch in St. Louis has a special
shape called a catenary, which looks a lot like a
parabola. It has a base width of 600 feet and is
630 feet high. Which is taller, this catenary arch
or a parabolic arch that has the same base width
but has a height of 450 feet at a point 150 feet
from one of the pillars? What is the height of the
parabolic arch?
Students extend their use of exponential functions
and equations to solve more complex problems.
They approximate solutions with graphs or tables,
check solutions numerically, and when possible, solve
problems exactly.
Examples:
•
E. coli bacteria reproduce by a simple process
called binary fission—each cell increases in size
and divides into two cells. In the laboratory, E. coli
bacteria divide approximately every 15 minutes. A
new E. coli culture is started with 1 cell.
a. Find a function that models the E. coli
population size at the end of each 15-minute
interval. Justify the function you found.
b. State an appropriate domain for the model
based on the context.
c. After what 15-minute interval will you have at
least 500 bacteria?
•
196
Estimate the solution to 2x = 16,384
July 2008
Washington State K–12 Mathematics Standards
Math 2
Performance Expectations
Explanatory Comments and Examples
Students are expected to:
M2.1.E Solve problems involving combinations
and permutations.
July 2008
Washington State K–12 Mathematics Standards
Examples:
•
The company Ali works for allows her to invest
in her choice of 10 different mutual funds, 6 of
which grew by at least 5% over the last year. Ali
randomly selected 4 of the 10 funds in which to
invest. What is the probability that 3 of Ali’s funds
grew by 5%?
•
Four points (A, B, C, and D) lie on one straight
line, n, and five points (E, F, G, H, and J) lie on
another straight line, m, that is parallel to n. What
is the probability that three points, selected at
random, will form a triangle?
197
Math 2
Mathematics 2
M2.2. Core Content: Quadratic functions, equations, and relationships
(Algebra)
S
tudents learn that exponential and quadratic functions can be used to model some situations
where linear functions may not be the best model. They use graphical and numerical methods with
exponential functions of the form y = abx and quadratic functions to analyze relationships, represent and
model problems, and answer questions. Students extend their algebraic skills and learn various methods
of solving quadratic equations over real or complex numbers, including using the quadratic formula,
factoring, graphing, and completing the square. They learn to translate between forms of quadratic
equations, applying the vertex form to evaluate maximum and minimum values and find symmetry of
the graph, and they learn to identify which form should be used in a particular situation. They interpret
the meaning of problem solutions and explain their limitations. Students recognize common examples
of situations that can be modeled by quadratic functions, such as maximizing area or the height of an
object moving under the force of gravity. They compare the characteristics of quadratic functions to those
of linear and exponential functions. The understanding of these particular types of functions, together
with students’ understanding of linear functions, provides students with a powerful set of tools to use
mathematical models to deal with problems and situations in advanced mathematics courses, in the
workplace, and in everyday life.
Performance Expectations
Explanatory Comments and Examples
Students are expected to:
M2.2.A Represent a quadratic function with a symbolic
expression, as a graph, in a table, and with a
description, and make connections among
the representations.
Example:
•
Kendre and Tyra built a tennis ball cannon that
launches tennis balls straight up in the air at an
initial velocity of 50 feet per second. The mouth of
the cannon is 2 feet off the ground. The function
h(t) = -16t2 + 50t + 2 describes the height, h, in
feet, of the ball t seconds after the launch.
Make a table from the function. Then use the table
to sketch a graph of the height of the tennis ball
as a function of time into the launch. Give a verbal
description of the graph. How high was the ball after
1 second? When does it reach this height again?
M2.2.B Sketch the graph of a quadratic function,
describe the effects that changes in the
parameters have on the graph, and interpret the
x-intercepts as solutions to a quadratic equation.
Note that in Mathematics 2, the parameter b in the
term bx in the quadratic form ax2 + bx + c is not
often used to provide useful information about the
characteristics of the graph.
Parameters considered most useful are:
198
•
a and c in f(x) = ax2 + c
•
a, h, and k in f(x) = a(x – h)2 + k, and
•
a, r, and s in f(x) = a(x – r)(x – s)
July 2008
Washington State K–12 Mathematics Standards
Math 2
Performance Expectations
Explanatory Comments and Examples
Students are expected to:
M2.2.B cont.
Example:
•
A particular quadratic function can be expressed in
the following two ways:
f(x) = -(x – 3)2 + 1
f(x) = -(x – 2)(x – 4)
What information about the graph can be directly
inferred from each of these forms? Explain your
reasoning.
Sketch the graph of this function, showing the roots.
M2.2.C Translate between the standard form of a
quadratic function, the vertex form, and the
factored form; graph and interpret the meaning
of each form.
Students translate among forms of a quadratic function
to convert to one that is appropriate—e.g., vertex
form—to solve specific problems.
Students learn about the advantages of the standard
form (f(x) = ax2 + bx + c), the vertex form
(f(x) = a(x – h)2 + d), and the factored form
(f(x) = a(x – r)(x – s)). They produce the vertex form
by completing the square on the function in standard
form, which allows them to see the symmetry of the
graph of a quadratic function as well as the maximum
or minimum. This opens up a whole range of new
problems students can solve using quadratics.
Students continue to find the solutions of the equation,
which can be either real or complex.
Example:
•
Find the minimum, the line of symmetry, and
the roots for the graphs of each of the following
functions:
f(x) = x2 – 4x + 3
f(x) = x2 – 4x + 4
f(x) = x2 – 4x + 5
M2.2.D Solve quadratic equations that can be factored
as (ax + b)(cx + d) where a, b, c, and d are
integers.
Students learn to efficiently solve quadratic equations
by recognizing and using the simplest factoring
methods, including recognizing special quadratics as
squares and differences of squares.
Examples:
July 2008
Washington State K–12 Mathematics Standards
•
2x2 + x – 3 = 0; (x – 1)(2x + 3) = 0; x = 1, −-
•
4x2 + 6x = 0; 2x(2x + 3) = 0; x = 0, −-
•
36x2 – 25 = 0; (6x + 5)(6x – 5) = 0; x = ±
•
x2 + 6x + 9 = 0; (x + 3)2 = 0; x = -3
3
2
3
2
5
6
199
Math 2
Performance Expectations
Explanatory Comments and Examples
Students are expected to:
M2.2.E Determine the number and nature of the roots
of a quadratic function.
Students should be able to recognize and interpret the
discriminant.
Students should also be familiar with the Fundamental
Theorem of Algebra, i.e., that all polynomials, not just
quadratics, have roots over the complex numbers. This
concept becomes increasingly important as students
progress through mathematics.
Example:
•
M2.2.F Solve quadratic equations that have real roots
by completing the square and by using the
quadratic formula.
For what values of a does f(x) = x2 – 6x + a have 2
real roots, 1 real root, and no real roots?
Students solve those equations that are not easily
factored by completing the square and by using the
quadratic formula. Completing the square should also
be used to derive the quadratic formula.
Students learn how to determine if there are two, one,
or no real solutions.
Examples:
•
Complete the square to solve x2 + 4x = 13.
x2 + 4x – 13 = 0
x2 + 4x + 4 = 17
(x + 2)2 = 17
x + 2 = ± 17
x = -2 ± 17
x ≈ 2.12, -6.12
•
Use the quadratic formula to solve 4x2 – 2x = 5.
x=
x=
- b ± b 2 – 4 ac
2a
- (- 2 )
x=
2 ± 84
8
x=
2 ± 2 21
8
( - 2)2 – 4(4 -5)
2(4)
1 ± 21
4
x ≈ 1.40, -0.90
x=
200
July 2008
Washington State K–12 Mathematics Standards
Math 2
Performance Expectations
Explanatory Comments and Examples
Students are expected to:
M2.2.G Solve quadratic equations and inequalities,
including equations with complex roots.
Students solve equations that are not easily factored by
completing the square and by using the quadratic formula.
Examples:
•
x2 – 10x + 34 = 0
•
3x2 + 10 = 4x
•
Wile E. Coyote launches an anvil from 180 feet
above the ground at time t = 0. The equation that
models this situation is given by
h = -16t2 + 96t + 180, where t is time measured
in seconds and h is height above the ground
measured in feet.
a. What is a reasonable domain restriction for t in
this context?
b. Determine the height of the anvil two seconds
after it was launched.
c. Determine the maximum height obtained by
the anvil.
d. Determine the time when the anvil is more
than 100 feet above ground.
•
M2.2.H Determine if a bivariate data set can be better
modeled with an exponential or a quadratic
function and use the model to make predictions.
July 2008
Washington State K–12 Mathematics Standards
Farmer Helen wants to build a pigpen. With 100
feet of fence, she wants a rectangular pen with
one side being a side of her existing barn. What
dimensions should she use for her pigpen in order
to have the maximum number of square feet?
In high school, determining a formula for a curve
of best fit requires a graphing calculator or similar
technological tool.
201
Math 2
Mathematics 2
M2.3. Core Content: Conjectures and proofs (Algebra, Geometry/Measurement)
S
tudents extend their knowledge of two-dimensional geometric figures and their properties to include
quadrilaterals and other polygons, with special emphasis on necessary and sufficient conditions
for triangle congruence. They work with geometric constructions, using dynamic software as a tool for
exploring geometric relationships and formulating conjectures and using compass-and-straightedge and
paper-folding constructions as contexts in which students demonstrate their knowledge of geometric
relationships. Students define the basic trigonometric ratios and use them to solve problems in a variety of
applied situations. They formalize the reasoning skills they have developed in previous grades and solidify
their understanding of what it means to mathematically prove a geometric statement. Students encounter
the concept of formal proof built on definitions, axioms, and theorems. They use inductive reasoning to
test conjectures about geometric relationships and use deductive reasoning to prove or disprove their
conclusions. Students defend their reasoning using precise mathematical language and symbols. Finally,
they apply their knowledge of linear functions to make and prove conjectures about geometric figures on
the coordinate plane.
Performance Expectations
Explanatory Comments and Examples
Students are expected to:
M2.3.A Use deductive reasoning to prove that a valid
geometric statement is true.
Valid proofs may be presented in paragraph, twocolumn, or flow-chart formats. Proof by contradiction is
a form of deductive reasoning.
Example:
•
Prove that the diagonals of a rhombus are
perpendicular bisectors of each other.
M2.3.B Identify errors or gaps in a mathematical
argument and develop counterexamples to
refute invalid statements about geometric
relationships.
Example:
M2.3.C Write the converse, inverse, and contrapositive
of a valid proposition and determine their validity.
Examples:
M2.3.D Distinguish between definitions and undefined
geometric terms and explain the role of
definitions, undefined terms, postulates
(axioms), and theorems.
•
Identify errors in reasoning in the following proof:
Given ∠ABC ≅ ∠PRQ, AB ≅ PQ, and BC ≅ QR,
then ∆ABC ≅ ∆PQR by SAS.
•
If m and n are odd integers, then the sum of m
and n is an even integer. State the converse and
determine whether it is valid.
•
If a quadrilateral is a rectangle, the diagonals have
the same length. State the contrapositive and
determine whether it is valid.
Students sketch points and lines (undefined terms) and
define and sketch representations of other common
terms. They use definitions and postulates as they
prove theorems throughout geometry. In their work
with theorems, they identify the hypothesis and the
conclusion and explain the role of each.
Students describe the consequences of changing
assumptions or using different definitions for
subsequent theorems and logical arguments.
202
July 2008
Washington State K–12 Mathematics Standards
Math 2
Performance Expectations
Explanatory Comments and Examples
Students are expected to:
M2.3.D cont.
Example:
•
There are two definitions of trapezoid that can be
found in books or on the web. A trapezoid is either
— a quadrilateral with exactly one pair of
parallel sides or
— a quadrilateral with at least one pair of
parallel sides.
Write some theorems that are true when applied
to one definition but not the other, and explain
your answer.
M2.3.E Know, explain, and apply basic postulates and
theorems about triangles and the special lines,
line segments, and rays associated with a
triangle.
M2.3.F Determine and prove triangle congruence and
other properties of triangles.
Examples:
•
Prove that the sum of the angles of a triangle is 180°.
•
Prove and explain theorems about the incenter,
circumcenter, orthocenter, and centroid.
•
The rural towns of Atwood, Bridgeville, and
Carnegie are building a communications tower to
serve the needs of all three towns. They want to
position the tower so that the distance from each
town to the tower is equal. Where should they
locate the tower? How far will it be from each town?
Students extend their work with similarity in
Mathematics 1 to proving theorems about congruence
and other properties of triangles.
Students should identify necessary and sufficient
conditions for congruence in triangles, and use these
conditions in proofs.
Examples:
M2.3.G Know, prove, and apply the Pythagorean
Theorem and its converse.
•
Prove that congruent triangles are similar.
•
Show that not all SSA triangles are congruent.
Students extend their work with the Pythagorean
Theorem from previous grades to include formal proof.
Examples:
•
Consider any right triangle with legs a and b
and hypotenuse c. The right triangle is used to
create Figures 1 and 2. Explain how these figures
constitute a visual representation of a proof of the
Pythagorean Theorem.
P
c
b
Q
a
Figure 1
July 2008
Washington State K–12 Mathematics Standards
Figure 2
203
Math 2
Performance Expectations
Explanatory Comments and Examples
Students are expected to:
M2.3.G cont.
•
A juice box is 6 cm by 8 cm by 12 cm. A straw is
inserted into a hole in the center of the top of the
box. The straw must stick out 2 cm so you can drink
from it. If the straw must be long enough to touch
each bottom corner of the box, what is the minimum
length the straw must be? (Assume the diameter of
the straw is 0 for the mathematical model.)
12 cm
M2.3.H Solve problems involving the basic trigonometric ratios of sine, cosine, and tangent.
8 cm
6 cm
•
In ∆ABC, with right angle at C, draw the altitude
CD from C to AB . Name all similar triangles in the
diagram. Use these similar triangles to prove the
Pythagorean Theorem.
•
Apply the Pythagorean Theorem to derive the
distance formula in the (x, y) plane.
•
Determine the length of the altitude of an
equilateral triangle whose side lengths measure
5 units.
Students apply their knowledge of the Pythagorean
Theorem from Grade 8 to define the basic
trigonometric ratios. They formally prove the
Pythagorean Theorem in Mathematics 2.
Examples:
204
•
A 12-foot ladder leans against a wall to form a 63°
angle with the ground. How many feet above the
ground is the point on the wall at which the ladder
is resting?
•
Use the Pythagorean Theorem to establish that
sin2ø + cos2 ø = 1 for ø between 0° and 90°.
July 2008
Washington State K–12 Mathematics Standards
Math 2
Performance Expectations
Explanatory Comments and Examples
Students are expected to:
M2.3.I Use the properties of special right triangles
(30°–60°–90° and 45°–45°–90°) to solve
problems.
Examples:
•
If one leg of a right triangle has length 5 and the
adjacent angle is 30°, what is the length of the
other leg and the hypotenuse?
•
If one leg of a 45°–45°–90° triangle has length 5,
what is the length of the hypotenuse?
•
The pitch of a symmetrical roof on a house 40 feet
wide is 30º. What is the length of the rafter, r, exactly
and approximately?
r
M2.3.J Know, prove, and apply basic theorems about
parallelograms.
40
‘
30º
Properties may include those that address symmetry
and properties of angles, diagonals, and angle sums.
Students may use inductive and deductive reasoning
and counterexamples.
Examples:
July 2008
Washington State K–12 Mathematics Standards
•
Are opposite sides of a parallelogram always
congruent? Why or why not?
•
Are opposite angles of a parallelogram always
congruent? Why or why not?
•
Prove that the diagonals of a parallelogram bisect
each other.
•
Explain why if the diagonals of a quadrilateral
bisect each other, then the quadrilateral is a
parallelogram.
•
Prove that the diagonals of a rectangle are
congruent. Is this true for any parallelogram?
Justify your reasoning.
205
Math 2
Performance Expectations
Explanatory Comments and Examples
Students are expected to:
M2.3.K Know, prove, and apply theorems about
properties of quadrilaterals and other polygons.
Examples:
•
What is the length of the apothem of a regular
hexagon with side length 8? What is the area of
the hexagon?
•
If the shaded pentagon were removed, it could be
replaced by a regular n-sided polygon that would
exactly fill the remaining space. Find the number of
sides, n, of a replacement polygon that makes the
three polygons fit perfectly.
P
M2.3.L Determine the coordinates of a point that is
described geometrically.
206
Examples:
•
Determine the coordinates for the midpoint of a
given line segment.
•
Given the coordinates of three vertices of a
parallelogram, determine all possible coordinates
for the fourth vertex.
•
Given the coordinates for the vertices of a
triangle, find the coordinates for the center of the
circumscribed circle and the length of its radius.
July 2008
Washington State K–12 Mathematics Standards
Math 2
Performance Expectations
Explanatory Comments and Examples
Students are expected to:
M2.3.M Verify and apply properties of triangles and
quadrilaterals in the coordinate plane.
Examples:
•
Given four points in a coordinate plane that are the
vertices of a quadrilateral, determine whether the
quadrilateral is a rhombus, a square, a rectangle,
a parallelogram, or none of these. Name all the
classifications that apply.
•
Given a parallelogram on a coordinate plane,
verify that the diagonals bisect each other.
•
Given the line with y-intercept 4 and x-intercept 3,
find the area of a square that has one corner on the
origin and the opposite corner on the line described.
•
Below is a diagram of a miniature golf hole as
drawn on a coordinate grid. The dimensions of
the golf hole are 4 feet by 12 feet. Players must
start their ball from one of the three tee positions,
located at (1, 1), (1, 2), or (1, 3). The hole is
located at (10, 3). A wall separates the tees from
the hole. At which tee should the ball be placed to
create the shortest “hole-in-one” path? Sketch the
intended path of the ball, find the distance the ball
will travel, and describe your reasoning. (Assume
the diameters of the golf ball and the hole are 0 for
the mathematical model.)
..
.
July 2008
Washington State K–12 Mathematics Standards
Wall
Tees
.
Hole
207
Math 2
Mathematics 2
M2.4. Core Content: Probability (Data/Statistics/Probability)
S
tudents formalize their study of probability, computing both combinations and permutations to
calculate the likelihood of an outcome in uncertain circumstances and applying the binominal theorem
to solve problems. They apply their understanding of probability to a wide range of practical situations,
including those involving permutations and combinations. Understanding probability helps students
become knowledgeable consumers who make sound decisions about high-risk games, financial
issues, etc.
Performance Expectations
Explanatory Comments and Examples
Students are expected to:
M2.4.A Apply the fundamental counting principle
and the ideas of order and replacement to
calculate probabilities in situations arising from
two-stage experiments (compound events).
M2.4.B Given a finite sample space consisting of
equally likely outcomes and containing
events A and B, determine whether A and B
are independent or dependent, and find the
conditional probability of A given B.
Example:
M2.4.C Compute permutations and combinations, and
use the results to calculate probabilities.
Example:
M2.4.D Apply the binomial theorem to solve problems
involving probability.
The binominal theorem is also applied when computing
with polynomials.
•
•
What is the probability of drawing a heart from
a standard deck of cards on a second draw,
given that a heart was drawn on the first draw
and not replaced?
Two friends, Abby and Ben, are among five
students being considered for three student council
positions. If each of the five students has an equal
likelihood of being selected, what is the probability
that Abby and Ben will both be selected?
Examples:
•
Use Pascal’s triangle and the binomial theorem
to find the number of ways six objects can be
selected four at a time.
•
In a survey, 33% of adults reported that they
preferred to get the news from newspapers rather
than television. If you survey 5 people, what is the
probability of getting exactly 2 people who say they
prefer news from the newspaper?
Write an equation that can be used to solve
the problem.
Create a histogram of the binomial distribution of
the probability of getting 0 through 5 responders
saying they prefer the newspaper.
208
July 2008
Washington State K–12 Mathematics Standards
Math 2
Mathematics 2
M2.5. Additional Key Content
(Algebra, Measurement)
S
tudents grow more proficient in their use of algebraic techniques as they use these techniques to
write equivalent expressions in various forms. They build on their understanding of computation using
arithmetic operations and properties and expand this understanding to include the symbolic language of
algebra. Students understand the role of units in measurement, convert among units within and between
different measurement systems as needed, and apply what they know to solve problems. They use
derived measures such as those used for speed (e.g., feet per second) or determining automobile gas
consumption (e.g., miles per gallon).
Performance Expectations
Explanatory Comments and Examples
Students are expected to:
M2.5.A Use algebraic properties to factor and combine
like terms in polynomials.
Algebraic properties include the commutative,
associative, and distributive properties.
Factoring includes:
•
Factoring a monomial from a polynomial, such as
4x2 + 6x = 2x(2x + 3);
•
Factoring the difference of two squares, such as
36x2 – 25y2 = (6x + 5y)(6x – 5y) and
x4 – y4 = (x + y)(x – y)(x2 + y2);
•
Factoring perfect square trinomials, such as
x2 + 6xy + 9y2 = (x + 3y)2;
•
Factoring quadratic trinomials, such as
x2 + 5x + 4 = (x + 4)(x + 1); and
•
Factoring trinomials that can be expressed as the
product of a constant and a trinomial, such as
0.5x2 – 2.5x – 7 = 0.5(x + 2)(x – 7).
M2.5.B Use different degrees of precision in
measurement, explain the reason for using
a certain degree of precision, and apply
estimation strategies to obtain reasonable
measurements with appropriate precision for a
given purpose.
Example:
M2.5.C Solve problems involving measurement
conversions within and between systems,
including those involving derived units, and
analyze solutions in terms of reasonableness
of solutions and appropriate units.
This performance expectation is intended to build on
students’ knowledge of proportional relationships.
Students should understand the relationship between
scale factors and their inverses as they relate to
choices about when to multiply and when to divide in
converting measurements.
•
The U.S. Census Bureau reported a national
population of 299,894,924 on its Population Clock
in mid-October of 2006. One can say that the
U.S. population is 3 hundred million (3 × 108) and
be precise to one digit. Although the population
had surpassed 3 hundred million by the end of
that month, explain why 3 × 108 remained precise
to one digit.
Derived units include those that measure speed,
density, flow rates, population density, etc.
July 2008
Washington State K–12 Mathematics Standards
209
Math 2
Performance Expectations
Explanatory Comments and Examples
Students are expected to:
M2.5.C cont.
Example:
•
M2.5.D Find the terms and partial sums of arithmetic
and geometric series and the infinite sum for
geometric series.
A digital camera takes pictures that are 3.2
megabytes in size. If the pictures are stored on a
1-gigabyte card, how many pictures can be taken
before the card is full?
Students build on the knowledge gained in
Mathematics 1 to find specific terms in a sequence and
to express arithmetic and geometric sequences in both
explicit and recursive forms.
Examples:
•
A ball is dropped from a height of 10 meters.
Each time it hits the ground, it rebounds
3
of the
4
distance it has fallen. What is the total sum of the
distances it falls and rebounds before coming to
rest?
•
Show that the sum of the first 10 terms of the
geometric series 1 +
1 1
1
+ +
+ ... is twice the
3 9 27
sum of the first 10 terms of the geometric series
1
3
1
9
1 – + – 210
1
+ ...
27
July 2008
Washington State K–12 Mathematics Standards
Math 2
Mathematics 2
M2.6. Core Processes: Reasoning, problem solving, and communication
S
tudents formalize the development of reasoning in Mathematics 2 as they use algebra, geometry, and
probability to make and defend generalizations. They justify their reasoning with accepted standards
of mathematical evidence and proof, using correct mathematical language, terms, and symbols in all
situations. They extend the problem-solving practices developed in earlier grades and apply them to
more challenging problems, including problems related to mathematical and applied situations. Students
formalize a coherent problem-solving process in which they analyze the situation to determine the
question(s) to be answered, synthesize given information, and identify implicit and explicit assumptions
that have been made. They examine their solution(s) to determine reasonableness, accuracy, and
meaning in the context of the original problem. The mathematical thinking, reasoning, and problemsolving processes students learn in high school mathematics can be used throughout their lives as they
deal with a world in which an increasing amount of information is presented in quantitative ways and more
and more occupations and fields of study rely on mathematics.
Performance Expectations
Explanatory Comments and Examples
Students are expected to:
M2.6.A Analyze a problem situation and represent it
mathematically.
M2.6.B Select and apply strategies to solve problems.
M2.6.C Evaluate a solution for reasonableness, verify
its accuracy, and interpret the solution in the
context of the original problem.
Examples:
•
AB is the diameter of the semicircle and the radius
of the quarter circle shown in the figure below. BC is
the perpendicular bisector of AB.
M2.6.D Generalize a solution strategy for a single
problem to a class of related problems, and
apply a strategy for a class of related problems
to solve specific problems.
M2.6.E Read and interpret diagrams, graphs, and
text containing the symbols, language, and
conventions of mathematics.
M2.6.F Summarize mathematical ideas with precision
and efficiency for a given audience and purpose.
M2.6.G Synthesize information to draw conclusions
and evaluate the arguments and conclusions
of others.
E
D
F
A
C
B
Imagine all of the triangles formed by AB and any
arbitrary point lying in the region bounded by AC,
CD, and AD , seen in bold below.
E
D
M2.6.H Use inductive reasoning to make conjectures,
and use deductive reasoning to prove or
disprove conjectures.
F
A
C
B
Use inductive reasoning to make conjectures about
what types of triangles are formed based upon
the region where the third vertex is located. Use
deductive reasoning to verify your conjectures.
July 2008
Washington State K–12 Mathematics Standards
211
Math 2
Performance Expectations
Explanatory Comments and Examples
Students are expected to:
M2.6 cont.
•
Rectangular cartons that are 5 feet long need to
be placed in a storeroom that is located at the end
of a hallway. The walls of the hallway are parallel.
The door into the hallway is 3 feet wide and the
width of the hallway is 4 feet. The cartons must be
carried face up. They may not be tilted. Investigate
the width and carton top area that will fit through
the doorway.
C
S
3
4
T
5
A
R
Generalize your results for a hallway opening of x
feet and a hallway width of y feet if the maximum
carton dimensions are c feet long and x2 + y2 = c2.
212
•
Prove (a + b)2 = a2 + 2ab + b2.
•
A student writes (x + 3)2 = x2 + 9. Explain why this
is incorrect.
July 2008
Washington State K–12 Mathematics Standards
Mathematics 3
July 2008
Washington State K–12 Mathematics Standards
213
Mathematics 3
In Mathematics 3, students develop a more coherent and formal view of mathematics, going beyond
specific rules and procedures to emphasize generalizations. Students extend their knowledge of
number systems to include complex numbers, and they evaluate possible solutions to algebraic
equations. The application and visualization of geometry extends to three-dimensional figures as
students study the effects of changes in one dimension on various attributes and properties of a
figure. Students study the composition of transformations on geometric figures. They generalize the
relationship of changes in the symbolic form of functions to transformations of their corresponding
graphs. They extend their study of functions to include logarithmic, radical, and cubic functions and
are introduced to the concept of inverse functions. Students study variability of data and examine the
validity of generalizing results to an entire population.
214
July 2008
Washington State K–12 Mathematics Standards
Math 3
Mathematics 3
M3.1. Core Content: Solving problems
(Algebra)
T
he first core content area highlights the types of problems students will be able to solve by the end of
Mathematics 3, as they extend their ability to solve problems with additional functions and equations.
Additionally, they deepen their understanding of and skills related to functions they encountered in
Mathematics 1 and 2, and they use these functions to solve more complex problems. When presented
with a contextual problem, students identify a function or equation that models the problem and use
that information to write an equation to solve the problem. For example, in addition to using graphs to
approximate solutions to problems modeled by exponential functions, they use knowledge of logarithms to
solve exponential equations. Turning word problems into equations that can be solved is a skill students
hone throughout the course.
Performance Expectations
Explanatory Comments and Examples
Students are expected to:
M3.1.A Select and justify functions and equations to
model and solve problems.
Example:
M3.1.B Solve problems that can be represented by
systems of equations and inequalities.
Examples:
•
•
A manufacturer wants to design a cylindrical soda
can that will hold 500 milliliters (mL) of soda.
The manufacturer’s research has determined
that an optimal can height is between 10 and 15
centimeters. Find a function for the radius in terms
of the height, and use it to find the possible range
of radius measurements in centimeters. Explain
your reasoning. Mr. Smith uses the following formula to calculate
students’ final grades in his Mathematics 3 class:
0.4E + 0.6T = C, where E represents the score
on the final exam, and T represents the average
score of all tests given during the grading period.
All tests and the final exam are worth a maximum
of 100 points. The minimum passing score on
tests, the final exam, and the course is 60.
Determine the inequalities that describe the
following situation and sketch a system of graphs
to illustrate it. When necessary, round scores to
the nearest tenth.
— Is it possible for a student to have a failing
test score average (i.e., T < 60 points)
and still pass the course?
— If you answered “yes,” what is the
minimum test score average a student can
have and still pass the course? What final
exam score is needed to pass the course
with a minimum test score average?
— A student has a particular test score
average. How can (s)he figure out the
minimum final exam score needed to pass
the course?
July 2008
Washington State K–12 Mathematics Standards
215
Math 3
Performance Expectations
Explanatory Comments and Examples
Students are expected to:
M3.1.C Solve problems that can be represented by
quadratic functions, equations, and inequalities.
In addition to solving area and velocity problems by
factoring and applying the quadratic formula to the
quadratic equation, students use the vertex form of
the equation to solve problems about maximums,
minimums, and symmetry.
Examples:
•
Fireworks are launched upward from the ground
with an initial velocity of 160 feet per second. The
formula for vertical motion is h(t) = 0.5at2 + vt + s,
where the gravitational constant, a, is -32 feet per
square second, v represents the initial velocity, and
s represents the initial height. Time t is measured
in seconds, and height h is measured in feet.
For the ultimate effect, the fireworks must explode
after they reach the maximum height. For the
safety of the crowd, they must explode at least 256
ft. above the ground. The fuses must be set for the
appropriate time interval that allows the fireworks
to reach this height. What range of times, starting
from initial launch and ending with fireworks
explosion, meets these conditions?
M3.1.D Solve problems that can be represented
by exponential and logarithmic functions
and equations.
M3.1.E Solve problems that can be represented by
a
inverse variations of the forms f(x) = + b,
x
f(x) = a2 + b, and f(x) = a .
x
(bx + c)
Examples:
•
If you need $15,000 in 4 years to start college,
how much money would you need to invest
now? Assume an annual interest rate of 4%
compounded monthly for 48 months.
•
The half-life of a certain radioactive substance is
65 days. If there are 4.7 grams initially present,
how long will it take for there to be less than 1
gram of the substance remaining?
Examples:
•
At the You’re Toast, Dude! toaster company, the
weekly cost to run the factory is $1400, and the
cost of producing each toaster is an additional $4
per toaster.
— Find a function to represent the weekly
cost in dollars, C(x), of producing x
toasters. Assume either unlimited
production is possible or set a maximum
per week. — Find a function to represent the total
production cost per toaster for a week.
— How many toasters must be produced
within a week to have a total production
cost per toaster of $8? 216
July 2008
Washington State K–12 Mathematics Standards
Math 3
Performance Expectations
Explanatory Comments and Examples
Students are expected to:
M3.1.E cont.
•
A person’s weight varies inversely as the square of
his distance from the center of the earth. Assume
the radius of the earth is 4000 miles. How much
would a 200-pound man weigh
— 1000 miles above the surface of the earth?
— 2000 miles above the surface of the earth?
July 2008
Washington State K–12 Mathematics Standards
217
Math 3
Mathematics 3
M3.2. Core Content: Transformations and functions
(Algebra, Geometry/Measurement)
S
tudents formalize their previous study of geometric transformations, focusing on the effect of
such transformations on the attributes of geometric figures. They study techniques for using
transformations to determine congruence and similarity. Students extend their study of transformations
to include transformations of many types of functions, including quadratic and exponential functions.
Performance Expectations
Explanatory Comments and Examples
Students are expected to:
M3.2.A Sketch results of transformations and
compositions of transformations for a given
two-dimensional figure on the coordinate
plane, and describe the rule(s) for performing
translations or for performing reflections about
the coordinate axes or the line y = x.
Transformations include translations, rotations,
reflections, and dilations.
M3.2.B Determine and apply properties of
transformations.
Students make and test conjectures about
compositions of transformations and inverses of
transformations, the commutativity and associativity of
transformations, and the congruence and similarity of
two-dimensional figures under various transformations.
Example:
•
Line m is described by the equation y = 2x + 3.
Graph line m and reflect line m across the line
y = x. Determine the equation of the image of the
reflection. Describe the relationship between the
line and its image.
Examples:
M3.2.C Given two congruent or similar figures in a
coordinate plane, describe a composition of
translations, reflections, rotations, and dilations
that superimposes one figure on the other.
218
•
Identify transformations (alone or in composition)
that preserve congruence.
•
Determine whether the composition of two
reflections of a line is commutative.
•
Determine whether the composition of two
rotations about the same point of rotation is
commutative.
•
Find a rotation that is equivalent to the composition
of two reflections over intersecting lines.
•
Find the inverse of a given transformation.
Examples:
•
Find a sequence of transformations that
superimposes the segment with endpoints
(0, 0) and (2, 1) on the segment with endpoints
(4, 2) and (3, 0).
•
Find a sequence of transformations that
superimposes the triangle with vertices (0, 0),
(1, 1), and (2, 0) on the triangle with vertices (0, 1),
(2, -1), and (0, -3).
July 2008
Washington State K–12 Mathematics Standards
Math 3
Performance Expectations
Explanatory Comments and Examples
Students are expected to:
M3.2.D Describe the symmetries of two-dimensional
figures and describe transformations, including
reflections across a line and rotations about
a point.
Although the expectation only addresses twodimensional figures, classroom activities can easily
extend to three-dimensional figures. Students can also
describe the symmetries, reflections across a plane,
and rotations about a line for three-dimensional figures.
M3.2.E Construct new functions using the
transformations f(x – h), f(x) + k, cf(x), and by
adding and subtracting functions, and describe
the effect on the original graph(s).
Students perform simple transformations on functions,
including those that contain the absolute value of
expressions, quadratic expressions, square root
expressions, and exponential expressions, to make
new functions.
Examples:
•
What sequence of transformations changes
f(x) = x2 to g(x) = -5(x – 3)2 + 2 ?
•
Carly decides to earn extra money by making
glass bead bracelets. She purchases tools for
$40.00. Elastic bead cord for each bracelet costs
$0.10. Glass beads come in packs of 10 beads,
and one pack has enough beads to make one
bracelet. Base price for the beads is $2.00 per
pack. For each of the first 100 packs she buys, she
gets $0.01 off each of the packs. (For example, if
she purchases three packs, each pack costs $1.97
instead of $2.00.) Carly plans to sell each bracelet
for $4.00. Assume Carly will make a maximum of
100 bracelets.
— Find a function C(b) that describes
Carly’s costs.
— Find a function R(b) that describes
Carly’s revenue.
Carly’s profit is described by P(b) = R(b) – C(b).
— Find P(b).
— What is the minimum number of bracelets
that Carly must sell in order to make a profit?
— To make a profit of $100?
July 2008
Washington State K–12 Mathematics Standards
219
Math 3
Mathematics 3
M3.3. Core Content: Functions and modeling
(Algebra)
S
tudents extend their understanding of exponential functions from Mathematics 2 with an emphasis on
inverse functions. This leads to a natural introduction of logarithms and logarithmic functions. They
learn to use the basic properties of exponential and logarithmic functions, graphing both types of functions
to analyze relationships, represent and model problems, and answer questions. Students apply these
functions in many practical situations, such as applying exponential functions to determine compound
interest and applying logarithmic functions to determine the pH of a liquid. In addition, students extend
their study of functions to include polynomials of higher degree and those containing radical expressions.
They formalize and deepen their understanding of real-valued functions, their defining characteristics
and uses, and the mathematical language used to describe them. They compare and contrast the
types of functions they have studied and their basic transformations. Students learn the practical and
mathematical limitations that must be considered when working with functions or when using functions to
model situations.
Performance Expectations
Explanatory Comments and Examples
Students are expected to:
M3.3.A Know and use basic properties of exponential
and logarithmic functions and the inverse
relationship between them.
Examples:
•
Given f(x) = 4x, write an equation for the inverse
of this function. Graph the functions on the same
coordinate grid.
— Find f(-3).
— Evaluate the inverse function at 7.
•
Derive the formulas:
— logba ⋅ logab = 1
— logaN = logbN ⋅ logab
•
Find the exact value of x in:
— logx16 =
4
3
— log381 = x
•
Solve for y in terms of x:
— loga
y
=x
x
— 100 = x ⋅ 10y
M3.3.B Graph an exponential function of the form
f(x) = abx and its inverse logarithmic function.
Students expand on the work they did in Mathematics
2 with functions of the form y = abx. Although the
concept of inverses is not fully developed until
Precalculus, there is an emphasis in Mathematics 3 on
students recognizing the inverse relationship between
exponential and logarithmic functions and how this is
reflected in the shapes of the graphs.
Example:
•
220
Find the equation for the inverse function of y = 3x.
Graph both functions. What characteristics of each
of the graphs indicate they are inverse functions?
July 2008
Washington State K–12 Mathematics Standards
Math 3
Performance Expectations
Explanatory Comments and Examples
Students are expected to:
M3.3.C Solve exponential and logarithmic equations.
Examples:
•
A recommended adult dosage of the cold
medication NoMoreFlu is 16 ml. NoMoreFlu
causes drowsiness when there are more than 4 ml
in one’s system, making it unsafe to drive, operate
machinery, etc. The manufacturer wants to print a
warning label telling people how long they should
wait after taking NoMoreFlu for the drowsiness
to pass. If the typical metabolic rate is such that
one quarter of the NoMoreFlu is lost every four
hours, and a person takes the full dosage, how
long should adults wait after taking NoMoreFlu to
ensure that there will be
— Less than 4 ml of NoMoreFlu in their
system?
— Less than 1 ml in their system?
— Less than 0.1 ml in their system?
M3.3.D Plot points, sketch, and describe the graphs of
functions of the form f (x) = a x − c + d, and solve
related equations.
2
•
Solve for x in 256 = 2 x
•
Solve for x in log5(x – 4) = 3.
−1
.
Students solve algebraic equations that involve
the square root of a linear expression over the
real numbers. Students should be able to identify
extraneous solutions and explain how they arose.
Students should view the function g(x) = x as the
inverse function of f(x) = x2, recognizing that the
functions have different domains for x greater than or
equal to 0.
Example:
•
Analyze the following equations and tell what
you know about the solutions. Then solve the
equations.
—
M3.3.E Plot points, sketch, and describe the graphs
of functions of the form f ( x) = a2 + b , and
x
a
f (x) =
, and solve related equations.
( bx + c )
—
5 x − 6 = −2
—
2 x + 15 = x
—
2x − 5 = x + 7
Examples:
a
•
Sketch the graphs of the four functions f ( x) = 2 + b
x
when a = 4 and 8 and b = 0 and 1.
•
Sketch the graphs of the four functions
f (x) =
July 2008
Washington State K–12 Mathematics Standards
2 x+5 =7
4
when b = 1 and 4 and c = 2 and 3.
( bx + c )
221
Math 3
Performance Expectations
Explanatory Comments and Examples
Students are expected to:
M3.3.F Plot points, sketch, and describe the graphs of
cubic polynomial functions of the form
f(x) = ax3 + d as an example of higher order
polynomials and solve related equations.
Example:
M3.3.G Solve systems of three equations with
three variables.
Students solve systems of equations using algebraic
and numeric methods.
•
Solve for x in 60 = -2x3 + 6.
Examples:
•
Jill, Ann, and Stan are to inherit $20,000. Stan is
to get twice as much as Jill, and Ann is to get twice
as much as Stan. How much does each get?
•
Solve the following system of equations.
2x – y – z = 7
3x + 5y + z = -10
4x – 3y + 2z = 4
222
July 2008
Washington State K–12 Mathematics Standards
Math 3
Mathematics 3
M3.4. Core Content: Quantifying variability
(Data/Statistics/Probability)
S
tudents extend their use of statistics as they graph bivariate data and analyze the graph to make
predictions. They calculate and interpret measures of variability, confidence intervals, and margins of
error for population proportions. Dual goals underlie the content in the section: Students prepare for the
further study of statistics and also become thoughtful consumers of data.
Performance Expectations
Explanatory Comments and Examples
Students are expected to:
M3.4.A Calculate and interpret measures of variability
and standard deviation and use these
measures and the characteristics of the normal
distribution to describe and compare data sets.
Students should be able to identify unimodality,
symmetry, standard deviation, spread, and the shape
of a data curve to determine whether the curve could
reasonably be approximated by a normal distribution.
Given formulas, students should be able to calculate
the standard deviation for a small data set, but
calculators ought to be used if there are very many
points in the data set. It is important that students
be able to describe the characteristics of the normal
distribution and identify common examples of data that
are and are not reasonably modeled by it. Common
examples of distributions that are approximately
normal include physical performance measurements
(e.g., weightlifting, timed runs), heights, and weights.
Apply the Empirical Rule (68–95–99.7 Rule) to
approximate the percentage of the population meeting
certain criteria in a normal distribution.
Example:
•
M3.4.B Calculate and interpret margin of error and
confidence intervals for population proportions.
Which is more likely to be affected by an outlier
in a set of data, the interquartile range or the
standard deviation?
Students will use technology based on the complexity
of the situation.
Students use confidence intervals to critique various
methods of statistical experimental design, data collection,
and data presentation used to investigate important
problems, including those reported in public studies.
Example:
•
July 2008
Washington State K–12 Mathematics Standards
In 2007, 400 of the 500 10th graders in Local High
School passed the WASL. In 2008, 375 of the 480
10th graders passed the test. The Local Gazette
headline read “10th Grade WASL Scores Decline
in 2008!”
223
Math 3
Performance Expectations
Explanatory Comments and Examples
Students are expected to:
M3.4.B cont.
In response, the Superintendent of Local School
District wrote a letter to the editor claiming that,
in fact, WASL performance was not significantly
lower in 2008 than it was in 2007. Who is correct,
the Local Gazette or the Superintendent?
Use mathematics to find the margin of error to justify
your conclusion. (Formula for the margin of error
(E): E = zc
p(1−p)
; z95 = 1.96, where n is the sample
n
size, p is the proportion of the sample with the trait
of interest, c is the confidence level, and zc is the
multiplier for the specified confidence interval.)
224
July 2008
Washington State K–12 Mathematics Standards
Math 3
Mathematics 3
M3.5. Core Content: Three-dimensional geometry (Geometry/Measurement)
S
tudents formulate conjectures about three-dimensional figures. They use deductive reasoning to
establish the truth of conjectures or to reject them on the basis of counterexamples. They extend
and formalize their work with perimeter, area, surface area, and volume of two- and three-dimensional
figures, focusing on mathematical derivations of these formulas and their applications in complex
problems. They use properties of geometry and measurement to solve both purely mathematical and
applied problems. They also extend their knowledge of distance and angle measurements in a plane to
measurements on a sphere.
Performance Expectations
Explanatory Comments and Examples
Students are expected to:
M3.5.A Describe the intersections of lines in the plane
and in space, of lines and planes, and of
planes in space.
Example:
M3.5.B Describe prisms, pyramids, parallelepipeds,
tetrahedra, and regular polyhedra in terms of
their faces, edges, vertices, and properties.
Examples:
M3.5.C Analyze cross-sections of cubes, prisms,
pyramids, and spheres and identify the
resulting shapes.
July 2008
Washington State K–12 Mathematics Standards
•
Describe all the ways that three planes can
intersect in space.
•
Given the number of faces of a regular polyhedron,
derive a formula for the number of vertices.
•
Describe symmetries of three-dimensional
polyhedra and their two-dimensional faces.
•
Describe the lateral faces that are required for a
pyramid to be a right pyramid with a regular base.
Describe the lateral faces required for an oblique
pyramid that has a regular base.
Examples:
•
Start with a regular tetrahedron with edges of unit
length 1. Find the plane that divides it into two
congruent pieces and whose intersection with
the tetrahedron is a square. Find the area of the
square. (Requires no pencil or paper.)
•
Start with a cube with edges of unit length 1. Find
the plane that divides it into two congruent pieces
and whose intersection with the cube is a regular
hexagon. Find the area of the hexagon.
•
Start with a cube with edges of unit length 1.
Find the plane that divides it into two congruent
pieces and whose intersection with the cube is a
rectangle that is not a face and contains four of the
vertices. Find the area of the rectangle.
•
Which has the larger area, the above rectangle or
the above hexagon?
225
Math 3
Performance Expectations
Explanatory Comments and Examples
Students are expected to:
M3.5.D Apply formulas for surface area and volume of
three-dimensional figures to solve problems.
Problems include those that are purely mathematical
as well as those that arise in applied contexts.
Three-dimensional figures include right and oblique
prisms, pyramids, cylinders, cones, spheres, and
composite three-dimensional figures.
Examples:
•
As Pam scooped ice cream into a cone, she began
to formulate a geometry problem in her mind. If the
ice cream was perfectly spherical with diameter
2.25'' and sat on a geometric cone that also had
diameter 2.25'' and was 4.5'' tall, would the cone
hold all the ice cream as it melted (without her eating
any of it)? She figured the melted ice cream would
have the same volume as the unmelted ice cream.
Find the solution to Pam’s problem and justify
your reasoning.
•
M3.5.E Predict and verify the effect that changing
one, two, or three linear dimensions has on
perimeter, area, volume, or surface area of
two- and three-dimensional figures.
M3.5.F Analyze distance and angle measures on a
sphere and apply these measurements to the
geometry of the earth.
226
A rectangle is 5 inches by 10 inches. Find the
volume of a cylinder that is generated by rotating
the rectangle about the 10-inch side.
The emphasis in high school should be on verifying the
relationships between length, area, and volume and on
making predictions using algebraic methods.
Examples:
•
What happens to the volume of a rectangular
prism if four parallel edges are doubled in length?
•
The ratio of a pair of corresponding sides in
two similar triangles is 5:3. The area of the
smaller triangle is 108 in2. What is the area
of the larger triangle?
Examples:
•
Use a piece of string to measure the distance
between two points on a ball or globe; verify that
the string lies on an arc of a great circle.
•
On a globe, show with examples why airlines use
polar routes instead of flying due east from Seattle
to Paris.
•
Show that the sum of the angles of a triangle on a
sphere is greater than 180 degrees.
July 2008
Washington State K–12 Mathematics Standards
Math 3
Mathematics 3
M3.6. Core Content: Algebraic properties
(Numbers, Algebra)
S
tudents continue to use variables and expressions to solve both purely mathematical and applied
problems, and they broaden their understanding of the real number system to include complex
numbers. Students extend their use of algebraic techniques to include manipulations of expressions with
rational exponents, operations on polynomials and rational expressions, and solving equations involving
rational and radical expressions.
Performance Expectations
Explanatory Comments and Examples
Students are expected to:
M3.6.A Explain how whole, integer, rational, real, and
complex numbers are related, and identify
the number system(s) within which a given
algebraic equation can be solved.
Example:
•
Within which number system(s) can each of the
following be solved? Explain how you know.
— 3x + 2 = 5
— x2 = 1
— x2 =
1
4
— x2 = 2
— x2 = -2
—
M3.6.B Use the laws of exponents to simplify and
evaluate numeric and algebraic expressions
that contain rational exponents.
x
=π
7
Examples:
•
Convert the following from a radical to exponential
form or vice versa.
1
— 24 3
—
—
—
•
M3.6.C Add, subtract, multiply, and divide polynomials.
5
16
x2 + 1
x2
x
Evaluate x-2/3 for x = 27.
Write algebraic expressions in equivalent forms using
algebraic properties to perform the four arithmetic
operations with polynomials.
Students should recognize that expressions are
essentially sums, products, differences, or quotients.
For example, the sum 2x2 + 4x can be written as a
product, 2x(x + 2).
July 2008
Washington State K–12 Mathematics Standards
227
Math 3
Performance Expectations
Explanatory Comments and Examples
Students are expected to:
M3.6.C cont.
M3.6.D Add, subtract, multiply, divide, and simplify
rational and more general algebraic expressions.
Examples:
•
(3x2 – 4x + 5) + (-x2 + x – 4) + (2x2 + 2x + 1)
•
(2x2 – 4) – (x2 + 3x – 3)
•
2x2
6
•
9 2x4
•
x 2 – 2x – 3
x +1
In the same way that integers were extended to
fractions, polynomials are extended to rational
expressions. Students must be able to perform the
four basic arithmetic operations on more general
expressions that involve exponentials.
The binomial theorem is useful when raising
expressions to powers, such as (x + 3)5.
Examples:
•
x + 1 3x − 3
−
( x + 1)2 x 2 − 1
•
Divide
3/2
228
( x + 2)
x +1
by
x+ 2
x 2− 1
July 2008
Washington State K–12 Mathematics Standards
Math 3
Mathematics 3
M3.7. Additional Key Content
(Geometry/Measurement)
S
tudents formulate conjectures about circles. They use deductive reasoning to establish the truth of
conjectures or to reject them on the basis of counterexamples. Students explain their reasoning using
precise mathematical language and symbols. They apply their knowledge of geometric figures and their
properties to solve a variety of both purely mathematical and applied problems.
Performance Expectations
Explanatory Comments and Examples
Students are expected to:
M3.7.A Know, prove, and apply basic theorems
relating circles to tangents, chords, radii,
secants, and inscribed angles.
M3.7.B Determine the equation of a circle that is
described geometrically in the coordinate
plane and, given equations for a circle
and a line, determine the coordinates of
their intersection(s).
M3.7.C Explain and perform constructions related to
the circle.
Examples:
•
Given a line tangent to a circle, know and explain
that the line is perpendicular to the radius drawn to
the point of tangency.
•
Prove that two chords equally distant from the
center of a circle are congruent.
•
Prove that if one side of a triangle inscribed in a circle
is a diameter, then the triangle is a right triangle.
•
Prove that if a radius of a circle is perpendicular to a
chord of a circle, then the radius bisects the chord.
Examples:
•
Write an equation for a circle with a radius of 2
units and center at (1, 3).
•
Given the circle x2 + y2 = 4 and the line y = x, find
the points of intersection.
•
Write an equation for a circle given a line segment
as a diameter.
•
Write an equation for a circle determined by a
given center and tangent line.
Students perform constructions using straightedge
and compass, paper folding, and dynamic geometry
software. What is important is that students understand
the mathematics and are able to justify each step in a
construction.
Example:
•
In each case, explain why the constructions work:
a. Construct the center of a circle from two chords.
b. Construct a circumscribed circle for a triangle.
c. Inscribe a circle in a triangle.
July 2008
Washington State K–12 Mathematics Standards
229
Math 3
Performance Expectations
Explanatory Comments and Examples
Students are expected to:
M3.7.D Derive and apply formulas for arc length and
area of a sector of a circle.
Example:
•
Find the area and perimeter of the Reuleaux
triangle below.
The Reuleaux triangle is constructed with
three arcs. The center of each arc is located at
the vertex of an equilateral triangle. Each arc
extends between the two opposite vertices of the
equilateral triangle.
The figure below is a Reuleaux triangle that
circumscribes equilateral triangle ABC. ∆ABC has
side length of 5 inches. AB has center C, BC has
center A, and CA has center B, and all three arcs
have the same radius equal to the length of the
sides of the triangle.
A
C
230
B
July 2008
Washington State K–12 Mathematics Standards
Math 3
Mathematics 3
M3.8. Core Processes: Reasoning, problem solving, and communication
S
tudents formalize the development of reasoning in Mathematics 3 as they use algebra, geometry, and
statistics to make and defend generalizations. They justify their reasoning with accepted standards
of mathematical evidence and proof, using correct mathematical language, terms, and symbols in all
situations. They extend the problem-solving practices developed in earlier grades and apply them to
more challenging problems, including problems related to mathematical and applied situations. Students
formalize a coherent problem-solving process in which they analyze the situation to determine the
question(s) to be answered, synthesize given information, and identify implicit and explicit assumptions
that have been made. They examine their solution(s) to determine reasonableness, accuracy, and
meaning in the context of the original problem. The mathematical thinking, reasoning, and problemsolving processes students learn in high school mathematics can be used throughout their lives as they
deal with a world in which an increasing amount of information is presented in quantitative ways and more
and more occupations and fields of study rely on mathematics.
Performance Expectations
Explanatory Comments and Examples
Students are expected to:
M3.8.A Analyze a problem situation and represent
it mathematically.
M3.8.B Select and apply strategies to solve problems.
M3.8.C Evaluate a solution for reasonableness, verify
its accuracy, and interpret the solution in the
context of the original problem.
M3.8.D Generalize a solution strategy for a single
problem to a class of related problems and
apply a strategy for a class of related problems
to solve specific problems.
Examples:
•
Show that a + b ≠ a + b , for all positive real
values of a and b.
•
Show that the product of two odd numbers is
always odd.
•
Leo is painting a picture on a canvas that
measures 32 inches by 20 inches. He has divided
the canvas into four different rectangles, as shown
in the diagram.
M3.8.E Read and interpret diagrams, graphs, and
text containing the symbols, language, and
conventions of mathematics.
M3.8.F Summarize mathematical ideas with precision
and efficiency for a given audience and purpose.
M3.8.G Synthesize information to draw conclusions
and evaluate the arguments and conclusions
of others.
M3.8.H Use inductive reasoning and the properties
of numbers to make conjectures, and use
deductive reasoning to prove or disprove
conjectures.
He would like the upper right corner to be a
rectangle that has a length 1.6 times its width. Leo
wants the area of the larger rectangle in the lower
left to be at least half the total area of the canvas.
Describe all the possibilities for the dimensions of
the upper right rectangle to the nearest hundredth,
and explain why the possibilities are valid.
July 2008
Washington State K–12 Mathematics Standards
231
Math 3
Performance Expectations
Explanatory Comments and Examples
Students are expected to:
If Leo uses the largest possible dimensions for
the smaller rectangle:
M3.8 cont.
— What will the dimensions of the larger
rectangle be?
— Will the larger rectangle be similar to the
rectangle in the upper right corner? Why
or why not?
— Is the original canvas similar to the
rectangle in the upper right corner?
(A rectangle whose length and width are in the ratio
1+ 5
(approximately equal to 1.6) is called a “golden
2
rectangle” and is often used in art and architecture.)
•
A relationship between variables can be
represented with a table, a graph, an equation, or
a description in words.
— How can you decide from a table whether
a relationship is linear, quadratic,
or exponential?
— How can you decide from a graph whether
a relationship is linear, quadratic,
or exponential?
— How can you decide from an equation
whether a relationship is linear, quadratic,
or exponential?
232
July 2008
Washington State K–12 Mathematics Standards
Acknowledgments
These K–12 mathematics standards have been developed by a team of Washington educators, mathematics
faculty, and citizens with support from staff of the Office of the Superintendent of Public Instruction and invited
national consultants, and facilitated by staff of the Charles A. Dana Center at The University of Texas at
Austin. In addition we would like to acknowledge Strategic Teaching, who was contracted by the State Board
of Education to conduct a final review and analysis of the draft K–12 Standards, as per 2008 Senate Bill
6534. The individuals who have played key roles in this project are listed below.
Washington Educators
and Community Leaders
Dana Anderson, Stanwood-Camano School District
Tim Bartlett, Granite Falls School District
Millie Brezinski, Nine Mile Falls School District
Jane Broom, Microsoft
Jewel Brumley, Yakima School District
Bob Dean, Evergreen Schools
John Burke, Gonzaga University
Shannon Edwards, Chief Leschi School
Andrea English, Arlington School District
John Firkins, Gonzaga University (retired)
Gary Gillespie, Spokane Public Schools
Russ Gordon, Whitman College
Katherine Hansen, Bethel School District
Tricia Hukee, Sumner School District
Michael Janski, Cascade School District
Russ Killingsworth, Seattle Pacific University
James King, University of Washington
Art Mabbott, Seattle Schools
Kristen Maxwell, Educational Service District 105
Rosalyn O’Donnell, Ellensburg School District
M. Cary Painter, Chehalis School District
Patrick Paris, Tacoma School District
Tom Robinson, Lake Chelan School District
Terry Rose, Everett School District
Allen Senear, Seattle Schools
Lorna Spear, Spokane Schools
David Thielk, Central Kitsap School District
Johnnie Tucker, retired teacher
Kimberly Vincent, Washington State University
Virginia Warfield, University of Washington
Sharon Young, Seattle Pacific University
Dana Center Facilitators
P. Uri Treisman
Cathy Seeley
Susan Hudson Hull
National Consultants
Mary Altieri, Consultant (retired teacher)
Angela Andrews, National-Louis University
Diane Briars, Pittsburgh Schools (retired)
Cathy Brown, Oregon Department of Education (retired)
Dinah Chancellor, Consultant
Philip Daro, Consultant
Bill Hopkins, Dana Center
Barbara King, Dana Center
Kurt Krieth, University of California at Davis
Bonnie McNemar, Consultant
David D. Molina, Consultant
Susan Eddins, Illinois Math and Science Academy (retired)
Wade Ellis, West Valley College, CA (retired)
Margaret Myers, The University of Texas at Austin
Lynn Raith, Pittsburgh Schools (retired)
Jane Schielack, Texas A&M University
Carmen Whitman, Consultant
OSPI Project Support
George W. Bright, Special Assistant to the Superintendent
Greta Bornemann, Teaching and Learning
Barbara Chamberlain, Interim Director
Larry Davison, Math Helping Corps Administrator
Lexie Domaradzki, Assistant Superintendent, Teaching ..
and Learning
Ron Donovan, Teaching & Learning
Dorian “Boo” Drury, Teaching & Learning
Lynda Eich, Assessment
Karen Hall, Assessment
Robert Hodgman, Assessment
Mary Holmberg, Assessment
Anton Jackson, Assessment
Yoonsun Lee, Assessment
Karrin Lewis, Teaching & Learning
Jessica Vavrus, Teaching and Learning
Joe Willhoft, Assistant Superintendent, Assessment
Strategic Teaching Reviewers
Linda Plattner
Andrew Clark
W. Stephen Wilson
July 2008
Washington State K–12 Mathematics Standards
Math 3
Office of Superintendent of Public Instruction Old Capitol Building P.O. Box 47200 Olympia, WA 98504‐7200 This document is available for purchase from the Department of Printing at: www.prt.wa.gov or by calling (360) 586‐6360. This document may also be downloaded, duplicated, and distributed as needed from our Web site at: www.k12.wa.us/CurriculumInstruct For more information about the contents of the document, please contact: Greta Bornemann, OSPI e‐mail: [email protected] Phone: 360‐725‐0437 OSPI Document Number: 09‐0013 This material is available in alternative format upon request. Contact the Resource Center at 1‐888‐59 LEARN (1‐888‐595‐3276), TTY (360) 664‐3631 Office of Superintendent of Public Instruction
Old Capitol Building
P.O. Box 47200
Olympia, WA 98504-7200
2009