Math Unit Five

Grade 4 Module 5: Fraction Equivalence, Ordering, and Operations

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.

New or Recently Introduced Terms

 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)

Familiar Terms and Symbols

 =, <, > (equal to, less than, greater than)

 Compose (change a smaller unit for an equivalent of a larger unit, e.g., 2 fourths = 1 half, 10 ones = 1 ten; combining 2 or more numbers, e.g., 1 fourth + 1 fourth = 2 fourths, 2 + 2 + 1 = 5)

 Decompose (change a larger unit for an equivalent of a smaller unit, e.g., 1 half = 2 fourths, 1 ten = 10 ones; partition a number into 2 or more parts, e.g., 2 fourths = 1 fourth + 1 fourth, 5 = 2 + 2 + 1)

 Equivalent fractions (fractions that name the same size or amount)

 Fraction (e.g., 1 3 , 2 3 , 3 3 , 4 3 )

 Fractional unit (e.g., half, third, fourth)

 Multiple (product of a given number and any other whole number)

 Non-unit fraction (fractions with numerators other than 1)

 Unit fraction (fractions with numerator 1)

 Unit interval (e.g., the interval from 0 to 1, measured by length)

 Whole (e.g., 2 halves, 3 thirds, 4 fourths)

A. Decomposition and Fraction Equivalence

Lessons 1–2: Decompose fractions as a sum of unit fractions using tape diagrams. Lesson 3: Decompose non-unit fractions and represent them as a whole number times a unit fraction using tape diagrams. Lesson 4: Decompose fractions into sums of smaller unit fractions using tape diagrams. Lesson 5: Decompose unit fractions using area models to show equivalence. Lesson 6: Decompose fractions using area models to show equivalence.

B Fraction Equivalence Using Multiplication and Division

Lessons 7–8: Use the area model and multiplication to show the equivalence of two fractions. Lessons 9–10: Use the area model and division to show the equivalence of two fractions. Lesson 11: Explain fraction equivalence using a tape diagram and the number line, and relate that to the use of multiplication and division.

C Fraction Comparison

Lessons 12–13: Reason using benchmarks to compare two fractions on the number line. Lessons 14–15: Find common units or number of units to compare two fractions.

D Fraction Addition and Subtraction

Lesson 16: Use visual models to add and subtract two fractions with the same units. Lesson 17: Use visual models to add and subtract two fractions with the same units, including subtracting from one whole. Lesson 18: Add and subtract more than two fractions. Lesson 19: Solve word problems involving addition and subtraction of fractions. Lessons 20–21: Use visual models to add two fractions with related units using the denominators 2, 3, 4, 5, 6, 8, 10, and 12.

E Extending Fraction Equivalence to Fractions Greater Than 1

Lesson 22: Add a fraction less than 1 to, or subtract a fraction less than 1 from, a whole number using decomposition and visual models. Lesson 23: Add and multiply unit fractions to build fractions greater than 1 using visual models. Lessons 24–25: Decompose and compose fractions greater than 1 to express them in various forms. Lesson 26: Compare fractions greater than 1 by reasoning using benchmark fractions. Lesson 27: Compare fractions greater than 1 by creating common numerators or denominators. Lesson 28: Solve word problems with line plots.

F Addition and Subtraction of Fractions by Decomposition

Lesson 29: Estimate sums and differences using benchmark numbers. Lesson 30: Add a mixed number and a fraction. Lesson 31: Add mixed numbers. Lesson 32: Subtract a fraction from a mixed number. Lesson 33: Subtract a mixed number from a mixed number. Lesson 34: Subtract mixed numbers.

G Repeated Addition of Fractions as Multiplication

Lessons 35–36: Represent the multiplication of n times a/b as (n × a)/b using the associative property and visual models. Lessons 37–38: Find the product of a whole number and a mixed number using the distributive property. Lesson 39: Solve multiplicative comparison word problems involving fractions. Lesson 40: Solve word problems involving the multiplication of a whole number and a fraction including those involving line plots.

H Exploring a Fraction Pattern

Lesson 41: Find and use a pattern to calculate the sum of all fractional parts between 0 and 1. Share and critique peer strategies.