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Common Core Math Units

Grade 4

Adapted from NGA Center CCSS are the sole owners and developers of the Common Core State Standards.

© Copyright 2010 National Governors Association Center for Best Practices and Council of Chief State School Officers. All rights reserved.

The Charles A. Dana Center at The University of Texas at Austin January 13, 2013

Page 1

Sequenced Units for the Common Core State Standards in Mathematics Grade 4 In the years prior to Grade 4, students gained an understanding of multiplication and division of whole numbers, generalized strategies for addition and

subtraction to multi-digit numbers, developed understanding of fractions as numbers, and reasoned with shapes and their attributes. They used arrays and

the concept of area to develop computational strategies for multiplication and division.

Throughout Grade 4, students continue to develop their understanding of number. They generalize their understanding of place value to

1,000,000. Students extend their understanding of the four operations to include multiplicative compare problems, operations with multi-digit numbers, and

multiplying fractions by whole numbers. Students further develop their understanding of fractions to include addition of fractions with like denominators and

comparison and ordering of fractions with either like numerators or like denominators. The geometry focus in Grade 4 is on reasoning about angle measurement

and lines.

This document reflects our current thinking related to the intent of the Common Core State Standards for Mathematics (CCSSM) and assumes 160 days for instruction,

divided among 14 units. The number of days suggested for each unit assumes 45-minute class periods and is included to convey how instructional time should be

balanced across the year. The units are sequenced in a way that we believe best develops and connects the mathematical content described in the CCSSM; however,

the order of the standards included in any unit does not imply a sequence of content within that unit. Some standards may be revisited several times during the course;

others may be only partially addressed in different units, depending on the focus of the unit. Strikethroughs in the text of the standards are used in some cases in an

attempt to convey that focus, and comments are included throughout the document to clarify and provide additional background for each unit.

Throughout Grade 4, students should continue to develop proficiency with the Common Core's eight Standards for Mathematical Practice:

1. Make sense of problems and persevere in solving them. S. Use appropriate tools strategically. 2. Reason abstractly and quantitatively. 6. Attend to precision. 3. Construct viable arguments and critique the reasoning of others. 7. Look for and make use of structure. 4. Model with mathematics. 8. Look for and express regularity in repeated reasoning.

These practices should become the natural way in which students come to understand and do mathematics. While, depending on the content to be understood or

on the problem to be solved, any practice might be brought to bear, some practices may prove more useful than others. Opportunities for highlighting certain

practices are indicated in different units in this document, but this highlighting should not be interpreted to mean that other practices should be neglected in

those units.

When using this document to help in planning your district's instructional program, you will also need to refer to the CCSSM document, relevant progressions

documents for the CCSSM, and the appropriate assessment consortium framework.


Grade 4




Page 2

Unit 1: Exploring multiples and factors

Suggested number of days: 10

I Can Statements Notes/Comments Unit Materials and Resources

In this unit students develop understanding of multiples and factors, applying their understanding of multiplication from the previous year. This understanding lays a strong foundation

for generalizing strategies learned in previous grades to develop, discuss, and use efficient, accurate, and generalizable computational strategies involving multi-digit numbers. These

concepts and the terms "prime" and "composite" are new to Grade 4, so they are introduced early in the year to give students ample time to develop and apply this understanding.

Common Core State Standards for

Mathematical Content

0perations and Algebraic Thinking - 4.0A

B. Gain familiarity with factors and multiples. (S)

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.

C. Generate and analyze patterns. (A)

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.


Mathematical Practice

3. Construct viable arguments and critique the

reasoning of others.

7. Look for and make use of structure.

4.OA.4.1 Determine if a whole number

(1-100) is a multiple of a

given 1 digit number (ex. – Is

56 a multiple of 7? Is 45 a

multiple of 2) (MS)

4.OA.4.2 Find all factor pairs for a

whole number up to 100 (ex.

56 = __ x __) (MS)

4.OA.4.3 Determine if a whole number

(1-100) is prime or

composite.

4.OA.4.4 Recognize that a whole

number (1-100) is a multiple

of each of its factors. (MS)

4.OA.5.1 Use a rule to create a number or shape

pattern. (MS)

4.OA.5.2 Determine if there are other

relationships within a pattern (ex.4, 8, 16,

32… - all even. 5, 12, 19, 26… -

odd/even).

4.OA.5.3 Express a pattern using a formula.

While working on 4.0A.C.5,

students use manipulatives to

determine whether a number is

prime or composite. Although

there are shape patterns in arrays,

the focus of this unit is number

patterns. 4.0A.C.5 is repeated in

unit 13, where the focus will be on

identifying shape patterns.

The focus of this unit is not

necessarily to become fluent in

finding all factor pairs, but to use

student's understanding of the

concept and language to discuss the

structure of multiples and factors

(MP.3, MP.7).


Grade 4




Page 3

Unit 2: Using multiplication and division

strategies with larger numbers



In this unit students continue using computational and problem-solving strategies, with a focus on building conceptual understanding of multiplication of larger numbers and division

with remainders. Area and perimeter of rectangles provide one context for developing such understanding.



Operations and Algebraic Thinking - 4.OA

A. Use the four operations with whole numbers to

solve problems. (M)

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.

Number and Operations in Base Ten2 - 4.NBT

B. Use place value understanding and properties

of operations to perform multi-digit arithmetic.

(M)

5. Multiply a whole number of up to four digits

4.OA.3.1 Solve multi-step word problems using

the four operations with whole

numbers (3 or 4 digits by a 1 digit

number or two two-digit numbers)

4.0A.3.2 Interpret remainders in various

situations.

4.OA.3.3 Find whole number quotients without

remainders (3 digit dividends and one

digit divisor). (MS)

4.OA.3.4 Find whole number quotients with



4.OA.3.5 Find whole number quotients without



4.OA.3.6 Find whole number quotients with



4.OA.3.7 Justify an answer based upon the

interpretation of remainders.

4.OA.3.8

Justify an answer using mental math

and estimation.

4.NBT.5.1 Multiply a 4 digit number by a 1 digit

number.

4.NBT.5.2 Illustrate and explain multiplication using

rectangular arrays.

4.OA.A.3 is the first time students

are expected to interpret

remainders based upon the

context. All four operations will be

addressed in unit 8, and the

standard will be finalized in unit

14.


Grade 4




Page 4

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.

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.

NOTE: 2Grade 4 expectations in this domain are

limited to whole numbers less than or equal to

1,000,000.

Measurement and Data - 4.MD

A. Solve problems involving measurement and

conversion of measurements from a larger unit

to a smaller unit. (S)

3. Apply the area and perimeter formulas for

4.NBT.5.3 Illustrate and explain multiplication using

area models.

4.NBT.5.4 Apply the properties of operations to

multiply numbers. (MS)

4.NBT.5.5 Multiply 2, two digit numbers (ex. 23 x

45). (MS)

4.NBT.5.6 Multiply numbers using written

equations. (MS)

4.NBT.6.1 Divide up to 4 digit number by a 1 digit

divisor. (MS)


divide 4 digit numbers. (MS)

4.NBT.6.3 Apply strategies based on place value to

divide up to 4 digit number by a 1 digit

divisor. (MS)

4.NBT.6.4 Explore different strategies for the

division of 4 digit dividends and 1 digit

divisors. (MS)

4.NBT.6.5 Illustrate and explain division with a

rectangular array.

4.NBT.6.6 Illustrate and explain division with an

area model.


equation.

4.NBT.6.8 Explore the relationship between

multiplication and division. (MS)

4.MD.3.1

4.MD.3.2

Calculate the area of a rectangle using

the formula A=L x W or A= B x H when

side lengths are given.

Solve for the missing side length of a

rectangle using the formula A=L x W or

4.MD.A.3 provides the context of

area and perimeter of rectangles

to use for problem solving.

Students are first introduced to

formulas in this unit and make


Grade 4




Page 5

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.



1. Make sense of problems and persevere in solving

them.

2. Reason abstractly and quantitatively.

8. Look for and express regularity in repeated

reasoning.

A=B x H when the area is given along

with one other dimension.

4.MD.3.3

4.MD.3.4

Calculate the perimeter of a rectangle

using the formula P=S+S+S+S or P=2L +

2W when side lengths are given.

Solve for the missing side length of a

rectangle using the formula P=S+S+S+S

or P=2L + 2W when the perimeter is

given along with one other dimension.

4.MD.3.5 Apply the area and perimeter formula

to solve real-world problems.

sense of the formulas using their

prior work with area and

perimeter.

Students make sense of multi-step

problems (MP.1) and reason about

how the formulas connect to the

context (MP.2). The use of

generalized strategies and formulas

provides an opportunity to

investigate and use regularity in

repeated reasoning (MP.8).


Grade 4




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Unit 3: Decomposing and composing fractions

for addition and subtraction



In this unit students extend their prior knowledge of unit fractions with denominators of 2, 3, 4, 6, and 8 from Grade 3 to include denominators of 5, 10, 12, and 100. In Grade 4, they

use their understanding of partitioning to find unit fractions to compose and decompose fractions in order to add fractions with like denominators. This is foundational for further work

with fractions later in the year, such as comparing fractions and multiplying fractions by a whole number.



Number and Operations-Fractions3 - 4.NF

B. Build fractions from unit fractions by applying

and extending previous understandings of

operations on whole numbers. (M)

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.


limited to fractions with denominators 2, 3, 4, 5,

6, 8, 10, 12, and 100.



4. Model with mathematics.

4.NF.3a.1 Add or subtract fractions with like

denominators using manipulatives or

visual models. (MS)

4.NF.3a.2 Add and subtract improper fractions

with like denominators using

manipulatives or visual models.

4.NF.3b.1 Decompose a fraction into a sum of

fractions with the same denominator in

more than one way.

4.NF.3b.2 Create a visual model to justify

decompositions.

Students use visual and concrete

models to represent a fractional

situation in order to add and subtract

fractions (MP.4).


Grade 4




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Unit 4: Applying place value concepts in whole

number addition and subtraction



The focus of this unit is to provide students time to develop and practice efficient addition and subtraction of multi-digit whole numbers while developing place value concepts.




A. Generalize place value understanding for multi-

digit whole numbers. (M)

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.

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.

3. Use place value understanding to round

multi-digit whole numbers to any place.

4.NBT.1.1

4.NBT.1.2

4.NBT.1.3

Recognize a digit in one place

represents 10 times as much as it

represents in the place to the right (3

digit numbers).



represents in the place to the right (4

digit numbers).



represents in the place to the right

(multi-digit numbers).

4.NBT.2.1

4.NBT.2.2

Read, write, and compare multi-digit

numbers in expanded form. (MS)

Read, write, and compare multi-digit

numbers using base ten numerals

(standard form). (MS)

4.NBT.2.3 Read, write, and compare multi-digit

numbers in word form. (MS)

4.NBT.2.4 Compare multi-digit numbers using <, >,

=. (MS)

4.NBT.3.1 Round multi-digit numbers up to the

millions place. (MS)

4.NBT.A.1 will be revisited in unit

6 connected to conversions

within the metric system of

measurement.

4.NBT.A.3 will be revisited

in unit 7 with multiplication

and division as a context.


Grade 4




Page 8



(M)

4. Fluently add and subtract multi-digit whole

numbers using the standard algorithm.

NOTE: 2Grade 4 expectations in this domain

are limited to whole numbers less than or

equal to 1,000,000.



6. Attend to precision.


reasoning.

4.NBT.4.1 Fluently add multi-digit numbers up to

millions place. (MS)

4.NBT.4.2 Fluently subtract multi-digit numbers

up to millions place. (MS)

4.NBT.B.4 will be revisited in

unit 8 and finalized in unit 14 for

fluency in addition and

subtraction of multi-digit whole

numbers.

Students use the structure of the

base-ten system to generalize their

strategies and to discuss

reasonableness of their

computations and work towards

fluency (MP.6, MP.8).


Grade 4




Page 9

Unit 5: Understanding fraction equivalence

and comparison



In this unit students develop an understanding of fraction equivalence and various methods for comparing fractions. Students should understand that when comparing fractions, it is

not always necessary to generate equivalent fractions. Other methods, such as comparing fractions to a benchmark, can be used to discuss relative sizes. The justification of

comparing or generating equivalent fractions using visual models is an emphasis of this unit.1



Number and Operations-Fractions3 - 4.NF

A. Extend understanding of fraction equivalence

and ordering. (M)

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.

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.

NOTE: 3

Grade 4 expectations in this domain

are limited to fractions with denominators 2, 3,

4, 5, 6, 8, 10, 12, and 100.


4.NF.1.1 Identify equivalent fractions.

4.NF.1.2 Create visual fraction models to

show why fractions are equal.

4.NF.1.3 Explain why two fractions are

equivalent, but have different

denominators.

4.NF.1.4 Create equivalent fractions in

number form (ie. ½ = 6/12) by

multiplying or dividing the

numerator and denominator by the

same number.

4.NF.2.1 Compare a fraction to a benchmark

fraction such as 1/2, using a visual

model.

4.NF.2.2 Compare fractions to a benchmark

fraction such as 1/2, using numerical

comparison. (ie. 3/6 ____ 7/12)

4.NF.2.3 Use multiples to find a LCD.

4.NF.2.4 Compare fractions using symbols (<,

>, =).

4.NF.2.5 Compare two fractions with different

numerators (like denominators).

4.NF.2.6 Compare two unlike fractions by

creating like denominators.

Students justify their methods for

generating equivalent fractions and

comparing fractions by using their

conceptual understanding and

models (MP.3, MP.5).


Grade 4




Page 10




5. Use appropriate tools strategically.

4.NF.2.7 Explain that the size of the whole

matters when comparing fractions

(ie. ½ of a medium pizza is not equal

to ½ of a large pizza).

4.NF.2.8 Justify comparisons by using a visual

fraction model.

4.NF.2.9 Create a visual model to explain the

comparison of fractions.

4.NF.2.10 Compare two unlike fractions using a

variety of methods.


Grade 4




Page 11

Unit 6: Introducing measurement conversions



In this unit students build a conceptual understanding of the relative sizes of units of measure within a single system of measurement. Measurement conversions are used to

introduce multiplication as a comparison. The concepts in this unit are foundational for the concepts in unit 7 and unit 8.





solve problems. (M)

1. Interpret a multiplication equation as a

comparison, e.g., interpret 35 = 5 x 7 as a

statement that 35 is 5 times as many as 7 and 7

times as many as 5. Represent verbal statements

of multiplicative comparisons as multiplication

equations.




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.



1,000,000.




4.OA.1.1 Multiply two given numbers (one and

two digits).

4.OA.1.2 Write/compose a multiplication

equation.

4.OA.1.3 Interpret a verbal comparison into an

equation.

4.OA.1.4 Compare amounts using

multiplication. (for example – 5 times

as many…)

4.OA.1.5 Change a number sentence into a

word sentence.

4.OA.1.6 Translate (give an example of) verbal

statements as multiplication

statement.

4.NBT.1.1 Identify place values up to 1,000,000.

4.NBT.1.2 Use multiplication by a power of 10 to

determine the value of a digit in a

multi-digit whole number.

4.OA.A.l is repeated in unit 11,

in which the focus is on

multiplication of fractions.2

4.NBT.A.l was addressed in unit 4,

in which the focus was on addition

and subtraction. In this unit, metric

measurement provides an

opportunity to deepen the

students' understanding of place

value in relation to multiples of 10.

4.MD.A.l introduces units of measure

new to

Grade 4.

In this unit students look for patterns

in different measurement systems

(MP.2, MP.7) and discuss precisely

how many times larger one unit is

than another (MP.6)


Grade 4




Page 12


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), ...






4.MD.1.1 Identify and associate units of

measurements used to measure

length.



capacity.



weight.


measurements used to measure time.

4.MD.1.5 Compare units of measurement within

a given system (ie. 1 inch < 1 foot).

4.MD.1.6 Convert (change) from a larger unit to

a smaller unit.

4.MD.1.7 Create a table to record equivalent

measures listing number pairs.


Grade 4




Page 13

Unit 7: Solving problems using multiplicative

comparison



In this unit students are introduced to multiplicative compare problems, extending their conceptual work with multiplicative comparison from unit 6. For students to develop this

concept, they must be provided rich problem situations that encourage them to make sense of the relationships among the quantities involved, model the situation, and check their

solution using a different method. CCSSM Table 2 is an important resource for understanding multiplicative comparison problems, which are new to Grade 4 students.3





solve problems. (M)

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.




3. Use place value understanding to round multi-

digit whole numbers to any place.



1,000,000

4.OA.2.1 Identify variables, symbols, &

unknown numbers.

4.OA.2.2 Identify key words and relate words to

operations.

4.OA.2.3 Represent word problems and/or

equations with pictures and symbols.

4.OA.2.4 Compare amounts with multiplication.

(for example – 7 times as many as…)

*this is not repeated addition.

4.OA.2.5 Compare amounts with addition (7

more than).

4.OA.2.6 Solve word problems using

multiplication (4 digits by 1 digit or 2

digits by 2 digits).

4.OA.2.7 Solve word problems using division (4

digit dividends by 1 digit divisor).

4.OA.2.8 Identify multiplicative comparisons

from additive comparisons.

4.NBT.3.1 Round numbers up to the millions

place.

4.NBT.3.2 Explain why a number is rounded to a

given place.

4.NBT.3.3 Demonstrate understanding of place

value using a drawing, chart, table,

diagram, etc…

4.OA.A.2 is also addressed in unit

14 because of the time it takes to

master the concepts and its

importance to future

mathematics.4

4.NBT.A.3 was addressed in unit 4

with a focus on addition and

subtraction. In this unit, the focus

is on multiplication and division.

4.MD.A.2 is used as a context for

multiplicative compare problems

with whole numbers only. This

standard is revisited in unit 8 to

include the four operations, and

addressed in unit 12 with decimal

fractions.

Students use charts and diagrams to

explain their own methods as well

make sense of approaches taken by

others (MP.1).


Grade 4




Page 14





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.




them.

4.MD.2.1 Represent measurement quantities

using diagrams with a measurement

scale.

4.MD.2.2 Apply the four operations to solve

word problems involving distance.


word problems involving elapsed time.


word problems involving liquid

volume.


word problems involving mass.


word problems involving money.

*Note:

− These problems are limited to converting

larger to smaller units.

− These problems include whole numbers,

fractions, and decimals.


Grade 4




Page 15

Unit 8: Solving measurement problems using

the four operations



In this unit students combine competencies from different domains to solve measurement problems using the four operations.S

Measurement is included in this unit to provide a

context for problem solving. All of the problem types in Table 1 and Table 2 on pages 88 and 89 of the Common Core State Standards for Mathematics should be addressed in this unit.





solve problems. (M)



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.




(M)



4.OA.3.1 Add, subtract, multiply and divide with

or without remainders.

4.OA.3.2 Identify key words to decide which

operation(s) to use to solve a word

problem.

4.OA.3.3 Explain what a remainder is and how it

will affect a given problem.

4.OA.3.4 Determine if an answer makes sense,

based on the problem.

4.OA.3.5 Justify my answers using mental math

and estimation.

4.OA.3.6 Write an equation to solve the word

problem using a letter to represent the

missing number

4.OA.3.7 Solve multistep word problems with

whole numbers.

4.OA.3.8 Calculate long division with

remainders.

4.NBT.4.1 Add numbers up to millions place

value.

4.NBT.4.2 Subtract numbers up to millions place

value.

4.NBT.4.3 Justify an answer by using the

relationship between addition and

subtraction (inverse operations).

4.OA.A.3 and 4.NBT.B.4 are

repeated here to include all four

operations and will be finalized in

unit 14. Repeating these standards

throughout the year provides

students multiple opportunities to

develop these skills-which

are major areas of focus for this

grade level.

4.MD.A.2 is repeated from the

previous unit, but in this unit the

emphasis is on using the four

operations and all problem types.

This standard will be finalized in

unit 12 to include decimal

fractions.

Students use various diagrams and

precise language to solve

measurement problems and explain

their strategies (MP.1, MP.6). They

make connections between abstract

representations and the problem

situations (MP.2).


Grade 4




Page 16


















them.





scale.







volume.





*Note:






Grade 4




Page 17

Unit 9: Solving addition and subtraction word

problems involving fractions and mixed

numbers



In this unit students will use their understanding of adding and subtracting fractions and generating equivalent fractions to solve problems involving fractions and mixed numbers.

Students rely on their previous work with whole numbers as fractions to compose and decompose whole numbers into fractional quantities.6

Data is used in this unit to support

students' understanding of fractional quantities both smaller and larger than 1.7



Number and Operations-Fractions - 4.NF




3. Understand a fraction a/b with a > 1 as a sum

of fractions 1/b.

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.

NOTE: Grade 4 expectations in this domain are

limited to fractions with denominators 2, 3, 4,

5, 6, 8, 10, 12, and 100.

4.NF.3c.1 Justify that a mixed number is a whole

number and a fraction by using a visual

model.

4.NF.3c.2 Change a mixed number into an

improper fraction.

4.NF.3c.3 Change an improper fraction into a

mixed number.

4.NF.3d.1 Solve word problems using addition

and subtraction of fractions with like

denominators using visual models and

equations.

4.MD.B.4 extends students' work

from Grade 3 with simple fractions

on a line plot (3.MD.B.4) to include

eighths and to now solve addition

and subtraction problems using the

data.

Students reason about fractions by

using abstract models to represent

both the data and the fractional

quantities (MP.2, MP.4).


Grade 4




Page 18


B. Represent and interpret data. (S)

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.





4.MD.4.1 Answer questions about data

measured on a line plot.

4.MD.4.2 Create a line plot to display (show) a

data set that includes fractions or

mixed numbers.

4.MD.4.3 Add and subtract fractions using

information from a line plot.


Grade 4




Page 19

Unit 10: Angle measurement



This unit is an introduction to angles and angle measurement. Students start this unit drawing points, lines, segments, rays and angles since it is foundational to the other standards in

this unit. Students use their understanding of equal partitioning and unit measurement to understand angle and turn measure.




C. Geometric measurement: understand concepts

of angle and measure angles. (A)

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 "one-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.

6. Measure angles in whole-number degrees

using a protractor. Sketch angles of

specified measure.

4.MD.5.1 Identify an angle.

4.MD.5a.1 Recognize that a circle has 360

degrees.

4.MD.5a.2 Explain that an angle measurement is

a fraction of a circle.

4.MD.5b.1 Recognize that angles are measured

in degrees within a circle.

4.MD.6.1 Identify benchmark angles (90º, 180º,

270º, 360º).

4.MD.6.2 Measure angles using a protractor.

4.MD.6.3 Sketch angles of a given

measurement (degree) using a

protractor.

In this unit, 4.G.A.1 focuses on

drawing points, lines, line segments,

rays, and different types of angles.

The standard will be addressed in its

entirety in unit 13.

Students select and use a protractor

to measure angles and represent the

angles with drawings (MP.4, MP.S).


Grade 4




Page 20

Geometry - 4.G

A. Draw and identify lines and angles, and classify

shapes by properties of their lines and angles. (A)

1. Draw points, lines, line segments, rays,

angles (right, acute, obtuse), and

perpendicular and parallel lines. Identify

these in two-dimensional figures.





4.G.1.1 Identify points, lines, line segments,

rays, angles, perpendicular, and

parallel lines in two dimensional

figures.

4.G.1.2 Draw points, lines, line segments,



figures

4.G.1.3 Identify types of angles (right, acute,

obtuse) in two-dimensional figures.


Grade 4




Page 21

Unit 11: Multiplying fractions by whole

numbers



In this unit students apply their understanding of composing and decomposing fractions to develop a conceptual understanding of multiplication of a fraction by a whole number.

Students also use and extend their previous understandings of operations with whole numbers and relate that understanding to fractions.





solve problems. (M)

1. Interpret a multiplication equation as a

comparison, e.g., interpret 35 = 5 x 7 as a

statement that 35 is 5 times as many as 7 and 7

times as many as 5. Represent verbal statements

of multiplicative comparisons as multiplication

equations.





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

1lb. For example, use a visual fraction

model to represent 5/4 as the product 5 x

(1/4), recording the conclusion by the

equation 5/4 = 5 x (1/4).

4.OA.1.1 Multiply two given numbers (one and

two digits).

4.OA.1.2 Write/compose a multiplication

equation.

4.OA.1.3 Interpret a verbal comparison into an

equation.

4.OA.1.4 Compare amounts using

multiplication. (for example – 5 times

as many…)

4.OA.1.5 Change a number sentence into a

word sentence.

4.OA.1.6 Translate (give an example of) verbal

statements as multiplication

statement.

4.NF.4a.1 Multiply a whole number by a fraction

by changing the whole number into a

fraction (ie. 5 x ¾ = 5/1 x ¾).

4.OA.A.1 is readdressed in this

unit to include multiplication of

fractions and apply the

understanding of "times as much"

(i.e. multiplication as comparison)

to multiplying a fraction by a

whole number.

Students use precise language to

communicate their comprehension

of the problem situations and defend

their various solution methods

(MP.1, MP.6)


Grade 4




Page 22

b. Understand a multiple of alb as a

multiple of 1lb, and use this

understanding to multiply a fraction by

a whole number. For example, use a

visual fraction model to express 3 x

(2/5) as 6 x (1/5), recognizing this

product as 6/5. (In general, n x (a/b) =

(n x a)/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?

NOTE: Grade 4 expectations in this domain

are limited to fractions with denominators

2, 3, 4, 5, 6, 8, 10, 12, and 100.




them.


4.NF.4b.1 Create a fraction from a whole number

by placing the whole number over 1

(ie. 5 = 5/1).

4.NF.4b.2 Use models to represent a fraction

times a whole number.

4.NF.4c.1 Solve multiplication problems by

multiplying the whole number by the

numerator.

4.NF.4c.2 Solve multiplication word problems

involving fractions and whole numbers

using visual models.

4.NF.4c.3 Solve multiplication word problems

involving fractions and whole numbers

using equations.


Grade 4




Page 23

Unit 12: Comparing decimal fractions and

understanding notation



In this unit of study students use their previous work with fractions to represent special fractions in a new way. Students use their understanding of equivalent fractions to begin to use

decimal notation-however, it is not the intent at this grade level to connect this notation to the base-ten system. The focus is on solving word problems involving simple fractions or

decimals. Work with money can support this work with decimal fractions.




C. Understand decimal notation for fractions, and

compare decimal fractions. (M)

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.

NOTE: Students who can generate

equivalent fractions can develop

strategies for adding fractions with

unlike denominators in general. But

addition and subtraction with unlike

denominators in general is not a

requirement at this grade.

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.

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

4.NF.5.1 Use base ten models to represent

fractions.

4.NF.5.2 Convert unlike denominators to like

denominators (10,100) and add

fractions.

4.NF.5.3 Change a fraction with a denominator

of 10 into a fraction with a

denominator of 100 by multiplying the

numerator and denominator by 10.

4.NF.6.1 Write a fraction as a decimal to

represent its place value.

4.NF.6.2 Convert between decimals and

fractions (ie. 0.62 = 62/100).

4.NF.6.3 Locate fractions and decimals on a

number line (tenths and hundredths).

4.NF.7.1 Compare two decimals to the

hundredth place using a hundreds grid

and using symbols (<,>,=).

4.MD.A.2 was addressed in unit 7.

It is important to note that students

are not expected to do

computations with quantities in

decimal notation. Students can use

visual fraction models to solve

problems involving simple fractions

or decimals.

Students compare decimals fractions

and justify their comparisons using

either a fraction model or their

understanding of the notation (MP.3,

MP.7).


Grade 4




Page 24

results of comparisons with the symbols >, =,

or <, and justify the conclusions, e.g., by using

a visual model.


limited to fractions with denominators 2, 3, 4,

5, 6, 8, 10, 12, and 100.




to a smaller unit. (M)
















4.NF.7.2 Recognize that in order to compare

decimals, they must have the same

place value.

4.NF.7.3 Create a model to justify an answer.



scale.







volume.





*Note:






Grade 4




Page 25

Unit 13: Recognizing and analyzing attributes

of 2-dimensional shapes



In this unit students develop their spatial reasoning skills by using a wide variety of attributes to talk about 2-dimensional shapes. Students analyze geometric figures based on angle

measurement, parallel and perpendicular lines, and symmetry.



0perations and Algebraic Thinking - 4.0A

C. Generate and analyze patterns. (A)

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.


C. Geometric measurement: understand concepts

of angle and measure angles. (A)

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.

4.OA.5.1 Use rules to explain a pattern.

4.OA.5.2 Create a number or shape pattern.

4.OA.5.3 Create and explain a number or shape

pattern.

4.OA.5.4 Identify a pattern within a pattern.

(ex. 1,4,7,10)

4.MD.7.1 Decompose (separate) angles into

smaller angles.

4.MD.7.2 Add angle measures to make a larger

angle.

4.MD.7.3 Use addition and subtraction to find

unknown angles in real-world and

mathematical problems.

4.MD.7.4 Use an equation with a symbol for the

unknown angle measure.

In this unit, 4.0A.C.5 includes

repeated and growing shape

patterns.10

4.G.A.1 was first addressed in unit

10, and is addressed in its entirety

in this unit to include

perpendicular and parallel lines.

The concepts in this unit lend

themselves to using technology

applications (MP.S).

Students understand that geometric

figures can be classified by analyzing

various properties (MP.7) and justify

their conclusions by using viable

arguments (MP.3).


Grade 4




Page 26

Geometry - 4.G

A. Draw and identify lines and angles, and classify

shapes by properties of their lines and angles. (A)

1. Draw points, lines, line segments, rays,

angles (right, acute, obtuse), and

perpendicular and parallel lines. Identify

these in two-dimensional figures.

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.

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.







4.G.1.1 Identify points, lines, line segments,



figures.

4.G.1.2 Draw points, lines, line segments, rays,

angles, perpendicular, and parallel

lines in two dimensional figures

4.G.1.3 Identify types of angles (right, acute,

obtuse) in two-dimensional figures.

4.G.2.1 Identify two dimensional shapes that

have parallel or perpendicular lines.

4.G.2.2 Identify two dimensional shapes that

have angles of a specific size.

4.G.2.3 Recognize and label a right triangle.

4.G.2.4 Classify two dimensional shapes on the

presence or absence of designated

lines and angles.

4.G.3.1 Recognize a line of symmetry.

4.G.3.2 Draw a line of symmetry.

4.G.3.3 Recognize figures that have lines of

symmetry.

4.G.3.4 Draw figures that have lines of

symmetry.


Grade 4




Page 27

Unit 14: Problem solving with whole numbers



This is a culminating unit in which students focus on problem solving in order to demonstrate fluency with the standard algorithms in addition and subtraction. They demonstrate

computational fluency with all problem types.

All standards in this unit have been addressed in prior units. These concepts require greater emphasis due to the depth of the ideas, the time they take to master, and/or their

importance to future mathematics.





solve problems. (M)

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. 1

NOTE: 1See Glossary, Table 2.



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

4.OA.2.1 Identify variables, symbols, & unknown

numbers.

4.OA.2.2 Identify key words and relate words to

operations.

4.OA.2.3 Represent word problems and/or

equations with pictures and symbols.

4.OA.2.4 Compare amounts with multiplication.

(for example – 7 times as many as…)

*this is not repeated addition.

4.OA.2.5 Compare amounts with addition (7

more than).

4.OA.2.6 Solve word problems using

multiplication (4 digits by 1 digit or 2

digits by 2 digits).

4.OA.2.7 Solve word problems using division (4

digit dividends by 1 digit divisor).

4.OA.2.8 Identify multiplicative comparisons

from additive comparisons.

4.OA.3.1 Add, subtract, multiply and divide with

or without remainders.

4.OA.3.2 Identify key words to decide which

operation(s) to use to solve a word

problem.

4.OA.3.3 Explain what a remainder is and how it

will affect a given problem.

In demonstrating fluency, students

explain and flexibly use properties of

operations and place value to solve

problems, looking for shortcuts and

applying generalized strategies

(MP.2, MP.8).


Grade 4




Page 28

strategies including rounding.

Number and Operations in Base Ten - 4.NBT



(M)



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.

4.OA.3.4 Determine if an answer makes sense,

based on the problem.

4.OA.3.5 Justify my answers using mental math

and estimation.

4.OA.3.6 Write an equation to solve the word

problem using a letter to represent the

missing number

4.OA.3.7 Solve multistep word problems with

whole numbers.

4.OA.3.8 Calculate long division with

remainders.

4.NBT.4.1 Add numbers up to millions place

value.

4.NBT.4.2 Subtract numbers up to millions place

value.

4.NBT.4.3 Justify an answer by using the

relationship between addition and

subtraction (inverse operations).

4.NBT.5.1 Multiply a 4 digit number by a 1 digit

number.

4.NBT.5.2 Illustrate and explain multiplication

using rectangular arrays.

4.NBT.5.3 Illustrate and explain multiplication

using area models.


multiply numbers.

4.NBT.5.5 Multiply 2, two digit numbers (ex. 23

x 45).

4.NBT.5.6 Multiply numbers using written

equations.


Grade 4




Page 29

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.



1,000,000.





reasoning.

4.NBT.6.1 Divide up to 4 digit number by a 1

digit divisor.


divide 4 digit numbers.

4.NBT.6.3 Apply strategies based on place value

to divide up to 4 digit number by a 1

digit divisor.

4.NBT.6.4 Explore different strategies for the

division of 4 digit dividends and 1

digit divisors.

4.NBT.6.5 Illustrate and explain division with a

rectangular array.


area model.


equation.

4.NBT.6.8 Explore the relationship between

multiplication and division.