June 2012
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Standards for Mathematical Practice in Kindergarten
The Common Core State Standards for Mathematical Practice are expected to be integrated into every mathematics lesson for all students
Grades K-12. Below are a few examples of how these Practices may be integrated into tasks that students complete.
Practice
1. Make Sense and
Persevere in
Solving
Problems.
2. Reason
abstractly and
quantitatively.
3. Construct viable
arguments and
critique the
reasoning of
others.
4. Model with
mathematics.
Explanation and Examples
Mathematically proficient students in Kindergarten begin to develop effective dispositions toward problem solving. In rich
settings in which informal and formal possibilities for solving problems are numerous, young children develop the ability to
focus attention, test hypotheses, take reasonable risks, remain flexible, try alternatives, exhibit self-regulation, and persevere
(Copley, 2010). Using both verbal and nonverbal means, kindergarten students begin to explain to themselves and others the
meaning of a problem, look for ways to solve it, and determine if their thinking makes sense or if another strategy is needed.
As the teacher uses thoughtful questioning and provides opportunities for students to share thinking, kindergarten students
begin to reason as they become more conscious of what they know and how they solve problems.
Mathematically proficient students in Kindergarten begin to use numerals to represent specific amount (quantity). For
example, a student may write the numeral “11” to represent an amount of objects counted, select the correct number card
“17” to follow “16” on the calendar, or build a pile of counters depending on the number drawn. In addition, kindergarten
students begin to draw pictures, manipulate objects, use diagrams or charts, etc. to express quantitative ideas such as a joining
situation (Mary has 3 bears. Juanita gave her 1 more bear. How many bears does Mary have altogether?), or a separating
situation (Mary had 5 bears. She gave some to Juanita. Now she has 3 bears. How many bears did Mary give Juanita?). Using
the language developed through numerous joining and separating scenarios, kindergarten students begin to understand how
symbols (+, -, =) are used to represent quantitative ideas in a written format.
In Kindergarten, mathematically proficient students begin to clearly express, explain, organize and consolidate their math
thinking using both verbal and written representations. Through opportunities that encourage exploration, discovery, and
discussion, kindergarten students begin to learn how to express opinions, become skillful at listening to others, describe their
reasoning and respond to others’ thinking and reasoning. They begin to develop the ability to reason and analyze situations as
they consider questions such as, “Are you sure…?” , “Do you think that would happen all the time…?”, and “I wonder
why…?”
Mathematically proficient students in Kindergarten begin to experiment with representing real-life problem situations in
multiple ways such as with numbers, words (mathematical language), drawings, objects, acting out, charts, lists, and number
sentences. For example, when making toothpick designs to represent the various combinations of the number “5”, the student
writes the numerals for the various parts (such as “4” and “1”) or selects a number sentence that represents that particular
situation (such as 5 = 4 + 1)*.
*According to CCSS, “Kindergarten students should see addition and subtraction equations, and student writing of equations in
kindergarten in encouraged, but it is not required”. However, please note that it is not until First Grade when “Understand the meaning of
the equal sign” is an expectation (1.OA.7).
5. Use
appropriate
tools
In Kindergarten, mathematically proficient students begin to explore various tools and use them to investigate mathematical
concepts. Through multiple opportunities to examine materials, they experiment and use both concrete materials (e.g. 3dimensional solids, connecting cubes, ten frames, number balances) and technological materials (e.g., virtual manipulatives,
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strategically.
6. Attend to
precision
7. Look for and
make use of
structure
8. Look for and
express
regularity in
repeated
reasoning.
calculators, interactive websites) to explore mathematical concepts. Based on these experiences, they become able to decide
which tools may be helpful to use depending on the problem or task. For example, when solving the problem, “There are 4
dogs in the park. 3 more dogs show up in the park. How many dogs are in the park?”, students may decide to act it out using
counters and a story mat; draw a picture; or use a handful of cubes.
Mathematically proficient students in Kindergarten begin to express their ideas and reasoning using words. As their
mathematical vocabulary increases due to exposure, modeling, and practice, kindergarteners become more precise in their
communication, calculations, and measurements. In all types of mathematical tasks, students begin to describe their actions
and strategies more clearly, understand and use grade-level appropriate vocabulary accurately, and begin to give precise
explanations and reasoning regarding their process of finding solutions. For example, a student may use color words (such as
blue, green, light blue) and descriptive words (such as small, big, rough, smooth) to accurately describe how a collection of
buttons is sorted.
Mathematically proficient students in Kindergarten begin to look for patterns and structures in the number system and other
areas of mathematics. For example, when searching for triangles around the room, kindergarteners begin to notice that some
triangles are larger than others or come in different colors- yet they are all triangles. While exploring the part-whole
relationships of a number using a number balance, students begin to realize that 5 can be broken down into sub-parts, such as
4 and 1 or 4 and 2, and still remain a total of 5.
In Kindergarten, mathematically proficient students begin to notice repetitive actions in geometry, counting, comparing, etc.
For example, a kindergartener may notice that as the number of sides increase on a shape, a new shape is created (triangle has
3 sides, a rectangle has 4 sides, a pentagon has 5 sides, a hexagon has 6 sides). When counting out loud to 100, kindergartners
may recognize the pattern 1-9 being repeated for each decade (e.g., Seventy-ONE, Seventy-TWO, Seventy- THREE…
Eighty-ONE, Eighty-TWO, Eighty-THREE…). When joining one more cube to a pile, the child may realize that the new
amount is the next number in the count sequence.
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Kindergarten Critical Areas
The Critical Areas are designed to bring focus to the standards at each grade by describing the big ideas that
educators can use to build their curriculum and to guide instruction.
The Critical Areas for Kindergarten can be found on page 9 in the Common Core State Standards for Mathematics.
1. Representing, relating, and operating on whole numbers, initially with sets of objects.
Students use numbers, including written numerals, to represent quantities and to solve quantitative problems, such as counting objects in a
set; counting out a given number of objects; comparing sets or numerals; and modeling simple joining and separating situations with sets
of objects, or eventually with equations such as 5 + 2 = 7 and 7 – 2 = 5. (Kindergarten students should see addition and subtraction
equations, and student writing of equations in kindergarten is encouraged, but it is not required.) Students choose, combine, and apply
effective strategies for answering quantitative questions, including quickly recognizing the cardinalities of small sets of objects, counting
and producing sets of given sizes, counting the number of objects in combined sets, or counting the number of objects that remain in a set
after some are taken away.
2. Describing shapes and space.
Students describe their physical world using geometric ideas (e.g., shape, orientation, spatial relations) and vocabulary. They identify,
name, and describe basic two-dimensional shapes, such as squares, triangles, circles, rectangles, and hexagons, presented in a variety of
ways (e.g., with different sizes and orientations), as well as three-dimensional shapes such as cubes, cones, cylinders, and spheres. They
use basic shapes and spatial reasoning to model objects in their environment and to construct more complex shapes.
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Counting and Cardinality
• Know number names and the count sequence.
Count to 100 by ones and tens
Student Friendly/”I Can” statements
1. Recite numbers from 0 – 100, increasing
by ones.
2. Recite numbers from 0 – 100, increasing
by tens.
Resources
Assessments
http://nlvm.usu.edu
Count forward beginning from a given number within the known sequence (instead of having to begin at 1)
Student Friendly/”I Can” statements
1. Count by ones, starting at one.
2. Count by ones, starting at a number other
than one.
Resources
Assessments
Write numbers from 0 to 20. Represent a number of objects with a written numeral 0-20 (with 0 representing a count of no objects.)
Student Friendly/”I Can” statements
1. Write numbers from 0 – 20.
2. Count, with 1-1 correspondence, up to 10
objects.
3. Demonstrate, when shown a written
number from 0 – 20, how many objects
are represented by that number.
4. Represent the number of objects with a
written number.
Resources
5
Assessments
• Count to tell the number of objects.
Understand the relationship between numbers and quantities; connect counting to cardinality.
Student Friendly/”I Can” statements
1. Count objects saying the number name in
standard order.
Resources
Assessments
http://nlvm.usu.edu
When counting objects, say the number names in standard order, pairing each object with one and only one number name and each
number name with one and only one object.
Student Friendly/”I Can” statements
1. When given a group of objects, will count
using 1:1 correspondence.
2. When given a number, will present that
number of objects to represent the
number.
Resources
Assessments
Understand the last number name said tells the number of objects counted. The number of objects is the same regardless of their
arrangement or the order in which they were counted.
Student Friendly/”I Can” statements
1. Understand the last number named is the
number of objects counted.
Resources
Assessments
Understand that each successive number name refers to a quantity that is one larger.
Student Friendly/”I Can” statements
1. Understand that each successive number
name is one larger.
Resources
Assessments
Count to answer "how many?" questions about as many as 20 things arranged in a line, in a rectangular array, or a circle, or as many
as 10 things in a scattered configuration; given a number from 1-20, count out that many objects.
Student Friendly/”I Can” statements
1. Count up to 20 objects that have been
arranged in a line, rectangular array, or
circle
2. Count as many as 10 items in a scattered
configuration
3. Match each object with one and only one
Resources
6
Assessments
number name and each number with one
and only one object
4. Conclude that the last number of the
counted sequence signifies the quantity of
the counted collection.
5. Given a number from 1-20, count out that
many objects.
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• Compare numbers.
Identify whether the number of objects in one group is greater than, less than, or equal to the number of objects in another group,
e.g. by using matching and counting strategies.
Student Friendly/”I Can” statements
Resources
1. Describe greater than, less than, or equal
http://nlvm.usu.edu
to.
2. Determine whether a group of 10 or
fewer objects is greater than, less than, or
equal to another group of 10 or fewer
objects.
Compare two numbers between 1 and 10 presented as written numerals.
Student Friendly/”I Can” statements
Resources
1. Know the quantity of each numeral.
2. Determine whether a written number is
greater than, less than, or equal to
another written number.
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Assessments
Assessments
Operations and Algebraic Thinking
• Understand addition as putting together and adding to, and understand subtraction as taking apart and
taking from.
Represent addition and subtraction with objects, fingers, mental images, drawings, sounds (e.g., claps), acting out situations, verbal
explanations, expressions, or equations.
Student Friendly/”I Can” statements
1. Know adding is putting together parts to
make the whole.
2. Know subtracting is taking apart or taking
away from the whole to find the other
part.
Resources
Assessments
http://nlvm.usu.edu
3. Know the symbols (+, -, =) and the
words (plus, minus, equal) for adding
and subtracting.
4. Analyze addition or subtraction problem
to determine whether to ‘put together’ or
‘take apart’.
5. Model an addition/subtraction problem
given a real-life story.
6. Represent addition and subtraction with
objects, fingers, mental images, drawings,
sounds, acting out situations, verbal
explanations, expressions, or equations in
multiple ways, e.g., 2+3=5, 5=2+3, ||+|||
=|||||, and vertically.
(Writing equations in kindergarten is not
required but encouraged.)
Solve addition and subtraction word problems, and add and subtract within 10, e.g., by using objects or drawings to represent the
problem.
Student Friendly/”I Can” statements
1. Add and subtract within 10 (Maximum
sum and minuend is 10).
Resources
9
Assessments
2. Solve addition and subtraction word
problems within 10.
3. Use objects/drawings to represent an
addition and subtraction word problem.
Decompose numbers less than or equal to 10 into pairs in more than one way, e.g., by using objects or drawings, and record each
decomposition by a drawing or equation (e.g., 5 = 2 + 3 and 5 = 4 + 1).
Student Friendly/”I Can” statements
1. Solve addition number sentences within
10.
2. Decompose numbers less than or equal
to 10 into pairs in more than one way.
Resources
Assessments
3. Use objects or drawings then record
each composition by a drawing or
writing an equation.
For any number from 1 to 9, find the number that makes 10 when added to the given number, e.g., by using objects or drawings,
and record the answer with a drawing or equation.
Student Friendly/”I Can” statements
1. Know that two numbers can be added
together to make ten
2. Using materials or representations, find
the number that makes 10 when added to
the given number for any number from 1
to 9, and record the answer using
materials, representations, or equations.
Resources
Assessments
Fluently add and subtract within 5.
Student Friendly/”I Can” statements
1. Fluently with speed and accuracy add and
subtract within 5.
Resources
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Timed facts tests
Assessments
Number and Operations in Base Ten
• Work with numbers 11–19 to gain foundations for place value.
Compose and decompose numbers from 11 to 19 into ten ones and some further ones, e.g., by using objects or drawings, and
record each composition or decomposition by a drawing or equation (e.g., 18 = 10 + 8); understand that these numbers are
composed of ten ones and one, two, three, four, five, six, seven, eight, or nine ones.
Student Friendly/”I Can” statements
1. Know that a (spoken) number (11-19)
represents a quantity.
2. Understand that numbers 11-19 are
composed of 10 ones and one, two, three,
four, five, six, seven, eight, or nine ones.
3. Represent compositions or
decompositions by a drawing or equation.
4. Compose numbers 11-19 into ten ones
and some further ones using objects and
drawings.
5. Decompose numbers 11-19 into ten ones
and some further ones using objects and
drawings.
Resources
http://nlvm.usu.edu
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Assessments
Measurement and Data
• Describe and compare measurable attributes.
Describe measurable attributes of objects, such as length or weight. Describe several measurable attributes of a single object.
1.
2.
3.
4.
Student Friendly/”I Can” statements
Understand the meaning of attribute.
Identify one attribute of an object.
Identify attributes of various objects.
Identify multiple attributes of a single
object.
Resources
Assessments
http://nlvm.usu.edu
Directly compare two objects with a measurable attribute in common, to see which object has “more of”/“less of” the attribute, and
describe the difference. For example, directly compare the heights of two children and describe one child as taller/shorter.
Student Friendly/”I Can” statements
1. Know the meaning of the following words:
more/less, taller/shorter, etc.
2. Know that two objects can be compared
using a particular attribute.
3. Compare two objects and determine
which has more and which has less of the
measureable attribute to describe the
difference.
Resources
Assessments
• Classify objects and count the number of objects in categories.
Classify objects into given categories; count the numbers of objects in each category and sort the categories by count.
Student Friendly/”I Can” statements
1. Recognize non-measurable attributes such
as shape, color
2. Recognize measurable attributes such as
length, weight, height
3. Know what classify means
4. Know what sorting means
Resources
http://nlvm.usu.edu
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Assessments
5. Know that a category is the group that an
object belongs to according to a
particular, selected attribute
6. Understand one to one correspondence
with ten or less objects. Note: This target
being included here depends on the ordering
and grouping of content standards from
Counting and Cardinality.
7. Classify objects into categories by
particular attributes
8. Count objects in a given group. Note: This
is addressed in another content standard.
K.CC.5. It is important to integrate
standards to assist students with making
connections and building deeper
understanding.
9. Sort objects into categories then
determine the order by number of objects
in each category (limit category counts to
be less than or equal to ten) For example,
if m&m’s are categorized by the attribute
of color, then are “sorted” or ordered by
the number in each group (there are more
red than green, the blue group has fewer
than the green.)
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Geometry
• Identify and describe shapes.
Describe objects in the environment using names of shapes, and describe the relative positions of these objects using terms such as
above, below, beside, in front of, behind, and next to.
1.
2.
3.
4.
5.
Student Friendly/”I Can” statements
Identify objects.
Name objects.
Identify objects as 2- or 3- dimensional.
Describe positions such as above, below,
beside, in front of, behind, and next to.
Determine the relative position of the 2dimensional or 3-dimensional shapes
within the environment, using the
appropriate positional words.
Resources
Assessments
http://nlvm.usu.edu
Correctly name shapes regardless of their orientations or overall size.
Student Friendly/”I Can” statements
1. Know that size does not affect the name
of the shape.
2. Know that orientation does not affect the
name of the shape
Resources
Assessments
Identify shapes as two-dimensional (lying in a plane, “flat”) or three-dimensional (“solid”).
Student Friendly/”I Can” statements
1. Identify 2-dimensional shapes as lying in a
plane and flat
2. Identify 3-dimensional shapes as a solid
Resources
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Assessments
• Analyze, compare, create, and compose shapes.
Analyze and compare two- and three-dimensional shapes, in different sizes and orientations, using informal language to describe
their similarities, differences, parts (e.g., number of sides and vertices/“corners”) and other attributes (e.g., having sides of equal
length).
Student Friendly/”I Can” statements
1. Identify and count number of sides,
vertices/”corners”, and other attributes of
shapes
2. Describe similarities of various two- and
three-dimensional shapes
3. Describe differences of various two- and
three-dimensional shapes
4. Analyze and compare two-dimensional
shapes, in different sizes and orientations,
using informal language to describe their
similarities, differences, and other
attributes (e.g. having sides of equal
length).
5. Analyze and compare three-dimensional
shapes, in different sizes and orientations,
using informal language to describe their
similarities, differences, parts (e.g.
number of sides and vertices/”corners”)
and other attributes (e.g. having sides of
equal length).
6. Create shapes.
7. Make larger shapes from simple
shapes.
Resources
Assessments
Model shapes in the world by building shapes from components (e.g. sticks and clay balls) and drawing shapes.
Student Friendly/”I Can” statements
1. Recognize and identify (square, circles,
triangles, rectangles, hexagons, cubes,
Resources
15
Assessments
cones, cylinders, spheres)
2. Identify shapes in the real world
3. Analyze the attributes of real world
objects to identify shapes.
4. Construct shapes from components
(e.g., sticks and clay balls)
5. Draw shapes
Compose simple shapes to form larger shapes. For example, “Can you join these two triangles with full sides touching to make a
rectangle?”
Student Friendly/”I Can” statements
Resources
Assessments
1. Identify simple shapes (squares, triangles,
rectangles, hexagons)
2. Analyze how to put simple shapes
together to compose a new or larger
shape.
3. Compose a new or larger shape using
more than one simple shape.
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Standards for Mathematical Practice in First Grade
The Common Core State Standards for Mathematical Practice are practices expected to be integrated into every mathematics lesson for all students
Grades K-12. Below are a few examples of how these Practices may be integrated into tasks that students complete.
Mathematical Practice Explanations and Examples
Mathematically proficient students in First Grade continue to develop the ability to focus attention, test hypotheses, take
1 ) Make Sense
and Persevere in
reasonable risks, remain flexible, try alternatives, exhibit self-regulation, and persevere (Copley, 2010). As the teacher uses
thoughtful questioning and provides opportunities for students to share thinking, First Grade students become conscious of
Solving
Problems.
what they know and how they solve problems. They make sense of task-type problems, find an entry point or a way to
begin the task, and are willing to try other approaches when solving the task. They ask themselves, “Does this make
sense?” First Grade students’
conceptual understanding builds from their experiences in Kindergarten as they continue to rely on concrete manipulatives
and pictorial representations to solve a problem, eventually becoming fluent and flexible with mental math as a result of
these experiences.
2) Reason
Mathematically proficient students in First Grade recognize that a number represents a specific quantity. They use numbers
and symbols to represent a problem, explain thinking, and justify a response. For example, when solving the problem:
abstractly and
quantitatively.
“There are 60 children on the playground. Some children line up. There are 20 children still on the playground.
How many children lined up?” first grade students may write 20 + 40 = 60 to indicate a Think-Addition strategy. Other
students may illustrate a counting-on by tens strategy by writing 20 + 10 + 10 + 10 + 10 = 60. The numbers and equations
written illustrate the students’ thinking and the strategies used, rather than how to simply compute, and how the story is
decontextualized as it is represented abstractly with symbols.
Mathematically proficient students in First Grade continue to develop their ability to clearly express, explain, organize and
3) Construct
viable arguments
consolidate their math thinking using both verbal and written representations. Their understanding of grade appropriate
vocabulary helps them to construct viable arguments about mathematics. For example, when justifying why a particular
and critique the
reasoning of
shape isn’t a square, a first grade student may hold up a picture of a rectangle, pointing to the various parts, and reason, “It
can’t be a square because, even though it has 4 sides and 4 angles, the sides aren’t all the same size.” In a classroom where
others
risk-taking and varying perspectives are
encouraged, mathematically proficient students are willing and eager to share their ideas with others, consider other ideas
proposed by classmates, and question ideas that don’t seem to make sense.
Mathematically proficient students in First Grade model real-life mathematical situations with a number sentence or an
4) Model with
mathematics.
equation, and check to make sure that their equation accurately matches the problem context. They also use tools, such as
tables, to help collect information, analyze results, make conclusions, and review their conclusions to see if the results
make sense and revising as needed.
Mathematically proficient students in First Grade have access to a variety of concrete (e.g. 3-dimensional solids, ten
5) Use
appropriate tools
frames, number balances, number lines) and technological tools (e.g., virtual manipulatives, calculators, interactive
websites) and use them to investigate mathematical concepts. They select tools that help them solve and/or illustrate
strategically.
solutions to a problem. They recognize that multiple tools can be used for the same problem- depending on the strategy
used. For example, a child who is in the counting stage may choose connecting cubes to solve a problem. While, a student
who understands parts of number, may solve the same problem using ten-frames to decompose numbers rather than using
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6) Attend to precision.
7) Look for and
make use of structure.
8) Look for and
express regularity in
repeated reasoning.
individual connecting cubes. As the teacher provides numerous opportunities for students to use educational materials, first
grade students’ conceptual understanding and higher-order thinking skills are developed.
Mathematically proficient students in First Grade attend to precision in their communication, calculations, and
measurements. They are able to describe their actions and strategies clearly, using grade-level appropriate vocabulary
accurately. Their explanations and reasoning regarding their process of finding a solution becomes more precise. In varying
types of mathematical tasks, first grade students pay attention to details as they work. For example, as students’ ability to
attend to position and direction develops, they begin to notice reversals of numerals and self-correct when appropriate.
When measuring an object, students check to make sure that
there are not any gaps or overlaps as they carefully place each unit end to end to measure the object (iterating length units).
Mathematically proficient first grade students understand the symbols they use (=, >,<) and use clear explanations in
discussions with others. For example, for the sentence 4 > 3, a proficient student who is able to attend to precision states,
“Four is more than 3” rather than “The alligator eats the four. It’s bigger.”
Mathematically proficient students in First Grade carefully look for patterns and structures in the number system and other
areas of mathematics. For example, while solving addition problems using a number balance, students recognize that
regardless whether you put the 7 on a peg first and then the 4, or the 4 on first and then the 7, they both equal 11
(commutative property). When decomposing two-digit numbers, students realize that the number of tens they have
constructed ‘happens’ to coincide with the digit in the tens place. When exploring geometric properties, first graders
recognize that certain attributes are critical (number of sides, angles), while other properties are not (size, color,
orientation).
Mathematically proficient students in First Grade begin to look for regularity in problem structures when solving
mathematical tasks. For example, when adding three one-digit numbers and by making tens or using doubles, students
engage in future tasks looking for opportunities to employ those same strategies. Thus, when solving 8+7+2, a student may
say, “I know that 8 and 2 equal 10 and then I add 7 more. That makes 17. It helps to see if I can make a 10 out of 2 numbers
when I start.” Further, students use repeated reasoning while solving a task with multiple correct answers. For example, in
the task “There are 12 crayons in the box. Some are red and some are blue. How many of each could there be?” First Grade
students realize that the 12 crayons could include 6 of each color (6+6 = 12), 7 of one color and 5 of another (7+5 = 12),
etc. In essence, students repeatedly find numbers that add up to 12.
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Grade 1 Critical Areas
The Critical Areas are designed to bring focus to the standards at each grade by describing the big ideas that
educators can use to build their curriculum and to guide instruction.
The Critical Areas for First Grade can be found on page 13 in the Common Core State Standards for Mathematics.
1. Developing understanding of addition, subtraction, and strategies for addition and subtraction within 20.
Students develop strategies for adding and subtracting whole numbers based on their prior work with small numbers. They use
a variety of models, including discrete objects and length-based models (e.g., cubes connected to form lengths), to model addto, take-from,
put-together, take-apart, and compare situations to develop meaning for the operations of addition and
subtraction, and to develop strategies to solve arithmetic problems with these operations. Students understand connections
between counting and addition and subtraction (e.g., adding two is the same as counting on two). They use properties of
addition to add whole numbers and to create and use increasingly sophisticated strategies based on these properties (e.g.,
“making tens”) to solve addition and subtraction problems within 20. By comparing a variety of solution strategies, children
build their understanding of the relationship between addition and subtraction.
2. Developing understanding of whole number relationships and place value, including grouping in tens and ones.
Students develop, discuss, and use efficient, accurate, and generalizable methods to add within 100 and subtract multiples of
10. They compare whole numbers (at least to 100) to develop understanding of and solve problems involving their relative
sizes. They think of whole numbers between 10 and 100 in terms of tens and ones (especially recognizing the numbers 11 to
19 as composed of a ten and some ones). Through activities that build number sense, they understand the order of the counting numbers and
their relative magnitudes.
3. Developing understanding of linear measurement and measuring lengths as iterating length units.
Students develop an understanding of the meaning and processes of measurement, including underlying concepts such as
iterating (the mental activity of building up the length of an object with equal-sized units) and the transitivity principle for
indirect measurement.
4. Reasoning about attributes of, and composing and decomposing geometric shapes.
Students compose and decompose plane or solid figures (e.g., put two triangles together to make a quadrilateral) and build
understanding of part-whole relationships as well as the properties of the original and composite shapes. As they combine
shapes, they recognize them from different perspectives and orientations, describe their geometric attributes, and determine
how they are alike and different, to develop the background for measurement and for initial understandings of properties such
as congruence and symmetry.
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Operations and Algebraic Thinking
• Represent and solve problems involving addition and subtraction.
Use addition and subtraction within 20 to solve word problems involving situations of adding to, taking from, putting together,
taking apart, and comparing, with unknowns in all positions, e.g., by using objects, drawings, and equations with a symbol for the unknown
number to represent the problem.
Student Friendly/”I Can” statements
1. Use a symbol for an unknown number in
an addition or subtraction problem within
20
2. Add and subtract to solve word problems
within 20.
3. Interprets situations to solve word
problems with unknowns in all positions
within 20 using addition and subtraction
4. Determines appropriate representations
for solving word problems involving
different situations using addition and
subtraction
Resources
Assessments
http://nlvm.usu.edu
Solve word problems that call for addition of three whole numbers whose sum is less than or equal to 20, e.g., by using objects,
drawings, and equations with a symbol for the unknown number to represent the problem.
Student Friendly/”I Can” statements
1. Add three whole numbers whose sum is
less than or equal to 20.
2. Solve addition word problems that require
adding three whole numbers whose sum
is less than or equal to 20.
Resources
20
Assessments
• Understand and apply properties of operations and the relationship between addition and subtraction.
Apply properties of operations as strategies to add and subtract. Examples: If 8 + 3 = 11 is known, then 3 + 8 = 11 is also known.
(Commutative property of addition.) To add 2 + 6 + 4, the second two numbers can be added to make a ten, so 2 + 6 + 4 = 2 + 10 = 12.
(Associative property of addition.)
Student Friendly/”I Can” statements
1. Know the commutative property.
2. Know the associative property.
3. Understand subtraction as the unknown
addend.
4. Explain how properties of operation
strategies work.
5. Apply strategies using properties of
operations to solve addition and
subtraction problems
Resources
Assessments
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Understand subtraction as an unknown-addend problem. For example, subtract 10 – 8 by finding the number that makes 10 when added to
8.
1.
2.
3.
4.
Student Friendly/”I Can” statements
Identify the unknown in a subtraction
problem
Understand subtraction as the unknown
addend.
Solve subtraction problems to find the
missing addend.
Explain the relationship of addition and
subtraction.
Resources
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Assessments
• Add and subtract within 20.
Relate counting to addition and subtraction (e.g., by counting on 2 to add 2).
Student Friendly/”I Can” statements
1. Know how to count on and count back.
2. Explain how counting on and counting
back relate to addition and subtraction.
Resources
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Assessments
Add and subtract within 20, demonstrating fluency for addition and subtraction within 10. Use strategies such as counting on;
making ten (e.g., 8 + 6 = 8 + 2 + 4 = 10 + 4 = 14) ; decomposing a number leading to a ten (e.g., 13 – 4 = 13 – 3 – 1 = 10 – 1 = 9) ; using the
relationship between addition and subtraction (e.g., knowing that 8 + 4 = 12, one knows 12 – 8= 4) ; and creating equivalent but easier or
known sums (e.g., adding 6 +7 by creating the known equivalent 6 + 6 + 1 = 12 + 1 = 13).
Student Friendly/”I Can” statements
1. Add fluently within 10.
2. Subtract fluently within 10.
3. Apply strategies to add and subtract
within 20.
Resources
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22
Assessments
• Work with addition and subtraction equations.
Understand the meaning of the equal sign, and determine if equations involving addition and subtraction are true or false. For
example, which of the following equations are true and which are false? 6 = 6, 7 = 8 – 1, 5 + 2 = 2 + 5, 4 + 1 = 5 + 2.
Student Friendly/”I Can” statements
1. Explain the meaning of an equal sign (the
quantity on each side of the equality
symbol is the same).
2. Compare the values on each side of an
equal sign.
3. Determine if the equation is true or false.
Resources
Assessments
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Determine the unknown whole number in an addition or subtraction equation relating to three whole numbers. For example,
determine the unknown number that makes the equation true in each of the equations 8 +? = 11, 5 = � – 3, 6 + 6 = �.
Student Friendly/”I Can” statements
1. Recognize part-part-whole relationships
of three whole numbers
Example:
+5=8
5=
-3
In each instance the 3 and 5 represent
the parts and the 8 would be
representative of the whole.
2. Determine the missing value in an
addition or subtraction equation by using
a variety of strategies.
Resources
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Assessments
Number and Operations in Base Ten
• Extend the counting sequence.
Count to 120, starting at any number less than 120. In this range, read and write numerals and represent a number of objects with a
written numeral.
1.
2.
3.
4.
Student Friendly/”I Can” statements
Write numerals up to 120.
Represent a number of objects up to
120 with a written numeral.
Count (saying the number sequence) to
120, starting at any number less than
120
Read the numerals up to 120.
Resources
Assessments
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• Understand place value.
Understand that the two digits of a two-digit number represent amounts of tens and ones.
Student Friendly/”I Can” statements
Resources
1. Explain what each digit of a two-digit
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number represents
Understand the following as special cases:
a. 10 can be thought of as a bundle of ten ones — called a “ten.”
Student Friendly/”I Can” statements
Resources
1. Identify a bundle of 10 ones as a “ten”.
Assessments
Assessments
b. The numbers from 11 to 19 are composed of a ten and one, two, three, four, five, six, seven, eight, or nine ones.
Student Friendly/”I Can” statements
Resources
Assessments
1. Represent numbers 11 to 19 as
composed of a ten and correct
number of ones.
c. The numbers 10, 20, 30, 40, 50, 60, 70, 80, 90 refer to one, two, three, four, five, six, seven, eight, or nine tens (and 0 ones).
Student Friendly/”I Can” statements
1. Represent the numbers 20, 30, 40, 50, 60,
Resources
24
Assessments
70, 80, and 90 as composed of the correct
number of tens.
Compare two two-digit numbers based on meanings of the tens and ones digits, recording the results of comparisons with the
symbols >, =, and <.
Student Friendly/”I Can” statements
1. Identity the value of each digit
represented in the two-digit number.
2. Know what each symbol represents >, <,
and =.
3. Compare two two-digit numbers based on
meanings of the tens and ones digits.
Resources
4. Use >, =, and < symbols to record the
results of comparisons.
25
Assessments
• Use place value understanding and properties of operations to add and subtract.
Add within 100, including adding a two-digit number and a one-digit number, and adding a two-digit number and a multiple of 10,
using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between
addition and subtraction; relate the strategy to a written method and explain the reasoning used. Understand that in adding twodigit numbers, one adds tens and tens, ones and ones; and sometimes it is necessary to compose a ten.
1.
2.
3.
4.
Student Friendly/”I Can” statements
Identify the value of each digit of a
number within 100.
Decompose any number within one
hundred into ten(s) and one(s).
Choose an appropriate strategy for solving
an addition or subtraction problem within
100.
Relate the chosen strategy (using concrete
models or drawings and strategies based
on place value, properties of operations,
and/or the relationship between addition
and subtraction) to a written method
(equation) and explain the reasoning
used.
Resources
Assessments
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5. Use composition and decomposition
of tens when necessary to add and
subtract within 100.
Given a two-digit number, mentally find 10 more or 10 less than the number, without having to count; explain the reasoning used.
Student Friendly/”I Can” statements
1. Identify the value of each digit in a
number within 100.
2. Apply knowledge of place value to
mentally add or subtract 10 to/from a
given two digit number.
3. Explain how to mentally find 10 more or
Resources
26
Assessments
10 less than the given two-digit number.
Subtract multiples of 10 in the range 10-90 from multiples of 10 in the range 10-90 (positive or zero differences), using concrete models
or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and
subtraction; relate the strategy to a written method and explain the reasoning used.
Student Friendly/”I Can” statements
1. Identify the value of each digit of a
number within 100.
2. Subtract multiples of 10 in the range of
10-90 from multiples of 10 in the range of
10-90 (positive or zero differences).
3. Choose appropriate strategy (concrete
models or drawings and strategies based
on place value, properties of operations,
and/or the relationship between addition
and subtraction) for solving subtraction
problems with multiples of 10.
Resources
4. Relate the chosen strategy to a written
method (equation) and explain the
reasoning used.
27
Assessments
Measurement and Data
• Measure lengths indirectly and by iterating length units.
Order three objects by length; compare the lengths of two objects indirectly by using a third object.
1.
2.
3.
4.
Student Friendly/”I Can” statements
Identify the measurement known as the
length of an object
Directly compare the length of three
objects.
Order three objects by length
Compare the lengths of two objects
indirectly by using a third object to
compare them (e.g., if the length of object
A is greater than the length of object B,
and the length of object B is greater than
the length of object C, then the length of
object A is greater than the length of
object C.)
Resources
Assessments
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Express the length of an object as a whole number of length units, by laying multiple copies of a shorter object (the length unit) end to
end; understand that the length measurement of an object is the number of same-size length units that span it with no gaps or
overlaps. Limit to contexts where the object being measured is spanned by a whole number of length units with no gaps or overlaps.
1.
2.
3.
4.
Student Friendly/”I Can” statements
Knows to use the same size nonstandard objects as iterated (repeating)
units
Know that length can be measured with
various units
Compare a smaller unit of measurement
to a larger object
Determine the length of the measured
object to be the number of smaller
Resources
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Assessments
iterated (repeated) objects that equal its
length
5. Demonstrate the measurement of an
object using non-standard units (e.g.
paper clips, unifix cubes, etc.) by laying
the units of measurement end to end with
no gaps or overlaps
•Tell and write time.
Tell and write time in hours and half-hours using analog and digital clocks.
Student Friendly/”I Can” statements
1. Recognize that analog and digital clocks
are objects that measure time.
2. Know hour hand and minute hand and
distinguish between the two.
3. Determine where the minute hand must
be when the time is to the hour (o’clock).
4. Determine where the minute hand must
be when the time is to the half hour
(thirty).
5. Tell/Write the time to the hour and half
hour correctly using analog and digital
clocks – for instance when it is 3:30 the
hour hand is between the 3 and the 4; the
minute hand is on the 6.
Resources
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29
Assessments
•Represent and interpret data.
Organize, represent, and interpret data with up to three categories; ask and answer questions about the total number of data
points, how many in each category, and how many more or less are in one category than in another.
Student Friendly/”I Can” statements
Resources
1. Recognize different methods to organize http://nlvm.usu.edu
data
2. Recognize different methods to
represent data
3. Organize data with up to three
categories
4. Represent data with up to three
categories
5. Interpret data representation by asking
and answering questions about the data.
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Assessments
Geometry
• Reason with shapes and their attributes.
Distinguish between defining attributes (e.g., triangles are closed and three-sided) versus non-defining attributes (e.g., color, orientation,
overall size); build and draw shapes to possess defining attributes.
1.
2.
3.
4.
5.
Student Friendly/”I Can” statements
Identify defining attributes of shapes.
Identify non-defining attributes of
shapes.
Distinguish between (compare/contrast)
defining and non-defining attributes of
shapes.
Build shapes to show defining
attributes.
Draw shapes to show defining
attributes.
Resources
Assessments
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Compose two-dimensional shapes (rectangles, squares, trapezoids, triangles, half-circles, and quarter-circles) or three-dimensional
shapes (cubes, right rectangular prisms, right circular cones, and right circular cylinders) to create a composite shape, and compose
new shapes from the composite shape.
Student Friendly/”I Can” statements
1. Know that shapes can be composed and
decomposed to make new shapes
2. Describe properties of original and
composite shapes
3. Determine how the original and created
composite shapes are alike and different
4. Create composite shapes
5. Compose new shapes from a composite
shape
Resources
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Assessments
Partition circles and rectangles into two and four equal shares, describe the shares using the words halves, fourths, and quarters,
and use the phrases half of, fourth of, and quarter of. Describe the whole as two of, or four of the shares. Understand for these
examples that decomposing into more equal shares creates smaller shares.
1.
2.
3.
4.
Student Friendly/”I Can” statements
Identify when shares are equal
Identify two and four equal shares
Describe equal shares using vocabulary:
halves, fourths and quarters, half of,
fourth of, and quarter of
Describe the whole as two of two or four
of four equal shares
Resources
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Assessments
Standards for Mathematical Practice in Second Grade
The Common Core State Standards for Mathematical Practice are practices expected to be integrated into every mathematics lesson for all students
Grades K-12. Below are a few examples of how these Practices may be integrated into tasks that Grade 2 students complete.
Mathematical Practice Explanations and Examples
Mathematically proficient students in Second Grade examine problems and tasks, can make sense of the meaning of the
1) Make Sense and
Persevere in Solving
task and find an entry point or a way to start the task. Second Grade students also develop a foundation for problem solving
strategies and become independently proficient on using those strategies to solve new tasks. In Second Grade, students’
Problems.
work continues to use concrete manipulatives and pictorial representations as well as mental mathematics. Second Grade
students also are expected to persevere while solving tasks; that is, if students reach a point in which they are stuck, they
can reexamine the task in a different way and continue to solve the task. Lastly, mathematically proficient students
complete a task by asking themselves the question, “Does my answer make sense?”
2) Reason abstractly
Mathematically proficient students in Second Grade make sense of quantities and relationships while solving tasks. This
involves two processes- decontexualizing and contextualizing. In Second Grade, students represent situations by
and quantitatively.
decontextualizing tasks into numbers and symbols. For example, in the task, “There are 25 children in the cafeteria and
they are joined by 17 more children. How many students are in the cafeteria? ” Second Grade students translate that
situation into an equation, such as: 25 + 17 = __ and then solve the problem. Students also contextualize situations during
the problem solving process. For example, while solving the task above, students can refer to the context of the task to
determine that they need to subtract 19 since 19 children leave. The processes of reasoning also other areas of mathematics
such as determining the length of quantities when measuring with standard units.
3) Construct viable
Mathematically proficient students in Second Grade accurately use definitions and previously established solutions to
construct viable arguments about mathematics. During discussions about problem solving strategies, students
arguments and
critique the
constructively critique the strategies and reasoning of their classmates. For example, while solving 74 - 18, students may
use a variety of strategies, and after working on the task, can discuss and critique each others’ reasoning and strategies,
reasoning of others.
citing similarities and differences between strategies.
Mathematically proficient students in Second Grade model real-life mathematical situations with a number sentence or an
4) Model with
mathematics.
equation, and check to make sure that their equation accurately matches the problem context. Second Grade students use
concrete manipulatives and pictorial representations to provide further explanation of the equation. Likewise, Second Grade
students are able to create an appropriate problem situation from an equation. For example, students are expected to create
a story problem for the equation 43 + 17 = ___ such as “There were 43 gumballs in the machine. Tom poured in 17 more
gumballs. How many gumballs are now in the machine?”
Mathematically proficient students in Second Grade have access to and use tools appropriately. These tools may include
5) Use appropriate
tools strategically.
snap cubes, place value (base ten) blocks, hundreds number boards, number lines, rulers, and concrete geometric shapes
(e.g., pattern blocks, 3-d solids). Students also have experiences with educational technologies, such as calculators and
virtual manipulatives, which support conceptual understanding and higher-order thinking skills. During classroom
instruction, students have access to various mathematical tools as well as paper, and determine which tools are the most
appropriate to use. For example, while measuring the length of the hallway, students can explain why a yardstick is more
appropriate to use than a ruler.
6) Attend to
Mathematically proficient students in Second Grade are precise in their communication, calculations, and
33
precision.
7) Look for and
make use of
structure.
8) Look for and
express regularity in
repeated reasoning.
measurements. In all mathematical tasks, students in Second Grade communicate clearly, using grade-level appropriate
vocabulary accurately as well as giving precise explanations and reasoning regarding their process of finding solutions. For
example, while measuring an object, care is taken to line up the tool correctly in order to get an accurate measurement.
During tasks involving number sense, students consider if their answer is reasonable and check their work to ensure the
accuracy of solutions.
Mathematically proficient students in Second Grade carefully look for patterns and structures in the number system and
other areas of mathematics. For example, students notice number patterns within the tens place as they connect skip count
by 10s off the decade to the corresponding numbers on a 100s chart. While working in the Numbers in Base Ten domain,
students work with the idea that 10 ones equals a ten, and 10 tens equals 1 hundred. In addition, Second Grade students also
make use of structure when they work with subtraction as missing addend problems, such as 50- 33 = __ can be written as
33+ __ = 50 and can be thought of as,” How much more do I need to add to 33 to get to 50?”
Mathematically proficient students in Second Grade begin to look for regularity in problem structures when solving
mathematical tasks. For example, after solving two digit addition problems by decomposing numbers (33+ 25 = 30 + 20 +
3 +5), students may begin to generalize and frequently apply that strategy independently on future tasks. Further, students
begin to look for strategies to be more efficient in computations, including doubles strategies and making a ten. Lastly,
while solving all tasks, Second Grade students accurately check for the reasonableness of their solutions during and after
completing the task.
34
Grade 2 Critical Areas
The Critical Areas are designed to bring focus to the standards at each grade by describing the big ideas that
educators can use to build their curriculum and to guide instruction.
The Critical Areas for Second Grade can be found on page 17 in the Common Core State Standards for Mathematics.
1. Extending understanding of base-ten notation
Students extend their understanding of the base-ten system. This includes ideas of counting in fives, tens, and multiples of
hundreds, tens, and ones, as well as number relationships involving these units, including comparing. Students understand
multi-digit numbers (up to 1000) written in base-ten notation, recognizing that the digits in each place represent amounts of thousands,
hundreds, tens, or ones (e.g., 853 is 8 hundreds + 5 tens + 3 ones).
2. Building fluency with addition and subtraction.
Students use their understanding of addition to develop fluency with addition and subtraction within 100. They solve problems
within 1000 by applying their understanding of models for addition and subtraction, and they develop, discuss, and use
efficient, accurate, and generalizable methods to compute sums and differences of whole numbers in base-ten notation, using
their understanding of place value and the properties of operations. They select and accurately apply methods that are
appropriate for the context and the numbers involved to mentally calculate sums and differences for numbers with only tens or
only hundreds.
3. Using standard units of measure.
Students recognize the need for standard units of measure (centimeter and inch) and they use rulers and other measurement
tools with the understanding that linear measure involves an iteration of units. They recognize that the smaller the unit, the
more iterations they need to cover a given length.
4. Describing and analyzing shapes.
Students describe and analyze shapes by examining their sides and angles. Students investigate, describe, and reason about
decomposing and combining shapes to make other shapes. Through building, drawing, and analyzing two- and three-dimensional shapes,
students develop a foundation for understanding area, volume, congruence, similarity, and symmetry in
later grades.
35
Operations and Algebraic Thinking
• Represent and solve problems involving addition and subtraction.
Use addition and subtraction within 100 to solve one- and two-step word problems involving situations of adding to, taking from,
putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using drawings and equations with a symbol
for the unknown number to represent the problem.
1.
2.
3.
4.
5.
6.
Student Friendly/”I Can” statements
Identify the unknown in an addition or
subtraction word problem
Write an addition and subtraction
equation with a symbol for the unknown
Use drawings or equations to represent
one- and two-step word problems
Add and subtract within 100 to solve onestep word problems with unknowns in all
positions
Add and subtract within 100 to solve twostep word problems with unknowns in all
positions
Determine operation needed to solve
addition and subtraction problems in
situations including add to, take from, put
together, take apart, and compare
Resources
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36
Assessments
• Add and subtract within 20.
Fluently add and subtract within 20 using mental strategies. By end of Grade 2, know from memory all sums of two one-digit
numbers.
Student Friendly/”I Can” statements
1. Know mental strategies for addition and
subtraction
2. Know from memory all sums of two onedigit numbers
3. Apply mental strategies to add and
subtract fluently within 20.
Resources
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37
Assessments
• Work with equal groups of objects to gain foundations for multiplication.
Determine whether a group of objects (up to 20) has an odd or even number of members, e.g., by pairing objects or counting them
by 2s; write an equation to express an even number as a sum of two equal addends.
Student Friendly/”I Can” statements
1. Count a group of objects up to 20 by 2s.
2. Recognize in groups that have even
numbers objects will pair up evenly.
3. Recognize in groups of odd numbers
objects will not pair up evenly.
4. Determine whether a group of objects is
odd or even, using a variety of strategies.
5. Generalize the fact that all even numbers
can be formed from the addition of 2
equal addends.
6. Write an equation to express a given even
number as a sum of two equal addends.
Resources
Assessments
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Use addition to find the total number of objects arranged in rectangular arrays with up to 5 rows and up to 5 columns; write an
equation to express the total as a sum of equal addends.
Student Friendly/”I Can” statements
1. Write an equation with repeated equal
addends from an array.
2. Generalize the fact that arrays can be
written as repeated addition problems.
3. Solve repeated addition problems to find
the number of objects using rectangular
arrays.
Resources
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Assessments
Number and Operations in Base Ten
• Understand place value.
Understand that the three digits of a three-digit number represent amounts of hundreds, tens, and ones; e.g., 706 equals 7
hundreds, 0 tens, and 6 ones.
Student Friendly/”I Can” statements
1. Identify the ones, tens, and hundreds
place.
2. Regroup ten ones into the tens places
value.
3. Regroup ten tens into the hundreds place
value.
Resources
Assessments
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4. Explain the value of each digit in a 3digit number.
Understand the following as special cases:
a. 100 can be thought of as a bundle of ten tens — called a “hundred.”
Student Friendly/”I Can” statements
1. Identify a bundle of 10 tens as a
“hundred.”
Resources
Assessments
b. The numbers 100, 200, 300, 400, 500, 600, 700, 800, 900 refer to one, two, three, four, five, six, seven, eight, or nine hundreds
(and 0 tens and 0 ones).
Student Friendly/”I Can” statements
1. Represents a three digit number with
hundreds, tens, and ones.
Resources
2. Represent 200, 300, 400, 500, 600,
700, 800, 900 with one, two, three,
four, five, six, seven, eight, or nine
hundreds and 0 tens and 0 ones
39
Assessments
Count within 1000; skip-count by 5s, 10s, and 100s
Student Friendly/”I Can” statements
Resources
Assessments
1. Count to 1000 by hundreds.
2. Count to 1000 by tens.
3. Count to 1000 by fives.
Read and write numbers to 1000 using base-ten numerals, number names, and expanded form.
1.
2.
3.
4.
5.
6.
7.
8.
Student Friendly/”I Can” statements
Know what expanded form means.
Recognize that the digits in each place
represent amounts of thousands,
hundreds, tens, or ones.
Read numbers to 1000 using base ten
numerals.
Write numbers to 1000 using base ten
numerals.
Read numbers to 1000 using number
names.
Write numbers to 1000 using number
names.
Read numbers to 1000 using expanded
form.
Write numbers to 1000 using expanded
form.
Resources
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Assessments
Compare two three-digit numbers based on meanings of the hundreds, tens, and ones digits, using >, =, and < symbols to record the
results of comparisons.
Student Friendly/”I Can” statements
1. Understand the meaning of the hundreds,
tens, and ones place values.
2. Identify the meaning of comparison
symbols, >, < or =.
3. Record comparisons of numbers up to
three digits using the appropriate symbol.
Resources
Assessments
4. Compare two three-digit numbers
based on place value of each digit.
• Use place value understanding and properties of operations to add and subtract.
Fluently add and subtract within 100 using strategies based on place value, properties of operations, and/or the relationship
between addition and subtraction.
Student Friendly/”I Can” statements
1. Know strategies for adding and
subtracting based on place value.
2. Know strategies for adding and
subtracting based on properties of
operations.
3. Know strategies for adding and
subtracting based on the relationship
between addition and subtraction.
Resources
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4. Choose a strategy (place value,
properties of operations, and /or the
relationship between addition and
subtraction) to fluently add and
subtract within 100.
41
Assessments
Add up to four two-digit numbers using strategies based on place value and properties of operations.
Student Friendly/”I Can” statements
1. Know strategies for adding two digit
numbers based on place value and
properties of operations.
2. Use strategies to add up to four two-digit
numbers.
Resources
Assessments
Add and subtract within 1000, using concrete models or drawings and strategies based on place value, properties of operations,
and/or the relationship between addition and subtraction; relate the strategy to a written method. Understand that in adding or
subtracting three digit numbers, one adds or subtracts hundreds and hundreds, tens and tens, ones and ones; and sometimes it is
necessary to compose or decompose tens or hundreds.
1.
2.
3.
4.
5.
Student Friendly/”I Can” statements
Understand place value within 1000.
Decompose any number within 1000 into
hundred(s), ten(s), and one(s).
Choose an appropriate strategy for solving
an addition or subtraction problem within
1000.
Relate the chosen strategy (using concrete
models or drawings and strategies based
on place value, properties of operations,
and/or the relationship between addition
and subtraction) to a written method
(equation) and explain the reasoning
used.
Use composition and decomposition of
hundreds and tens when necessary to add
and subtract within 1000.
Resources
42
Assessments
Mentally add 10 or 100 to a given number 100–900, and mentally subtract 10 or 100 from a given number 100–900.
Student Friendly/”I Can” statements
1. Know place value within 1000.
2. Apply knowledge of place value to
mentally add or subtract 10 or 100
to/from a given number 100-900.
Resources
Assessments
Explain why addition and subtraction strategies work, using place value and the properties of operations.
Student Friendly/”I Can” statements
1. Know addition and subtraction strategies
using place value and properties of
operations related to addition and
subtraction.
2. Explain why addition and subtraction
strategies based on place value and
properties of operations work.
Resources
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Assessments
Measurement and Data
• Measure and estimate lengths in standard units.
Measure the length of an object by selecting and using appropriate tools such as rulers, yardsticks, meter sticks, and measuring
tapes.
1.
2.
3.
4.
Student Friendly/”I Can” statements
Identify tools that can be used to
measure length.
Identify the unit of length for the tool
used (inches, centimeters, feet, meters).
Determine which tool to use to measure
the length of an object.
Measure the length of objects by using
appropriate tools.
Resources
Assessments
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Measure the length of an object twice, using length units of different lengths for the two measurements; describe how the two
measurements relate to the size of the unit chosen.
Student Friendly/”I Can” statements
1. Know how to measure the length of
objects with different units.
2. Compare measurements of an object
taken with two different units.
3. Describe why the measurements of an
object taken with two different units are
different.
4. Explain the length of an object in
relation to the size of the units used to
measure it.
Resources
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Assessments
Estimate lengths using units of inches, feet, centimeters, and meters.
1.
2.
3.
4.
Student Friendly/”I Can” statements
Know strategies for estimating length.
Recognize the size of inches, feet,
centimeters, and meters.
Estimate lengths in units of inches, feet,
centimeters, and meters.
Determine if estimate is reasonable.
Resources
Assessments
Measure to determine how much longer one object is than another, expressing the length difference in terms of a standard length
unit.
Student Friendly/”I Can” statements
1. Name standard length units.
2. Compare lengths of two objects.
3. Determine how much longer one object
is than another in standard length units.
Resources
Assessments
• Relate addition and subtraction to length.
Use addition and subtraction within 100 to solve word problems involving lengths that are given in the same units, e.g., by using
drawings (such as drawings of rulers) and equations with a symbol for the unknown number to represent the problem.
Student Friendly/”I Can” statements
1. Add and subtract lengths within 100.
2. Solve word problems involving lengths
that are given in the same units.
3. Solve word problems involving length
that have equations with a symbol for
the unknown number.
Resources
http://nlvm.usu.edu
45
Assessments
Represent whole numbers as lengths from 0 on a number line diagram with equally spaced points corresponding to the numbers 0,
1, 2, ..., and represent whole-number sums and differences within 100 on a number line diagram.
Student Friendly/”I Can” statements
1. Represent whole numbers from 0 on a
number line with equally spaced points.
2. Explain length as the distance between
zero and another mark on the number
line diagram.
3. Use a number line to represent the
solution of whole-number sums and
differences related to length within 100.
Resources
Assessments
• Work with time and money.
Tell and write time from analog and digital clocks to the nearest five minutes, using a.m. and p.m.
1.
2.
3.
4.
5.
6.
Student Friendly/”I Can” statements
Tell time using analog clocks to the
nearest 5 minutes
Tell time using digital clocks to the nearest
5 minutes
Write time using analog clocks and digital
clocks
Identify the hour and minute hand on an
analog clock
Identify and label when a.m. and p.m.
occur
Determine what time is represented by
the combination of the number on the
clock face and the position of the hands.
Resources
Assessments
http://nlvm.usu.edu
Touch math money
Everyday Math
Study Island
Solve word problems involving dollar bills, quarters, dimes, nickels, and pennies, using $ and ¢ symbols appropriately. Example: If
you have 2 dimes and 3 pennies, how many cents do you have?
46
Student Friendly/”I Can” statements
1. Identify and recognize the value of dollar
bills, quarters, dimes, nickels, and
pennies.
2. Identify the $ and ¢ symbol.
3. Solve word problems involving dollar bills,
quarters, dimes, nickels, and pennies
using $ and ¢ symbols appropriately.
Resources
Assessments
• Represent and interpret data objects and count the number of objects in categories.
Generate measurement data by measuring lengths of several objects to the nearest whole unit, or by making repeated
measurements of the same object. Show the measurements by making a line plot, where the horizontal scale is marked off in wholenumber units.
1.
2.
3.
4.
5.
Student Friendly/”I Can” statements
Read tools of measurement to the nearest
unit.
Represent measurement data on a line
plot.
Measure lengths of several objects to the
nearest whole unit.
Measure lengths of objects by making
repeated measurements of the same
object.
Create a line plot with a horizontal scale
marked in whole numbers using
measurements.
Resources
http://nlvm.usu.edu
47
Assessments
Draw a picture graph and a bar graph (with single-unit scale) to represent a data set with up to four categories. Solve simple puttogether, take-apart, and compare problems4 using information presented in a bar graph.
1.
2.
3.
4.
5.
6.
Student Friendly/”I Can” statements
Recognize and Identify picture graphs and
bar graphs.
Identify and label the components of a
picture graph and bar graph.
Solve problems relating to data in graphs
by using addition and subtraction
Make comparisons between categories in
the graph using more than, less than, etc.
Draw a single-unit scale picture graph to
represent a given set of data with up to
four categories
Draw a single-unit scale bar graph to
represent a given set of data with up to
four categories
Resources
48
Assessments
Geometry
• Reason with shapes and their attributes.
Recognize and draw shapes having specified attributes, such as a given number of angles or a given number of equal faces. Identify
triangles, quadrilaterals, pentagons, hexagons, and cubes.
1.
2.
3.
4.
5.
Student Friendly/”I Can” statements
Identify the attributes of triangles,
quadrilaterals, pentagons, hexagons,
and cubes (e.g. faces, angles, sides,
vertices, etc).
Identify triangles, quadrilaterals,
pentagons, hexagons, and cubes based
on the given attributes.
Describe and analyze shapes by examining
their sides and angles, not by measuring.
Compare shapes by their attributes (e.g.
faces, angles).
Draw shapes with specified attributes
Resources
Assessments
http://nlvm.usu.edu
Partition a rectangle into rows and columns of same-size squares and count to find the total number of them.
1.
2.
3.
4.
5.
Student Friendly/”I Can” statements
Counts to find the total number of
same-size squares.
Defines partition.
Identify a row.
Identify a column.
Determines how to partition a rectangle
into same-size squares.
Resources
49
Assessments
Partition circles and rectangles into two, three, or four equal shares, describe the shares using the words halves, thirds, half of, a
third of, etc., and describe the whole as two halves, three thirds, four fourths. Recognize that equal shares of identical wholes need
not have the same shape.
1.
2.
3.
4.
Student Friendly/”I Can” statements
Identify two , three and four equal
shares of a whole
Describe equal shares using vocabulary:
halves, thirds, fourths half of, third of
etc.
Describe the whole as two halves , three
thirds, or four fourths
Justify why equal shares of identical
wholes need not have the same shape.
Resources
50
Assessments
Standards for Mathematical Practices – Grade 3
The Common Core State Standards for Mathematical Practice are expected to be integrated into every mathematics lesson for all students Grades K-12.
Below are a few examples of how these Practices may be integrated into tasks that students complete.
Mathematic Practices
Explanations and Examples
In third grade, mathematically proficient students know that doing mathematics involves solving problems and discussing
1. Make sense of problems
how they solved them. Students explain to themselves the meaning of a problem and look for ways to solve it. Third graders
and persevere in solving
may use concrete objects or pictures to help them conceptualize and solve problems. They may check their thinking by
them.
asking themselves, “Does this make sense?” They listen to the strategies of others and will try different approaches. They
often will use another method to check their answers.
Mathematically proficient third graders should recognize that a number represents a specific quantity. They connect the
2. Reason abstractly and
quantity to written symbols and create a logical representation of the problem at hand, considering both the appropriate units
quantitatively.
involved and the meaning of quantities.
In third grade, mathematically proficient students may construct arguments using concrete referents, such as objects,
3. Construct viable
arguments and critique the pictures, and drawings. They refine their mathematical communication skills as they participate in mathematical discussions
involving questions like “How did you get that?” and “Why is that true?” They explain their thinking to others and respond
reasoning of others.
to others’ thinking
4. Model with mathematics. Mathematically proficient students experiment with representing problem situations in multiple ways including
numbers, words (mathematical language), drawing pictures, using objects, acting out, making a chart, list, or graph, creating
equations, etc. Students need opportunities to connect the different representations and explain the connections. They should
be able to use all of these representations as needed. Third graders should evaluate their results in the context of the situation
and reflect on whether the results make sense.
Mathematically proficient third graders consider the available tools (including estimation) when solving a
5. Use appropriate tools
mathematical problem and decide when certain tools might be helpful. For instance, they may use graph paper to find all the
strategically.
possible rectangles that have a given perimeter. They compile the possibilities into an organized list or a table, and determine
whether they have all the possible rectangles.
Mathematically proficient third graders develop their mathematical communication skills, they try to use clear and
6. Attend to precision.
precise language in their discussions with others and in their own reasoning. They are careful about specifying units of
measure and state the meaning of the symbols they choose. For instance, when figuring out the area of a rectangle they
record their answers in square units.
In third grade mathematically proficient students look closely to discover a pattern or structure. For instance, students use
7. Look for and make use
properties of operations as strategies to multiply and divide (commutative and distributive properties).
of
structure.
Mathematically proficient students in third grade should notice repetitive actions in computation and look for more
8. Look for and express
shortcut methods. For example, students may use the distributive property as a strategy for using products they know to solve
regularity in repeated
products that they don’t know. For example, if students are asked to find the product of 7 x 8, they might decompose 7 into 5
reasoning.
and 2 and then multiply 5 x 8 and 2 x 8 to arrive at 40 + 16 or 56. In addition, third graders continually evaluate their work
by asking themselves, “Does this make sense?”
51
Grade 3 Critical Areas
The Critical Areas are designed to bring focus to the standards at each grade by describing the big ideas that educators can use
to build their curriculum and to guide instruction.
The Critical Areas for third grade can be found on page 21 in the Common Core State Standards for Mathematics.
1. Developing understanding of multiplication and division and strategies for multiplication and division within 100.
Students develop an understanding of the meanings of multiplication and division of whole numbers through activities and problems
involving equal-sized groups, arrays, and area models; multiplication is finding an unknown product, and division is finding an unknown
factor in these situations. For equal-sized group situations, division can require finding the unknown number of groups or the unknown
group size. Students use properties of operations to calculate products of whole numbers, using increasingly sophisticated strategies based on these
properties to solve multiplication and division problems involving single-digit factors. By comparing a variety of solution
strategies, students learn the relationship between multiplication and division.
2. Developing understanding of fractions, especially unit fractions (fractions with numerator 1).
Students develop an understanding of fractions, beginning with unit fractions. Students view fractions in general as being built out of unit
fractions, and they use fractions along with visual fraction models to represent parts of a whole. Students understand that the size of a
fractional part is relative to the size of the whole. For example, 1/2 of the paint in a small bucket could be less paint than 1/3 of the paint
in a larger bucket, but 1/3 of a ribbon is longer than 1/5 of the same ribbon because when the ribbon is divided into 3 equal parts, the parts
are longer than when the ribbon is divided into 5 equal parts. Students are able to use fractions to represent numbers equal to, less than,
and greater than one. They solve problems that involve comparing fractions by using visual fraction models and strategies based on
noticing equal numerators or denominators.
3. Developing understanding of the structure of rectangular arrays and of area.
Students recognize area as an attribute of two-dimensional regions. They measure the area of a shape by finding the total number of same size units of
area required to cover the shape without gaps or overlaps, a square with sides of unit length being the standard unit for
measuring area. Students understand that rectangular arrays can be decomposed into identical rows or into identical columns. By
decomposing rectangles into rectangular arrays of squares, students connect area to multiplication, and justify using multiplication to
determine the area of a rectangle.
4. Describing and analyzing two-dimensional shapes.
Students describe, analyze, and compare properties of two-dimensional shapes. They compare and classify shapes by their sides and
angles, and connect these with definitions of shapes. Students also relate their fraction work to geometry by expressing the area of part of
a shape as a unit fraction of the whole.
52
Operations and Algebraic Thinking
• Represent and solve problems involving multiplication and division.
Interpret products of whole numbers, e.g., interpret 5 × 7 as the total number of objects in 5 groups of 7 objects each. For example,
describe a context in which a total number of objects can be expressed as 5 × 7.
Student Friendly/”I Can” statements
1. Define product.
2. Construct multiplication problems using
manipulatives.
3. Identify the difference between 3 x 2 and
2 x 3 to define Commutative property.
4. Create sentences or phrases that
demonstrate understanding of products
of whole numbers that identify with real
world connections.
Resources
http://nlvm.usu.edu
Cuisennaire rods
Ten Base Blocks
Graph paper for drawing out arrays
Sets of items such as paper clips, erasers,
beans, etc.
Blank paper to create visual dictionary.
Assessments
Performance task with rubric
Create a multiplication word story with
detailed solution.
Ongoing visual dictionary with rubric.
Interpret whole-number quotients of whole numbers, e.g., interpret 56 ÷ 8 as the number of objects in each share when 56 objects
are partitioned equally into 8 shares, or as a number of shares when 56 objects are partitioned into equal shares of 8 objects each.
For example, describe a context in which a number of shares or a number of groups can be expressed as 56 ÷ 8.
Student Friendly/”I Can” statements
1. Define quotient.
2. Perform the operation of division on
whole numbers to determine quotient.
3. Demonstrate an understanding of whole
numbers being partitioned into equal
parts to determine quotient.
Resources
http://nlvm.usu.edu
Cuisennaire rods
Ten Base Blocks
Graph paper for drawing out arrays
Sets of items such as paper clips, erasers,
beans, etc.
Blank paper for visual dictionary
53
Assessments
Performance task with rubric
Create a division word story with detailed
solution
Ongoing visual dictionary with rubric
Use multiplication and division within 100 to solve word problems in situations involving equal groups, arrays, and measurement
quantities, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem.
Student Friendly/”I Can” statements
1. Identify signal words for operations in
word problems.
2. Break apart word problem by drawing out
each sentence.
3. Perform correct operations to solve word
problems.
Resources
http://nlvm.usu.edu
Cuisennaire rods
Ten Base Blocks
Graph paper for drawing out arrays
Sets of items such as paper clips, erasers,
beans, etc.
Blank paper for signal word list and drawing
out problems.
Assessments
Standardized test open items with rubric
Determine the unknown whole number in a multiplication or division equation relating three whole numbers. For example,
determine the unknown number that makes the equation true in each of the equations 8 × ? = 48, 5 = � ÷ 3, 6 × 6 = ?.
Student Friendly/”I Can” statements
1. Demonstrate knowledge of multiplication
facts and understand that division is the
inverse operation.
2. Identify the missing number in
multiplication and division problems.
Resources
http://nlvm.usu.edu
Cuisennaire rods
Ten Base Blocks
Graph paper for drawing out arrays
Sets of items such as paper clips, erasers,
beans, etc.
Assessments
• Understand properties of multiplication and the relationship between multiplication and division.
Apply properties of operations as strategies to multiply and divide. Examples: If 6 × 4 = 24 is known, then 4 × 6 = 24 is also known.
(Commutative property of multiplication.) 3 × 5 × 2 can be found by 3 × 5 = 15, then 15 × 2 = 30, or by 5 × 2 = 10, then 3 × 10 = 30.
(Associative property of multiplication.) Knowing that 8 × 5 = 40 and 8 × 2 = 16, one can find 8 × 7 as 8 × (5 + 2) = (8 × 5) + (8 × 2) = 40
+ 16 = 56. (Distributive property.)
Student Friendly/”I Can” statements
1. Define, understand, and apply
Commutative property of multiplication.
2. Define, understand, and apply Associative
property of multiplication.
3. Define, understand, and apply Distributive
property of multiplication.
Resources
http://nlvm.usu.edu
Cuisennaire rods
Ten Base Blocks
Graph paper for drawing out arrays
Sets of items such as paper clips, erasers,
beans, etc.
54
Assessments
Understand division as an unknown-factor problem. For example, find 32 ÷ 8 by finding the number that makes 32 when multiplied
by 8.
Student Friendly/”I Can” statements
1. Demonstrate knowledge of multiplication
facts and understand that division is the
inverse operation.
2. Identify the missing number in
multiplication and division problems.
Resources
http://nlvm.usu.edu
Cuisennaire rods
Ten Base Blocks
Graph paper for drawing out arrays
Sets of items such as paper clips, erasers,
beans, etc.
Assessments
• Multiply and divide within 100.
Fluently multiply and divide within 100, using strategies such as the relationship between multiplication and division (e.g., knowing
that 8 × 5 = 40, one knows 40 ÷ 5 = 8) or properties of operations. By the end of Grade 3, know from memory all products of two
one-digit numbers.
Student Friendly/”I Can” statements
1. Multiply and divide 40 problems in one
minute.
Resources
http://nlvm.usu.edu
Assessments
One or two minute timed multiplication and
division timed tests
55
• Solve problems involving the four operations, and identify and explain patterns in arithmetic.
Solve two-step word problems using the four operations. Represent these problems using equations with a letter standing for the
unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding.
Student Friendly/”I Can” statements
1. Write equations as a plan to solve word
problems.
2. Know that letters are variables that stand
for numbers.
3. Solve equation with appropriate
operations.
4. Check reasonable of answer by
estimating, mentally computing, and/or
rounding.
Resources
http://nlvm.usu.edu
Assessments
Develop a problem solving plan, write a two
step word problem, explain in detail the
solution with rubric.
Identify arithmetic patterns (including patterns in the addition table or multiplication table), and explain them using properties of
operations. For example, observe that 4 times a number is always even, and explain why 4 times a number can be decomposed into
two equal addends.
Student Friendly/”I Can” statements
1. Identify patterns in tables.
2. Explain the pattern using the rule with the
properties of operations.
Resources
Addition tables
Multiplication tables
Input/output tables
56
Assessments
Number and Operations in Base Ten
• Use place value understanding and properties of operations to perform multi-digit arithmetic.
Use place value understanding to round whole numbers to the nearest 10 or 100.
Student Friendly/”I Can” statements
1. Understand that ten ones equal ten and
ten tens equal one hundred, etc.
2. Understand the position of numbers is
important to naming the value of the
number.
Resources
http://nlvm.usu.edu
Cuisennaire rods
Ten Base Blocks
Sets of items such as paper clips, erasers,
beans, etc.
Assessments
Fluently add and subtract within 1000 using strategies and algorithms based on place value, properties of operations, and/or the
relationship between addition and subtraction.
Student Friendly/”I Can” statements
1. Add and subtract 40 problems in one
minute for demonstrating fluency.
2. Demonstrate which addends equal 10,
100, and 1000.
3. Know that addition and subtraction are
inverse operations.
Resources
http://nlvm.usu.edu
Cuisennaire rods
Ten Base Blocks
Sets of items such as paper clips, erasers,
beans, etc.
Assessments
Addition and subtraction timed tests
Multiply one-digit whole numbers by multiples of 10 in the range 10–90 (e.g., 9 × 80, 5 × 60) using strategies based on place value
and properties of operations.
Student Friendly/”I Can” statements
1. Multiply one-digit whole numbers by
multiples of 10.
Resources
57
Assessments
Number and Operations—Fractions
• Develop understanding of fractions as numbers.
Understand a fraction 1/b as the quantity formed by 1 part when a whole is partitioned into b equal parts; understand a fraction a/b
as the quantity formed by a parts of size 1/b.
Student Friendly/”I Can” statements
1. Divide a whole into equal and even parts.
2. Name the parts of the whole by counting
the number of the equal pieces that make
up the whole and identifying it as 1/
(number of equal parts).
Resources
Assessments
http://nlvm.usu.edu
fraction bars
graph paper
Understand a fraction as a number on the number line; represent fractions on a number line diagram.
a. Represent a fraction 1/b on a number line diagram by defining the interval from 0 to 1 as the whole and partitioning it into b
equal parts. Recognize that each part has size 1/b and that the endpoint of the part based at 0 locates the number 1/b on the
number line. b. Represent a fraction a/b on a number line diagram by marking off a lengths 1/b from 0. Recognize that the resulting
interval has size a/b and that its endpoint locates the number a/b on the number line.
Student Friendly/”I Can” statements
1. Place simple fractions on a number line.
2. Know that the denominator represents
the number of lines from 0 to 1 on the
number line.
Resources
http://nlvm.usu.edu
fraction bars
graph paper
clothes line or similar rope
clothes pins
number lines
58
Assessments
Explain equivalence of fractions in special cases, and compare fractions by reasoning about their size.
a. Understand two fractions as equivalent (equal) if they are the same size, or the same point on a number line.
Student Friendly/”I Can” statements
1. Identify when two fractions are equivalent
because they are the same size.
2. Identify when two fractions are equivalent
because they are the same point on the
number line.
Resources
http://nlvm.usu.edu
fraction bars
graph paper
clothes line or similar rope
clothes pins
number lines
Assessments
b. Recognize and generate simple equivalent fractions, e.g., 1/2 = 2/4, 4/6 = 2/3). Explain why the fractions are equivalent, e.g., by
using a visual fraction model.
Student Friendly/”I Can” statements
1. Create equivalent fractions with different
denominators.
2. Explain why two fractions are equivalent.
Resources
http://nlvm.usu.edu
fraction bars
graph paper
clothes line or similar rope
clothes pins
number lines
Assessments
c. Express whole numbers as fractions, and recognize fractions that are equivalent to whole numbers. Examples: Express 3 in the
form 3 = 3/1; recognize that 6/1 = 6; locate 4/4 and 1 at the same point of a number line diagram.
Student Friendly/”I Can” statements
1. Know that a whole number can be
represented by the number over one.
2. Know that a whole number over itself
represents one.
Resources
http://nlvm.usu.edu
fraction bars
graph paper
clothes line or similar rope
clothes pins
number lines
plain paper to draw out wholes
59
Assessments
d. Compare two fractions with the same numerator or the same denominator by reasoning about their size. Recognize that
comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with the symbols >, =,
or <, and justify the conclusions, e.g., by using a visual fraction model.
Student Friendly/”I Can” statements
1. Compare two fractions with the same
denominator and explain which is
larger/smaller.
2. Compare two fractions with the same
numerators and explain which is
larger/smaller.
3. Explain which comparisons were easier to
do by using visual models.
Resources
http://nlvm.usu.edu
fraction bars
graph paper
clothes line or similar rope
clothes pins
number lines
plain paper
60
Assessments
Fraction book project with rubric
Measurement and Data
• Solve problems involving measurement and estimation of intervals of time, liquid volumes, and masses of
objects.
Tell and write time to the nearest minute and measure time intervals in minutes. Solve word problems involving addition and
subtraction of time intervals in minutes, e.g., by representing the problem on a number line diagram.
Student Friendly/”I Can” statements
1. Tell time to the nearest minute.
2. Add and subtract time intervals in
minutes.
3. Solve time word problems.
4. Graph time intervals on a number line (xaxis).
Resources
Assessments
http://nlvm.usu.edu
analog clock
number lines
graph paper
Measure and estimate liquid volumes and masses of objects using standard units of grams (g), kilograms (kg), and liters (l). Add,
subtract, multiply, or divide to solve one-step word problems involving masses or volumes that are given in the same units, e.g., by
using drawings (such as a beaker with a measurement scale) to represent the problem.
Student Friendly/”I Can” statements
1. Measure liquids to the nearest liter.
2. Measure mass to the nearest gram and
kilogram.
3. Estimate liquids to the nearest liter.
4. Estimate mass to the nearest gram or
kilogram.
5. Add and subtract same measure word
problems involving liquids and mass using
models or drawings.
6. Multiply and divide same measure word
problems involving mass and liquids using
models or drawings.
Resources
Scales
Grams and kilogram weights
Liter bottles
Paper for drawing out problems
61
Assessments
• Represent and interpret data.
Draw a scaled picture graph and a scaled bar graph to represent a data set with several categories. Solve one- and two-step “how
many more” and “how many less” problems using information presented in scaled bar graphs. For example, draw a bar graph in
which each square in the bar graph might represent 5 pets
Student Friendly/”I Can” statements
1. Create a picture graph for data collected
that represents several categories with
scales that represent more than one unit.
2. Create a bar graph for data collected that
represents several categories with scales
that represent more than one unit.
3. Analyze and interpret graphs to solve oneand two-step problems.
Resources
Assessments
http://nlvm.usu.edu
graph paper
pictures for pictographs
Excel spreadsheet
Colored pencils
Generate measurement data by measuring lengths using rulers marked with halves and fourths of an inch. Show the data by making
a line plot, where the horizontal scale is marked off in appropriate units— whole numbers, halves, or quarters.
Student Friendly/”I Can” statements
1. Use rulers to measure objects to the
nearest half and quarter of an inch.
2. Graph measurement data on a line plot
with appropriate measures.
Number lines
Rulers
Various objects
Resources
Assessments
• Geometric measurement: understand concepts of area and relate area to multiplication and to addition.
Recognize area as an attribute of plane figures and understand concepts of area measurement.
a. A square with side length 1 unit, called “a unit square,” is said to have “one square unit” of area, and can be used to measure
area.
Student Friendly/”I Can” statements
1. Recognize one square unit of area as a 1 x
1 array.
Resources
http://nlvm.usu.edu
squares of various sizes
graph paper
62
Assessments
b. A plane figure which can be covered without gaps or overlaps by n unit squares is said to have an area of n square units.
Student Friendly/”I Can” statements
1. Measure and calculate area based on one
square unit.
2. Draw various areas by using arrays.
Resources
http://nlvm.usu.edu
squares of various sizes
graph paper
Assessments
Measure areas by counting unit squares (square cm, square m, square in, square ft, and improvised units).
Student Friendly/”I Can” statements
1. Label areas with the appropriate label.
Resources
http://nlvm.usu.edu
squares of various sizes
graph paper
Assessments
Relate area to the operations of multiplication and addition.
a. Find the area of a rectangle with whole-number side lengths by tiling it, and show that the area is the same as would be found by
multiplying the side lengths.
Student Friendly/”I Can” statements
1. Find areas using models.
Resources
http://nlvm.usu.edu
tiles or centimeter blocks
graph paper
Assessments
b. Multiply side lengths to find areas of rectangles with whole number side lengths in the context of solving real world and
mathematical problems, and represent whole-number products as rectangular areas in mathematical reasoning.
Student Friendly/”I Can” statements
1. Solve real world problems of area by using
arrays, multiplication, and models.
2. Understand that areas of rectangles can
be solved with multiplying the side
lengths.
Resources
http://nlvm.usu.edu
tiles or centimeter blocks
graph paper
63
Assessments
c. Use tiling to show in a concrete case that the area of a rectangle with whole-number side lengths a and b + c is the sum of a × b
and a × c. Use area models to represent the distributive property in mathematical reasoning.
Student Friendly/”I Can” statements
1. Use models to show how to use the
distributive property to find the area of a
rectangle.
Resources
http://nlvm.usu.edu
tiles or centimeter blocks
graph paper
Assessments
d. Recognize area as additive. Find areas of rectilinear figures by decomposing them into non-overlapping rectangles and adding the
areas of the non-overlapping parts, applying this technique to solve real world problems.
Student Friendly/”I Can” statements
1. Find areas of multiple rectangular figures
by separating the rectangles and add the
areas of the individual rectangles to get
the area of the entire figure.
Resources
http://nlvm.usu.edu
tiles or centimeter blocks
graph paper
Assessments
• Geometric measurement: recognize perimeter as an attribute of plane figures and distinguish between linear
and area measures.
Solve real world and mathematical problems involving perimeter of polygons, including finding the perimeter given the side lengths,
finding an unknown side length, and exhibiting rectangles with the same perimeter and different areas or with the same area and
different perimeters.
1.
2.
3.
4.
5.
Student Friendly/”I Can” statements
Find the perimeter of a rectangle by
deconstructing the rectangle into a
continuous line to understand that a
perimeter is linear.
Understand that a perimeter of a
rectangle is found by adding all the side
lengths of a rectangle.
Find a missing length of a rectangle if
given one side length and the area of the
rectangle.
Find rectangles with the same area and
different perimeters.
Find rectangles with different areas and
the same perimeter.
Resources
http://nlvm.usu.edu
tiles or centimeter blocks
graph paper
number lines
rectangles of different sizes
64
Assessments
Geometry
• Reason with shapes and their attributes.
Understand that shapes in different categories (e.g., rhombuses, rectangles, and others) may share attributes (e.g., having four
sides), and that the shared attributes can define a larger category (e.g., quadrilaterals). Recognize rhombuses, rectangles, and
squares as examples of quadrilaterals, and draw examples of quadrilaterals that do not belong to any of these subcategories.
Student Friendly/”I Can” statements
1. Understand the nomenclature of
quadrilaterals refer to the number of
sides.
2. Understand that specific shapes belong to
the larger category of quadrilaterals.
Resources
Assessments
http://nlvm.usu.edu
graph paper
quadrilaterals
rectangles
squares
trapezoids
rhombuses
graphic organizer (double bubble or Venn
diagram) to explain similarities and
differences of various shapes
Partition shapes into parts with equal areas. Express the area of each part as a unit fraction of the whole. For example, partition a
shape into 4 parts with equal area, and describe the area of each part as 1/4 of the area of the shape.
Student Friendly/”I Can” statements
1. Partition shapes into equal parts.
2. Name the parts of a shape as the number
of parts over the total number of equal
parts that make up the shape.
Resources
http://nlvm.usu.edu
pattern blocks
graph paper
dot paper
65
Assessments
Standards for Mathematical Practices – Grade 4
The Common Core State Standards for Mathematical Practice are expected to be integrated into every mathematics lesson for all students
Grades K-12. Below are a few examples of how these Practices may be integrated into tasks that students complete.
Mathematic Practices
1. Make sense of
problems and
persevere in solving
them.
2. Reason abstractly
and quantitatively.
3. Construct viable
arguments and
critique the
reasoning of others.
4. Model with
mathematics.
5. Use appropriate
tools
strategically.
6. Attend to precision.
7. Look for and make
use of
structure.
8. Look for and
express
regularity in repeated
reasoning.
Explanations and Examples
Mathematically proficient students in grade 4 know that doing mathematics involves solving problems and discussing
how they solved them. Students explain to themselves the meaning of a problem and look for ways to solve it. Fourth
graders may use concrete objects or pictures to help them conceptualize and solve problems. They may check their
thinking by asking themselves, “Does this make sense?” They listen to the strategies of others and will try different
approaches. They often will use another method to check their answers.
Mathematically proficient fourth graders should recognize that a number represents a specific quantity. They connect the quantity
to written symbols and create a logical representation of the problem at hand, considering both the appropriate units involved and
the meaning of quantities. They extend this understanding from whole numbers to their work with fractions and decimals. Students
write simple expressions, record calculations with numbers, and represent or round numbers using place value concepts.
In fourth grade mathematically proficient students may construct arguments using concrete referents, such as objects,
pictures, and drawings. They explain their thinking and make connections between models and equations. They refine
their mathematical communication skills as they participate in mathematical discussions involving questions like “How
did you get that?” and “Why is that true?” They explain their thinking to others and respond to others’ thinking.
Mathematically proficient fourth grade students experiment with representing problem situations in multiple ways
including numbers, words (mathematical language), drawing pictures, using objects, making a chart, list, or graph,
creating equations, etc. Students need opportunities to connect the different representations and explain the connections. They
should be able to use all of these representations as needed. Fourth graders should evaluate their results in the context of the
situation and reflect on whether the results make sense.
Mathematically proficient fourth graders consider the available tools (including estimation) when solving a mathematical
problem and decide when certain tools might be helpful. For instance, they may use graph paper or a number line to
represent and compare decimals and protractors to measure angles. They use other measurement tools to understand the relative
size of units within a system and express measurements given in larger units in terms of smaller units.
As fourth graders develop their mathematical communication skills, they try to use clear and precise language in their
discussions with others and in their own reasoning. They are careful about specifying units of measure and state the
meaning of the symbols they choose. For instance, they use appropriate labels when creating a line plot.
In fourth grade mathematically proficient students look closely to discover a pattern or structure. For instance, students
use properties of operations to explain calculations (partial products model). They relate representations of counting
problems such as tree diagrams and arrays to the multiplication principal of counting. They generate number or shape
patterns that follow a given rule.
Students in fourth grade should notice repetitive actions in computation to make generalizations Students use models to explain
calculations and understand how algorithms work. They also use models to examine patterns and generate their own algorithms.
For example, students use visual fraction models to write equivalent fractions.
Grade 4 Critical Areas
66
The Critical Areas are designed to bring focus to the standards at each grade by describing the big ideas that educators can use
to build their curriculum and to guide instruction.
The Critical Areas for fourth grade can be found on page 27 in the Common Core State Standards for Mathematics.
1. Developing understanding and fluency with multi-digit multiplication, and developing understanding of dividing to
find quotients involving multi-digit dividends.
Students generalize their understanding of place value to 1,000,000, understanding the relative sizes of numbers in each place. They apply
their understanding of models for multiplication (equal-sized groups, arrays, area models), place value, and properties of operations, in
particular the distributive property, as they develop, discuss, and use efficient, accurate, and generalizable methods to compute products of
multi-digit whole numbers. Depending on the numbers and the context, they select and accurately apply appropriate methods to estimate or
mentally calculate products. They develop fluency with efficient procedures for multiplying whole numbers; understand and explain why
the procedures work based on place value and properties of operations; and use them to solve problems. Students apply their understanding
of models for division, place value, properties of operations, and the relationship of division to multiplication as they develop, discuss, and
use efficient, accurate, and generalizable procedures to find quotients involving multi-digit dividends. They select and accurately apply
appropriate methods to estimate and mentally calculate quotients, and interpret remainders based upon the context.
2. Developing an understanding of fraction equivalence, addition and subtraction of fractions with like denominators,
and multiplication of fractions by whole numbers.
Students develop understanding of fraction equivalence and operations with fractions. They recognize that two different fractions can be
equal (e.g., 15/9 = 5/3), and they develop methods for generating and recognizing equivalent fractions. Students extend previous
understandings about how fractions are built from unit fractions, composing fractions from unit fractions, decomposing fractions into unit
fractions, and using the meaning of fractions and the meaning of multiplication to multiply a fraction by a whole number.
3. Understanding that geometric figures can be analyzed and classified based on their properties, such as having
parallel sides, perpendicular sides, particular angle measures, and symmetry.
Students describe, analyze, compare, and classify two-dimensional shapes. Through building, drawing, and analyzing two-dimensional
shapes, students deepen their understanding of properties of two-dimensional objects and the use of them to solve problems involving
symmetry.
67
Operations and Algebraic Thinking
• Use the four operations with whole numbers to solve problems.
Interpret a multiplication equation as a comparison, e.g., interpret 35 = 5 × 7 as a statement that 35 is 5 times as many as 7 and 7
times as many as 5. Represent verbal statements of multiplicative comparisons as multiplication equations.
Student Friendly/”I Can” statements
1. Model multiplication facts by visually
representing the equation.
2. Understand that a multiplication equation
is a comparison of quantities using the
product is x times as much as y
(xy=product) or that the product is y times
as much as x.
3. Write verbal statements of multiplicative
comparisons.
Resources
Assessments
http://nlvm.usu.edu
Cuisennaire rods
Graph paper
Lined paper
Multiply or divide to solve word problems involving multiplicative comparison, e.g., by using drawings and equations with a symbol
for the unknown number to represent the problem, distinguishing multiplicative comparison from additive comparison.
1.
2.
3.
4.
Student Friendly/”I Can” statements
Identify the operation of a word problem.
Create drawings and equations (number
model) to represent a word problem.
Solve a multiplication or division word
problem.
Write the solution as a comparison.
Resources
http://nlvm.usu.edu
Cuisennaire rods
Graph paper
Lined paper
68
Assessments
Solve multistep word problems posed with whole numbers and having whole-number answers using the four operations, including
problems in which remainders must be interpreted. Represent these problems using equations with a letter standing for the
unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding.
Student Friendly/”I Can” statements
1. Identify the operation of a word problem.
2. Create drawings and equations (number
model) to represent a word problem.
3. Determine what the solution means
including that of remainders to division
problems.
4. Check reasonableness of answers by using
mental computation and estimation with
rounding.
Resources
http://nlvm.usu.edu
Cuisennaire rods
Graph paper
Lined paper
Ten base blocks
Assessments
• Gain familiarity with factors and multiples.
Find all factor pairs for a whole number in the range 1–100. Recognize that a whole number is a multiple of each of its factors.
Determine whether a given whole number in the range 1–100 is a multiple of a given one-digit number. Determine whether a given
whole number in the range 1–100 is prime or composite.
1.
2.
3.
4.
Student Friendly/”I Can” statements
Define factors and multiples.
Explain that a whole number is a multiple
of each of its factors.
Determine that a whole number in the 1100 range is a multiple of one digit
numbers.
Define prime and composite.
Resources
http://nlvm.usu.edu
Cuisennaire rods
Graph paper
Lined paper
Ten base blocks
69
Assessments
• Generate and analyze patterns.
Generate a number or shape pattern that follows a given rule. Identify apparent features of the pattern that were not explicit in the
rule itself. For example, given the rule “Add 3” and the starting number 1, generate terms in the resulting sequence and observe that
the terms appear to alternate between odd and even numbers. Explain informally why the numbers will continue to alternate in this
way.
1.
2.
3.
4.
5.
Student Friendly/”I Can” statements
Identify a number pattern.
Identify a shape pattern.
Identify the characteristics of the pattern.
Explain how the pattern continues.
Create number and shape patterns.
Resources
http://nlvm.usu.edu
variety of number patterns
variety of shape patterns
pattern blocks
70
Assessments
Number and Operations in Base Ten
• Generalize place value understanding for multi-digit whole numbers.
Recognize that in a multi-digit whole number, a digit in one place represents ten times what it represents in the place to its right.
For example, recognize that 700 ÷ 70 = 10 by applying concepts of place value and division.
Student Friendly/”I Can” statements
1. Define multi-digit whole number.
2. Identify the factors of ten that make up
the multi-digit whole number.
3. Apply concept of place value to find
factors.
Resources
http://nlvm.usu.edu
http://dabbleboards.com/draw
mathsolutions.com
http:://mathplayground
http://www.shodor.org/interactive/
http://nrich.maths.org/public/
http://www.mathsisfun.com/definitions/letterp.htm
l
Ten base blocks
Assessments
Study Island
Read and write multi-digit whole numbers using base-ten numerals, number names, and expanded form. Compare two multi-digit
numbers based on meanings of the digits in each place, using >, =, and < symbols to record the results of comparisons.
Student Friendly/”I Can” statements
1. Define numerals, number names, and
expanded form.
2. Identify >, =, and <.
3. Compare multi-digit numbers using <,
=, and >.
4. Explain comparison based on place
value.
Resources
http://nlvm.usu.edu
http://dabbleboards.com/draw
mathsolutions.com
http:://mathplayground
http://www.shodor.org/interactive/
http://nrich.maths.org/public/
http://www.mathsisfun.com/definitions/letterp.htm
l
Ten base blocks
71
Assessments
Use place value understanding to round multi-digit whole numbers to any place.
Student Friendly/”I Can” statements
1. Use place value to round multi-digit
whole numbers.
Resources
http://nlvm.usu.edu
http://dabbleboards.com/draw
mathsolutions.com
http:://mathplayground
http://www.shodor.org/interactive/
http://nrich.maths.org/public/
http://www.mathsisfun.com/definitions/letterp.htm
l
Ten base blocks
Assessments
• Use place value understanding and properties of operations to perform multi-digit arithmetic.
Fluently add and subtract multi-digit whole numbers using the standard algorithm.
Student Friendly/”I Can” statements
1. Define standard algorithm.
2. Add single digit numbers without
counting (fluently).
3. Fluently subtract numbers.
4. Understand the trading (10 ones for
one ten) in both addition and
subtraction.
5. Add and subtract multi-digit whole
numbers using the standard algorithm.
Resources
http://nlvm.usu.edu
http://dabbleboards.com/draw
mathsolutions.com
http:://mathplayground
http://www.shodor.org/interactive/
http://nrich.maths.org/public/
http://www.mathsisfun.com/definitions/letterp.htm
l
Ten base blocks
72
Assessments
Timed tests of addition facts
Timed tests of subtraction facts
Multiply a whole number of up to four digits by a one-digit whole number, and multiply two two-digit numbers, using strategies
based on place value and the properties of operations. Illustrate and explain the calculation by using equations, rectangular arrays,
and/or area models.
Student Friendly/”I Can” statements
1. Represent multiplication problems by
arrays.
2. Represent multiplication problems by
area models.
3. Represent multiplication problems by
equations.
4. Multiply four-digit by one digit whole
numbers using place value and
properties of operations.
5. Multiply two-digit by two digit whole
numbers using place value and
properties of operations.
6. Explain products with equations, arrays
and/or area models.
Resources
Assessments
Timed tests for multiplication facts
http://nlvm.usu.edu
graph paper
Cuisennaire rods
Find whole-number quotients and remainders with up to four-digit dividends and one-digit divisors, using strategies based on place
value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the
calculation by using equations, rectangular arrays, and/or area models.
1.
2.
3.
4.
Student Friendly/”I Can” statements
Define quotient, remainder, dividend,
and divisor.
Find quotients and remainders of four
digit dividends and one digit divisors.
Use equations, arrays, and area models
to explain the quotient.
Explain the relationship between
multiplication and division.
http://nlvm.usu.edu
graph paper
Cuisennaire rods
Resources
73
Assessments
Number and Operations—Fractions
• Extend understanding of fraction equivalence and ordering.
Explain why a fraction a/b is equivalent to a fraction (n × a)/(n × b) by using visual fraction models, with attention to how the
number and size of the parts differ even though the two fractions themselves are the same size. Use this principle to recognize and
generate equivalent fractions.
Student Friendly/”I Can” statements
1. Understand that n/n equals one and that
the multiplicative identity applies to
fractions as well as whole numbers.
2. Understand that any whole number can
be expressed as n/n to indicate one
whole.
3. Understand that you can rename any
fraction by multiplying the numerator and
the denominator by one (in terms of any
whole number over itself n/n).
4. Explain with a visual model how even
though the number of parts when
multiplied by n/n the fractions have the
same value.
Resources
Assessments
http://nlvm.usu.edu
youtubefractions.com
fraction bars
graph paper
plain paper to draw out fraction bars
Compare two fractions with different numerators and different denominators, e.g., by creating common denominators or
numerators, or by comparing to a benchmark fraction such as 1/2. Recognize that comparisons are valid only when the two fractions
refer to the same whole. Record the results of comparisons with symbols >, =, or <, and justify the conclusions, e.g., by using a visual
fraction model.
Student Friendly/”I Can” statements
1. Use multiplicative identity in the form of
n/n to create common denominators to
compare fractions.
2. Use benchmark fractions to compare
fractions.
3. Explain the comparison of two fractions
using visual models.
Resources
http://nlvm.usu.edu
youtubefractions.com
fraction bars
graph paper
plain paper to draw out fraction bars
74
Assessments
• Build fractions from unit fractions by applying and extending previous understandings of operations on
whole numbers.
Understand a fraction a/b with a > 1 as a sum of fractions 1/b.
a. Understand addition and subtraction of fractions as joining and separating parts referring to the same whole.
1.
2.
3.
4.
Student Friendly/”I Can” statements
Understand that equal fractional parts of
a whole make up sum of the whole.
Understand that the whole can be broken
down to its equal fractional parts.
Understand that partial sums of the
fractional equal parts of a whole can be
combined to make the whole.
Understand that the whole can be broken
into a variety of the fractional parts.
Resources
Assessments
http://nlvm.usu.edu
youtubefractions.com
fraction bars
graph paper
plain paper to draw out fraction bars
b. Decompose a fraction into a sum of fractions with the same denominator in more than one way, recording each decomposition by
an equation. Justify decompositions, e.g., by using a visual fraction model. Examples: 3/8 = 1/8 + 1/8 + 1/8 ; 3/8 = 1/8 + 2/8 ; 2 1/8 =
1 + 1 + 1/8 = 8/8 + 8/8 + 1/8.
Student Friendly/”I Can” statements
1. Create sums of fractions using the same
denominator and various numerators.
2. Justify the sums by using models.
Resources
http://nlvm.usu.edu
youtubefractions.com
fraction bars
graph paper
plain paper to draw out fraction bars
75
Assessments
c. Add and subtract mixed numbers with like denominators, e.g., by replacing each mixed number with an equivalent fraction,
and/or by using properties of operations and the relationship between addition and subtraction.
1.
2.
3.
4.
5.
Student Friendly/”I Can” statements
Define mixed numbers.
Define improper fractions.
Show using models how to turn mixed
numbers into improper fractions and
improper fractions into mixed numbers.
Add mixed numbers with the same
denominator with the appropriate trading
to express the sum as a proper mixed
number.
Subtract mixed numbers with the same
denominator by converting the whole
numbers into fractions with the same
denominator, make the appropriate
trading to subtract the mixed numbers
and express the difference as a proper
mixed number.
Resources
http://nlvm.usu.edu
youtubefractions.com
fraction bars
graph paper
plain paper to draw out fraction bars
Assessments
d. Solve word problems involving addition and subtraction of fractions referring to the same whole and having like denominators,
e.g., by using visual fraction models and equations to represent the problem.
Student Friendly/”I Can” statements
1. Solve word problems for adding and
subtracting fractions by drawing out the
problem.
2. Solve word problems for adding and
subtracting fractions by writing the
equations to represent the problem.
Resources
http://nlvm.usu.edu
youtubefractions.com
fraction bars
graph paper
plain paper to draw out fraction bars
76
Assessments
Apply and extend previous understandings of multiplication to multiply a fraction by a whole number.
a. Understand a fraction a/b as a multiple of 1/b. For example, use a visual fraction model to represent 5/4 as the product 5 × (1/4),
recording the conclusion by the equation 5/4 = 5 × (1/4).
Student Friendly/”I Can” statements
1. Use a visual model to show that
multiplying a whole number by a fraction
is repeated addition of the fraction (1/b)
by the number of times of the value of the
whole number.
Resources
http://nlvm.usu.edu
youtubefractions.com
fraction bars
graph paper
plain paper to draw out fraction bars
Assessments
b. Understand a multiple of a/b as a multiple of 1/b, and use this understanding to multiply a fraction by a whole number. For
example, use a visual fraction model to express 3 × (2/5) as 6 × (1/5), recognizing this product as 6/5. (In general, n × (a/b) = (n ×
a)/b.)
Student Friendly/”I Can” statements
1. Use a visual model to show that
multiplying a whole number (x) by a
fraction a/b is repeated addition of the
fraction by the number of times of the
value of the whole number (xa) and can
be renamed to a whole number multiplied
by 1/b.
Resources
http://nlvm.usu.edu
youtubefractions.com
fraction bars
graph paper
plain paper to draw out fraction bars
Assessments
c. Solve word problems involving multiplication of a fraction by a whole number, e.g., by using visual fraction models and equations
to represent the problem. For example, if each person at a party will eat 3/8 of a pound of roast beef, and there will be 5 people at
the party, how many pounds of roast beef will be needed? Between what two whole numbers does your answer lie?
Student Friendly/”I Can” statements
1. Solve word problems for multiplying
fractions by a whole number by drawing
out the problem.
2. Solve word problems for multiplying
fractions by a whole number by writing
the equations to represent the problem.
Resources
http://nlvm.usu.edu
youtubefractions.com
fraction bars
graph paper
plain paper to draw out fraction bars
77
Assessments
• Understand decimal notation for fractions, and compare decimal fractions.
Express a fraction with denominator 10 as an equivalent fraction with denominator 100, and use this technique to add two fractions
with respective denominators 10 and 100.4 For example, express 3/10 as 30/100, and add 3/10 + 4/100 = 34/100.
Student Friendly/”I Can” statements
1. Use place value, ten base blocks and/or 10
x 10 grid to understand that 1/10 it the
same as 10/100.
2. Add fractions with 10 and 100 in the
denominator and express as n/100.
Resources
http://nlvm.usu.edu
10 base blocks
Place value charts
Graph paper
Colored pencils
Assessments
Use decimal notation for fractions with denominators 10 or 100. For example, rewrite 0.62 as 62/100; describe a length as 0.62
meters; locate 0.62 on a number line diagram.
Student Friendly/”I Can” statements
1. Write fractions with denominators of 10
and 100 as decimals.
2. Place decimals on a number line.
Resources
http://nlvm.usu.edu
10 base blocks
Place value charts
Graph paper
Colored pencils
Assessments
Compare two decimals to hundredths by reasoning about their size. Recognize that comparisons are valid only when the two
decimals refer to the same whole. Record the results of comparisons with the symbols >, =, or <, and justify the conclusions, e.g., by
using a visual model.
Student Friendly/”I Can” statements
1. Use area models of different areas and
the same area to compare decimals to
hundredths.
2. Understand that decimal comparison only
applies when the area is the same.
3. Explain comparisons of <, >, and = with
visual models.
Resources
http://nlvm.usu.edu
10 base blocks
Place value charts
Graph paper
Colored pencils
78
Assessments
Measurement and Data
• Solve problems involving measurement and conversion of measurements from a larger unit to a smaller unit.
1. Know relative sizes of measurement units within one system of units including km, m, cm; kg, g; lb, oz.; l, ml; hr, min, sec. Within a
single system of measurement, express measurements in a larger unit in terms of a smaller unit. Record measurement equivalents in
a two-column table. For example, know that 1 ft is 12 times as long as 1 in. Express the length of a 4 ft snake as 48 in. Generate a
conversion table for feet and inches listing the number pairs (1, 12), (2, 24), (3, 36), ...
Student Friendly/”I Can” statements
1. Convert larger units of the metric system
to larger units of the metric system.
2. Record metric system conversions in a
two-column table.
3. Convert larger customary units of
measurement to smaller customary units
of measurement.
4. Record conversions of customary
measurements in a two-column table.
5. Convert larger time measurements into
smaller time measurements.
6. Record time measurement conversions in
a two-column table.
Resources
http://nlvm.usu.edu
meter stick with inches and feet on reverse
side
rulers with metric and customary
measurements
scale with metric units and customary units
analog clocks with second hands.
Paper for two-column notes
79
Assessments
2. Use the four operations to solve word problems involving distances, intervals of time, liquid volumes, masses of objects, and
money, including problems involving simple fractions or decimals, and problems that require expressing measurements given in a
larger unit in terms of a smaller unit. Represent measurement quantities using diagrams such as number line diagrams that feature a
measurement scale.
Student Friendly/”I Can” statements
1. Solve distance word problems for four
operations that include simple fractions or
decimals that require conversion from
larger unit to smaller unit.
2. Represent solutions to distance word
problems using a diagram with the correct
scale.
3. Solve intervals of time word problems for
four operations that include simple
fractions or decimals that require
conversion from larger unit to smaller
unit.
4. Represent solutions to time word
problems using a diagram with the correct
scale.
5. Solve liquid volumes word problems for
four operations that include simple
fractions or decimals that require
conversion from larger unit to smaller
unit.
6. Represent solutions to liquid volumes
word problems using a diagram with the
correct scale.
7. Solve masses of objects word problems
for four operations that include simple
fractions or decimals that require
conversion from larger unit to smaller
unit.
8. Represent solutions to mass word
problems using a diagram with the correct
Resources
http://nlvm.usu.edu
meter stick with inches and feet on reverse
side
rulers with metric and customary
measurements
scale with metric units and customary units
analog clocks with second hands.
Paper
80
Assessments
scale.
9. Solve money word problems for four
operations that include simple fractions or
decimals that require conversion from
larger unit to smaller unit.
10. Represent solutions to money word
problems using a diagram with the correct
scale.
3. Apply the area and perimeter formulas for rectangles in real world and mathematical problems. For example, find the width of a
rectangular room given the area of the flooring and the length, by viewing the area formula as a multiplication equation with an
unknown factor.
Student Friendly/”I Can” statements
1. Determine when a rectangular real world
problem or mathematical problem is the
application of area or perimeter formulas.
2. Solve real world applications for area of
rectangles.
3. Solve real world applications for
perimeter of rectangle.
Graph paper
Resources
Assessments
• Represent and interpret data.
Make a line plot to display a data set of measurements in fractions of a unit (1/2, 1/4, 1/8). Solve problems involving addition and
subtraction of fractions by using information presented in line plots. For example, from a line plot find and interpret the difference in
length between the longest and shortest specimens in an insect collection.
Student Friendly/”I Can” statements
1. Create line plots to display fractional data
of measurements.
2. Solve problems of addition and
subtraction of fractions based on the data
of the line plot.
Resources
http://nlvm.usu.edu
unnumbered number lines
81
Assessments
• Geometric measurement: understand concepts of angle and measure angles.
Recognize angles as geometric shapes that are formed wherever two rays share a common endpoint, and understand concepts of
angle measurement:
a. An angle is measured with reference to a circle with its center at the common endpoint of the rays, by considering the fraction of
the circular arc between the points where the two rays intersect the circle. An angle that turns through 1/360 of a circle is called a
“one-degree angle,” and can be used to measure angles.
Student Friendly/”I Can” statements
1. Define ray, endpoints, arc, and angle.
2. Understand that an angle is measured
from the center of a circle as the endpoint
of two rays and that the measure of the
angle is the distance of the two points
between where the rays intersect the
diameter of the circle (arc).
3. Understand that circles are 360 degrees.
4. Understand that 1 degree angle is 1/360
of a circle.
Resources
Assessments
http://nlvm.usu.edu
circular protractors
various size circles
b. An angle that turns through n one-degree angles is said to have an angle measure of n degrees.
Student Friendly/”I Can” statements
1. Understand that angles are measured in
protractors
Resources
Assessments
one-degree angles so that the number
represented in angle measurements are
the number of one degree angles.
Measure angles in whole-number degrees using a protractor. Sketch angles of specified measure.
Student Friendly/”I Can” statements
1. Measure angles with a protractor.
2. Create angles with a certain measure.
Resources
Protractors
Unlined paper
82
Assessments
Recognize angle measure as additive. When an angle is decomposed into non-overlapping parts, the angle measure of the whole is
the sum of the angle measures of the parts. Solve addition and subtraction problems to find unknown angles on a diagram in real
world and mathematical problems, e.g., by using an equation with a symbol for the unknown angle measure.
1.
2.
3.
4.
5.
6.
Student Friendly/”I Can” statements
Understand that half circles are 180
degrees and that the rays form a straight
line so the angle measure of a straight
angle is 180 degrees.
Understand that ¼ of a circle has an angle
measurement of 90 degrees and forms a
right angle.
Understand that 1/8 of a circle has an
angle measurement of 45 degrees.
Understand when angles overlap their
angle measures the sum of the angles can
be found by decomposing the angles into
non-overlapping parts.
Solve addition and subtraction problems
that from diagrams representing real
world or mathematical problems with
missing angle measures.
Use the symbolic representation of angles
and of missing angle measures in problem
solutions.
Resources
Various diagrams of examples of real world
application of angle measurements with
unknown measures and overlapping angles.
83
Assessments
Geometry
• Draw and identify lines and angles, and classify shapes by properties of their lines and angles.
Draw points, lines, line segments, rays, angles (right, acute, obtuse), and perpendicular and parallel lines. Identify these in twodimensional figures.
1.
2.
3.
4.
Student Friendly/”I Can” statements
Define points, lines, line segments, rays,
acute angles, obtuse angles,
perpendicular lines and parallel lines.
Draw points, lines, line segments, rays,
right, acute, and obtuse angles,
perpendicular and parallel lines.
Know the symbolic representation of
lines, line segments, rays, angles,
perpendicular and parallel lines.
Identify points, lines, line segments, rays,
right angles, acute angles, obtuse angles,
perpendicular and parallel lines in two
dimensional figures.
Resources
Assessments
http://nlvm.usu.edu
rulers
plain paper
Various two dimensional figures
Classify two-dimensional figures based on the presence or absence of parallel or perpendicular lines, or the presence or absence of
angles of a specified size. Recognize right triangles as a category, and identify right triangles.
Student Friendly/”I Can” statements
1. Classify two-dimensional figures based on
angle size (acute, obtuse) and parallel or
perpendicular lines (right).
2. Identify triangles by angle measure.
3. Recognize that a triangle and be both
acute and right.
Resources
Assessments
protractors
various two-dimensional figures
Recognize a line of symmetry for a two-dimensional figure as a line across the figure such that the figure can be folded along the line
into matching parts. Identify line-symmetric figures and draw lines of symmetry.
Student Friendly/”I Can” statements
1. Define symmetry.
2. Identify lines of symmetry.
3. Draw lines of symmetry.
Resources
various two-dimensional figures
rulers
84
Assessments
Standards for Mathematical Practices – Grade 5
The Common Core State Standards for Mathematical Practice are expected to be integrated into every mathematics lesson for all students Grades K-12.
Below are a few examples of how these Practices may be integrated into tasks that students complete.
Mathematical Practice
Explanations and Examples
1. Make sense of problems
Mathematically proficient students in grade 5 should solve problems by applying their understanding of
operations with whole numbers, decimals, and fractions including mixed numbers. They solve problems
and persevere in solving
them.
related to volume and measurement conversions. Students seek the meaning of a problem and look for efficient
ways to represent and solve it. They may check their thinking by asking themselves, “What is the most
efficient way to solve the problem?”, “Does this make sense?”, and “Can I solve the problem in a different
way?”.
2. Reason abstractly and
Mathematically proficient students in grade 5should recognize that a number represents a specific quantity.
They connect quantities to written symbols and create a logical representation of the problem at hand,
quantitatively.
considering both the appropriate units involved and the meaning of quantities. They extend this understanding
from whole numbers to their work with fractions and decimals. Students write simple expressions that record
calculations with numbers and represent or round numbers using place value concepts.
In fifth grade mathematical proficient students may construct arguments using concrete referents, such as
3. Construct viable
arguments and critique the
objects, pictures, and drawings. They explain calculations based upon models and properties of operations and
rules that generate patterns. They demonstrate and explain the relationship between volume and multiplication.
reasoning of others.
They refine their mathematical communication skills as they participate in mathematical discussions involving
questions like “How did you get that?” and “Why is that true?” They explain their thinking to others and
respond to others’ thinking.
4. Model with mathematics. Mathematically proficient students in grade 5 experiment with representing problem situations in multiple
ways including numbers, words (mathematical language), drawing pictures, using objects, making a chart, list,
or graph, creating equations, etc. Students need opportunities to connect the different representations and
explain the connections. They should be able to use all of these representations as needed. Fifth graders should
evaluate their results in the context of the situation and whether the results make sense. They also evaluate the
utility of models to determine which models are most useful and efficient to solve problems.
Mathematically proficient fifth graders consider the available tools (including estimation) when solving a
5. Use appropriate tools
strategically.
mathematical problem and decide when certain tools might be helpful. For instance, they may use unit cubes to
fill a rectangular prism and then use a ruler to measure the dimensions. They use graph paper to accurately
create graphs and solve problems or make predictions from real world data.
Mathematically proficient students in grade 5 continue to refine their mathematical communication skills by
6. Attend to precision.
using clear and precise language in their discussions with others and in their own reasoning. Students use
appropriate terminology when referring to expressions, fractions, geometric figures, and coordinate grids. They
are careful about specifying units of measure and state the meaning of the symbols they choose. For instance,
when figuring out the volume of a rectangular prism they record their answers in cubic units
7. Look for and make use of In fifth grade mathematically proficient students look closely to discover a pattern or structure. For instance,
students use properties of operations as strategies to add, subtract, multiply and divide with whole numbers,
structure.
85
8. Look for and express
regularity in repeated
reasoning.
fractions, and decimals. They examine numerical patterns and relate them to a rule or a graphical
representation.
Mathematically proficient fifth graders use repeated reasoning to understand algorithms and make
generalizations about patterns. Students connect place value and their prior work with operations to understand
algorithms to fluently multiply multi-digit numbers and perform all operations with decimals to hundredths.
Students explore operations with fractions with visual models and begin to formulate generalizations.
86
Grade 5 Critical Areas
The Critical Areas are designed to bring focus to the standards at each grade by describing the big ideas that educators can use
to build their curriculum and to guide instruction.
The Critical Areas for fifth grade can be found on page 33 in the Common Core State Standards for Mathematics.
1. Developing fluency with addition and subtraction of fractions, and developing understanding of the multiplication of fractions
and of division of fractions in limited cases (unit fractions divided by whole numbers and whole numbers divided by unit
fractions).
Students apply their understanding of fractions and fraction models to represent the addition and subtraction of fractions with unlike denominators as
equivalent calculations with like denominators. They develop fluency in calculating sums and differences of fractions, and make reasonable estimates of
them. Students also use the meaning of fractions, of multiplication and division, and the relationship between multiplication and division to understand
and explain why the procedures for multiplying and dividing fractions make sense. (Note: this is limited to the case of dividing unit fractions by whole
numbers and whole numbers by unit fractions.)
2. Extending division to 2-digit divisors, integrating decimal fractions into the place value system and developing understanding
of operations with decimals to hundredths, and developing fluency with whole number and decimal operations.
Students develop understanding of why division procedures work based on the meaning of base-ten numerals and properties of operations. They finalize
fluency with multi-digit addition, subtraction, multiplication, and division. They apply their understandings of models for decimals, decimal notation,
and properties of operations to add and subtract decimals to hundredths. They develop fluency in these computations, and make reasonable estimates of
their results. Students use the relationship between decimals and fractions, as well as the relationship between finite decimals and whole numbers (i.e., a
finite decimal multiplied by an appropriate power of 10 is a whole number), to understand and explain why the procedures for multiplying and dividing
finite decimals make sense. They compute products and quotients of decimals to hundredths efficiently and accurately.
3. Developing understanding of volume.
Students recognize volume as an attribute of three-dimensional space. They understand that volume can be measured by finding the total number of
same-size units of volume required to fill the space without gaps or overlaps. They understand that a 1-unit by 1-unit by 1-unit cube is the standard unit
for measuring volume. They select appropriate units, strategies, and tools for solving problems that involve estimating and measuring volume. They
decompose three-dimensional shapes and find volumes of right rectangular prisms by viewing them as decomposed into layers of arrays of cubes. They
measure necessary attributes of shapes in order to determine volumes to solve real world and mathematical problems.
87
Operations and Algebraic Thinking
• Write and interpret numerical expressions.
Use parentheses, brackets, or braces in numerical expressions, and evaluate expressions with these symbols
Student Friendly/”I Can” statements
Resources
Assessments
1. Know the order of operations.
http://nlvm.usu.edu
2. Use parentheses, brackets, and/or braces
in numerical expression and evaluate
them to find the solution.
Write simple expressions that record calculations with numbers, and interpret numerical expressions without evaluating them. For example,
express the calculation “add 8 and 7, then multiply by 2” as 2 × (8 + 7). Recognize that 3 × (18932 + 921) is three times as large as 18932 + 921,
without having to calculate the indicated sum or product.
Student Friendly/”I Can” statements
Resources
Assessments
1. Write simple word expressions from
numerical expressions.
2. Write numerical expressions from word
expressions.
3. Interpret the relationship of the numbers
in the expressions.
88
• Analyze patterns and relationships.
Generate two numerical patterns using two given rules. Identify apparent relationships between corresponding terms. Form ordered pairs
consisting of corresponding terms from the two patterns, and graph the ordered pairs on a coordinate plane. For example, given the rule “Add
3” and the starting number 0, and given the rule “Add 6” and the starting number 0, generate terms in the resulting sequences, and observe that
the terms in one sequence are twice the corresponding terms in the other sequence. Explain informally why this is so.
Student Friendly/”I Can” statements
1. Define coordinate plane, coordinates,
axes, ordered pairs, corresponding terms.
2. Graph coordinate-pairs on a coordinate
plane.
3. Given two rules, create a chart with both
patterns on the chart.
4. Identify the relationship between the
corresponding terms in the chart.
5. Graph the ordered pairs from the chart on
a coordinate plane.
6. Determine the relationship of the
corresponding terms from a coordinate
graph.
7. Identify how the relationship would
continue.
Resources
http://nlvm.usu.edu
coordinate planes
graph paper
89
Assessments
Number and Operations in Base Ten
• Understand the place value system.
Recognize that in a multi-digit number, a digit in one place represents 10 times as much as it represents in the place to its right and 1/10 of what
it represents in the place to its left.
Student Friendly/”I Can” statements
Resources
Assessments
1. Use place value to multiply decimals by
http://nlvm.usu.edu
powers of ten to convert them to whole
place value charts
numbers.
2. Understand that multiplying by 1/10 is the
same as dividing by 10.
Explain patterns in the number of zeros of the product when multiplying a number by powers of 10, and explain patterns in the placement of the
decimal point when a decimal is multiplied or divided by a power of 10. Use whole-number exponents to denote powers of 10.
Student Friendly/”I Can” statements
Resources
Assessments
1. Define exponents.
Place value charts
2. Explain that the number of zeros in a
Graph paper
product represents the exponent of 10.
3. Explain that since a positive exponent
denotes multiplication of 10, a negative
exponent denotes division of 10.
4. Use whole-number exponents to denote
powers of ten.
Read, write, and compare decimals to thousandths.
a. Read and write decimals to thousandths using base-ten numerals, number names, and expanded form, e.g., 347.392 = 3 × 100 + 4 × 10 + 7 × 1
+ 3 × (1/10) + 9 × (1/100) + 2 × (1/1000).
Student Friendly/”I Can” statements
Resources
Assessments
Place value charts
1. Read and write (words and numerical)
Mathplayground.com – Decention – Decimals,
decimals to the thousandths.
percents, fractions
2. Read and write the expanded form of
Base 10 blocks
decimals with fractional notation.
Fraction bars
Decimal bars
NLVM
Mathisfun.com/definitions
90
b. Compare two decimals to thousandths based on meanings of the digits in each place, using >, =, and < symbols to record the results of
comparisons.
Student Friendly/”I Can” statements
Resources
Assessments
1. Compare two decimals to the thousandths Place value charts
using <, >, and = for recording.
Use place value understanding to round decimals to any place.
Student Friendly/”I Can” statements
Resources
Assessments
1. Round decimals to any place.
Place value charts
• Perform operations with multi-digit whole numbers and with decimals to hundredths.
Fluently multiply multi-digit whole numbers using the standard algorithm.
Student Friendly/”I Can” statements
1. Multiply multi-digit whole numbers using
the standard algorithm.
Resources
Assessments
http://nlvm.usu.edu
graph paper for arrays
youtube.com/watch
Find whole-number quotients of whole numbers with up to four-digit dividends and two-digit divisors, using strategies based on place value, the
properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations,
rectangular arrays, and/or area models.
Student Friendly/”I Can” statements
Resources
Assessments
1. Divide four-digit dividends and two-digit
Graph paper
divisors.
2. Explain quotient by using equations,
rectangular arrays, and/or area models.
91
Number and Operations—Fractions
• Use equivalent fractions as a strategy to add and subtract fractions.
Add and subtract fractions with unlike denominators (including mixed numbers) by replacing given fractions with equivalent fractions in such a
way as to produce an equivalent sum or difference of fractions with like denominators. For example, 2/3 + 5/4 = 8/12 + 15/12 = 23/12. (In
general, a/b + c/d = (ad + bc)/bd.)
Student Friendly/”I Can” statements
Resources
Assessments
1. Rename fractions as equivalent fractions.
http://nlvm.usu.edu
2. Rename two fractions with unlike
Use arrays to show simple unlike denominator
denominators as equivalent fractions with fractions and how additional divisions of the
the same denominators.
arrays are necessary to get common
3. Recognize that the product of two unlike
denominators in order to add or subtract.
denominators will create a common
denominator.
Show that it might take multiple divisions to
4. Recognize that the common denominator get the common denominator.
created by the product of the two unlike
denominators may not be the smallest
Once students have an understanding of the
denominator.
arrays, write the numeric equivalent to the
5. Add two fractions with unlike
steps taken in explaining the arrays.
denominators after renaming the
fractions to equivalent fractions with like
denominators.
6. Subtract two fractions with unlike
denominators after renaming the
fractions to equivalent fractions with like
denominators.
92
Solve word problems involving addition and subtraction of fractions referring to the same whole, including cases of unlike denominators, e.g., by
using visual fraction models or equations to represent the problem. Use benchmark fractions and number sense of fractions to estimate
mentally and assess the reasonableness of answers. For example, recognize an incorrect result 2/5 + 1/2 = 3/7, by observing that 3/7 < 1/2.
Student Friendly/”I Can” statements
Resources
Assessments
1. Solve real world addition and subtraction
Plain paper
fraction problems.
Graph paper
2. Use benchmark fractions to mentally
Fraction bars
estimate the solutions.
3. Assess reasonableness of answers.
4. Recognize mistakes.
• Apply and extend previous understandings of multiplication and division to multiply and divide fractions.
Interpret a fraction as division of the numerator by the denominator (a/b = a ÷ b). Solve word problems involving division of whole numbers
leading to answers in the form of fractions or mixed numbers, e.g., by using visual fraction models or equations to represent the problem. For
example, interpret 3/4 as the result of dividing 3 by 4, noting that 3/4 multiplied by 4 equals 3, and that when 3 wholes are shared equally
among 4 people each person has a share of size 3/4. If 9 people want to share a 50-pound sack of rice equally by weight, how many pounds of
rice should each person get? Between what two whole numbers does your answer lie?
Student Friendly/”I Can” statements
Resources
Assessments
1. Interpret fractions as division problems.
http://nlvm.usu.edu
2. Solve word problems involving division of plain paper
whole numbers in which the solution is a
graph paper
fraction or mixed number.
3. Use models to explain result.
93
Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction.
a. Interpret the product (a/b) × q as a parts of a partition of q into b equal parts; equivalently, as the result of a sequence of operations a × q ÷ b.
For example, use a visual fraction model to show (2/3) × 4 = 8/3, and create a story context for this equation. Do the same with (2/3) × (4/5) =
8/15. (In general, (a/b) × (c/d) = ac/bd.)
Student Friendly/”I Can” statements
Resources
Assessments
1. Understand that a fraction multiplied by a Graph paper
whole number is the same as the
Fraction tiles
numerator of the fraction multiplied by
Plain paper
the whole number and the product
Lined paper
divided by the denominator.
2. Create a model showing the product of a
fraction and a whole number.
3. Create a word story for the equation of
product of whole number and a fraction.
4. Understand that when two fractions are
multiplied, the product is the result of the
product of the two numerators over the
product of the two denominators.
5. Create a model showing the product of
two fractions.
6. Create a word story for the equation of
the product of two fractions.
b. Find the area of a rectangle with fractional side lengths by tiling it with unit squares of the appropriate unit fraction side lengths, and show
that the area is the same as would be found by multiplying the side lengths. Multiply fractional side lengths to find areas of rectangles, and
represent fraction products as rectangular areas.
Student Friendly/”I Can” statements
Resources
Assessments
1. Find the area of fractional side lengths by
Graph paper
tiling with the appropriate unit size.
Fraction tiles
2. Show that the area model is the same as
multiplying the fractional side lengths.
3. Create fractional rectangular areas to
represent fraction products.
94
Interpret multiplication as scaling (resizing), by:
a. Comparing the size of a product to the size of one factor on the basis of the size of the other factor, without performing the indicated
multiplication.
Student Friendly/”I Can” statements
Resources
Assessments
1. Understand that when one factor stays
Graph paper
the same and the other is changed by a
fractional equivalent of the original the
product of the new terms will be equal to
the fractional equivalent of the new term
when compared to the original product by
using visual models.
2. Compare products when one factor
changes without multiplying.
b. Explaining why multiplying a given number by a fraction greater than 1 results in a product greater than the given number (recognizing
multiplication by whole numbers greater than 1 as a familiar case); explaining why multiplying a given number by a fraction less than 1 results in
a product smaller than the given number; and relating the principle of fraction equivalence a/b = (n×a)/(n×b) to the effect of multiplying a/b by
1.
Student Friendly/”I Can” statements
Resources
Assessments
Graph paper
1. Explain why a product is greater than 1
when a number is multiplied by a fraction Fraction tiles
greater than 1.
2. Explain why a product is less than one
when a number is multiplying by a
fraction less than 1.
95
Solve real world problems involving multiplication of fractions and mixed numbers, e.g., by using visual fraction models or equations to
represent the problem.
Student Friendly/”I Can” statements
Resources
Assessments
Graph paper
1. Solve real world problems involving
Fraction tiles
multiplication of fractions using visual
models.
2. Solve real world problems involving
multiplication of fractions using
equations.
3. Solve real world problems involving
multiplication of mixed numbers by using
visual models.
4. Solve real world problems involving
multiplication of mixed numbers using
equations.
Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions.
a. Interpret division of a unit fraction by a non-zero whole number, and compute such quotients. For example, create a story context for (1/3) ÷
4, and use a visual fraction model to show the quotient. Use the relationship between multiplication and division to explain that (1/3) ÷ 4 = 1/12
because (1/12) × 4 = 1/3.
Student Friendly/”I Can” statements
Resources
Assessments
Graph paper
1. Use a visual fraction model to find the
Fraction tiles
quotient of a unit fraction (numerator of
1) divided by a whole number.
2. Create a word story for the quotient of a
unit fraction divided by a whole number.
b. Interpret division of a whole number by a unit fraction, and compute such quotients. For example, create a story context for 4 ÷ (1/5), and use
a visual fraction model to show the quotient. Use the relationship between multiplication and division to explain that 4 ÷ (1/5) = 20 because 20 ×
(1/5) = 4.
Student Friendly/”I Can” statements
Resources
Assessments
1. Use a visual fraction model to find the
Graph paper
quotient of a whole number divided by a
Fraction tiles
unit fraction (numerator of 1).
2. Create a word story for the quotient of a
whole number divided by a unit fraction.
96
c. Solve real world problems involving division of unit fractions by non-zero whole numbers and division of whole numbers by unit fractions, e.g.,
by using visual fraction models and equations to represent the problem. For example, how much chocolate will each person get if 3 people share
1/2 lb of chocolate equally? How many 1/3-cup servings are in 2 cups of raisins?
Student Friendly/”I Can” statements
1. Solve real world problems involving
division of unit fractions by whole
numbers and division of whole numbers
by unit fractions using visual models.
2. Solve real world problems involving
division of unit fractions by whole
numbers and division of whole numbers
by unit fractions using equations.
Resources
Graph paper
Fraction tiles
97
Assessments
Measurement and Data
• Convert like measurement units within a given measurement system.
Convert among different-sized standard measurement units within a given measurement system (e.g., convert 5 cm to 0.05 m), and use these
conversions in solving multi-step, real world problems.
Student Friendly/”I Can” statements
Resources
Assessments
http://nlvm.usu.edu
1. Recognize units of measurement within
metric and customary ruler
the same system.
meter stick and yard stick
2. Divide and multiply to change units.
measuring cups and pint, quart, gallon
3. Convert units of measurement within
containers
the same system.
scales with metric weights and ounces and
4. Solve multi-step, real world problems
pounds
that involve converting.
analog clock with second hand
post-it notes
Learner.org (Math in Daily Life)
Movement of decimals – what do we see in
the world?
98
• Represent and interpret data.
Make a line plot to display a data set of measurements in fractions of a unit (1/2, 1/4, 1/8). Use operations on fractions for this grade to solve
problems involving information presented in line plots. For example, given different measurements of liquid in identical beakers, find the amount
of liquid each beaker would contain if the total amount in all the beakers were redistributed equally.
Student Friendly/”I Can” statements
Resources
Assessments
1. Identify benchmark fractions (1/2, 1/4,
1/8)
http://nlvm.usu.edu
2. Collect fractional data.
number lines
3. Make a line plot to display a data set of
fraction tiles
measurements in fractions of a unit (1/2,
graph paper
1/4, 1/8).
4. Solve problems involving information
presented in line plots which use
fractions of a unit (1/2, 1/4, 1/8) by
adding, subtracting, multiplying, and
dividing fractions.
• Geometric measurement: understand concepts of volume and relate volume to multiplication and to addition.
Recognize volume as an attribute of solid figures and understand concepts of volume measurement.
a. A cube with side length 1 unit, called a “unit cube,” is said to have “one cubic unit” of volume, and can be used to measure volume.
Student Friendly/”I Can” statements
Resources
Assessments
1. Recognize that volume is the
http://nlvm.usu.edu
measurement of the space inside a solid
centimeter cubes
three-dimensional figure.
cubes of various sizes
2. Recognize a unit cube has 1 cubic unit of
Clear three dimensional cube that can be
volume and is used to measure volume of opened to fill with centimeter cubes
three-dimensional shapes.
b. A solid figure which can be packed without gaps or overlaps using n unit cubes is said to have a volume of n cubic units.
Student Friendly/”I Can” statements
Resources
Assessments
1. Recognize any solid figure packed without http://nlvm.usu.edu
centimeter cubes
gaps or overlaps and filled with (n) “unit
cubes of various sizes
cubes” indicates the total cubic units or
Clear three dimensional cube that can be
volume.
opened to fill with centimeter cubes
99
Measure volumes by counting unit cubes, using cubic cm, cubic in, cubic ft, and improvised units.
Student Friendly/”I Can” statements
Resources
Assessments
1. Measure volumes by counting unit cubes, http://nlvm.usu.edu
using cubic cm, cubic in, cubic ft, and
centimeter cubes
improvised units.
cubes of various sizes
Clear three dimensional cube that can be
opened to fill with centimeter cubes
Relate volume to the operations of multiplication and addition and solve real world and mathematical problems involving volume.
a. Find the volume of a right rectangular prism with whole-number side lengths by packing it with unit cubes, and show that the volume is the
same as would be found by multiplying the edge lengths, equivalently by multiplying the height by the area of the base. Represent threefold
whole-number products as volumes, e.g., to represent the associative property of multiplication.
Student Friendly/”I Can” statements
Resources
Assessments
1. Identify a right rectangular prism.
http://nlvm.usu.edu
centimeter cubes
2. Find the volume of a right rectangular
right rectangular prisms
prism with whole number side lengths by
Clear three dimensional prisms that can be
packing it with unit cubes.
opened to fill with centimeter cubes
b. Apply the formulas V = l × w × h and V = b × h for rectangular prisms to find volumes of right rectangular prisms with whole number edge
lengths in the context of solving real world and mathematical problems.
Student Friendly/”I Can” statements
1. Develop volume formula for a
rectangle prism by comparing volume
when filled with cubes to volume by
multiplying the height by the area of
the base, or when multiplying the
edge lengths (LxWxH).the three
Resources
http://nlvm.usu.edu
centimeter cubes
right rectangular prisms
Clear three dimensional prisms that can be
opened to fill with centimeter cubes
dimensions in any order to calculate
volume (Commutative and associative
properties).
2. Find the volume of a right rectangular
prism by the volume formula.
100
Assessments
c. Recognize volume as additive. Find volumes of solid figures composed of two non-overlapping right rectangular prisms by adding the volumes
of the non-overlapping parts, applying this technique to solve real world problems.
Student Friendly/”I Can” statements
1. Recognize that volume is additive in that a
complex 3-d figure can be broken down
into understandable three-dimensional
figures.
2. Solve real world problems by
decomposing a solid figure into two nonoverlapping right rectangular prisms and
adding their volumes.
Resources
http://nlvm.usu.edu
centimeter cubes
right rectangular prisms
Clear three dimensional prisms that can be
opened to fill with centimeter cubes
101
Assessments
Geometry
• Graph points on the coordinate plane to solve real-world and mathematical problems.
Use a pair of perpendicular number lines, called axes, to define a coordinate system, with the intersection of the lines (the origin) arranged to
coincide with the 0 on each line and a given point in the plane located by using an ordered pair of numbers, called its coordinates. Understand
that the first number indicates how far to travel from the origin in the direction of one axis, and the second number indicates how far to travel in
the direction of the second axis, with the convention that the names of the two axes and the coordinates correspond (e.g., x-axis and xcoordinate, y-axis and y-coordinate).
Student Friendly/”I Can” statements
Resources
Assessments
1. Define the coordinate system,
http://nlvm.usu.edu
perpendicular, coordinates
coordinate graphs
2. Identify the x- and y-axis
3. Locate the origin on the coordinate
system
4. Identify coordinates of a point on a
coordinate system
5. Recognize and describe the
connection between the ordered pair
and the x- and y-axis (from the origin)
Represent real world and mathematical problems by graphing points in the first quadrant of the coordinate plane, and interpret coordinate
values of points in the context of the situation.
Student Friendly/”I Can” statements
1. Graph points in the first quadrant
2. Represent real world and mathematical
problems by graphing points in the first
quadrant
3. Interpret coordinate values of points
in real world context and
mathematical problems
Resources
Coordinate planes
Directions for maps
Climbing rope
Learners.org
102
Assessments
• Classify two-dimensional figures into categories based on their properties.
Understand that attributes belonging to a category of two-dimensional figures also belong to all subcategories of that category. For example, all
rectangles have four right angles and squares are rectangles, so all squares have four right angles.
Student Friendly/”I Can” statements
Resources
Assessments
Variety of two dimensional shapes.
1. Recognize that some two-dimensional
shapes can be classified into more than
one category based on their attributes.
2. Describe common attributes.
3. Name categories and determine which
two-dimensional shapes go into which
categories.
Classify two-dimensional figures in a hierarchy based on properties.
Student Friendly/”I Can” statements
Resources
1. Recognize if a two-dimensional shape is
Variety of two dimensional shapes.
classified into a category, that it belongs
to all subcategories of that category.
2. Classify two-dimensional shape according
to categories and subcategories.
103
Assessments
Standards for Mathematical Practices – Grade 6
Standards for Mathematical
Practice
1. Make sense of problems
and persevere in solving
them.
2. Reason abstractly and
quantitatively.
3. Construct viable arguments
and critique the reasoning of
others.
4. Model with mathematics.
5. Use appropriate tools
strategically.
6. Attend to precision.
7. Look for and make use of
Explanations and Examples
In grade 6, students solve real world problems through the application of algebraic and geometric concepts.
These problems involve ratio, rate, area and statistics. Students seek the meaning of a problem and look for
efficient ways to represent and solve it. They may check their thinking by asking themselves, “What is the
most efficient way to solve the problem?”, “Does this make sense?”, and “Can I solve the problem in a
different way?”. Students can explain the relationships between equations, verbal descriptions, tables and
graphs. Mathematically proficient students check answers to problems using a different method.
In grade 6, students represent a wide variety of real world contexts through the use of real numbers and
variables in mathematical expressions, equations, and inequalities. Students contextualize to understand the
meaning of the number or variable as related to the problem and decontextualize to manipulate symbolic
representations by applying properties of operations.
In grade 6, students construct arguments using verbal or written explanations accompanied by expressions,
equations, inequalities, models, and graphs, tables, and other data displays (i.e. box plots, dot plots,
histograms, etc.). They further refine their mathematical communication skills through mathematical
discussions in which they critically evaluate their own thinking and the thinking of other students. They pose
questions like “How did you get that?”, “Why is that true?” “Does that always work?” They explain their
thinking to others and respond to others’ thinking.
In grade 6, students model problem situations symbolically, graphically, tabularly, and contextually. Students
form expressions, equations, or inequalities from real world contexts and connect symbolic and graphical
representations. Students begin to explore covariance and represent two quantities simultaneously. Students
use number lines to compare numbers and represent inequalities. They use measures of center and variability
and data displays (i.e. box plots and histograms) to draw inferences about and make comparisons between data
sets. Students need many opportunities to connect and explain the connections between the different
representations. They should be able to use all of these representations as appropriate to a problem context.
Students consider available tools (including estimation and technology) when solving a mathematical problem
and decide when certain tools might be helpful. For instance, students in grade 6 may decide to represent
figures on the coordinate plane to calculate area. Number lines are used to understand division and to create
dot plots, histograms and box plots to visually compare the center and variability of the data. Additionally,
students might use physical objects or applets to construct nets and calculate the surface area of threedimensional figures.
In grade 6, students continue to refine their mathematical communication skills by using clear and precise
language in their discussions with others and in their own reasoning. Students use appropriate terminology
when referring to rates, ratios, geometric figures, data displays, and components of expressions, equations or
inequalities.
Students routinely seek patterns or structures to model and solve problems. For instance, students recognize
104
structure.
8. Look for and express
regularity in repeated
reasoning.
patterns that exist in ratio tables recognizing both the additive and multiplicative properties. Students apply
properties to generate equivalent expressions (i.e. 6 + 2x = 3 (2 + x) by distributive property) and solve
equations (i.e. 2c + 3 = 15, 2c = 12 by subtraction property of equality, c=6 by division property of equality).
Students compose and decompose two- and three-dimensional figures to solve real world problems involving
area and volume.
In grade 6, students use repeated reasoning to understand algorithms and make generalizations about patterns.
During multiple opportunities to solve and model problems, they may notice that a/b ÷ c/d = ad/bc and
construct other examples and models that confirm their generalization. Students connect place value and their
prior work with operations to understand algorithms to fluently divide multi-digit numbers and perform all
operations with multi-digit decimals. Students informally begin to make connections between covariance,
rates, and representations showing the relationships between quantities.
105
Grade 6 Critical Areas
The Critical Areas are designed to bring focus to the standards at each grade by describing the big ideas that educators can use
to build their curriculum and to guide instruction.
The Critical Areas for sixth grade can be found beginning on page 39 in the Common Core State Standards for Mathematics.
1. Connecting ratio and rate to whole number multiplication and division and using concepts of ratio and rate to solve problems.
Students use reasoning about multiplication and division to solve ratio and rate problems about quantities. By viewing equivalent ratios and rates as
deriving from, and extending, pairs of rows (or columns) in the multiplication table, and by analyzing simple drawings that indicate the relative size of
quantities, students connect their understanding of multiplication and division with ratios and rates. Thus students expand the scope of problems for
which they can use multiplication and division to solve problems, and they connect ratios and fractions. Students solve a wide variety of problems
involving ratios and rates.
2. Completing understanding of division of fractions and extending the notion of number to the system of rational numbers,
which includes negative numbers.
Students use the meaning of fractions, the meanings of multiplication and division, and the relationship between multiplication and division to
understand and explain why the procedures for dividing fractions make sense. Students use these operations to solve problems. Students extend their
previous understandings of number and the ordering of numbers to the full system of rational numbers, which includes negative rational numbers, and in
particular negative integers. They reason about the order and absolute value of rational numbers and about the location of points in all four quadrants of
the coordinate plane.
3. Writing, interpreting, and using expressions and equations.
Students understand the use of variables in mathematical expressions. They write expressions and equations that correspond to given situations, evaluate
expressions, and use expressions and formulas to solve problems. Students understand that expressions in different forms can be equivalent, and they use
the properties of operations to rewrite expressions in equivalent forms. Students know that the solutions of an equation are the values of the variables that
make the equation true. Students use properties of operations and the idea of maintaining the equality of both sides of an equation to solve simple onestep equations. Students construct and analyze tables, such as tables of quantities that are in equivalent ratios, and they use equations (such as 3x = y) to
describe relationships between quantities.
4. Developing understanding of statistical thinking.
Building on and reinforcing their understanding of number, students begin to develop their ability to think statistically. Students recognize that a data
distribution may not have a definite center and that different ways to measure center yield different values. The median measures center in the sense that
it is roughly the middle value. The mean measures center in the sense that it is the value that each data point would take on if the total of the data values
were redistributed equally, and also in the sense that it is a balance point. Students recognize that a measure of variability (interquartile range or mean
absolute deviation) can also be useful for summarizing data because two very different sets of data can have the same mean and median yet be
distinguished by their variability. Students learn to describe and summarize numerical data sets, identifying clusters, peaks, gaps, and symmetry,
considering the context in which the data were collected.
106
Expressions and Equations
• Apply and extend previous understandings of arithmetic to algebraic expressions.
Write and evaluate numerical expressions involving whole-number exponents.
Student Friendly/”I Can” statements
Resources
Assessments
1. Write numerical expressions involving
http://nlvm.usu.edu
whole number exponents
Ex. 34 = 3x3x3x3
2. Evaluate numerical expressions involving
whole number exponents
Ex. 34= 3x3x3x3 = 81
3. Solve order of operation problems that
contain exponents
Ex. 3 + 22 – (2 + 3) = 2
Write, read, and evaluate expressions in which letters stand for numbers.
a. Write expressions that record operations with numbers and with letters standing for numbers. For example, express the calculation “Subtract
y from 5” as 5 – y.
Student Friendly/”I Can” statements
1. Use variables to stand in for numbers in
expressions.
2. Translate written phrases into algebraic
expressions.
Resources
Assessments
Algebra tiles
Cups
Centimeter cubes
3. Translate algebraic expressions into
written phrases.
b. Identify parts of an expression using mathematical terms (sum, term, product, factor, quotient, coefficient); view one or more parts of an
expression as a single entity. For example, describe the expression 2 (8 + 7) as a product of two factors; view (8 + 7) as both a single entity and a
sum of two terms.
Student Friendly/”I Can” statements
Resources
Assessments
1. Identify parts of an expression using
mathematical terms (sum, term, product, Algebra tiles
factor, quotient, coefficient, constant,
Cups
monomial)
Centimeter cubes
2. Identify parts of an expression as a single
entity, even if not a monomial.
107
c. Evaluate expressions at specific values of their variables. Include expressions that arise from formulas used in real-world problems. Perform
arithmetic operations, including those involving whole number exponents, in the conventional order when there are no parentheses to specify a
particular order (Order of Operations). For example, use the formulas V = s3 and A = 6 s2 to find the volume and surface area of a cube with sides
of length s = 1/2.
Student Friendly/”I Can” statements
1. Substitute specific values for variables.
2. Evaluate algebraic expressions including
those that arise from real-world problems.
3. Apply order of operations when there
are no parentheses for expressions
that include whole number exponents.
Resources
Assessments
Algebra tiles
Cups
Centimeter cubes
Equality mats
Apply the properties of operations to generate equivalent expressions. For example, apply the distributive property to the expression 3 (2 + x) to
produce the equivalent expression 6 + 3x; apply the distributive property to the expression 24x + 18y to produce the equivalent expression 6 (4x +
3y); apply properties of operations to y + y + y to produce the equivalent expression 3y.
Student Friendly/”I Can” statements
Resources
Assessments
1. Generate equivalent expressions using the Algebra tiles
properties of operations. (e.g. distributive Cups
property, associative property, adding like Centimeter cubes
terms with the addition property of
Equality mats
equality, etc.)
2. Apply the properties of operations to
generate equivalent expressions.
Identify when two expressions are equivalent (i.e., when the two expressions name the same number regardless of which value is substituted
into them). For example, the expressions y + y + y and 3y are equivalent because they name the same number regardless of which number y
stands for.
Student Friendly/”I Can” statements
Resources
Assessments
1. Recognize when two expressions are
Algebra tiles
equivalent.
Cups
Centimeter cubes
2. Prove (using various strategies) that two
Equality mats
equations are equivalent no matter what
number is substituted.
108
• Reason about and solve one-variable equations and inequalities.
Understand solving an equation or inequality as a process of answering a question: which values from a specified set, if any, make the equation
or inequality true? Use substitution to determine whether a given number in a specified set makes an equation or inequality true.
Student Friendly/”I Can” statements
Resources
Assessments
Algebra
tiles
1. Recognize solving an equation or
Cups
inequality as a process of answering
Centimeter cubes
“which values from a specified set, if any,
Equality mats
make the equation or inequality true?”
2. Know that the solutions of an equation or
inequality are the values that make the
equation or inequality true.
3. Use substitution to determine whether a
given number in a specified set makes an
equation or inequality true.
Use variables to represent numbers and write expressions when solving a real-world or mathematical problem; understand that a variable can
represent an unknown number, or, depending on the purpose at hand, any number in a specified set.
Student Friendly/”I Can” statements
1. Recognize that a variable can represent an
unknown number, or, depending on the
purpose at hand, any number in a
specified set.
2. Relate variables to a context.
3. Write expressions when solving a realworld or mathematical problem
Resources
Algebra tiles
Cups
Centimeter cubes
Equality mats
109
Assessments
Solve real-world and mathematical problems by writing and solving equations of the form x + p = q and px = q for cases in which p, q and x are all
nonnegative rational numbers.
Student Friendly/”I Can” statements
Resources
Assessments
1. Define inverse operation.
2. Know how inverse operations can be used Algebra tiles
in solving one-variable equations.
Cups
Centimeter cubes
3. Apply rules of the form x + p = q and px =
Equality mats
q, for cases in which p, q and x are all
nonnegative rational numbers, to solve
real world and mathematical problems.
(There is only one unknown quantity.)
4. Develop a rule for solving one-step
equations using inverse operations with
nonnegative rational coefficients.
5. Solve and write equations for real-world
mathematical problems containing one
unknown.
Write an inequality of the form x > c or x < c to represent a constraint or condition in a real-world or mathematical problem. Recognize that
inequalities of the form x > c or x < c have infinitely many solutions; represent solutions of such inequalities on number line diagrams.
Student Friendly/”I Can” statements
Resources
Assessments
1. Identify the constraint or condition in a
Algebra tiles
real-world or mathematical problem in
Cups
order to set up an inequality.
Centimeter cubes
2. Recognize that inequalities of the form x > Inequality mats
Number lines
c or x < c have infinitely many solutions.
3. Write an inequality of the form x > c or x <
c to represent a constraint or condition in
a real-world or mathematical problem.
4. Represent solutions to inequalities or the
form x > c or x < c, with infinitely many
solutions, on number line diagrams.
110
• Represent and analyze quantitative relationships between dependent and independent variables.
Student Friendly/”I Can” statements
Resources
1. Define independent and dependent
variables.
http://nlvm.usu.edu
Assessments
Use variables to represent two quantities in a real-world problem that change in relationship to one another; write an equation to express one
quantity, thought of as the dependent variable, in terms of the other quantity, thought of as the independent variable. Analyze the relationship
between the dependent and independent variables using graphs and tables, and relate these to the equation. For example, in a problem
involving motion at constant speed, list and graph ordered pairs of distances and times, and write the equation d = 65t to represent the
relationship between distance and time
Student Friendly/”I Can” statements
Resources
Assessments
1. Use variables to represent two quantities
in a real-world problem that change in
Graph paper
relationship to one another.
2. Write an equation to express one quantity
(dependent) in terms of the other
quantity (independent).
3. Analyze the relationship between the
dependent variable and independent
variable using tables and graphs
4. Relate the data in a graph and table to the
corresponding equation.
111
The Number System
• Apply and extend previous understandings of multiplication and division to divide fractions by fractions.
Interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions, e.g., by using visual fraction
models and equations to represent the problem. For example, create a story context for (2/3) ÷ (3/4) and use a visual fraction model to show the
quotient; use the relationship between multiplication and division to explain that (2/3) ÷ (3/4) = 8/9 because 3/4 of 8/9 is 2/3. (In general, (a/b) ÷
(c/d) = ad/bc.) How much chocolate will each person get if 3 people share 1/2 lb of chocolate equally? How many 3/4-cup servings are in 2/3 of a
cup of yogurt? How wide is a rectangular strip of land with length 3/4 mi and area 1/2 square mi?
Student Friendly/”I Can” statements
1. Compute quotients of fractions divided
by fractions (including mixed numbers).
2. Interpret quotients of fractions
3. Solving word problems involving division
of fractions by fractions, e.g., by using
visual fraction models and equations to
represent the problem.
Resources
http://nlvm.usu.edu
fraction tiles
graph paper
Study Island lesson
Plain paper
Cuisennarie rods
DOE Released items
M & M Book
Sports Activity Books
Mathwarehouse.com
• Compute fluently with multi-digit numbers and find common factors and multiples.
Fluently divide multi-digit numbers using the standard algorithm.
Student Friendly/”I Can” statements
Resources
http://nlvm.usu.edu
1. Fluently divide multi-digit numbers
using the standard algorithm with speed Cuisennarie rods
and accuracy.
Assessments
Assessments
Multiplication facts timed tests
Division facts timed tests
Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation.
Student Friendly/”I Can” statements
Resources
1. Fluently add, subtract, multiply, and
Graph paper
divide multi-digit decimals using the
standard algorithm for each operation
with speed and accuracy.
112
Assessments
Find the greatest common factor of two whole numbers less than or equal to 100 and the least common multiple of two whole numbers less
than or equal to 12. Use the distributive property to express a sum of two whole numbers 1–100 with a common factor as a multiple of a sum of
two whole numbers with no common factor. For example, express 36 + 8 as 4 (9 + 2).
Student Friendly/”I Can” statements
Resources
Assessments
1. Identify the factors of two whole numbers
less than or equal to 100 and determine
the Greatest Common Factor.
2. Identify the multiples of two whole
numbers less than or equal to 12 and
determine the Least Common Multiple.
3. Apply the Distributive Property to rewrite
addition problems by factoring out the
Greatest Common Factor.
• Apply and extend previous understandings of numbers to the system of rational numbers.
Understand that positive and negative numbers are used together to describe quantities having opposite directions or values (e.g., temperature
above/below zero, elevation above/below sea level, credits/debits, positive/negative electric charge); use positive and negative numbers to
represent quantities in real-world contexts, explaining the meaning of 0 in each situation.
Student Friendly/”I Can” statements
Resources
Assessments
1. Identify an integer and its opposite
Two colored counters
2. Use integers to represent quantities in
Algebra tiles
real world situations (above/below sea
Number lines
level, etc)
Magnetic number line with blow up integer
3. Explain where zero fits into a situation
cubes
represented by integers
Dolphin game
+, _ basic mat
Integer App (iPad)
Poker chips (different colors)
In relation to money – purchasing, checkbook
Understand a rational number as a point on the number line. Extend number line diagrams and coordinate axes familiar from previous grades to
represent points on the line and in the plane with negative number coordinates.
a. Recognize opposite signs of numbers as indicating locations on opposite sides of 0 on the number line; recognize that the opposite of the
opposite of a number is the number itself, e.g., –(–3) = 3, and that 0 is its own opposite.
Student Friendly/”I Can” statements
Resources
Assessments
1. Identify a rational number as a point on
Two colored counters
the number line.
Algebra tiles
2. Identify the location of zero on a number
Number lines
113
line in relation to positive and negative
numbers
3. Recognize opposite signs of numbers as
locations on opposite sides of 0 on the
number line
4. Reason that the opposite of the opposite
of a number is the number itself.
b. Understand signs of numbers in ordered pairs as indicating locations in quadrants of the coordinate plane; recognize that when two ordered
pairs differ only by signs, the locations of the points are related by reflections across one or both axes.
1.
2.
3.
4.
5.
6.
Student Friendly/”I Can” statements
Recognize the signs of both numbers in an
ordered pair indicate which quadrant of
the coordinate plane the ordered pair will
be located
Find and position integers and other
rational numbers on a horizontal or
vertical number line diagram
Find and position pairs of integers and
other rational numbers on a coordinate
plane
Reason that when only the x value in a set
of ordered pairs are opposites, it creates a
reflection over the y axis, e.g., (x,y) and (x,y)
Recognize that when only the y value in a
set of ordered pairs are opposites, it
creates a reflection over the x axis, e.g.,
(x,y) and (x, -y)
Reason that when two ordered pairs differ
only by signs, the locations of the points
are related by reflections across both
axes, e.g., (-x, -y) and (x,y)
Resources
Coordinate planes
114
Assessments
c. Find and position integers and other rational numbers on a horizontal or vertical number line diagram; find and position pairs of
integers and other rational numbers on a coordinate plane.
Student Friendly/”I Can” statements
1. Find and position integers and other
rational numbers on a horizontal or
vertical number line diagram
2. Find and position pairs of integers and
other rational numbers on a coordinate
plane
Resources
Assessments
Coordinate Planes
Understand ordering and absolute value of rational numbers.
a. Interpret statements of inequality as statements about the relative position of two numbers on a number line diagram. For example, interpret
–3 > –7 as a statement that –3 is located to the right of –7 on a number line oriented from left to right.
Student Friendly/”I Can” statements
Resources
Assessments
1. Interpret statements of inequality as
statements about relative position of two Number lines
numbers on a number line diagram.
b. Write, interpret, and explain statements of order for rational numbers in real-world contexts. For example, write –3 oC > –7 oC to express the
fact that –3 oC is warmer than –7 oC.
Student Friendly/”I Can” statements
Resources
Assessments
1. Order rational numbers on a number line
Number lines
115
c. Understand the absolute value of a rational number as its distance from 0 on the number line; interpret absolute value as magnitude for a
positive or negative quantity in a real-world situation. For example, for an account balance of –30 dollars, write |–30| = 30 to describe the size of
the debt in dollars.
Student Friendly/”I Can” statements
Resources
Assessments
1. Identify absolute value of rational
numbers
2. Interpret statements of inequality as
statements about relative position of two
numbers on a number line diagram.
3. Interpret absolute value as magnitude
for a positive or negative quantity in a
real-world situation
d. Distinguish comparisons of absolute value from statements about order. For example, recognize that an account balance less than –30 dollars
represents a debt greater than 30 dollars.
Student Friendly/”I Can” statements
Resources
Assessments
1. Write, interpret, and explain statements
of order for rational numbers in realNumber lines
world contexts
2. Distinguish comparisons of absolute value
from statements about order and apply to
real world contexts
Solve real-world and mathematical problems by graphing points in all four quadrants of the coordinate plane. Include use of coordinates and
absolute value to find distances between points with the same first coordinate or the same second coordinate.
Student Friendly/”I Can” statements
1. Calculate absolute value.
2. Graph points in all four quadrants of the
coordinate plane.
3. Solve real-world problems by graphing
points in all four quadrants of a
coordinate plane.
4. Given only coordinates, calculate the
distances between two points with the
same first coordinate or the same second
Resources
Coordinate planes
116
Assessments
coordinate using absolute value.
117
Statistics and Probability
• Develop understanding of statistical variability.
Recognize a statistical question as one that anticipates variability in the data related to the question and accounts for it in the answers. For
example, “How old am I?” is not a statistical question, but “How old are the students in my school?” is a statistical question because one
anticipates variability in students’ ages.
Student Friendly/”I Can” statements
1. Recognize that data can have variability.
2. Recognize a statistical question
(examples versus non-examples).
Resources
Assessments
Understand that a set of data collected to answer a statistical question has a distribution which can be described by its center, spread, and
overall shape.
Student Friendly/”I Can” statements
Resources
Assessments
1. Know that a set of data has a distribution.
2. Describe a set of data by its center, e.g.,
mean and median.
3. Describe a set of data by its spread and
overall shape, e.g. by identifying data
clusters, peaks, gaps and symmetry
118
Recognize that a measure of center for a numerical data set summarizes all of its values with a single number, while a measure of variation
describes how its values vary with a single number.
Student Friendly/”I Can” statements
1. Recognize there are measures of central
tendency for a data set, e.g., mean,
median, mode.
2. Recognize there are measures of
variances for a data set, e.g., range,
interquartile range, mean absolute
deviation.
3. Recognize measures of central tendency
for a data set summarizes the data with a
single number.
4. Recognize measures of variation for a data
set describes how its values vary with a
single number.
Resources
http://nlvm.usu.edu
Assessments
• Summarize and describe distributions.
Display numerical data in plots on a number line, including dot plots, histograms, and box plots.
Student Friendly/”I Can” statements
Resources
http://nlvm.usu.edu
1. Identify the components of dot plots,
histograms, and box plots.
2. Analyze a set of data to determine its
variance.
3. Create a dot plot to display a set of
numerical data.
4. Create a histogram to display a set of
numerical data.
5. Find the median, quartile and
interquartile range of a set of data.
6. Create a box plot to display a set of
numerical data.
119
Assessments
Summarize numerical data sets in relation to their context, such as by:
a. Reporting the number of observations.
Student Friendly/”I Can” statements
Resources
1. Organize and display data in tables and
graphs.
2. Report the number of observations in a
data set or display.
Assessments
b. Describing the nature of the attribute under investigation, including how it was measured and its units of measurement.
Student Friendly/”I Can” statements
1. Describe the data being collected,
including how it was measured and its
units of measurement.
Resources
Assessments
c. Giving quantitative measures of center (median and/or mean) and variability (interquartile range and/or mean absolute deviation), as well as
describing any overall pattern and any striking deviations from the overall pattern with reference to the context in which the data were
gathered.
Student Friendly/”I Can” statements
1. Calculate quantitative measures of center,
e.g., mean, median, mode.
2. Calculate quantitative measures of
variance, e.g., range, interquartile range,
mean absolute deviation.
3. Identify outliers
4. Determine the effect of outliers on
quantitative measures of a set of data,
e.g., mean, median, mode, range,
interquartile range, mean absolute
deviation.
5. Choose the appropriate measure of
central tendency to represent the data.
Resources
Assessments
d. Relating the choice of measures of center and variability to the shape of the data distribution and the context in which the data were
gathered.
120
Student Friendly/”I Can” statements
1. Analyze the shape of the data distribution
and the context in which the data were
gathered to choose the appropriate
measures of central tendency and
variability and justify why this measure is
appropriate in terms of the context
Resources
121
Assessments
Ratios and Proportional Relationships
• Understand ratio concepts and use ratio reasoning to solve problems.
Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities. For example, “The ratio of
wings to beaks in the bird house at the zoo was 2:1, because for every 2 wings there was 1 beak.” “For every vote candidate A received,
candidate C received nearly three votes.”
Student Friendly/”I Can” statements
Resources
Assessments
1. Write ratio notationhttp://nlvm.usu.edu
__:__, __ to __, __/__
Study Island lesson
2. Know order matters when writing a ratio
Shodor.org
3. Know ratios can be simplified
Learner.org
4. Know ratios compare two quantities; the
Mathgoodies
quantities do not have to be the same unit A Math Curse by John Sueszka
of measure
5. Recognize that ratios appear in a variety
of different contexts; part-to-whole, partto-part, and rates
6. Generalize that all ratios relate two
quantities or measures within a given
situation in a multiplicative relationship.
7. Analyze your context to determine which
kind of ratio is represented.
Understand the concept of a unit rate a/b associated with a ratio a:b with b ≠ 0, and use rate language in the context of a ratio relationship. For
example, “This recipe has a ratio of 3 cups of flour to 4 cups of sugar, so there is 3/4 cup of flour for each cup of sugar.” “We paid $75 for 15
hamburgers, which is a rate of $5 per hamburger.”
Student Friendly/”I Can” statements
1. Identify and calculate a unit rate.
2. Use appropriate math terminology as
related to rate.
3. Analyze the relationship between a ratio
a:b and a unit rate a/b where b ≠ 0.
Resources
http://nlvm.usu.edu
Study Island lesson
Shodor.org
Learner.org
Mathgoodies
A Math Curse by John Sueszka
122
Assessments
Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams,
double number line diagrams, or equations.
a. Make tables of equivalent ratios relating quantities with whole number measurements, find missing values in the tables, and plot the pairs of
values on the coordinate plane. Use tables to compare ratios.
Student Friendly/”I Can” statements
Resources
Assessments
1. Make a table of equivalent ratios using
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whole numbers.
Study Island lesson
2. Find the missing values in a table of
Shodor.org
equivalent ratios.
Learner.org
3. Plot pairs of values that represent
Mathgoodies
equivalent ratios on the coordinate plane. A Math Curse by John Sueszka
4. Use tables to compare proportional
quantities.
b. Solve unit rate problems including those involving unit pricing and constant speed. For example, if it took 7 hours to mow 4 lawns, then at that
rate, how many lawns could be mowed in 35 hours? At what rate were lawns being mowed?
Student Friendly/”I Can” statements
1. Solve real-world and mathematical
problems involving ratio and rate, e.g., by
reasoning about tables of equivalent
ratios, tape diagrams, double number line
diagrams, or equations.
2. Apply the concept of unit rate to solve
real-world problems involving unit pricing.
3. Apply the concept of unit rate to solve
real-world problems involving constant
speed.
Resources
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Study Island lesson
Shodor.org
Learner.org
Mathgoodies
A Math Curse by John Sueszka
123
Assessments
c. Find a percent of a quantity as a rate per 100 (e.g., 30% of a quantity means 30/100 times the quantity); solve problems involving finding the
whole, given a part and the percent.
Student Friendly/”I Can” statements
1. Know that a percent is a ratio of a number
to 100.
2. Find a % of a number as a rate per 100.
3. Solve real-world problems involving
finding the whole, given a part and a
percent.
Resources
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Study Island lesson
Shodor.org
Learner.org
Mathgoodies
A Math Curse by John Sueszka
Assessments
d. Use ratio reasoning to convert measurement units; manipulate and transform units appropriately when multiplying or dividing quantities.
Student Friendly/”I Can” statements
Resources
Assessments
1. Apply ratio reasoning to convert
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measurement units in real-world and
Study Island lesson
mathematical problems.
Shodor.org
2. Apply ratio reasoning to convert
Learner.org
measurement units by multiplying or
Mathgoodies
dividing in real-world and mathematical
A Math Curse by John Sueszka
problems.
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Geometry
• Solve real-world and mathematical problems involving area, surface area, and volume.
Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles
and other shapes; apply these techniques in the context of solving real-world and mathematical problems.
Student Friendly/”I Can” statements
Resources
Assessments
1. Recognize and know how to compose http://nlvm.usu.edu
www.learner.org/interactives/geometry/index.html
and decompose polygons into
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triangles and rectangles.
2. Compare the area of a triangle to the Various polygons
Dot paper
area of the composted rectangle.
(Decomposition addressed in previous Graph paper
grade.)
3. Apply the techniques of composing
and/or decomposing to find the area
of triangles, special quadrilaterals and
polygons to solve mathematical and
real world problems.
4. Discuss, develop and justify formulas
for triangles and parallelograms (6th
grade introduction)
125
Find the volume of a right rectangular prism with fractional edge lengths by packing it with unit cubes of the appropriate unit fraction edge
lengths, and show that the volume is the same as would be found by multiplying the edge lengths of the prism. Apply the formulas V = l w h and
V = b h to find volumes of right rectangular prisms with fractional edge lengths in the context of solving real-world and mathematical problems.
Student Friendly/”I Can” statements
Resources
Assessments
1. Know how to calculate the volume of
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a right rectangular prism.
www.learner.org/interactives/geometry/index.html
2. Apply volume formulas for right
rectangular prisms to solve real-world softchalkconnect.com
and mathematical problems involving Various right prisms –open to input centimeter
cubes
rectangular prisms with fractional
Centimeter cubes
edge lengths.
Dot paper
3. Model the volume of a right
rectangular prism with fractional edge Graph paper
lengths by packing it with unit cubes
of the appropriate unit fraction edge
lengths.
Draw polygons in the coordinate plane given coordinates for the vertices; use coordinates to find the length of a side joining points with the
same first coordinate or the same second coordinate. Apply these techniques in the context of solving real-world and mathematical problems.
Student Friendly/”I Can” statements
1. Draw polygons in the coordinate
plane.
2. Use coordinates (with the same xcoordinate or the same y-coordinate)
to find the length of a side of a
polygon.
3. Apply the technique of using
coordinates to find the length of a
side of a polygon drawn in the
coordinate plane to solve real-world
and mathematical problems.
Resources
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www.learner.org/interactives/geometry/index.html
softchalkconnect.com
coordinate planes
126
Assessments
Represent three-dimensional figures using nets made up of rectangles and triangles, and use the nets to find the surface area of these figures.
Apply these techniques in the context of solving real-world and mathematical problems.
Student Friendly/”I Can” statements
1. Know that 3-D figures can be
represented by nets.
2. Represent three-dimensional figures
using nets made up of rectangles and
triangles.
3. Apply knowledge of calculating the area
of rectangles and triangles to a net, and
combine the areas for each shape into
one answer representing the surface
area of a 3-dimensional figure.
4. Solve real-world and mathematical
problems involving surface area using
nets.
Resources
http://nlvm.usu.edu
www.learner.org/interactives/geometry/index.html
softchalkconnect.com
dot paper
graph paper
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Assessments
Standards for Mathematical
Practice
1. Make sense of problems
and persevere in solving
them.
2. Reason abstractly and
quantitatively.
3. Construct viable arguments
and critique the reasoning of
others.
4. Model with mathematics.
5. Use appropriate tools
Strategically.
6. Attend to precision.
7. Look for and make use of
structure.
Standards for Mathematical Practice – Grade 7
Explanations and Examples
In grade 7, students solve problems involving ratios and rates and discuss how they solved the problems.
Students solve real world problems through the application of algebraic and geometric concepts. Students seek
the meaning of a problem and look for efficient ways to represent and solve it. They may check their thinking
by asking themselves, “What is the most efficient way to solve the problem?”, “Does this make sense?”, and
“Can I solve the problem in a different way?”.
In grade 7, students represent a wide variety of real world contexts through the use of real numbers and
variables in mathematical expressions, equations, and inequalities. Students contextualize to understand the
meaning of the number or variable as related to the problem and decontextualize to manipulate symbolic
representations by applying properties of operations.
In grade 7, students construct arguments using verbal or written explanations accompanied by expressions,
equations, inequalities, models, and graphs, tables, and other data displays (i.e. box plots, dot plots,
histograms, etc.). The students further refine their mathematical communication skills through mathematical
discussions in which they critically evaluate their own thinking and the thinking of other students. They pose
questions like “How did you get that?”, “Why is that true?”, “Does that always work?”. They explain their
thinking to others and respond to others’ thinking.
In grade 7, students model problem situations symbolically, graphically, tabularly, and contextually. Students
form expressions, equations, or inequalities from real world contexts and connect symbolic and graphical
representations. Students explore covariance and represent two quantities simultaneously. They use measures
of center and variability and data displays (i.e. box plots and histograms) to draw inferences, make
comparisons and formulate predictions. Students use experiments or simulations to generate data sets and
create probability models. Students need many opportunities to connect and explain the connections between
the different representations. They should be able to use all of these representations as appropriate to any
problem’s context.
Students consider available tools (including estimation and technology) when solving a mathematical problem
and decide when certain tools might be helpful. For instance, students in grade 7 may decide to represent
similar data sets using dot plots with the same scale to visually compare the center and variability of the data.
Students might use physical objects or applets to generate probability data and use graphing calculators or
spreadsheets to manage and represent data in different forms.
In grade 7, students continue to refine their mathematical communication skills by using clear and precise
language in their discussions with others and in their own reasoning. Students define variables, specify units of
measure, and label axes accurately. Students use appropriate terminology when referring to rates, ratios,
probability models, geometric figures, data displays, and components of expressions, equations or inequalities.
Students routinely seek patterns or structures to model and solve problems. For instance, students recognize
patterns that exist in ratio tables making connections between the constant of proportionality in a table with the
slope of a graph. Students apply properties to generate equivalent expressions (i.e. 6 + 2x = 3 (2 + x) by
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8. Look for and express
regularity in repeated
reasoning.
distributive property) and solve equations (i.e. 2c + 3 = 15, 2c = 12 by subtraction property of equality), c = 6
by division property of equality). Students compose and decompose two- and three-dimensional figures to
solve real world problems involving scale drawings, surface area, and volume. Students examine tree diagrams
or systematic lists to determine the sample space for compound events and verify that they have listed all
possibilities.
In grade 7, students use repeated reasoning to understand algorithms and make generalizations about patterns.
During multiple opportunities to solve and model problems, they may notice that a/b ÷ c/d = ad/bc and
construct other examples and models that confirm their generalization. They extend their thinking to include
complex fractions and rational numbers. Students formally begin to make connections between covariance,
rates, and representations showing the relationships between quantities. They create, explain, evaluate, and
modify probability models to describe simple and compound events.
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Grade 7 Critical Areas (from CCSS pg. 46)
The Critical Areas are designed to bring focus to the standards at each grade by describing the big ideas that educators can use
to build theircurriculum and to guide instruction. The Critical Areas for seventh grade can be found on page 46 in the Common Core State
Standards for Mathematics.
1. Developing understanding of and applying proportional relationships
Students extend their understanding of ratios and develop understanding of proportionality to solve single- and multi-step problems. Students use their
understanding of ratios and proportionality to solve a wide variety of percent problems, including those involving discounts, interest, taxes, tips, and
percent increase or decrease. Students solve problems about scale drawings by relating corresponding lengths between the objects or by using the fact
that relationships of lengths within an object are preserved in similar objects. Students graph proportional relationships and understand the unit rate
informally as a measure of the steepness of the related line, called the slope. They distinguish proportional relationships from other relationships.
2. Developing understanding of operations with rational numbers and working with expressions and linear equations
Students develop a unified understanding of number, recognizing fractions, decimals (that have a finite or a repeating decimal representation), and
percents as different representations of rational numbers. Students extend addition, subtraction, multiplication, and division to all rational numbers,
maintaining the properties of operations and the relationships between addition and subtraction, and multiplication and division. By applying these
properties, and by viewing negative numbers in terms of everyday contexts (e.g., amounts owed or temperatures below zero), students explain and
interpret the rules for adding, subtracting, multiplying, and dividing with negative numbers. They use the arithmetic of rational numbers as they
formulate expressions and equations in one variable and use these equations to solve problems.
3. Solving problems involving scale drawings and informal geometric constructions, and working with two- and threedimensional shapes to solve problems involving area, surface area, and volume
Students continue their work with area from Grade 6, solving problems involving the area and circumference of a circle and surface area of
threedimensional objects. In preparation for work on congruence and similarity in Grade 8 they reason about relationships among two-dimensional
figures using scale drawings and informal geometric constructions, and they gain familiarity with the relationships between angles formed by intersecting
lines. Students work with three-dimensional figures, relating them to two-dimensional figures by examining cross-sections. They solve real-world and
mathematical problems involving area, surface area, and volume of two- and three-dimensional objects composed of triangles, quadrilaterals, polygons,
cubes and right prisms.
4. Drawing inferences about populations based on samples
Students build on their previous work with single data distributions to compare two data distributions and address questions about differences between
populations. They begin informal work with random sampling to generate data sets and learn about the importance of representative samples for drawing
inferences.
Expressions and Equations
130
• Use properties of operations to generate equivalent expressions.
Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients.
Student Friendly/”I Can” statements
1. Define like terms, coefficients, constants,
Resources
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pattern blocks to describe like terms,
coefficients
Assessments
linear expressions, and rational numbers.
2. Combine like terms with rational
coefficients.
3. Factor and expand linear expressions with
rational coefficients using the distributive
property.
4. Apply properties of operations as
strategies to add, subtract, factor, and
expand linear expressions with rational
coefficients.
Understand that rewriting an expression in different forms in a problem context can shed light on the problem and how the quantities in it are
related. For example, a + 0.05a = 1.05a means that “increase by 5%” is the same as “multiply by 1.05.”
Student Friendly/”I Can” statements
Resources
Assessments
1. Write equivalent expressions with
fractions, decimals, percents, and
integers.
2. Rewrite an expression in an equivalent
form in order to provide insight about
how quantities are related in a problem
context
131
• Solve real-life and mathematical problems using numerical and algebraic expressions and equations.
Solve multi-step real-life and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions,
and decimals), using tools strategically. Apply properties of operations to calculate with numbers in any form; convert between forms as
appropriate; and assess the reasonableness of answers using mental computation and estimation strategies. For example: If a woman making
$25 an hour gets a 10% raise, she will make an additional 1/10 of her salary an hour, or $2.50, for a new salary of $27.50. If you want to place a
towel bar 9 3/4 inches long in the center of a door that is 27 1/2 inches wide, you will need to place the bar about 9 inches from each edge; this
estimate can be used as a check on the exact computation.
Student Friendly/”I Can” statements
Resources
Assessments
1. Recognize key vocabulary in word
problems.
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2. Identify the problem.
3. Predict the outcome.
4. Solve multi-step real-life and
mathematical problems posed with
positive and negative rational numbers in
any form (whole numbers, fractions, and
decimals), using tools strategically.
5. Apply properties of operations to
calculate with numbers in any form.
6. Convert between numerical forms as
appropriate.
7. Assess the reasonableness of answers
using mental computation and estimation
strategies (compare answer to
prediction).
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Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve
problems by reasoning about the quantities.
a. Solve word problems leading to equations of the form px + q = r and p(x + q) = r, where p, q, and r are specific rational numbers. Solve
equations of these forms fluently. Compare an algebraic solution to an arithmetic solution, identifying the sequence of the operations used in
each approach. For example, the perimeter of a rectangle is 54 cm. Its length is 6 cm. What is its width?
1.
2.
3.
4.
5.
Student Friendly/”I Can” statements
Fluently solve equations of the form px +
q = r and p(x + q) = r with speed and
accuracy.
Identify the sequence of operations used
to solve an algebraic equation of the form
px + q = r and p(x + q) = r.
Use variables and construct equations to
represent quantities of the form px + q = r
and p(x + q) = r from real-world and
mathematical problems.
Solve word problems leading to equations
of the form px + q = r and p(x + q) = r,
where p, q, and r are specific rational
numbers.
Compare an algebraic solution to an
arithmetic solution by identifying the
sequence of the operations used in each
approach. For example, the perimeter of a
rectangle is 54 cm. Its length is 6 cm.
What is its width? This can be answered
algebraically by using only the formula for
perimeter (P=2l+2w) to isolate w or by
finding an arithmetic solution by
substituting values into the formula.
Resources
133
Assessments
b. Solve word problems leading to inequalities of the form px + q > r or px + q < r, where p, q, and r are specific rational numbers. Graph the
solution set of the inequality and interpret it in the context of the problem. For example: As a salesperson, you are paid $50 per week plus $3 per
sale. This week you want your pay to be at least $100. Write an inequality for the number of sales you need to make, and describe the solutions.
Student Friendly/”I Can” statements
Resources
Assessments
1. Solve word problems leading to
inequalities of the form px + q > r or px + q
< r, where p, q, and r are specific rational
numbers.
2. Graph the solution set of the inequality of
the form px + q > r or px + q < r, where p,
q, and r are specific rational numbers.
3. Interpret the solution set of an inequality
in the context of the problem.
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The Number System
• Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers.
Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction
on a horizontal or vertical number line diagram.
a. Describe situations in which opposite quantities combine to make 0. For example, a hydrogen atom has 0 charge because its two constituents
are oppositely charged.
Student Friendly/”I Can” statements
Resources
Assessments
1. Describe situations in which opposite
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quantities combine to make zero (sports
number lines
and money examples).
b. Understand p + q as the number located a distance |q| from p, in the positive or negative direction depending on whether q is positive or
negative. Show that a number and its opposite have a sum of 0 (are additive inverses). Interpret sums of rational numbers by describing realworld contexts.
Student Friendly/”I Can” statements
Resources
Assessments
1. Understand addition and subtraction of
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positives and negatives and describe real
number lines
world contexts.
c. Understand subtraction of rational numbers as adding the additive inverse, p – q = p + (–q). Show that the distance between two rational
numbers on the number line is the absolute value of their difference, and apply this principle in real-world contexts.
Student Friendly/”I Can” statements
Resources
1. Identify properties of addition and
subtraction when adding and subtracting
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rational numbers.
number lines
d. Apply properties of operations as strategies to add and subtract rational numbers.
Student Friendly/”I Can” statements
1. Apply properties of operations as
strategies to add and subtract rational
numbers.
Resources
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number lines
135
Assessments
Assessments
Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.
a. Understand that multiplication is extended from fractions to rational numbers by requiring that operations continue to satisfy the properties
of operations, particularly the distributive property, leading to products such as (–1)(–1) = 1 and the rules for multiplying signed numbers.
Interpret products of rational numbers by describing real-world contexts.
Student Friendly/”I Can” statements
1. Recognize that the process for multiplying
fractions can be used to multiply rational
numbers including integers.
2. Know and describe the rules when
multiplying signed numbers.
3. Apply the properties of operations,
particularly distributive property, to
multiply rational numbers.
4. Interpret the products of rational
numbers by describing real-world
contexts.
Resources
Assessments
b. Understand that integers can be divided, provided that the divisor is not zero, and every quotient of integers (with non-zero divisor) is a
rational number. If p and q are integers, then –(p/q) = (–p)/q = p/(–q). Interpret quotients of rational numbers by describing real-world contexts.
1.
2.
3.
4.
5.
Student Friendly/”I Can” statements
Explain why integers can be divided
except when the divisor is 0.
Describe why the quotient is always a
rational number .
Know and describe the rules when
dividing signed numbers, integers.
Recognize that –(p/q) = -p/q = p/-q.
Interpret the quotient of rational numbers
by describing real-world contexts.
Resources
136
Assessments
c. Apply properties of operations as strategies to multiply and divide rational numbers.
Student Friendly/”I Can” statements
Resources
1. Identify how properties of operations can
be used to multiply and divide rational
numbers (such as distributive property,
multiplicative inverse property,
multiplicative identity, commutative
property for multiplication, associative
property for multiplication, etc.)
2. Apply properties of operations as
strategies to multiply and divide rational
numbers.
Assessments
d. Convert a rational number to a decimal using long division; know that the decimal form of a rational number terminates in 0s or eventually
repeats.
Student Friendly/”I Can” statements
Resources
Assessments
1. Convert a rational number to a decimal
using long division.
2. Explain that the decimal form of a rational
number terminates (stops) in zeroes or
repeats.
Solve real-world and mathematical problems involving the four operations with rational numbers
Student Friendly/”I Can” statements
Resources
1. Add rational numbers.
2. Subtract rational numbers.
3. Multiply rational numbers.
4. Divide rational numbers.
5. Solve real-world mathematical problem
by adding, subtracting, multiplying, and
dividing rational numbers, including
complex fractions.
137
Assessments
Statistics and Probability
• Use random sampling to draw inferences about a population.
Understand that statistics can be used to gain information about a population by examining a sample of the population; generalizations about a
population from a sample are valid only if the sample is representative of that population. Understand that random sampling tends to produce
representative samples and support valid inferences.
Student Friendly/”I Can” statements
Resources
Assessments
1. Know statistics terms such as population,
sample, sample size, random sampling,
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generalizations, valid, biased and
US Census Bureau
unbiased.
2. Recognize sampling techniques such as
convenience, random, systematic, and
voluntary.
3. Know that generalizations about a
population from a sample are valid only if
the sample is representative of that
population
4. Apply statistics to gain information about
a population from a sample of the
population.
5. Generalize that random sampling tends to
produce representative samples and
support valid inferences.
138
Use data from a random sample to draw inferences about a population with an unknown characteristic of interest. Generate multiple samples
(or simulated samples) of the same size to gauge the variation in estimates or predictions. For example, estimate the mean word length in a
book by randomly sampling words from the book; predict the winner of a school election based on randomly sampled survey data. Gauge how
far off the estimate or prediction might be.
Student Friendly/”I Can” statements
Resources
Assessments
1. Define random sample.
2. Identify an appropriate sample size.
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3. Analyze & interpret data from a random
sample to draw inferences about a
population with an unknown
characteristic of interest.
4. Generate multiple samples (or simulated
samples) of the same size to determine
the variation in estimates or predictions
by comparing and contrasting the
samples.
139
Draw informal comparative inferences about two populations.
Informally assess the degree of visual overlap of two numerical data distributions with similar variabilities, measuring the difference between
the centers by expressing it as a multiple of a measure of variability. For example, the mean height of players on the basketball team is 10 cm
greater than the mean height of players on the soccer team, about twice the variability (mean absolute deviation) on either team; on a dot plot,
the separation between the two distributions of heights is noticeable.
Student Friendly/”I Can” statements
1. Identify measures of central tendency
(mean, median, and mode) in a data
distribution.
2. Identify measures of variation including
upper quartile, lower quartile, upper
extreme-maximum, lower extrememinimum, range, interquartile range, and
mean absolute deviation (i.e. box-andwhisker plots, line plot, dot plots, etc.).
3. Compare two numerical data distributions
on a graph by visually comparing data
displays, and assessing the degree of
visual overlap.
4. Compare the differences in the measure
of central tendency in two numerical data
distributions by measuring the difference
between the centers and expressing it as a
multiple of a measure of variability.
Resources
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US Census Bureau
140
Assessments
Use measures of center and measures of variability for numerical data from random samples to draw informal comparative inferences about
two populations. For example, decide whether the words in a chapter of a seventh-grade science book are generally longer than the words in a
chapter of a fourth-grade science book.
Student Friendly/”I Can” statements
1. Find measures of central tendency (mean,
median, and mode) and measures of
variability (range, quartile, etc.).
2. Analyze and interpret data using
measures of central tendency and
variability.
3. Draw informal comparative inferences
about two populations from random
samples.
Resources
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US Census Bureau
Assessments
Investigate chance process and develop, use, and evaluate probability models.
Understand that the probability of a chance event is a number between 0 and 1 that expresses the likelihood of the event occurring. Larger
numbers indicate greater likelihood. A probability near 0 indicates an unlikely event, a probability around 1/2 indicates an event that is neither
unlikely nor likely, and a probability near 1 indicates a likely event.
Student Friendly/”I Can” statements
Resources
Assessments
1. Know that probability is expressed as a
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number between 0 and 1.
2. Know that a random event with a
probability of ½ is equally likely to happen
3. Know that as probability moves closer to 1
it is increasingly likely to happen
4. Know that as probability moves closer to 0
it is decreasingly likely to happen
5. Draw conclusions to determine that a
greater likelihood occurs as the number of
favorable outcomes approaches the total
number of outcomes.
141
Approximate the probability of a chance event by collecting data on the chance process that produces it and observing its long-run relative
frequency, and predict the approximate relative frequency given the probability. For example, when rolling a number cube 600 times, predict
that a 3 or 6 would be rolled roughly 200 times, but probably not exactly 200 times.
Student Friendly/”I Can” statements
Resources
Assessments
1. Determine relative frequency
(experimental probability) is the number
of times an outcome occurs divided by the
total number of times the experiment is
completed
2. Determine the relationship between
experimental and theoretical probabilities
by using the law of large numbers
3. Predict the relative frequency
(experimental probability) of an event
based on the (theoretical) probability
Develop a probability model and use it to find probabilities of events. Compare probabilities from a model to observed frequencies; if the
agreement is not good, explain possible sources of the discrepancy.
a. Develop a uniform probability model by assigning equal probability to all outcomes, and use the model to determine probabilities of events.
For example, if a student is selected at random from a class, find the probability that Jane will be selected and the probability that a girl will be
selected.
Student Friendly/”I Can” statements
Resources
Assessments
1. Recognize uniform (equally likely)
probability.
2. Use models to determine the probability
of events
3. Develop a uniform probability model
and use it to determine the probability
of each outcome/event.
142
b. Develop a probability model (which may not be uniform) by observing frequencies in data generated from a chance process. For example, find
the approximate probability that a spinning penny will land heads up or that a tossed paper cup will land open-end down. Do the outcomes for
the spinning penny appear to be equally likely based on the observed frequencies?
Student Friendly/”I Can” statements
Resources
Assessments
1. Develop a probability model (which may
not be uniform) by observing
frequencies in data generated from a
chance process.
2. Analyze a probability model and justify
why it is uniform or explain the
discrepancy if it is not.
Find probabilities of compound events using organized lists, tables, tree diagrams, and simulation.
a. Understand that, just as with simple events, the probability of a compound event is the fraction of outcomes in the sample space for which
the compound event occurs.
Student Friendly/”I Can” statements
Resources
Assessments
1. Define and describe a compound event.
2. Know that the probability of a compound
event is the fraction of outcomes in the
sample space for which the compound
event occurs.
b. Represent sample spaces for compound events using methods such as organized lists, tables and tree diagrams. For an event described in
everyday language (e.g., “rolling double sixes”), identify the outcomes in the sample space which compose the event.
Student Friendly/”I Can” statements
1. Choose the appropriate method such as
organized lists, tables and tree diagrams
to represent sample spaces for compound
events
2. Find probabilities of compound events
using organized lists, tables, tree
diagrams, etc. and analyze the outcomes.
3. Identify the outcomes in the sample space
for an everyday event.
Resources
143
Assessments
c. Design and use a simulation to generate frequencies for compound events. For example, use random digits as a simulation tool to
approximate the answer to the question: If 40% of donors have type A blood, what is the probability that it will take at least 4 donors to find one
with type A blood?
Student Friendly/”I Can” statements
Resources
Assessments
1. Define simulation.
2. Design and use a simulation to generate
frequencies for compound events.
144
Ratios and Proportional Relationships
• Analyze proportional relationships and use them to solve real-world and mathematical problems.
Compute unit rates associated with ratios of fractions, including ratios of lengths, areas and other quantities measured in like or different units.
For example, if a person walks 1/2 mile in each 1/4 hour, compute the unit rate as the complex fraction 1/2/1/4 miles per hour, equivalently 2
miles per hour.
Student Friendly/”I Can” statements
Resources
Assessments
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1. Compute unit rates associated with
ratios of fractions in like or different
units.
Recognize and represent proportional relationships between quantities.
a. Decide whether two quantities are in a proportional relationship, e.g., by testing for equivalent ratios in a table or graphing on a coordinate
plane and observing whether the graph is a straight line through the origin.
Student Friendly/”I Can” statements
Resources
Assessments
1. Know that a proportion is a statement of
equality between two ratios.
2. Analyze two ratios to determine if they
are proportional to one another with a
variety of strategies. (e.g. using tables,
graphs, pictures, etc.)
b. Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional
relationships.
Student Friendly/”I Can” statements
Resources
Assessments
1. Define constant of proportionality as a
unit rate.
2. Analyze tables, graphs, equations,
diagrams, and verbal descriptions of
proportional relationships to identify the
constant of proportionality.
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c. Represent proportional relationships by equations. For example, if total cost t is proportional to the number n of items purchased at a
constant price p, the relationship between the total cost and the number of items can be expressed as t = pn.
Student Friendly/”I Can” statements
Resources
Assessments
1. Represent proportional relationships by
writing equations.
d. Explain what a point (x, y) on the graph of a proportional relationship means in terms of the situation, with special attention to the points (0,
0) and (1, r) where r is the unit rate.
Student Friendly/”I Can” statements
Resources
Assessments
1. Recognize what (0, 0) represents on the
graph of a proportional relationship.
2. Recognize what (1, r) on a graph
represents, where r is the unit rate.
3. Explain what the points on a graph of a
proportional relationship means in terms
of a specific situation.
Use proportional relationships to solve multistep ratio and percent problems. Examples: simple interest, tax, markups and markdowns,
gratuities and commissions, fees, percent increase and decrease, percent error.
Student Friendly/”I Can” statements
1. Recognize situations in which percentage
proportional relationships apply.
2. Apply proportional reasoning to solve
multistep ratio and percent problems,
e.g., simple interest, tax, markups,
markdowns, gratuities, commissions, fees,
percent increase and decrease, percent
error, etc.
Resources
146
Assessments
Geometry
• Draw, construct and describe geometrical figures and describe the relationships between them.
Solve problems involving scale drawings of geometric figures, including computing actual lengths and areas from a scale drawing and
reproducing a scale drawing at a different scale.
Student Friendly/”I Can” statements
Resources
Assessments
1. Use ratios and proportions to create scale
drawing
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2. Identify corresponding sides of scaled
mathopenref
geometric figures
Cut-the-Knot
3. Compute lengths and areas from scale
Graph paper
drawings using strategies such as
proportions.
4. Solve problems involving scale drawings
of geometric figures using scale factors.
5. Reproduce a scale drawing that is
proportional to a given geometric
figure using a different scale.
147
Draw (freehand, with ruler and protractor, and with technology) geometric shapes with given conditions. Focus on constructing triangles from
three measures of angles or sides, noticing when the conditions determine a unique triangle, more than one triangle, or no triangle.
Student Friendly/”I Can” statements
Resources
Assessments
1. Know which conditions create unique
Dot paper
triangles, more than one triangle, or no
Graph paper
triangle.
Ruler
2. Analyze given conditions based on the
Protractor
three measures of angles or sides of a
triangle to determine when there is a
unique triangle, more than one triangle,
or no triangle.
3. Construct triangles from three given angle
measures to determine when there is a
unique triangle, more than one triangle or
no triangle using appropriate tools
(freehand, rulers, protractors, and
technology).
Describe the two-dimensional figures that result from slicing three-dimensional figures, as in plane sections of right rectangular prisms and right
rectangular pyramids.
Student Friendly/”I Can” statements
1. Define slicing as the cross-section of a 3D
figure.
2. Describe the two-dimensional figures that
result from slicing a three-dimensional
figure such as a right rectangular prism or
pyramid.
3. Analyze three-dimensional shapes by
examining two dimensional crosssections.
Resources
148
Assessments
• Solve real-life and mathematical problems involving angle measure, area, surface area, and volume.
Know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship
between the circumference and area of a circle.
Student Friendly/”I Can” statements
Resources
Assessments
1. Know the parts of a circle including
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radius, diameter, area, circumference,
center, and chord.
2. Identify Pi.
3. Know the formulas for area and
circumference of a circle
4. Given the circumference of a circle, find
its area.
5. Given the area of a circle, find its
circumference.
6. Justify that Pi can be derived from the
circumference and diameter of a circle.
7. Apply circumference or area formulas to
solve mathematical and real-world
problems
8. Justify the formulas for area and
circumference of a circle and how they
relate to π
9. Informally derive the relationship
between circumference and area of a
circle.
149
Use facts about supplementary, complementary, vertical, and adjacent angles in a multi-step problem to write and solve simple equations for an
unknown angle in a figure.
Student Friendly/”I Can” statements
Resources
Assessments
1. Identify and recognize types of angles:
supplementary, complementary,
vertical, adjacent.
2. Determine complements and
supplements of a given angle.
3. Determine unknown angle measures by
writing and solving algebraic equations
based on relationships between angles.
Solve real-world and mathematical problems involving area, volume and surface area of two- and three-dimensional objects composed of
triangles, quadrilaterals, polygons, cubes, and right prisms.
Student Friendly/”I Can” statements
1. Know the formulas for area and volume
and then procedure for finding surface
area and when to use them in realworld and math problems for two- and
three-dimensional objects composed of
triangles, quadrilaterals, polygons,
cubes, and right prisms.
2. Solve real-world and math problems
involving area, surface area and volume
of two- and three-dimensional objects
composed of triangles, quadrilaterals,
polygons, cubes, and right prisms.
Resources
150
Assessments
Standards for Mathematical Practice – Grade 8
Standards for Mathematical
Explanations and Examples
Practice
1. Make sense of problems
In grade 8, students solve real world problems through the application of algebraic and geometric concepts.
and persevere in solving
Students seek the meaning of a problem and look for efficient ways to represent and solve it. They may
them.
check their thinking by asking themselves, “What is the most efficient way to solve the problem?”, “Does
this make sense?”, and “Can I solve the problem in a different way?”
2. Reason abstractly and
In grade 8, students represent a wide variety of real world contexts through the use of real numbers and
quantitatively.
variables in mathematical expressions, equations, and inequalities. They examine patterns in data and assess
the degree of linearity of functions. Students contextualize to understand the meaning of the number or
variable as related to the problem and decontextualize to manipulate symbolic representations by applying
properties of operations.
3. Construct viable arguments
In grade 8, students construct arguments using verbal or written explanations accompanied by expressions,
and critique the reasoning of
equations, inequalities, models, and graphs, tables, and other data displays (i.e. box plots, dot plots,
others.
histograms, etc.). They further refine their mathematical communication skills through mathematical
discussions in which they critically evaluate their own thinking and the thinking of other students. They pose
questions like “How did you get that?”, “Why is that true?” “Does that always work?” They explain their
thinking to others and respond to others’ thinking.
4. Model with mathematics.
In grade 8, students model problem situations symbolically, graphically, tabularly, and contextually.
Students form expressions, equations, or inequalities from real world contexts and connect symbolic and
graphical representations. Students solve systems of linear equations and compare properties of functions
provided in different forms. Students use scatterplots to represent data and describe associations between
variables. Students need many opportunities to connect and explain the connections between the different
representations. They should be able to use all of these representations as appropriate to a problem context.
5. Use appropriate tools
Students consider available tools (including estimation and technology) when solving a mathematical
strategically.
problem and decide when certain tools might be helpful. For instance, students in grade 8 may translate a set
of data given in tabular form to a graphical representation to compare it to another data set. Students might
draw pictures, use applets, or write equations to show the relationships between the angles created by a
transversal.
6. Attend to precision.
In grade 8, students continue to refine their mathematical communication skills by using clear and precise
language in their discussions with others and in their own reasoning. Students use appropriate terminology
when referring to the number system, functions, geometric figures, and data displays.
7. Look for and make use of
Students routinely seek patterns or structures to model and solve problems. In grade 8, students apply
structure.
properties to generate equivalent expressions and solve equations. Students examine patterns in tables and
graphs to generate equations and describe relationships. Additionally, students experimentally verify the
effects of transformations and describe them in terms of congruence and similarity.
8. Look for and express
In grade 8, students use repeated reasoning to understand algorithms and make generalizations about
regularity in repeated
patterns. Students use iterative processes to determine more precise rational approximations for irrational
151
reasoning.
numbers. They analyze patterns of repeating decimals to identify the corresponding fraction. During multiple
opportunities to solve and model problems, they notice that the slope of a line and rate of change are the
same value. Students flexibly make connections between covariance, rates, and representations showing the
relationships between quantities.
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Grade 8 Critical Areas
The Critical Areas for eighth grade can be found on page 52 in the Common Core State Standards for Mathematics.
1. Formulating and reasoning about expressions and equations, including modeling an association in bivariate data with a linear
equation, and solving linear equations and systems of linear equations
Students use linear equations and systems of linear equations to represent, analyze, and solve a variety of problems. Students recognize equations for
proportions (y/x = m or y = mx) as special linear equations (y = mx + b), understanding that the constant of proportionality (m) is the slope, and the
graphs are lines through the origin. They understand that the slope (m) of a line is a constant rate of change, so that if the input or x-coordinate changes
by an amount A, the output or y-coordinate changes by the amount m·A. Students also use a linear equation to describe the association between two
quantities in bivariate data (such as arm span vs. height for students in a classroom). At this grade, fitting the model, and assessing its fit to the data are
done informally. Interpreting the model in the context of the data requires students to express a relationship between the two quantities in question and to
interpret components of the relationship (such as slope and y-intercept) in terms of the situation. Students strategically choose and efficiently implement
procedures to solve linear equations in one variable, understanding that when they use the properties of equality and the concept of logical equivalence,
they maintain the solutions of the original equation. Students solve systems of two linear equations in two variables and relate the systems to pairs of
lines in the plane; these intersect, are parallel, or are the same line. Students use
linear equations, systems of linear equations, linear functions, and their understanding of slope of a line to analyze situations and solve problems.
2. Grasping the concept of a function and using functions to describe quantitative relationships
Students grasp the concept of a function as a rule that assigns to each input exactly one output. They understand that functions describe situations where
one quantity determines another. They can translate among representations and partial representations of functions (noting that tabular and graphical
representations may be partial representations), and they describe how aspects of the function are reflected in the different representations.
3. Analyzing two- and three-dimensional space and figures using distance, angle, similarity, and congruence, and understanding
and applying the Pythagorean Theorem
Students use ideas about distance and angles, how they behave under translations, rotations, reflections, and dilations, and ideas about congruence and
similarity to describe and analyze two-dimensional figures and to solve problems. Students show that the sum of the angles in a triangle is the angle
formed by a straight line, and that various configurations of lines give rise to similar triangles because of the angles created when a transversal cuts
parallel lines. Students understand the statement of the Pythagorean Theorem and its converse, and can explain why the Pythagorean Theorem holds, for
example, by decomposing a square in two different ways. They apply the Pythagorean Theorem to find distances between points on the coordinate plane,
to find lengths, and to analyze polygons. Students complete their work on volume by solving problems involving cones, cylinders, and spheres.
Expressions and Equations
• Work with radicals and integer exponents.
Know and apply the properties of integer exponents to generate equivalent numerical expressions. For example, 32 × 3–5 = 3–3 = 1/33 = 1/27
153
Student Friendly/”I Can” statements
1. Explain the properties of integer
exponents to generate equivalent
numerical expressions. For example,
3² x 3-5 = 3-3 = 1/33 = 1/27.
2. Apply the properties of integer
exponents to produce equivalent
numerical expressions.
Resources
Assessments
http://nlvm.usu.edu
Use square root and cube root symbols to represent solutions to equations of the form x2 = p and x3 = p, where p is a positive rational number.
Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that √2 is irrational.
Student Friendly/”I Can” statements
Resources
Assessments
1. Explain square root and cube root.
2. Explain the rule for multiplying and
dividing integers.
3. Explain the difference between rational
and irrational.
4. Use square root and cube root symbols to
represent solutions to equations of the
form x2 = p and x3 = p, where p is a positive
rational number.
5. Evaluate square roots of small perfect
squares.
6. Evaluate cube roots of small perfect
cubes.
7. Know that the square root of 2 is
irrational.
Use numbers expressed in the form of a single digit times an integer power of 10 to estimate very large or very small quantities, and to express
how many times as much one is than the other. For example, estimate the population of the United States as 3 × 108 and the population of the
world as 7 × 109, and determine that the world population is more than 20 times larger.
Student Friendly/”I Can” statements
Resources
Assessments
1. Express numbers as a single digit times an
integer power of 10.
2. Use scientific notation to estimate very
154
large and/or very small quantities.
3. Compare quantities to express how much
larger one is compared to the other.
Perform operations with numbers expressed in scientific notation, including problems where both decimal and scientific notation are used. Use
scientific notation and choose units of appropriate size for measurements of very large or very small quantities (e.g., use millimeters per year for
seafloor spreading). Interpret scientific notation that has been generated by technology.
Student Friendly/”I Can” statements
Resources
Assessments
1. Choose appropriate units of measure
when using scientific notation.
2. Use scientific notation to express very
large and very small quantities.
3. Perform operations using numbers
expressed in scientific notations.
4. Interpret scientific notation that has
been generated by technology.
155
• Understand the connections between proportional relationships, lines, and linear equations.
Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships
represented in different ways. For example, compare a distance-time graph to a distance-time equation to determine which of two moving
objects has greater speed.
Student Friendly/”I Can” statements
Resources
Assessments
1. Graph proportional relationships.
2. Interpret the unit rate of proportional
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relationships as the slope of the graph.
graph paper
3. Compare two different proportional
relationships represented in different
ways. (For example, compare a distancetime graph to a distance-time equation to
determine which of two moving objects
has greater speed.)
156
Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane;
derive the equation y = mx for a line through the origin and the equation y = mx + b for a line intercepting the vertical axis at b.
Student Friendly/”I Can” statements
1. Identify characteristics of similar triangles.
2. Find the slope of a line.
3. Determine the y-intercept of a line.
(Interpreting unit rate as the slope of the
graph is included in 8.EE.)
4. Analyze patterns for points on a line
through the origin.
5. Derive an equation of the form y = mx for
a line through the origin.
6. Analyze patterns for points on a line that
do not pass through or include the origin.
7. Derive an equation of the form y=mx + b
for a line intercepting the vertical axis at b
(the y-intercept).
8. Use similar triangles to explain why the
slope m is the same between any two
distinct points on a non-vertical line in the
coordinate plane.
Resources
Coordinate Planes
157
Assessments
• Analyze and solve linear equations and pairs of simultaneous linear equations.
Solve linear equations in one variable.
a. Give examples of linear equations in one variable with one solution, infinitely many solutions, or no solutions. Show which of these
possibilities is the case by successively transforming the given equation into simpler forms, until an equivalent equation of the form x = a, a = a,
or a = b results (where a and b are different numbers).
Student Friendly/”I Can” statements
Resources
Assessments
1. Give examples of linear equations in one
variable with one solution and show that
the given example equation has one
solution by successively transforming the
equation into an equivalent equation of
the form x = a.
2. Give examples of linear equations in one
variable with infinitely many solutions and
show that the given example has infinitely
many solutions by successively
transforming the equation into an
equivalent equation of the form a = a.
3. Give examples of linear equations in one
variable with no solution and show that
the given example has no solution by
successively transforming the equation
into an equivalent equation of the form b
= a, where a and b are different numbers.
b. Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the
distributive property and collecting like terms.
Student Friendly/”I Can” statements
Resources
Assessments
1. Solve linear equations with rational
number coefficients.
2. Solve equations whose solutions
require expanding expressions using
the distributive property and/ or
collecting like terms.
158
Analyze and solve pairs of simultaneous linear equations.
a. Understand that solutions to a system of two linear equations in two variables correspond to points of intersection of their graphs, because
points of intersection satisfy both equations simultaneously.
Student Friendly/”I Can” statements
Resources
Assessments
1. Identify the solution(s) to a system of two
linear equations in two variables as the
point(s) of intersection of their graphs.
2. Describe the point(s) of intersection
between two lines as points that satisfy
both equations simultaneously.
b. Solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations. Solve simple cases by
inspection. For example, 3x + 2y = 5 and 3x + 2y = 6 have no solution because 3x + 2y cannot simultaneously be 5 and 6.
Student Friendly/”I Can” statements
1. Define “inspection”.
2. Identify cases in which a system of two
equations in two unknowns has no
solution
3. Identify cases in which a system of two
equations in two unknowns has an infinite
number of solutions.
4. Solve a system of two equations (linear) in
two unknowns algebraically.
5. Estimate the point(s) of intersection for a
system of two equations in two unknowns
by graphing the equations.
6. Solve simple cases of systems of two
linear equations in two variables by
inspection.
Resources
159
Assessments
c. Solve real-world and mathematical problems leading to two linear equations in two variables. For example, given coordinates for two pairs of
points, determine whether the line through the first pair of points intersects the line through the second pair.
Student Friendly/”I Can” statements
1. Give equations and context that include
whole number and/or decimals/fractions.
2. Define variables and create a system of
linear equations in two variables.
3. Recognize that graphed lines with one
point of intersection (different slopes) will
have one solution, parallel lines (same
slope, different y-intercepts) have no
solutions, and lines that are the same
(same slope, same y-intercept) will have
infinitely many solutions.
4. Connect algebraic and graphical solutions
and the context of the system of linear
equations.
5. Make sense of solutions.
Resources
160
Assessments
Functions
• Define, evaluate, and compare functions.
Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of
an input and the corresponding output.
Student Friendly/”I Can” statements
Resources
Assessments
1. Understand rules that take x as input
http://nlvm.usu.edu/en/nav/category_g_4_t_2.html
and gives y as output is a function.
lineplotter – draws lines with slopes
2. Identify when functions occur as
shodor.org
when there is exactly one y-value is
associated with any x-value.
3. Use y to represent the output to
represent this function with the
equations.
4. Identify functions from equations,
graphs, and tables/ordered pairs.
5. Recognize graphs as a function using
the vertical line test, showing that
each x-value has only one y-value.
6. Recognize when graphs are not
functions when there are 2 y-values
for multiple x-value.
161
Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal
descriptions). For example, given a linear function represented by a table of values and a linear function represented by an algebraic expression,
determine which function has the greater rate of change.
Student Friendly/”I Can” statements
Resources
Assessments
1. Identify functions algebraically
http://nlvm.usu.edu/en/nav/category_g_4_t_2.html
including slope and y intercept.
lineplotter – draws lines with slopes
2. Identify functions using graphs.
shodor.org
3. Identify functions using tables.
4. Identify functions using verbal
descriptions.
5. Compare and Contrast 2 functions
with different representations.
6. Draw conclusions based on different
representations of functions.
Interpret the equation y = mx + b as defining a linear function, whose graph is a straight line; give examples of functions that are not linear. For
example, the function A = s2 giving the area of a square as a function of its side length is not linear because its graph contains the points (1,1),
(2,4) and (3,9), which are not on a straight line.
Student Friendly/”I Can” statements
1. Recognize that a linear function is
graphed as a straight line.
2. Recognize the equation y=mx+b is the
equation of a function whose graph is
a straight line where m is the slope
and b is the y-intercept.
3. Provide examples of nonlinear
functions using multiple
representations.
4. Compare the characteristics of linear
and nonlinear functions using various
representations.
Resources
http://nlvm.usu.edu/en/nav/category_g_4_t_2.html
lineplotter – draws lines with slopes
shodor.org
Graph paper
Coordinate planes
162
Assessments
• Use functions to model relationships between quantities.
Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a
description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and
initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values.
Student Friendly/”I Can” statements
Resources
Assessments
1. Recognize that slope is determined by http://nlvm.usu.edu/en/nav/category_g_4_t_2.html
the constant rate of change.
lineplotter – draws lines with slopes
2. Recognize that the y-intercept is the
shodor.org
initial value where x=0.
graph paper
3. Determine the rate of change from
Coordinate planes
two (x,y) values, a verbal description,
values in a table, or graph.
4. Determine the initial value from two
(x,y) values, a verbal description,
values in a table, or graph.
5. Construct a function to model a linear
relationship between two quantities.
6. Relate the rate of change and initial
value to real world quantities in a
linear function in terms of the
situation modeled and in terms of its
graph or a table of values.
163
Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or
decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally.
Student Friendly/”I Can” statements
1. Analyze a graph and describe the
functional relationship between two
quantities using the qualities of the
graph.
2. Sketch a graph given a verbal
description of its qualitative features.
Resources
http://nlvm.usu.edu/en/nav/category_g_4_t_2.html
lineplotter – draws lines with slopes
shodor.org
Graph paper
Coordinate planes
3. Interpret the relationship between
x and y values by analyzing a
graph.
164
Assessments
The Number System
• Know that there are numbers that are not rational, and approximate them by rational numbers.
Understand informally that every number has a decimal expansion; the rational numbers are those with decimal expansions that terminate in 0s
or eventually repeat. Know that other numbers are called irrational.
Student Friendly/”I Can” statements
Resources
Assessments
1. Define irrational numbers
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2. Show that the decimal expansion of
graph paper
rational numbers repeats eventually.
number lines
3. Convert a decimal expansion which
repeats eventually into a rational
number.
4. Show informally that every number
has a decimal expansion
Use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line
diagram, and estimate the value of expressions (e.g., ð2). For example, by truncating the decimal expansion of √2, show that √2 is between 1 and
2, then between 1.4 and 1.5, and explain how to continue on to get better approximations.
1. Student Friendly/”I Can” statements
Resources
Assessments
2. Approximate irrational numbers as
Number lines
rational numbers.
3. Approximately locate irrational
numbers on a number line.
4. Estimate the value of expressions
involving irrational numbers using
rational approximations. (For example,
by truncating the decimal expansion of
2 , show that 2 is between 1 and 2,
then between 1.4 and 1.5, and explain
how to continue on to get better
approximations.)
5. Compare the size of irrational numbers
using rational approximations.
165
Statistics and Probability
• Investigate patterns of association in bivariate data.
Construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities. Describe
patterns such as clustering, outliers, positive or negative association, linear association, and nonlinear association.
Student Friendly/”I Can” statements
Resources
Assessments
1. Describe patterns such as clustering,
outliers, positive or negative association,
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linear association, and nonlinear
graph paper
association
2. Construct scatter plots for bivariate
measurement data
3. Interpret scatter plots for bivariate
(two different variables such as
distance and time) measurement data
to investigate patterns of association
between two quantities
Know that straight lines are widely used to model relationships between two quantitative variables. For scatter plots that suggest a linear
association, informally fit a straight line, and informally assess the model fit by judging the closeness of the data points to the line.
Student Friendly/”I Can” statements
1. Know straight lines are used to model
relationships between two quantitative
variables
2. Informally assess the model fit by judging
the closeness of the data points to the
line.
Resources
Graph paper
Coordinate planes
rulers
3. Fit a straight line within the plotted
data.
166
Assessments
Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept. For
example, in a linear model for a biology experiment, interpret a slope of 1.5 cm/hr as meaning that an additional hour of sunlight each day is
associated with an additional 1.5 cm in mature plant height.
Student Friendly/”I Can” statements
1. Find the slope and intercept of a linear
equation.
2. Interpret the meaning of the slope and
intercept of a linear equation in terms of
the situation. (For example, in a linear
model for a biology experiment,
interpret a slope of 1.5 cm/hr as
meaning that an additional hour of
sunlight each day is associated with an
additional 1.5 cm in mature plant
height.)
3. Solve problems using the equation of a
linear model.
Resources
Graph paper
Coordinate planes
167
Assessments
Understand that patterns of association can also be seen in bivariate categorical data by displaying frequencies and relative frequencies in a
two-way table. Construct and interpret a two-way table summarizing data on two categorical variables collected from the same subjects. Use
relative frequencies calculated for rows or columns to describe possible association between the two variables. For example, collect data from
students in your class on whether or not they have a curfew on school nights and whether or not they have assigned chores at home. Is there
evidence that those who have a curfew also tend to have chores?
Student Friendly/”I Can” statements
Resources
Assessments
1. Recognize patterns shown in comparison
of two sets of data.
Graph paper
2. Know how to construct a two-way table.
Coordinate planes
3. Interpret the data in the two-way table to
recognize patterns. (For example, collect
data from students in your class on
whether or not they have a curfew on
school nights and whether or not they
have assigned chores at home. Is there
evidence that those who have a curfew
also tend to have chores?)
4. Use relative frequencies of the data to
describe relationships (positive, negative,
or no correlation)
168
Geometry
• Understand congruence and similarity using physical models, transparencies, or geometry software.
Verify experimentally the properties of rotations, reflections, and translations:
a. Lines are taken to lines, and line segments to line segments of the same length.
Student Friendly/”I Can” statements
Resources
1. Define & identify rotations, reflections,
http://nlvm.usu.edu 6-8th grade Geometry –
and translations.
Congruent triangles
2. Use physical models, transparencies, or
geometry software to verify the
Mathwarehouse – similar figures, triangles,
properties of rotations, reflections, and
angels, circles
translations (ie. Lines are taken to lines
and line segments to line segments of the Mimio resources – Geometry Sketchpad
same length, angles are taken to angles of
the same measure, & parallel lines are
Cuttheknot – area of a triangle
taken to parallel lines.)
3. Identify corresponding sides &
corresponding angles.
4. Understand prime notation to describe an
image after a translation, reflection, or
rotation.
5. Identify line of reflection.
169
Assessments
b. Angles are taken to angles of the same measure.
Student Friendly/”I Can” statements
Resources
1. Use physical models, transparencies, or
http://nlvm.usu.edu 6-8th grade Geometry –
geometry software to verify the
Congruent triangles
properties of rotations, reflections, and
translations (ie. Lines are taken to lines
Mathwarehouse – similar figures, triangles,
and line segments to line segments of the angels, circles
same length, angles are taken to angles of
the same measure, & parallel lines are
Mimio resources – Geometry Sketchpad
taken to parallel lines.)
2. Identify corresponding sides &
Cuttheknot – area of a triangle
corresponding angles.
3. Identify center of rotation.
4. Identify direction and degree of rotation.
c. Parallel lines are taken to parallel lines.
Student Friendly/”I Can” statements
1. Use physical models, transparencies, or
geometry software to verify the
properties of rotations, reflections, and
translations (ie. Lines are taken to lines
and line segments to line segments of the
same length, angles are taken to angles of
the same measure, & parallel lines are
taken to parallel lines.)
2. Identify corresponding sides &
corresponding angles.
3. Understand prime notation to describe an
image after a translation, reflection, or
rotation.
Resources
http://nlvm.usu.edu 6-8th grade Geometry –
Congruent triangles
Mathwarehouse – similar figures, triangles,
angels, circles
Mimio resources – Geometry Sketchpad
Cuttheknot – area of a triangle
4. Identify line of reflection.
170
Assessments
Assessments
Understand that a two-dimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations,
reflections, and translations; given two congruent figures, describe a sequence that exhibits the congruence between them.
Student Friendly/”I Can” statements
Resources
Assessments
th
1. Define congruency.
http://nlvm.usu.edu 6-8 grade Geometry –
2. Identify symbols for congruency.
Congruent triangles
3. Apply the concept of congruency to write
congruent statements.
Mathwarehouse – similar figures, triangles,
4. Reason that a 2-D figure is congruent to
angels, circles
another if the second can be obtained by
a sequence of rotations, reflections,
Mimio resources – Geometry Sketchpad
translation.
Cuttheknot – area of a triangle
5. Describe the sequence of rotations,
reflections, translations that exhibits the
congruence between 2-D figures using
words.
Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates.
Student Friendly/”I Can” statements
Resources
Assessments
th
http://nlvm.usu.edu
6-8
grade
Geometry
–
1. Define dilations as a reduction or
Congruent triangles
enlargement of a figure.
2. Identify scale factor of the dilation.
Mathwarehouse – similar figures, triangles,
3. Describe the effects of dilations,
angels, circles
translations, rotations, & reflections on
2-D figures using coordinates.
Mimio resources – Geometry Sketchpad
Cuttheknot – area of a triangle
171
Understand that a two-dimensional figure is similar to another if the second can be obtained from the first by a sequence of rotations,
reflections, translations, and dilations; given two similar two-dimensional figures, describe a sequence that exhibits the similarity between them.
Student Friendly/”I Can” statements
Resources
Assessments
1. Define similar figures as corresponding
http://nlvm.usu.edu 6-8th grade Geometry –
angles are congruent and corresponding
Congruent triangles
sides are proportional.
2. Recognize symbol for similar.
Mathwarehouse – similar figures, triangles,
3. Apply the concept of similarity to write
angels, circles
similarity statements.
4. Reason that a 2-D figure is similar to
Mimio resources – Geometry Sketchpad
another if the second can be obtained by
a sequence of rotations, reflections,
Cuttheknot – area of a triangle
translation, or dilation.
5. Describe the sequence of rotations,
reflections, translations, or dilations that
exhibits the similarity between 2-D figures
using words and/or symbols.
172
Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles
created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles. For example, arrange three copies of
the same triangle so that the sum of the three angles appears to form a line, and give an argument in terms of transversals why this is so.
Student Friendly/”I Can” statements
Resources
Assessments
th
1. Define similar triangles
http://nlvm.usu.edu 6-8 grade Geometry –
2. Define and identify transversals
Congruent triangles
3. Identify angles created when parallel line
is cut by transversal (alternate interior,
Mathwarehouse – similar figures, triangles,
alternate exterior, corresponding, vertical, angels, circles
adjacent, etc.)
4. Justify that the sum of interior angles
Mimio resources – Geometry Sketchpad
equals 180. (For example, arrange three
copies of the same triangle so that the
Cuttheknot – area of a triangle
three angles appear to form a line.)
5. Justify that the exterior angle of a triangle
is equal to the sum of the two remote
interior angles.
6. Use Angle-Angle Criterion to prove
similarity among triangles. (Give an
argument in terms of transversals why
this is so.)
• Understand and apply the Pythagorean Theorem.
Explain a proof of the Pythagorean Theorem and its converse.
Student Friendly/”I Can” statements
Resources
Assessments
1. Define key vocabulary: square root,
http://nlvm.usu.edu
Pythagorean Theorem, right triangle, legs
a & b, hypotenuse, sides, right angle,
Mimio resources – Geometry Sketchpad
converse, base, height, proof.
2. Be able to identify the legs and
Cuttheknot – Pythagorean Triples, Find
hypotenuse of a right triangle.
Hypotenuse, Find Leg
3. Explain a proof of the Pythagorean
Theorem.
4. Explain a proof of the converse of the
Pythagorean Theorem.
173
Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three
dimensions.
Student Friendly/”I Can” statements
Resources
Assessments
http://nlvm.usu.edu
1. Recall the Pythagorean Theorem and
its converse.
Mimio resources – Geometry Sketchpad
2. Solve basic mathematical Pythagorean
Theorem problems and its converse to
find missing lengths of sides of triangles Cuttheknot – Pythagorean Triples, Find
Hypotenuse, Find Leg
in two and three-dimensions.
3. Apply Pythagorean theorem in solving
real-world problems dealing with two
and three-dimensional shapes.
Apply the Pythagorean Theorem to find the distance between two points in a coordinate system.
Student Friendly/”I Can” statements
1. Recall the Pythagorean Theorem and its
converse.
2. Determine how to create a right triangle
from two points on a coordinate graph.
3. Use the Pythagorean Theorem to solve for
the distance between the two points.
Resources
http://nlvm.usu.edu
Mimio resources – Geometry Sketchpad
Cuttheknot – Pythagorean Triples, Find
Hypotenuse, Find Leg
174
Assessments
Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems.
Student Friendly/”I Can” statements
1. Identify and define vocabulary:
cone, cylinder, sphere, radius,
diameter, circumference, area,
volume, pi, base, height
2. Know formulas for volume of cones,
cylinders, and spheres
3. Compare the volume of cones, cylinders,
and spheres.
4. Determine and apply appropriate volume
formulas in order to solve mathematical
and real-world problems for the given
shape.
5. Given the volume of a cone, cylinder, or
sphere, find the radii, height, or
approximate for π.
Resources
http://nlvm.usu.edu
learner.org/interactives/geometry – 3D
shapes, volume
learner.org/interactives/geometry/index.html
softschalk
nsdl.org – link to CCSS and Science
175
Assessments
NUMBER AND QUANTITY
Real Number System
CC.9-12.N.RN.1 Extend the properties of exponents to rational exponents. Explain how the definition of the meaning of rational exponents follows
from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents. For example,
we define 51/3 to be the cube root of 5 because we want 5( 1/3)3 = 5(1/3) 3 to hold, so 5(1/3)3 must equal 5.
Student Friendly/”I Can” statements
1. Define radical notation as a convention
used to represent rational exponents.
2. Explain the properties of operations of
rational exponents as an extension of the
properties of integer exponents.
3. Explain how radical notation, rational
exponents, and properties of integer
exponents relate to one another.
Note from Appendix A: In implementing the
standards in curriculum, these standards
should occur before discussing exponential
functions with continuous domains.
Resources
Assessments
http://nlvm.usu.edu
CC.9-12.N.RN.2 Extend the properties of exponents to rational exponents. Rewrite expressions involving radicals and rational exponents using
the properties of exponents.
Student Friendly/”I Can” statements
1. Using the properties of exponents, rewrite
a radical expression as an expression with
a rational exponent.
2. Using the properties of exponents, rewrite
an expression with a rational exponent as
a radical expression.
Notes from Appendix A: In implementing the
standards in curriculum, these standards
should occur before discussing exponential
functions with continuous domains.
Resources
Assessments
CC.9-12.N.RN.3 Use properties of rational and irrational numbers. Explain why the sum or product of rational numbers is rational; that the sum of
a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational.
176
Student Friendly/”I Can” statements
1. Define rational and irrational numbers
2. Evaluate sums/products of rational
numbers and irrational numbers.
3. Explain why the sum or product of a
rational number is rational.
4. Explain why the sum or product of an
irrational number is irrational.
Note from Appendix A: Connect N.RN.3 to
physical situations, e.g., finding the perimeter
of a square of area 2.
Resources
177
Assessments
Quantities
CC.9-12.N.Q.1 Reason quantitatively and use units to solve problems. Use units as a way to understand problems and to guide the solution of
multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data
displays.*
1.
2.
3.
4.
5.
6.
Student Friendly/”I Can” statements
Calculate unit conversions.
Recognize units given or needed to solve
problem.
Use given units and the context of a
problem as a way to determine if the
solution to a multi-step problem is
reasonable (e.g. length problems dictate
different units than problems dealing with
a measure such as slope)
Choose appropriate units to represent a
problem when using formulas or graphing.
Interpret units or scales used in formulas
or represented in graphs.
Use units as a way to understand
problems and to guide the solution of
multi-step problems.
Resources
http://nlvm.usu.edu
Assessments
CC.9-12.N.Q.2 Reason quantitatively and use units to solve problems. Define appropriate quantities for the purpose of descriptive modeling.*
Student Friendly/”I Can” statements
1. Define descriptive modeling.
2. Determine appropriate quantities for the
purpose of descriptive modeling.
Resources
178
Assessments
CC.9-12.N.Q.3 Reason quantitatively and use units to solve problems. Choose a level of accuracy appropriate to limitations on measurement
when reporting quantities.*
1.
2.
3.
4.
Student Friendly/”I Can” statements
Identify appropriate units of
measurement to report quantities.
Determine the limitations of different
measurement tools.
Choose and justify a level of accuracy
and/or precision appropriate to
limitations on measurement when
reporting quantities.
Identify important quantities in a problem
or real-world context.
Resources
179
Assessments
The Complex Number System
CC.9-12.N.CN.1 Perform arithmetic operations with complex numbers. Know there is a complex number i such that i2 = −1, and every complex
number has the form a + bi with a and b real.
Student Friendly/”I Can” statements
1. Define i as √-1 or i2 = -1.
2. Define complex numbers.
3. Write complex numbers in the form a +
bi with a and b being real numbers.
Resources
http://nlvm.usu.edu
Assessments
CC.9-12.N.CN.2 Perform arithmetic operations with complex numbers. Use the relation i2 = –1 and the commutative, associative, and distributive
properties to add, subtract, and multiply complex numbers.
Student Friendly/”I Can” statements
1. Know that the commutative, associative,
and distributive properties extend to the
set of complex numbers over the
operations of addition and
multiplication.
2. Use the relation i2 = -1 and the
commutative, associative, and
distributive properties to add, subtract,
and multiply complex numbers.
Resources
Assessments
CC.9-12.N.CN.3 (+) Perform arithmetic operations with complex numbers. Find the conjugate of a complex number; use conjugates to find moduli
and quotients of complex numbers.
1.
2.
3.
4.
Student Friendly/”I Can” statements
Perform arithmetic operations with
complex numbers.
Find the conjugate of a complex number.
Use the conjugate to find quotients of
complex numbers.
Find the magnitude(length),
modulus(length) or absolute
value(length), of the vector
representation of a complex number.
Resources
180
Assessments
CC.9-12.N.CN.4 (+) Represent complex numbers and their operations on the complex plane. Represent complex numbers on the complex plane
in rectangular and polar form (including real and imaginary numbers), and explain why the rectangular and polar forms of a given complex number
represent the same number.
Student Friendly/”I Can” statements
1. Transform complex numbers in a complex
plane from rectangular to polar form and
vise versa,
2. Know and explain why both forms,
rectangular and polar, represent the same
number.
Resources
Assessments
CC.9-12.N.CN.5 (+) Represent complex numbers and their operations on the complex plane. Represent addition, subtraction, multiplication, and
conjugation of complex numbers geometrically on the complex plane; use properties of this representation for computation. For example, (-1 +
√3i)3 = 8 because (-1 + √3i) has modulus 2 and argument 120°.
Student Friendly/”I Can” statements
1. Geometrically show addition, subtraction,
and multiplication of complex numbers on
the complex coordinate plane.
2. Geometrically show that the conjugate of
complex numbers in a complex plane is
the reflection across the x-axis.
3. Evaluate the power of a complex number,
in rectangular form, using the polar form
of that complex number.
Resources
Assessments
CC.9-12.N.CN.6 (+) Represent complex numbers and their operations on the complex plane. Calculate the distance between numbers in the
complex plane as the modulus of the difference, and the midpoint of a segment as the average of the numbers at its endpoints.
Student Friendly/”I Can” statements
1. Calculate the distance between values in
the complex plane as the magnitude,
modulus, of the difference, and the
midpoint of a segment as the average of
the coordinates of its endpoints.
Resources
181
Assessments
CC.9-12.N.CN.7 Use complex numbers in polynomial identities and equations. Solve quadratic equations with real coefficients that have complex
solutions.
Student Friendly/”I Can” statements
1. Solve quadratic equations with real
coefficients that have solutions of the
form a + bi and a – bi.
Note from Appendix A: Limit to polynomials
with real coefficients.
Resources
Assessments
CC.9-12.N.CN.8 (+) Use complex numbers in polynomial identities and equations. Extend polynomial identities to the complex numbers. For
example, rewrite x^2 + 4 as (x + 2i)(x – 2i).
Student Friendly/”I Can” statements
Resources
Assessments
1. Use polynomial identities to write
equivalent expressions in the form of
complex numbers.
CC.9-12.N.CN.9 (+) Use complex numbers in polynomial identities and equations. Know the Fundamental Theorem of Algebra; show that it is true
for quadratic polynomials.
Student Friendly/”I Can” statements
1. Understand The Fundamental Theorem of
Algebra, which says that the number of
complex solutions to a polynomial
equation is the same as the degree of the
polynomial. Show that this is true for a
quadratic polynomial.
Resources
182
Assessments
Vector and Matrix Quantities
CC.9-12.N.VM.1 (+) Represent and model with vector quantities. Recognize vector quantities as having both magnitude and direction. Represent
vector quantities by directed line segments, and use appropriate symbols for vectors and their magnitudes (e.g., v(bold), |v|, ||v||, v(not bold)).
Student Friendly/”I Can” statements
1. Know that a vector is a directed line
segment representing magnitude and
direction.
Resources
Assessments
http://nlvm.usu.edu
2. Use the appropriate symbol
representation for vectors and their
magnitude.
CC.9-12.N.VM.2 (+) Represent and model with vector quantities. Find the components of a vector by subtracting the coordinates of an initial point
from the coordinates of a terminal point.
Student Friendly/”I Can” statements
1. Find the component form of a vector by
subtracting the coordinates of an initial
point from the coordinates of a terminal
point, therefore placing the initial point of
the vector at the origin.
Resources
Assessments
CC.9-12.N.VM.3 (+) Represent and model with vector quantities. Solve problems involving velocity and other quantities that can be represented
by vectors.
Student Friendly/”I Can” statements
Resources
1. Solve problems such as velocity and
other quantities that can be
represented using vectors.
183
Assessments
CC.9-12.N.VM.4 (+) Perform operations on vectors. Add and subtract vectors.
CC.9-12.N.VM.4a (+) Add vectors end-to-end, component-wise, and by the parallelogram rule. Understand that the magnitude of a sum of two
vectors is typically not the sum of the magnitudes.
Student Friendly/”I Can” statements
1. Know how to add vectors head to tail,
using the horizontal and vertical
components, and by finding the
diagonal formed by the parallelogram.
Resources
www.phet.colorado.edu/en/simulation/vector
-addition
Assessments
CC.9-12.N.VM.4b (+) Given two vectors in magnitude and direction form, determine the magnitude and direction of their sum.
Student Friendly/”I Can” statements
1. Understand that the magnitude of a sum
of two vectors is not the sum of the
magnitudes unless the vectors have the
same heading or direction.
Resources
www.phet.colorado.edu/en/simulation/vector
-addition
Assessments
CC.9-12.N.VM.4c (+) Understand vector subtraction v – w as v + (–w), where (–w) is the additive inverse of w, with the same magnitude as w and
pointing in the opposite direction. Represent vector subtraction graphically by connecting the tips in the appropriate order, and perform vector
subtraction component-wise.
Student Friendly/”I Can” statements
1. Know how to subtract vectors and that
vector subtraction is defined much like
subtraction of real numbers, in that v – w
is the same as v + (–w), where –w is the
additive inverse of w. The opposite of w,
-w, has the same magnitude, but the
direction of the angle differs by 180.
2. Represent vector subtraction on a graph
by connecting the vectors head to tail in
the correct order and using the
components of those vectors to find the
difference.
Resources
www.phet.colorado.edu/en/simulation/vector
-addition
184
Assessments
CC.9-12.N.VM.5 (+) Perform operations on vectors. Multiply a vector by a scalar.
CC.9-12.N.VM.5a (+) Represent scalar multiplication graphically by scaling vectors and possibly reversing their direction; perform scalar
multiplication component-wise, e.g., as c(v(sub x), v(sub y)) = (cv(sub x), cv(sub y)).
Student Friendly/”I Can” statements
1. Represent scalar multiplication of vectors
on a graph by increasing or decreasing
the magnitude of the vector by the factor
of the given scalar. If the scalar is less
than zero, the new vector’s direction is
opposite the original vector’s direction.
Resources
Assessments
2. Represent scalar multiplication of
vectors using the component form,
such as c(vx, vy) = (cvx, cvy).
CC.9-12.N.VM.5b (+) Compute the magnitude of a scalar multiple cv using ||cv|| = |c|v. Compute the direction of cv knowing that when |c|v ≠ 0, the
direction of cv is either along v (for c > 0) or against v (for c < 0).
Student Friendly/”I Can” statements
Resources
Assessments
1. Find the magnitude of a scalar multiple,
cv, is the magnitude of v multiplied by the
factor of the |c|. Know when c > 0, the
direction is the same, and when c < 0,
then the direction of the vector is
opposite the direction of the original
vector.
CC.9-12.N.VM.6 (+) Perform operations on matrices and use matrices in applications. Use matrices to represent and manipulate
data, e.g., to represent payoffs or incidence relationships in a network.
Student Friendly/”I Can” statements
Resources
Assessments
1. Represent and manipulate data using
matrices, e.g., to organize merchandise,
keep total sales, costs, and using graph
theory and adjacency matrices to make
predictions.
185
CC.9-12.N.VM.7 (+) Perform operations on matrices and use matrices in applications. Multiply matrices by scalars to produce new matrices, e.g.,
as when all of the payoffs in a game are doubled.
Student Friendly/”I Can” statements
1. Multiply matrices by a scalar, e.g., when
the inventory of jeans for July is twice
that for January.
Resources
Assessments
CC.9-12.N.VM.8 (+) Perform operations on matrices and use matrices in applications. Add, subtract, and multiply matrices of appropriate
dimensions.
Student Friendly/”I Can” statements
1. Know that the dimensions of a matrix are
based on the number of rows and
columns.
2. Add, subtract, and multiply matrices of
appropriate dimensions.
Resources
http://math.liu.se/whalun/matrix
Assessments
CC.9-12.N.VM.9 (+) Perform operations on matrices and use matrices in applications. Understand that, unlike multiplication of numbers, matrix
multiplication for square matrices is not a commutative operation, but still satisfies the associative and distributive properties.
Student Friendly/”I Can” statements
Resources
1. Understand that matrix multiplication
is not commutative, AB ≠ BA, however
it is associative and satisfies the
distributive properties.
186
Assessments
CC.9-12.N.VM.10 (+) Perform operations on matrices and use matrices in applications. Understand that the zero and identity matrices play a role
in matrix addition and multiplication similar to the role of 0 and 1 in the real numbers. The determinant of a square matrix is nonzero if and only if
the matrix has a multiplicative inverse.
Student Friendly/”I Can” statements
1. Identify a zero matrix and understand
that it behaves in matrix addition,
subtraction, and multiplication, much like
0 in the real numbers system.
2. Identify an identity matrix for a square
matrix and understand that it behaves in
matrix multiplication much like the
number 1 in the real number system.
3. Find the determinant of a square matrix,
and know that it is a nonzero value if the
matrix has an inverse.
4. Know that if a matrix has an inverse, then
the determinant of a square matrix is a
nonzero value.
Resources
Assessments
CC.9-12.N.VM.11 (+) Perform operations on matrices and use matrices in applications. Multiply a vector (regarded as a matrix with one column)
by a matrix of suitable dimensions to produce another vector. Work with matrices as transformations of vectors.
Student Friendly/”I Can” statements
1. Translate the vector

AB
Resources
, where A(1,3)
and B(4,9), 2 units to the right and 5 units
up, perform the following matrix
multiplication.
 1 0 2 
 0 1 5 


 0 0 1  
1 4  3 6
3 9  =  8 14
1 1   1 1




187
Assessments
CC.9-12.N.VM.12 (+) Perform operations on matrices and use matrices in applications. Work with 2 X 2 matrices as transformations of the plane,
and interpret the absolute value of the determinant in terms of area.
Student Friendly/”I Can” statements
1. Given the coordinates of the vertices of a
parallelogram in the coordinate plane,
find the vector representation for two
adjacent sides with the same initial point.
Write the components of the vectors in a
2x2 matrix and find the determinant of
the 2x2 matrix. The absolute value of the
determinant is the area of the
parallelogram. (This is called the dot
product of the two vectors.)
Resources
188
Assessments
ALGEBRA
Seeing Structure in Expressions
CC.9-12.A.SSE.1 Interpret the structure of expressions. Interpret expressions that represent a quantity in terms of its context.*
Student Friendly/”I Can” statements
1. Define and recognize parts of an
expression, such as terms, factors, and
coefficients.
Notes from Appendix A: limit to linear
expressions and to exponential expressions
with integer exponents.
Resources
http://nlvm.usu.edu
manipulatives
onemathematicalcat.org – identifying variable
parts and coefficients of terms
ixl.com (no constant)
Assessments
CC.9-12.A.SSE.1a Interpret parts of an expression, such as terms, factors, and coefficients.*
Student Friendly/”I Can” statements
1. Interpret parts of an expression, such as
terms, factors, and coefficients in terms of
the context.
Notes from Appendix A: limit to linear
expressions and to exponential expressions
with integer exponents.
Resources
http://nlvm.usu.edu
manipulatives
onemathematicalcat.org – identifying variable
parts and coefficients of terms
ixl.com (no constant)
Assessments
CC.9-12.A.SSE.1b Interpret complicated expressions by viewing one or more of their parts as a single entity. For example, interpret P(1+r)^n as
the product of P and a factor not depending on P.*
Student Friendly/”I Can” statements
1. Interpret complicated expressions, in
terms of the context, by viewing one or
more of their parts as a single entity.
Notes from Appendix A: Limit to linear
expressions with integer exponents
Resources
http://nlvm.usu.edu
manipulatives
onemathematicalcat.org – identifying variable
parts and coefficients of terms
ixl.com (no constant)
189
Assessments
CC.9-12.A.SSE.2 Interpret the structure of expressions. Use the structure of an expression to identify ways to rewrite it. For example, see x 4 – y4
as (x2)2 – (y2)2, thus recognizing it as a difference of squares that can be factored as (x 2 – y2)(x2 + y2).
Student Friendly/”I Can” statements
1. Identify ways to rewrite expressions,
such as difference of squares, factoring
out a common monomial, regrouping,
etc.
2. Identify various structures of expressions
(e.g. an exponential monomial multiplied
by a scalar of the same base, difference of
squares in terms other than just x)
3. Use the structure of an expression to
identify ways to rewrite it.
4. Classify expressions by structure and
develop strategies to assist in
classification.
Notes from Appendix A: Focus on quadratics
and exponential expressions
Resources
Assessments
CC.9-12.A.SSE.3 Write expressions in equivalent forms to solve problems. Choose and produce an equivalent form of an expression to reveal
and explain properties of the quantity represented by the expression.*
Student Friendly/”I Can” statements
1. Choose and produce an equivalent form
of a quadratic expression to reveal and
explain properties of the quantity
represented by the original expression.
Notes from Appendix A: It is important to
balance conceptual understanding and
procedural fluency in work with equivalent
expressions. For example, development of
skill in factoring and completing the square
goes hand-in-hand with understanding what
different forms of a quadratic expression
reveal.
Resources
190
Assessments
CC.9-12.A.SSE.3a Factor a quadratic expression to reveal the zeros of the function it defines.*
Student Friendly/”I Can” statements
1. Factor a quadratic expression to
produce an equivalent form of the
original expression
2. Explain the connection between the
factored form of a quadratic expression
and the zeros of the function it defines.
3. Explain the properties of the quantity
represented by the quadratic
expression.
Resources
Assessments
CC.9-12.A.SSE.3b Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines.*
Student Friendly/”I Can” statements
1. Complete the square on a quadratic
expression to produce an equivalent form
of an expression.
2. Explain the connection between the
completed square form of a quadratic
expression and the maximum or minimum
value of the function it defines.
3. Explain the properties of the quantity
represented by the expression.
4. Choose and produce an equivalent form
of a quadratic expression to reveal and
explain properties of the quantity
represented by the original expression.
Notes from Appendix A: It is important to
balance conceptual understanding and
procedural fluency in work with equivalent
expressions. For example, development of
skill in factoring and completing the square
goes hand-in-hand with understanding what
different forms of a quadratic expression
reveal.
Resources
191
Assessments
CC.9-12.A.SSE.3c Use the properties of exponents to transform expressions for exponential functions. For example the expression 1.15^t can be
rewritten as [1.15^(1/12)]^(12t) ≈ 1.012^(12t) to reveal the approximate equivalent monthly interest rate if the annual rate is 15%.*
1.
2.
3.
4.
Student Friendly/”I Can” statements
Use the properties of exponents to
transform simple expressions for
exponential functions.
Use the properties of exponents to
transform expressions for exponential
functions.
Choose and produce an equivalent form
of an exponential expression to reveal
and explain properties of the quantity
represented by the original expression.
Explain the properties of the quantity or
quantities represented by the
transformed exponential expression.
Resources
192
Assessments
CC.9-12.A.SSE.4 Write expressions in equivalent forms to solve problems. Derive the formula for the sum of a finite geometric series (when the
common ratio is not 1), and use the formula to solve problems. For example, calculate mortgage payments.*
Student Friendly/”I Can” statements
1. Find the first term in a geometric
sequence given at least two other terms.
2. Define a geometric series as a series with
a constant ratio between successive
terms.
Resources
(1 − r n )
3. Use the formula S = a
or an
(1 − r )
equivalent form to solve problems.
4. Derive a formula (i.e. equivalent to the
(1 − r n )
S
=
a
formula
) for the sum of a
(1 − r )
finite geometric series (when the common
ratio is not 1).
Note from Appendix A: Consider extending
A.SSE.4 to infinite geometric series in
curricular implementations of this course
description.
193
Assessments
Arithmetic with Polynomials and Rational Expressions
CC.9-12.A.APR.1 Perform arithmetic operations on polynomials. Understand that polynomials form a system analogous to the integers, namely,
they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials.
Student Friendly/”I Can” statements
1. Identify that the sum, difference, or
product of two polynomials will always be
a polynomial, which means that
polynomials are closed under the
operations of addition, subtraction, and
multiplication.
2. Define “closure”.
3. Apply arithmetic operations of addition,
subtraction, and multiplication to
polynomials.
Note from Appendix A: Focus on polynomial
expressions that simplify to forms that are
linear or quadratic in a positive integer
power of x.
Resources
Assessments
Manipulatives
Nlvm.usu.edu
CC.9-12.A.APR.2 Understand the relationship between zeros and factors of polynomial. Know and apply the Remainder Theorem: For a
polynomial p(x) and a number a, the remainder on division by x – a is p(a), so p(a) = 0 if and only if (x – a) is a factor of p(x).
Student Friendly/”I Can” statements
1. Define the remainder theorem for
polynomial division and divide
polynomials.
2. Given a polynomial p(x) and a number a,
divide p(x) by (x – a) to find p(a) then
apply the remainder theorem and
conclude that p(x) is divisible by x – a if
and only if p(a) = 0.
Resources
194
Assessments
CC.9-12.A.APR.3 Understand the relationship between zeros and factors of polynomials. Identify zeros of polynomials when suitable factorizations
are available, and use the zeros to construct a rough graph of the function defined by the polynomial.
Student Friendly/”I Can” statements
1. When suitable factorizations are
available, factor polynomials using any
available methods.
2. Create a sign chart for a polynomial f(x)
using the polynomial’s x-intercepts and
testing the domain intervals for which
f(x) greater than and less than zero.
3. Use the x-intercepts of a polynomial
function and the sign chart to construct
a rough graph of the function.
Resources
Assessments
CC.9-12.A.APR.4 Use polynomial identities to solve problems. Prove polynomial identities and use them to describe numerical relationships. For
example, the polynomial identity (x^2 + y^2)^2 = (x^2 – y^2)^2 + (2xy)^2 can be used to generate Pythagorean triples.
Student Friendly/”I Can” statements
1. Explain that an identity shows a
relationship between two quantities, or
expressions, that is true for all values of
the variables, over a specified set.
2. Prove polynomial identities.
3. Use polynomial identities to describe
numerical relationships.
Resources
195
Assessments
CC.9-12.A.APR.5 (+) Know and apply the Binomial Theorem for the expansion of (x + y) n in powers of x and y for a positive integer n, where x and
y are any numbers, with coefficients determined for example by Pascal’s Triangle.1
Student Friendly/”I Can” statements
1. Define the Binomial Theorem and
compute combinations.
2. Apply the Binomial theorem to expand
(x+y)n, when n is a positive integer and x
and y are any number, rather than
expanding by multiplying.
3. Explain the connection between Pascal’s
Triangle and the determination of the
coefficients in the expansion of (x+y)n,
when n is a positive integer and x and y
are any number.
Resources
196
Assessments
CC.9-12.A.APR.6 Rewrite rational expressions. Rewrite simple rational expressions in different forms; write a(x)/b(x) in the form q(x) + r(x)/b(x),
where a(x), b(x), q(x), and r(x) are polynomials with the degree of r(x) less than the degree of b(x), using inspection, long division, or, for the more
complicated examples, a computer algebra system.
Student Friendly/”I Can” statements
Resources
Assessments
1. Use inspection to rewrite simple rational
expressions in different forms; write
a(x)/b(x) in the form q(x) + r(x)/b(x),
where a(x), b(x), q(x), and r(x) are
polynomials with the degree of r(x) less
than the degree of b(x).
2. Use long division to rewrite simple
rational expressions in different forms;
write a(x)/b(x) in the form q(x) +
r(x)/b(x), where a(x), b(x), q(x), and r(x)
are polynomials with the degree of r(x)
less than the degree of b(x).
3. Use a computer algebra system to
rewrite complicated rational expressions
in different forms; write a(x)/b(x) in the
form q(x) + r(x)/b(x), where a(x), b(x),
q(x), and r(x) are polynomials with the
degree of r(x) less than the degree of
b(x).
CC.9-12.A.APR.7 (+) Rewrite rational expressions. Understand that rational expressions form a system analogous to the rational numbers, closed
under addition, subtraction, multiplication, and division by a nonzero rational expression; add, subtract, multiply, and divide rational expressions.
Student Friendly/”I Can” statements
1. Add, subtract, multiply, and divide
rational expressions.
2. Informally verify that rational expressions
form a system analogous to the rational
numbers, closed under addition,
subtraction, multiplication, and division
by a nonzero rational expression.
Resources
197
Assessments
198
Creating Equations
CC.9-12.A.CED.1 Create equations that describe numbers or relationship. Create equations and inequalities in one variable and use them to
solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions.*
Student Friendly/”I Can” statements
1. Solve all available types of equations &
inequalities, including root equations &
inequalities, in one variable.
2. Describe the relationships between the
quantities in the problem (for example,
how the quantities are changing or
growing with respect to each other);
express these relationships using
mathematical operations to create an
appropriate equation or inequality to
solve.
3. Create equations and inequalities in one
variable and use them to solve
problems.
4. Create equations and inequalities in one
variable to model real-world situations.
5. Compare and contrast problems that
can be solved by different types of
equations.
Note from Appendix A: Use all available
types of functions to create such equations,
including root functions, but constrain to
simple cases.
Resources
http://nlvm.usu.edu
manipulatives – homemade algebra tiles
models – create models of equations to solve
problems (eg. model of rectangular prism and
use area formula to find the surface area of
rectangular prism).
Lesson resources from Paula Turgeon
199
Assessments
CC.9-12.A.CED.2 Create equations that describe numbers or relationship. Create equations in two or more variables to represent relationships
between quantities; graph equations on coordinate axes with labels and scales.*
Student Friendly/”I Can” statements
1. Identify the quantities in a
mathematical problem or real-world
situation that should be represented by
distinct variables and describe what
quantities the variables represent.
2. Graph one or more created equation on
a coordinate axes with appropriate
labels and scales.
3. Create at least two equations in two or
more variables to represent
relationships between quantities
4. Justify which quantities in a
mathematical problem or real-world
situation are dependent and
independent of one another and which
operations represent those
relationships.
5. Determine appropriate units for the
labels and scale of a graph depicting the
relationship between equations created
in two or more variables.
Note from Appendix A: (While functions
used in A.CED.2will often be linear,
exponential, or quadratic the types of
problems should draw from more complex
situations than those addressed in Algebra I.
For example, finding the equation of a line
through a given point perpendicular to
another line allows one to find the distance
from a point to a line.)
Resources
200
Assessments
CC.9-12.A.CED.3 Create equations that describe numbers or relationship. Represent constraints by equations or inequalities, and by systems
of equations and/or inequalities, and interpret solutions as viable or non-viable options in a modeling context. For example, represent
inequalities describing nutritional and cost constraints on combinations of different foods.*
Student Friendly/”I Can” statements
1. Recognize when a modeling context
involves constraints.
2. Interpret solutions as viable or
nonviable options in a modeling
context.
3. Determine when a problem should be
represented by equations, inequalities,
systems of equations and/ or
inequalities.
4. Represent constraints by equations or
inequalities, and by systems of
equations and/or inequalities.
Note from Appendix A: While functions
used will often be linear, exponential, or
quadratic the types of problems should
draw from more complex situations than
those addressed in Algebra I. For example,
finding the equation of a line through a
given point perpendicular to another line
allows one to find the distance from a point
to a line.
Resources
201
Assessments
CC.9-12.A.CED.4 Create equations that describe numbers or relationship. Rearrange formulas to highlight a quantity of interest, using the same
reasoning as in solving equations. For example, rearrange Ohm’s law V = IR to highlight resistance R.*
Student Friendly/”I Can” statements
Resources
1. Define a “quantity of interest” to
mean any numerical or algebraic
a
quantity (e.g. 2( ) = d , in which 2 is
b
the quantity of interest showing that d
π r 2h
must be even;
= Vcone and
3
π r 2 h = Vcylinder showing that
Vcylinder = 3 ∗ Vcone )
2. Rearrange formulas to highlight a
quantity of interest, using the same
reasoning as in solving equations. (e.g.
π r2 can be re-written as (π r)r which
makes the form of this expression
resemble bh. The quantity of interest
could also be (a +b)n = a n b0 + a(n-1)b1 +
… + a0b n).
Note from Appendix A: While functions
used will often be linear, exponential, or
quadratic the types of problems should
draw from more complex situations
than those addressed in Algebra I. For
example, finding the equation of a line
through a given point perpendicular to
another line allows one to find the
distance from a point to a line. Note
that the example given for A.CED.4
applies to earlier instances of this
standard, not to the current course.
202
Assessments
Reasoning with Equations and Inequalities
CC.9-12.A.REI.1 Understand solving equations as a process of reasoning and explain the reasoning. Explain each step in solving a simple
equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a
solution. Construct a viable argument to justify a solution method.
Student Friendly/”I Can” statements
1. Know that solving an equation means
that the equation remains balanced
during each step.
2. Recall the properties of equality.
3. Explain why, when solving equations, it
is assumed that the original equation is
equal.
4. Determine if an equation has a solution.
5. Choose an appropriate method for
solving the equation.
6. Justify solution(s) to equations by
explaining each step in solving a simple
equation using the properties of
equality, beginning with the assumption
that the original equation is equal.
7. Construct a mathematically viable
argument justifying a given, or selfgenerated, solution method.
From Appendix A: Students should focus on
and master A.REI.1 for linear equations and
be able to extend and apply their reasoning
to other types of equations in future
courses.
Resources
http://nlvm.usu.edu
203
Assessments
CC.9-12.A.REI.2 Understand solving equations as a process of reasoning and explain the reasoning. Solve simple rational and
radical equations in one variable, and give examples showing how extraneous solutions may arise.
Student Friendly/”I Can” statements
Resources
Assessments
1. Determine the domain of a rational
function.
2. Determine the domain of a radical
function.
3. Solve radical equations in one variable.
4. Solve rational equations in one variable.
5. Give examples showing how extraneous
solutions may arise when solving
rational and radical equations.
CC.9-12.A.REI.3 Solve equations and inequalities in one variable. Solve linear equations and inequalities in one variable, including
equations with coefficients represented by letters.
Student Friendly/”I Can” statements
Resources
Assessments
1. Recall properties of equality
2. Solve multi-step equations in one
variable
3. Solve multi-step inequalities in one
variable
4. Determine the effect that rational
coefficients have on the inequality
symbol and use this to find the solution
set.
5. Solve equations and inequalities with
coefficients represented by letters.
204
CC.9-12.A.REI.4 Solve equations and inequalities in one variable. Solve quadratic equations in one variable.
Student Friendly/”I Can” statements
Resources
Assessments
1. Solve quadratic equations in one
variable.
Notes from Appendix A: Students should
learn of the existence of the complex
number system, but will not solve
quadratics with complex solutions until
Algebra II.
CC.9-12.A.REI.4a Use the method of completing the square to transform any quadratic equation in x into an equation of the form
(x – p)2 = q that has the same solutions. Derive the quadratic formula from this form.
Student Friendly/”I Can” statements
Resources
Assessments
1. Use the method of completing the
square to transform any quadratic
equation in x into an equation of the
form (x-p)2 = q that has the same
solutions.
2. Derive the quadratic formula by
completing the square on a quadratic
equation in x.
205
CC.9-12.A.REI.4b Solve quadratic equations by inspection (e.g., for x2 = 49), taking square roots, completing the square, the
quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives
complex solutions and write them as a ± bi for real numbers a and b.
Student Friendly/”I Can” statements
Resources
Assessments
1. Solve quadratic equations by inspection
(e.g., for x2 = 49), taking square roots,
completing the square, the quadratic
formula and factoring
2. Express complex solutions as a ± bi for
real numbers solutions as a and b.
3. Determine appropriate strategies (see
first knowledge target listed) to solve
problems involving quadratic equations,
as appropriate to the initial form of the
equation.
4. Recognize when the quadratic formula
gives complex solutions.
Note from Appendix A: Students should
learn of the existence of the complex
number system, but will not solve
quadratics with complex solutions until
Algebra II.
CC.9-12.A.REI.5 Solve systems of equations. Prove that, given a system of two equations in two variables, replacing one equation
by the sum of that equation and a multiple of the other produces a system with the same solutions.
Student Friendly/”I Can” statements
Resources
Assessments
1. Solve systems of equations using the
elimination method (sometimes called
linear combinations).
2. Solve a system of equations by
substitution (solving for one variable in
the first equation and substitution it
into the second equation).
206
CC.9-12.A.REI.6 Solve systems of equations. Solve systems of linear equations exactly and approximately (e.g., with graphs),
focusing on pairs of linear equations in two variables.
Student Friendly/”I Can” statements
Resources
Assessments
1. Solve systems of equations using any
method.
2. Justify the method used to solve
systems of linear equations exactly and
approximately focusing on pairs of
linear equations in two variables.
Notes from Appendix A: Build on student
experiences graphing and solving systems of
linear equations from middle school to
focus on justification of the methods used.
Include cases where the two equations
describe the same line (yielding infinitely
many solutions) and cases where two
equations describe parallel lines (yielding no
solution); connect to GPE.5 when it is
taught in Geometry, which requires
students to prove the slope criteria for
parallel lines.
207
CC.9-12.A.REI.7 Solve systems of equations. Solve a simple system consisting of a linear equation and a quadratic equation in
two variables algebraically and graphically. For example, find the points of intersection between the line y = –3x and the circle x^2
+ y^2 = 3.
Student Friendly/”I Can” statements
Resources
Assessments
1. Transform a simple system consisting of
a linear equation and a quadratic
equation in 2 variables so that a
solution can be found algebraically and
graphically.
2. Explain the correspondence between
the algebraic & graphical solutions to a
simple system consisting of a linear
equation and a quadratic equation in 2
variables.
Notes from Appendix A: Include systems
consisting of one linear and one quadratic
equation. Include systems that lead to work
with fractions. For example, finding the
intersections between x 2 + y 2 = 1 and y =
x+ 1
3 4
leads to the point ( , ) on the unit
2
5 5
circle, corresponding to the Pythagorean
triple of 3 2 + 4 2 = 5 2
CC.9-12.A.REI.8 (+) Solve systems of equations. Represent a system of linear equations as a single matrix equation in a vector variable.
Student Friendly/”I Can” statements
Resources
1. Write a system of linear equations as
a single matrix equation.
208
Assessments
CC.9-12.A.REI.9 (+) Solve systems of equations. Find the inverse of a matrix if it exists and use it to solve systems of linear equations (using
technology for matrices of dimension 3 × 3 or greater).
Student Friendly/”I Can” statements
1. Find the inverse of the coefficient
matrix in the equation, if it exits. Use
the inverse of the coefficient matrix to
solve the system. Use technology for
matrices with dimensions 3 by 3 or
greater.
2. Find the dimension of matrices.
3. Understand when matrices can be
multiplied.
4. Understand that matrix multiplication is
not commutative.
5. Understand the concept of an identity
matrix.
6. Understand why multiplication by the
inverse of the coefficient matrix yields a
solution to the system (if it exists).
Resources
Assessments
CC.9-12.A.REI.10 Represent and solve equations and inequalities graphically. Understand that the graph of an equation in two variables is the
set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line).
Student Friendly/”I Can” statements
1. Recognize that the graphical
representation of an equation in two
variables is a curve, which may be a
straight line.
2. Explain why each point on a curve is a
solution to its equation.
Notes from Appendix A: For A.REI.10, focus
on linear and exponential equations and be
able to adapt and apply that learning to
other types of equations in future courses.
Resources
209
Assessments
CC.9-12.A.REI.11 Represent and solve equations and inequalities graphically. Explain why the x-coordinates of the points where the graphs of
the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, e.g., using technology
to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial,
rational, absolute value, exponential, and logarithmic functions.*
Student Friendly/”I Can” statements
1. Recognize and use function notation to
represent linear and exponential
equations
2. Recognize that if (x1, y1) and (x2, y2)
share the same location in the
coordinate plane that x1 = x2 and y1 = y2.
3. Recognize that f(x) = g(x) means that
there may be particular inputs of f and g
for which the outputs of f and g are
equal.
4. Explain why the x-coordinates of the
points where the graph of the equations
y=f(x) and y=g(x) intersect are the
solutions of the equations f(x) = g(x) .
(Include cases where f(x) and/or g(x) are
linear and exponential equations)
5. Approximate/find the solution(s) using
an appropriate method for example,
using technology to graph the functions,
make tables of values or find successive
approximations (Include cases where
f(x) and/or g(x) are linear and
exponential equations).
Notes from Appendix A: For A.REI.11, focus
on cases where f(x) and g(x) are linear or
exponential.
Resources
210
Assessments
CC.9-12.A.REI.12 Represent and solve equations and inequalities graphically. Graph the solutions to a linear inequality in two variables as a
half-plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables
as the intersection of the corresponding half-planes.
Student Friendly/”I Can” statements
1. Identify characteristics of a linear
inequality and system of linear
inequalities, such as: boundary line
(where appropriate), shading, and
determining appropriate test points to
perform tests to find a solution set.
2. Explain the meaning of the intersection
of the shaded regions in a system of
linear inequalities.
3. Graph a line, or boundary line, and
shade the appropriate region for a two
variable linear inequality.
4. Graph a system of linear inequalities
and shade the appropriate overlapping
region for a system of linear
inequalities.
Resources
211
Assessments
FUNCTIONS
Interpreting Functions
CC.9-12.F.IF.1 Understand the concept of a function and use function notation. Understand that a function from one set (called the domain) to
another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its
domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x).
Student Friendly/”I Can” statements
1. Identify the domain and range of a
function.
2. Determine if a relation is a function.
3. Determine the value of the function with
proper notation (i.e. f(x)=y, the y value is
the value of the function at a particular
value of x)
4. Evaluate functions for given values of x.
Note from Appendix A: Students should
experience a variety of types of situations
modeled by functions. Detailed analysis of any
particular class of functions at this stage is not
advised. Students should apply these concepts
throughout their future mathematics courses.
Resources
http://nlvm.usu.edu
212
Assessments
CC.9-12.F.I Understand the concept of a function and use function notation. Use function notation, evaluate functions for inputs in their domains,
and interpret statements that use function notation in terms of a context.
Student Friendly/”I Can”
statements
1. Identify
mathematical
relationships and
express them using
function notation.
2. Define a reasonable
domain, which
depends on the
context and/or
mathematical
situation, for a
function focusing on
linear and
exponential
functions.
3. Evaluate functions at
a given input in the
domain, focusing on
linear and
exponential
functions.
4. Interpret statements
that use functions in
terms of real world
situations, focusing
on linear and
exponential
functions.
Note from Appendix A:
Students should
experience a variety of
types of situations
Resources
http://Algebralab.org/lessons/lesson.aspx?file=Algebra_functionsRelationsEvaluation.xml
213
Assessments
modeled by functions.
Detailed analysis of any
particular class of
functions at this stage is
not advised. Students
should apply these
concepts throughout
their future mathematics
courses.
214
CC.9-12.F.IF.3 Understand the concept of a function and use function notation. Recognize that sequences are functions, sometimes defined
recursively, whose domain is a subset of the integers. For example, the Fibonacci sequence is defined recursively by f(0) = f(1) = 1, f(n+1) = f(n) +
f(n-1) for n ≥ 1 (n is greater than or equal to 1).
Student Friendly/”I Can” statements
Resources
1. Recognize that sequences are functions,
sometimes defined recursively, whose
domain is a subset of the integers. For
example, the Fibonacci sequence is
defined recursively by f(0) = f(1) = 1, f(n
+ 1) = f(n) + f(n - 1) for n ≥ 1.
Notes from Appendix A: Students should
experience a variety of types of situations
modeled by functions. Detailed analysis of
any particular class of functions at this stage
is not advised. Students should apply these
concepts throughout their future
mathematics courses. Draw examples from
linear and exponential functions. In F.IF.3,
draw connection to F.BF.2, which requires
students to write arithmetic and geometric
sequences. Emphasize arithmetic and
geometric sequences as examples of linear
and exponential functions.
215
Assessments
CC.9-12.F.IF.4 Interpret functions that arise in applications in terms of the context. For a function that models a relationship between two
quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description
of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums
and minimums; symmetries; end behavior; and periodicity.*
Student Friendly/”I Can” statements
1. Define and recognize the key features in
tables and graphs of linear and
exponential functions: intercepts;
intervals where the function is increasing,
decreasing, positive, or negative, and end
behavior.
2. Identify whether the function is linear or
exponential, given its table or graph.
3. Interpret key features of graphs and
tables of functions in the terms of the
contextual quantities the function
represents.
4. Sketch graphs showing key features of a
function that models a relationship
between two quantities from a given
verbal description of the relationship.
Notes from Appendix A: Focus on linear and
exponential.
Resources
216
Assessments
CC.9-12.F.IF.5 Interpret functions that arise in applications in terms of the context. Relate the domain of a function to its graph and, where
applicable, to the quantitative relationship it describes. For example, if the function h(n) gives the number of person-hours it takes to assemble n
engines in a factory, then the positive integers would be an appropriate domain for the function.*
Student Friendly/”I Can” statements
1. Given the graph or a verbal/written
description of a function, identify and
describe the domain of the function.
2. Identify an appropriate domain based on
the unit, quantity, and type of function it
describes.
3. Relate the domain of the function to its
graph and, where applicable, to the
quantitative relationship it describes.
4. Explain why a domain is appropriate for a
given situation.
Resources
Notes from Appendix A: For F.IF.4 and 5, focus on
linear and exponential functions in Algebra 1 unit
2.
217
Assessments
CC.9-12.F.IF.6 Interpret functions that arise in applications in terms of the context. Calculate and interpret the average rate of change of a
function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.*
Student Friendly/”I Can” statements
1. Recognize slope as an average rate of
change.
2. Calculate the average rate of change of a
function (presented symbolically or as a
table) over a specified interval.
3. Estimate the rate of change from a linear
or exponential graph.
4. Interpret the average rate of change of a
function (presented symbolically or as a
table) over a specified interval.
Notes from Appendix A: Focus on linear
functions and exponential functions whose
domain is a subset of the integers. Unit 5 of
the Traditional Algebra 1 Pathway and the
Traditional Algebra II Pathway address other
types of functions.
Resources
Assessments
CC.9-12.F.IF.7 Analyze functions using different representations. Graph functions expressed symbolically and show key features of the graph, by
hand in simple cases and using technology for more complicated cases.*
Student Friendly/”I Can” statements
1. Graph linear functions by hand in simple
cases or using technology for more
complicated cases and show/label
intercepts of the graph.
Note from Appendix A: Focus linear functions.
Include comparisons of two functions
presented algebraically. For example,
compare two linear functions.
Resources
TI_8x graphing calculators – show students
graph various changes in y=mx+b in that
functions effects their graphs and determine
what they are. y=x2
Y=ln(x)
y=sin(x)
x
Y=a
http://nlvm.usu.edu function transformations
218
Assessments
CC.9-12.F.IF.7a Graph linear and quadratic functions and show intercepts, maxima, and minima.*
Student Friendly/”I Can” statements
1. Graph linear functions by hand in simple
cases or using technology for more
complicated cases and show/label
intercepts of the graph.
Note from Appendix A: Focus linear functions.
Include comparisons of two functions
presented algebraically. For example,
compare two linear functions.
Resources
Assessments
CC.9-12.F.IF.7b Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions.*
Student Friendly/”I Can” statements
1. Graph square root, cube root, and
piecewise-defined functions, including
step functions and absolute value
functions, by hand in simple cases or using
technology for more complicated cases,
and show/label key features of the graph.
2. Determine the difference between simple
and complicated linear, quadratic, square
root, cube root, and piecewise-defined
functions, including step functions and
absolute value functions and know when
the use of technology is appropriate.
3. Compare and contrast the domain and
range of absolute value, step and piecewise defined functions with linear,
quadratic, and exponential.
Notes from Appendix A: Compare and
contrast absolute value, step and piece-wise
defined functions with linear, quadratic, and
exponential functions. Highlight issues of
domain, range, and usefulness when
examining piece-wise defined functions.
Resources
219
Assessments
CC.9-12.F.IF.7c Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior.*
Student Friendly/”I Can” statements
1. Graph polynomial functions, by hand in
simple cases or using technology for more
complicated cases, and show/label
maxima and minima of the graph, identify
zeros when suitable factorizations are
available, and show end behavior.
2. Determine the difference between simple
and complicated polynomial functions,
and know when the use of technology is
appropriate.
3. Relate the relationship between zeros of
quadratic functions and their factored
forms to the relationship between
polynomial functions of degrees greater
than two.
Notes from Appendix A: Relate F.IF.7c to the
relationship between zeros of quadratic
functions and their factored forms.
Resources
Assessments
CC.9-12.F.IF.7d (+) Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end
behavior.*
Student Friendly/”I Can” statements
Resources
Assessments
1. Identify zeros and asymptotes in
rational functions when factorable,
and showing end behavior.
CC.9-12.F.IF.7e Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period,
midline, and amplitude.*
Student Friendly/”I Can” statements
1. Exponential and logarithmic functions,
showing intercepts and end behavior.
2. Trigonometric functions, showing period,
midline, and amplitude.
Resources
220
Assessments
CC.9-12.F.IF.8 Analyze functions using different representations. Write a function defined by an expression in different but equivalent forms to
reveal and explain different properties of the function.
Student Friendly/”I Can” statements
1. Identify different forms of a quadratic
expression.
2. Write functions in equivalent forms using
the process of factoring
Resources
Assessments
CC.9-12.F.IF.8a Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of
the graph, and interpret these in terms of a context.
Student Friendly/”I Can” statements
1. Identify zeros, extreme values, and
symmetry of the graph of a quadratic
function
2. Interpret different but equivalent forms of
a function defined by an expression in
terms of a context
3. Use the process of factoring and
completing the square in a quadratic
function to show zeros, extreme values,
and symmetry of the graph, and interpret
these in terms of a context.
Note from Appendix A: Extend work with
quadratics to include the relationship
between coefficients and roots, and that once
roots are known, a quadratic equation can be
factored.
Resources
221
Assessments
CC.9-12.F.IF.8b Use the properties of exponents to interpret expressions for exponential functions. For example, identify percent rate of change in
functions such as y = (1.02)^t, y = (0.97)^t, y = (1.01)^(12t), y = (1.2)^(t/10), and classify them as representing exponential growth and decay.
Student Friendly/”I Can” statements
1. Classify the exponential function as
exponential growth or decay by examining
the base.
2. Use the properties of exponents to
interpret expressions for exponential
functions in a real-world context.
Note from Appendix A: Note this unit extends
the work begun in Unit 2 on exponential
functions with integer exponents.
Resources
222
Assessments
CC.9-12.F.IF.9 Analyze functions using different representations. Compare properties of two functions each represented in a different way
(algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a graph of one quadratic function and an algebraic
expression for another, say which has the larger maximum.
Student Friendly/”I Can” statements
1. Identify types of functions based on verbal
, numerical, algebraic, and graphical
descriptions and state key properties (e.g.
intercepts, maxima, minima, growth rates,
average rates of change, and end
behaviors)
2. Differentiate between exponential, linear,
and quadratic functions using a variety of
descriptors (graphically, verbally,
numerically, and algebraically)
3. Use a variety of function representations
(algebraically, graphically, numerically in
tables, or by verbal descriptions) to
compare and contrast properties of two
functions
Note from Appendix A: Focus on expanding
the types of functions considered to include,
linear, exponential, and quadratic. Extend
work with quadratics to include the
relationship between coefficients and roots,
and that once roots are known, a quadratic
equation can be factored.
Resources
223
Assessments
Building Functions
CC.9-12.F.BF.1 Build a function that models a relationship between two quantities. Write a function that describes a relationship between two
quantities.*
Student Friendly/”I Can” statements
1. Write a function that describes a
relationship between two quantities by
determining an explicit expression, a
recursive process, or steps for
calculation from a context.
Resources
http://nlvm.usu.edu
Assessments
CC.9-12.F.BF.1a Determine an explicit expression, a recursive process, or steps for calculation from a context.
Student Friendly/”I Can” statements
1. Define “explicit function” and “recursive
process”.
Note from Appendix A: Focus on situations
that exhibit a quadratic relationship. This
standard builds from Algebra 1 Unit 2.
Resources
Assessments
CC.9-12.F.BF.1b Combine standard function types using arithmetic operations. For example, build a function that models the temperature of a
cooling body by adding a constant function to a decaying exponential, and relate these functions to the model.
Student Friendly/”I Can” statements
1. Combine two functions using the
operations of addition, subtraction,
multiplication, and division
2. Evaluate the domain of the combined
function.
Given a real-world situation or mathematical
problem:
3. build standard functions to represent
relevant relationships/ quantities
4. determine which arithmetic operation
should be performed to build the
appropriate combined function
5. relate the combined function to the
context of the problem
Resources
224
Assessments
Note from Appendix: Focus on situations that
exhibit a quadratic relationship.
CC.9-12.F.BF.1c (+) Compose functions. For example, if T(y) is the temperature in the atmosphere as a function of height, and h(t) is the height of
a weather balloon as a function of time, then T(h(t)) is the temperature at the location of the weather balloon as a function of time.
Student Friendly/”I Can” statements
Resources
1. Compose functions.
225
Assessments
CC.9-12.F.BF.2 Build a function that models a relationship between two quantities. Write arithmetic and geometric sequences both recursively and
with an explicit formula, use them to model situations, and translate between the two forms.*
Student Friendly/”I Can” statements
1. Identify arithmetic and geometric
patterns in given sequences.
2. Generate arithmetic and geometric
sequences from recursive and explicit
formulas.
3. Given an arithmetic or geometric
sequence in recursive form, translate into
the explicit formula.
4. Given an arithmetic or geometric
sequence as an explicit formula, translate
into the recursive form.
5. Use given and constructed arithmetic and
geometric sequences, expressed both
recursively and with explicit formulas, to
model real-life situations.
6. Determine the recursive rule given
arithmetic and geometric sequences.
7. Determine the explicit formula given
arithmetic and geometric sequences.
8. Justify the translation between the
recursive form & explicit formula for
arithmetic and geometric sequences.
Notes from Appendix A: Connect arithmetic
sequences to linear functions and geometric
sequences to exponential functions.
Resources
226
Assessments
CC.9-12.F.BF.3 Build new functions from existing functions. Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for
specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the
effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them.
Student Friendly/”I Can” statements
1. Given a single transformation on a
function (symbolic or graphic) identify the
effect on the graph.
2. Using technology, identify effects of single
transformations on graphs of functions.
3. Graph a given function by replacing f(x) by
f(x) + k, k f(x), f(kx), and f(x + k) for specific
values of k (both positive and negative).
4. Describe the differences and similarities
between a parent function and the
transformed function.
5. Find the value of k, given the graphs of a
parent function, f(x), and the transformed
function: f(x) + k, k f(x), f(kx), or f(x + k).
6. Recognize even and odd functions from
their graphs and from their equations.
7. Experiment with cases and illustrate an
explanation of the effects on the graph
using technology.
Notes from Appendix A: Focus on vertical
translations of graphs of linear and
exponential functions. Relate the vertical
translation of a linear function to its yintercept. While applying other
transformations to a linear graph is
appropriate at this level, it may be difficult for
students to identify or distinguish between
the effects of the other transformations
included in this standard.
Resources
TI-83 and TI-84 Calculators
http://colorado.sims/sims/curve-fitting/curvefitting_en.html
http://classzone
227
Assessments
CC.9-12.F.BF.4 Build new functions from existing functions. Find inverse functions.
Student Friendly/”I Can” statements
1. Solve an equation of the form f(x) = c for
a simple function f that has an inverse
and write an expression for the inverse.
Notes from Appendix A: Focus on linear
functions but consider simple situations
where the domain of the function must be
restricted in order for the inverse to exist,
such as f(x) = x2, x>0.
Resources
Assessments
CC.9-12.F.BF.4a Solve an equation of the form f(x) = c for a simple function f that has an inverse and write an expression for the inverse. For
example, f(x) =2(x^3) or f(x) = (x+1)/(x-1) for x ≠ 1 (x not equal to 1).
Student Friendly/”I Can” statements
1. Solve an equation of the form f(x) = c for
a simple function f that has an inverse
and write an expression for the inverse.
Notes from Appendix A: Focus on linear
functions but consider simple situations
where the domain of the function must be
restricted in order for the inverse to exist,
such as f(x) = x2, x>0.
Resources
Assessments
CC.9-12.F.BF.4b (+) Verify by composition that one function is the inverse of another.
Student Friendly/”I Can” statements
Resources
Assessments
1. Verify that one function is the inverse of
another by illustrating that f-1(f(x)) = f(f1
(x)) = x.
CC.9-12.F.BF.4c (+) Read values of an inverse function from a graph or a table, given that the function has an inverse.
Student Friendly/”I Can” statements
Resources
1. Read values of an inverse function
from a graph or table.
228
Assessments
CC.9-12.F.BF.4d (+) Produce an invertible function from a non-invertible function by restricting the domain.
Student Friendly/”I Can” statements
1. Find the inverse of a function that is not
one-to-one by restricting the domain.
Resources
Assessments
CC.9-12.F.BF.5 (+) Build new functions from existing functions. Understand the inverse relationship between exponents and logarithms and use
this relationship to solve problems involving logarithms and exponents.
Student Friendly/”I Can” statements
1. Understand the inverse relationship
between exponents and logarithms and
use this relationship to solve problems
involving logarithms and exponents.
Resources
229
Assessments
Linear and Exponential Models
CC.9-12.F.LE.1 Construct and compare linear, quadratic, and exponential models and solve problems. Distinguish between situations that can be
modeled with linear functions and with exponential functions.*
Student Friendly/”I Can” statements
1. Recognize that linear functions grow by
equal differences over equal intervals.
2. Recognize that exponential functions
grow by equal factors over equal intervals.
3. Distinguish between situations that can be
modeled with linear functions and with
exponential functions to solve
mathematical and real-world problems.
Resources
Assessments
http://nlvm.usu.edu
CC.9-12.F.LE.1a Prove that linear functions grow by equal differences over equal intervals and that exponential functions grow by equal factors
over equal intervals.*
Student Friendly/”I Can” statements
1. Prove that linear functions grow by equal
differences over equal intervals.
2. Prove that exponential functions grow by
equal factors over equal intervals.
Resources
Assessments
CC.9-12.F.LE.1b. Recognize situations in which one quantity changes at a constant rate per unit interval relative to another.*
Student Friendly/”I Can” statements
1. Recognize situations in which one
quantity changes at a constant rate per
unit (equal differences) interval relative
to another to solve mathematical and
real-world problems.
Resources
230
Assessments
CC.9-12.F.LE.1c Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another.*
Student Friendly/”I Can” statements
1. Recognize situations in which a quantity
grows or decays by a constant percent
rate per unit (equal factors) interval
relative to another to solve
mathematical and real-world problems.
Resources
Assessments
CC.9-12.F.LE.2 Construct and compare linear, quadratic, and exponential models and solve problems. Construct linear and exponential functions,
including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from
a table).*
Student Friendly/”I Can” statements
1. Recognize arithmetic sequences can be
expressed as linear functions.
2. Recognize geometric sequences can be
expressed as exponential functions.
3. Construct linear functions, including
arithmetic sequences, given a graph, a
description of a relationship, or two inputoutput pairs (include reading these from a
table).
4. Construct exponential functions, including
geometric sequences, given a graph, a
description of a relationship, or two inputoutput pairs (include reading these from a
table).
5. Determine when a graph, a description of
a relationship, or two input-output pairs
(include reading these from a table)
represents a linear or exponential
function in order to solve problems.
Resources
231
Assessments
CC.9-12.F.LE.3 Construct and compare linear, quadratic, and exponential models and solve problems. Observe using graphs and tables that a
quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function.*
Student Friendly/”I Can” statements
1. Informally define the concept of “end
behavior”.
2. Compare tables and graphs of linear and
exponential functions to observe that a
quantity increasing exponentially
exceeds all others to solve mathematical
and real-world problems.
Note from Appendix A: Limit to comparisons
between linear and exponential models.
Resources
Assessments
CC.9-12.F.LE.4 Construct and compare linear, quadratic, and exponential models and solve problems. For exponential models, express as a
logarithm the solution to abct = d where a, c, and d are numbers and the base b is 2, 10, or e; evaluate the logarithm using technology.*
Student Friendly/”I Can” statements
1. Recognize the laws and properties of
logarithms, including change of base.
2. Recognize and describe the key features
logarithmic functions.
3. Recognize and know the definition of
logarithm base b.
4. Evaluate a logarithm using technology
5. For exponential models, express as a
Resources
logarithm the solution to ct a ⋅ bct = d ,
where a, b, and d are numbers and the
base is 2, 10, or e.
232
Assessments
CC.9-12.F.LE.5 Interpret expressions for functions in terms of the situation they
model. Interpret the parameters in a linear or exponential function in terms of a context.*
Student Friendly/”I Can” statements
1. Recognize the parameters in a linear or
exponential function including: vertical
and horizontal shifts, vertical and
horizontal dilations.
2. Recognize rates of change and
intercepts as “parameters” in linear or
exponential functions.
3. Interpret the parameters in a linear or
exponential function in terms of a
context.
Resources
233
Assessments
Trigonometric Functions
CC.9-12.F.TF.1 Extend the domain of trigonometric functions using the unit circle. Understand radian measure of an angle as the length of the arc
on the unit circle subtended by the angle.
Student Friendly/”I Can” statements
1. Define a radian measure of an angle as
the length of the arc on the unit circle
subtended by the angle.
2. Define terminal and initial side of an
angle on the unit circle.
Resources
http://nlvm.usu.edu
http://teachingcommons.cdl.edu/mec/teacher
_resources/trigresources.htm
Assessments
CC.9-12.F.TF.2 Extend the domain of trigonometric functions using the unit circle. Explain how the unit circle in the coordinate plane enables the
extension of trigonometric functions to all real numbers, interpreted as radian measures of angles traversed counterclockwise around the unit
circle.
Student Friendly/”I Can” statements
1. Explain the relationship between a
counterclockwise radian measure of an
angle along the unit circle, terminal
coordinate on the unit circle of that
angle, and the associated real number.
2. Explain how radian measures of angles of
the unit circle in the coordinate plane
enable the extension of trigonometric
functions to all real numbers.
Resources
http://teachingcommons.cdl.edu/mec/teacher
_resources/trigresources.htm
Assessments
CC.9-12.F.TF.3 (+) Extend the domain of trigonometric functions using the unit circle. Use special triangles to determine geometrically the values
of sine, cosine, tangent for π/3, π/4 and π/6, and use the unit circle to express the values of sine, cosine, and tangent for π - x, π + x, and 2π - x in
terms of their values for x, where x is any real number.
Student Friendly/”I Can” statements
1. Know the definition of sine and cosine as
y- and x-coordinates of points on the unit
circle and are familiar with the graphs of
the sine and cosine functions.
Resources
http://teachingcommons.cdl.edu/mec/teacher_reso
urces/trigresources.htm
234
Assessments
CC.9-12.F.TF.4 (+) Extend the domain of trigonometric functions using the unit circle. Use the unit circle to explain symmetry (odd and even) and
periodicity of trigonometric functions.
Student Friendly/”I Can” statements
Resources
Assessments
http://teachingcommons.cdl.edu/mec/teacher_reso
1. Define sine and cosine as y- and xurces/trigresources.htm
coordinates of points on the unit circle
2. Use graphs of the sine and cosine
functions.
3. Use unit circle to explain symmetry.
4. Use unit circle to explain periodicity of
trigonometric functions.
CC.9-12.F.TF.5 Model periodic phenomena with trigonometric functions. Choose trigonometric functions to model periodic phenomena with
specified amplitude, frequency, and midline.*
Student Friendly/”I Can” statements
1. Define and recognize the amplitude,
frequency, and midline parameters in a
symbolic trigonometric function.
2. Interpret the parameters of a
trigonometric function (amplitude,
frequency, and midline) in the context
of real-world situations.
3. Explain why real-world or mathematical
phenomena exhibits characteristics of
periodicity.
Resources
http://teachingcommons.cdl.edu/mec/teacher_reso
urces/trigresources.htm
4. Choose trigonometric functions to
model periodic phenomena for which
the amplitude, frequency, and midline
are already specified.
235
Assessments
CC.9-12.F.TF.6 (+) Model periodic phenomena with trigonometric functions. Understand that restricting a trigonometric function to a domain on
which it is always increasing or always decreasing allows its inverse to be constructed.
Student Friendly/”I Can” statements
1. Using trigonometric functions model
periodic phenomena restricting domain
to always increasing.
2. Using trigonometric functions model
periodic phenomena restricting domain
to always decreasing.
3. Compare and contrast the two models.
Resources
http://teachingcommons.cdl.edu/mec/teacher_reso
urces/trigresources.htm
Assessments
CC.9-12.F.TF.7 (+) Model periodic phenomena with trigonometric functions. Use inverse functions to solve trigonometric equations that arise in
modeling contexts; evaluate the solutions using technology, and interpret them in terms of the context.*
Student Friendly/”I Can” statements
1. Evaluate the solutions of a problem using
trigonometric functions to model a
periodic phenomena and its inverse using
technology.
2. Interpret the solutions of a problem using
trigonometric functions to model a
periodic phenomena and its inverse in
terms of the context.
Resources
CC.9-12.F.TF.8 Prove and apply trigonometric identities. Prove the Pythagorean identity
or tan (Ø), given sin (Ø), cos (Ø), or tan (Ø), and the quadrant of the angle.
Student Friendly/”I Can” statements
1. Define trigonometric ratios as related to
the unit circle.
2. Prove the Pythagorean identity sin2(θ) +
cos2(θ) = 1
3. Use the Pythagorean identity, sin2(θ) +
cos2(θ) = 1, to find sin (θ), cos (θ), or tan
(θ), given sin (θ), cos (θ), or tan (θ), and
the quadrant of the angle.
Resources
236
Assessments
sin 2 (φ ) + cos 2 (φ ) = 1 and use it to find sin (Ø), cos (Ø),
Assessments
CC.9-12.F.TF.9 (+) Prove and apply trigonometric identities. Prove the addition and subtraction formulas for sine, cosine, and tangent and use
them to solve problems.
Student Friendly/”I Can” statements
1. Apply trigonometric identities to prove
solutions.
2. Prove addition and subtraction formulas
for sine, cosine, and tangent.
3. Use proofs to solve problems.
Resources
237
Assessments
GEOMETRY
Congruence
CC.9-12.G.CO.1 Experiment with transformations in the plane. Know precise definitions of angle, circle, perpendicular line, parallel line, and line
segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc.
Student Friendly/”I Can” statements
1. Describe the undefined terms: point, line,
and distance along a line in a plane.
2. Define perpendicular lines, parallel lines,
line segments, and angles.
3. Define circle and the distance around a
circular arc.
Resources
http://nlvm.usu.edu
238
Assessments
CC.9-12.G.CO.2 Experiment with transformations in the plane. Represent transformations in the plane using, e.g., transparencies and geometry
software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations
that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch).
Student Friendly/”I Can” statements
1. Describe the different types of
transformations including translations,
reflections, rotations and dilations.
2. Describe transformations as functions
that take points in the coordinate plane
as inputs and give other points as
outputs
3. Represent transformations in the plane
using, e.g., transparencies and geometry
software.
4. Write functions to represent
transformations.
5. Compare transformations that preserve
distance and angle to those that do not
(e.g., translation versus horizontal
stretch)
From Appendix A: Build on student
experience with rigid motions from earlier
grades. Point out the basis of rigid motions
in geometric concepts, e.g, translations
move points a specific distance along a line
parallel to a specified line; rotations move
objects along a circular arc with a specified
center through a specified angle.
Resources
239
Assessments
CC.9-12.G.CO.3 Experiment with transformations in the plane. Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the
rotations and reflections that carry it onto itself.
Student Friendly/”I Can” statements
1. Given a rectangle, parallelogram,
trapezoid, or regular polygon, describe
the rotations and/or reflections that carry
it onto itself.
From Appendix A: Build on student experience
with rigid motions from earlier grades. Point
out the basis of rigid motions in geometric
concepts, e.g, translations move points a
specific distance along a line parallel to a
specified line; rotations move objects along a
circular arc with a specified center through a
specified angle.
Resources
Assessments
CC.9-12.G.CO.4 Experiment with transformations in the plane. Develop definitions of rotations, reflections, and translations in terms of angles,
circles, perpendicular lines, parallel lines, and line segments.
Student Friendly/”I Can” statements
1. Recall definitions of angles, circles,
perpendicular and parallel lines and line
segments.
2. Develop definitions of rotations,
reflections and translations in terms of
angles, circles, perpendicular lines,
parallel lines and line segments.
From Appendix A: Build on student experience
with rigid motions from earlier grades. Point
out the basis of rigid motions in geometric
concepts, e.g., translations move points a
specific distance along a line parallel to a
specified line; rotations move objects along a
circular arc with a specified center through a
specified angle.
Resources
240
Assessments
CC.9-12.G.CO.5 Experiment with transformations in the plane. Given a geometric figure and a rotation, reflection, or translation, draw the
transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given
figure onto another.
Student Friendly/”I Can” statements
1. Given a geometric figure and a rotation,
reflection or translation, draw the
transformed figure using, e.g. graph
paper, tracing paper or geometry
software.
2. Draw a transformed figure and specify the
sequence of transformations that were
used to carry the given figure onto the
other.
From Appendix A: Build on student experience
with rigid motions from earlier grades. Point
out the basis of rigid motions in geometric
concepts, e.g., translations move points a
specific distance along a line parallel to a
specified line; rotations move objects along a
circular arc with a specified center through a
specified angle.
Resources
Assessments
CC.9-12.G.CO.6 Understand congruence in terms of rigid motions. Use geometric descriptions of rigid motions to transform figures and to predict
the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are
congruent.
Student Friendly/”I Can” statements
1. Use descriptions of rigid motion and
transformed geometric figures to predict
the effects rigid motion has on figures in
the coordinate plane.
2. Knowing that rigid transformations
preserve size and shape or distance and
angle, use this fact to connect the idea of
congruency and develop the definition of
congruent.
Resources
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Assessments
CC.9-12.G.CO.7 Understand congruence in terms of rigid motions. Use the definition of congruence in terms of rigid motions to show that two
triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent.
Student Friendly/”I Can” statements
1. Identify corresponding angles and sides of
two triangles.
2. Identify corresponding pairs of angles and
sides of congruent triangles after rigid
motions.
3. Use the definition of congruence in terms
of rigid motions to show that two
triangles are congruent if corresponding
pairs of sides and corresponding pairs of
angles are congruent.
4. Use the definition of congruence in terms
of rigid motions to show that if the
corresponding pairs of sides and
corresponding pairs of angles of two
triangles are congruent then the two
triangles are congruent.
5. Justify congruency of two triangles using
transformations.
From Appendix A: Rigid motions are at the
foundation of the definition of congruence.
Students reason from the basic properties of
rigid motions (that they preserve distance and
angle), which are assumed without proof.
Rigid motions and their assumed properties
can be used to establish the usual triangle
congruence criteria, which can then be used
to prove other theorems.
Resources
242
Assessments
CC.9-12.G.CO.8 Understand congruence in terms of rigid motions. Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow
from the definition of congruence in terms of rigid motions.
Student Friendly/”I Can” statements
Resources
1. Informally use rigid motions to take angles
to angles and segments to segments (from
8th grade).
2. Formally use dynamic geometry software
or straightedge and compass to take
angles to angles and segments to
segments.
3. Explain how the criteria for triangle
congruence (ASA, SAS, SSS) follows from
the definition of congruence in terms of
rigid motions (i.e. if two angles and the
included side of one triangle are
transformed by the same rigid motion(s)
then the triangle image will be congruent
to the original triangle).
From Appendix A: Rigid motions are at the
foundation of the definition of congruence.
Students reason from the basic properties of
rigid motions (that they preserve distance and
angle), which are assumed without proof.
Rigid motions and their assumed properties
can be used to establish the usual triangle
congruence criteria, which can then be used
to prove other theorems.
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Assessments
CC.9-12.G.CO.9 Prove geometric theorems. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a
transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular
bisector of a line segment are exactly those equidistant from the segment’s endpoints.
Student Friendly/”I Can”
statements
1. Identify and use
properties of; Vertical
angles, Parallel lines
with transversals, All
angle relationships,
Corresponding angles,
Alternate interior
angles, Perpendicular
bisector, and Equidistant
from endpoint.
2. Prove vertical angles are
congruent.
3. Prove corresponding
angles are congruent
when two parallel lines
are cut by a transversal
and converse.
4. Prove alternate interior
angles are congruent
when two parallel lines
are cut by a transversal
and converse.
5. Prove points are on a
perpendicular bisector
of a line segment are
exactly equidistant from
the segments endpoint.
From Appendix A:
Encourage multiple ways of
Resources
http://mathwarehouse.com/geometry/angle/interactive_transversal_angles.php
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Assessments
writing proofs, such as in
narrative paragraphs, using
flow diagrams, in twocolumn format, and using
diagrams without words.
Students should be
encouraged to focus on the
validity of the underlying
reasoning while exploring a
variety of formats for
expressing that reasoning.
245
CC.9-12.G.CO.10 Prove geometric theorems. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to
180 degrees; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side
and half the length; the medians of a triangle meet at a point.
Student Friendly/”I Can” statements
1. Identify the hypothesis and conclusion of
a theorem.
2. Design an argument to prove theorems
about triangles.
3. Analyze components of the theorem.
4. Prove theorems about triangles.
From Appendix A: Encourage multiple ways of
writing proofs, such as in narrative
paragraphs, using flow diagrams, in twocolumn format, and using diagrams without
words. Students should be encouraged to
focus on the validity of the underlying
reasoning while exploring a variety of formats
for expressing that reasoning.
Implementations of G.CO.10 may be extended
to include concurrence of perpendicular
bisectors and angle bisectors as preparation
for G.C.3 in Unit 5.
Resources
246
Assessments
CC.9-12.G.CO.11 Prove geometric theorems. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite
angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals.
Student Friendly/”I Can” statements
1. Classify types of quadrilaterals.
2. Explain theorems for parallelograms and
relate to figure.
3. Use the principle that corresponding
parts of congruent triangles are
congruent to solve problems.
4. Use properties of special quadrilaterals
in a proof.
From Appendix A: Encourage multiple ways of
writing proofs, such as in narrative
paragraphs, using flow diagrams, in twocolumn format, and using diagrams without
words. Students should be encouraged to
focus on the validity of the underlying
reasoning while exploring a variety of formats
for expressing that reasoning.
Resources
247
Assessments
CC.9-12.G.CO.12 Make geometric constructions. Make formal geometric constructions with a variety of tools and methods (compass and
straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a
segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line
parallel to a given line through a point not on the line.
Student Friendly/”I Can” statements
1. Explain the construction of geometric
figures using a variety of tools and
methods.
2. Apply the definitions, properties and
theorems about line segments, rays and
angles to support geometric
constructions.
3. Apply properties and theorems about
parallel and perpendicular lines to
support constructions.
4. Perform geometric constructions
including: Copying a segment; copying
an angle; bisecting a segment; bisecting
an angle; constructing perpendicular
lines, including the perpendicular
bisector of a line segment; and
constructing a line parallel to a given
line through a point not on the line,
using a variety of tools and methods
(compass and straightedge, string,
reflective devices, paper folding,
dynamic geometric software, etc.).
From Appendix A: Build on prior student
experience with simple constructions.
Emphasize the ability to formalize and explain
how these constructions result in the desired
objects. Some of these constructions are
closely related to previous standards and can
be introduced in conjunction with them.
Resources
248
Assessments
CC.9-12.G.CO.13 Make geometric constructions. Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.
Student Friendly/”I Can” statements
Note: Underpinning performance, reasoning,
and knowledge targets, if applicable, are
addressed in G.CO.12
1. Construct an equilateral triangle, a square
and a regular hexagon inscribed in a circle.
From Appendix A: Build on prior student
experience with simple constructions.
Emphasize the ability to formalize and explain
how these constructions result in the desired
objects.
Some of these constructions are closely
related to previous standards and can be
introduced in conjunction with them.
Resources
249
Assessments
Similarity, Right Triangles and Trigonometry
CC.9-12.G.SRT.1 Understand similarity in terms of similarity transformations. Verify experimentally the properties of dilations given by a center
and a scale factor:
-- a. A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center
unchanged.
-- b. The dilation of a line segment is longer or shorter in the ratio given by the scale factor.
Student Friendly/”I Can” statements
1. Define image, pre-image, scale factor,
center, and similar figures as they relate
to transformations.
2. Identify a dilation stating its scale factor
and center
3. Verify experimentally that a dilated image
is similar to its pre-image by showing
congruent corresponding angles and
proportional sides.
4. Verify experimentally that a dilation takes
a line not passing through the center of
the dilation to a parallel line by showing
the lines are parallel.
5. Verify experimentally that dilation leaves
a line passing through the center of the
dilation unchanged by showing that it is
the same line.
6. Explain that the scale factor represents
how many times longer or shorter a
dilated line segment is than its pre-image.
7. Verify experimentally that the dilation of a
line segment is longer or shorter in the
ratio given by the scale factor.
Resources
http://nlvm.usu.edu
250
Assessments
CC.9-12.G.SRT.2 Understand similarity in terms of similarity transformations. Given two figures, use the definition of similarity in terms of similarity
transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all
corresponding pairs of angles and the proportionality of all corresponding pairs of sides.
Student Friendly/”I Can” statements
1. By using similarity transformations,
explain that triangles are similar if all pairs
of corresponding angles are congruent
and all corresponding pairs of sides are
proportional.
2. Given two figures, decide if they are
similar by using the definition of similarity
in terms of similarity transformations.
Resources
Assessments
CC.9-12.G.SRT.3 Understand similarity in terms of similarity transformations. Use the properties of similarity transformations to establish the AA
criterion for two triangles to be similar.
Student Friendly/”I Can” statements
1. Recall the properties of similarity
transformations.
2. Establish the AA criterion for similarity of
triangles by extending the properties of
similarity transformations to the general
case of any two similar triangles.
Resources
Assessments
CC.9-12.G.SRT.4 Prove theorems involving similarity. Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle
divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity.
Student Friendly/”I Can” statements
1. Recall postulates, theorems, and
definitions to prove theorems about
triangles.
2. Prove theorems involving similarity about
triangles.
(Theorems include: a line parallel to one side of a
Resources
triangle divides the other two proportionally, and
conversely; the Pythagorean Theorem proved
using triangle similarity.)
251
Assessments
CC.9-12.G.SRT.5 Prove theorems involving similarity. Use congruence and similarity criteria for triangles to solve problems and to prove
relationships in geometric figures.
Student Friendly/”I Can” statements
1. Recall congruence and similarity criteria
for triangles.
2. Use congruency and similarity theorems
for triangles to solve problems.
3. Use congruency and similarity theorems
for triangles to prove relationships in
geometric figures.
Resources
Assessments
CC.9-12.G.SRT.6 Define trigonometric ratios and solve problems involving right triangles. Understand that by similarity, side ratios in right
triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles.
Student Friendly/”I Can” statements
1. Names the sides of right triangles as
related to an acute angle.
2. Recognize that if two right triangles have
a pair of acute, congruent angles that the
triangles are similar.
3. Compare common ratios for similar right
triangles and develop a relationship
between the ratio and the acute angle
leading to the trigonometry ratios.
Resources
Assessments
CC.9-12.G.SRT.7 Define trigonometric ratios and solve problems involving right triangles. Explain and use the relationship between the sine and
cosine of complementary angles.
Student Friendly/”I Can” statements
1. Use the relationship between the sine and
cosine of complementary angles.
2. Explain how the sine and cosine of
complementary angles are related to each
other.
Resources
252
Assessments
CC.9-12.G.SRT.8 Define trigonometric ratios and solve problems involving right triangles. Use trigonometric ratios and the Pythagorean Theorem
to solve right triangles in applied problems.
Student Friendly/”I Can” statements
1. Recognize which methods could be used
to solve right triangles in applied
problems.
2. Solve for an unknown angle or side of a
right triangle using sine, cosine, and
tangent.
3. Apply right triangle trigonometric ratios
and the Pythagorean Theorem to solve
right triangles in applied problems.
Resources
Assessments
CC.9-12.G.SRT.9 (+) Apply trigonometry to general triangles. Derive the formula A = (1/2)ab sin(C) for the area of a triangle by drawing an
auxiliary line from a vertex perpendicular to the opposite side.
Student Friendly/”I Can” statements
1. Recall right triangle trigonometry to solve
mathematical problems.
2. For a triangle that is not a right triangle,
draw an auxiliary line from a vertex,
perpendicular to the opposite side and
derive the formula, A=½ ab sin (C), for the
area of a triangle, using the fact that the
height of the triangle is, h=a sin(C).
Resources
253
Assessments
CC.9-12.G.SRT.10 (+) Apply trigonometry to general triangles. Prove the Laws of Sines and Cosines and use them to solve problems.
Student Friendly/”I Can” statements
1. Use the Laws of Sines and Cosines this
to find missing angles or side length
measurements.
2. Prove the Law of Sines
3. Prove the Law of Cosines
4. Recognize when the Law of Sines or Law
of Cosines can be applied to a problem
and solve problems in context using
them.
From Appendix A: With respect to the
general case of Laws of Sines and Cosines,
the definition of sine and cosine must be
extended to obtuse angles.
Resources
Assessments
CC.9-12.G.SRT.11 (+) Apply trigonometry to general triangles. Understand and apply the Law of Sines and the Law of Cosines to find unknown
measurements in right and non-right triangles (e.g., surveying problems, resultant forces).
Student Friendly/”I Can” statements
1. Determine from given measurements in
right and non-right triangles whether it
is appropriate to use the Law of Sines or
Cosines.
2. Apply the Law of Sines and the Law of
Cosines to find unknown measurements
in right and non-right triangles (e.g.,
surveying problems, resultant forces).
From Appendix A: With respect to the
general case of the Laws of Sines and
Cosines, the definition of sine and cosine
must be extended to obtuse angles.
Resources
254
Assessments
Circles
CC.9-12.G.C.1 Understand and apply theorems about circles. Prove that all circles are similar.
1.
2.
3.
4.
Student Friendly/”I Can” statements
Recognize when figures are similar. (Two
figures are similar if one is the image of
the other under a transformation from
the plane into itself that multiplies all
distances by the same positive scale
factor, k. That is to say, one figure is a
dilation of the other. )
Compare the ratio of the circumference of
a circle to the diameter of the circle.
Discuss, develop and justify this ratio for
several circles.
Determine that this ratio is constant for all
circles.
Resources
Assessments
http://nlvm.usu.edu
CC.9-12.G.C.2 Understand and apply theorems about circles. Identify and describe relationships among inscribed angles, radii, and chords.
Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a
circle is perpendicular to the tangent where the radius intersects the circle.
Student Friendly/”I Can” statements
1. Identify inscribed angles, radii, chords,
central angles, circumscribed angles,
diameter, tangent.
2. Recognize that inscribed angles on a
diameter are right angles.
3. Recognize that radius of a circle is
perpendicular to the radius at the point of
tangency.
4. Examine the relationship between central,
inscribed and circumscribed angles by
applying theorems about their measures.
Resources
255
Assessments
CC.9-12.G.C.3 Understand and apply theorems about circles. Construct the inscribed and circumscribed circles of a triangle, and prove properties
of angles for a quadrilateral inscribed in a circle.
Student Friendly/”I Can” statements
1. Define inscribed and circumscribed circles
of a triangle.
2. Recall midpoint and bisector definitions.
3. Define a point of concurrency.
4. Prove properties of angles for a
quadrilateral inscribed in a circle.
5. Construct inscribed circles of a triangle
6. Construct circumscribed circles of a
triangle.
Resources
Assessments
CC.9-12.G.C.4 (+) Understand and apply theorems about circles. Construct a tangent line from a point outside a given circle to the circle.
Student Friendly/”I Can” statements
1. Recall vocabulary: Tangent, Radius,
Perpendicular bisector , Midpoint, Identify
the center of the circle
2. Synthesize theorems that apply to circles
and tangents, such as:
Tangents drawn from a common
external point are congruent.
A radius is perpendicular to a tangent
at the point of tangency.
3. Construct the perpendicular bisector of
the line segment between the center C to
the outside point P.
4. Construct arcs on circle C from the
midpoint Q, having length of CQ.
5. Construct the tangent line.
Resources
256
Assessments
CC.9-12.G.C.5 Find arc lengths and areas of sectors of circles. Derive using similarity the fact that the length of the arc intercepted by an angle is
proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector.
Student Friendly/”I Can” statements
1. Recall how to find the area and
circumference of a circle.
2. Explain that 1° = Π/180 radians
3. Recall from G.C.1, that all circles are
similar.
4. Determine the constant of proportionality
(scale factor).
5. Justify the radii of any two circles (r 1 and
r2) and the arc lengths (s1 and s2)
determined by congruent central angles
are proportional, such that r1 /s1 = r2/s2
6. Verify that the constant of a proportion is
the same as the radian measure, Θ, of the
given central angle. Conclude s = r Θ
From Appendix A: Emphasize the similarity of
all circles. Note that by similarity of sectors
with the same central angle, arc lengths are
proportional to the radius. Use this as a basis
for introducing radian as a unit of measure. It
is not intended that it be applied to the
development of circular trigonometry in this
course.
Resources
257
Assessments
Expressing Geometric Properties with Equations
CC.9-12.G.GPE.1 Translate between the geometric description and the equation for a conic section. Derive the equation of a circle of given center
and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation.
Student Friendly/”I Can” statements
1. Define a circle.
2. Use Pythagorean Theorem.
3. Complete the square of a quadratic
equation.
4. Derive equation of a circle using the
Pythagorean Theorem – given
coordinates of the center and length of
the radius.
5. Determine the center and radius by
completing the square.
From Appendix A: Emphasize the similarity
of all circles. Note that by similarity of
sectors with the same central angle, arc
lengths are proportional to the radius. Use
this as a basis for introducing radian as a
unit of measure. It is not intended that it be
applied to the development of circular
trigonometry in this course.
Resources
Assessments
http://nlvm.usu.edu
CC.9-12.G.GPE.2 Translate between the geometric description and the equation for a conic section. Derive the equation of a parabola given a
focus and directrix.
Student Friendly/”I Can” statements
1. Define a parabola including the
relationship of the focus and the
equation of the directrix to the parabolic
shape.
2. Derive the equation of parabola given
the focus and directrix.
From Appendix A: The directrix should be
parallel to a coordinate axis.
Resources
258
Assessments
CC.9-12.G.GPE.3 (+) Translate between the geometric description and the equation for a conic section. Derive the equations of ellipses and
hyperbolas given the foci, using the fact that the sum or difference of distances from the foci is constant.
Student Friendly/”I Can” statements
1. Given the foci, derive the equation of an
ellipse, noting that the sum of the
distances from the foci to any fixed point
on the ellipse is constant, identifying the
major and minor axis.
2. Given the foci, derive the equation of a
hyperbola, noting that the absolute value
of the differences of the distances form
the foci to a point on the hyperbola is
constant, and identifying the vertices,
center, transverse axis, conjugate axis,
and asymptotes.
Resources
Assessments
CC.9-12.G.GPE.4 Use coordinates to prove simple geometric theorems algebraically. For example, prove or disprove that a figure defined by four
given points in the coordinate plane is a rectangle; prove or disprove that the point (1, √3) lies on the circle centered at the origin and containing
the point (0, 2).
Student Friendly/”I Can” statements
Resources
1. Recall previous understandings of
coordinate geometry (including, but not
limited to: distance, midpoint and slope
formula, equation of a line, definitions
of parallel and perpendicular lines, etc.)
2. Use coordinates to prove simple
geometric theorems algebraically.
For example, prove or disprove that
a figure defined by four given points
in the coordinate plane is a
rectangle; prove or disprove that the
point (1, √3) lies on the circle
centered at the origin and
containing the point (0, 2).
From Appendix A: Include simple proofs involving circles.
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Assessments
CC.9-12.G.GPE.5 Use coordinates to prove simple geometric theorems algebraically. Prove the slope criteria for parallel and perpendicular lines
and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given
point).
Student Friendly/”I Can” statements
1. Recognize that slopes of parallel lines are
equal.
2. Recognize that slopes of perpendicular
lines are opposite reciprocals (i.e, the
slopes of perpendicular lines have a
product of -1)
3. Find the equation of a line parallel to a
given line that passes through a given
point.
4. Find the equation of a line perpendicular
to a given line that passes through a given
point.
5. Prove the slope criteria for parallel and
perpendicular lines and use them to solve
geometric problems.
From Appendix A: Relate work on parallel
lines in G.GPE.5 to work on A.REI.5 in High
School Algebra 1 involving systems of
equations having no solution or infinitely
many solutions.
Resources
Assessments
CC.9-12.G.GPE.6 Use coordinates to prove simple geometric theorems algebraically. Find the point on a directed line segment between two given
points that partitions the segment in a given ratio.
Student Friendly/”I Can” statements
1. Recall the definition of ratio.
2. Recall previous understandings of
coordinate geometry.
3. Given a line segment (including those with
positive and negative slopes) and a ratio,
find the point on the segment that
partitions the segment into the given
ratio.
Resources
260
Assessments
CC.9-12.G.GPE.7 Use coordinates to prove simple geometric theorems algebraically. Use coordinates to compute perimeters of polygons and
areas of triangles and rectangles, e.g., using the distance formula.*
Student Friendly/”I Can” statements
Resources
1. Use the coordinates of the vertices of a
polygon to find the necessary dimensions
for finding the perimeter (i.e., the
distance between vertices).
2. Use the coordinates of the vertices of a
triangle to find the necessary dimensions
(base, height) for finding the area (i.e., the
distance between vertices by counting,
distance formula, Pythagorean Theorem,
etc.).
3. Use the coordinates of the vertices of a
rectangle to find the necessary
dimensions (base, height) for finding the
area (i.e., the distance between vertices
by counting, distance formula).
4. Determine the formula for distance.
5. Formulate a model of figures in contextual
problems to compute area and/or
perimeter.
From Appendix A: G.GPE.7 provides practice
with the distance formula and its connection
with the Pythagorean theorem.
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Assessments
Geometric Measurement and Dimensions
CC.9-12.G.GMD.1 Explain volume formulas and use them to solve problems. Give an informal argument for the formulas for the circumference of
a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri’s principle, and informal limit arguments.
Student Friendly/”I
Can” statements
1. Explain the
formulas for the
circumference of a
circle and the area
of a circle by
determining the
meaning of each
term or factor.
2. Explain the
formulas for the
volume of a
cylinder, pyramid
and cone by
determining the
meaning of each
term or factor.
3. Identify that the
cut piece of solid
figure has same
area as the base.
4. Identify what
attributes are
constant and
which would
change.
5. Show calculation
that proves
volumes are the
same when the
area of the
Resources
http://nlvm.usu.edu
http://softchalkconect.com/lesson/files/AG9dfynWh2101b/InformalArgument_formulas2.html
http://www.mathopenref.com/circumference.html
http://historyforkids.org/scienceforkids/math/geometry/circumferenceproof.htm
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Assessments
dissection is the
same.
CC.9-12.G.GMD.2 (+) Explain volume formulas and use them to solve problems. Give an informal argument using Cavalieri’s principle for the
formulas for the volume of a sphere and other solid figures.
Student Friendly/”I
Can” statements
1. Using Cavalieri’s
Principle, provide
informal
arguments to
develop the
formulas for the
volume of spheres
and other solid
figures.
Resources
263
Assessments
CC.9-12.G.GMD.3 Explain volume formulas and use them to solve problems. Use volume formulas for cylinders, pyramids, cones, and spheres to
solve problems.*
Student Friendly/”I Can” statements
1. Identify attributes of 3-D figures
2. Explore relationship between the net of
a solid and volume of a 3-D figure.
3. Identify when one dimension changes,
the volume changes.
4. Develop a formula for the volume for
each solid figure.
5. Utilize the appropriate formula for
volume depending on the figure.
6. Use volume formulas for cylinders,
pyramids, cones, and spheres to solve
contextual problems.
From Appendix A: Informal arguments for
area and volume formulas can make use of
the way in which area and volume scale
under similarity transformations: when one
figure in the plane results from another by
applying a similarity transformation with
scale factor K, its area is K2 times the area of
the first. Similarly, volumes of solid figures
scale by K3 under a similarity
transformations with scale factor K.
Resources
264
Assessments
CC.9-12.G.GMD.4 Visualize relationships between two-dimensional and three-dimensional objects. Identify the shapes of two-dimensional crosssections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects.
Student Friendly/”I Can” statements
1. Use strategies to help visualize
relationships between two-dimensional
and three dimensional objects
2. Relate the shapes of two-dimensional
cross-sections to their threedimensional objects
3. Discover three-dimensional objects
generated by rotations of twodimensional objects.
Resources
265
Assessments
Modeling with Geometry
CC.9-12.G.MG.1 Apply geometric concepts in modeling situations. Use geometric shapes, their measures, and their properties to describe objects
(e.g., modeling a tree trunk or a human torso as a cylinder).*
Student Friendly/”I Can” statements
1. Use measures and properties of geometric
shapes to describe real world objects
2. Given a real world object, classify the
object as a known geometric shape; use
this to solve problems in context.
From Appendix A: Focus on situations that
require relating two- and three-dimensional
objects, determining and using volume, and
the trigonometry of general triangles.
Resources
http://nlvm.usu.edu
Assessments
CC.9-12.G.MG.2 Apply geometric concepts in modeling situations. Apply concepts of density based on area and volume in modeling situations
(e.g., persons per square mile, BTUs per cubic foot).*
Student Friendly/”I Can” statements
1. Define density.
2. Apply concepts of density based on area
and volume to model real-life situations
(e.g., persons per square mile, BTUs per
cubic foot).
3. Only changing one variable, explain how
density changes.
4. Identify real-life applications when density
would be a factor and explain why it is a
factor.
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Assessments
CC.9-12.G.MG.3 Apply geometric concepts in modeling situations. Apply geometric methods to solve design problems (e.g., designing an object
or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios).*
Student Friendly/”I Can” statements
1. Identify the design problem
2. Identify the geometric methods used to
solve problem.
3. Describe a typographical grid system.
4. Apply geometric methods to solve design
problems (e.g., designing an object or
structure to satisfy physical constraints
or minimize cost; working with
typographic grid systems based on
ratios).
From Appendix A: Focus on situations well
modeled by trigonometric ratios for acute
angles.
Resources
267
Assessments
STATISTICS AND PROBABILITY
Categorical and Quantitative Data
CC.9-12.S.ID.1 Summarize, represent, and interpret data on a single count or measurement variable. Represent data with plots on the real
number line (dot plots, histograms, and box plots).*
Student Friendly/”I Can” statements
1. Summarize the data of a dot plot and
identify frequency.
2. Represent frequency of data by
creating a histogram.
3. Use measures of central tendency to
represent data in box plots.
4. Construct dot plots, histograms and
box plots for data on a real number
line.
5. Interpret the tendencies and the data
as represented in each of the graphs.
Resources
http://nlvm.usu.edu
Assessments
CC.9-12.S.ID.2 Summarize, represent, and interpret data on a single count or measurement variable. Use statistics appropriate to the shape of
the data distribution to compare center (median, mean) and spread (interquartile range, standard deviation) of two or more different data sets.*
Student Friendly/”I Can” statements
Resources
1. Calculate median and mean of each
data set.
2. Calculate interquartile range and
standard deviation.
3. Describe a distribution using center
and spread.
4. Use the correct measure of center and
spread to describe a distribution that is
symmetric or skewed.
5. Identify outliers (extreme data points) and
their effects on data sets.
6. Compare two or more different data sets
using the center and spread of each.
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CC.9-12.S.ID.3 Summarize, represent, and interpret data on a single count or measurement variable. Interpret differences in shape, center, and
spread in the context of the data sets, accounting for possible effects of extreme data points (outliers).*
Student Friendly/”I Can” statements
1. Interpret differences in different data sets
in context.
2. Interpret differences due to possible
effects of outliers.
Resources
Assessments
CC.9-12.S.ID.4 Summarize, represent, and interpret data on a single count or measurement variable. Use the mean and standard deviation of a
data set to fit it to a normal distribution and to estimate population percentages. Recognize that there are data sets for which such a procedure is
not appropriate. Use calculators, spreadsheets, and tables to estimate areas under the normal curve.*
Student Friendly/”I Can” statements
Resources
Assessments
1. Identify data sets as approximately
normal or not.
2. Calculate the mean and standard
deviation of a data set.
3. Use the mean and standard deviation to
find Z- score to fit it to a normal
distribution where appropriate.
4. Use calculators, spreadsheets, and tables
to estimate areas under the normal curve.
5. Interprets areas under a normal curve in
context.
CC.9-12.S.ID.5 Summarize, represent, and interpret data on two categorical and quantitative variables. Summarize categorical data for two
categories in two-way frequency tables. Interpret relative frequencies in the context of the data (including joint, marginal, and conditional relative
frequencies). Recognize possible associations and trends in the data.*
Student Friendly/”I Can” statements
1. Create a two-way table from two
categorical variables and read values from
two way table. Interpret joint, marginal,
and relative frequencies in context.
2. Recognize associations and trends in data
from a two-way table.
Resources
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Assessments
CC.9-12.S.ID.6 Summarize, represent, and interpret data on two categorical and quantitative variables. Represent data on two quantitative
variables on a scatter plot, and describe how the variables are related.*
Student Friendly/”I Can” statements
1. Create a scatter plot from two
quantitative variables.
2. Describe the form, strength and direction
of the relationship.
Resources
Assessments
CC.9-12.S.ID.6a Fit a function to the data; use functions fitted to data to solve problems in the context of the data. Use given functions or choose a
function suggested by the context. Emphasize linear, quadratic, and exponential models.*
1.
2.
3.
4.
5.
6.
Student Friendly/”I Can” statements
Categorize data as linear or not. Use
algebraic methods and technology to fit a
linear function to the data. Use the
function to predict values.
Explain the meaning of the slope and yintercept in context.
Categorize data as exponential. Use
algebraic methods and technology to fit
an exponential function to the data. Use
the function to predict values.
Explain the meaning of the growth rate
and y-intercept in context.
Categorize data as quadratic. Use
algebraic methods and technology to fit a
quadratic function to the data. Use the
function to predict values.
Explain the meaning of the constant and
coefficients in context.
Resources
Assessments
CC.9-12.S.ID.6b Informally assess the fit of a function by plotting and analyzing residuals.*
Student Friendly/”I Can” statements
Resources
1. Define residual.
2. Calculate a residual.
3. Create and analyze a residual plot.
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Assessments
CC.9-12.S.ID.6c Fit a linear function for a scatter plot that suggests a linear association.*
Student Friendly/”I Can” statements
1. Categorize data as linear or not.
2. Use algebraic methods and technology to
fit a linear function to the data.
3. Use the function to predict values.
Resources
Assessments
CC.9-12.S.ID.7 Interpret linear models. Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the
data.*
Student Friendly/”I Can” statements
1. Explain the meaning of the slope and yintercept in context.
Resources
Assessments
CC.9-12.S.ID.8 Interpret linear models. Compute (using technology) and interpret the correlation coefficient of a linear fit.*
Student Friendly/”I Can” statements
1. Use a calculator or computer to find the
correlation coefficient for a linear
association.
2. Interpret the meaning of the value in the
context of the data.
Resources
Assessments
CC.9-12.S.ID.9 Interpret linear models. Distinguish between correlation and causation.*
Student Friendly/”I Can” statements
Resources
1. Explain the difference between
correlation and causation.
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Assessments
Inferences and Justifying Conclusions
CC.9-12.S.IC.1 Understand and evaluate random processes underlying statistical experiments. Understand statistics as a process for making
inferences about population parameters based on a random sample from that population.*
Student Friendly/”I Can” statements
1. Explain in context the difference
between values describing a
population and a sample.
Resources
Assessments
http://nlvm.usu.edu
CC.9-12.S.IC.2 Understand and evaluate random processes underlying statistical experiments. Decide if a specified model is consistent with
results from a given data-generating process, e.g., using simulation. For example, a model says a spinning coin falls heads up with probability 0.
5. Would a result of 5 tails in a row cause you to question the model?*
Student Friendly/”I Can” statements
Resources
1. Explain how well and why a sample
represents the variable of interest
from a population.
2. Demonstrate understanding of the
different kinds of sampling methods.
3. Design simulations of random sampling:
assign digits in appropriate proportions
for events, carry out the simulation using
random number generators and random
number tables and explain the outcomes
in context of the population and the
known proportions.
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CC.9-12.S.IC.3 Make inferences and justify conclusions from sample surveys, experiments, and observational studies. Recognize the purposes of
and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each.*
Student Friendly/”I Can” statements
1. Identify situations as either a sample
survey, experiment, or observational
study.
2. Discuss the appropriateness of either a
sample survey’s, experiment’s, or
observational study’s use in contexts with
limiting factors.
3. Design or evaluate sample surveys,
experiments and observational studies
with randomization.
4. Discuss the importance of randomization
in these processes.
Resources
Assessments
CC.9-12.S.IC.4 Make inferences and justify conclusions from sample surveys, experiments, and observational studies. Use data from a sample
survey to estimate a population mean or proportion; develop a margin of error through the use of simulation models for random sampling.*
Student Friendly/”I Can” statements
Resources
1. Use sample means and sample
proportions to estimate population
values.
2. Conduct simulations of random sampling
to gather sample means and sample
proportions.
3. Explain what the results mean about
variability in a population and use results
to calculate margins of error for these
estimates.
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Assessments
CC.9-12.S.IC.5 Make inferences and justify conclusions from sample surveys, experiments, and observational studies. Use data from a
randomized experiment to compare two treatments; use simulations to decide if differences between parameters are significant.*
Student Friendly/”I Can” statements
1. Evaluate effectiveness and differences in
two treatments based on data from
randomized experiments.
2. Explain in context.
3. Use simulations to generate data
simulating application of two treatments.
4. Use results to evaluate significance of
differences.
Resources
Assessments
CC.9-12.S.IC.6 Make inferences and justify conclusions from sample surveys, experiments, and observational studies. Evaluate reports based on
data.*
Student Friendly/”I Can” statements
Resources
1. Read and explain in context data from
outside reports.
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Assessments
Conditional Probability and Rules of Probability
CC.9-12.S.CP.1 Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as
unions, intersections, or complements of other events (“or,” “and,” “not”).
Student Friendly/”I Can” statements
1. Define a sample space and events within
the sample space.
2. Identify subsets from sample space given
defined events, including unions,
intersections and complements of events.
Resources
Assessments
http://nlvm.usu.edu
CC.9-12.S.CP.2 Understand independence and conditional probability and use them to interpret data. Understand that two events A and B are
independent if the probability of A and B occurring together is the product of their probabilities, and use this characterization to determine if they
are independent.*
Student Friendly/”I Can” statements
1. Identify two events as independent or
not.
2. Explain properties of Independence and
Conditional Probabilities in context and
simple English.
Resources
Assessments
CC.9-12.S.CP.3 Understand independence and conditional probability and use them to interpret data. Understand the conditional probability of A
given B as P(A and B)/P(B), and interpret independence of A and B as saying that the conditional probability of A given B is the same as the
probability of A, and the conditional probability of B given A is the same as the probability of B.*
Student Friendly/”I Can” statements
1. Define and calculate conditional
probabilities.
2. Use the Multiplication Principal to decide
if two events are independent and to
calculate conditional probabilities.
Resources
275
Assessments
CC.9-12.S.CP.4 Construct and interpret two-way frequency tables of data when two categories are associated with each object being classified.
Use the two-way table as a sample space to decide if events are independent and to approximate conditional probabilities. For example, collect
data from a random sample of students in your school on their favorite subject among math, science, and English. Estimate the probability that a
randomly selected student from your school will favor science given that the student is in tenth grade. Do the same for other subjects and
compare the results.
Student Friendly/”I Can” statements
1. Construct and interpret two-way
frequency tables of data for two
categorical variables.
2. Calculate probabilities from the table. Use
probabilities from the table to evaluate
independence of two variables.
Resources
Assessments
CC.9-12.S.CP.5 Understand independence and conditional probability and use them to interpret data. Recognize and explain the concepts of
conditional probability and independence in everyday language and everyday situations. For example, compare the chance of having lung cancer
if you are a smoker with the chance of being a smoker if you have lung cancer.*
Student Friendly/”I Can” statements
1. Recognize and explain the concepts of
independence and conditional probability
in everyday situations.
Resources
Assessments
CC.9-12.S.CP.6 Use the rules of probability to compute probabilities of compound events in a uniform probability model. Find the conditional
probability of A given B as the fraction of B’s outcomes that also belong to A, and interpret the answer in terms of the model.*
Student Friendly/”I Can” statements
1. Calculate conditional probabilities using
the definition: “the conditional probability
of A given B as the fraction of B’s
outcomes that also belong to A”.
2. Interpret the probability in context.
Resources
276
Assessments
CC.9-12.S.CP.7 Use the rules of probability to compute probabilities of compound events in a uniform probability model. Apply the Addition Rule,
P(A or B) = P(A) + P(B) – P(A and B), and interpret the answer in terms of the model.*
Student Friendly/”I Can” statements
1. Identify two events as disjoint (mutually
exclusive).
2. Calculate probabilities using the Addition
Rule. Interpret the probability in context.
Resources
Assessments
CC.9-12.S.CP.8 (+) Use the rules of probability to compute probabilities of compound events in a uniform probability model. Apply the general
Multiplication Rule in a uniform probability model, P(A and B) = [P(A)]x[P(B|A)] =[P(B)]x[P(A|B)], and interpret the answer in terms of the model.*
Student Friendly/”I Can” statements
1. Calculate probabilities using the General
Multiplication Rule.
2. Interpret in context.
Resources
Assessments
CC.9-12.S.CP.9 (+) Use the rules of probability to compute probabilities of compound events in a uniform probability model. Use permutations and
combinations to compute probabilities of compound events and solve problems.*
Student Friendly/”I Can” statements
1. Identify situations as appropriate for use
of a permutation or combination to
calculate probabilities.
2. Use permutations and combinations in
conjunction with other probability
methods to calculate probabilities of
compound events and solve problems.
Resources
277
Assessments
Using Probability to Make Decisions
CC.9-12.S.MD.1 (+) Calculate expected values and use them to solve problems. Define a random variable for a quantity of interest by assigning a
numerical value to each event in a sample space; graph the corresponding probability distribution using the same graphical displays as for data
distributions.*
Student Friendly/”I Can” statements
1. Understand what a random variable is and
the properties of a random variable.
Resources
http://nlvm.usu.edu
Assessments
2. Given a probability situation
(theoretical or empirical), be able to
define a random variable, assign
probabilities to it’s sample space, create a
table and graph of the distribution of the
random variable.
CC.9-12.S.MD.2 (+) Calculate expected values and use them to solve problems. Calculate the expected value of a random variable; interpret it as
the mean of the probability distribution.*
Student Friendly/”I Can” statements
Resources
Assessments
1. Calculate and interpret in context the
expected value of a random variable.
CC.9-12.S.MD.3 (+) Calculate expected values and use them to solve problems. Develop a probability distribution for a random variable defined
for a sample space in which theoretical probabilities can be calculated; find the expected value. For example, find the theoretical probability
distribution for the number of correct answers obtained by guessing on all five questions of a multiple-choice test where each question has four
choices, and find the expected grade under various grading schemes.*
Student Friendly/”I Can” statements
Resources
1. Develop a theoretical probability
distribution and find the expected
value.
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Assessments
CC.9-12.S.MD.4 (+) Calculate expected values and use them to solve problems. Develop a probability distribution for a random variable defined
for a sample space in which probabilities are assigned empirically; find the expected value. For example, find a current data distribution on the
number of TV sets per household in the United States, and calculate the expected number of sets per household. How many TV sets would you
expect to find in 100 randomly selected households?*
Student Friendly/”I Can” statements
1. Develop an empirical probability
distribution and find the expected value.
Resources
Assessments
CC.9-12.S.MD.5 (+) Use probability to evaluate outcomes of decisions. Weigh the possible outcomes of a decision by assigning probabilities to
payoff values and finding expected values.*
Student Friendly/”I Can” statements
Resources
Assessments
1. Set up a probability distribution for a
random variable representing payoff
values in a game of chance.
CC.9-12.S.MD.5a (+) Find the expected payoff for a game of chance. For example, find the expected winnings from a state lottery ticket or a game
at a fast-food restaurant.*
Student Friendly/”I Can” statements
1. Find the expected payoff for a game of
chance.
Resources
Assessments
CC.9-12.S.MD.5b (+) Evaluate and compare strategies on the basis of expected values. For example, compare a high-deductible versus a lowdeductible automobile insurance policy using various, but reasonable, chances of having a minor or a major accident.*
Student Friendly/”I Can” statements
1. Evaluate and compare strategies on the
basis of expected values.
Resources
279
Assessments
CC.9-12.S.MD.6 (+) Use probability to evaluate outcomes of decisions. Use probabilities to make fair decisions (e.g., drawing by lots, using a
random number generator).*
Student Friendly/”I Can” statements
1. Make decisions based on expected values.
2. Use expected values to compare long
term benefits of several situations.
Resources
Assessments
CC.9-12.S.MD.7 (+) Use probability to evaluate outcomes of decisions. Analyze decisions and strategies using probability concepts (e.g., product
testing, medical testing, pulling a hockey goalie at the end of a game).*
Student Friendly/”I Can” statements
Resources
1. Explain in context decisions made
based on expected values.
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Assessments