Joliet Public Schools District 86
Mathematics Curriculum
Aligned with
The New Illinois State Standards Incorporating the Common Core
Grades 3-5
Charles E. Coleman, Ed.D.
Superintendent
June 2013
Joliet Public Schools District 86
Mathematics Curriculum
Aligned with
The New Illinois State Standards Incorporating the Common Core
Grades 3-5
Charles E. Coleman, Ed.D.
Superintendent
June 2013
Joliet Public Schools District 86
Mathematics Mission Statement
The mission of the Joliet Public Schools District 86 Mathematics Curriculum is to
develop a program of study that supports focused instruction of the concepts and
key skills at each grade level. This curriculum provides structured opportunities
for students to become mathematical thinkers in a global society.
Acknowledgements
Thank you to the members of the Grades 3-5 Mathematics Common Core Curriculum
Team for their collaborative work in creating this guide.
Karen Blaha, Jefferson Elementary School
Liz Cox, Sanchez Elementary School
Jeannie Dannenberg, Pershing Elementary School
Laura DiMartino, Marshall Elementary School
Megan Dowd, Eisenhower Academy
Annette Herrera, Cunningham Elementary School
Shawna McHugh, Thigpen Elementary School
Sheila Huckins, Taft Elementary School
Erica Stabrawa, Woodland Elementary School
Loretta Stuart, Culbertson Elementary School
Sean Joyce, Technology Department, John F. Kennedy Administrative Center
Jan Taylor, Curriculum & Instruction, John F. Kennedy Administrative Center
Table of Contents
THE STANDARDS FOR MATHEMATICAL PRACTICE ....................................................................................................................................... 6
MATHEMATICAL PRACTICE STANDARDS ......................................................................................................................................................... 7
HOW TO READ THE GRADE LEVEL STANDARDS ........................................................................................................................................... 8
Grade 3 Standards ...................................................................................................................................................................................................... 9
Grade 4 Standards........................................................................................................................................................................................................ 51
Grade 5 Standards...................................................................................................................................................................................................... 119
Common Core State Documents Referenced .................................................................................................................................................... 213
THE STANDARDS FOR MATHEMATICAL PRACTICE
The Standards for Mathematical Practice describe varieties of expertise that
mathematics educators at all levels should seek to develop in their students.
These practices rest on important ―pr
ocesses and proficiencies‖ with longstanding
importance in mathematics education. The first of these are the NCTM process
standards of problem solving, reasoning and proof, communication, representation,
and connections. The second are the strands of mathematical proficiency specified
in the National Research Council’s report Adding It Up: adaptive reasoning, strategic
competence, conceptual understanding (comprehension of mathematical concepts,
operations and relations), procedural fluency (skill in carrying out procedures flexibly,
accurately, efficiently and appropriately), and productive disposition (habitual inclination to
see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and
one’s own efficacy).
The Common Core State Standards, Standards for Mathematical Practice.
6
MATHEMATICAL PRACTICE STANDARDS
1. Make sense of problems and persevere in solving them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the reasoning of others.
4. Model with mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.
7. Look for and make use of structure.
8. Look for and express regularity and reasoning.
The Common Core State Standards, Standards for Mathematical Practice.
7
CONTENT STANDARDS
HOW TO READ THE GRADE LEVEL STANDARDS
Standards define what students should understand and be able to do.
Clusters summarize groups of related standards. Note that standards from different
clusters may sometimes be closely related, because mathematics is a connected subject.
Domains are larger groups of related standards. Standards from different domains may
sometimes be closely related.
8
Grade 3 Standards
Mathematics | Grade 3 – Critical Areas
In Grade 3, instructional time should focus on four critical areas: (1) developing understanding of multiplication
and division and strategies for multiplication and division within 100; (2) developing understanding of fractions,
especially unit fractions (fractions with numerator 1); (3) developing understanding of the structure of
rectangular arrays and of area; and (4) describing and analyzing two-dimensional shapes.
(1) Students develop an understanding of the meanings of multiplication and division of whole numbers through activities and
problems involving equal-sized groups, arrays, and area models; multiplication is finding an unknown product, and division is finding
an unknown factor in these situations. For equal-sized group situations, division can require finding the unknown number of groups or
the unknown group size. Students use properties of operations to calculate products of whole numbers, using increasingly
sophisticated strategies based on these properties to solve multiplication and division problems involving single-digit factors. By
comparing a variety of solution strategies, students learn the relationship between multiplication and division.
(2) Students develop an understanding of fractions, beginning with unit fractions. Students view fractions in general as being built out
of unit fractions, and they use fractions along with visual fraction models to represent parts of a whole. Students understand that the
size of a fractional part is relative to the size of the whole. For example, 1/2 of the paint in a small bucket could be less paint than 1/3
of the paint in a larger bucket, but 1/3 of a ribbon is longer than 1/5 of the same ribbon because when the ribbon is divided into 3
equal parts, the parts are longer than when the ribbon is divided into 5 equal parts. Students are able to use fractions to represent
numbers equal to, less than, and greater than one. They solve problems that involve comparing fractions by using visual fraction
models and strategies based on noticing equal numerators or denominators.
(3) Students recognize area as an attribute of two-dimensional regions. They measure the area of a shape by finding the total
number of same size units of area required to cover the shape without gaps or overlaps, a square with sides of unit length being the
standard unit for measuring area. Students understand that rectangular arrays can be decomposed into identical rows or into
identical columns. By decomposing rectangles into rectangular arrays of squares, students connect area to multiplication, and justify
using multiplication to determine the area of a rectangle.
(4) Students describe, analyze, and compare properties of two dimensional shapes. They compare and classify shapes by their sides
and angles, and connect these with definitions of shapes. Students also relate their fraction work to geometry by expressing the area
of part of a shape as a unit fraction of the whole.
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The Common Core State Standards
Grade 3 Overview
Operations and Algebraic Thinking
• Represent and solve problems involving multiplication and division.
• Understand properties of multiplication and the relationship between multiplication and division.
• Multiply and divide within 100.
• Solve problems involving the four operations, and identify and explain patterns in arithmetic.
Number and Operations in Base Ten
• Use place value understanding and properties of operations to perform multi-digit arithmetic.
Number and Operations—Fractions
• Develop understanding of fractions as numbers
Measurement and Data
• Solve problems involving measurement and estimation of intervals of time, liquid volumes, and masses of objects.
• Represent and interpret data.
• Geometric measurement: understand concepts of area and relate area to multiplication and to addition.
• Geometric measurement: recognize perimeter as an attribute of plane figures and distinguish between linear and
area measures.
Geometry
• Reason with shapes and their attributes.
The Common Core State Standards
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Mathematical Content Standards
Grade 3
Domain: Operations and Algebraic Thinking
Enduring Understanding(s):
 Mathematical operations are used in solving problems in which a new value is produced from one more values.
Algebraic thinking involves choosing, combining, and applying effective strategies for answering quantitative questions
Essential Questions:
 In what ways can operations affect numbers?

How can different strategies be helpful when solving a problem?
Major Cluster: Represent and solve problems involving multiplication and division.
Common Core Standard: 3.OA.1
Interpret products of whole numbers, e.g., interpret 5 × 7 as the total number of objects in 5 groups of 7 objects each.
What students will know and be able to do:
Students are required to think in terms of groups of things rather than individual things.
Learning Targets:
 I can interpret and describe multiplication as equal groups within groups.
Vocabulary:
array, row, column, multiplication, multiple, factor, product, shares, division, dividend, divisor, quotient, equal groups, divide
Sample Problem:

Jim purchased 5 packages of muffins. Each package contained 3 muffins. How many muffins did Jim purchase? 5 groups of 3, 5 x 3 =
15. Does he have enough muffins for 17 students?
Describe another situation where there would be 5 groups of 3 or 5 x 3.
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Resources:


National Council of Teachers of Mathematics, Illuminations: Exploring equal sets. This four-part lesson encourages students to
explore models for multiplication, the inverse of multiplication, and representing multiplication facts in equation form.
National Council of Teachers of Mathematics, Illuminations: All About Multiplication
In this four-lesson unit, students explore several meanings and representation of multiplications and learn

http://ccssmath.org/?page_id=59



about properties of operations for multiplication.
www.k-5mathteachingresources.com
Major Cluster: Represent and solve problems involving multiplication and division.
Common Core Standard: 3.OA.2
Interpret whole-number quotients of whole numbers, e.g., interpret 56 ÷ 8 as the number of objects in each share when 56
objects are partitioned equally into 8 shares, or as a number of shares when 56 objects are partitioned into equal shares of 8
objects each.
What students will know and be able to do:
Recognize the operation of division in two different types of situations. One situation requires determining how many groups
and the other situation requires sharing (determining how many in each group).
Use a variety of representations for creating and solving one-step word problems, i.e., numbers, words, pictures, physical objects, or
equations. They use multiplication and division of whole numbers up to 10 x10. Students explain their thinking, show their work by using
at least one representation, and verify that their answer is reasonable.
Learning Targets:
 I can interpret and describe division as a set of objects partitioned into an equal number of shares.
 I can interpret and describe division as groups of a whole set.
Vocabulary:
division, dividend, divisor, quotient, shares, partitioned, reasonable/reasonableness ÷,
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Sample Problem:
There are 12 cookies on the counter. If you are sharing the cookies equally among three bags, how many cookies will go in each bag?
Show all the ways to solve this problem and explain how you solved the problem.
Measurement (repeated subtraction) models focus on the question, ―How many groups can you make? A context or measurement
models would be: There are 12 cookies on the counter. If you put 3 cookies in each bag, how many bags will you fill? How do you know?
Resources:
 Hands-On Standards: Common Core Edition Grade 3 by: ETA hand2mind

http://ccssmath.org/?page_id=59
Major Cluster: Represent and solve problems involving multiplication and division.
Common Core Standard: 3.OA.3
Use multiplication and division within 100 to solve word problems in situations involving equal groups, arrays, and
measurement quantities, e.g., by using drawings and equations with a symbol for the unknown number to represent the
problem.
What students will know and be able to do:
Use a variety of representations for creating and solving one-step word problems, i.e., numbers, words, pictures, physical objects, or
equations. They use multiplication and division of whole numbers up to 10 x 10. Students explain their thinking, show their work by using
more than one representation, and verify that their answers are reasonable.
Learning Targets:
 I can solve multiplication and division word problems involving equal groups, arrays, and measurement quantities with
unknowns in all positions.
Vocabulary:
divide, division, dividend, divisor, quotient, shares, partitioned, unknown
13
Sample Problem:
There are 24 desks in the classroom. If the teacher puts 6 desks in each row, how many rows are there?
(This task can be solved by drawing an array by putting 6 desks in each row. This task can be solved by drawing pictures of
groups. This task can be solved by using repeated addition: 6+6 = 12, 12+12= 24; therefore there are 4 groups with 6 desks in
each group.) Could these desks be arranged in a different way? Explain your thinking.
Resources:
 Hands-On Standards: Common Core Edition Grade 3 by: ETA hand2mind

http://ccssmath.org/?page_id=59
Major Cluster: Represent and solve problems involving multiplication and division.
Common Core Standard: 3.OA.4
Determine the unknown whole number in a multiplication or division equation relating three whole numbers.
For example, determine the unknown number that makes the equation true in each of the equations 8 ×? = 48, 5 = _ ÷ 3, 6 ×
6 =?
What students will know and be able to do:
The focus of this standard goes beyond the traditional notion of fact families, by having students explore the inverse
relationship of multiplication and division. Students apply their understanding of the meaning of the equal sign as ‖the same
as‖ to interpret an equation with an unknown.
Equations in the form of a x b = c and c = a x b should be used interchangeably, with the unknown in different positions.
Learning Targets:
 I can determine the unknown number in multiplication and division equations.
Vocabulary:
division, dividend, divisor, quotient, shares, partitioned, reasonableness, related facts, unknown
14
Sample Problem:
Example 1:
Rachel has 3 bags. There are 4 marbles in each bag. How many marbles does Rachel have altogether? 3 x 4 = m
What if Rachel had 4 bags, how many marbles would be in each?
If she had 15 marbles, how many bags would she need to have groups of 3 marbles in each bag?
Example 2: (Mathematical Thinking)
When given 4 x ? = 40, they might think:
• 4 groups of some number is the same as 40
• 4 times some number is the same as 40
• I know that 4 groups of 10 is 40 so the unknown number is 10
• The missing factor is 10 because 4 times 10 equal 40
Resources:
 http://www.k-5mathteachingresources.com/3rd-grade-number-activities.html

http://ccssmath.org/?page_id=59
Major Cluster: Understand properties of multiplication and the relationship between multiplication and division.
Students use properties of operations to calculate products of whole numbers, using increasingly sophisticated strategies
based on these properties to solve multiplication and division problems involving single-digit factors. By comparing a variety of
solution strategies, students learn the relationship between multiplication and division.
Common Core Standard: 3.OA.5
Apply properties of operations as strategies to multiply and divide.
What students will know and be able to do:
Use properties of operations to calculate products of whole numbers
Use increasingly sophisticated strategies based on these properties (Students need not use formal terms for these
properties). to solve multiplication and division problems involving single-digit factors
Compare a variety of solution strategies, students learn the relationship between multiplication and division
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Learning Targets:
 I can apply the commutative, associative, and distributive properties of multiplication and division.
Vocabulary:
Multiplication, division, associative, distributive, commutative properties of multiplication
Sample Problem 1:
If 6 × 4 = 24 is known, then 4 × 6 = 24 is also known. (Commutative property of multiplication.)
3 × 5 × 2 can be found by 3 × 5 = 15, then 15 × 2 = 30, or by 5 × 2 = 10, then 3 × 10 = 30.
(Associative property of multiplication.)
Knowing that 8 × 5 = 40 and 8 × 2 = 16, one can find 8 × 7 as 8 × (5 + 2) = (8 × 5) + (8 × 2) = 40 +
16 = 56.
(Distributive property.)
Sample Problem 2:
Draw an illustration showing 4 x 5 = 5 x 4
Sample Problem 3:
Draw an illustration showing the how to break 8 x 7 apart into two simpler multiplication problems. For instance 8x7 can be
thought of as (2x8) + (5x8).
Resources:


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National Council of Teachers of Mathematics (NCTM); Illuminations: Multiplication: It„s in the Cards.
http://ccssmath.org/?page_id=59
Major Cluster: Understand properties of multiplication and the relationship between multiplication and division.
Students use properties of operations to calculate products of whole numbers, using increasingly sophisticated strategies
based on these properties to solve multiplication and division problems involving single-digit factors. By comparing a variety of
solution strategies, students learn the relationship between multiplication and division.
Common Core Standard: 3.OA.6
Understand division as an unknown-factor problem. For example, find 32 ÷ 8 by finding the number that makes 32 when
multiplied by 8.
What students will know and be able to do:
Use the knowledge that multiplication and division are inverse operations to find the unknown.
Learning Targets:
 I can explain the relationship between multiplication and division.
Vocabulary:
divide, division, dividend, divisor, quotient, shares, partitioned, reasonableness, related facts, unknown, multiplication, factor
Cluster: Multiply and divide within 100.
Common Core Standard: 3.OA.7
Fluently multiply and divide within 100, using strategies such as the relationship between multiplication and division (e.g.,
knowing that 8 × 5 = 40, one knows 40 ÷ 5 = 8) or properties of operations.
By the end of Grade 3, know from memory all products of two one-digit numbers.
17
What students will know and be able to do:
Internalize basic math facts with fluency (accuracy and efficiency- using a reasonable amount of steps, time and flexibility).
Students should have exposure to multiplication and division problems presented in both vertical and horizontal forms.
NOTE: The focus is not on timed tests and repetitive practice. To build fluency and automaticity, students should
have ample experiences working with manipulatives, pictures, arrays, word problems, and numbers to internalize the
basic facts (up to 9 x 9).
Learning Targets:
 I can multiply and divide with automaticity within 100 using different strategies.
Vocabulary:
divide, division, dividend, divisor, quotient, shares, partitioned, reasonableness, related facts, unknown, array, associative,
distributive, commutative property of multiplication, identity and zero properties
Sample Strategies:
• Multiplication by zeros and ones
• Doubles (2s facts), Doubling twice (4s), Doubling three times (8s)
• Tens facts (relating to place value, 5 x 10 is 5 tens or 50)
• Five facts (half of tens)
• Skip counting (counting groups of __ and knowing how many groups have been counted)
• Square numbers (ex: 3 x 3)
• Nines (10 groups less one group, e.g., 9 x 3 is 10 groups of 3 minus one group of 3)
• Decomposing into known facts (6 x 7 is 6 x 6 plus one more group of 6)
• Turn-around facts (Commutative Property)
• Fact families (Ex: 6 x 4 = 24; 24 ÷ 6 = 4; 24 ÷ 4 = 6; 4 x 6 = 24)
• Missing factors
Resources:


18
http://www.k-5mathteachingresources.com/3rd-grade-number-activities.html
http://ccssmath.org/?page_id=59
Cluster: Solve problems involving the four operations, and identify and explain patterns in arithmetic.
Common Core Standard: 3.OA.8
Solve two-step word problems using the four operations. Represent these problems using equations with a letter standing for
the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including
rounding. This standard is limited to problems posed with whole numbers and having whole-number answers; students should
know how to perform operations in the conventional order when there are no parentheses to specify a particular order.
What students will know and be able to do:
This standard calls for students to represent problems using equations with a letter to represent unknown quantities. The
focus in this standard is to have students use and discuss various strategies including estimating and rounding. Students
should estimate during problem solving, and then revisit their estimate to check for reasonableness.
Learning Targets:
 I can represent and solve two-step word problems using equations with a letter standing for the unknown quantity.
 I can decide if my answers are reasonable using mental math and estimation strategies including rounding.
Vocabulary:
arithmetic pattern, order of operations, operation, multiply, divide, factor, product, quotient, subtract, add, addend, sum,
difference, equation, unknown, strategies, reasonableness, mental computation, estimate, estimation, rounding, patterns,
(properties)
19
Sample Problem:
On a vacation, your family travels 267 miles on the first day, 194 miles on the second day and 34 miles on the third day.
How many total miles did they travel? Is your answer reasonable, and how do you know?
Here are some possible estimation strategies for the problem:
Student 1
I first thought about 267 and 34. I noticed that their sum is about 300. Then I knew that 194 is close to
200. When I put 300 and 200 together, I get 500.
Student 2
I first thought about 194. It is really close to 200. I also have 2 hundreds in 267. That gives me a total of 4 hundreds. Then I
have 67 in 267 and the 34. When I put 67 and 34 together that is really close to 100. When I add that hundred to the 4
hundreds that I already had, I end up with 500.
Resources:
http://www.k-5mathteachingresources.com/3rd-grade-number-activities.html
http://ccssmath.org/?page_id=59
20
Cluster: Solve problems involving the four operations, and identify and explain patterns in arithmetic.
Common Core Standard: 3.OA.9
Identify arithmetic patterns (including patterns in the addition table or multiplication table), and explain them using properties
of operations.
For example, observe that 4 times a number is always even, and explain why 4 times a number can be decomposed
into two equal addends.
What students will know and be able to do:
Examine arithmetic patterns involving both addition and multiplication. Arithmetic patterns are patterns that change by the
same rate, such as adding the same number. For example, the series 2, 4, 6, 8, 10 is an arithmetic pattern that increases by 2
between each term. This standard also mentions identifying patterns related to the properties of operations.
Students need ample opportunities to observe and identify important numerical patterns related to operations. They should
build on their previous experiences with properties related to addition and subtraction. Students investigate addition and
multiplication tables in search of patterns and explain why these patterns make sense mathematically.
Learning Targets:
 I can identify and describe arithmetic patterns including: number charts, addition tables, and multiplication tables.
 I can explain arithmetic patterns using properties of operations.
Vocabulary:
arithmetic pattern, order of operations, operation, multiply, multiples, divide, factor, product, quotient, subtract, add, addend,
sum, difference, equation, unknown, strategies, reasonableness, mental computation, estimate, estimation, rounding,
patterns, (properties), decompose
Sample Strategies:
• Any sum of two even numbers is even.
• Any sum of two odd numbers is even.
• Any sum of an even number and an odd number is odd.
• The multiples of 4, 6, 8, and 10 are all even because they can all be decomposed into two equal groups.
• The doubles (2 addends the same) in an addition table fall on a diagonal while the doubles (multiples of 2) in a multiplication
table fall on horizontal and vertical lines.
• The multiples of any number fall on a horizontal and a vertical line due to the commutative property.
21
• All the multiples of 5 end in a 0 or 5 while all the multiples of 10 end with 0. Every other multiple of 5 is a multiple of 10.
Resources:
 NCTM.org (Illuminations): Times. Students can also look for patterns in the table. National Council of Teachers of
Mathematics, Illuminations – Multiplication: It„s in the Cards
 http://ccssmath.org?page_id=59
Domain: Number and Operations in Base Ten
Enduring Understanding(s):
Understanding place value can lead to number sense and efficient strategies for computing with numbers.
Essential Questions:
How does a digit’s position affect its value?
Cluster: Use place value understanding and properties of operations to perform multi-digit arithmetic.1
1
A range of algorithms may be used.
Standard: NBT.1
Use place value understanding to round whole numbers to the nearest 10 or 100.
What students will know and be able to do:
Have a deep understanding of place value and number sense and can explain and reason about the answers they get when
they round. Students should have numerous experiences using a number line and a hundreds chart as tools to support their
work with rounding.
Know when and why to round numbers. They identify possible answers and halfway points. Then they narrow where the given
number falls between the possible answers and halfway points. They also understand that by convention if a number is
exactly at the halfway point of the two possible answers, at this level the number is rounded up.
Learning Targets


22
I can I can round whole numbers to the nearest 10 or 100.
I can explain when it is appropriate to round numbers and why it is appropriate.
Vocabulary:
estimate, digit, round, whole numbers, place value
Sample Problem(s):
I baked 178 cookies. About how many cookies did I bake?
How do you know this is true?
Possible pathway of mathematical thinking:
The answer is either 170 or 180 because the closest benchmark numbers are 170 and 180.
The halfway point is 175.
178 is between 175 and 180. Therefore, the rounded number is 180.
I know this because 178 on a number line is closer to 180 than 170
I know rounding is used to give an estimate.
Resources:
http://www.funbrain.com/tens/index.html
http://www.ehow.com/way_5182955_math-games-rounding.html
http://www.ixl.com/math/grade-3
http://ccssmath.org/?page_id=59
23
Cluster: Use place value understanding and properties of operations to perform multi-digit arithmetic.1
A range of algorithms may be used.
Standard: 3.NBT.2
Fluently add and subtract within 1000 using strategies and algorithms based on place value,
properties of operations, and/or the relationship between addition and subtraction
What students will know and be able to do:
(using a reasonable number of steps and time), and flexibility (using strategies such as the distributive property) add
and subtract problems with four digits. The word algorithm refers to a procedure or a series of steps. There are other
algorithms other than the standard/traditional algorithm. Students should have experiences beyond the
standard/traditional algorithm. Problems should include both vertical and horizontal forms, including opportunities for
students to apply the commutative and associative properties.
Students should explain their thinking and show work by using strategies and algorithms, and verify that their answer is
reasonable.
Learning Targets:

24
I can accurately complete vertical and horizontal addition and subtraction problems up to 1000
Vocabulary:
commutative property of addition, associative property of addition, inverse operation, algorithm, add, addend, sum,
difference, vertical, horizontal, place value
Sample Problem(s):
Example 1
There are 178 fourth graders and 225 fifth graders on the playground. What is the total number of students on the
playground?
Student 1
100 + 200 = 300
70 + 20 = 90
8 + 5 = 13
So, 300 + 90 + 13 = 403 students
Student 2
I added 2 to 178 to get 180.
I added 220 to get 400.
I added the 3 left over to get 403.
Student 3
I know the 75 plus 25 equals 100.
I then added 1 hundred from 178 and 2 hundreds from 275.
I had a total of 4 hundreds and I had 3 more left to add.
25
So I have 4 hundreds, plus 3 more, which is 403.
Resources:
http://www.amblesideprimary.com/ambleweb/mentalmaths/pyramid.html
http://www.tncurriculumcenter.org/concept/Place+Value
http://ccssmath.org/?page_id=59
Cluster: Use place value understanding and properties of operations to perform multi-digit arithmetic.1
A range of algorithms may be used.
Standard: 3.NBT.3
Multiply one-digit whole numbers by multiples of 10 in the range 10-90 (e.g. 9x80, 5x60) using strategies based on place
value and properties of operations.
What students will know and be able to do:
Accurately apply the commutative and associative properties to solve vertical and horizontal forms of multiplication
problems consisting of a whole number factor and a multiple of 10
Learning Targets:
 I can multiply horizontal and vertical multiplication problems that have a one-digit factor and multiple of 10.
Vocabulary:
Associative, Commutative, Identity and Zero properties of Multiplication, inverse operation, algorithm, multiply,
multiplication, digit, multiple, operation, vertical, horizontal, whole numbers, place value
Sample Problem(s):
For example, for the problem 50 x 4, students should think of this as 4 groups of 5 tens or 20 tens. Twenty tens is equal
to 200.
Students use base ten blocks, diagrams, or hundreds charts to multiply one-digit numbers by multiples of 10 from 10-90.
They apply their understanding of multiplication and the meaning of the multiples of 10. For example, 30 equals 3 tens
and 70 equals 7 tens. They can interpret 2 x 40 as 2 groups of 4 tens or 8 groups of ten. They understand that 5 x 60
equals 5 groups of 6 tens or 30 tens and know that 30 tens equals 300. After developing this understanding they begin
to recognize the patterns in multiplying by multiples of 10. Students may use manipulatives, drawings, document
26
camera, or interactive whiteboard to demonstrate understanding.
Resources:
http://ccssmath.org/?page_id=59
Domain: Number and Operations: Fractions
Enduring Understanding(s):
Fractions and decimals allow for quantities to be expressed with greater precision than with just whole numbers.
Essential Questions: Why express quantities, measurements, and number relationships in different ways?
Major Cluster: Develop understanding of fractions as numbers.
*Grade expectations in this domain are limited to fractions with denominators 2,3,4,6,8
Standard: 3.NF.1
Understand a fraction 1/b as the quantity formed by 1 part when a whole is partitioned into b equal parts; understand a
fraction a/b as the quantity formed by a parts of size 1/b.
What students will know and be able to do:
Develop understanding for the following:



27
Fractional parts must be equal-sized
The number of equal parts tell how many make a whole
When a whole is cut into equal parts, the denominator represents the total number of equal parts.


The numerator of a fraction is the number of equal parts begin evaluated
Explain the relationship of a numerator and a denominator of a fraction.
Construct an illustration and written explanation when given a fraction.
Learning Targets:


I can understand and can explain what the numerator and the denominator in a fraction represents.
I can explain, using an illustration and words, a fraction is part of a whole.
Vocabulary:
fraction, unit fraction, numerator, denominator, number line, whole, part, halves, thirds, fourths, sixths, eighths
Sample Problem(s):
Some important concepts related to developing understanding of fractions include:
Understand that fractional parts must be equal-sized.
Example



28
Non-example
The number of equal parts tells how many make a whole.
As the number of equal pieces in the whole increases, the size of the fractional pieces decreases.
The size of the fractional part is relative to the whole.




The number of children in one-half of a classroom is different than the number of children in one half of a school.
(the whole in each set is different therefore the half in each set will be different)
When a whole is cut into equal parts, the denominator represents the number of equal parts.
The numerator of a fraction is the count of the number of equal parts.
¾ means that there are 3 one-fourths.
Students can count one fourth, two fourths, three fourths.

Resources: http://ccssmath.org/?page_id=59
Major Cluster: Develop understanding of fractions as numbers.
*Grade expectations in this domain are limited to fractions with denominators 2,3,4,6,8
Standard: 3.NF.2
Understand a fraction as a number on the number line; represent fractions on a number line diagram.
a. Represent a fraction 1/b on a number line diagram by defining the interval from 0 to 1 as the whole and partitioning it
into b equal parts. Recognize that each part has size 1/b and that the endpoint of the part based at 0 locates the number
1/b on the number line.
b. Represent a fraction a/b on a number line diagram by marking off a lengths 1/b from 0. Recognize that the resulting
interval has size a/b and that its endpoint locates the number a/b on the number line.
What students will know and be able to do:
Label fractions on a number line (e.g., that, 1/4 is between 0 and 1).
Learning Targets:
 I can understand that a fraction is a number on the number line.
 I can represent any fraction as a location on a number line.
Vocabulary:
fraction, unit fraction, numerator, denominator, number line, represent, endpoint, point
29
Sample Problem(s):
Students transfer their understanding of parts of a whole to partition a number line into equal parts. There are two new
concepts addressed in this standard which students should have time to develop.
1. On a number line from 0 to 1, students can partition (divide) it into equal parts and recognize that each segmented
part represents the same length.
2. Students label each fractional part based on how far it is from zero to the endpoint.
30
Resources: http://ccssmath.org/?page_id=59
Cluster: Develop understanding of fractions as numbers.
*Grade expectations in this domain are limited to fractions with denominators 2,3,4,6,8
Standard: 3.NF3.
Explain equivalence of fractions in special cases, and compare fractions by reasoning about their size.
a. Understand two fractions as equivalent if they are the same size, or the same point on a number line.
b. Recognize and generate simple equivalent fractions, (e.g. ½=2/4, 4/6=2/3). Explain why the fractions are
equivalent
c. Express whole numbers as fractions, and recognize fractions that are equivalent to whole numbers.
Examples: Express 3 in the form 3 = 3/1; recognize that 6/1= 6; locate 4/4 and 1 at the same point of a number
line diagram.
d. Compare two fractions with the same numerator or the same denominator by reasoning about their size.
Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of
comparisons with the symbols >, =, or <, and justify the conclusions, e.g., by using a visual fraction model.
31
What students will know and be able to do:
Compare fractions by looking at the size (relationship) of the parts and the number of the parts. For example, 1/8 is
smaller than 1/2 because when 1 whole is cut into 8 pieces, the pieces are much smaller than when 1 whole is cut into 2
pieces.
Only explore equivalent fractions using models without using algorithms or procedures.
Understand that n/n is equal to 1 and n/1 is equal to n.
Write whole numbers as fractions (3/1 is 3 wholes divided into one group).
(This standard is the building block for later work where students divide a set of objects into a specific number of
groups).
Compare fractions with or without visual fraction models including number lines.
Reason that comparisons are only valid if the wholes are identical.
Learning Targets:
 I can compare fractions with the same denominator and different numerator using different strategies.
 I can compare fractions with the same numerator and different denominator using different strategies.
 I can explain and illustrate how fractions are equivalent and compare fractions using their size.
 I can create a fraction that represents a whole number.
Vocabulary:
fraction, unit fraction, numerator, denominator, greater than, less than, equal to, compare, equivalent fractions, number
line
Sample Problem(s):
Idea Box (d): Compare using <, =, >
Same numerator:
2/6
>
2/9
Same denominator:
4/10
32
<
6/10
Resources:
http://alex.state.al.us/ccrs/sites/alex.state.al.us.ccrs/files/Grade%203.pdf
http://ccssmath.org/?page_id=59
Domain: Measurement and Data
Enduring Understanding(s):
 Measurement processes are used in everyday life to describe and quantify the world.
 Data displays describe and represent data in alternate ways.
Essential Questions:
 Why does ―w
hat‖ we measure influence ―h
ow‖ we measure?
Major Cluster: Solve problems involving measurement and estimation of intervals of time, liquid volumes, and masses
of objects.
Standard: 3.MD.1
Tell and write time to the nearest minute and measure time intervals in minutes. Solve word problems involving addition
and subtraction of time intervals in minutes, e.g., by representing the problem on a number line diagram
What students will know and be able to do:
Solve elapsed time word problems using clock models or number lines to solve. On the number line, students should be
given the opportunities to determine the intervals and size of jumps on their number line. Students could use predetermined number lines (intervals every 5 or 15 minutes) or open number lines (intervals determined by students).
Learning Targets:



33
I can say and write time to the nearest minute.
I can measure the duration of time in minutes.
I can solve addition and subtraction word problems involving the duration of time measured in minutes.
Vocabulary:
estimate, time, time intervals, minute, hour, elapsed time, measure, number line, half-hour, quarter hour
Sample Problem(s):
Students make a schedule of their day. Students find the time spent doing each activity. Students identify the start time,
end time, and elapsed time using their clocks to model each. Students work with a partner. One creates the start time.
The other students create the end time. Students then determine elapsed time using their clocks to model.
Resources: http://ccssmath.org/?page_id=59
Major Cluster: Solve problems involving measurement and estimation of intervals of time, liquid volumes, and masses
of objects.
Standard: 3.MD.2.
Measure and estimate liquid volumes and masses of objects using standard units of grams (g), kilograms (kg), and
liters (l). Add, subtract, multiply, or divide to solve one-step word problems involving masses or volumes that are given
in the same units, e.g., by using drawings (such as a beaker with a measurement scale) to represent the problem.
1
Excludes compound units such as cm3 and finding the geometric volume of a container.
2
Excludes multiplicative comparison problems (problems involving notions of (times as much‖)
What students will know and be able to do:
Reason about the units of mass and volume. Explore multiple opportunities of weighing classroom objects and filling
containers to help them develop a basic understanding of the size and weight of a liter, a gram, and a kilogram.
NOTE: Students are not expected to do conversions between units, but reason as they estimate, using benchmarks to
measure weight and capacity.
34
Learning Targets:
I can describe the attributes of a unit (attributes are what makes the ―un
it ―sta
nd out‖ among other units).
I can partition larger units into smaller equivalent units.
I can repeatedly add the same unit to determine the measure (iteration).
I can explain the size of a unit and the number of units needed to measure another unit (compensatory
principal).
 I can solve one-step word problems involving masses or volumes that are given in the same units.
Vocabulary:




mass, volume, liter, gram, kilogram, add, subtract, multiply, divide, standard unit
Sample Problem(s):
Students identify 5 things that have a mass of about one gram. They record their findings with words and pictures. (Students can
repeat this for 5 grams and 10 grams.) This activity helps develop gram benchmarks. One large paperclip weighs about one gram.
A box of large paperclips (100 clips) has a mass of about 100 grams so 10 boxes would have a mass of one kilogram.
OR:
A paper clip has a mass of about a) a gram, b) 10 grams, c) 100 grams?
Students need multiple opportunities ―massing classroom objects and filling containers to help them develop a basic
understanding of the size and mass of a liter, a gram, and a kilogram. Students identify 5 things that have a mass of about one
gram. They record their findings with words and pictures.
Resources:
http://www.ixl.com/math/grade-3
http://www.ixl.com/math/grade-2
http://ccssmath.org/?page_id=59
35
Supporting Cluster: Represent and interpret data.
Standard: 3.MD.3
Draw a scaled picture graph and a scaled bar graph to represent a data set with several categories. Solve one-and
two-step ―ho
w many more‖ and ―ho
w many less‖ problems using information presented in scaled bar graphs.
For example, draw a bar graph in which each square in the bar graph might represent 5 pets.
What students will know and be able to do:
Read and solve problems using scaled graphs before being asked to draw one. (Note: students should be exposed to
different scales on graphs.)
While exploring data concepts, students should: Pose a question, Collect data, Analyze data, and Interpret data.
Students should be graphing data that is relevant to their lives.
Learning Targets:
I can draw the scale on a picture or bar graph to represent data.
I can read a graph using the titles, scale, labels, and data.
I can generate a question, conduct a survey, and organize data on a graph.
 I can analyze a bar graph to solve one- and two-step problems asking ―ho
w many more/less?‖
Vocabulary:



scaled picture graph, scaled bar graph, line plot, data, scale (on a graph)
Sample Problem(s):
Students should generate a question: ―H
ow many more kids like soccer than volleyball?‖
Collect and organize data such as a student survey.
Use questioning such as ―I
f a square is equal to 4 pets, how many squares would you need to represent 18 pets?‖
Picture graphs: Scaled picture graphs include symbols that represent multiple units. Below is an example of a picture
graph with symbols that represent multiple units. Graphs should include a title, scale, categories, category label, and
36
data. Students need to use both horizontal and vertical bar graphs.
If you were to purchase a book for the class library which would be the best genre? Why?
Example of Scaled Graph:
(cont. on next page)
 Single Bar Graphs: Students use both horizontal and vertical bar graphs. Bar graphs include a title, scale, scale label,
categories, category label, and data.
Analyze and Interpret data which could include:
• How many more nonfiction books were read than fantasy books?
37
• Did more people read biography and mystery books or fiction and fantasy books?
• About how many books in all genres were read?
• Using the data from the graphs, what type of book was read more often than a mystery but less often than a
fairytale?
• What interval was used for this scale?
• What can we say about types of books read? What is a typical type of book read?
• If you were to purchase a book for the class library which would be the best genre?
Resources: http://ccssmath.org/?page_id=59
Supporting Cluster: Represent and interpret data.
Students multiply and divide to solve problems using information presented in scaled bar graphs. Pictographs and
scaled bar graphs are a visually appealing context for one-and two-step word problems.
Standard: 3.MD.4
Generate measurement data by measuring lengths using rulers marked with halves and fourths of an inch. Show the
data by making a line plot, where the horizontal scale is marked off in appropriate units— whole numbers, halves, or
quarters
What students will know and be able to do:
Students in second grade measured length in whole units using both metric and U.S. customary systems. It’s
important to review with students how to read and use a standard ruler including details about halves and quarter
marks on the ruler. Students should connect their understanding of fractions to measuring to one-half and one-quarter
inch. Third graders need many opportunities measuring the length of various objects in their environment.
This standard provides a context for students to work with fractions by measuring objects to a quarter of an inch.
Learning Targets:
I can use a ruler to measure lengths to the nearest whole, half, and quarter inches.
I can generate and record measurements data using whole, half, and quarter inches,
 I can create a line plot marked with appropriate units to show measurement with the horizontal scale marked
off in whole number, half, or quarter units.
Vocabulary:


38
scaled picture graph, scaled bar graph, line plot, data, scale (on a graph), half/halves, quarters, fourths,
Sample Problem(s):
Measure objects in your desk to the nearest ¼ or ½ of an inch, display data collected on a line plot. How many
objects measured ¼ ? ½ ? etc…
Some important ideas related to measuring with a ruler are:


The starting point of where one places a ruler to begin measuring
Measuring is approximate. Items that students measure will not always measure exactly ¼, ½ or one inch.
Students will need to decide on an appropriate estimate length.
Making paper rulers and folding to find the half and quarter marks will help students develop a stronger understanding
of measuring length.
Students generate data by measuring and create a line plot to display their findings.
Resources:
Hands-On Standards: Common Core Edition Grade 3 by: ETA hand2mind
http://ccssmath.org/?page_id=59
Major Cluster: Geometric measurement: understand concepts of area and relate area to multiplication and to
addition.
Standard: 3.MD.5
Recognize area as an attribute of plane figures and understand concepts of area measurement.
a. A square with side length 1 unit, called ―aunit square,‖ is said to have ―on
e square unit‖ of area, and can be used
to measure area.
39
b. A plane figure which can be covered without gaps or overlaps by n unit squares is said to have an area of n square
units.
What students will know and be able to do:
Describe a square unit and why it is used to measure area.
Use square units to measure a plane figure’s area.
Learning Targets:
I can define a square unit and know when to use it to measure.
I can measure the area of a figure using square tiles or graph paper.
Vocabulary:


area, plane figure, unit square, multiplication, addition
Sample Problem(s):
Students develop understanding of using square units to measure area by:



40
Using different sized square units
Filling in an area with the same sized square units and counting the number of square units
An interactive whiteboard would allow students to see that square units can be used to cover a plane figure.
Resources:
Hands-On Standards: Common Core Edition Grade 3 by: ETA hand2mind
http://ccssmath.org/?page_id=59
Major Cluster: Geometric measurement: understand concepts of area and relate area to multiplication and to
addition.
Standard: 3.MD.6
Measure areas by counting unit squares (square cm, square m, square in, square ft, and improvised units).
What students will know and be able to do:
Count the square units to find the area in metric, customary, or improvised square units. Using different sized graph
paper, students can explore the areas measured in square centimeters and square inches.
Learning Targets:
 I can find the area using at least two different units of measurement
Vocabulary:
area, plane figure, square unit, square centimeter, square meter, square inch, square feet, square yards, metric and
customary units of measurement, improvised units of measurement
Sample Problem(s):
On a geoboard, create a figure that has an area of 6 square units, 10 square units, 12 square units, etc.
41
Resources:
http://ccssmath.org/?page_id=59
Major Cluster: Geometric measurement: understand concepts of area and relate area to multiplication and to
addition.
Standard: 3.MD.7
Relate area to the operations of multiplication and addition.
a. Find the area of a rectangle with whole-number side lengths by tiling it, and show that the area is the same as
would be found by multiplying the side lengths.
b. Multiply side lengths to find areas of rectangles with whole-number side lengths in the context of solving real world
and mathematical problems, and represent whole-number products as rectangular areas in mathematical reasoning.
c. Use tiling to show in a concrete case that the area of a rectangle with whole-number side lengths a and b + c is the
sum of a × b and a × c. Use area models to represent the distributive property in mathematical reasoning.
d. Recognize area as additive. Find areas of rectilinear figures by decomposing them into non-overlapping rectangles
and adding the areas of the non-overlapping parts, applying this technique to solve real world problems.
What students will know and be able to do:
Find the area of a figure using addition, multiplication or manipulation of the figure.
Decompose a rectilinear figure into different rectangles.
Students should solve real world applications.
Tile a rectangle then multiply the side lengths to show it is the same.
42
Learning Targets:
 I can find the area using units of measurement.
 I can find the area of a figure using repeated addition or multiplication.
 I can use strategies such as the distributive property to solve for area.
 I can solve real world and mathematical problems involving perimeters of polygons.
 I can draw a rectangular area model to represent a product.
Vocabulary:
area, area model, plane figure, square unit, tiling, multiplication, distributive property, addition, rectangle, rectilinear
Sample Problem(s):
Drew wants to tile the bathroom floor using 1 foot tiles. How many square foot tiles will he need?
6 ft
8 ft
Joe and John made a poster that was 4ft. by 3ft. Melisa and Barb made a poster that was 4ft. by 2ft. They placed
their posters on the wall side-by-side so that that there was no space between them. How much area will the two
posters cover?
Students use pictures, words, and numbers to explain their understanding of the distributive property in this context
43
Students can decompose a rectilinear figure into different rectangles. They find the area of the figure by adding the
areas of each of the rectangles together.
44
Resources:


http://www.bgfl.org/bgfl/custom/resources_ftp/client_ftp/ks2/maths/perimeter_and_area/index.html -md7
http://ccssmath.org/?page_id=59
Additional Cluster: Geometric measurement: recognize perimeter as an attribute of plane figures and distinguish
between linear and area measures.
Standard: 3.MD.8
Solve real world and mathematical problems involving perimeters of polygons, including finding the perimeter given
the side lengths, finding an unknown side length, and exhibiting rectangles with the same perimeter and different
areas or with the same area and different perimeters.
What students will know and be able to do:
Develop an understanding of the concept of perimeter, and the patterns that exist, through multiple experiences such
as kinesthetic activities, paper/pencil activities, and activities that involve technology.
They find the perimeter of objects; use addition to find perimeters
Use geoboards, tiles, and graph paper to find all the possible rectangles that have a given perimeter (e.g., find the
rectangles with a perimeter of 14 cm.), and organize/ compile the possibilities into an organized list or a table, and
determine whether they have all the possible rectangles.
Given a perimeter and a length or width, use objects or pictures to find the missing length or width. They justify and
communicate their solutions using words, diagrams, pictures, numbers, and/or an interactive whiteboard.
Use geoboards, tiles, and graph paper to find all the possible rectangles that have a given area (e.g find the
rectangles that have an area of 12 square units), and organize/ compile the possibilities into an organized list or a
table, and determine whether they have all the possible rectangles.
45
Learning Targets:
 I can define perimeter of a figure when I know all the side lengths and when a length is missing.
 I can describe how to find a missing length of a side by using other data given.
 I can create or draw different shapes/rectangles with the same perimeter and different areas and organize my
creations in a list or table.
 I can create or draw different shapes/rectangles with the different perimeter and same areas and organize my
creations in a list or table.
 I can solve real world and mathematical problems involving perimeters of polygons.
Vocabulary:
Perimeter, area, unknown, side length, rectangle, polygon, attribute, linear, plane figure
Sample Problem(s):
Using geoboards, students create a figure when given a perimeter.
Using any manipulative: Record and find the rectangle with the greatest perimeter when given the area is 12 inches.
Resources: http://ccssmath.org/?page_id=59
46
Domain: Number and Operations: Geometry
Enduring Understanding(s):
Geometric attributes (such as shapes, lines, angles, figures, and planes) provide descriptive information about an
object’s properties and position in space and support visualization and problem solving.
Essential Questions:
How does geometry better describe objects?
Supporting Cluster: Reason with shapes and their attributes.
i
Work should be positioned in support of area, measurement and understanding of fractions.
Standard: 3.G. 1
Understand that shapes in different categories (e.g., rhombuses, rectangles, and others) may share attributes (e.g.,
having four sides), and that the shared attributes can define a larger category (e.g., quadrilaterals). Recognize
rhombuses, rectangles, and squares as examples of quadrilaterals, and draw examples of quadrilaterals that do not
belong to any of these subcategories.
What students will know and be able to do:
Recognize shapes that are and are not quadrilaterals by examining the properties of the geometric figures. They
conceptualize that a quadrilateral must be a closed figure with four straight sides and begin to notice characteristics of
the angles and the relationship between opposite sides.
Provide details and use proper vocabulary when describing the properties of quadrilaterals. Students sort geometric
figures and identify squares, rectangles, and rhombuses as quadrilaterals based on the different attributes of each
shape.
47
Classify shapes by attributes and drawing shapes that fit specific categories.
Learning Targets:




I can identify shared attributes of shapes that are in different categories.
I can categorize figures by their attributes (characteristics).
I can recognize that a shape belongs in more than one category and has more than one name.
I can draw a polygon based on its attributes.
Vocabulary:
rhombus, rectangle, square, quadrilateral, parallelogram, trapezoid, parallel lines, attribute, side of a polygon, vertex
(vertices), closed figure
Sample Problem(s):
Sasha has been making quadrilaterals and she wants to make a new one. So far, she has made a rectangle, square,
and rhombus. Help Sasha find a quadrilateral she has not yet made.
Compare the properties of the new quadrilateral with the properties of the ones she has already made.
Compare the properties of the new quadrilaterals using a Venn Diagram.
Resources:


48
http://teams.lacoe.edu/documentation/classrooms/amy/geometry/6-8/activities/quad_quest/quad_quest.html -g1
http://ccssmath.org/?page_id=59
Supporting Cluster: Reason with shapes and their attributes.
3
Work should be positioned in support of area measurement and understanding of fractions.
Standard: 3.G. 2
Partition shapes into parts with equal areas. Express the area of each part as a unit fraction of the whole. For example
partition a shape into 4 parts with equal area and describe the area as of each as ¼ of the area of the shape.
What students will know and be able to do:
Understand and model that a whole shape can be divided into equal parts, and label each part as a fraction.
Given equal parts of a whole shape, students can reconstruct that shape.
Can explain their reasoning verbally or in written form.
Partition shapes into halves, thirds, fourths, sixths, and eighths.
Learning Targets:



I can divide shapes into parts with equal areas and explain their value in fraction form.
I can combine equal areas to make shapes and explain how.
I can label the area of each equal part using a unit fraction of the whole.
Vocabulary:
Square, triangle, rectangle, hexagon, octagon, trapezoid, (all other two-dimensional shapes), halves (1/2), thirds (1/3),
fourths (1/4), sixths (1/6), eighths (1/8), equal parts, area, partition, unit fraction
49
Sample Problem(s):
Partition a shape into 4
parts with equal area, and describe the
area of each part as 1/4 of the area of
the shape.
Resources:
http://schools.utah.gov/CURR/mathelem/Core-Curriculum/Operations-and-Algebraic-Thinking.aspx
50
Grade 4 Standards
Mathematics | Grade 4 – Critical Areas
In Grade 4, instructional time should focus on three critical areas: (1) developing understanding and fluency with
multi-digit multiplication, and developing understanding of dividing to find quotients involving multi-digit
dividends; (2) developing an understanding of fraction equivalence, addition and subtraction of fractions with
like denominators, and multiplication of fractions by whole numbers; (3) understanding that geometric figures
can be analyzed and classified based on their properties, such as having parallel sides, perpendicular sides,
particular angle measures, and symmetry.
(1) Students generalize their understanding of place value to 1,000,000 understanding the relative sizes of numbers in each place.
They apply their understanding of models for multiplication (equal-sized groups, arrays, area models), place value, and properties of
operations, in particular the distributive property, as they develop, discuss, and use efficient, accurate, and generalizable methods to
compute products of multi-digit whole numbers. Depending on the numbers and the context, they select and accurately apply
appropriate methods to estimate or mentally calculate products. They develop fluency with efficient procedures for multiplying whole
numbers; understand and explain why the procedures work based on place value and properties of operations; and use them to
solve problems. Students apply their understanding of models for division, place value, properties of operations, and the relationship
of division to multiplication as they develop, discuss, and use efficient, accurate, and generalizable procedures to find quotients
involving multi-digit dividends. They select and accurately apply appropriate methods to estimate and mentally calculate quotients,
and interpret remainders based upon the context.
(2) Students develop understanding of fraction equivalence and operations with fractions. They recognize that two different fractions
can be equal (e.g., 15/9 = 5/3), and they develop methods for generating and recognizing equivalent fractions. Students extend
previous understandings about how fractions are built from unit fractions, composing fractions
from unit fractions, decomposing fractions into unit fractions, and using the meaning of fractions and the meaning of multiplication to
multiply a fraction by a whole number.
(3) Students describe, analyze, compare, and classify two-dimensional shapes. Through building, drawing, and analyzing twodimensional shapes, students deepen their understanding of properties of two-dimensional objects and the use of them to solve
problems involving symmetry.
The Common Core State Standards
51
Grade 4 Overview
Operations and Algebraic Thinking
• Use the four operations with whole numbers to solve problems.
• Gain familiarity with factors and multiples.
• Generate and analyze patterns.
Number and Operations in Base Ten
• Generalize place value understanding for multi digit whole numbers.
• Use place value understanding and properties of operations to perform multi-digit arithmetic.
Number and Operations—Fractions
• Extend understanding of fraction equivalence and ordering.
• Build fractions from unit fractions by applying and extending previous understandings of operations on whole
numbers.
• Understand decimal notation for fractions, and compare decimal fractions.
Measurement and Data
• Solve problems involving measurement and conversion of measurements from a larger unit to a smaller unit.
• Represent and interpret data.
• Geometric measurement: understand concepts of angle and measure angles.
Geometry
• Draw and identify lines and angles, and classify shapes by properties of their lines and angles.
The Common Core State Standards
52
Mathematical Common Core Content State Standards
Grade 4
Domain: Operations and Algebraic Thinking
Enduring Understanding(s): Mathematical operations are used in solving problems in which a new value is
produced from one or more values. Algebraic thinking involves choosing, combining, and applying effective
strategies for answering quantitative questions.
Essential Questions: In what ways can operations affect numbers? How can different strategies be helpful when
solving a problem?
Major Cluster: Use the four operations with whole numbers to solve problems.
Standard: 4.OA.1
Interpret a multiplication equation as a comparison, e.g., interpret 35=5x7 as a statement that 35 is 5 times as many
as 7 and 7 times as many as 5. Represent verbal statements of multiplicative comparisons as multiplication
equations.
What students should know and be able to do:
A multiplicative comparison is a situation in which one quantity is multiplied by a specified number to get another
quantity (e.g., ―
a equals n times as much as b‖). Students will understand that the equal sign (=) means the same as
(e.g. 14 is the same as 7 times 2 or 7 groups of 2).
Learning Targets:


53
I can interpret a multiplication equation as a comparison of the factors to the product.
I can represent verbal statements of multiplicative comparisons with equations.
Vocabulary:
multiply, compare, multiplicative comparison, equation, represent, base ten, array, operation, factor, product,
expanded form, place value, decompose
Sample Problem(s):
Multiplicative Comparison- Create a multiplication equation for the following and explain how you got your answer.
Sally is five years old. Her mom is eight times older. How old is Sally’s Mom?
Sally has five times as many pencils as Mary. If Sally has 5 pencils, how many does Mary have?
Create a word problem using: 5 x 7 = 35
Resources:
54

http://Nlvm.usu.edu/en/nav/topic_t_1.html
Virtual Manipulative tool to build arrays

http://Mathstory.com/mathlessons/arrayrace.htm
A short lesson and a game for students to practice building arrays, writing equations, and solving for a product.

http://www.internet4classrooms.com/grade_level_help/solve_problems_math_fourth_4th_grade.htm
Locate the activity ―Gr
oups of Dogs‖ at the following site for students to look at arrays using objects.
Standard: 4.OA.2
Multiply or divide to solve word problems involving multiplicative comparison, e.g., by using drawings and equations
with a symbol for the unknown number to represent the problem, distinguishing multiplicative comparison from
additive comparison.
What students should know and be able to do:
Students will translate comparative situations into equations with an unknown and solve.
Learning Targets:
I can solve word problems involving multiplicative comparisons with a symbol for the unknown using
multiplication and division.
Vocabulary:

variable, inverse operations, multiplicative comparison, additive comparison, symbol
Sample Problem(s): See Table 2 that follows
55
56
57
Resources:

http://www.helpingwithmath.com
Look at Multiplication and Division Word Problems for examples to use with students of multiplicative
comparison.

http://www.mathplayground.com/wordproblems.html
Challenging examples of word problems using multiplicative comparison.

http://www.mathscore.com/math/standards/Common%Core/4th%20Grade/
Scroll down to the correct domain and standard to find a listing of online problems displaying multiplicative
comparison
Standard: 4.OA. 3
Solve multistep word problems posed with whole numbers and having whole-number answers using the four
operations, including problems in which remainders must be interpreted. Represent these problems using equations
with a letter standing for the unknown quantity. Assess the reasonableness of answers using mental computation.
and estimation strategies including rounding.
What students should know and be able to do:
Students will be able to solve multistep story problems using all four operations, including division problems with
remainders.
Students will use and discuss various strategies, including estimation strategies or rounding, and arrive at a
reasonable answer.
Students will be able to use a letter as an unknown quantity in equations.
58
Learning Targets:



I can represent word problems with an equation using a letter for an unknown.
I can solve multistep word problems with whole-number answers using the four operations, including problems
in which remainders must be interpreted.
I can assess the reasonableness of answers using mental math and estimation strategies.
Vocabulary:
mental math, estimation, rounding, remainder, variable (unknown), operations, equation, reasonableness
Sample Problem(s):
Solve the problem and explain your thinking.
59
Resources:

http://www.mathplayground.com
Click on the ―
Word Problems‖ tab and select ―W
ord Problems with Katie‖ for different types of multistep
problems.

http://www.mathscore.com/math/practice/Word%Problems%20With%20Remainders/
Multiple problems are listed that require students to solve different operations in steps to determine the answer.

http://www.ixl.com/math/grade-4/multi-step-word-problems
The site offers additional examples of word problems.

http://www.internet4classrooms.com/grade_level_help/solve_problems_math_fourth_4th_grade.htm
Look for the activity ―T
wo-step Computation‖ to play a game with multi-step operations.
Supporting Cluster: Gain familiarity with factors and multiples.
Standard: 4.OA.4
Find all factor pairs for a whole number in the range 1–100. Recognize that a whole number is a multiple of each of its
factors. Determine whether a given whole number in the range 1–100 is a multiple of a given one-digit number. Determine
whether a given whole number in the range 1–100 is prime or composite.
What students should know and be able to do:
Students will determine and explain multiples and factors of any whole number 1-100. They will also be able to determine
if a number is prime or composite.
60
Learning Targets:
 I can explain the relationship between a factor and a multiple.
 I can recognize if a number 1-100 is a multiple of a single digit number.
 I can recognize if a number 1-100 is prime or composite.
 I can explain why 1 is neither prime nor composite.
 I can find all the factor pairs of a whole number 1-100.
Vocabulary:
factor, multiple, prime, composite, whole number
61
Sample Problem(s):
Given a number, students can determine if it is prime or composite. Students also list numbers that are prime or
composite. (cont. on next page)
62
Resources:

http://illuminations.nctm.org/LessonDetail.aspx?ID=L620
Students distinguish between numbers with several factors and those with only a few factors.

http://illuminations.nctm.org/lessons/FactorGame/FactorGame-AS-Problems.pdf
This link offers a worksheet that assesses students’ knowledge after playing the Factor Game.

http://www.xpmath.com/forums/arcade.php?do=play&gameid=60
Students play King Kong by whacking him if he’s holding a prime number.

http://www.aaamath.com/fra63ax2.htm
The computer lists a number and the student identifies it as prime or composite.

http://www.mathplayground.com/howto_primenumbers.html
Watch a video that defines the terms from the standard, including factor, prime, and composite.

―
Sieve of Eratosthemes‖ on Teacher Tube Lesson for teachers to watch before having students use a one hundred grid to
find prime numbers.
Additional Cluster: Generate and analyze patterns.
Standard: 4.OA.5
Generate a number or shape pattern that follows a given rule. Identify apparent features of the pattern that were not
explicit in the rule itself. For example, given the rule “Add 3” and the starting number 1, generate terms in the resulting
sequence and observe that the terms appear to alternate between odd and even numbers. Explain informally why the
numbers will continue to alternate in this way.
63
What students should know and be able to do:
Students will create and extend number or shape patterns that follow a given rule. They will identify different features of a
pattern that are not stated in the rule itself.
Learning Targets:



I can create a number pattern that follows a given rule.
I can create a shape pattern that follows a given rule
I can identify features of patterns that are not stated in the rule itself.
Vocabulary:
number pattern, shape pattern, pattern rule, sequence, alternate
64
Sample Problem(s):
*#*#...
65
Resources:

http://www.uen.org/3-6interactives/math.shtml#patterns
A listing of online pattern games that students can play
Domain: Numbers and Operations in Base Ten
*Expectations in this domain are limited to whole numbers less than or equal to 1,000,000.
Enduring Understanding(s): Understanding place value can lead to number sense and efficient strategies for
computing with numbers.
Essential Questions: How does a digit’s position affect its value?
Major Cluster: Generalize place value understanding for multi-digit whole numbers.
Standard: 4.NBT.1
Recognize that in a multi-digit whole number, a digit in one place represents ten times what it represents in the place to
its right. For example, recognize that 700 ÷ 70 = 10 by applying concepts of place value and division.
What students should know and be able to do:
Students will extend their understanding of place value related to multiplying and dividing by multiples of ten and reason
about the value of digits in a number.
Learning Targets:

66
I can recognize that in a multi-digit number, up to one million, a digit in a place represents 10 times as much as
the digit in the place to its right.
Vocabulary:
inverse operation, base ten numeral (instead of standard form), value, place, and place value, digit
Sample Problem(s):
How is the digit 2 in the number 582 similar to and different from the digit 2 in the number 528?
Resources:
67

hand2mind.com/hosstudent
Student pages for Hands-On Standards Common Core Edition ETA hand2mind

DEA (various probes and resources)

www.IXL.com
(students can do up to 20 problems for free per session)

http://illuminations.nctm.org
Wealth of lessons and ideas for problem solving on every math topic: official NCTM site
Standard: 4.NBT.2.
Read and write multi-digit whole numbers using base-ten numerals, number names, and expanded form. Compare two
multi-digit numbers based on meanings of the digits in each place, using >, =, and < symbols to record the results of
comparisons.
What students should know and be able to do:
Students will read, write, and compare numbers less than or equal to 1,000,000.
Learning Targets:
 I can read and write multi-digit whole numbers using base-ten numerals, number names, and expanded form.
 I can compare two multi-digit whole numbers based on the value of the digits in each place.
 I can use what <, >, and = symbols to record my comparisons of two multi-digit whole numbers to any place.
Vocabulary:
base-ten numeral (formally known as standard form), number names (formally known as word form), expanded form,
greater than (>), less than (<), equal to (=)
Sample Problem(s):
68
Resources:

http://www.uen.org/Lessonplan/preview?LPid=18917
Reading and writing numbers

http://www.aaastudy.com/cmp.htm#topic3
Comparing numbers

http://nrich.maths.org/public/search.php?search=compare
Comparing numbers
Major Cluster: Generalize place value understanding for multi-digit whole numbers.
Standard: 4.NBT.3.
Use place value understanding to round multi-digit whole numbers to any place.
What students should know and be able to do:
Students will know how to round numbers to any place.
Students will have a deep understanding of place value and number sense and can explain and reason about the
answers they get when they round.
Learning Targets:

I can use what I know about place value to round multi-digit whole numbers up to 1,000,000.
Vocabulary:
estimate, round, about, close to, almost, exact, benchmark, place value, base-ten
69
Sample Problem(s):
Resources:

70
http://www.321know.com/grade4.htm#topic49
Number lines
Major Cluster: Use place value understanding and properties of operations to perform multi-digit arithmetic.
Standard: 4.NBT.4.
Fluently add and subtract multi-digit whole numbers using the standard algorithm.
What students should know and be able to do:
Students will accurately and efficiently add and subtract multi-digit whole numbers using the standard algorithm.
Learning Targets:
 I can fluently add multi-digit whole numbers using the standard algorithm.
 I can fluently subtract multi-digit whole numbers using the standard algorithm.
Vocabulary:
algorithm, sum, difference, total, addend
71
Sample Problem(s):
72
Resources:

http://nlvm.usu.edu/en/nav/frames_asid_154_g_2_t_1.html?from=category_g_2_t_1.html
Base 10 virtual manipulatives
Standard: 4.NBT.5.
Multiply a whole number of up to four digits by a one-digit whole number, and multiply two two-digit numbers, using
strategies based on place value and the properties of operations. Illustrate and explain the calculation by using
equations, rectangular arrays, and/or area models.
What students should know and be able to do:
Students will use base ten blocks, area models, partitioning, compensation strategies, etc. when multiplying whole
numbers and use words and diagrams to explain their thinking. Students will use the properties of operations to
decompose numbers in multi-digit multiplication.
Learning Targets:
I can multiply a number with up to two digits by two digit numbers using strategies based on place value and
properties of operation.
 I can multiply a number with up to four digits by a one-digit whole number using strategies based on place
value and properties of operation.
 I can illustrate and explain calculations using equations, rectangular arrays, and/or area models.
Vocabulary:

product, rectangular array, equation, area model, factors, properties of multiplication, rows, columns, partial products
73
Sample Problem(s):
Resources:
74

http://www.prongo.com/math/multiplication.html
―Ba
tters Up‖ multiplication game

http://eduplace.com/math/mthexp/g4/mathbkg/
Unit 3 Houghton Mifflin math Series

http://eduplace.com/math/mthexp/g5/mathbkg/
Unit 4 Houghton Mifflin Math Series
Standard: 4.NBT.6.
Find whole-number quotients and remainders with up to four-digit dividends and one-digit divisors, using strategies
based on place value, the properties of operations, and/or the relationship between multiplication and division.
Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.
What students should know and be able to do:
Students will use base ten blocks, area models, partitioning, compensation strategies, etc. when dividing whole
numbers and use words and diagrams to explain their thinking. Students will use the properties of operations to
decompose numbers with up to four-digit dividends and one-digit divisors.
Learning Targets:


I can find the whole number quotient of a division problem with up to four-digit dividends and one-digit divisors
using strategies based on place value, properties of operations, and/or the relationship between multiplication
and division.
I can illustrate and explain division calculations using equations, rectangular arrays, and/or area models.
Vocabulary:
divisor, dividend, quotient, remainder, array, area model, inverse operations, properties of operations, product, factor
75
Sample Problem(s):
(cont. on next page)
76
77
Resources:

http://nrich.maths.org/6402
―
The Remainders Game‖

http://www.kidsnumbers.com/long-division.php

http://eduplace.com/math/mthexp/g4/mathbkg/
Unit 4 Houghton Mifflin Math Series
Domain: Numbers and Operations-Fractions
*Expectations in this domain are limited to fractions with denominators 2, 3, 4, 5, 6, 8, 10, 12, and 100.
Enduring Understanding(s): Fractions and decimals allow for quantities to be expressed with greater precision than
with just whole numbers.
Essential Questions: Why express quantities, measurements, and number relationships in different ways?
Major Cluster: Extend understanding of fraction equivalence and ordering.
Standard: 4.NF.1.
Explain why a fraction a/b is equivalent to a fraction (n x a)/(n x b) by using visual fraction models, with attention to
how the number and size of the parts differ even though the two fractions themselves are the same size. Use this
principle to recognize and generate equivalent fractions.
78
What students should know and be able to do:
Students will use visual fraction models to generate equivalent fractions and explain why it is equivalent.
Learning Targets:


I can recognize and generate equivalent fractions by multiplying by a fractional equivalent of one (1/1, 2/2, etc.).
I can use visual fraction models to explain why fractions are equivalent.
Vocabulary:
fraction, equivalent fraction, numerator, denominator, equivalent, number line model, area model, identity property of
multiplication, n x 1 = n
Sample Problem:
(cont. on next page)
79
Resources:
80

hand2mind.com/hosstudent
Student pages for Hands-On Standards Common Core Edition ETA hand2mind

DEA (various probes and resources)

www.IXL.com
(Students can do up to 20 problems for free per session)

http://illuminations.nctm.org
Wealth of lessons and ideas for problem solving on every math topic: official NCTM site
Standard: 4.NF.2.
Compare two fractions with different numerators and different denominators, e.g., by creating common denominators
or numerators, or by comparing to a benchmark fraction such as 1/2. Recognize that comparisons are valid only
when the two fractions refer to the same whole. Record the results of comparisons with symbols >, =, or <, and justify
the conclusions, e.g., by using a visual fraction model.
What students should know and be able to do:
Students will compare fractions by creating visual fraction models or finding common denominators or numerators.
Students will draw fraction models to help them compare and use reasoning skills based on fraction benchmarks.
Learning Targets:
I can compare two fractions with different numerators and denominators by creating common denominators or
numerators.
 I can compare two fractions with different numerators and denominators by comparing them to a benchmark
fraction..
 I can explain that comparisons of fractions are valid only when the fractions refer to the same whole.
 I can record the results of comparisons with symbols >, =, or <, and justify the conclusions.
Vocabulary:

benchmark fractions (thirds, halves, fourths), numerator, denominator, >, =, <, equivalent fractions
81
Sample Problem(s):
(cont. on next page)
82
83
Resources:
84

http://nlvm.usu.edu/en/nav/frames_asid_159_g_2_t_1.html?from=category_g_2_t_1.html
Virtual manipulatives (National Library of Virtual Manipulatives, grades 3-5; Number and operations; Number
line bar)

http://illuminations.nctm.org/LessonDetail.aspx?id=U112
NCTM illuminations—comparing fractions, fractions lessons 4, 5, and 6
Major Cluster: Build fractions from unit fractions by applying and extending previous understandings of
operations on whole numbers.
Standard: 4.NF.3.
Understand a fraction a/b with a > 1 as a sum of fractions 1/b.
a. Understand addition and subtraction of fractions as joining and separating parts referring to the same whole.
b. Decompose a fraction into a sum of fractions with the same denominator in more than one way, recording each
decomposition by an equation. Justify decompositions, e.g., by using a visual fraction model.
Examples: 3/8=1/8+1/8+1/8 ; 3/8=1/8+2/8; 2 1/8=1 + 1+1/8=8/8+8/8 +1/8.
c. Add and subtract mixed numbers with like denominators, e.g., by replacing each mixed number with an equivalent
fraction, and/or by using properties of operations and the relationship between addition and subtraction.
d. Solve word problems involving addition and subtraction of fractions referring to the same whole and having like
denominators, e.g., by using visual fraction models and equations to represent the problem.
What students should know and be able to do:
Students will identify a unit fraction and decompose a non-unit fraction into a combination of several unit fractions.
Students will understand addition and subtraction of fractions as joining and separating parts referring to the same
whole. Students will add and subtract mixed numbers or convert mixed numbers into improper fractions and/or by
using the properties of operations. Students will solve word problems by using visual models and writing equations to
represent addition and subtraction of fractions.
85
Learning Targets:
I can add and subtract fractions by joining and separating parts that refer to the same whole.
I can decompose a fraction into the sum of smaller fractions with the same denominator in more than one way.
I can use my strategies to add and subtract mixed numbers with like denominators.
I can use my strategies to solve word problems involving addition and subtraction of fractions that refer to the
same whole and have like denominators.
Vocabulary:




decompose (decomposition), mixed numbers, fractional form, unit fraction
(Note: We have intentionally excluded the term ―im
proper fraction‖ and instead use the term ―
fractional form.‖)
Sample Problem(s):
(cont. on next page)
86
Resources:
87

http://studyjams.scholastic.com/studyjams/jams/math/fractions/add-sub-common-denom.htm
Study Jams: add and subtract with common denominators

http://www.ncpublicschools.org/docs/acre/standards/support-tools/unpacking/math/4th.pdf
Multiplying fractions
Standard: 4.NF.4.
Apply and extend previous understandings of multiplication to multiply a fraction by a whole number.
a. Understand a fraction a/b as a multiple of 1/b. For example, use a visual fraction model to represent 5/4 as the
product 5x(1/4), recording the conclusion by the equation 5/4 = 5x(1/4).
b. Understand a multiple of a/b as a multiple of 1/b, and use this understanding to multiply a fraction by a whole
number. For example, use a visual fraction model to express 3x(2/5) as 6x(1/5), recognizing this product as 6/5. (In
general, nx(a/b)=(nxa)/b.)
c. Solve word problems involving multiplication of a fraction by a whole number, e.g., by using visual fraction models
and equations to represent the problem. For example, if each person at a party will eat 3/8 of a pound of roast beef, and
there will be 5 people at the party, how many pounds of roast beef will be needed? Between what two whole numbers
does your answer lie?
What students should know and be able to do:
Students will demonstrate their understanding of multiplication as repeated addition using and creating visual fraction
models to multiply a whole number by a fraction. Students will use visual fraction models and equations to solve word
problems involving multiplication of a fraction by a whole number.
Learning Targets:




88
I can use what I know about multiplication to multiply a fraction by a whole number.
I can express any fraction as a multiple of a unit fraction.
I can multiply any fraction by a whole number by creating an equivalent multiplication expression involving a unit
fraction.
I can solve word problems involving multiplication of a fraction by a whole number using visual models and place
value strategies.
Vocabulary:
unit fractions, multiple, fractional form (5/4), mixed number
Sample Problem(s):
Resources:
89

http://nlvm.usu.edu/en/nav/frames_asid_194_g_2_t_1.html?from=category_g_2_t_1.html
Multiplying Fractions

http://www.homeschoolmath.net/teaching/f/multiplying_fractions_1.php
Multiplying Fractions by whole numbers: free video
Major Cluster: Understand decimal notation for fractions, and compare decimal fractions.
Standard: 4.NF.5.
Express a fraction with denominator 10 as an equivalent fraction with denominator 100, and use this technique to add
two fractions with respective denominators 10 and 100. For example, express 3/10 as 30/100, and add 3/10 + 4/100 =
34/100.
Students who can generate equivalent fractions can develop strategies for adding fractions with unlike denominators in
general.
What students should know and be able to do:
Students will change fractions with a 10 in the denominator into equivalent fractions that have a 100 in the denominator
and add them. But addition and subtraction with unlike denominators in general is not a requirement at this grade.
Learning Targets:
 I can express a fraction with a denominator of 10 as an equivalent fraction with a denominator of 100.
 I can add two fractions when one of the fractions has a denominator of 10 and the other has a denominator of 100.
Vocabulary:
common denominator, equivalent fraction
90
Sample Problem(s):
(cont. on next page)
91
Resources:
92

hand2mind.com/hosstudent
Student pages for Hands-On Standards Common Core Edition ETA hand2mind

DEA (various probes and resources)

www.IXL.com
(Students can do up to 20 problems for free per session)

http://illuminations.nctm.org
Wealth of lessons and ideas for problem solving on every math topic: official NCTM site
Standard: 4.NF.6.
Use decimal notation for fractions with denominators 10 or 100. For example, rewrite 0.62 as 62/100; describe a
length as 0.62 meters; locate 0.62 on a number line diagram.
What students should know and be able to do:
Students will convert fractions with denominators of 10 and 100 into a decimal and represent such values on a number
line.
Learning Targets:
 I can write a fraction with a denominator of 10 or 100 as a decimal.
 I can locate a decimal on a number line.
Vocabulary:
tenths, hundredths, decimal, equivalent fraction
93
Sample Problem(s):
Resources:
hand2mind.com/hosstudent
Student pages for Hands-On Standards Common Core Edition ETA hand2mind
 DEA (various probes and resources)
 www.IXL.com
(Students can do up to 20 problems for free per session)
 http://illuminations.nctm.org
Wealth of lessons and ideas for problem solving on every math topic: official NCTM site

94
Standard: 4.NF.7.
Compare two decimals to hundredths by reasoning about their size. Recognize that comparisons are valid only when
the two decimals refer to the same whole. Record the results of comparisons with the symbols >, =, or <, and justify
the conclusions, e.g., by using a visual model.
What students should know and be able to do:
Students will be able to understand that comparisons between decimals or fractions are only valid when the whole is
the same for both cases. Students will compare two decimals to the hundredths using the symbols >, =, or <, and
justify their conclusions by using a model.
Learning Targets:



I can compare two decimals to hundredths by reasoning about their size.
I can recognize that comparisons of decimals are valid only when they refer to the same whole.
I can use >, =, and < symbols to record my comparisons of two decimals and justify my conclusions.
Vocabulary:
decimal, tenth, hundredth, fraction, equivalent, >, =,<.
95
Sample Problem(s):
Resources:

96
http://nlvm.usu.edu/en/nav/frames_asid_264_g_2_t_1.html?from=category_g_2_t_1.html
Can be used to compare, although intended for addition and subtraction
Domain: Measurement and Data
Enduring Understanding(s): Measurement processes are used in everyday life to describe and quantify the world.
Data displays describe and represent data in alternative ways.
Essential Questions: Why does ―
what‖ we measure influence ―
how‖ we measure? Why display data in different ways?
Supporting Cluster: Solve problems involving measurement and conversion of measurements from a larger
unit to a smaller unit.
Standard: 4.MD.1.
Know relative sizes of measurement units within one system of units including km, m, cm; kg, g; lb, oz.; l, ml; hr, min,
sec. Within a single system of measurement, express measurements in a larger unit in terms of a smaller unit.
Record measurement equivalents in a two-column table. For example, know that 1 ft is 12 times as long as 1 in.
Express the length of a 4 ft snake as 48 in. Generate a conversion table for feet and inches listing the number pairs
(1, 12), (2, 24), (3, 36), …
What students should know and be able to do:
Students will know relative sizes of measurement units within one system of units and convert from a larger unit to a
smaller unit. Students should be able to record their measurement equivalence in a two column table.
Learning Targets:




97
I can describe the relationship between sizes of measurement units in the same measurement system.
I can convert measurements from larger units to smaller units within the same measurement system.
I can record equivalent measurements in a two-column table.
I can use a conversion table to create ordered pairs.
Vocabulary: Refer to vocabulary terms within the Standard description.
Sample Problem(s):
98
Resources:

Measurement tools including, but not limited to, rulers, balances, cups, weights, beakers and clocks.
http://www.harcourtschool.com/activity/con_math/g04c24.html
.
 http://www.jmathpage.com/JIMSMeasurementpage.html
Several measurement activities

Standard: 4.MD.2.
Use the four operations to solve word problems involving distances, intervals of time, liquid volumes, masses of
objects, and money, including problems involving simple fractions or decimals, and problems that require expressing
measurements given in a larger unit in terms of a smaller unit. Represent measurement quantities using diagrams
such as number line diagrams that feature a measurement scale.
What students should know and be able to do:
Students will be able to solve word problems using the four operations involving simple fractions or decimals and
conversions of measurement from a larger unit to a smaller unit. Word problems involve distance, intervals of time,
liquid volumes, masses of objects, and money and require students to represent measurement quantities using
diagrams.
Learning Targets:


I can use the four operations to solve word problems involving simple fractions, money with whole numbers and
decimals, liquid volumes, masses of objects, distances, and converting measurements from larger to smaller
units.
I can represent measurements with diagrams like number lines that have a measurement scale.
Vocabulary:
distance, time, interval, volume, mass, number line, scale, diagram
99
Sample Problem(s):
(cont. on next page)
100
Resources:
101

http://www.thatquiz.org/tq/previewtest?A/O/Z/P/83711291417153
Customary Measurement word problems

http://www.helpingwithmath.com/by_subject/word_problems/wor_measurement01_4md2.htm
Distance , time, volume, mass word problems
Standard: 4.MD.3.
Apply the area and perimeter formulas for rectangles in real world and mathematical problems. For example, find the
width of a rectangular room given the area of the flooring and the length, by viewing the area formula as a
multiplication equation with an unknown factor.
What students should know and be able to do:
Students will be able to understand and apply the area and perimeter formulas for rectangles in real world and
mathematical problems.
Learning Targets:

I can use a formula to find the perimeter and area of a rectangle in real world and mathematical problems.
Vocabulary:
rectangle, perimeter, area, array, unit, square units
Sample Problem(s):
The area of a room is 32 square units. The length is 4 units. What is the width?
A square room has a perimeter of 36 units. What are the lengths of the sides? How do you know?
Resources:
http://nrich.maths.org/6923
Growing rectangles
 http://nrich.maths.org/2663
Perimeters

102
Supporting Cluster: Represent and interpret data.
Standard: 4.MD.4.
Make a line plot to display a data set of measurements in fractions of a unit (1/2, 1/4, 1/8). Solve problems involving
addition and subtraction of fractions by using information presented in line plots. For example, from a line plot find
and interpret the difference in length between the longest and shortest specimens in an insect collection.
What students should know and be able to do:
Students will create a line plot to display measurements in fractions of a unit (1/2, ¼, 1/8) and add and subtract
fractions based on that data.
Learning Targets:


I can make a line plot that displays measurements in fractions of a unit (1/2, ¼, 1/8).
I can solve addition and subtraction problems using information from a line plot.
Vocabulary:
line plot, fraction, measurement, data, data set, unit
103
Sample Problem(s):
104
Resources:

http://www.uen.org/3-6interactives/math.shtml#fractions
Interactive fraction games
Additional Cluster: Geometric measurement: understand concepts of angle and measure angles.
Standard: 4.MD.5.
Recognize angles as geometric shapes that are formed wherever two rays share a common endpoint, and understand
concepts of angle measurement:
a. An angle is measured with reference to a circle with its center at the common endpoint of the rays, by considering
the fraction of the circular arc between the points where the two rays intersect the circle. An angle that turns through
1/360 of a circle is called a ―on
e-degree angle,‖ and can be used to measure angles.
b. An angle that turns through n one-degree angles is said to have an angle measure of n degrees.
What students should know and be able to do:
Students will recognize angles and describe how they are formed. Students will describe the measurement of the
angle in degrees and its relation to a fraction of a 360° circle.
105
Learning Targets:
 I can describe how angles are formed.
 I can explain how angles are measured.
 I can explain how the measurement of an angle relates to a fraction of a 360° circle.
 I can describe how a ―
one-degree angle‖ is used to measure angles.
 I can express an angle measurement in terms of the number of one-degree angles in that angle.
Vocabulary:
intersection, circular arc, angle, vertex, point of origin, circle, ray, degree, circular interior, circular exterior, end point
106
Sample Problem(s):
107
Resources:

http://www.teachertube.com/viewVideo.php?title=angles_in_a_circle&video_id=231281
Video showing angles growing by n degrees—good for teacher background ONLY

http://www.ixl.com/math/grade/4
Math quiz using angles and degrees

http://www.mathisfun.com/angles.html
Finding degrees of angles
Standard: 4.MD.6.
Measure angles in whole-number degrees using a protractor. Sketch angles of specified measure.
What students should know and be able to do:
Students will be able to create and measure angles using a protractor.
Learning Targets:
 I can use a protractor to measure an angle in whole-number degrees.
 I can sketch angles of a specified measure.
Vocabulary:
°
angle, degree, ray, degree symbol , protractor
108
Sample Problem(s):
Resources:

109
http://www.mathopenref.com/
Reference source for math vocabulary
Standard: 4.MD.7.
Recognize angle measure as additive. When an angle is decomposed into non-overlapping parts, the angle measure
of the whole is the sum of the angle measures of the parts. Solve addition and subtraction problems to find unknown
angles on a diagram in real world and mathematical problems, e.g., by using an equation with a symbol for the
unknown angle measure.
What students should know and be able to do:
Students will recognize that the angle measurement of a larger angle is the sum of the angle measures of its
decomposed parts. Students will add and subtract to solve real world and mathematical problems containing missing
angle measurements and write an equation with an unknown angle measurement.
Learning Targets:
I can find an angle measure by adding the measurements of the smaller angles that make up the larger angle.
I can use my addition and subtraction strategies to solve for an unknown angle on a diagram, in real-world and
mathematical problems.
Vocabulary:


angle, protractor, sum, degree (and symbol °), acute angle, obtuse angle, straight angle, right angle, angle measure,
perpendicular
110
Sample Problem(s):
(cont. on next page)
111
Resources:
http://www.mathsisfun.com/geometry/complementary-angles.html
Complimentary angles
 http://www.mathsisfun.com/geometry/supplementary-angles.html
Complimentary/supplementary angles
 http://www.khanacademy.org/video/complementary-and-supplementary-angles?playlist=Geometry
Complimentary/supplementary angles video
 Sir Cumference and the Great Knight of Angleland, by Cindy Neuschwander
(great book for introducing circumference and angles)

112
Domain: Geometry
Enduring Understanding(s): Geometric attributes (such as shapes, lines, angles, figures, and planes) provide
descriptive information about an object’s properties and position in space and support visualization and problem
solving.
Essential Questions: How does geometry better describe objects?
Additional Cluster: Draw and identify lines and angles, and classify shapes by
properties of their lines and angles.
Standard: 4.G.1.
Draw points, lines, line segments, rays, angles (right, acute, obtuse), and perpendicular and parallel lines. Identify
these in two-dimensional figures.
What students should know and be able to do:
Students will draw points, lines, line segments, rays, angles, and perpendicular and parallel lines and identify them in
two-dimensional figures.
Learning Targets:
I can draw points, lines, line segments, rays, right angles, acute angles, obtuse angles, perpendicular and
parallel lines.
 I can identify points, lines, line segments, rays, right angles, acute angles, obtuse angles, perpendicular and
parallel lines in two-dimensional figures.
Vocabulary:

point, line, line segment, ray, angle, obtuse, acute, right, parallel, perpendicular, two-dimensional, figure, ║, ┴, <
113
Sample Problem(s):
114
Resources:


http://nrich.maths.org/public/viewer.php?obj_id=312&part=
http://www.geogebra.org/cms/
―be
ndy sides‖ problem to find all possible values of a quadrilateral’s angles if its sides are of unequal length
Standard: 4.G.2.
Classify two-dimensional figures based on the presence or absence of parallel or perpendicular lines, or the presence
or absence of angles of a specified size. Recognize right triangles as a category, and identify right triangles.
What students should know and be able to do:
Students will classify two-dimensional figures and right triangles based on their characteristics.
Learning Targets:
 I can classify a two-dimensional figure based on whether or not it has perpendicular or parallel lines.
 I can classify a two-dimensional figure based on the size of its angles.
 I can classify triangles as right triangles based on their characteristics.
Vocabulary:
classify, right triangle, category, parallel line, perpendicular line, acute angle, obtuse angle, right angle, presence,
absence, two-dimensional figure
115
Sample Problem(s): (cont. on next page
116
Resources:

http://www.uen.org/Lessonplan/preview.cgi?LPid=11235
Lesson plan from Utah Education Network: Identify and describe attributes of two-dimensional shapes

http://illuminations.nctm.org/LessonDetail.aspx?ID=L270
―Po
lygon Capture‖ game including all instructions from NCTM illuminations site
Standard: 4.G.3.
Recognize a line of symmetry for a two-dimensional figure as a line across the figure such that the figure can be
folded along the line into matching parts. Identify line-symmetric figures and draw lines of symmetry.
What students should know and be able to do:
Students will define, explain how to identify, and draw lines of symmetry in two-dimensional figures.
Learning Targets:
 I can identify lines of symmetry on a two-dimensional figure.
 I can identify figures that have line symmetry.
 I can draw lines of symmetry on a two-dimensional figure.
 I can recognize a line of symmetry when a figure can be folded along the line into matching parts.
Vocabulary:
symmetry, two-dimensional, matching parts, symmetrical, line, line of symmetry, congruent
117
Sample Problem(s):
Resources:
118

http://www.linkslearning.org/Kids/1_Math/2_Illustrated_Lessons/4_Line_Symmetry/index.html
Line symmetry problem

http://www.innovationslearning.co.uk/subjects/maths/activities/year3/symmetry/shape_game.asp
Symmetry Game
Grade 5 Standards
Mathematics | Grade 5 – Critical Areas
In Grade 5, instructional time should focus on three critical areas: (1) developing fluency with addition and
subtraction of fractions, and developing understanding of the multiplication of fractions and of division of
fractions in limited cases (unit fractions divided by whole numbers and whole numbers divided by unit fractions);
(2) extending division to 2-digit divisors, integrating decimal fractions into the place value system and
developing understanding of operations with decimals to hundredths, and developing fluency with whole number
and decimal operations; and (3) developing understanding of volume.
(1) Students apply their understanding of fractions and fraction models to represent the addition and subtraction of fractions with
unlike denominators as equivalent calculations with like denominators. They develop fluency in calculating sums and differences of
fractions, and make reasonable estimates of them. Students also use the meaning of fractions, of multiplication and division, and the
relationship between multiplication and division to understand and explain why the procedures for multiplying and dividing fractions
make sense. (Note: this is limited to the case of dividing unit fractions by whole numbers and whole numbers by unit fractions.)
(2) Students develop understanding of why division procedures work based on the meaning of base-ten numerals and properties of
operations. They finalize fluency with multi-digit addition, subtraction, multiplication, and division. They apply their understandings of
models for decimals, decimal notation, and properties of operations to add and subtract decimals to hundredths. They develop
fluency in these computations, and make reasonable estimates of their results. Students use the relationship between decimals and
fractions, as well as the relationship between finite decimals and whole numbers (i.e., a finite decimal multiplied by an appropriate
power of 10 is a whole number), to understand and explain why the procedures for multiplying and dividing finite decimals make
sense. They compute products and quotients of decimals to hundredths efficiently and accurately.
(3) Students recognize volume as an attribute of three-dimensional space. They understand that volume can be measured by finding
the total number of same-size units of volume required to fill the space without gaps or overlaps. They understand that a 1-unit by 1unit by 1-unit cube is the standard unit for measuring volume. They select appropriate units, strategies, and tools for solving
problems that involve estimating and measuring volume. They decompose three-dimensional shapes and find volumes of right
rectangular prisms by viewing them as decomposed into layers of arrays of cubes. They measure necessary attributes of shapes in
order to determine volumes to solve real world and mathematical problems.
The Common Core State Standards
119
Grade 5 Overview
Operations and Algebraic Thinking
• Write and interpret numerical expressions.
• Analyze patterns and relationships.
Number and Operations in Base Ten
• Understand the place value system.
• Perform operations with multi-digit whole numbers and with decimals to hundredths.
Number and Operations—Fractions
• Use equivalent fractions as a strategy to add and subtract fractions.
• Apply and extend previous understandings of multiplication and division to multiply and divide fractions.
Measurement and Data
• Convert like measurement units within a given measurement system.
• Represent and interpret data.
• Geometric measurement: understand concepts of volume and relate volume to multiplication and to addition.
Geometry
• Graph points on the coordinate plane to solve real-world and mathematical problems.
• Classify two-dimensional figures into categories based on their properties.
The Common Core State Standards
120
Mathematical Common Core Content State Standards
Grade 5
Domain: Operations and Algebraic Thinking - 5.OA
Enduring Understanding(s):
Mathematical operations are used in solving problems in which a new value is produced from one or more values.
Algebraic thinking involves choosing, combining, and applying effective strategies for answering quantitative questions.
Essential Questions:
In what ways can operations affect numbers?
How can different strategies be helpful when solving a problem?
Additional Cluster: Write and interpret numerical expressions.
Standard: 5.OA.1
Use parentheses, brackets, or braces in numerical expressions, and evaluate expressions with these symbols.
121
What students should know and be able to do:
Students will understand:
 what numerical expressions are (e.g., 3 + a ).
 the steps of the order of operations with parentheses, brackets, or braces.
Students are able to:
 use expressions with symbols or models to represent the four operations.



use physical models, pictures, drawings, etc. to represent numerical expressions and their solutions.
evaluate expressions using the order of operations with parentheses, brackets, or braces.
use numerical expressions to evaluate problems.
Vocabulary:
expression, numeric expression, parentheses, brackets, braces, operation, order of operations, symbols, ( ), { }, [ ]
Sample Problem(s):
(From Illustrative Mathematics)
MATERIALS:
4 dice per team
122
Recording sheet
Two-minute timer for each turn
ACTION:
Have students work in groups of 2 - 4. Introduce the game with an example, and then have them play independently. Discussion of "challenging
rolls" afterwards can be productive.
Students roll the 4 dice to generate their seed numbers. They then use those 4 numbers to create as many numbers as they can (1 - 10).
Scoring is done as in bowling; numbered "pins" are "knocked down" by creating an expression equal to the number.
The game can be structured in two different ways to assure that students are checking each other's expressions and verifying that they are written
as intended:
a. During a student's turn, have them record just the expressions (not the intended result), and then pass the set to another student (a
judge). That judge then computes each expression as written and records which pins were knocked down.
b. Have the students play in teams. Each team tries to achieve a "strike" (knocking down all of the pins, which is almost always possible).
Striving for the strike encourages students to brainstorm strategies for the "difficult" numbers, which leads them to discuss parts of each
expression they have created already.
123
124
Commentary:
The purpose of this game is to help students think flexibly about numbers and operations and to record multiple operations using proper notation.
Students eager to knock down all of the pins quickly develop patterns in their expressions. They may re-use parts of an expression, perhaps
changing just the final operation; for example, if the dice showed 1, 2, 5, 5, they might write:
(2 + 1) + (5 ÷ 5) = 4
(2 + 1) − (5 ÷ 5) = 2
(2 + 1) × (5 ÷ 5) = 3
Or they might change one of the internal operations:
(2 + 1) × (5 ÷ 5) = 3
(2 − 1) × (5 ÷ 5) = 1
(2 × 1) × (5 ÷ 5) = 2
Similar-but-different expressions like these emphasize the importance of parentheses and the flexibility they give us in creating expressions with
very specific meanings.
125
Solution: 1 Example of a student worksheet that has been filled out:
126
Example:
127
Evaluate the expression 2{ 5[12 + 5(500 - 100) + 399]}
Students should have experiences working with the order of first evaluating terms in parentheses, then brackets, and then braces.
The first step would be to subtract 500 – 100 = 400. Then multiply 400 by 5 = 2,000.
Inside the bracket, there is now [12 + 2,000 + 399]. That equals 2,411.
Next multiply by the 5 outside of the bracket. 2,411 x 5 = 12,055.
Next multiply by the 2 outside of the braces. 12,055 x 2= 24,110.
Mathematically, there cannot be brackets or braces in a problem that does not have parentheses. Likewise, there cannot be braces in a problem
that does not have both parentheses and brackets.
128
Resources:

http://www.yuureka.com
manipulatives to use for each standards

http://illustrativemathematics.org/
sample problems

http://www.jefferson.kyschools.us/Departments/Gheens/Curriculum%20Maps/2012_2013/INS_Elem_Math.html
scroll to math planning 1st unit of study.

http://illuminations.nctm.org
lessons and activities

http://www.graniteschools.org/depart/teachinglearning/curriculuminstruction/math/Pages/MathematicsVocabulary.aspx
vocabulary
Standard: 5.OA.2
Write simple expressions that record calculations with numbers, and interpret numerical expressions without evaluating them. For example,
express the calculation ―
add 8 and 7, then multiply by 2‖ as 2 x (8 + 7). Recognize that 3 x (18932 + 921) is three times as large as 18932 + 921,
without having to calculate the indicated sum or product.
129
What students should know and be able to do:
Students will understand:
 the word ―th
en‖ implies one operation happens after another and parentheses are used to indicate the order of operations.
Students will be able to:
 write a real-world problem as an expression.
 write an expression and apply it to a real world problem recognize that 3 × (18,932 + 921) is three times as large as 18,932 +
921, without having to solve.
 recognize that 3(18,932 + 921) means the same thing as 3 x (18, 932 + 921).
 write expressions using the correct numerical and symbolic notation in the proper order.

use numerical and symbolic notation to represent an expression from a problem.
Learning Targets:



I can write simple expressions to record calculations and real world problems.
I can interpret expressions without evaluating them.
I can compare expressions without solving them.
130
Vocabulary:
expression, parentheses, bracket, brace, order of operations, terminology for operations (e.g. sum, add, multiply, difference), ―t
hen‖
Sample Problem(s):

Sample 1:
Write an expression for the steps ―
double five and then add 26.‖
Student response: (2 x 5) + 26

Sample 2:
Describe how the expression 5 (10 x 10) relates to 10 x 10.
Student response:
The expression 5 (10 x 10) is 5 times larger than the expression 10 x 10 since I know that 5 (10 x 10) means that I have 5 groups of (10 x 10).
131
Resources:

http://www.yuureka.com
manipulatives to use for each standards

http://illustrativemathematics.org/
sample problems

http://www.jefferson.kyschools.us/Departments/Gheens/Curriculum%20Maps/2012_2013/INS_Elem_Math.html
scroll to math planning 1st unit of study.

http://illuminations.nctm.org
lessons and activities

http://www.graniteschools.org/depart/teachinglearning/curriculuminstruction/math/Pages/MathematicsVocabulary.aspx
vocabulary
132
Additional Cluster: Analyze patterns and relationships.
Standard: 5.OA.3
Generate two numerical patterns using two given rules. Identify apparent relationships between corresponding terms. Form ordered pairs
consisting of corresponding terms from the two patterns, and graph the ordered pairs on a coordinate plane. For example, given the rule ―
Add
3‖ and the starting number 0, and given the rule ―
Add 6‖ and the starting number 0, generate terms in the resulting sequences, and observe
that the terms in one sequence are twice the corresponding terms in the other sequence. Explain informally why this is so.
What students should know and be able to do:
Students will understand:
 how a coordinate plane is formed (e.g., origin, x-axis, y-axis).
 numerical patterns.
 conclusions based on identified patterns.
 how patterns can be represented on a coordinate plane.
 how numeric patterns can be represented with ordered pairs on a coordinate plane.
Students are able to:
 graph points in Quadrant I.
 plot points on a coordinate plane.
 perform basic operations within numeric patterns .
 record numerical patterns.
 generate ordered pairs from two numerical patterns.
133
Learning Targets:



I can generate two numerical patterns using two given rules.
I can identify apparent relationships between corresponding terms.
I can form ordered pairs consisting of corresponding terms from the two numeric patterns, and graph the ordered pairs on a
coordinate plane.
Vocabulary:
corresponding terms, coordinate plane, ordered pair, coordinates, numeric pattern, relationship, graph, origin, x-axis, y-axis
Sample Problem(s):
134
Example:
Describe the pattern:
Since Terri catches 4 fish each day, and Sam catches 2 fish, the amount of Terri’s fish is always greater. Terri’s fish is also always twice as
much as Sam’s fish. Today, both Sam and Terri have no fish. They both go fishing each day. Sam catches 2 fish each day. Terri catches 4 fish
each day. How many fish do they have after each of the five days? Make a graph of the number of fish.
Resources:

http://www.jefferson.kyschools.us/Departments/Gheens/Curriculum%20Maps/2012_2013/INS_Elem_Math.html

http://katm.org/wp/wp-content/uploads/flipbooks/5th-Flipbookedited2.pdf
Domain: Number and Operations in Base Ten
Enduring Understanding(s):
Understanding place value can lead to number sense and efficient strategies for computing with numbers.
Essential Questions:
How does a digit’s position affect its value?
135
Major Cluster: Understand the place value system.
Standard: 5.NBT.1
Recognize that in a multi-digit number, a digit in one place represents 10 times as much as it represents in the place to its right and 1/10 of
what it represents in the place to its left.
What students should know and be able to do:
Students will understand:


the value of each digit in the base 10 system.
know that the value of a digit within a number increases or decreases when multiplied or divided by ten in the base ten system.
Students will be able to:
 make number line representations of numbers, including decimal values.
 model whole numbers and parts of whole numbers with drawings, base ten blocks, and other concrete models.
 read and name the place value of digits in multi-digit numbers including decimals.
 recognize the change in place value position when multiplying and dividing by 10.
Learning Targets:



I can explain how the value of a digit in a multi-digit number relates to the value of the digits around it.
I can recognize that a digit in one place represents 10 times as much as it represents in the place to its right.
I can recognize that a digit in one place represents 1/10 of what it represents in the place to its left.
136
Vocabulary:
base ten system, decimal, names of the place values (tenth, hundredth, thousandth), digit, numeral
Sample Problem(s):

Sample 1
Steve and Dominique were having a discussion with Maria about the number 3,109.
Steve said, ―
There are no tens, so I just need to go to the ones place to get a ten.‖
Dominique said, ―
You can’t do that. There has to be ten ones.‖
Who do you agree with? Why? Support your answer with evidence.

Sample 2
Have students find the difference in place value between the digit 2 in the number 0.542 and the digit 2 in the number 0.324.
What is the difference in place value between the digit 2 in the number 0.542 and the digit 2 in the number 0.324?
Have students choose their manipulative to show the difference in the digit 2 for each numeral.
137
Resources:

http://www.jefferson.kyschools.us/Departments/Gheens/Curriculum%20Maps/2012_2013/INS_Elem_Math.html

http://intermediateelem.wikispaces.com/Fifth+Grade+Math+Resource+Backup

Refer to Hands on Standards, Common Core Edition Grade 5 pg. 8-11.

National Library of Virtual Manipulatives, number line with decimals.

http://nlvm.usu.edu/en/nav/frames_asid_334_g_2_t_1.html?from=category_g_2_t_1.htm
Standard: 5.NBT.2
Explain patterns in the number of zeros of the product when multiplying a number by powers of 10, and explain patterns in the
placement of the decimal point when a decimal is multiplied or divided by a power of 10. Use whole number exponents to denote
powers of 10.
138
What students should know and be able to do:
Students will understand:
 relationships between digits in multi-digit numbers.
 the value of a digit within a number increases when moved to the left and decreases as the number moves to the
right in the base ten system.
 an exponent indicates the number of times a base is multiplied by itself.
Students are able to:
 represent numbers with base ten blocks or other representations.
 read and name the place value of digits in multi-digit numbers, including decimals.
 determine the value of a digit if it is moved left or right. For example, the number 19 can be written as 19 ones or 190
tenths.
Learning Targets:



I can multiply and divide by powers of ten.
I can explain patterns in the placement of the decimal point when a decimal is multiplied or divided by a power of ten.
I can use whole number exponents to denote powers of 10.
Vocabulary:
139
base ten system, names of the place value (tenths, hundredths, thousandths), powers of 10, whole number exponents, product
Sample Problem(s):

Sample 1
1(10,000) + 2(1,000) + 4(100) + 3(10) + 2(1) + 5(1/10) + 3(1/100).
Which number below is one-tenth of the expanded form above?
A 12422.53
B 1243.253
C 12432.53
D 12432.43
Retrieved from www.p12.nysed.gov

Sample 2
High Roller Revisited
Directions:
• Players will work in groups to play the game ―
High Roller Revisited.‖
• Roll the die 4 times. After each roll, decide where to record the digit on the place value chart.
• Use the 4 digits to make the greatest number possible.
140
• Once a digit is recorded, players may not make changes to their number.
• Pass the die to the next player and continue to play.
• Compare numbers. The player with the highest number wins the round.
• Play 5 rounds. The player who wins the most rounds wins the game.
Round
Ones
.
Tenths
1.
.
2.
.
3.
.
4.
.
5.
.
Hundredths Thousandths
Retrieved from https://www.georgiastandards.org/CommonCore/Common%20Core%20Frameworks/CCGPS_Math_5_Unit2FrameworkSE.pdf

Sample 3
Martha earned $4.20 each day for ten days of babysitting. Over a year’s time, she worked ten times ten days. Write an
141
expression using exponents of 10 to show how much she earned in ten days, then in 10 times ten days. How much did she
earn in one year? Justify your answer.
Resources:

http://www.mathplayground.com/howto_dividedecimalspower10.html

Printable worksheet for additional practice or quick mastery assessment

http://mrmaffesoli.com/Printables/5NBT2/index.html
Standard: 5.NBT.3
Read, write, and compare decimals to thousandths.
a. Read and write decimals to thousandths using base-ten numerals, number names, and
expanded form, e.g., 347.392 = 3 × 100 + 4 × 10 + 7 × 1 + 3 × (1/10) + 9 × (1/100) + 2 ×
(1/1000).
b. Compare two decimals to thousandths based on meanings of the digits in each place, using
>, =, and < symbols to record the results of comparisons.
142
What students should know and be able to do:
Students will understand:
 the value of decimal numbers as compared to whole numbers such as 0, 0.5 and 1.
 the value of digits to the thousandths place.
Students are able to:


compare decimal numbers to the thousandths place using the symbols (<, =, >).
write numbers in expanded form to the thousandths place.

compare decimals to thousandths place by using value charts, grids, manipulatives and technology, etc.

represent multi-digit numbers in expanded form (such as 435 as 400 + 30 + 5).
Learning Targets:


I can read and write decimals to the thousandths place using base ten numerals, number word names and expanded
form.
I can compare decimal numbers to the thousandths place using the symbols (<, =, >).
Sample Problems:
Example:
Some equivalent forms of 0.72 are:
143








72/100
7/10 + 2/100
7 x (1/10) + 2 x (1/100)
0.70 + 0.02
70/100 + 2/100
0.720
7 x (1/10) + 2 x (1/100) + 0 x (1/1000)
720/1000
Students need to understand the size of decimal numbers and relate them to common benchmarks such as 0, 0.5
(0.50 and 0.500), and 1. Comparing tenths to tenths, hundredths to hundredths, and thousandths to thousandths is simplified if students
use their understanding of fractions to compare decimals.
Example:
Comparing 0.25 and 0.17, a student might think, ―
25 hundredths is more than 17 hundredths‖. They may also think that it is 8
hundredths more. They may write this comparison as 0.25 > 0.17 and recognize that 0.17 < 0.25 is another way to express this
comparison. Comparing 0.207 to 0.26, a student might think, ―
Both numbers have 2 tenths, so I need to compare the hundredths. The
second number has 6 hundredths and the first number has no hundredths so the second number must be larger. Another student might
think while writing fractions, ―
I know that 0.207 is 207 thousandths (and
may write 207/1000). 0.26 is 26 hundredths (and may write 26/100) but I can also think of it as 260 thousandths (260/1000). So, 260
thousandths is more than 207 thousandths.
144
Vocabulary:
base ten numerals, decimals, number word names, expanded form, symbols <, =, >, tenth, hundredth, thousandth
Resources:

www.k-5mathteachingresources.com
Match the standard form to the expanded form using activities such as those found on the website for ―
Expanded
notation games.‖ (See resources for ―
Expanded notation games.‖)
 http://www.ehow.com/list_5880741_math-games-expanded-notation.html
Expanded notation games:
Standard: 5.NBT.4
Use place value understanding to round decimals to any place.
What students should know and be able to do:
Students will understand:
 the rules and procedures for rounding decimals to any targeted value.
Students are able to:
 model rounding of decimals with pictures or manipulatives.
 place decimal values on number lines.
 round decimals to any place value.
145
Learning Targets:
I can use place value understanding to round decimals to any place.

Vocabulary:
rounding, base ten system, decimals
Sample Problem(s):
Sample 1

Ellen wanted to buy the following items: A DVD player for $49.95, a DVD holder for $19.95 and a personal stereo for $21.95. Does
Ellen have enough money to buy all three items if she has $90 with her?

Sample 2
If the rounded number is 9.6, what could the original number have been?
146
Resources:

https://sites.google.com/a/.../nbt-3-read-write-compare-decimals

http://www.numbernut.com/advanced/activities/estimate_mem 20_round10th.shtm
Major Cluster: Perform operations with multi-digit whole numbers and with decimals to hundredths.
Standard: 5.NBT.5
Fluently multiply multi-digit whole numbers using the standard algorithm.
What students should know and be able to do:
Students will understand:
 that multiplication is repeated addition
Students are able to:
 model multiplication of multi-digit multiplication using set, array, and area models
 multiply two-digit numbers using strategies based on place value and the properties of operations
 apply multiplication facts fluently
 calculate multi-digit addition and multiplication to solve multi-digit multiplication problems
147
Learning Targets:


I can represent multi-digit multiplication with arrays, diagrams, area models and manipulatives, etc.
I can use the standard algorithm to calculate the product of multi-digit whole numbers.
Vocabulary:
multiply, multi-digit whole numbers, algorithm, factor, product, multiple, array, product, factor, area model, equal groups, multiples,
.
algorithm, partial products, multiplication symbols [3 x a, 3 a, 3 * a, 3(a), 3a]
148
Sample Problem(s):
149
Resources:

www.internet4classrooms.com › ... › Mathematics › 5th Grade

http://emccss.everydaymathonline.com/g_login.html
Standard: 5.NBT.6
Find whole-number quotients of whole numbers with up to four-digit dividends and two-digit divisors, using strategies based on place
value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by
using equations, rectangular arrays, and/or area models.
What students should know and be able to do:
Students will understand:
 properties of operations. For example, commutative, associative, identity, etc.
 both quotative (repeated subtraction of the divisor) and partitive (partition into groups) division.
 the significance of place value.
Students are able to:
 model division with equations, arrays, and area models.
 find the whole-number quotient of whole numbers with up to four-digit dividends and two-digit divisors.
150
Learning Targets:



I can divide up to a four-digit by a two-digit number using place value strategies.
I can divide up to a four-digit by a two-digit number using the properties of operations
I can divide up to a four-digit by a two-digit number using the relationship between multiplication and division.
Vocabulary:
quotient, dividend, divisor, properties of operations, multiplication, division, rectangular arrays, area models, equations, partitive,
quotative, arrays, division notation (�� , ���, � ÷ �, b) a )
151
Sample Problem(s):
152
Resources:

www.k-5mathteachingresources.com

Common Core State Standards document page 89 ―Comm
on Multiplication and Division Situations‖ Understanding
Division (Section 2):

http://www.conceptualstudy.org/Elementary Math/Understanding Division.htm
Standard: 5.NBT.7.
Add, subtract, multiply, and divide decimals to hundredths, using concrete models or drawings and strategies based on place value,
properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain
the reasoning used.
153
What students should know and be able to do:
Students will understand:


the relationship between all operations (+ - x ÷) and their place value when working with decimals
the properties of operations in relationship to adding, subtracting, multiplying, and dividing decimals.
Students are able to:




add, subtract, multiply and divide decimals using concrete models or drawings.
fluently add, subtract, multiply, and divide decimals using standard algorithms.
accurately represent multi-digit numbers with concrete models and drawings.
relate the strategy to a written method and explain the reasoning used.
Learning Targets:




154
I can add, subtract, multiply, and divide decimals to hundredths.
I can divide decimals by 0.1 and 0.01 to build understanding of the place value significance in division of decimal numbers.
I can use concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship
between addition and subtraction.
I can relate the strategy to a written method and explain the reasoning used.
Vocabulary:
.
add, subtract, multiply, divide, decimal, place value, properties of operations, operation notations, multiplication symbols [3 x a, 3 a,
3 * a, 3(a), 3a] b) a ), division notation (�� , ���, � ÷ �, a÷b )
Sample Problem(s):
1. I divided 6.12 by 3 and got the quotient 2.4. What did I do wrong? Give a similar problem where I might make the same error.
Calculate the following and show your work.
1.
2.
3.
4.
3.4 + 6.2
7.7 – 4.1
5.6 × 2.4
8.4 ÷ 2.1
Examples:
 3.6 + 1.7
A student might estimate the sum to be larger than 5 because 3.6 is more than 3 . and 1.7 is more than 1 ..
 5.4 – 0.8
A student might estimate the answer to be a little more than 4.4 because a number less than 1 is being subtracted.

155
6 x 2.4
A student might estimate an answer between 12 and 18 since 6 x 2 is 12 and 6 x 3 is 18. Another student might give an estimate of a
little less than 15 because s/he figures the answer to be very close, but smaller than 6 x 2 and think of 2 groups of 6 as 12 (2 groups
of 6) + 3 (. of a group of 6).
Students should be able to express that when they add decimals they add tenths to tenths and hundredths to hundredths. So, when
they are adding in a vertical format (numbers beneath each other), it is important that they write numbers with the same place value
beneath each other. This understanding can be reinforced by connecting addition of decimals to their understanding of addition of
fractions. Adding fractions with denominators of 10 and
100 is a standard in fourth grade.
Example:
A recipe for a cake requires 1.25 cups of milk, 0.40 cups of oil, and 0.75 cups of water. How much liquid is in the
mixing bowl?
(cont. on next page)
156
157
Resources:
158

Refer to Hands-On Standards, Common Core Edition, p. 34-45

www.mathgoodies.com/lessons/decimals/solve_word_problems.html

fractionanddecimal.weebly.com/lesson-9-real-world-problems-involv
Domain: Number and Operations - Fractions
Enduring Understanding(s):
Fractions and decimals allow for quantities to be expressed with greater precision than with just whole numbers.
Essential Questions:
Why express quantities, measurements, and number relationships in different ways?
Major Cluster: Use equivalent fractions as a strategy to add and subtract fractions.
Standard: 5.NF.1
Add and subtract fractions with unlike denominators (including mixed numbers) by replacing given fractions with equivalent fractions in
such a way as to produce an equivalent sum or difference of fractions with like denominators. For example, 2/3 + 5/4 = 8/12 + 15/12 =
23/12. (In general, a/b + c/d = (ad + bc)/bd.)
159
What students should know and be able to do:
Students will understand:






recognize that fractions with unlike denominators cannot be added or subtracted.
recognize when to convert fractions with unlike denominators.
mixed numbers as a whole number and a fraction or mixed numbers as whole numbers and parts of a whole.
a proper fraction as a fraction less than 1.
an improper fraction as a fraction greater than 1.
that all addition and subtraction of fractions requires like denominators.
Students are able to:



recognize equivalent fractions.
convert fractions to equivalent fractions with like denominators as needed.
find the sum or difference of fractions with like denominators.

add and subtract fractions and mixed numbers with like denominators.

rename improper fractions and mixed numbers.
Learning Targets:



160
I can rename improper fractions and mixed numbers, by using strategies such as decomposing.
I can add and subtract fractions and mixed numbers with unlike denominators by finding equivalent fractions with like
denominators
I can represent fractions with models, including set models, area models, and linear models and connect them to numerical
representations.
Vocabulary:
fractions, unlike denominators, numerators, mixed numbers, equivalent fractions, like denominators, whole numbers, parts of whole
numbers proper fraction, improper fraction, common denominators, equivalent fractions, number lines, fraction bar, sum, difference,
decompose
Sample Problem(s):
161
Resources:

Extensions:
http://www.k-5mathteachingresources.com/supportfiles/magicsquaresadditionfractions.pdf

Equivalent Fractions:
http://illuminations.nctm.org/ActivityDetail.aspx?ID=80

Fraction Game:
http://illuminations.nctm.org/ActivityDetail.aspx?ID=18
Standard: 5.NF.2.
Solve word problems involving addition and subtraction of fractions referring to the same whole, including cases of unlike
denominators, e.g., by using visual fraction models or equations to represent the problem. Use benchmark fractions and number
sense of fractions to estimate mentally and assess the reasonableness of answers. For example, recognize an incorrect result 2/5 +
1/2 = 3/7, by observing that 3/7 < 1/2.
162
What students should know and be able to do:
Students will understand :
 problem solving strategies involving addition and subtraction of fractions.
 proper fractions, improper fractions, and mixed numbers.
 how to compare a fraction to a benchmark fraction.
 strategies to interpret and understand word problems with fractions.
Students will be able to:
 represent fractions with a variety of models such as: set, area, or linear models, etc..
 convert fractions with unlike denominators to equivalent fractions with like denominators.
 use benchmark fractions and number sense to estimate the reasonableness of answers.
Learning Targets:



I can determine the reasonableness of an answer using estimation and benchmark fractions.
I can set up an equation to represent a word problem involving addition and subtraction of fractions.
I can convert fractions to equivalent fractions with like denominators.
Vocabulary:
visual fraction models, benchmark fraction, estimate, equations, reasonableness, sum, difference, numerator, denominator, like
denominators, unlike denominators, estimation, equivalent fractions
163
Sample Problem(s):
Julia was doing a science experiment in which she was comparing the heights of a bean plant at the start and end of a
two-week time period. At the end of the two-week time period, the plant was 2/3 of a foot tall. At the beginning of the time
period, it was 1/12 of a foot tall. How much did the plant grow over the two-week time period?
Ask the students to work in groups and estimate the answer. Then allow them to use either a linear model (with a number
line) or a bar model to find the answer.
5. NF. 2
Using an area model to subtract
This model shows 1, subtracted from 3 1/6 leaving 1 + . = 1/6 which a student can then change
to 1 + 3/12 + 2/12 = 1 5/12. 3 1/6 can be expressed with a denominator of 12. Once this is done a
student can complete the problem, 2 14/12 – 1 9/12 = 1 5/12.
This diagram models a way to show how 3 1/6 and 1 can be expressed with a denominator of 12.
Once this is accomplished, a student can complete the problem, 2 14/12 – 1 9/12 = 1 5/12
164
Resources:
165

8 Step Model Drawing: Singapore‟s Best Problem-Solving Math Strategies by Bob Hogan and Char Forsten

http://nrich.maths.org/2312

Extensions:
http://nrich.maths.org/2312
Major Cluster: Apply and extend previous understandings of multiplication and division to multiply and divide fractions.
Standard: 5.NF.3.
Interpret a fraction as division of the numerator by the denominator (a/b = a ÷ b). Solve word problems involving division of whole
numbers leading to answers in the form of fractions or mixed numbers, e.g., by using visual fraction models or equations to represent
the problem. For example, interpret 3/4 as the result of dividing 3 by 4, noting that 3/4 multiplied by 4 equals 3, and that when 3 wholes
are shared equally among 4 people each person has a share of size 3/4. If 9 people want to share a 50-pound sack of rice equally by
weight, how many pounds of rice should each person get? Between what two whole numbers does your answer lie?
What students should know and be able to do:
Students will understand:
 a fraction as division of the numerator by the denominator (a / b = a ÷ b).
 a fraction (a / b) x b = a.
 problem solving strategies involving fractions.
 that a quotient may be a whole number, mixed number, or a fraction.
 that a remainder can be written as a fraction.
Students will be able to:




rewrite division expressions as fractions.
solve word problems with remainders written as a fraction.
determine between which two whole numbers a fraction lies.
use concrete and pictorial models to show fractions represented by whole number division.
Learning Targets:



166
I can rename a fraction as a division problem and I can rename a division problem as a fraction.
I can illustrate fractions with pictorial representations.
I can solve word problems with fraction models or equations to represent the problem.
Vocabulary:
numerator, denominator, division, fraction, mixed number, whole number, visual fraction models, equations, quotient, remainder, proper
fraction, improper fraction, divisor, dividend, fair sharing
Sample Problem(s):
Have students suggest how to divide 2 pizzas equally among 3 students (2/3 = 2 ÷ 3).
Explain how to divide the pizzas in smaller pieces using fraction circles. Show that each pizza is divided equally into the
number of parts which represent the number of students. So each pizza is divided into 3 smaller pieces. Show that dividing 6
smaller pieces of pizza among 3 students means each student gets 2 pieces each. Because a pizza comprises 3 equal
pieces, each student gets 2/3 of a pizza.
If 9 people want to share a 50-pound sack of rice equally by weight, how many pounds of rice should each person get? This can be
solved in two ways. First, they might partition each pound among the 9 people, so that each person gets 50 x 1/9 = 50/9 pounds.
Second, they might use the equation 9 x 5= 45 to see that each person can be given 5 pounds, with 5 pounds remaining. Partitioning
the remainder gives 5 5/9 pounds for each person.
167
If 9 people want to share a 50-pound sack of rice equally by weight, how many pounds of rice should each person get? This can be
solved in two ways. First, they might partition each pound among the 9 people, so that each person gets 50 x 1/9 = 50/9 pounds.
Second, they might use the equation 9 x 5= 45 to see that each person can be given 5 pounds, with 5 pounds remaining. Partitioning
the remainder gives 5 5/9 pounds for each person.
168
Resources:

http://www.k-5mathteachingresources.com/
Go to 5th grade number activities: Number and Operations- Fractions 5.NF3 Relating Fractions to Division Problems
Standard: 5.NF.4.
Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction.
a. Interpret the product (a/b) x q as a parts of a partition of q into b equal parts; equivalently, as the result of a sequence of operations
a x q ÷ b. For example, use a visual fraction model to show (2/3) x 4 = 8/3, and create a story context for this equation. Do the same
with (2/3) x (4/5) = 8/15. (In general, (a/b) x (c/d) = ac/bd.)
b. Find the area of a rectangle with fractional side lengths by tiling it with unit squares of the appropriate unit fraction side lengths,
and show that the area is the same as would be found by multiplying the side lengths. Multiply fractional side lengths to find areas of
rectangles, and represent fraction products as rectangular areas.
169
What students should know and be able to do:
Students will understand:
 a whole number multiplied by a fraction can be represented as repeated addition.
 a fraction can be multiplied by a fraction and the product will be smaller value.
 fraction products as rectangular areas.
Students will be able to:
 represent a whole number as a fraction. For example, 12 = 12/1.
 represent problems with visual fraction models.
 understand a story context that can be represented as the format of (a/b) x q .
 understand that area is the number of square units in a figure.
 understand the meaning of multiplication of a whole number by a fraction.
Learning Targets:






I can compare the product of two fractions to the product of two other fractions based upon the size of the unit fraction.
I can create a story context for a situation involving multiplication of a fraction or a whole number by a fraction.
I can multiply a fraction by a fraction including improper fractions and mixed numbers
I can use area models to represent multiplication of a fraction by a whole number and a fraction by a fraction
I can find area using the dimensions of a rectangle with fractional side lengths.
I can use tiles and arrays to calculate area with fractional side lengths.
Vocabulary:
product, equal parts, equivalent, equation, area, unit fraction, multiplication, rectangle, fractional side lengths,
partition, factors, numerator, denominator, fraction, whole number
Sample Problem(s):
170
5.NF 4a.
1. Isabel had 6 feet of wrapping paper. She used 3/5 of the paper to wrap some presents. How much does she have
left?
2. Tim ran 3/5 of a mile every day. How far did he run after 6 days? (Interpreting this as 6x3/5).
171
172
5.NF 4b
1. Mr. Brown is building a sandbox that is 6-1/2 feet by 4-1/2 feet. Draw a model of the sandbox, labeling all
dimensions. Find the total area of the sandbox. Explain your answer.
2. Find the area of a rug that is 3-1/2 feet by 2-1/2 feet.
173
Resources:


174
3-5 Fractions-Rectangle Multiplication:
http://nlvm.usu.edu/en/nav/frames_asid_194_g_2_t_1.html?from=topic_t_1.html
http://www.ncpublicschools.org/docs/acre/standards/common-core-tools/unpacking/math/5th.pdf
Standard: 5.NF.5.
Interpret multiplication as scaling (resizing), by:
a. Comparing the size of a product to the size of one factor on the basis of the size of the other factor, without performing the
indicated multiplication.
b. Explaining why multiplying a given number by a fraction greater than 1 results in a product greater than the given number
(recognizing multiplication by whole numbers greater than 1 as a familiar case); explaining why multiplying a given number by a
fraction less than 1 results in a product smaller than the given number; and relating the principle of fraction equivalence a/b = (n x a) /
(n x b) to the effect of multiplying a/b by 1.
What students should know and be able to do:
Students will understand:
 that multiplying a fraction greater than 1 and a given number results in a product greater than the given number.
 that multiplying a fraction less than 1 and a given number results in a product smaller than the given number.
Examples: 3 x 1/4 = 3/4 3/4 < 3
 that multiplying a fraction less than 1 times a fraction less than 1 results in a number less than either fraction.
Examples: 1/2 x 1/2 = 1/4 1/4 < 1/2
Students are able to:
 explain why multiplying a given number by a fraction greater than 1 results in a product greater than the given
number.
 explain why multiplying a given number by a fraction less than 1 results in a product smaller than the given number.
 compare the size of a product to the size of one factor on the basis of the size of the other factor, without
performing the indicated multiplication.
175
Learning Targets:




I can use scaling and resizing to compare the result of a product without performing the indicated multiplication.
I can explain why multiplying a quantity by a number smaller than one produces a smaller quantity.
I can explain why multiplying a quantity by a number larger than one produces a larger quantity.
I can explain why multiplying a quantity by a form of one produces an equivalent quantity.
Vocabulary:
scaling (resizing), multiplication, product, factor, fraction equivalence, array, products, x means ―
of‖,
increase, decrease, fraction greater than 1, improper fraction, fraction less than 1, proper fraction, mixed number
176
Sample Problem(s):
1. Mrs. Jones teaches in a room 60 feet wide and 40 feet long. Mr. Thomas teaches in a room that is half as wide, but
has the same length. How do the dimensions and area of Mr. Thomas’ classroom compare to Mrs. Jones’ room?
Draw a picture to prove your answer.
2. How does the product of 225 x 60 compare to the product of 225 x 30? How do you know? Possible strategy/answer:
Since 30 is half of 60, the product of 22 5 x 60 will be double or twice as large as the product of 225 x 30.
3. Joey has a bedroom that is 12 feet by 8 feet long. His sister, Mary, has a bedroom that is 10 feet by 12 feet long.
Which bedroom has the greatest area? Justify your reasoning with a model and an equation
4. Mrs. Bennett is planting two flowerbeds. The first flowerbed is 5 feet long and 1-1/5 feet wide. The second flowerbed
is 5 feet long and 5/6 feet wide. How do the areas of these two flowerbeds compare? Is the value of the area larger or
smaller than 5 square feet for each flowerbed? Draw pictures to prove your answer.
Resources:
177

Numbers and Operations 3-5— Rectangular Multiplication:
http://nlvm.usu.edu/en/nav/grade_g_ 1.html

Numbers and Operations 3-5— Fractions Rectangular Multiplication:
http://nlvm.usu.edu/en/nav/frames_a sid_194_g_2_t_1.html?from=categor y_g_2_t_1.html
Standard: 5.NF.6.
Solve real world problems involving multiplication of fractions and mixed numbers, e.g. by using visual fraction models or equations
to represent the problem.
What students should know and be able to do:
Students will understand:
 strategies to interpret word problems involving multiplication of fractions.
Students are able to:
 make sense of a real-world problem by using visual fraction models or equations to represent.
Learning Targets:


I can use visual fraction models to solve real-world problems involving multiplication of fractions and mixed
numbers.
I can use visual equations to solve real-world problems involving multiplication of fractions and mixed numbers.
Vocabulary:
multiplication, fraction, mixed number, visual fraction model, equation, factor, product, fraction
Sample Problem(s):
1. Evan bought 6 roses for his mother. Two-thirds of them were red. How many red roses were there?
178
2. Tasha finished a job in 3/4 hour. Megan finished the same job in 4/5 of the time Tasha took. How long did Megan
finish the job?
3. Given the problem 3/5 x 1-1/2, write a real-world problem to represent this expression and solve.
179
take to
180
Resources:
181

http://www.k5mathteachingresources .com/supportfiles/fractionxfractionwo rdproblems.pdf

http://www.k5mathteachingresources .com/supportfiles/adjustingarecipepr ojectfinal.pd
Standard: 5.NF.7.
Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions.
a. Interpret division of a unit fraction by a non-zero whole number, and compute such quotients. For example, create a
story context for (1/3) ÷ 4, and use a visual fraction model to show the quotient. Use the relationship between
multiplication and division to explain that (1/3) ÷ 4 = 1/12 because (1/12) x 4 = 1/3.
b. Interpret division of a whole number by a unit fraction, and compute such quotients. For example, create a story
context for 4 ÷ (1/5), and use a visual fraction model to show the quotient. Use the relationship between multiplication
and division to explain that 4÷ (1/5) = 20 because 20 x (1/5) = 4.
c. Solve real world problems involving division of unit fractions by non-zero whole numbers and division of whole
numbers by unit fractions, e.g., by using visual fraction models and equations to represent the problem. For example,
how much chocolate will each person get if 3 people share 1/2 lb of chocolate equally? How many 1/3-cup servings
are in 2 cups of raisins?
4
Students able to multiply fractions in general can develop strategies to divide fractions in general, by reasoning about
the relationship between multiplication and division. But division of a fraction by a fraction is not a requirement at this
grade.
182
What students should know and be able to do:
Students will understand:
 how to interpret division of a unit fraction.
 how to interpret division of a whole number by a unit fraction.
 how to solve real world problems involving division of unit fractions.
Students are able to:
 interpret division of a unit fraction by a non-zero whole number, and compute such quotients.
 interpret division of a whole number by a unit fraction, and compute such quotients.
 solve real world problems involving division of unit fractions by non-zero whole numbers and division of whole numbers
by unit fractions.
Learning Targets:



I can solve an equation to represent the division of a unit fraction by a non-zero whole number or division of a whole
number by a unit fraction.
I can solve real-world problems involving division of unit fractions.
I can use visual fraction models to illustrate division of a unit fraction by a non-zero whole number or division of a
whole number by a unit fraction.
Vocabulary:
division, unit fractions, whole numbers, quotients, dividend, divisor, equation, inverse operations
183
Sample Problem(s):
1. One-third of a pan of brownies is on the table. Eight friends want to share the brownies. How much of the total pan does
each person get?
2. Bill is going to run an 8-mile race. There are check points every 1/3 of a mile. How many check points are there in the
race?
3. One-half of a pie is shared between 3 friends. How much of the original whole pie does each person get?
4. Jacob has a 40-gallon gas tank. If it fills up 1/8 of a gallon every minute, how long will it take to fill an entire tank?
184
Resources:

http://www.ixl.com/math/grade-5/divide-fractions-by-whole-numbers\
Domain: Measurement and Data
Enduring Understanding(s):
Measurement processes are used in everyday life to describe and quantify the world.
Data displays describe and represent data in alternative ways.
Essential Questions:
How does geometry better describe objects?
Why display data in different ways?
Supporting Cluster: Convert like measurement units within a given measurement system.
Standard: 5.MD.1
Convert among different-sized standard measurement units within a given measurement system (e.g., convert 5 cm to 0.05 m), and use
these conversions in solving multi-step, real world problems.
185
What students should know and be able to do:
Students will understand:


the relationship within measurement systems (capacity, length and mass).
appropriate use of measurement tools in solving multi-step real-world problems.
Students are able to:


use conversions within a measurement system involving multi-step and real-world problems.
recognize uses of measurement in everyday situations.
Learning Targets:

I can convert among different size standard measurement units within the same system and solve multi-step and real-world
problems.
Vocabulary:
conversions, standard measurement units, measurement system
186
Sample Problem(s):
1. Hannah uses clay to make vases. She uses 750 grams of clay for each vase. How many kilograms of clay does she need to
make 10 vases?
2. Alicia’s aunt is 1.72 meters tall. Alicia’s mother is 175 centimeters tall. Who is taller? Explain.
3. To ride the Daring Dipper, you need to be 42 inches tall. Terrence’s little sister is 3 feet 4 inches. Is she tall enough to go on the
ride? Explain.
Resources:
Refer to Investigations Unit 6
Supporting Cluster: Represent and interpret data.
Standard: 5.MD.2
Make a line plot to display a data set of measurements in fractions of a unit (1/2, 1/4, 1/8). Use operations on fractions for this grade
to solve problems involving information presented in line plots. For example, given different measurements of liquid in identical
beakers, find the amount of liquid each beaker would contain if the total amount in all the beakers were redistributed equally.
187
What the students should know and be able to do:
Students will understand:
 line plots, fractions, and operations with fractions.
 equal redistribution of measurements.
Students will be able to:




measure in fractional units and create a line plot.
analyze data to perform operations with fractional units.
measure in fractional units with appropriate tools.
create line plots with collected data.
Learning Targets:
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I can measure in fractions of a unit and graph the results on a line plot.
I can draw conclusions and communicate results from data found on a line plot.
I can use operations to solve problems involving information presented in line plots which use fractions of a unit (1/2, ¼, 1/8)
I can redistribute measurements equally on a line plot.
Vocabulary:
line plot, data set, fractions of a unit (1/2, ¼, 1/8), operations
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Sample Problem(s):
1.
The line plot shows the amount of water that will be used by a clown to make water balloons for a birthday party. If the water was
redistributed equally, how much would be in each balloon?
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2.
Resources:
http://www.mrmaffesoli.com/Printables/5MD2/5MD2-A.pdf (Resources categorized by standards)
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Major Cluster: Geometric measurement: understand concepts of volume and relate volume to multiplication and to
addition.
Standard: 5.MD.3
Recognize volume as an attribute of solid figures and understand concepts of volume measurement.
a. A cube with side length 1 unit, called a ―
unit cube,‖ is said to have ―
one cubic unit‖ of volume, and can be used to measure volume.
b. A solid figure which can be packed without gaps or overlaps using n unit cubes is said to have a volume of n cubic units.
What students should know and be able to do:
Students will understand:
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attributes of solid figures (cubes).
that volume is the measurement of three dimensional space.
a cubic unit is the measurement label for volume.
Students are able to:
 recognize volume as an attribute of solid figures
 understand concepts of volume measurement.
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Learning Targets:
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I can use volume as one characteristic to describe a solid figure.
I can explain different ways volume can be measured.
I can identify a unit cube and explain how it can be used to measure volume.
I can explain the relationships the number of cubes it takes to fill a solid figure and the volume of that figure.
I can identify pictures and objects of unit cubes.
Vocabulary:
volume, solid figures, unit cube, one cubic unit
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Sample Problem(s):
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Resources:
www.mheonline.com (McGraw-Hill standard based lessons)
Standard: 5.MD.4
Measure volumes by counting unit cubes, using cubic cm, cubic in, cubic ft, and improvised units.
What students should know and be able to do:
Students will understand:

how to measure volumes by counting unit cubes, using cubic cm, cubic in, cubic ft, and improvised units.
Students are able to:

count various cubic units.
Learning Targets:

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I can measure volume by using cubic cm, cubic in, cubic ft, and improvised units by counting cubes.
Vocabulary:
volume, unit cubes, cubic cm, cubic inches, cubic feet
Sample Problem(s):
1. Which of these situations is about volume?
a. determine the amount of fencing it takes to go around a square garden
b. determining how many tiles it will take to cover the kitchen floor
c. determining how many rectangular containers of food will fit into a freezer
2. Vanessa wants to find the volume of her lunchbox. Which of these units should she use?
a. cubic feet
b. cubic inches
c. cubic yards
3. The volume of this rectangular solid is 40 cubic feet. What is its height? Show your work.
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2ft.
5ft.
Resources:
Refer to Investigations Unit 2
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Standard: 5.MD.5
Relate volume to the operations of multiplication and addition and solve real world and mathematical problems involving volume.
a. Find the volume of a right rectangular prism with whole-number side lengths by packing it with unit cubes, and show that the volume
is the same as would be found by multiplying the edge lengths, equivalently by multiplying the height by the area of the base.
Represent threefold whole-number products as volumes,
e.g., to represent the associative property of multiplication.
b. Apply the formulas V = l × w × h and V = b × h for rectangular prisms to find volumes of right rectangular prisms with whole number
edge lengths in the context of solving real world and mathematical problems.
c. Recognize volume as additive. Find volumes of solid figures composed of two non-overlapping right rectangular prisms by adding
the volumes of the non-overlapping parts, applying this technique to solve real world problems.
What students should know and be able to do:
Students will understand:
 the base is equal to length x width.
 the formulas for volume are equal.
 attributes of solid figures.
Students are able to:
 represent threefold whole-number products as volumes.
 solve real world problems by applying the formulas of volume.
 apply the associative property of multiplication to find volume.
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Learning Targets:
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I can pack a right rectangular prism with unit cubes and discover the numerical representation for the volume using the
associative property of multiplication and addition.
I can explain how to relate counting cubes to the formula for finding volume.
I can discover the threefold (three edge lengths) whole-number products as volume.
I can apply the formulas V = l × w × h and V = b × h.
I can separate the shape into rectangular prisms to find volume.
Vocabulary:
volume, operations of multiplication and addition, right rectangular prism, unit cubes, lengths, height, area of the base, products, solid
figures, prisms
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Sample Problem(s):
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Resources:
Refer to Investigations Unit 2
Domain: Geometry
Enduring Understanding(s):
Geometric attributes (such as shapes, lines, angles, figures, and planes) provide descriptive information about an object’s properties
and position in space and support visualization and problem solving.
Essential Questions:
How does geometry better describe objects?
Additional Cluster: Graph points on the coordinate plane to solve real-world and mathematical problems.
Standard: 5.G.1
Use a pair of perpendicular number lines, called axes, to define a coordinate system, with the intersection of the lines (the origin)
arranged to coincide with the 0 on each line and a given point in the plane located by using an ordered pair of numbers, called its
coordinates. Understand that the first number indicates how far to travel from the origin in the direction of one axis, and the second
number indicates how far to travel in the direction of the second axis, with the convention that the names of the two axes and the
coordinates correspond (e.g., x-axis and x-coordinate, y-axis and y-coordinate).
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What students should know and be able to do:
Students will understand:
 that perpendicular number lines are called axes and define a coordinate system.
 that the intersection of two number lines is called the origin and coincides with the zero on the number lines.
 that points on a plane are located by using ordered pairs.
 ordered pairs have an x and y coordinate.
 the coordinates indicate how far to travel from the origin on either axes.
Students are able to:
 explain how a number line works along both the x and y axis.
 locate points on a coordinate plane.
Learning Targets:
I can identify the number in an ordered pair that is the x-coordinate (which coincides with the x-axis) and the number that is
the y-coordinate (which coincides with the y-axis).
 I can match a coordinate pair with a given point within quadrant I.
Vocabulary:

perpendicular number lines, axes, coordinate system, the origin, ordered pair, coordinates, x-axis, x-coordinate, y-axis, y-coordinate,
coordinate plane grid
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Sample Problem(s):
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Resources:

Refer to Hands-On Standards, Common Core Edition, p. 96-99 and p. 16-19(connect graphing to coordinates on
a grid).

http://www.mrmaffesoli.com/5thGrade/5thGradeCCS.html (Resources categorized by standard)
Standard: 5.G.2
Represent real world and mathematical problems by graphing points in the first quadrant of the coordinate plane, and interpret
coordinate values of points in the context of the situation.
What students should know and be able to do:
Students will understand:
 how to represent real world and mathematical problems within the first quadrant within coordinate plane
Students are able to:
 graph points in the first quadrant of the coordinate plane.
 interpret data from the coordinate plane and represent that data within the context of the situation.
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Learning Targets:



I can interpret coordinate points in the context of real-world situations (points on a map, or data on a line graph).
I can interpret what the axes represent in different situations (in a line graph the x-axis may represent time while the y-axis
may represent temperature).
I can correctly interpret real-world data and plot that data in the first quadrant of a coordinate plane.
Vocabulary:
graphing points, first quadrant, coordinate plane, coordinate values, points
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Sample Problems:
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Resources:
Refer to Hands-On Standards, Common Core Edition, p. 100-103
http://www.mrmaffesoli.com/5thGrade/5thGradeCCS.html (Resources categorized by standards)
Additional Cluster: Classify two-dimensional figures into categories based on their properties.
Standard: 5.G.3
Understand that attributes belonging to a category of two-dimensional figures also belong to all subcategories of that category. For
example, all rectangles have four right angles and squares are rectangles, so all squares have four right angles.
What students should know and be able to do:
Students will understand:
 the attributes of two-dimensional shapes.
Students are able to:
 identify properties of two dimensional figures based on characteristics and definitions.
 represent a variety of basic two-dimensional figures.
 identify the relationship between general two dimensional figures and their subcategories.
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Learning Targets:
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
I can classify and define two-dimensional figures based on their attributes (e.g., a rhombus is a quadrilateral with four equal
sides).
I can draw or construct specific two-dimensional figures according to the definitions provided, attributes described, or
categories given.
Vocabulary:
two dimensional figures, category, subcategory
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Sample Problem(s):
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Resources:
Google: Math Common Core Resource Site
Standard: 5.G.4
Classify two-dimensional figures in a hierarchy based on properties.
What students should know and be able to do:
Students will understand:
 identify the relevant properties of a two-dimensional figures
Students are able to:
 construct various two dimensional figures, showing an understanding of the properties that define the figures
Learning Targets:

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I can classify two-dimensional figures in a hierarchy based on properties.
Vocabulary:
two dimensional figures, properties, hierarchy
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Sample Problem(s):
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Resources:
Refer to Investigations Unit 5
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Common Core State Documents Referenced
In addition to the resources listed within the curriculum guide, the following state websites
were utilized:
Arizona Department of Education – www.azed.gov/azcommoncore
Delaware Department of Education – http://www.doe.k12.de.us
Illinois State Board of Education – http://www.isbe.net
New York Department of Education – www.engageny.org
South Carolina Department of Education – http://ed.sc.gove/core
Utah Department of Education – www.schools.utah.gov/core
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