3rd/4th Math

In third grade mathematics, students

- Develop understanding of multiplication and division and strategies for multiplication and division within 100,

- Develop understanding of fractions, especially unit fractions (fractions with numerator 1),

- Develop understanding of the structure of rectangular arrays and of area,

- Describe and analyze two-dimensional shapes.

Fourth grade mathematics is about

- Developing understanding and fluency with multi-digit multiplication, and developing understanding of dividing to find quotients involving multi-digit dividends

- Developing an understanding of fraction equivalence, addition and subtraction of fractions with like denominators, and multiplication of fractions by whole numbers

- 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

3/4 Pacing Guide SY 22-23

Module 1: The Meaning of Multiplication and Division, Developing Fluency

(30 Hours)

Standards

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. For example, describe a context in which a total number of objects can be expressed as 5 × 7.

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. For example, describe a context in which a number of shares or a number of groups can be expressed as 56 ÷ 8.

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.

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 = ?.

3.OA.5 Apply properties of operations as strategies to multiply and divide. (Students need not use formal terms for these properties.) Examples: If 6 × 4 = 24 is known, then 4 × 6 = 24 is also known. (Commutative property of multiplication.) 3 × 5 × 2 can be found by 3 × 5 = 15, then 15 × 2 = 30, or by 5 × 2 = 10, then 3 × 10 = 30. (Associative property of multiplication.) Knowing that 8 × 5 = 40 and 8 × 2 = 16, one can find 8 × 7 as 8 × (5 + 2) = (8 × 5) + (8 × 2) = 40 + 16 = 56. (Distributive property.)

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.

Overview

Students build knowledge of addition and arrays, allowing them to use that model and skip-counting strategies in multiplication and division problems. Students learn the remaining multiplication and division facts to continue to develop understanding of multiplication and division strategies within 100. They use those strategies to solve two-step word problems.

This unit is a combination of two THIRD GRADE units. Fourth grade students are expected to have increased fluency and better application (i.e. word problems) in relation to operations.

Unit Vocab

Array (arrangement of objects in rows and columns)

Commutative property/commutative (e.g., rotate a rectangular array 90 degrees to demonstrate that factors in a multiplication sentence can switch places)

Equal groups (with reference to multiplication and division; one factor is the number of objects in a group and the other is a multiplier that indicates the number of groups)

Distribute (with reference to the distributive property, e.g., in 12 × 3 = (10 × 3) + (2 × 3) the 3 is the multiplier for each part of the decomposition)

Divide/division (partitioning a total into equal groups to show how many equal groups add up to a specific number, e.g., 15 ÷ 5 = 3)

Factors (numbers that are multiplied to obtain a product) Multiplication/multiply (an operation showing how many times a number is added to itself, e.g., 5 × 3 =15)

Number of groups (factor in a multiplication problem that refers to the total equal groups)

Parentheses (symbols ( ) used around an expression or numbers within an equation)

Quotient (the answer when one number is divided by another)

Rotate (turn, used with reference to turning arrays 90 degrees)

Row/column (in reference to rectangular arrays)

Size of groups (factor in a multiplication problem that refers to how many in a group)

Unit (one segment of a partitioned tape diagram)

Unknown (the missing factor or quantity in multiplication or division)

Topics A - E Only: All Materials

Topics F-I Only: All Materials

Topic A: Multiplication and the Meaning of Factors

Standards

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. For example, describe a context in which a total number of objects can be expressed as 5 × 7

Overview

Students use multiplication as they understand what factors are and differentiate between the size of groups and the number of groups within a given context.

Lessons

Topic Assessment

Topic B: Division as an Unknown Factor Problem

Standards

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. For example, describe a context in which a number of shares or a number of groups can be expressed as 56 ÷ 8.

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.

Overview

Students understand division as an unknown factor problem, and relate the meaning of unknown factors to either the number or the size of groups.

Lessons

Topic Assessment

Topic C: Multiplication using Units of 2 and 3

Standards

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. For example, describe a context in which a total number of objects can be expressed as 5 × 7.

3.OA.5 Apply properties of operations as strategies to multiply and divide. Examples: If 6 × 4 = 24 is known, then 4 × 6 = 24 is also known. (Commutative property of multiplication.) 3 × 5 × 2 can be found by 3 × 5 = 15, then 15 × 2 = 30, or by 5 × 2 = 10, then 3 × 10 = 30. (Associative property of multiplication.) Knowing that 8 × 5 = 40 and 8 × 2 = 16, one can find 8 × 7 as 8 × (5 + 2) = (8 × 5) + (8 × 2) = 40 + 16 = 56. (Distributive property.)

Overview

Begin to build fluency with facts of 2 and 3 using the array model and familiar skip counting strategies. Students manipulate arrays to study the commutative property. They rotate arrays 90 degrees, distinguishing rows from columns, the meaning of factors changes based upon orientation.

Lessons

Topic Assessment

Topic D: Division Using Units of 2 and 3

Standards

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. For example, describe a context in which a number of shares or a number of groups can be expressed as 56 ÷ 8.

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 = ?

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.

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

Overview

Model division as the unknown factor in multiplication and interpret the quotient using units or 2 and 3. Students solve two types of division situations—partitive (group size unknown) and measurement (number of groups unknown)—using factors of 2 and 3

Lessons

Topic Assessment

Topic E: Multiplication and Division Using Units of 4, 5 and 10

Standards

3.OA.5 Apply properties of operations as strategies to multiply and divide. Examples: If 6 × 4 = 24 is known, then 4 × 6 = 24 is also known. (Commutative property of multiplication.) 3 × 5 × 2 can be found by 3 × 5 = 15, then 15 × 2 = 30, or by 5 × 2 = 10, then 3 × 10 = 30. (Associative property of multiplication.) Knowing that 8 × 5 = 40 and 8 × 2 = 16, one can find 8 × 7 as 8 × (5 + 2) = (8 × 5) + (8 × 2) = 40 + 16 = 56. (Distributive property.)

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.

Overview

Students shift from simple understanding to analyzing the relationship between multiplication and division. Practice of both operations is combined—this time using units of 4. Students apply the multiplication and division strategies and model relationships between factors analyzing the patterns that emerge using factors of 5 and 10.

Lessons

Topic Assessment

Topic F:Multilpication and Division Using Units of 6 and 7

Standards

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.

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 = ?

3.OA.5 Apply properties of operations as strategies to multiply and divide. (Students need not use formal terms for these properties.) Examples: If 6 × 4 = 24 is known, then 4 × 6 = 24 is also known. (Commutative property of multiplication.) 3 × 5 × 2 can be found by 3 × 5 = 15, then 15 × 2 = 30, or by 5 × 2 = 10, then 3 × 10 = 30. (Associative property of multiplication.) Knowing that 8 × 5 = 40 and 8 × 2 = 16, one can find 8 × 7 as 8 × (5 + 2) = (8 × 5) + (8 × 2) = 40 + 16 = 56. (Distributive property.)

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

Overview

Students learn the units 6, 7, and how to compose up to, then over the next decade. They apply the distributive property as a strategy to multiply and divide. They continue using letters to represent the unknown.

Lessons

Topic Assessment

Topic G:Multilpication and Division Using Units up to 8

Standards

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.

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 = ?

3.OA.5 Apply properties of operations as strategies to multiply and divide. (Students need not use formal terms for these properties.) Examples: If 6 × 4 = 24 is known, then 4 × 6 = 24 is also known. (Commutative property of multiplication.) 3 × 5 × 2 can be found by 3 × 5 = 15, then 15 × 2 = 30, or by 5 × 2 = 10, then 3 × 10 = 30. (Associative property of multiplication.) Knowing that 8 × 5 = 40 and 8 × 2 = 16, one can find 8 × 7 as 8 × (5 + 2) = (8 × 5) + (8 × 2) = 40 + 16 = 56. (Distributive property.)

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.

Overview

Students learn the conventional order for operations when parentheses are and are not present. They learn about the associative property and make use of structure to solve problems.

Lessons

Topic Assessment

Topic H:Multilpication and Division Using Units up to 9

Standards

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.

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 = ?

3.OA.5 Apply properties of operations as strategies to multiply and divide. (Students need not use formal terms for these properties.) Examples: If 6 × 4 = 24 is known, then 4 × 6 = 24 is also known. (Commutative property of multiplication.) 3 × 5 × 2 can be found by 3 × 5 = 15, then 15 × 2 = 30, or by 5 × 2 = 10, then 3 × 10 = 30. (Associative property of multiplication.) Knowing that 8 × 5 = 40 and 8 × 2 = 16, one can find 8 × 7 as 8 × (5 + 2) = (8 × 5) + (8 × 2) = 40 + 16 = 56. (Distributive property.)

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.

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.

Overview

Students explore a variety of arithmetic patterns with multiplication and division using units of 9 that become engaging strategies for quickly learning facts with automaticity. They continue to interpret the unknown factor to solve multiplication and division problems.

Lessons

Topic Assessment

Topic I:Multilpication of Single Digit Factors and Multiples of 10

Standards

3.OA.5 Apply properties of operations as strategies to multiply and divide. (Students need not use formal terms for these properties.) Examples: If 6 × 4 = 24 is known, then 4 × 6 = 24 is also known. (Commutative property of multiplication.) 3 × 5 × 2 can be found by 3 × 5 = 15, then 15 × 2 = 30, or by 5 × 2 = 10, then 3 × 10 = 30. (Associative property of multiplication.) Knowing that 8 × 5 = 40 and 8 × 2 = 16, one can find 8 × 7 as 8 × (5 + 2) = (8 × 5) + (8 × 2) = 40 + 16 = 56. (Distributive property.)

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, i.e., Order of Operations.)

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.

3.NBT.3 Multiply one-digit whole numbers by multiples of 10 in the range 10–90 (e.g., 9 × 80, 5 × 60) using strategies based on place value and properties of operations

Overview

Students multiply by multiples of 10. Students use place values to notice that the product shifts one place value to the left when multiplied by 10. They also model place value strategies using the associative property.

Lessons

Topic Assessment

Module 2: Place Value and Problem Solving with Units of Measure

(15 Hours)

Standards

3.NBT.1 Use place value understanding to round whole numbers to the nearest 10 or 100.

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

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.

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.

Overview

Module 2 uses place value to unify measurement, rounding skills, and the standard algorithms for addition and subtraction. The module begins with plenty of hands-on experience using a variety of tools to build practical measurement skills and conceptual understanding of metric and time units. Estimation naturally surfaces through application; this transitions students into rounding. In the module’s final topics students round to assess whether or not their solutions to problems solved using the standard algorithms are reasonable.

This is a THIRD GRADE UNIT. Fourth graders will need extension with numbers up to the 10,000s while ensuring they master the basic

Unit Vocab

About (with reference to rounding and estimation, an answer that is not precise)

Addend (the numbers that are added together in an addition equation, e.g., in 4 + 5, the numbers 4 and 5 are the addends)

Capacity (the amount of liquid that a particular container can hold)

Continuous (with reference to time as a continuous measurement)

Endpoint (used with rounding on the number line; the numbers that mark the beginning and end of a given interval)

Gram (g, unit of measure for weight)

Interval (time passed or a segment on the number line)

Halfway (with reference to a number line, the midpoint between two numbers, e.g., 5 is halfway between 0 and 10)

Kilogram (kg, unit of measure for mass)

Liquid volume (the space a liquid takes up) Liter (L, unit of measure for liquid volume)

Milliliter (mL, unit of measure for liquid volume)

Plot (locate and label a point on a number line)

Point (a specific location on the number line)

Reasonable (with reference to how plausible an answer is, e.g., “Is your answer reasonable?”)

Round (estimate a number to the nearest 10 or 100 using place value)

Second (a unit of time)

Standard algorithm (for addition and subtraction)

≈ (symbol used to show that an answer is approximate)

Module 2 - All Materials Here

Topic A: Time Measurement and Problem Solving

Standards

3.NBT.2 Fluently add and subtraction within 1000 using strategies and algorithms based on place value, properties of operations, and/or the relationship between addition and subtraction.

3.MC.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.

Overview

Students learn to tell time to the nearest minute, understanding time as a continuous measurement. They use the clock to solve addition and subtraction problems. Tell and write time to the nearest minute using analog and digital clocks.

Lessons

Topic Assessment

Topic B: Rounding to the Nearest 10 and 100

Standards

3.NBT.1 Use place value understanding to round whole numbers to the nearest 10 or 100.

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.

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 onestep 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.

Overview

Students measure and then use place value understandings and the number line as tools to round two-, three-, and four-digit measurements to the nearest ten or hundred

Lessons

Topic Assessment

Topic C: Two and Three Digit Measurement addition using standard algorithm

Standards

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

Overview

Students add two- and three-digit metric measurements and intervals of minutes within 1 hour. They use estimations to test the reasonableness of sums precisely calculated using standard algorithms.

Lessons

Topic Assessment

Topic D: Two and Three Digit Measurement subtraction using standard algorithm

Standards

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.

Overview

Students use two- and three-digit metric measurements and intervals of minutes within 1 hour to subtract. They use estimations to test the reasonableness of differences precisely calculated using standard algorithms.

Lessons

Topic Assessment

Module 3: Fractions on the Number Line

(25 Hours)

Standards

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.

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.

3.NF.3 Explain equivalence of fractions in special cases, and compare fractions by reasoning about their size.

a. Understand two fractions as equivalent (equal) if they are the same size, or the same point on a number line.
b. Recognize and generate simple equivalent fractions, e.g., 1/2 = 2/4, 4/6 = 2/3. Explain why the fractions are equivalent, e.g., by using a visual fraction model.
c. Express whole numbers as fractions, and recognize fractions that are equivalent to whole numbers. Examples: Express 3 in the form of 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

Overview

Students understand fractions as equal partitions of a whole and think of fractions as being constructed out of unit fractions. They work with area models and the number line. Understand fractions as equal partitions of a whole.

Though this is traditionally the final THIRD GRADE UNIT, fractions are the major focus of math work in 4th grade. It is important to ensure strong conceptual understanding of fractions and working with them formally for all 4th graders. Third graders are expected to know halves, thirds, fourths, sixths, and eights. Fourth graders denominators should include 2, 3, 4, 5, 6, 8, 10, 12, and 100).

Unit Vocab

Copies (refers to the number of unit fractions in 1 whole)

Equivalent fractions (fractions that name the same size or the same point on the number line)

Fractional unit (half, third, fourth, etc.) Non-unit fraction (fraction with numerator other than 1)

Unit fraction (fraction with numerator 1)

Unit interval (the interval from 0 to 1, measured by length)

Module 3: All Materials Here

Topic A: Partitioning A Whole into Equal Parts

Standards

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 of each part as 1/4 of the area of the shape.

Overview

Students partition a whole to measure equal parts. They identify and count equal parts as 1 half, 1 fourth, 1 third, 1 sixth, and 1 eighth in unit form before an introduction to the unit fraction 1/b.

Lessons

Topic Assessment

Topic B: Unit Fractions and their Relation to the Whole

Standards

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.

Overview

Students compare unit fractions and learn to build non-unit fractions with unit fractions as basic building blocks.

Lessons

Topic Assessment

Topic C: Comparing Unit Fractions and Specifying the Whole

Standards

3.NF.3 Explain equivalence of fractions in special cases, and compare fractions by reasoning about their size.

d. Compare two fractions with the same numerator or the same denominator by reasoning about their size. Recognize that comparisons are valid only when 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.

Overview

Students compare unit fractions and label fractions in relation to the number of equal parts in that whole. They also learn that redefining the whole can change the unit fraction of the shaded part.

Lessons

Topic Assessment

Topic D: Fractions on the Number line

Standards

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.

3.NF.3 Explain equivalence of fractions in special cases, and compare fractions by reasoning about their size.

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 >, =, <, and justify the conclusions, e.g., by using a visual fraction model.

Overview

Students apply their understanding of unit fractions and the importance of specifying the whole to the number line.

Lessons

Topic Assessment

Topic E: Equivalent Fractions

Standards

3.NF.3 Explain equivalence of fractions in special cases, and compare fractions by reasoning about their size.

a. Understand two fractions as equivalent (equal) if they are the same size, or the same point on a number line.
b. Recognize and generate simple equivalent fractions, e.g., 1/2 = 2/4, 4/6 = 2/3. Explain why the fractions are equivalent, e.g., by using a visual fraction model.
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.

Overview

Students compare fractions on the number line to recognize that equivalent fractions refer to the same whole, and also the same point on the line.

Lessons

Topic Assessment

Topic F: Comparison, Order, and Size of Fractions

Standards

3.NF.3 Explain equivalence of fractions in special cases, and compare fractions by reasoning about their size.

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

Overview

Students compare fractions by reasoning about their size and understand that fractions with the same numerator and a larger denominator are actually smaller pieces of the whole.

Lessons

Topic Assessment

Module 4: Place Value, Rounding

(15 Hours)

Standards

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.

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.

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. 4.NBT.3 Use place value understanding to round multi-digit whole numbers to any place.

Overview

Extended work with whole numbers. Utilizing large numbers, familiar units to develop their understanding of millions base ten system on the place value chart.

This is a FOURTH GRADE unit. Third graders may need additional support. All students should be working with rounding and solving word problems using all four operations. Third graders need to engage in this work to 1,000. Fourth graders need to engage in this work up to 1,000,000. This is a quick unit that can be condensed.

Unit Vocab

Ten thousands, hundred thousands (as places on the place value chart)

Millions, ten millions, hundred millions (as places on the place value chart)

Variable (letters that stand for numbers and can be added, subtracted, multiplied, and divided as numbers are)

Module 4: All Materials Here

Topic A: Place Value off Multi-Digit Whole Numbers

Standards

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.

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.

Overview

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.

Lessons

Topic B: Comparing Multi-Digit Whole Numbers Using Place Value

Standards

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.

Overview

Students compare Multi-Digit Whole Numbers initially using the place value chart, then compare the value of each digit to surmise greater value. Moving toward fluency with numbers, students compare numbers by observing across the entire number and noticing value differences.

Lessons

Topic C: Rounding Multi-Digit Numbers

Standards

4.NBT.3 Use place value understanding to round multi-digit whole numbers to any place.

Overview

Comparison leads directly into rounding, where their skill with isolating units is applied and extended. Rounding to the nearest ten and hundred was mastered with three-digit numbers in Grade 3. Now, Grade 4 students moving into Topic C learn to round to any place value (4.NBT.3), initially using the vertical number line though ultimately moving away from the visual model altogether. Topic C also includes word problems where students apply rounding to real life situations.

Lessons

Topic D: Multi-Digit Whole Number Addition

Standards

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.

4.NBT.4 Fluently add and subtract multi-digit whole numbers using the standard algorithm.

Overview

In Topics D and E, students focus on single like-unit calculations (ones with ones, thousands with thousands, etc.), at times requiring the composition of greater units when adding (10 hundreds are composed into 1 thousand) and decomposition into smaller units when subtracting (1 thousand is decomposed into 10 hundreds) (4.NBT.4).

Lessons

Topic F: Multi-Digit Whole Number Subtraction

Standards

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.

4.NBT.4 Fluently add and subtract multi-digit whole numbers using the standard algorithm.

Overview

Students focus on single like-unit calculations (ones with ones, thousands with thousands, etc.), at times requiring the composition of greater units when adding (10 hundreds are composed into 1 thousand) and decomposition into smaller units when subtracting (1 thousand is decomposed into 10 hundreds) (4.NBT.4). Throughout these topics, students apply their algorithmic knowledge to solve word problems. Students also use a variable to represent the unknown quantity.

Lessons

Module 5: Multi-Digit Multiplication and Division

(35 Hours)

Standards

4.OA.1 Interpret a multiplication equation as a comparison, e.g., interpret 35 = 5 × 7 as a statement that 35 is 5 times as many as 7 and 7 times as many as 5. Represent verbal statements of multiplicative comparisons as multiplication equations.

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.

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.

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.

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.

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.

Overview

Students use place value understanding and visual representations to solve multiplication and division problems with multi-digit numbers. As a key area of focus for Grade 4, this module moves slowly but comprehensively to develop students’ ability to reason about the methods and models chosen to solve problems with multi-digit factors and dividends.

This is a major FOURTH GRADE unit. All students are to be solving two step word problems using all 4 operations. At 3rd grade, this is expected within 100 - including multiplication and division under 12. At fourth grade, this gets extended into 4 digit by 1 digit and 2 digit by 2 digit multiplication and division.

Unit Vocab

Associative property (e.g., 96 = 3 × (4 × 8) = (3 × 4) × 8)

Composite number (positive integer having three or more whole number factors)

Distributive property (e.g., 64 × 27 = (60 × 20) + (60 × 7) + (4 × 20) + (4 × 7))

Divisible

Divisor (the number by which another number is divided)

Formula (a mathematical rule expressed as an equation with numbers and/or variables)

Long division (process of dividing a large dividend using several recorded steps)

Partial product (e.g., 24 × 6 = (20 × 6) + (4 × 6) = 120 + 24)

Prime number (positive integer greater than 1 having whole number factors of only 1 and itself)

Remainder (the number left over when one integer is divided by another)

Module 5: All Materials Here

Topic A: Multiplicative Comparison Word Problems

Standards

4.OA.1 Interpret a multiplication equation as a comparison, e.g., interpret 35 = 5 × 7 as a statement that 35 is 5 times as many as 7 and 7 times as many as 5. Represent verbal statements of multiplicative comparisons as multiplication equations.

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.

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.

Overview

Students begin in Topic A by investigating the formulas for area and perimeter. They then solve multiplicative comparison problems including the language of times as much as with a focus on problems using area and perimeter as a context (e.g., “A field is 9 feet wide. It is 4 times as long as it is wide. What is the perimeter of the field?”). Students create diagrams to represent these problems as well as write equations with symbols for the unknown quantities (4.OA.1). This is foundational for understanding multiplication as scaling in Grade 5 and sets the stage for proportional reasoning in Grade 6. This Grade 4 module, beginning with area and perimeter, allows for new and interesting word problems as students learn to calculate with larger numbers and interpret more complex problems (4.OA.2, 4.OA.3, 4.MD.3).

Lessons

Topic B: Multiplication by 10, 100, and 1,000

Standards

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.

Overview

In Topic B, students use place value disks to multiply single-digit numbers by multiples of 10, 100, and 1,000 and two-digit multiples of 10 by two-digit multiples of 10 (4.NBT.5). Reasoning between arrays and written numerical work allows students to see the role of place value units in multiplication (as pictured below). Students also practice the language of units to prepare them for multiplication of a single-digit factor by a factor with up to four digits and multiplication of two two-digit factors.

Lessons

Topic C: Multiplication Four Digits by Single-Digit Numbers

Standards

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.

Overview

Building on their work in Topic B, students begin in Topic C decomposing numbers into base ten units in order to find products of single-digit by multi-digit numbers. Students use the distributive property and multiply using place value disks to model. Practice with place value disks is used for two-, three-, and four-digit by one-digit multiplication problems with recordings as partial products. Students bridge partial products to the recording of multiplication via the standard algorithm.1 Finally, the partial products method, the standard algorithm, and the area model are compared and connected by the distributive property (4.NBT.5).

Lessons

Topic D: Multiplication Word Problems

Standards

4.OA.1 Interpret a multiplication equation as a comparison, e.g., interpret 35 = 5 × 7 as a statement that 35 is 5 times as many as 7 and 7 times as many as 5. Represent verbal statements of multiplicative comparisons as multiplication equations.

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.

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.

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

Overview

Topic D gives students the opportunity to apply their new multiplication skills to solve multi-step word problems (4.OA.3, 4.NBT.5) and multiplicative comparison problems (4.OA.2). Students write equations from statements within the problems (4.OA.1) and use a combination of addition, subtraction, and multiplication to solve

Lessons

Topic E: Division of Tens and Ones

Standards

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.

Overview

In Topic E, students synthesize their Grade 3 knowledge of division types (group size unknown and number of groups unknown) with their new, deeper understanding of place value. Students focus on interpreting the remainder within division problems, both in word problems and long division (4.OA.3).

Lessons

Topic F: Divisibility

Standards

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.

Overview

In Topic F, armed with an understanding of remainders, students explore factors, multiples, and prime and composite numbers within 100 (4.OA.4), gaining valuable insights into patterns of divisibility as they test for primes and find factors and multiples. This prepares them for Topic G’s work with multi-digit dividends.

Lessons

Topic G: Thousands, Hundreds, Tens and Ones

Standards

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.

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.

Overview

Topic G extends the practice of division with three- and four-digit dividends using place value understanding. A connection to Topic B is made initially with dividing multiples of 10, 100, and 1,000 by single-digit numbers. Place value disks support students visually as they decompose each unit before dividing. Students then practice using the standard algorithm to record long division. They solve word problems and make connections to the area model as was done with two-digit dividends (4.NBT.6, 4.OA.3).

Lessons

Topic H: Multiply Two-Digit by Two-Digits

Standards

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.

Overview

Topic H culminates at the most abstract level by explicitly connecting the partial products appearing in the area model to the distributive property and recording the calculation vertically (4.NBT.5). Students see that partial products written vertically are the same as those obtained via the distributive property: 4 twenty-sixes + 30 twenty-sixes = 104 + 780 = 884. As students progress through this module, they are able to apply the multiplication and division algorithms because of their in-depth experience with the place value system and multiple conceptual models. This helps to prepare them for fluency with the multiplication algorithm in Grade 5 and the division algorithm in Grade 6. Students are encouraged in Grade 4 to continue using models to solve when appropriate.

Lessons

Module 6: Fraction Equivalence, Ordering, and Operations

(40 Hours)

Standards

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.

4.NF.1 Explain why a fraction a/b is equivalent to a fraction (n × a)/(n × b) by using visual fraction models, with attention to how the number and size of the parts differ even though the two fractions themselves are the same size. Use this principle to recognize and generate equivalent fractions.

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.

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.

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 5 × (1/4), recording the conclusion by the equation 5/4 = 5 × (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 3 × (2/5) as 6 × (1/5), recognizing this product as 6/5. (In general, n × (a/b) = (n × 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?

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.

Overview

In this 40-day module, students build on their Grade 3 work with unit fractions as they explore fraction equivalence and extend this understanding to mixed numbers. This leads to the comparison of fractions and mixed numbers and the representation of both in a variety of models. Benchmark fractions play an important part in students’ ability to generalize and reason about relative fraction and mixed number sizes. Students then have the opportunity to apply what they know to be true for whole number operations to the new concepts of fraction and mixed number operations.

This is a FOURTH GRADE UNIT as fractions are the major focus of math work in 4th grade. It is important to ensure strong conceptual understanding of fractions and working with them formally for all 4th graders. Third graders are also expected to know fractions - but only halves, thirds, fourths, sixths, and eights. Fourth graders denominators should include 2, 3, 4, 5, 6, 8, 10, 12, and 100).

Unit Vocab

Benchmark (standard or reference point by which something is measured)

Common denominator (when two or more fractions have the same denominator)

Denominator (e.g., the 5 in 3 /5 names the fractional unit as fifths)

Fraction greater than 1 (a fraction with a numerator that is greater than the denominator)

Line plot (display of data on a number line, using an x or another mark to show frequency)

Mixed number (number made up of a whole number and a fraction)

Numerator (e.g., the 3 in 3/ 5 indicates 3 fractional units are selected)

Module 6: All Materials Here

Topic A: Decomposition and Fraction Equivalence

Standards

4.NF.3b Understand a fraction a/b with a > 1 as a sum of fractions 1/b.

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.

4.NF.4a Apply and extend previous understandings of multiplication to multiply a fraction by a whole number.

a. Understand a fraction a/b as a multiple of 1/b. For example, use a visual fraction model to represent 5/4 as the product 5 × (1/4), recording the conclusion by the equation 5/4 = 5 × (1/4).

Overview

Students begin Topic A by decomposing fractions and creating tape diagrams to represent them as sums of fractions with the same denominator in different ways (4.NF.3b). They proceed to see that representing a fraction as the repeated addition of a unit fraction is the same as multiplying that unit fraction by a whole number. The introduction of multiplication as a record of the decomposition of a fraction (4.NF.4a) early in the module allows students to become familiar with the notation before they work with more complex problems. As students continue working with decomposition, they represent familiar unit fractions as the sum of smaller unit fractions. A folded paper activity allows them to see that, when the number of fractional parts in a whole increases, the size of the parts decreases. They proceed to investigate this concept with the use of tape diagrams and area models. Reasoning enables them to explain why two different fractions can represent the same portion of a whole (4.NF.1).

Lessons

Topic B: Fraction Equivalence Using Multiplication and Division

Standards

4.NF.1 Explain why a fraction a/b is equivalent to a fraction (n × a)/(n × b) by using visual fraction models, with attention to how the number and size of the parts differ even though the two fractions themselves are the same size. Use this principle to recognize and generate equivalent fractions.

Overview

In Topic B, students use tape diagrams and area models to analyze their work from earlier in the module and begin using multiplication to create an equivalent fraction that comprises smaller units (4.NF.1). Based on the use of multiplication, they reason that division can be used to create a fraction that comprises larger units (or a single unit) equivalent to a given fraction. Their work is justified using area models and tape diagrams and, conversely, multiplication is used to test for and/or verify equivalence.

Students use the tape diagram to transition to modeling equivalence on the number line. They see that, by multiplying, any unit fraction length can be partitioned into n equal lengths and that doing so multiplies both the total number of fractional units (the denominator) and number of selected units (the numerator) by n. They also see that there are times when fractional units can be grouped together, or divided, into larger fractional units. When that occurs, both the total number of fractional units and number of selected units are divided by the same number.

Lessons

Topic C: Fraction Comparison

Standards

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.

Overview

In Grade 3, students compared fractions using fraction strips and number lines with the same denominators. In Topic C, they expand on comparing fractions by reasoning about fractions with unlike denominators. Students use the relationship between the numerator and denominator of a fraction to compare to a known benchmark on the number line.

Alternatively, students compare using the same numerators. They find that the fraction with the greater denominator is the lesser fraction since the size of the fractional unit is smaller as the whole is decomposed into more equal parts. Throughout the process, their reasoning is supported using tape diagrams and number lines in cases where one numerator or denominator is a factor of the other. When the units are unrelated, students use area models and multiplication, the general method pictured below to the left, whereby two fractions are expressed in terms of the same denominators. Students also reason that comparing fractions can only be done when referring to the same whole, and they record their comparisons using the comparison symbols <, >, and = (4.NF.2).

Lessons

Topic D: Fraction Addition and Subtraction

Standards

4.NF.3ad 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.
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.

Overview

In Topic D, students apply their understanding of whole number addition (the combining of like units) and subtraction (finding an unknown part) to work with fractions (4.NF.3a). They see through visual models that, if the units are the same, computation can be performed immediately, e.g., 2 bananas + 3 bananas = 5 bananas and 2 eighths + 3 eighths = 5 eighths. They see that, when subtracting fractions from one whole, the whole is decomposed into the same units as the part being subtracted, e.g., 1 – 3/5 = 5/5 – 3/5 = 2/5 . Students practice adding more than two fractions and model fractions in word problems using tape diagrams (4.NF.3d).

Lessons

Topic E: Fractions Greater than 1

Standards

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.

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.

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.

Overview

At the beginning of Topic E, students use decomposition and visual models to add and subtract fractions less than 1 to or from whole numbers. They use addition and multiplication to build fractions greater than 1 and represent them on the number line. Students then use these visual models and decompositions to reason about the various forms in which a fraction greater than or equal to 1 may be presented, both as fractions and mixed numbers. They practice converting between these forms and begin understanding the usefulness of each form in different situations. Next, students compare fractions greater than 1, building on their rounding skills and using understanding of benchmarks to reason about which of two fractions is greater (4.NF.2). Students progress to finding and using like denominators or numerators to compare and order mixed numbers. They apply their skills of comparing numbers greater than 1 by solving word problems (4.NF.3d) requiring the interpretation of data presented in line plots (4.MD.4).

Lessons

Topic F: Additional and Subtraction of Fractions by Decomposition

Standards

4.NF.3c 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.

Overview

In Topic F, students estimate sums and differences of mixed numbers, rounding before performing the actual operation to determine what a reasonable outcome is. They proceed to use decomposition to add and subtract mixed numbers (4.NF.3c). This work builds on their understanding of a mixed number being the sum of a whole number and fraction.

Using unit form, students add and subtract like units first (e.g., ones and ones, fourths and fourths). Students use decomposition, shown with number bonds, in mixed number addition to make one from fractional units before finding the sum. When subtracting, students learn to decompose the minuend or subtrahend when there are not enough fractional units from which to subtract. Alternatively, students can rename the subtrahend, giving more units to the fractional units, which connects to whole number subtraction when renaming 9 tens 2 ones as 8 tens 12 ones.

Lessons

Topic G: Repeated addition of Fractions as Multiplication

Standards

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 5 × (1/4), recording the conclusion by the equation 5/4 = 5 × (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 3 × (2/5) as 6 × (1/5), recognizing this product as 6/5. (In general, n × (a/b) = (n × 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?

Overview

In Topic G, students build on the concept of representing repeated addition as multiplication, applying this familiar concept to work with fractions (4.NF.4a, 4.NF.4b). They use the associative property and their understanding of decomposition. Just as with whole numbers, the unit remains unchanged. This understanding connects to students’ work with place value and whole numbers. Students proceed to explore the use of the distributive property to multiply a whole number by a mixed number. They recognize that they are multiplying each part of a mixed number by the whole number and use efficient strategies to do so. The topic closes with solving multiplicative comparison word problems involving fractions (4.NF.4c) as well as problems involving the interpretation of data presented on a line plot.

Lessons