M.Duffy
Current as of August 2006
The Madison Metropolitan School District does not discriminate in its education programs,
related activities (including School-Community Recreation) and employment practices as
required by applicable local, state, and federal laws.
Document History
Draft approved:
First release:
First revision:
Second release:
Second revision:
Third release:
Fourth revision:
September 2001
October 2001
September 2003
October 2003
September 2004
October 2004
August 2006
Standards distributed to all elementary teachers.
Expanded measurement and geometry strands.
Standards distributed to all elementary teachers.
Alignment with WSAS Criterion Referenced Test Framework
Standards distributed to all elementary teachers.
Standards update.
Table of Contents
Overview..............................................................................................................................p. 2 white
MMSD Principles for School Mathematics.........................................................................p. 3 white
MMSD Process Standards for School Mathematics............................................................p. 4 white
MMSD K-5 Mathematics Content Standards for
Number, Operations & Algebraic Relationships .................................................................p. 5 yellow
MMSD Grade level standards for money is included in this section.
by Thread…p.17
MMSD K-5 Mathematics Content Standards for Geometry................................................p.25 pink
by Thread…p.31
MMSD K-5 Mathematics Content Standards for Measurement..........................................p.33 goldenrod
MMSD Grade level standards for time is included in this section.
by Thread…p.39
MMSD K-5 Mathematics Content Standards for Data Analysis & Probability .................p.41 blue
by Thread…p.47
Appendix A: CGI Problem Type Chart ...............................................................................white
Appendix B: Standards as they appear on the MMSD Elementary Report Card ................white
Acknowledgements..............................................................................................................p.51 white
Each set of MMSD K-5 Mathematical Content Standards begins with a summary of the WMAS and
NCTM Content Standards for the corresponding MMSD standards. The summaries provide an
explanation of the scope of mathematics that children should explore in each strand.
Revisions to the original document include combining standards for Number & Operations with
Algebraic Relationships, creating grade specific standards for Geometry and Measurement, more
specificity, clarity and examples across strands, fifth grade connections to MMSD Middle Grades Math
Standards, and alignment to the Wisconsin Student Assessment System Criterion Reference Test
Framework.
Boldface Type indicates new material or grade-level changes in a standard addressed at previous grade
levels.
indicates a standard that appears in modified family-friendly language that appears on the MMSD
Elementary Report Card.
indicates a standard or a set of standards tested and the grade (3-5) the standard is tested
on the WKCE beginning November 2005. Read these standards for both the grade(s) that you currently
teach and the previous grade-level(s) to become fully aware of all standards that are tested.
“WKCE in … grade”
Overview
The Elementary Mathematics Standards Update Committee convened in July of 2001. The committee of
eighteen teachers revised the 1998 MMSD Mathematics Standards to align the new document with the
1998 Wisconsin Model Academic Standards (WMAS) for Mathematics and to reflect the recent changes
in the National Council of Teachers of Mathematics, Principles and Standards for School Mathematics
(PSSM), 2000.
This document represents the collective vision of three groups of professional: the K-5 teachers who
served as Update Committee members; the professionals in the study of elementary mathematics
education; and the MMSD Math Resource team.
The result of the committee’s efforts are standards meant to be used as a guide for instruction, a guide
for evaluating curriculum materials, a guide for assessment of children’s mathematical understanding,
and a guide for communication to parents.
These mathematics standards remain a working document intended to promote discussion about how
children learn mathematics and how best to teach them. We hope you will find the standards beneficial.
Please direct feedback and any questions about the standards to your school’s elementary math resource
teacher listed in the MMSD Staff Directory.
Further information about school math standards and the WMAS test framework can be accessed at the
following:
•
•
•
•
www.madison.k12.wi.us/tnl/standards/math/
standards.nctm.org/
www.dpi.state.wi.us/standards/matintro.html
dpi.wi.gov/pubsales/math_1.html
2
MMSD Principles for School Mathematics
MMSD has adopted the following six NCTM principles for school mathematics. Decisions about the
content and nature of mathematics education have important consequences for students and society. The
following principles should be followed when developing curriculum frameworks, selecting curriculum
materials, planning instructional units or lessons, designing assessments, assigning teachers and students
to classes, making instructional decisions in the classroom, and establishing supportive professional
development programs for teachers.
Equity.
Excellence in mathematics education requires equity—high expectations and strong
support for all students.
Curriculum. Curriculum is more than a collection of activities. The curriculum must be coherent,
focused on important mathematics, and well articulated across the grades.
Teaching.
Effective mathematics teaching requires understanding what students know and need to
learn and then challenging and supporting them to learn it well.
Learning.
Students must learn mathematics with understanding, actively building new knowledge
from experience and prior knowledge.
Assessment.
Assessment should support the learning of important mathematics and furnish useful
information to both teachers and students.
Technology. Technology is essential in teaching and learning mathematics; it influences the
mathematics that is taught and enhances students’ learning
3
MMSD Process Standards for School Mathematics
WKCE in 3rd, 4th, & 5th grade
MMSD has adopted the following five NCTM Process Standards for School Mathematics (PSSM, pp. 52, 56,
60, 64, 67) which align with Wisconsin Model Academic Standard A: Mathematical Processes, 1998 (p. 4):
Students in Wisconsin will draw on a broad body of mathematical knowledge and apply
a variety of mathematical skills and strategies, including reasoning, oral and written
communication, and the use of appropriate technology, when solving mathematical,
real-world and non-routine problems.
In order to participate fully as a citizen and a worker in our contemporary world, a person should be
mathematically powerful. Mathematical power is the ability to explore, to conjecture, to reason logically, and to
apply a wide repertoire of methods to solve problems. Because no one lives and works in isolation, it is important
to have the ability to communicate mathematical ideas clearly and effectively.
Problem Solving Standard
In grades K-5 all MMSD students will:
• Build new mathematical knowledge through problem solving
• Solve problems that arise in mathematics and in other contexts WMAS A.4.4
• Apply and adapt a variety of appropriate strategies to solve problems (Ex. illustrate, simplify, look for
•
•
patterns and relationships, test reasonableness of results, generalize) WMAS A.4.1
Monitor and reflect on the process of mathematical problem solving
Formulate questions for further explorations WMAS A.4.1
Reasoning and Proof Standard
In grades K-5 all MMSD students will:
• Recognize reasoning and justification as fundamental aspects of mathematics WMAS A.4.1
• Make and investigate mathematical conjectures
• Develop and analyze problem solving strategies, mathematical arguments, and justifications WMAS A.4.5
Communication Standard
In grades K-5 all MMSD students will:
• Organize and consolidate mathematical thinking
• Communicate mathematical thinking coherently to peers, teachers, and others using the language of
•
mathematics to express mathematical ideas precisely (Ex. words, physical objects, numbers, symbols,
pictures, charts, graphs, tables, diagrams, and models) WMAS A.4.2, WMAS A.4.5
Analyze and compare the mathematical thinking and strategies of others
Representation Standard
In grades K-5 all MMSD students will:
• Create and use representations to organize, record, and communicate mathematical ideas WMAS 4.4
• Select, apply, and translate among mathematical representations to solve problems
• Use representations to model and interpret physical, social, and mathematical phenomena
Connections Standard
In grades K-5 all MMSD students will:
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•
•
•
•
Read and understand mathematical texts and other instructional materials and recognize mathematical
ideas as they appear in other contexts
Recognize and use connections among mathematical ideas
See relationships between various kinds of problems and actual events WMAS A.4.1, WMAS A.4.3
Understand how mathematical ideas interconnect and build on one another to produce a coherent whole
Recognize and apply mathematics in contexts outside of mathematics including other subjects,
personal experiences, current events, and personal interests WMAS A.4.3
4
MMSD K-5 Mathematics Content Standards for
Number, Operations & Algebraic Relationships
Wisconsin’s Model Academic Standards (WMAS) Standard B: Number, Operations and Relationships, 1998 (p.6)
Students in Wisconsin will use numbers effectively for various purposes, such as counting, measuring,
estimating, and problem solving.
People use numbers to quantify, describe, and label things in the world around them. It is important to know the various ways of
representing them. Number sense is a matter of necessity, not only in one’s occupation but also in the conduct of daily life, such as
shopping, cooking, planning a budget, or analyzing information printed in the media. When computing, an educated person needs to
know which operations (e.g., addition, multiplication), which procedures (e.g., mental techniques, algorithms), or which technological
aids (e.g., calculator, spreadsheet) are appropriate.
National Council of Teachers of Mathematics (NCTM) Principles and Standards for School Mathematics (PSSM)
Number & Operations, 2000 (pp.78-79, 148-149)
Instructional programs from pre-kindergarten through grade 12 should enable all students to:
•
•
•
Understand numbers, ways of representing numbers, relationships among numbers, and number
systems.
Understand meanings of operations and how they relate to one another.
Compute fluently and make reasonable estimates.
The concepts and skills related to number and operations are a major emphasis of mathematics instruction in kindergarten through
grade 2. Over this span, the small child who holds up two fingers in response to the question “How many is two?” grows to become
the second grader who solves more sophisticated problems using multi-digit computation strategies. In these years, children’s
understanding of number develops significantly. Children come to school with rich and varied informal knowledge of number
(Baroody 1992; Fuson 1988; Gelman 1994). During the early years teachers must help students strengthen their sense of number,
moving from the initial development of basic counting techniques to more sophisticated understandings of the size of numbers,
number relationships, patterns, operations, and place value.
Students’ work with numbers should be connected to their work with other mathematics. For example, computational fluency (having
and using efficient and accurate methods for computing) can both enable and be enabled by students’ investigations of data, a
knowledge of patterns supports the development of skip-counting and algebraic thinking; and experiences with shape, space, and
number help students develop estimation skills related to quantity and size.
As they work with numbers, students should develop efficient and accurate strategies that they understand, whether learning basic
addition and subtraction number combinations or computing with multi-digit numbers. They should explore numbers into the
hundreds and solve problems with a particular focus on two-digit numbers. Although good judgement must be used about which
numbers are important for students of a certain age to work with, teachers should be careful not to underestimate what young students
can learn about number. Students are often surprisingly adept when they encounter numbers, even large numbers, in problem contexts.
Therefore, teachers should regularly encourage students to demonstrate and deepen their understanding of numbers and operations by
solving interesting, contextualized problems and by discussing the representations and strategies they use.
In grades 3-5 students’ development of number sense should continue, with a focus on multiplication and division. Their
understanding of the meanings of these operations should grow deeper as they encounter a range of representations and problem
situations, learn about the properties of these operations, and develop fluency in whole number computation. An understanding of the
base-ten number system should be extended through continued work with larger numbers as well as decimals. Through the study of
various meanings and models of fractions—how fractions are related to each other and to the unit whole and how they are
represented—students can gain facility in comparing fractions, often by using benchmarks such as ½ or 1. They also should consider
numbers less than zero through familiar models such as a thermometer or a number line.
When students leave grade 5, they should be able to solve problems involving whole-number computation and should recognize that
each operation will help them solve many different types of problems. They should be able to solve many problems mentally, to
estimate a reasonable result for a problem, to efficiently recall or derive the basic number combinations for each operation, and to
compute fluently with multi-digit whole numbers. They should understand the equivalence of fractions, decimals, and percents and the
information each type of representation conveys. With these understandings and skills, they should be able to develop strategies for
computing with familiar fractions and decimals.
5
Wisconsin’s Model Academic Standards (WMAS) Standard F: Algebraic Relationships, 1998 (p. 14)
Students in Wisconsin will discover, describe, and generalize simple and complex patterns and
relationships. In the context of real-world problem situations, the student will use algebraic techniques
to define and describe the problem to determine and justify appropriate solutions.
Algebra is the language of mathematics. Much of the observable world can be characterized as having patterned regularity where a
change in one quantity results in changes in other quantities. Through algebra and the use of variables and functions, mathematical
models can be built which are essential to personal, scientific, economic, social, medical, artistic, and civic fields of inquiry.
National Council of Teachers of Mathematics (NCTM) Principles and Standards for School Mathematics (PSSM)
2000 Algebraic Relationships Standard (pp. 90-91, 93, 158-159)
Instructional programs from pre-kindergarten through grade 12 should enable all students to:
•
•
•
•
Understand patterns, relations, and functions
Represent and analyze mathematical situations and structures using algebraic symbols
Use mathematical models to represent and understand quantitative relationships
Analyze change in various contexts.
Algebraic concepts can evolve and continue to develop during pre-kindergarten through grade 2. They will be manifested through
work with classification, patterns, and relations, operations with whole numbers, explorations of function, and step-by-step processes.
When students notice that operations seem to have particular properties, they are beginning to think algebraically. For example, they
realize that changing the order in which two numbers are added does not change the result or that adding zero to a number leaves that
number unchanged. Students’ observations and discussions of how quantities relate to one another lead to initial experiences with
function relationships, and their representations of mathematical situations using concrete objects, pictures, and symbols are the
beginnings of mathematical modeling.
Two central themes of algebraic thinking are appropriate for young students. The first involves making generalizations and using
symbols to represent mathematical ideas, and the second is representing and solving problems (Carpenter and Levi, 1999). For
example, adding pairs of numbers in different orders such as 3 + 5 and 5 + 3 can lead students to infer that when two numbers are
added, the order does not matter. As students generalize from observations about number and operations, they are forming the basis of
algebraic thinking.
Similarly, when students decompose numbers in order to compute, they often use the associative property for the computation. For
instance, they may compute 8 + 5, saying, “8 + 2 is 10, and 3 more is 13.” Students often discover and make generalizations about
other properties. Although it is not necessary to introduce vocabulary such as commutativity or associativity, teachers must be aware
of the algebraic properties used by students at this age. They should build students’ understanding of the importance of their
observations about mathematical situations and challenge them to investigate whether specific observations and conjectures hold for
all cases.
Although algebra is a word that has not commonly been heard in grades 3-5 classrooms, the mathematical investigations and
conversations of students in these grades frequently include elements of algebraic reasoning. These experiences and conversations
provide rich contexts for advancing mathematical understanding and are also an important precursor to the more formalized study of
algebra in the middle and secondary grades. In grades 3-5, algebraic ideas should emerge and be investigated as students:
• identify or build numerical and geometric patterns;
• describe patterns verbally and represent them with tables or symbols;
• look for and apply relationships between varying quantities to make predictions;
• make and explain generalizations that seem to always work in particular situations;
• use graphs to describe patterns and make predictions;
• explore number properties;
• use invented notation, standard symbols, and variables to express a pattern, generalization, or situation.
6
MMSD Math Content Standards for Number, Operations & Algebraic Relationships – Kindergarten
Achievement of the following grade-level standards supports achievement of Wisconsin Model Academic Standards.
Note: Meeting these standards builds the foundation for developing number sense and an understanding of operations.
By the end of kindergarten MMSD students will:
Supports Standard:
WMAS B.4.2
Count sets with up to 30 objects by 1s.
Read, write, order, and model whole numbers up to 20. Child:
•
•
•
•
•
counts forward from any number (1-20)
names the number directly after any number (1-20)
counts backward from any number (10-1)
names the number directly before any number (2-10)
compares two quantities to determine which is “bigger or smaller”
Solve simple (result unknown) story problems involving joining, separating, grouping, and
partitioning (measurement division) situations. Child:
•
WMAS B.4.3
2 items on RC
WMAS B.4.5
directly models solutions
Investigate the concept of ‘half’ by solving equal sharing problems.
WMAS B.4.1
Create and tell a simple math story problem.
WMAS A.4.3
Explain solution strategies and listen to others during class discussions about problem
solving including comparisons and connections between solution strategies.
WMAS B.4.5
Show mathematical thinking using:
•
•
•
•
WMAS B.4.1
WMAS F.4.5
number lines (pre-constructed)
physical objects and drawings 1s
oral descriptions
technology
Recognize, extend, describe, create, and translate patterns from one representation to
another (Ex. sequences of shapes or sounds to simple alphabetic or numeric patterns).
7
WMAS F.4.3
MMSD Math Content Standards for Number, Operations & Algebraic Relationships – Grade 1
Achievement of the following grade-level standards supports achievement of Wisconsin Model Academic Standards.
Note: Meeting these standards builds the foundation for developing number sense and an understanding of operations
By the end of first grade MMSD students will:
Supports Standard:
WMAS B.4.2
Count sets with up to 100 objects by 1s, 2s, 5s, and 10s.
Read, write, order, and model whole numbers up to 100. Child:
•
•
•
counts backward and forward from any number (1-100)
names the number directly before or after any number (1-100)
compares two numbers to determine “which is closer to”,” “before or after,” “bigger or smaller”
Solve story problems involving joining (JRU, JCU), separating (SRU, SCU), part-part wholewhole unknown (PPW-WU), comparing (CDU), grouping, and partitioning* situations. Child:
•
•
•
WMAS B.4.3
WMAS B.4.5
counts on and counts back by 1s
mentally adds a single-digit number to a decade number without counting by 1s (Ex: 20 + 5, 30 + 6)
directly models solutions (Ex. ten frames, base ten blocks)
Investigate fraction concepts [unit fractions (1/2, 1/3, 1/4) and non-unit fractions (2/3, 3/4)]
through:
•
•
•
•
solving partitioning or equal sharing story problems where the solution has a fractional part (Ex: 3
children share 2 cookies equally. How much does each child get?)
reasoning qualitatively about the inverse relationship between the number of partitions (number of
sharers) and the size of a partitioned piece (size of a share)
drawing fractional parts of a set of objects or a single unit
naming fractional parts using words (Ex. “one half” instead of the symbolic representation 1/2)
WMAS B.4.1
WMAS B.4.6
Write a simple story problem and solve it.
WMAS A.4.3
Explain solution strategies and listen to others during class discussions about problem solving
including comparisons and connections between solution strategies.
WMAS A.4.5
Demonstrate fluency with addition facts for:
WMAS B.4.5
•
•
‘within ten’ facts (sums less than 10)
combinations to make 10
Show mathematical thinking using:
•
•
•
•
•
•
renaming (43 is the same as 4 tens and 3 ones)
hundreds charts and number lines (‘pre-constructed’)
ten frames
physical objects and drawings (10s and 1s)
oral and written descriptions
technology
WMAS B.4.1
WMAS F.4.5
Demonstrate an understanding that the “=” sign means “the same as” by solving true/false
and open number sentences such as:
3+1 = 4
5 = 4+1
8+2 = 2+8
4+2 = F +2
7=7
30 = 3+0 (False)
Recognize, extend, describe, create, and translate patterns from one representation to another. (Ex.
5, 7, 9, 11…)
Name and give monetary value of commonly used coins.
8
WMAS F.4.2
WMAS F.4.3
WMAS D.4.4
MMSD Math Content Standards for Number, Operations & Algebraic Relationships – Grade 2
Achievement of the following grade-level standards supports achievement of Wisconsin Model Academic Standards.
Note: Meeting these standards builds the foundation for developing number sense and an understanding of operations.
By the end of second grade MMSD students will:
Supports Standard:
Count sets with up to 1,000 objects by grouping in 100s, 10s, 1s.
Read, write, order, and model whole numbers up to 1,000. Child:
•
•
•
counts backward and forward from any number (1-100)
names the number directly before or after any number (1-100)
compares two numbers to determine “which is closer to,” “before or after,” “bigger or smaller”
WMAS B.4.2
WKCE in 3 rd grade
WMAS B.4.3
WKCE in 3 rd grade
Solve* number (in horizontal and vertical format) and story problems involving joining (JRU,
JCU), separating (SRU, SCU), part whole (PPW-WU and PPW-WU), comparing (CDU),
grouping, and partitioning* situations. Child:
•
•
•
•
•
•
•
•
•
computes sums of two numbers (2-digit or smaller) using place value understanding (beyond direct
modeling)
computes the difference between two numbers (2-digit or smaller) (Ex. incrementing, compensating, or
10s and 1s)
multiplies (in a story context) with products up to 50 (Ex. repeated addition or grouping)
divides (in a story context) with a dividend up to 30 and a divisor up to 5 (Ex. repeated subtraction,
partitioning/sharing or measuring)
estimates
•
to predict a possible solution for a sum or differences using nearest 10 or 100
•
to judge the reasonableness of the results of a computation
mentally composes and decomposes quantities
derives or recalls facts
counts on and counts back by 10s and 1s from any number
directly models solutions using 100s, 10s, and 1s
WMAS B.4.5
WKCE in 3 rd grade
2 items on RC
*Students should be able to solve one-step story problems using more than one strategy.
Investigate fraction concepts [unit fractions (1/2, 1/3, 1/4, 1/6, 1/8) and non-unit fractions (2/3,
3/4)] through:
•
•
•
•
solving partitioning or equal sharing story problems where the solution has a fractional part (Ex: 4 children
share 5 cookies equally. How much does each child get?)
solving fractions as ‘operators’ problems (Ex: How many cookies in ½ bag of 10 cookies?)
drawing fractional parts of a set of objects or a single unit (Ex. cookies, rectangles)
naming fractional parts using words and symbolic notation (1/2)
WMAS B.4.1
WMAS B.4.6
Write a story problem and solve it.
WMAS A.4.3
Explain solution strategies and listen to others during class discussions about problem solving
including comparisons and connections between solution strategies.
WMAS A.4.5
Demonstrate fluency with all addition facts:
•
•
•
•
•
doubles
doubles ± 1
‘across ten’ facts (sums greater than ten)
‘with in ten’ facts (sums less than 10)
combinations to make 10
WMAS B.4.5
Use part-whole relationships, comparison, the concept of difference, or recall to determine
the results of subtraction for related addition facts.
WMAS B.4.5
WMAS D.4.4
Read, write, count, compare, and make combinations of coins in amounts up to $1.00.
9
WKCE in 3 rd grade
Explain mathematical thinking using:
symbolic notation
Notation includes:
ƒ
ƒ
ƒ
ƒ
the equal “= ” sign (a relational symbol meaning “the same as”)
operation symbols
letter or box (variable)
“arrow” language
empty number line
ƒ
For example:
Correct use of “arrow language” to solve 4x7:
+7
+7
+7
7 + 7 → 14 + 7 → 21 + 7 → 28 or 7 ⎯⎯→ 14 ⎯⎯→ 21 ⎯⎯→ 28
Incorrect use of “=” sign to show a solution strategy:
7 + 7 = 14 + 7 = 21 + 7 = 28
Correct use of = sign to show a solution strategy:
7 + 7 = 14
14 + 7 = 21
21 + 7 = 28
•
•
•
•
•
•
WMAS B.4.1
WMAS F.4.2
WKCE in 3 rd grade
symbolic renaming of 2-digit numbers (Ex. 43 = 40 + 3, 65 = 50 + 10 +5)
pictorial or graphical (arrays, charts, graphs, tables, diagrams, tens frames)
number lines (‘empty’ and ‘pre-constructed’)
physical objects and drawings (base ten blocks using 100s, 10s, 1s)
oral and written descriptions
technology
Demonstrate an understanding that the “=” sign means “the same as” by solving true/false or
open number sentences.
For example: A child uses knowledge of facts, basic properties, and relational thinking as opposed to
computation to reason about T/F or open number sentences (equations) such as the following:
* Denotes a false number sentence
WMAS F.4.5
10 = 6+4
6+4 = 10+0
6+4 = 10+1*
6+4 = 5+5
6+4 = F +5
6+7 = 6+6 + F
98+6 - 6 = F
9+8 = 9+9 - F
36+5 = 5 + F
22+37 = 20+30+20*
WKCE in 3 rd grade
Make and discuss conjectures about basic number properties (zero property, commutative
property) that emerge from discussions about T/F or open number sentences.
For example, given this series of T/F or open number sentences:
(* Denotes a false number sentence)
967 + 0 = 967
23 + 7 = 23*
40 + 0 = 400*
89 = c + 89
A child’s conjecture may be:
When you add zero to a number, you get the number you
started with. (a+0 = a)
WMAS F.4.6
Recognize, describe, create, extend, and translate patterns including attribute, number, and
geometric patterns in tables or other sets of data. Child:
•
•
•
counts by 2s, 3s, 5s, 10s, 25s, and 100s
describes relationships within patterns involving addition or subtraction (Ex. ‘What’s my rule?’,
‘Function machines’, ‘16, 26, 36….’)
determines ‘ odd’ or ‘even’ with a total set of 20 or less
10
WMAS F.4.3
WKCE in 3 rd grade
MMSD Math Content Standards for Number, Operations & Algebraic Relationships – Grade 3
Achievement of the following grade-level standards supports achievement of Wisconsin Model Academic Standards.
Note: Meeting these standards builds the foundation for developing number sense and an understanding of operations.
By the end of third grade MMSD students will:
Read, write, and order whole numbers up to 10,000. Child:
•
•
Supports Standard:
compares two whole numbers up to 100 to determine their difference
uses words such as closer to, between, after or before, smaller or bigger
WMAS B.4.3
WKCE in 4th grade
Solve* number (in horizontal and vertical format) and story problems** involving joining,
separating, part-part whole, comparing, grouping, and partitioning situations. Child:
•
•
•
•
•
•
•
•
efficiently computes the sum of two numbers (3-digit or smaller) using place value understanding (Ex:
incrementing, compensating, standard or other algorithm)
efficiently computes the difference between two 2-digit or smaller numbers using place value understanding
3-digit on WKCE in 4th grade
(Ex: incrementing, compensating, 10s-1s)
multiplies 1-digit × 2-digit numbers (Ex. repeated addition, decomposing/composing, doubling, 10s and 1s)
divides with a dividend up to 45 and a divisor up to 5 (Ex. repeated subtraction, partitioning/sharing or
measuring)
estimates
•
to predict possible solutions for sums and differences using nearest 10, 100 or 1,000
•
to judge the reasonableness of the results of a computation
mentally composes and decomposes quantities
counts on and counts back by 100s, 10s, and 1s from any number
directly models solutions using 100s, 10s, and 1s
WMAS B.4.5
WKCE in 4th grade
2 items on RC
*Students should be able to solve story problems using more than one strategy.
**Including multi-step problems and all CGI problem types.
Investigate fraction concepts [unit fractions (1/2, 1/3, 1/4, 1/6, 1/8), non-unit fractions (5/6, 3/8)
improper fractions (3/2, 6/4) and mixed numbers (2½, 1⅔)] through:
•
•
•
•
•
•
•
reasoning about basic equivalencies (Ex: 1/3 = 2/6, 1/2 = 2/4) to solve problems
relating fractions to benchmarks of 0, whole numbers, 1/2s (order simple fractions)
solving joining and separating story problems involving commonly used fractions (with like
denominators)
solving partitioning or equal sharing story problems where the solution has a fractional part (Ex: 4 children
share 5 cookies equally. How much does each child get?)
solving fractions as ‘operators’ problems (Ex: How many eggs in 1/4 of a dozen eggs?)
drawing fractional parts of a set of objects or a single unit (Ex. cookies, rectangles)
naming and using fraction notation for a set of objects or a single unit for 1/4 and 1/2
WMAS B.4.1
WMAS B.4.6
WKCE in 4th grade
Write a story problem and solve it..
WMAS A.4.3
Explain solution strategies and listen to others during class discussions about problem solving
including comparisons and connections between solution strategies.
WMAS F.4.3
Demonstrate fluency using part-whole relationships, comparison, the concept of difference,
or recall to determine the results of subtraction for facts.
Demonstrate fluency with multiplication facts with 2, 5, 4, and 3 as multiplier or multiplicand.
WMAS B.4.5
WMAS B.4.5
WMAS D.4.4
Read, write, count, compare, & make change using a collection of coins up to $1.00 and dollar bills.
11
WKCE in 3rd grade
Explain mathematical thinking using:
symbolic notation
Notation includes:
ƒ
ƒ
ƒ
ƒ
the equal “= ” sign (a relational symbol meaning “the same as”)
operation symbols
letter or box (variable)
“arrow” language
empty number line
ƒ
For example:
Correct use of “arrow language” to solve 4x7:
+7
+7
+7
7 + 7 → 14 + 7 → 21 + 7 → 28 or 7 ⎯⎯→ 14 ⎯⎯→ 21 ⎯⎯→ 28
Incorrect use of “=” sign to show a solution strategy:
7 + 7 = 14 + 7 = 21 + 7 = 28
Correct use of = sign to show a solution strategy:
7 + 7 = 14
14 + 7 = 21
21 + 7 = 28
•
•
•
•
•
•
WMAS B.4.1
WMAS F.4.2
WKCE in 4th grade
symbolic renaming of 3-digit numbers (Ex. 359 = 300+50+9 = 300 + 59)
pictorial or graphical (arrays, charts, graphs, tables, diagrams, tens frames)
number lines (‘empty’ and ‘pre-constructed’)
physical objects and drawings (base ten blocks using 100s, 10s, 1s)
oral and written descriptions
technology
Demonstrate an understanding that the “=” sign means “the same as” by solving true/false or
open number sentences.
For example: A child uses knowledge of facts, basic properties, and relational thinking as opposed to
computation to reason about T/F or open number sentences (equations) such as the following:
* Denotes a false number sentence
192+32-32 = F
376+84 - F = 376
10+10+10 = 3×9 +3
6×4 = 2×4 +4 +8*
15 = 8+7
15 = 15
8+7 = 15+1*
8+7 = 6+9
6+9 = F +5
2×6 = F
WMAS F.4.2
WMAS F.4.6
WKCE in 4th grade
3×6 = 2×6 + F
3×6 +3 = 4×6*
3×7 = 7+7+7
4×7 = 7+7+7+F
14+14 = 4×7
Make and discuss conjectures about basic number properties (zero property, commutative, base
ten) that emerge from discussions about T/F or open number sentences.
For example, given this series of T/F or open number sentences:
(* Denotes a false number sentence)
536 – 0 = 536
48 – 9 = 48*
570 – 0 = 57*
c – 0 = 654
A child’s conjecture may be:
When you subtract zero from a number, you get the number
you started with. (a – 0 = a)
WMAS F.4.6
Recognize, describe, create, extend, and translate patterns including attribute, number, and
geometric patterns in tables or other sets of data. Child:
•
•
•
counts by 2s, 3s, 5s, 10s, 25s and 100s
describes relationships within patterns involving addition or subtraction (Ex. ‘What’s my rule?’,
‘Function machines’, ‘12, 24, 36….’)
determine ‘odd’ or ‘even’ for a set of 20 or less
12
WMAS F.4.3
WKCE in 3rd grade
MMSD Math Content Standards for Number, Operations & Algebraic Relationships – Grade 4
Achievement of the following grade-level standards supports achievement of Wisconsin Model Academic Standards.
Note: Meeting these standards builds the foundation for developing number sense and an understanding of operations.
By the end of fourth grade MMSD students will:
Supports Standard:
Read, write, order and compare, whole numbers up to 100,000 and decimals (money context)
WMAS B.4.3
WKCE in 5th grade
Solve* number (in horizontal and vertical format) and story problems** involving joining,
separating, part-part whole, comparing, grouping, and partitioning situations. Child:
•
•
•
•
•
efficiently computes the sum of two numbers up to 4-digit or smaller and decimals in a money context (Ex:
incrementing, compensating, standard or other algorithm)
efficiently computes the difference between two numbers (3-digit or smaller) (Ex: incrementing,
4-digit or smaller WKCE in 5th grade
compensating, standard or other algorithm)
efficiently multiplies 1-digit × 2-digit numbers (Ex: doubling, ratio, partial products, standard or other
algorithm)
divides a 2-digit number by itself or a single-digit whole number except zero (Ex: ratio table, subtracting
groups of the divisor)
estimates
• addition and subtraction of decimals using money
• multiplication of 2-digit × 1-digit numbers
• division in a story context
• to predict a possible solution for a sum or differences using nearest 10, 100 or 1,000
• to judge the reasonableness of the results of a computation
• mentally composes and decomposes quantities
WMAS B.4.5
WKCE in 5th grade
2 items on RC
*Students should be able to solve story problems using more than one strategy.
**Including multi-step, multi-operation problems, money and all CGI problem types.
Demonstrate an understanding of fraction concepts [unit fractions (1/2, 1/3, 1/4, 1/6, 1/8,
1/10), non-unit fractions (Ex. 5/6, 3/8, 2/10), improper fractions (Ex. 3/2, 6/4), and mixed
numbers (Ex. 3⅛, 1⅔)]. Child:
•
•
•
•
•
•
names and uses fraction notation for a set of objects or a single unit (Ex. ¼, ½)
renames improper fractions
compares two fractions by relating them to benchmarks of 0, whole numbers, 1/2s solves joining and
separating problems involving fractions with like denominators
draws fractional parts of a set of objects or a single unit (Ex. cookies, rectangles)
solves partitioning or equal sharing story problems where the solution has a fractional part (Ex: 4
children share 5 cookies equally. How much does each child get?)
solves fractions as ‘operators’ problems (Ex: How many eggs in 1/4 of a dozen eggs?)
WMAS B.4.1
WMAS B.4.6
WKCE in 5th grade
Investigate fraction concepts [unit fractions (1/2, 1/3, 1/4, 1/6, 1/8), non-unit fractions (5/6, 3/8)
improper fractions (3/2, 6/4) and mixed numbers (2½, 1⅔)] and decimals through:
•
•
•
•
determining the approximate location of fractions (halves, thirds, fourths, and tenths), and decimals
(tenths, hundredths) on a number line
exploring the connections between operations with whole numbers and operations with fractions
relating commonly-used fractions benchmark percents of 25%, 50%, 75%, 100 %
reasons about equivalencies (Ex: 1/3 = 2/6, 1/2 = 2/4)
WMAS B.4.1
WMAS B.4.6
WKCE in 5th grade
Write a story problem and solve it.
WMAS A.4.3
Explain solution strategies and listen to others during class discussions about problem solving
including comparisons and connections between solution strategies.
WMAS F.4.5
WMAS B.4.5
Demonstrate fluency with all multiplication facts.
Use the inverse relationship to determine the results of a division using multiplication facts.
WMAS B.4.5
Read, write, count, compare, and make change using a collection of coins and bills up to $10.00
WKCE in 4th grade
WMAS D.4.4
13
Explain mathematical thinking using:
symbolic notation
Notation includes:
ƒ
ƒ
ƒ
ƒ
ƒ
parenthesis ( )
the equal “= ” sign (a relational symbol meaning “the same as”)
operation symbols
letter or box (variable)
“arrow” language
empty number line
ƒ
For example:
Correct use of “arrow language” to solve 4 x 72:
+72
+72
+72
72 + 72 → 144 + 72 → 216 + 72 → 288 or 72 ⎯⎯→ 144 ⎯⎯→ 216 ⎯⎯→ 288
•
•
•
•
•
•
WMAS B.4.1
WMAS F.4.2
WKCE in 5th grade
Incorrect use of “=” sign to show a solution strategy:
72 + 72 = 144 + 74 = 216 + 74 = 288
Correct use of = sign to show a solution strategy:
72 + 72 = 144
144 + 72 = 216
216 + 72 = 288
symbolic renaming of 4-digit numbers (Ex. 9,473 = 9,000+400+70+3 = 9,400 + 73)
pictorial or graphical (ratio tables, arrays, charts, graphs, tables, diagrams, tens frames)
number lines (‘empty’ and ‘pre-constructed’)
physical objects and drawings (base ten blocks using 100s, 10s, 1s)
oral and written descriptions
technology
Demonstrate an understanding that the “=” sign means “the same as” by solving true/false or
open number sentences.
For example: A child uses knowledge of facts, basic properties, and relational thinking as opposed to
computation to reason about T/F or open number sentences (equations) such as the following
WMAS F.4.2
WMAS F.4.6
* Denotes a false number sentence
31 = 18+13
13 = h+h+3
18+13 = 31+1*
13+18 = 14+19*
13+18 = h +12
456+267−267 = j
342+38−37 = p
53+8−z = 54
36 = 3 + u + u + 3
c+c+c+4 = 16
q+q+8 = 16
2×9 = w+w – 2
42 + b + b = 22 + b
6×4 = 5×4 + F
8+F=3x6
7×9 = 7×10 - F
8×5 + F = 8×6
Make and discuss conjectures about basic number properties that emerge from discussions about
T/F or open number sentences.
For example, given this series of T/F or open number sentences:
(* Denotes a false number sentence)
A child’s conjecture may be:
5,467 × 1 = 5,467
48 × 1 = 49*
8.4 – 10 = 8.4*
c × 1 = 76
WMAS F.4.6
When you a multiply a number times 1, you get the number you
started with (a×1 = a)
Recognize, describe, create, extend, and translate patterns including attribute, number, and
geometric patterns in tables or other sets of data. Child:
•
•
•
counts by 2s, 3s, 5s, 10s, 25s (by any multiple) and 100s (from any number)
describes relationships within patterns involving addition or subtraction (Ex. ‘What’s my rule?’, ‘function
machines’, ‘12, 24, 36….’)
determine ‘odd or even’ for numbers up to 100
14
WMAS F.4.3
WKCE in 4th grade
MMSD Math Content Standards for Number, Operations & Algebraic Relationships – Grade 5
Achievement of the following grade-level standards supports achievement of Wisconsin Model Academic Standards.
Note: Meeting these standards builds the foundation for developing number sense and an understanding of operations.
By the end of fifth grade MMSD students will:
Read, write, and order and compare, whole numbers up to 100,000 and decimals (money context).
Supports Standard:
WMAS B.8.1
WKCE in 6th grade
Solve* number (in horizontal and vertical format) and story problems** (whole numbers and
decimals in money context) involving joining, separating, part-part whole, comparing, grouping,
and partitioning situations. Child:
•
•
•
•
•
efficiently computes the sum of two numbers (4-digit or smaller) and decimals in money context (Ex:
incrementing, compensating, standard or other algorithm)
efficiently computes the difference between two numbers (4-digit or smaller) (Ex: incrementing,
compensating, standard or other algorithm)
efficiently multiplies 2-digit × 3-digit numbers (Ex: ratio table, partial products, standard or other algorithm)
WKCE in 6th grade
efficiently divides up to a 4 digit or smaller number by itself, single digit number except zero, or a decade
(Ex: ratio table, subtracting groups of the divisor)
estimates
• addition and subtraction of decimals using money
• multiplication of 2-digit × 2-digit problems
• division in context
• to predict a possible solution for a sum or differences using nearest 10, 100 or 1,000
• to judge the reasonableness of the results of a computation
• mentally composes and decomposes quantities
WMAS B.8.2
WMAS B.8.7
WKCE in 6th grade
2 items on RC
*Students should be able to solve story problems using more than one strategy.
**Including multi-step, multi-operation problems, and all CGI problem types.
Demonstrate an understanding of fraction concepts [unit fractions (1/2, 1/3, 1/4, 1/6, 1/8, 1/10),
non-unit fractions (Ex. 5/6, 3/8, 2/10), improper fractions (Ex. 3/2, 6/4), and mixed numbers (Ex.
3⅛, 1⅔)].Child:
•
•
•
•
•
•
•
generates and justifies equivalencies
names and uses fraction notation for a set of objects or a single unit (Ex. ¼, ½)
renames improper fractions
compares two fractions by relating them to benchmarks of 0, whole numbers, 1/2s solves joining and separating
problems involving fractions with like denominators
draws fractional parts of a set of objects or a single unit (Ex. cookies, rectangles)
solves partitioning or equal sharing story problems where the solution has a fractional part (Ex: 4 children share
5 cookies equally. How much does each child get?)
solves fractions as ‘operators’ problems (Ex: How many eggs in 1/4 of a dozen eggs?)
WMAS B.4.1
WMAS B.4.6
WKCE in 6th grade
Investigate fraction concepts [unit fractions (1/2, 1/3, 1/4, 1/6, 1/8), non-unit fractions (Ex. 5/6, 3/8)
improper fractions (Ex. 3/2, 6/4) and mixed numbers (Ex. 2½, 1⅔)] and decimals through:
•
•
•
•
determining the approximate location of fractions (halves, thirds, fourths, and tenths), and decimals (tenths,
hundredths) on a number line
exploring the connections between operations with whole numbers and operations with fractions
relating commonly-used fractions benchmark percents of 25%, 50%, 75%, 100 %
reasons about equivalencies (Ex: 1/3 = 2/6, 1/2 = 2/4)
WMAS B.4.1
WMAS B.4.6
WKCE in 6th grade
Write a story problem and solve it.
WMAS A.8.3
Explain solution strategies and listen to others during class discussions about problem solving
including comparisons and connections between solution strategies.
WMAS F.8.5
Know division facts and the first ten multiples of 2-10 and 25.
Use the inverse relationship to determine the results of a division using multiplication facts.
WMAS B.8.6
WKCE in 6th grade
WMAS B.8.6
Determine factors for numbers 1-100.
WKCE in 6th grade
WMAS D.4.4
Read, write, order, count, and make change using a collection of coins and bills up to $100.00.
15
WKCE in 5th grade
Explain mathematical thinking using:
symbolic notation
Notation includes:
ƒ
ƒ
ƒ
ƒ
ƒ
parenthesis ( )
the equal “= ” sign (a relational symbol meaning “the same as”)
operation symbols
letter or box (variable)
empty number line
“arrow” language
ƒ
For example:
Correct use of “arrow language” to solve 4 x 72:
+72
+72
+72
72 + 72 → 144 + 72 → 216 + 72 → 288 or 72 ⎯⎯→ 144 ⎯⎯→ 216 ⎯⎯→ 288
•
•
•
•
•
•
Incorrect use of “=” sign to show a solution strategy:
72 + 72 = 144 + 74 = 216 + 74 = 288
Correct use of = sign to show a solution strategy:
72 + 72 = 144
144 + 72 = 216
216 + 72 = 288
WMAS B.8.1
WMAS F.8.2
WKCE in 6th grade
symbolic renaming of 4-digit numbers (Ex. 9,473 = 9,000+400+70+3 = 9,400 + 73)
pictorial or graphical (ratio tables, arrays, charts, graphs, tables, diagrams, tens frames)
number lines (‘empty’ and ‘pre-constructed’)
physical objects and drawings (base ten blocks using 100s, 10s, 1s)
oral and written descriptions
technology
Demonstrate an understanding that the “=” sign means “the same as” by solving true/false or open
number sentences.
For example: A child uses knowledge of facts, basic properties, and relational thinking as opposed to
computation to reason about T/F or open number sentences (equations) such as the following:
WMAS F.8.1
WMAS B.8.5
* Denotes a false number sentence
23+48 = 24+47
23+48 = t+22
53+68 = 52+67+s
39+73 = 74+38 − b
358+130+2 = 400+80+9*
11 = b+b
k+k-k = 18
m+m+m+12 = m+ 24
75-5+j = 70+j+j
42+b-b = 22+b
23×9 = (20×9) + (c×9)
c×5 = 40+5
15×9 = 9×15
4×10 – 4 = 4×8*
4+3×8 = 4×8-4
WKCE in 6th grade
Make and discuss conjectures about basic number properties that emerge from discussions about
T/F or open number sentences.
For example, given this series of T/F or open number sentences:
(* Denotes a false number sentence):
A child’s conjecture may be:
345 × 0 = 0
28 × 0 = 28*
94 – c = 0
c×0=c
WMAS F.8.5
When you a multiply a number times 0, you get zero. (a×0 = 0)
Recognize, describe, create, extend, and translate patterns including attribute, number, and
geometric patterns in tables or other sets of data (Ex: 12, 24, 36….,)
•
•
•
determine a the eighth item in a pattern when given the first five
determine a rule that describes a functional relationship or pattern using addition, subtraction multiplication rules
determine ‘odd or even’ for numbers up to 100
16
WMAS F.4.3
WKCE in 5th grade
Threads for:
Number, Operations & Algebraic Relationships
Counting sets
By the end of
grade:
Students will:
K
Count sets with up to 30 objects by 1s
1
Count sets with up to 100 objects by 1s, 2s, 5s, and 10s.
2
Count sets with up to 1,000 objects by grouping in 100s, 10s, 1s
WKCE in 3 rd grade
Reading, writing, and ordering whole numbers
By the end of
grade:
Students will:
K
Read, write, order, and model whole numbers up to 20 (See grade level standards for details.)
1
Read, write, order, and model whole numbers up to 100 (See grade level standards for details.)
2
Read, write, order, and model whole numbers up to 1,000 (See grade level standards for details.)
3
Read, write, and order whole numbers up to 10,000
4-5
WKCE in 3 rd grade
WKCE in 4th & 5th grade
Read, write, and order whole numbers up to 100,000 and decimals (money)
Comparing quantities
By the end of
grade:
Students will:
K
•
compare two quantities to determine which is “bigger or smaller’
1
•
compare two numbers to determine which is “closer to”, “between”, “after or before”
2
•
compare two numbers to determine their difference
WKCE in 3 rd grade
Solving story problems
By the end of
grade:
Students will:
K
Solve simple (result unknown) story problems involving joining, separating, grouping, and
partitioning (measurement division) situations.
1
Solve story problems involving joining (JRU, JCU), separating (SRU, SCU), part- part whole wholeunknown (PPW-WU) comparing (CDU), grouping, and partitioning situations.
2
Solve* story problems involving joining, separating, part-part whole (part unknown), comparing,
WKCE in 3 rd grade
grouping, and partitioning situations.
*Students should be able to solve story problems using more than one strategy
3
4-5
**Including multi-step problems and all CGI problem types.
WKCE in 4th grade
**Including multi-step, multi-operation problems, money and all CGI problem types
WKCE in 5h grade
17
Writing story problems
By the end of
grade:
Students will:
K
Create and verbalize simple math story problem.
1
Write a simple story problem.
2-5
Write a story problem and solve it.
Computing sums
By the end of
grade:
Child:
K
•
directly models solutions
•
•
counts on by 1s
mentally adds a single-digit number to a decade number without counting by 1s (Ex: 20 + 5,
30 + 6)
directly models solutions (Ex. ten frames, base ten blocks, tally marks)
1
•
•
2
computes sums of any two numbers (2-digit or smaller) using place value understanding (in
WKCE in 3 rd grade
horizontal format)
mentally composes and decomposes quantities
derives or recalls facts
counts on by 10s and 1s from any number
directly models solutions using 100s, 10s, and 1s
•
•
•
•
•
3
4-5
efficiently computes the sum of any two numbers (3-digit or smaller) using place value
understanding (Ex: incrementing, compensating, standard or other algorithm) WKCE in 4th grade
counts on by 100s , 10s, and 1s from any number
•
•
efficiently computes the sum of any two numbers up to (4-digit or smaller) and decimals in a
WKCE in 5h grade
money context
Computing differences
By the end of
grade:
Child:
K
•
directly models solutions
1
•
•
counts back by 1s
directly models solutions (Ex. ten frames, base ten blocks, tally marks)
•
computes the difference between two numbers (2-digits or smaller) in horizontal
format using incrementing, compensating, or 10s & 1s
WKCE in 3 rd grade
mentally composes and decomposes quantities
derives or recalls facts
counts on and counts back by 10s and 1s from any number
directly models solutions using 100s, 10s, and 1s
2
•
•
•
•
•
3-5
•
efficiently computes the difference between numbers 2-digits or smaller (4-digit in 5th grade)
using place value understanding (Ex: incrementing, 10s-1s, compensating) WKCE in 4th & 5th grade
counts back by 100s, 10s, and 1s from any number
18
Multiplying
By the end of
grade:
K-1
Child:
directly models solutions
2
multiplies (in a story context) with products up to 50 (Ex. repeated addition or grouping
3
multiplies 1-digit × 2-digit numbers (Ex: repeated addition, decomposing/composing, doubling, 10s
WKCE in 4th grade
and 1s)
4
efficiently multiplies 1-digit × 2-digit numbers (Ex: doubling, ratio, partial products, standard or other
WKCE in 5h grade
algorithm)
5
efficiently multiplies 2-digit × 3-digit numbers
WKCE in 3 rd grade
Dividing
By the end of
grade:
K-1
Child:
directly models solutions
2
divides (in a story context) with a dividend up to 30 and a divisor up to 5 (Ex. repeated subtraction,
WKCE in 3 rd grade
partitioning/sharing or measuring)
3
divides with a dividend up to 45 and a divisor up to 5
4
divides a 2-digit number by itself or a single-digit whole number except zero (Ex: ratio table,
WKCE in 5th grade
subtracting groups of the divisor)
5
efficiently divides a 4-digit or smaller number by itself, a single digit number except zero, or a decade
number (10, 20, 30…) (Ex: ratio table, subtracting groups of the divisor)
WKCE in 4th grade
Estimating
By the end of
grade:
Child estimates:
•
2-3
4-5
•
to predict a possible solution for a sum or differences using nearest 10, 100 (1,000 in 5th
WKCE in 3rd and 4th grade
grade)
to judge the reasonableness of the results of a computation
WKCE in 3rd and 4th grade
•
•
•
addition and subtraction of decimals using money
multiplication of 2-digit × 1-digit problems
division in a story context
WKCE in 5h grade
WKCE in 5h grade
WKCE in 5h grade
Money
By the end of
grade:
1
2
3-5
Students will:
Name and gives monetary value of commonly used coins.
Read, write, count, compare, and combine coins in amounts up to $1.00.
WKCE in 3 rd grade
Read write, count, compare, and make change. (amounts to $1.00 in 3rd grade, $10.00 in 4th grade,
WKCE in 3rd, 4th and 5th grade
$100.00 in 5th grade)
19
Investigating Fractions
By the end of
grade:
Investigate fraction concepts through:
1
[unit fractions (1/2, 1/3, 1/4) and non-unit fractions (2/3, 3/4)]
• reason qualitatively about the inverse relationship between the number of partitions (number
of sharers) and the size of a partitioned piece (size of a share)
• drawing fractional parts of a set of objects or a single unit
• naming fractional parts using words (Ex. “one half” instead of the symbolic representation
1/2)
2
[unit fractions (1/2, 1/3, 1/4, 1/6, 1/8) and non-unit fractions (2/3, 3/4)]
• naming a set of objects or a single unit and using fraction notation for 1/2
3
[unit fractions (1/2, 1/3, 1/4, 1/6, 1/8), non-unit fractions (5/6, 3/8) improper fractions (3/2, 6/4) and mixed
numbers (2½, 1⅔)]
•
reasoning about equivalencies (Ex: 1/3 = 2/6, 1/2 = 2/4)
•
relating fractions to benchmarks of 0, whole numbers, 1/2s (order simple fractions)
• naming for a set of objects or a single unit and using fraction notation for ½ & ¼
WKCE in 3rd & 4th grade
4-5
[unit fractions (1/2, 1/3, 1/4, 1/6, 1/8), non-unit fractions (5/6, 3/8) improper fractions (3/2, 6/4) and mixed
numbers (2½, 1⅔) and decimals]
• determining the approximate location of fractions (halves, thirds, fourths, and tenths), and
decimals (tenths, hundredths) on a number line
• exploring the connections between operations with whole numbers and operations with
fractions
• reasons about equivalencies (Ex: 1/3 = 2/6, 1/2 = 2/4)
• relating commonly-used fractions to benchmark percents of 25%, 50%, 75%, 100 %
Solving fraction story problems
By the end of
grade:
K
1
2
3
Investigate fractions concepts by:
•
solving equal sharing problems involving ‘half.’
•
solving partitioning or equal sharing story problems where the solution has a fractional part
WKCE in 3rd grade
(Ex: 3 children share 2 cookies equally. How much does each child get?)
•
•
solve fractions as ‘operators’ problems (Ex: How many cookies in ½ bag?)
solving partitioning or equal sharing story problems where the solution has a fractional part (Ex: 4
WKCE in 3rd & 4th grade
children share 5 cookies equally. How much does each child get?)
•
solving joining and separating story problems involving commonly used fractions with like
WKCE in 5th grade
denominators
solving fractions as ‘operators’ problems (Ex: How many eggs in 1/4 of a dozen eggs?)
•
•
4-5
WKCE in 6th grade
•
•
•
determining the approximate location of fractions (halves, thirds, fourths, and tenths), and
decimals (tenths, hundredths) on a number line
exploring the connections between operations with whole numbers and operations with
fractions
relating commonly-used fractions benchmark percents of 25%, 50%, 75%, 100 %
solving joining and separating story problems involving commonly used fractions with like and
unlike (5th grade) denominators
20
Understanding fraction concepts
By the end of
grade:
4
WKCE in 4th& 5th
grade
5
Child:
[unit fractions (1/2, 1/3, 1/4, 1/6, 1/8, 1/10), non-unit fractions (Ex. 5/6, 3/8, 2/10), improper
fractions (Ex. 3/2, 6/4), and mixed numbers (Ex. 3⅛, 1⅔)]
•
names and uses fraction notation for a set of objects or a single unit (Ex. ¼, ½)
•
renames improper fractions
•
compares two fractions by relating them to benchmarks of 0, whole numbers, 1/2s
•
solves joining and separating problems involving fractions with like denominators
•
draws fractional parts of a set of objects or a single unit (Ex. cookies, rectangles)
•
solves partitioning or equal sharing story problems where the solution has a fractional
part (Ex: 4 children share 5 cookies equally. How much does each child get?)
•
solves fractions as ‘operators’ problems (Ex: How many eggs in 1/4 of a dozen eggs?)
[unit fractions (1/2, 1/3, 1/4, 1/6, 1/8, 1/10), non-unit fractions (Ex. 5/6, 3/8, 2/10), improper fractions
(Ex. 3/2, 6/4), and mixed numbers (Ex. 3⅛, 1⅔)]
•
generates and justifies equivalencies
Showing mathematical thinking
By the end of
grade:
Using:
K
•
•
•
•
number lines (pre-constructed)
physical objects and drawings (1s)
oral descriptions
technology
1
•
•
•
•
•
renaming (43 is the same as 4 tens and 3 ones)
hundreds charts and number lines (‘pre-constructed’)
ten frames
physical objects and drawings (10s and 1s)
oral or written descriptions
•
symbolic notation
Notation includes:
ƒ
parenthesis (WKCE 4th and 5th grade)
ƒ the equal “= ” sign (a relational symbol meaning “the same as”)
ƒ operation symbols
ƒ letter or box (variable)
ƒ “arrow” language
ƒ empty number line
symbolic renaming of numbers (Ex. 43 = 40 + 3, 65 = 50 + 10 +5) WKCE in 3rd, 4th, & 5th grade
pictorial or graphical (ratio tables 4th &5th grade) arrays, charts, graphs, tables, diagrams, tens
frames)
number lines (‘empty’ and ‘pre-constructed’)
physical objects and drawings (base ten blocks using 100s, 10s, 1s)
oral and written descriptions
technology
2-5
WKCE in 3rd, 4th, &
5th grade
•
•
•
•
•
•
Discussing mathematics
By the end of
grade:
K-5
Students will:
Explain solution strategies and listen to others during class discussions about problem solving
including comparisons and connections between solution strategies.
21
Knowing facts
By the end of
grade:
Child will:
1
Demonstrate fluency with addition facts for:
• ‘within ten’ facts (sums less than 10)
• combinations to make 10
2
Demonstrate fluency with all addition facts:
• doubles
• doubles ± 1
• ‘across ten’ facts (sums greater than ten)
3
Demonstrate fluency using part-whole relationships, comparison, the concept of difference, and
recall to determine the results of subtraction for facts.
Demonstrate fluency with multiplication facts with 2, 5, 4, and 3 as multiplier or multiplicand.
4
Demonstrate fluency with all multiplication facts.
Use the inverse relationship to determine the results of a division using multiplication facts.
5
Knows division facts and the first ten multiples of 2-10, and 25.
Determine factors for numbers 1-100.
Recognizing patterns
By the end of
grade:
Child will recognize, extend, describe, create, and translate patterns from one
representation to another including attribute, numeric, or geometric patterns
K
Ex: sequences of shapes or sounds to simple alphabetic or numeric patterns.
1
Ex. 5, 7, 9, 11, …
•
•
2-3
•
counts by 2s, 3s, 5s, 10s, 25s, and 100s
WKCE in 3rd grade
describes relationships within patterns involving addition or subtraction (Ex. ‘What’s my
WKCE in 3rd grade
rule?’, ‘Function machines’, ‘16, 26, 36….’)
determines ‘ odd’ or ‘even’ with a total set of up to 12 (20) using grouping by 2s
WKCE in 3rd & 4th grade
4
5
•
counts by 2s, 3s, 5s, 10s, and 25s (by any multiple) and 100s (from any number)
WKCE in 4th grade
•
determine a the eighth item in a pattern when given the first five in a series
•
determine a rule that describes a functional relationship or pattern using addition,
WKCE in 5th grade
subtraction multiplication rules
22
WKCE in 5th grade
Using the equal sign (=) as a relational symbol meaning “the same as”
By the end of
grade:
1
A child uses knowledge of facts, basic properties, and relational thinking as opposed to
computation to reason about T/F or open number sentences (equations) such as the
following: * Denotes a false number sentence
3+1 = 4
2
10 = 6+4
6+4 = 10+0
6+4 = 10+1*
6+4 = 5+5
6+4 = F +5
6+7 = 6+6 + F
3
15 = 8+7
15 = 15
8+7 = 15+1*
8+7 = 6+9
6+9 = F +5
WKCE in 3rd grade
4
WKCE in 4th grade
5
WKCE in 5th grade
31 = 18+13
31 = 31
18+13 = 31+1*
13+18 = 14+19
13+18 = h +12
13 = h+h+3
5 = 4+1
8+2 = 2+8
7=7
4+2 = F +2
30 = 3+0 *
98+6 - 6 = F
9+8 = 9+9 - F
22+37 = 20+30+20*
36+5 = 5 + F
192+32-32 = F
376+84 - F = 376
10+10+10 = 3×9 +3
6×4 = 2×4 +4 +8*
456+267−267 = j
342+38−37 = p
53+8−z = 54
36 = 3 + u + u + 3
23+48 = 24+47
23+48 = t+22
53+68 = 52+67+s
39+73 = 74+38 − b
358+130+2 = 400+80+9*
c+c+c+4 = 16
q+q+8 = 16
2×9 = w+w – 2
42 + b + b = 22 + b
11 = b+b
k+k-k = 18
m+m+m+12 = m+ 24
75-5+j = 70+j+j
42+b-b = 22+b
23
2×6 = F
3×6 = 2×6 + F
3×6 +3 = 4×6*
3×7 = 7+7+7
4×7 = 7+7+7+F
14+14 = 4×7
6×4 = 5×4 + F
8+F=3x6
7×9 = 7×10 - F
8×5 + F = 8×6
23×9 = (20×9) + (c×9)
c×5 = 40+5
15×9 = 9×15
4×10 – 4 = 4×8*
4+3×8 = 4×8-4
Making and discussing conjectures about basic number properties
By the end of
grade:
Given this series of T/F or open number sentences:
967 + 0 = 967
23 + 7 = 23*
40 + 0 = 400*
89 = c + 89
2
A child’s conjecture may be:
When you add zero to a number, you get the
number you started with. (a + 0 = a)
536 – 0 = 536
48 – 9 = 48*
570 – 0 = 57*
c – 0 = 654
3
A child’s conjecture may be:
When you subtract zero from a number, you get the
number you started with. (a – 0 = a)
5,467 × 1 = 5,467
48 × 1 = 49*
8.4 – 10 = 8.4*
c × 1 = 76
4
A child’s conjecture may be:
When you a multiply a number times 1, you get the
number you started with. (a × 1 = a)
345 × 0 = 0
28 × 0 = 28*
94 – c = 0
c×0=c
5
A child’s conjecture may be:
When you a multiply a number times 0, you get zero. (a × 0 = 0)
24
MMSD K-5 Mathematics Content Standard for
Geometry
Wisconsin Model Academic Standards (WMAS) Standard C: Geometry, 1998, p. 8
Students will be able to use geometric concepts, relationships and procedures to interpret,
represent, and solve problems.
Geometry and its study of shapes and relationships is an effort to understand the nature and beauty of the world. While
the need to understand our environment is still with us, the rapid advance of technology has created another need - to
understand ideas communicated visually through electronic media. For these reasons, educated people in the 21st
century need a well-developed sense of spatial order to visualize and model real world problem situations.
National Council of Teachers of Mathematics (NCTM) Principles and Standards for School
Mathematics (PSSM) 2000 Geometry Standard, pp. 96-97, 164-165
Instructional programs from kindergarten through grade 12 should enable all students to:
•
•
•
•
Analyze characteristics and properties of two- and three- dimensional geometric shapes and
develop mathematical arguments about geometric relationships.
Specify locations and describe spatial relationships using coordinate geometry and other
representational systems.
Apply transformations and use symmetry to analyze mathematical situations.
Use visualizations, spatial reasoning and geometric modeling to solve problems.
The geometric and spatial knowledge children bring to school should be expanded by explorations, investigations, and
discussions of shapes and structures in the classroom. Students should use their notions of geometric ideas to become
more proficient in describing, representing, and navigating their environment. They should learn to represent two- and
three-dimensional shapes through drawings, block constructions, dramatizations, and words. They should explore
shapes by decomposing them and creating new ones. Their knowledge of direction and position should be refined
through the use of spoken language to locate objects by giving and following multi-step directions.
.
Geometry offers students an aspect of mathematical thinking that is different from, but connected to, the world of
number. As students become familiar with shape, structure, location, and transformations and as they develop spatial
reasoning, they lay the foundation for understanding not only their spatial world but also other topics in mathematics
and in art, science, and social studies. Some students’ capabilities with geometric and spatial concepts exceed their
numerical skills. Building on these strengths fosters enthusiasm for mathematics and provides a context in which to
develop number and other mathematical concepts (Razel and Eylon 1991).
The reasoning skills that students develop in grades 3-5 allow them to investigate geometric problems of increasing
complexity and to study geometric properties. As they move from grade 3 to grade 5, they should develop clarity and
precision in describing the properties of geometric objects and then classifying them by these properties into categories
such as rectangle, triangle, pyramid, or prism. They can develop knowledge about how geometric shapes are related to
one another and begin to articulate geometric arguments about the properties of these shapes. They should also explore
motion, location, and orientation by, for example, creating paths on a coordinate grid or defining a series of flips and
turns to demonstrate that two shapes are congruent. As students investigate geometric properties and relationships,
their work can be closely connected with other mathematical topics, especially measurement and number.
The study of geometry in grades 3-5 requires thinking and doing. As students sort, build, draw, model, trace, measure,
and construct, their capacity to visualize geometric relationships will develop. At the same time they are learning to
reason and to make, test, and justify conjectures about these relationships. This exploration requires access to a variety
of tools, such as graph paper, rulers, pattern blocks, geoboards, and geometric solids, and is greatly enhanced by
electronic tools that support exploration, such as dynamic geometry software.
25
MMSD Math Content Standards for Geometry – Kindergarten
Achievement of the following grade-level standards supports achievement of Wisconsin Model Academic Standards.
By the end of kindergarten MMSD students will:
Supports Standard:
Investigate circles, polygons (square, rectangle, triangle), polyhedrons (cube,
rectangular prism) and other solids (sphere, cylinder, cone). Child:
• names basic figures and shapes
• builds with geometric shapes (Ex: Geoblocks, pattern blocks)
• talks about the results of putting together and taking apart shapes
• sorts shapes according to attributes
• talks about attributes
• matches geometric models to shapes in the environment
WMAS C.4.1
Draw:
•
•
•
an object in the environment (Ex: a face, a flower pot)
a two-dimensional figure (square, circle) from an example
front- or top- view of a three-dimensional object
Talk about locations and spatial relationships. Child:
• uses words such as far, near, over, under, next to, and between
WMAS C.4.1
WMAS C.4.3
MMSD Math Content Standards for Geometry – Grade 1
Achievement of the following grade-level standards supports achievement of Wisconsin Model Academic Standards.
By the end of first grade MMSD students will:
Supports Standard:
Investigate circles, polygons (hexagon, rhombus, trapezoid, square, rectangle, triangle),
polyhedrons (square pyramid, cube, triangular-, square- &rectangular prism) and other
solids (sphere, cylinder, cone). Child:
•
•
•
•
•
•
names basic figures and shapes
builds with geometric shapes or computer models (Ex: ‘Shapes’ software, square tiles,
Geoblocks, pattern blocks)
talks about the results of putting together and taking apart shapes
sorts shapes according to attributes
compares and talks about attributes
matches geometric models to shapes in the environment
WMAS C.4.1
Draw:
•
•
•
an object in the environment (Ex: a face, a flower pot)
a two-dimensional figure (triangle, square, circle)
front- or top- view of a three-dimensional object
WMAS C.4.1
Investigate the symmetry of two-dimensional shapes by:
•
•
folding paper to make a shape with mirror symmetry
using a mirror to find lines of symmetry
Talk about locations, spatial relationships and movement. Child:
•
•
solves simple problems involving shape, movement, and space
uses words such as slide, flip, turn (rotation), far, near, over, under, next to, and between
26
WMAS C.4.2
WMAS C.4.3
MMSD Math Content Standards for Geometry – Grade 2
Achievement of the following grade-level standards supports achievement of Wisconsin Model Academic Standards.
By the end of second grade MMSD students will:
Supports Standard:
Investigate circles, polygons (hexagon, rhombus, trapezoid, parallelogram, square, rectangle,
triangle), polyhedrons (square pyramid, cube, triangular-, square- & rectangular prism) and
other solids (sphere, cylinder, cone). Child:
•
•
•
•
•
•
names basic figures and shapes
builds with geometric shapes or computer models (Ex: Geoboards, pentominoes, square tiles,
Geoblocks, pattern blocks)
predicts the results of putting together and taking apart shapes
sorts and classifies shapes according to attributes (face, side, corner, edge)
compares and talks about attributes
matches geometric models to shapes in the environment
WMAS C.4.1
WKCE in 3rd grade
Draw:
•
•
•
the geometric shapes of objects in the environment
a two-dimensional figure (rectangle, triangle, square, circle)
front or top view of a three-dimensional object
WMAS C.4.1
Investigate the symmetry of two-dimensional shapes by:
•
•
•
determining which movements leave a plain (un-patterned) shape unchanged (sliding,
flipping, or rotating)
folding paper to make a shape with mirror symmetry
using a mirror to find lines of symmetry
Investigate the nets (flat patterns) of a rectangular prisms (Ex. cereal boxes, milk
cartons) by:
•
•
constructing and deconstructing shapes
drawing nets
WMAS C.4.2
WKCE in 3rd grade
WMAS C.4.1
Talk about locations, spatial relationships and movement. Child:
•
•
•
•
locates points on maps and simple coordinate grids with numbers and letters
represents points and simple figures on maps and grids with coordinates of letters and
numbers
solves problems involving shape, movement, and space
uses words such as, right, left, near, far, forward, back, near, far, slide, flip, turn (rotation),
over, under, next to, and between
27
WMAS C.4.3
WKCE in 3rd grade
MMSD Math Content Standards for Geometry – Grade 3
Achievement of the following grade-level standards supports achievement of Wisconsin Model Academic Standards.
By the end of third grade MMSD students will:
Supports Standard:
Investigate circles, polygons (octagon, hexagon, rhombus, trapezoid, parallelogram, square,
rectangle, triangle), polyhedrons (pyramids, cube, hexagonal-, octagonal-, triangular-,
square- & rectangular prism) and other solids (hemisphere, sphere, cylinder, cone). Child:
•
•
•
•
•
•
•
compares attributes of a classification
identifies properties of shapes
names shapes and uses geometric language (Ex. side, face, vertex, edge)
builds with geometric shapes or computer models (Ex: Geo-logo, tetrominoes, Geoboards,
pentominoes, square tiles, Geoblocks, pattern blocks)
predicts the results of putting together and taking apart shapes
sorts and classifies shapes according to attributes
matches geometric models to shapes in the environment
WMAS C.4.1
WKCE in 4th grade
Draw:
•
•
•
the geometric shapes of objects in the environment
a two-dimensional figure (rectangle, triangle, square, circle)
front or top view of a three-dimensional object
WMAS C.4.1
Investigate the symmetry of two-dimensional shapes by:
•
•
•
determining which movements leave plain (un-patterned) shapes unchanged (Ex. sliding,
rotating half-turn, quarter-turn, up-down flip, sideways flip)
folding paper to make a shape with mirror symmetry
using a mirror to identify all lines of symmetry for a given object
Determine multiple nets(flat patterns)of a cube and square pyramids by:
•
•
constructing and deconstructing shapes
drawing nets
WMAS C.4.2
WKCE in 4th grade
WMAS C.4.1
WKCE in 4th grade
Specify locations, spatial relationships, and movement. Child:
•
•
•
•
locates points on maps and simple coordinate grids with letters and numbers
represents points and simple figures on maps and simple coordinates grids with letters and
numbers
solves problems involving shape, movement, and space
uses words such as ½ turn, full turn, parallel, perpendicular, intersection, adjacent to,
interior of, forward, back, right, left, near, far, over, under, next to, and between
Use geometric models to solve problems in other areas of mathematics such as number
and measurement. (Ex. area model of multiplication or fractional parts, filling an open
box to determine volume)
28
WMAS C.4.3
WKCE in 4th grade
WMAS A.8.6
MMSD Math Content Standards for Geometry – Grade 4
Achievement of the following grade-level standards supports achievement of Wisconsin Model Academic Standards.
By the end of fourth grade MMSD students will:
Supports Standard:
Investigate circles, polygons (octagon, hexagon, rhombus, trapezoid, parallelogram, square,
rectangle, triangle), polyhedrons (square pyramid, cube, hexagonal-, octagonal-, triangular-,
square- & rectangular prism) and other solids (hemisphere, sphere, cylinder, cone). Child:
•
•
•
•
•
•
•
identifies and describes 3-dimensional shapes from multiple perspectives
determines the number of faces, edges, and vertices (corners) given an illustration of a 3dimensional figure
compares attributes of a classification
• parallel sides
• number of sides (two-dimensional shapes)
• number of faces (three-dimensional shapes)
identifies properties of shapes
names shapes and uses geometric language (Ex. lines, line segment, parallel, perpendicular,
right angle, side, face, vertex, edge)
builds with geometric shapes or computer models (Ex: Geoboards or dot paper, Geo-logo,
tetrominoes, drawings of pentominoes, square tiles, Geoblocks, pattern blocks)
predicts the results of subdividing putting together and taking apart shapes
WMAS C.4.1
WKCE in 5th grade
Draw:
•
•
•
the geometric shapes of objects in the environment
a two-dimensional figure (rectangle, triangle, square, circle)
front-, top-, or side-view of a three-dimensional object
WMAS C.4.1
Demonstrate an understanding of symmetry of two-dimensional shapes. Child:
•
•
determines which movements leave plain (un-patterned) shapes unchanged (Ex. slides, rotating
half-turn-180 degree turn, quarter-turn – 90 degrees, up-down flip, sideways flip)
identifies all lines of symmetry for a given shape
Determine multiple nets (flat patterns) of geometric solids (cubes, rectangular- and triangularprisms, and rectangular- and triangular-pyramids) by:
•
•
WMAS C.4.2
WKCE in 5th grade
WMAS C.4.1
WKCE in 5th grade
constructing and deconstructing shapes
drawing nets
Specify locations, describe spatial relationships, and movements. Child:
•
•
•
•
states the coordinates of points, objects and simple figures on maps or one-quadrant
coordinate grids
locates and plots points on maps and one-quadrant coordinate grids
solves problems involving shape, movement, and space
uses words such as congruent, similar, ½ turn, full turn, parallel, perpendicular, intersection,
adjacent to, interior of, forward, back, right, left, near, far, over, under, next to, and between
Use geometric models to solve problems in other areas of mathematics such as number and
measurement. (Ex. area model of multiplication or fractional parts, filling an open box to
determine volume)
29
WMAS C.4.3
WMAS C.4.4
WKCE in 5th grade
WMAS A.8.6
MMSD Math Content Standards for Geometry – Grade 5
Achievement of the following grade-level standards supports achievement of Wisconsin Model Academic Standards.
By the end of fifth grade MMSD students will:
Supports Standard:
Investigate understanding of irregular and regular two-dimensional polygons (3-, 4-, 5-, 6-, 8sides) and regular three-dimensional polyhedrons. Child:
•
•
•
•
•
•
•
•
classifies plane figures by characteristics of angles
identifies and describes 3-dimensional shapes from multiple perspectives
determines the number of faces, edges, and vertices (corners) given an illustration of a 3dimensional figure
compares attributes of a classification
• parallel sides
• number of sides (two-dimensional shapes)
• number of faces (three-dimensional shapes)
identifies properties of shapes
names shapes and uses geometric language (Ex. ray, line, line segment, parallel, perpendicular,
acute-, obtuse-, right-angle, side, face, vertex, edge)
builds with geometric shapes or computer models (Ex: Geoboards or dot paper, Geo-logo,
tetrominoes, pentominoes, square tiles, Geoblocks, pattern blocks)
predicts the results of subdividing putting together and taking apart shapes
WMAS C.8.1
WKCE in 6th grade
Draw:
•
•
•
the geometric shapes of objects in the environment
a two-dimensional figure (rectangle, triangle, square, circle)
front-, top-, or side-view of a three-dimensional object
WMAS C.8.1
Demonstrate an understanding of symmetry of two-dimensional shapes. Child:
•
•
•
•
design shapes that have at least one-line of symmetry
determines congruency and similarity of shapes
determines which movements leave patterned shapes unchanged (Ex. sliding, rotating half-turn180 degree turn, quarter-turn – 90 degrees, up-down flip, sideways flip)
identifies all lines of symmetry for a given shape
Determine multiple nets (flat patterns) of geometric solids (cubes, rectangular- and triangularprisms, and rectangular- and triangular-pyramids) by:
•
•
WMAS C.4.2
WKCE in 6th grade
WMAS C.4.1
WKCE in 6th grade
constructing and deconstructing shapes
drawing nets
Specify locations, describe spatial relationships, and movements. Child:
•
•
•
•
•
locates the fourth coordinate pair when given three vertices of a quadrilateral on a onequadrant coordinate grid
states the coordinates of points, objects and simple figures on maps or one-quadrant coordinate
grids
locates and plots points on maps and one-quadrant coordinate grids
solves problems involving shape, movement, and space
uses words such as congruent, similar, ½ turn, full turn, parallel, perpendicular, intersection,
between, adjacent to, interior of, forward, back, right, left, near, far, over, under, next to, and
between
Use geometric models to solve problems in other areas of mathematics such as number and
measurement. (Ex. area model of multiplication or fractional parts, filling an open box to
determine volume)
30
WMAS C.8.5
WKCE in 6th grade
WMAS A.8.6
Threads for:
Geometry
Investigating Shapes
By the end of
grade:
Students will:
K
Investigate circles, polygons (square, rectangle, triangle), polyhedrons (cube,
rectangular prism) and other solids (sphere, cylinder, cone). Child:
• names basic figures and shapes
• builds with geometric shapes (Ex: Geoblocks, pattern blocks)
• talks about the results of putting together and taking apart shapes
• sorts shapes according to attributes
• talks about attributes
• matches geometric models to shapes in the environment
1
Investigate all of the shapes mentioned in kindergarten including hexagons rhombus,
trapezoid and, square pyramid and square-& triangular- prisms. Child:
• compares attributes
2
Investigate all of the shapes mentioned in K-1 including parallelogram: Child:
• predict the results of putting together and taking apart shapes
3
WKCE in 3 rd grade
Investigate all of the shapes mentioned in K-2 including octagon, other pyramids,
hexagonal- & octagonal- prisms and hemispheres:
• compare attributes of a classification
• identify properties of shapes
• use geometric language (Ex. side, face, vertex, edge)
•
•
4
•
WKCE in 4th grade
•
5
•
•
WKCE in 5th grade
identify and describe 3-dimensional shapes from multiple perspectives
determine the number of faces, edges, and vertices (corners) given an illustration of a
3-dimensional figure
use geometric language (Ex. lines, line segment, parallel, perpendicular, right angle,
side, face, vertex, edge)
WKCE in 5th grade
compares attributes of a classification
•
parallel sides
•
number of sides (two-dimensional shapes)
•
number of faces (three-dimensional shapes)
classify plane figures by characteristics of angles
use geometric language (Ex. ray, line, line segment, parallel, perpendicular, acute-, obtuse& right-angle, side, face, vertex, edge)
Drawing objects and figures
By the end of
grade:
K-5
Students will:
Draw:
• an object in the environment (Ex. a face, a flower pot)
• a two-dimensional figure (Ex. square, circle, triangle)
• front-, top- or side-view (4th grade)of a three-dimensional object
31
Specifying locations, spatial relationships, and movements
By the end
of grade:
K
Students will:
Talk about locations and spatial relationships using words such as:
far, near, over, under, next to, and between
1
2
•
•
solves simple problems involving shape, movement, and space
uses words such as slide, flip, turn (rotation)
•
•
locates points on maps and simple coordinate grids with letters and numbers WKCE in 3 rd grade
represents points and simple figures on maps with and simple coordinate grids with letters
WKCE in 3 rd grade
and numbers
uses words such as right, left, near, far, forward, back
•
3
Specify locations and describe spatial relationships, and movements using words such as:
• uses words such as ½ turn, full turn, parallel, perpendicular, intersection, between, adjacent
to, interior of
•
•
4
locates and plots points on maps and one-quadrant coordinate grids
WKCE in 4th grade
states the coordinates of points, objects and simple figures on maps or one-quadrant
WKCE in 4t grade
coordinate grids
uses words such as congruent and similar
•
Specify locations and describe spatial relationships, and movements. Child:
•
5
locates the fourth coordinate pair when given three vertices of a quadrilateral on a oneWKCE in 5th grade
quadrant coordinate grid
Understanding symmetry
By the end
of grade:
1
2-3
Students will:
Investigate the symmetry of two-dimensional shapes by:
• folding paper to make a shape with mirror symmetry
• using a mirror to find lines of symmetry
•
determining which movements leave plain (un-patterned) shapes unchanged (Ex. sliding,
rotating half-turn, quarter-turn, up-down flip, sideways flip)
WKCE in 3rd, 4th, & 5th grade
Demonstrate an understanding identifying all lines of symmetry for a given object
4
5
determines which movements leave plain (un-patterned) shapes unchanged (Ex. slides, rotating half-turn180 degree turn, quarter-turn – 90 degrees, up-down flip, sideways flip)
•
•
design shapes that have at least one-line of symmetry
determines congruency and similarity of shapes
WKCE in 5th grade
WKCE in 5th grade
Working with nets
By the end
of grade:
Students will:
2
Investigate the nets of a rectangular prisms (Ex. cereal boxes, milk cartons) by:
• constructing and deconstructing shapes
• drawing nets
3
Determine multiple nets(flat patterns)of a cube and square pyramids.
WKCE in 4th grade
Determine multiple nets of geometric solids.
WKCE in 5th grade
4/5
32
MMSD K-5 Mathematics Content Standard for Measurement
Wisconsin's Model Academic Standards (WMAS) Standard D: Measurement, 1998, p. 10
Students in Wisconsin will select and use appropriate tools (including technology) and techniques
to measure things to a specified degree of accuracy. They will use measurements in problemsolving situations.
Measurement is the foundation upon which much technological, scientific, economic, and social inquiry rests. Before
things can be analyzed and subjected to scientific investigation or mathematical modeling, they must first be quantified
by appropriate measurement principles. Measurable attributes include such diverse concepts as voting preferences,
consumer price indices, speed and acceleration, length, monetary value, duration of an Olympic race, or probability of
contracting a fatal disease.
National Council of Teachers of Mathematics (NCTM) Principles and Standards for School
Mathematics (PSSM) 2000 Measurement Standard, pp. 102-103, 170-171
Instructional programs from kindergarten through grade 12 should enable all students to•
•
Understand measurable attributes of objects and the units, systems, and processes of
measurement.
Apply appropriate techniques, tools, and formulas to determine measurements.
Measurement is one of the most widely used applications of mathematics. It bridges two main areas of school
mathematics - geometry and number. Measurement activities can simultaneously teach important everyday skills,
strengthen students’ knowledge of other important topics in mathematics, and develop measurement concepts and
processes that will be formalized and expanded in later years.
Teaching that builds on students’ intuitive understandings and informal measurement experiences helps them
understand the attributes to be measured as well as what it means to measure. A foundation in measurement concepts
that enables students to use measurement systems, tools, and techniques should be established through direct
experiences with comparing objects, counting units, and making connections between spatial concepts and number.
Measurement is a process that students in grades 3-5 use every day as they explore questions related to their school or
home environment. For example, how much catsup is used in the school cafeteria each day? What is the distance from
my house to the school? What is the range of heights of players on the basketball team? Such questions require
students to use concepts and tools of measurement to collect data and to describe and quantify their world. In grades 35, measurement helps connect ideas within areas of mathematics and between mathematics and other disciplines. It can
serve as a context to help students understand important mathematical concepts such as fractions, geometric shapes,
and ways of describing data.
Prior to grade 3, students should have begun to develop an understanding of what it means to measure an object, that
is, identifying an attribute to be measured, choosing an appropriate unit, and comparing that unit to the object being
measured. They should have had many experiences with measuring length and should also have explored ways to
measure liquid volume, weight, and time. In grades 3-5, students should deepen and expand their understanding and
use of measurement. For example, they should measure other attributes such as area and angle. They need to begin
paying closer attention to the degree of accuracy when measuring and use a wider variety of measurement tools. They
should also begin to develop and use formulas for the measurement of certain attributes such as area.
In learning about measurement and learning how to measure, students should be actively involved, drawing on familiar
and accessible contexts. For example, students in grades 3-5 should measure objects and space in their classroom or
use maps to determine locations and distances around their community. They should determine an appropriate unit and
use it to measure the area of their classroom's floor, estimate the time it takes to do various tasks, and measure and
represent change in the size of attributes, such as their height.
33
MMSD Math Content Standards for Measurement – Kindergarten
Achievement of the following grade-level standards supports achievement of Wisconsin Model Academic Standards.
By the end of kindergarten MMSD students will:
Supports Standard:
WMAS D.4.1
Name, discuss, compare, and order objects according to attributes of capacity or length.
WMAS D.4.4
Investigate measuring length or capacity of object by:
•
measuring with multiple copies of units of the same-size (Ex: Unifix cubes laid end to end)
Time
Recognize temporal concepts such as before, after, sooner, or later.
WMAS D.4.4
MMSD Math Content Standards for Measurement– Grade 1
Achievement of the following grade-level standards supports achievement of Wisconsin Model Academic Standards.
By the end of first grade MMSD students will:
Supports Standard:
Name, discuss, compare, and order objects according to attributes of weight, capacity, or length.
WMAS D.4.1
Talk about measurement concepts including:
•
•
the necessity for identical units
conventions for communicating measurements by identifying the quantity and the name of
the unit (Ex. 12 cups)
Investigate measuring length, capacity and weight. Child:
•
WMAS D.4.2
WMAS D.4.4
measures with multiple copies of units of the same-size such (Ex: tiles or washers on a pan
balance)
Time
Associate the time of day with everyday events.
WMAS D.4.4
34
MMSD Math Content Standards for Measurement– Grade 2
Achievement of the following grade-level standards supports achievement of Wisconsin Model Academic Standards.
By the end of second grade MMSD students will:
Supports Standard:
Name, discuss, compare, and order objects according to attributes of, weight, capacity, area
length (perimeter), and temperature through observation or actual measurement.
WMAS D.4.1
WKCE in 3rd grade
Talk about measurement concepts including:
•
•
•
•
•
•
•
zero point (any point can act as the starting point of a measurement)
iteration (repeatedly laying one unit next to an object to measure its length or area)
subdividing units to increase the precision of a measurement
the relationship between the size of the unit and the number of units needed to make a
measurement
estimating measurements using non-standard units
the necessity for identical units
conventions for communicating measurements by identifying the quantity and the name of the
unit (Ex. 12 washers)
Measure length, area, capacity, mass, weight, and temperature. Child:
•
•
•
•
•
solves problems involving measurement
selects appropriate measurement tools and units (standard and non-standard)
reads a thermometer to the nearest 5 degrees F/C (WKCE in 3rd grade)
measures with accuracy to the nearest ½″, cm (WKCE in 3rd grade)
iterates (Ex: measuring the length of a room with a single meter stick)
WMAS D.4.2
WKCE in 3rd grade
WMAS D.4.1
WMAS D.4.3
WMAS D.4.4
Time
WMAS D.4.3
Tell and record time to the nearest minute using analog and digital clocks.
WKCE in 3rd grade
Identify increments of time:
•
•
•
•
seconds, minutes, days, months, years
minutes grouped by fives
benchmarks of 15, 30, 45
twelve numbers indicate 12 hours
WMAS D.4.3
WKCE in 3rd grade
35
MMSD Math Content Standards for Measurement– Grade 3
Achievement of the following grade-level standards supports achievement of Wisconsin Model Academic Standards.
By the end of third grade MMSD students will:
Supports Standard
Name, discuss, compare, and order objects according to attributes of, weight, capacity, area,
length (perimeter), and temperature through observation or actual measurement.
WMAS D.4.1
WKCE in 3rd grade
Demonstrate an understanding of measurement concepts including:
•
•
•
•
•
•
•
•
choosing an appropriate tool and unit (Ex. inches, centimeters, miles, feet, yards,
millimeters, cups, quarts, gallons, liters, pounds, ounces, grams, degrees F/C)
apply estimation techniques using non-standard measure
zero point (any point can act as the starting point of a measurement)
iteration (repeatedly laying one unit next to an object to measure its length)
subdividing units to increase the precision of a measurement
the relationship between the size of the unit and the number of units needed to make a
measurement
the necessity for identical units
conventions for communicating measurements by identifying the quantity and the name of the
unit (Ex. 12 strips of paper)
Measure length (perimeter), area, capacity, mass, weight, and temperature. Child:
•
•
•
•
•
solves problems involving measurement
selects appropriate measurement tools and units (standard and non-standard)
measures with accuracy to the nearest ½″, cm (WKCE in 3rd grade)
measures area by iteration (Ex: square tiles covering a surface) (WKCE in 3rd grade)
reads a thermometer to the nearest 5 degrees F/C (WKCE in 3rd grade)
Estimate measurements using:
•
WMAS D.4.2
WKCE in 3rd grade
WMAS D.4.1
WMAS D.4.3
WMAS D.4.4
WMAS D.4.4
non-standards units (Ex: estimation jars, paper clips, square tiles)
WKCE in 3rd grade
Time
Tell time to the nearest minute using analog and digital clocks. Child:
•
•
WMAS D.4.3
WKCE in 3rd grade
translates time between analog and digital clocks
records time
Identify increments of time:
•
•
•
•
seconds, minutes, days, months, years
minutes grouped by fives
benchmarks of 15, 30, 45
twelve numbers indicate 12 hours
WMAS D.4.3
WKCE in 3rd grade
36
MMSD Math Content Standards for Measurement– Grade 4
Achievement of the following grade-level standards supports achievement of Wisconsin Model Academic Standards.
By the end of fourth grade MMSD students will:
Supports Standard
Name, discuss, compare, and order objects according to attributes of, weight, volume and liquid
capacity, area (regular and irregular), length (perimeter), and temperature through observation
or actual measurement.
WMAS D.4.1
WKCE in 4th grade
Demonstrate an understanding of measurement concepts including:
•
•
•
•
•
•
•
•
converting measurement units (inches/feet/yards, cups/pints/quarts/gallons)
choosing an appropriate unit (Ex. inches, feet, yards, miles, millimeters, centimeters, meters,
quarts, cups, gallons, liters, grams, ounces, pounds, degrees F/C)
zero point (any point can act as the starting point of a measurement)
iteration (repeatedly laying one unit next to an object to measure length)
subdividing units to increase the precision of a measurement
knowing how changing the size of the unit changes the number of units needed to make a
measurement
the necessity to use identical units to make a measurement
conventions for communicating measurements by identifying the quantity and the name of the
unit (Ex. 12 square-inch tiles)
Measures, length (perimeter), area, capacity, mass, weight, and temperature. Child:
•
•
•
•
•
solves problems involving measurement
selects appropriate measurement tools and units (standard and non-standard)
measures length and perimeter to the nearest ¼″, cm (WKCE in 4th grade)
measures area and perimeter by iteration (Ex: square tiles covering a surface) (WKCE in 4th grade)
reads a thermometer to the nearest 5 degrees F/C (WKCE in 4th grade)
Estimate measurements using:
•
•
WMAS D.4.2
WKCE in 4th grade
WMAS D.4.1
WMAS D.4.2
WMAS D.4.3
WMAS D.4.4
WMAS D.4.4
WMAS D.4.5
proportional contexts (Ex. using map scales)
non-standards units (Ex: estimation jars)
WKCE in 4th grade
Time
Tell time to the nearest minute using analog and digital clocks Child:
•
•
•
compares elapsed time in problem solving situations (across two adjacent hours in quarterhour increments)
translates time between analog and digital clocks
records time
Convert units (minutes/hours/days/months/years).
WMAS D.4.3
WKCE in 4th grade
WMAS D.4.4
Identify increments of time:
•
•
•
•
seconds, minutes, days, months, years
minutes grouped by fives
benchmarks of 15, 30, 45
twelve numbers indicate 12 hours
WMAS D.4.3
WKCE in 4th grade
37
MMSD Math Content Standards for Measurement– Grade 5
Achievement of the following grade-level standards supports achievement of Wisconsin Model Academic Standards.
By the end of fifth grade MMSD students will:
Supports Standard
Name, discuss, compare, and order objects according to attributes of, weight, volume and liquid
capacity, area (regular and irregular), length (perimeter), and temperature through observation or
actual measurement.
WMAS D.8.1
Demonstrate an understanding of measurement concepts including:
•
•
•
•
•
•
•
•
•
•
•
additivity (the measurement of the whole is equal to the sum of the measure of the parts)
knowing that all measurements are approximations
knowing how differences in unit size affects precision
converting measurement units (millimeters/centimeters/meters, grams/kilograms,
inches/feet/yards, pints/ cups/quarts/gallons)
choosing an appropriate unit (Ex. inches, feet, yards, miles, millimeters, centimeters, meters,
quarts, cups, gallons, liters, grams, ounces, pounds, degrees F/C)
zero point (any point can act as the starting point of a measurement)
iteration (repeatedly laying one unit next to an object to measure length)
subdividing units to increase the precision of a measurement
knowing how changing the size of the unit changes the number of units needed to make a
measurement
the necessity to use identical units to make a measurement
conventions for communicating measurements by identifying the quantity and the name of the
unit (Ex. 12 square-inch tiles)
Measure angles, length (perimeter of regular and irregular shapes), area (rectangles and
irregular shapes on a grid), volume, capacity, mass, weight, and temperature. Child:
•
•
•
•
•
solves problems involving measurement
selects appropriate measurement tools and units (standard and non-standard)
measures length and perimeter to the nearest ⅛″, mm (WKCE in 5th grade
measures area and perimeter by iteration (Ex: square tiles covering a surface) (WKCE in 5th grade)
reads a thermometer to the nearest 1 degree F/C (WKCE in 5th grade)
Estimate measurements using:
•
•
•
•
common benchmarks (Ex. a paperclip has a mass of about one gram)
U.S. customary measurements
proportional contexts (Ex. using map scales)
non-standard units (Ex. estimation jars)
WMAS D.4.2
WKCE in 5th grade
WMAS D.4.1
WMAS D.4.3
WMAS D.4.4
WMAS D.4.4
WMAS D.4.5
WKCE in 5th grade
Time
Tell time to the nearest minute using analog and digital clocks Child:
•
•
•
WMAS D.4.3
WKCE in 5th grade
compares elapsed time in problem solving situations
translates time between analog and digital clocks
records time
Convert units (minutes/hours/days months/years).
WMAS D.4.4
38
Threads for:
Measurement
Attribute awareness
By the end
of grade:
Students will:
K
Name, discuss, compare, and order objects according to attributes (length and capacity).
1
Name, discuss, compare, and order objects according to attributes of weight, capacity, or length.
2-3
Name, discuss, compare, and order objects according to attributes of, weight, capacity, area length
WKCE in 3rd & 4th
(perimeter), and temperature through observation or actual measurement.
grade
Name, discuss, compare, and order objects according to attributes of, weight, volume and liquid capacity,
area (regular and irregular), length (perimeter), and temperature through observation or actual
WKCE in 5th grade
measurement.
4
Measurement concepts
By the end
of grade:
Students will:
Talk about measurement concepts including:
• the necessity for identical units
• conventions for communicating measurements by identifying the quantity and the name of
the unit (Ex. 12 cups)
1
•
•
•
•
2
•
3-4
WKCE in 3rd & 4th
grade
5
th
WKCE in 5 grade
zero point (any point can act as the starting point of a measurement)
iteration (repeatedly laying one unit next to an object to measure its length or area)
subdividing units to increase the precision of a measurement
the relationship between the size of the unit and the number of units needed to make a
measurement
estimating measurements using non-standard units
Demonstrate an understanding of measurement concepts including:
•
choosing an appropriate tools and units (Ex. inches, centimeters, miles, feet yards,
millimeters, cups, quarts, gallons, liters, pounds, ounces, grams, degrees F/C)
•
apply estimation techniques using non-standard measure
•
•
•
additivity (the measurement of the whole is equal to the sum of the measure of the parts)
knowing that all measurements are approximations
knowing how differences in unit size affects precision
Unit conversions
By the end
of grade:
Students will:
4
•
convert measurement units (inches/feet/yards, quarts/gallons)
WKCE in 4th grade
5
•
convert measurement units (millimeters/centimeters/meters, grams/kilograms)
WKCE in 5th grade
39
Measuring
By the end
of grade:
K-1
Students will:
Investigate measuring length or capacity (weight in 1st grade) of object by:
• measuring with multiple copies of units of the same-size (Ex: Unifix cubes laid end to end,
tiles or washers on a pan balance)
2-5
WKCE in 3rd, 4th, &
5th grade
5
Measure length (perimeter in 3rd grade), area, capacity, mass, weight, and temperature.
• solve problems involving measurement
• select appropriate measurement tools and units (standard and non-standard)
• read a thermometer to the nearest 5 degrees F/C (nearest 1 degree in 5th grade)
• measure with accuracy to the nearest ½″ (¼″ in 4th grade) cm (mm in 5th grade)
• iterate (Ex: measuring the length of a room with a single meter stick)
Also measure angles, perimeter of regular and irregular shapes, area rectangles and
WKCE in 5th grade
irregular shapes on a grid and volume
Estimation
By the end
of grade:
3
Students will:
Estimate measurements using:
• non-standards units (Ex: estimation jars, paper clips, square tiles)
WKCE in 3rd grade
4
•
proportional contexts (Ex. using map scales)
5
•
•
common benchmarks (Ex. a paperclip has a mass of about one gram)
U.S. customary measurements
WKCE in 4th grade
WKCE in 3rd grade
WKCE in 3rd grade
Time
By the end
of grade:
Students will:
K
Recognize temporal concepts such as before, after, sooner, or later.
1
Associate the time of day with everyday events.
Tell and record time to the nearest minute using analog and digital clocks.
2
3-4
4-5
WKCE in 3rd grade
Identify increments of minutes and hours represented on an analog clock:
• seconds, minutes, days, months/years
• minutes grouped by fives
• benchmarks of 15, 30, 45
• twelve numbers indicate 12 hours
•
translates time between analog and digital clocks
•
compares elapsed time in problem solving situations (across two adjacent hours in quarterhour increments)
WKCE in 3rd grade
40
WKCE in 3rd&4th grade
MMSD K-5 Mathematics Content Standards for
Data Analysis (Statistics) & Probability
Wisconsin Model Academic Standards (WMAS) Standard E: Data Analysis and Probability, 1998 (p. 12)
Students will use data collection and analysis, statistics and probability in problem-solving
situations, employing technology where appropriate.
Dramatic advances in technology have launched the world into the Information Age, when data are used to describe past events or
predict future events. Whether in the business place or in the home, as producers or consumers of information, citizens need to be
well versed in the concepts and procedures of data analysis in order to make informed decisions.
National Council of Teachers of Mathematics (NCTM) Principles and Standards for School
Mathematics (PSSM) 2000 Data Analysis and Probability Standard (pp. 108-109, 176-177)
Instructional programs from pre-kindergarten through grade 12 should enable all students to:
•
•
•
•
Formulate questions that can be addressed with data and collect, organize, and display relevant
data to answer them.
Select and use appropriate statistical methods to analyze data.
Develop and evaluate inferences and predictions that are based on data.
Understand and apply basic concepts of probability.
Informal comparing, classifying and counting activities can provide the mathematical beginnings for developing young
learners’ understanding of data, analysis of data, and statistics. The types of activities needed and appropriate for
kindergartners vary greatly from those for second graders; however, throughout the pre-K-2 years, students should
pose questions to investigate, organize the responses, and create representations of their data. Through data
investigations, teachers should encourage students to think clearly and to check new ideas against what they already
know in order to develop concepts for making informed decisions.
Ideas about probability at this level should be informal and focus on judgements that children make because of their
experiences. Activities that underlie experimental probability, such as tossing number cubes or dice, should occur at
this level (K-2), but the primary purpose for these activities is focused on other strands, such as number.
In grades 3-5, students should move toward seeing a set of data as a whole, describing its shape, and using statistical
characteristics of the data such as range and measures of center to compare data sets. Much of this work emphasizes
the comparison of related data sets. As students learn to describe the similarities and differences between data sets,
they will have an opportunity to develop clear descriptions of the data and to formulate conclusions and arguments
based on the data. They should consider how the data sets they collect are samples from larger populations and should
learn how to use language and symbols to describe simple situations involving probability.
Investigations involving data should happen frequently during grades 3-5. These can range from quick class surveys to
projects that take several days. Frequent work with brief surveys (How many brothers and sisters do people in our class
have? What’s the farthest you have ever been from home?) can acquaint students with particular aspects of collecting,
representing, summarizing, comparing, and interpreting data. More extended projects can engage students in a cycle of
data analysis—formulating questions, collecting and representing the data, and considering whether their data are
giving them the information they need to answer their question. Students in these grades are also becoming more
aware of the work beyond themselves and are ready to address some questions that have the potential to influence
decisions. For example, one class that studied playground injuries at their school gathered evidence that led to the
conclusion that the bars on one piece of playground equipment were too large for the hands of most students below
third grade. This finding resulted in a new policy for playground safety.
41
MMSD Math Content Standards for Data Analysis (Statistics) & Probability - Kindergarten
Participation in the following activities will support achievement of the Wisconsin Model Academic Standards. Children at this
grade-level should have many experiences with these activities, but will not be formally assessed at this grade-level.
By the end of kindergarten MMSD students will:
Supports Standard:
Pose questions that lead to data collection and analysis.
WMAS E.4.1
Collect and organize data to address questions.
WMAS E.4.1
Represent data using physical objects, drawings, or graphs.
WMAS E.4.1
Talk about possible conclusions based on data.
WMAS E.4.1
MMSD Math Content Standards for Data Analysis (Statistics) & Probability – Grade 1
Participation in the following activities will support achievement of the Wisconsin Model Academic Standards.
By the end of first grade MMSD students will:
Supports Standard:
Pose questions that lead to data collection and analysis.
WMAS E.4.1
Collect and organize data to address the questions.
WMAS E.4.1
Represent data using physical objects, drawings, or graphs.
WMAS E.4.1
Talk about possible conclusions based on data.
WMAS E.4.1
WMAS E.4.3
Use data presented in simple graphs (picture and bar graphs) to answer questions.
MMSD Math Content Standards for Data Analysis (Statistics) & Probability – Grade 2
Participation in the following activities will support achievement of the Wisconsin Model Academic Standards.
By the end of second grade MMSD students will:
Supports Standard:
Pose questions that lead to data collection and analysis.
WMAS E.4.1
Collect and organize data to address questions
• decide what data to collect, when and how to collect it
WMAS E.4.1
Represent data using concrete objects, drawings, or simple bar graphs
WMAS E.4.1
WMAS E.4.1
Discuss possible conclusions based on data.
WKCE in 3rd grade
Use data presented in Venn diagrams, tables, charts, and graphs (picture or bar) to answer
questions.
WMAS E.4.3 WKCE
in 3rd grade
Probability
Describe familiar events as impossible or certain (more, less, or equally likely) to occur.
•
WMAS E.4.4
WKCE in 3rd grade
choose a fair or unfair spinner
42
MMSD Math Content Standards for Data Analysis (Statistics) & Probability – Grade 3
Participation in the following activities will support achievement of the Wisconsin Model Academic Standards.
By the end of third grade MMSD students will:
Design investigations to address questions that will lead to data collection and analysis.
•
•
Supports Standard:
determine what data to collect, when and how to collect it
predict possible results and their implications
Collect and organize data from:
•
•
•
WMAS E.4.1
observations
surveys
experiments
Create appropriate representations of data such as:
•
•
WMAS E.4.1
WKCE in 4th grade
WMAS E.4.1
tables and charts
bar graphs
Describe the important features of a set of data including:
•
•
•
•
shape
high and low values (minimum and maximum)
difference between the high and low values (range)
most frequent value (mode)
WMAS E.4.2
Discuss possible conclusions and implications based on the data.
WKCE in 4th grade
Use data presented in Venn diagrams, tables, charts, and graphs (picture and bar) to answer
questions.
WKCE in 4th grade
WMAS E.4.3
WMAS E.4.3
Probability
Describe familiar events as impossible or certain (more, less, or equally likely) to occur.
•
•
describe the likely outcome of a simple event (Ex. one toss of a coin, one role of a number
cube)
design fair and unfair spinners
43
WMAS E.4.4
WKCE in 4th grade
MMSD Math Content Standards for Data Analysis (Statistics) & Probability – Grade 4
Participation in the following activities will support achievement of the Wisconsin Model Academic Standards.
By the end of fourth grade MMSD students will:
Design investigations to address questions that will lead to data collection and analysis:
•
•
WMAS E.4.1
determine what data to collect and how to collect it
predict possible results and their implications
Collect and organize data from:
•
•
•
WMAS E.4.1
observations
surveys
experiments
Create appropriate representations of data such as:
•
•
Supports Standard:
WMAS E.4.1
tables and charts
bar graphs
Determine the important features of a set of data (7 items or fewer) including:
•
•
•
•
•
middle value (median)
shape
high and low values (minimum and maximum)
difference between the high and low values (range)
most frequent value (mode)
WMAS E.4.2
WKCE in 5th grade
Discuss possible conclusions and implications based on data.
Use data presented data presented in Venn diagrams, tables, charts, graphs (picture and bar),
line plots, and to answer questions
WMAS E.4.1
WKCE in 5th grade
WMAS E.4.3
WKCE in 5th grade
Probability
Describe familiar events as impossible or certain (more, less, or equally likely) to occur.
•
•
•
•
test predictions using data from a variety of sources
use words to express probability
describe the likely outcome of a simple event (Ex. one toss of a coin, one role of a number
cube)
design fair and unfair spinners
44
WMAS E.4.4
WKCE in 5th grade
MMSD Math Content Standards for Data Analysis & Probability – Grade 5
Participation in the following activities will support achievement of the Wisconsin Model Academic Standards.
By the end of fifth grade MMSD students will:
Supports Standard:
Design investigations to address questions that will lead to data collection and analysis:
•
•
WMAS E.4.1
determine what data to collect and how to collect it
predict possible results and their implications
Collect and organize data from:
•
•
•
WMAS E.4.1
observations
surveys
experiments
Create appropriate representations of data such as:
•
•
•
WMAS E.4.1
line plots
tables and charts
bar graphs
Determine the important features of a set of data (10 or fewer items) including:
•
•
•
•
•
•
average value (mean)
middle value (median)
shape
high and low values (minimum and maximum)
difference between the high and low values (range)
most frequent value (mode)
WMAS E.4.2
WKCE in 6th grade
WMAS E.4.1
Discuss possible conclusions and implications based on data.
WKCE in 6th grade
Use data presented in Venn diagrams, tables, charts, graphs (picture and bar) and line plots to
answer questions.
WMAS E.4.3
WKCE in 6th grade
WMAS E.4.4
Predict outcomes or trends from graphs and tables.
WKCE in 6th grade
Probability
Determine and describe the possible combinations of three items
WMAS E.4.4
WKCE in 6th grade
Describe familiar events as impossible or certain (more, less, or equally likely) to occur.
•
•
•
•
test predictions using data from a variety of sources
use words, percents, and fractions to express probability
describe the likely outcome of a simple event (Ex. one toss of a coin, one role of a number
cube)
design fair and unfair spinners
45
WMAS E.4.4
WKCE in 6th grade
Threads for:
Data Analysis and Probability
Data collection K-2
By the
end of
grade:
Students will:
K-2
Pose questions that lead to data collection and analysis.
K-2
Collect and organize data to address questions.
• decide what data to collect, when and how to collect it (in 2nd
K-2
Represent data using physical objects, drawings, or graphs
grade)
Data collection 3-5
By the
end of
grade:
Students will:
3-5
Design investigations to address questions that will lead to data collection and analysis.
rd
th
th
• determine what data to collect, when and how to collect it
WKCE in 3 , 4 , & 5 grade
• predict possible results and their implications
WKCE in 3 , 4 , & 5 grade
3-5
Collect and organize data from:
• observations
• surveys
• experiments
WKCE in 3 , 4 , & 5 grade
3-5
Create appropriate representations of the data such as:
• bar graphs
• tables and charts
• line plots (grade 5)
47
rd
th
th
rd
th
th
th
WKCE in 5 grade
Describing data
By the end
of grade:
1
2-5
3
Students will:
Reads data presented in simple graphs (picture and bar graphs).
Use data presented in Venn diagrams, tables, charts, and graphs (picture and bar) to
rd
th
th
answer questions, line plots (in 5th grade)
WKCE in 3 , 4 , & 5 grade
Describe the important features of a set of data including:
• shape
• high and low values (minimum and maximum)
• difference between the high and low values (range)
• most frequent value (mode)
4
Describe the important features of a set of data including:
• middle value (median)
5
Describe the important features of a set of data including:
• average value (mean)
th
WKCE in 5 grade
th
WKCE in 5 grade
th
WKCE in 5 grade
Drawing conclusions based on data
By the end
of grade:
2-4
5
1-5
Students will:
Discuss possible conclusions and implications based on data.
Predict outcomes or trends from graphs and tables.
rd
th
th
WKCE in 3 , 4 , & 5 grade
th
WKCE in 5 grade
Use data presented in Venn diagrams, tables, charts, and graphs (picture and bar in 1st
rd
th
th
WKCE in 3 , 4 , & 5 grade
grade) to answer questions.
Probability
By the end
of grade:
2
Students will:
Describe familiar events as impossible or certain (more, less, or equally likely) to occur.
rd
• choose a fair or unfair spinner
WKCE in 3 grade
•
3-4
5
•
describe the likely outcome of a simple event (Ex. one toss of a coin, one role of a number
th
cube)
WKCE in 4 grade
th
design fair and unfair spinners
WKCE in 4 grade
•
•
test predictions using data from a variety of sources
use words, percents, and fractions to express probability
48
th
WKCE in 5 grade
th
WKCE in 5 grade
Appendix A: CGI Story Problem Framework
(JRU) JOIN RESULT UNKNOWN
JOIN
Connie had 35 marbles. Juan gave her 18
more marbles. How many marbles does
Connie have all together?
(JCU) JOIN CHANGE UNKNOWN
35 +
(SRU) SEPARATE RESULT UNKNOWN
Connie had some marbles. Juan gave her 18
more marbles. Now she has 53 marbles. How
many marbles did Connie have to start with?
= 53
+ 18 = 53
(SCU) SEPARATE CHANGE UNKNOWN
(SSU) SEPARATE START UNKNOWN
Connie had 53 marbles. She gave 35 to Juan. Connie had 53 marbles. She gave some to Juan. Connie had some marbles. She gave 35 to
How many marbles does she have now?
Now she has 18 left. How many did she give to Juan. Now she has 18 marbles left. How many
Juan?
marbles did she have to start with?
53
=
18
53 – 35 =
- 35 = 18
PART-PARTWHOLE
(PPW-WU) PART-PART-WHOLE (WHOLE UNKNOWN)
(PPW-PU) PART-PART-WHOLE (PART-UNKNOWN)
Connie has 35 red marbles and 18 blue marbles. How many marbles
does she have?
Connie has 53 marbles. 35 are red and the rest are blue. How many
blue marbles does Connie have?
35 + 18 =
(CDU) COMPARE DIFFERENCE UNKNOWN
COMPARE
(JSU) JOIN START UNKNOWN
Connie has 35 marbles. How many more
marbles does she need to have to have 53 all
together?
35 + 18 =
SEPARATE
(Choose contexts and number sizes appropriate for your students)
53 – 35 =
(CQU) COMPARE QUANTITY UNKNOWN
Connie has 53 marbles. Juan has 35 marbles. Juan has 35 marbles. Connie has 18 more than
How many more marbles does Connie have Juan. How many marbles does Connie have?
than Juan?
53 – 35 =
35 +
= 53
(M) MULTIPLICATION
Connie has 3 bags of cookies. There are 15
cookies in each bag. How many cookies
does Connie have all together?
3 × 15 =
35 + 18 =
(MD) MEASUREMENT DIVISION
Connie has 45 cookies. She wants to put 15
cookies in each bag. How many bags can she
fill?
45/15 =
35 +
= 53
(CRU) COMPARE REFERENT UNKNOWN
Connie has 53 marbles. She has 35 more
marbles than Juan. How many marbles does
Juan have?
53 – 35 =
35 +
= 53
(PD) PARTITIVE DIVISION
Connie has 45 cookies. She wants to put the
cookies into 3 bags with the same number in
each bag. How many cookies are in each bag?
45/3 =
Adapted with permission from: Carpenter, T.P., Fennema E., Franke, M.L., Levi, L., Empson, S.B. 1999. Children’s Mathematical Thinking. Cognitively Guided Instruction. Portsmouth, NH: Heinemann.
Appendix B: Standards as they appear on the MMSD Elementary Report Card
Standard
Number,
Operations, &
Algebraic
Relationships
Kindergarten
Grade 1
Grade 2
•
•
•
•
•
•
•
•
•
•
•
•
•
•
Counts a group of up to 30 objects accurately
Reads and orders numbers up to 20
Writes numbers up to 20
Solves simple story problems
Explains solutions to simple story problems
Extends and creates patterns
•
•
Counts a group of objects by 2s, 5s, and 10s
Reads, writes, and orders numbers up to 100
Solves story problems
Explains mathematical thinking using words, objects,
drawings and symbols
Knows addition facts for sums up to 10
Identifies coins and their values
•
•
Reads, writes, and orders numbers up to 1,000
Solves story problems
Solves addition and subtraction problems
Explains mathematical thinking using words, objects,
drawings and symbols
Knows all addition facts
Combines coins for values up to $1
Builds, sorts, and describes shapes and uses
words such as over, under, near, far, and between
to describe position.
Builds, sorts, describes and names (basic) figures and
shapes; solves simple problems involving shape,
movement, and space.
Measurement
Describes and compares the size of objects and
measures objects with units such as paperclips,
square tiles, or cubes.
Measures length, capacity, and weight using nonstandard units.
Data Analysis &
Probability
This strand is not formally assessed and does not
appear on the Kindergarten report card.
Reads simple bar graphs to answer questions.
Reads and solves problems about simple bar graphs.
Standard
Grade 3
Grade 4
Grade 5
•
•
•
•
•
Geometry
Number,
Operation, &
Algebraic
Relationships
•
•
•
•
•
Geometry
Measurement
Data Analysis &
Probability
Reads, writes, and orders numbers up to
10,000
Solves story problems
Solves addition, subtraction, and simple
multiplication problems
Explains mathematical thinking using words,
objects, drawings and symbols
Knows subtraction facts and multiplication facts
(for ×2s, ×3s, ×4s, ×5s)
Solves problems involving money
•
•
•
•
Solves story problems
Solves addition, subtraction, multiplication, and
simple division problems
Explains mathematical thinking using words, objects,
drawings and symbols.
Knows all multiplication facts
Solves problems involving fractions
Solves problems involving money
Describes, compares, and names shapes; and solves
problems involving shape, movement, and space.
•
•
•
•
•
•
•
Selects and uses appropriate measurement units and
tools; and solves problems involving measurement.
Tells and records time to the nearest minute (analog
and digital).
Solves story problems
Solves addition, subtraction, multiplication and
division problems
Explains mathematical thinking using words,
drawings, and symbols
Knows division facts and multiples of 2-10 and 25
Reads, writes, and orders decimals and fractions
Solves problems involving fractions
Solves problems involving money
Describes, compares, and names shapes; and
solves problems involving shape, movement, and
space
Describes and compares attributes of shapes and
figures; solves problems involving shape, movement,
space, and location on maps and grids.
Solves problems involving shape, movement, and space;
and plots and describes locations on maps and grids.
•
•
•
•
Selects and uses appropriate measurement
units and tools; and solves problems involving
measurement (including time).
Translates time between analog and digital
clocks.
Makes bar graphs and interprets information in
tables, charts, and graphs
•
•
•
Measures attributes using standard units of
measurement and carries out simple conversions
within a measurement system.
Solves elapsed time in problem solving situations
(across adjacent hours in quarter-hour increments).
Collects, organizes, represents and interprets
information in tables, charts, and graphs;
Describes the important features of a set of data.
See grade-level standards for details.
•
•
•
Selects and uses appropriate measurement units and
tools; and solves problems involving measurement.
Solves elapsed time in number and problem solving
situations (for any time increment).
Collects, organizes, represents and interprets
information in tables, charts, and graphs;
Describes the important features of a set of data. See
grade-level standards for details.
Special thanks to:
Members of the 2001 MMSD Mathematics Standards Update Committee who brought expertise,
devotion, and patience to create and revise and revise and revise the content for our first draft, provide
feedback on subsequent revisions, and write report card descriptors to match the standards:
Nancy Baumgardner
Mary Brand
Jude Bremer
Pam Buskus
Karen Falkner
Janice Gratch *
Marsha Gregg
Margaret Jensen *
Ginny Koberstein
Carol Murphy
Betty Overland
* MMSD math resource teacher
Lucia Rowley
Marilyn Smith
Kathy Statz *
Carrie Turner
Barb Wiesner
Daithi Wolfe
Mary Jo Yttri
Laura Huber* (2004 Revision)
Members of the 1994 MMSD Mathematics Teaching and Learning Standards Writing Committee who
provided the first math curriculum standards and benchmark indicators from which the 2001 math
standards are derived:
Karen Falkner
Janice Gratch
Mazie Jenkins
Glenn Johnson
Mary Kay Johnson
Annie Keith
Barbara Marten
Pat Reisdorf
Mary Riley
Shirley Steinbach
Mark Wagler
Fran Wong
Professionals engaged in the study of Elementary Mathematics Education who assisted with getting
started, clarifying, prioritizing, and organizing the content of the Standards to reflect current
understanding of how children understand mathematical ideas:
Dr. Thomas Carpenter, professor, University of Wisconsin, Madison
Dr. Donald Chambers, Mathematics Consultant (Retired)
Dr. Susan Empson, University of Texas, Austin
Dr. Vicki Jacobs, professor, San Diego State University, CA
Julie Koehler, PhD Candidate, C&I, University of Wisconsin, Madison
Dr. Linda Levi, associate researcher, University of Wisconsin, Madison
Dr. Richard Lehrer, professor, University of Wisconsin, Madison
Dr. Nancy Mack, Grand Valley State University, Allendale
James Moser, DPI Mathematics Consultant (Retired), author Suggested School District
Mathematics Standards Levels to be Compatible with State of Wisconsin Model
Academic Standards.
Sincerely,
Carrie Valentine
MMSD math resource teacher
2001 Update Committee Chair
51