# Fraction Equivalence and Ordering

## Objective

Recognize and generate equivalent fractions with larger units using division.

## Common Core Standards

### Core Standards

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• 4.NF.A.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.

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• 3.NF.A.3

• 4.OA.A.1

• 4.OA.A.2

## Criteria for Success

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1. Develop a general method for finding equivalent fractions by dividing both the numerator and the denominator by the same non-zero value.
2. Understand how the numbers and sizes of parts differ even though the two fractions are the same size, and connect this idea to the general method of using division to find an equivalent fraction (MP.7).
3. Generate equivalent fractions with larger units using the general method.
4. Find an equivalent fraction with the largest possible unit using the general method.
5. Determine whether two fractions are equivalent using division and support that work with a visual model (MP.3, MP.5).

## Tips for Teachers

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• The term “simplify”/“simplification” is intentionally excluded from CCSS since “it is possible to over-emphasize the importance of simplifying fractions in this way. There is no mathematical reason why fractions must be written in simplified form, although it may be convenient to do so in some cases” (NF Progression, p. 6). It is important for students to understand the term at some point, but because the focus of this unit is purely on equivalent fractions rather than some of those fractions being “simpler” than others, the term is excluded from the unit. Instead, the conversation centers on the number and sizes of the parts, in keeping with the language of the standard (4.NF.1). Students are prompted to find fractions in the largest possible terms so that they have practice doing so for when it makes sense to do so, but they aren’t expected to do so in every task in the Anchor Tasks and Problem Set.
• The supporting work of gaining familiarity with factors and multiples (4.OA.4) supports this major work of generating equivalent fractions with larger units using division, since students must find a common factor of both the numerator and denominator in order to be able to divide.
• As a supplement to the Problem Set, students can play a game to practice finding equivalent fractions with larger units with increasing efficiency, such as “Fraction Scattergories" or “Fraction Taboo” and “SlapFrac” (both of which would likely involve practice of finding equivalent fractions with both larger AND smaller units) from Games with Fraction Strips and Fraction Cards on The Max Ray Blog

#### Fishtank Plus

• Problem Set
• Student Handout Editor
• Vocabulary Package

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### Problem 1

Jamari says that because ${{1\over2}={{1\times3}\over{2\times3}}={3\over6}}$, that must also mean ${{3\over6} = {{3\div3}\over{6\div3}} = {1\over2}}$. Do you agree with Jamari? Why or why not?

### Problem 2

Find an equivalent fraction with larger units for each of the following:

a.      ${{6\over9}}$

b.     ${{15\over10}}$

### Problem 3

Find an equivalent fraction with the largest unit for each of the following:

a.   ${{4\over8}}$

b.   $\frac{24}{18}$

## Problem Set & Homework

#### Discussion of Problem Set

• Look at #1a and #1c. Both fractions have a denominator of 15, but their equivalent fractions have different denominators. Why?
• Look at #2. What happened to the size of the fractional pieces? What happened to the number of them? How is that related to the computational way of finding an equivalent fraction?
• In #4, how is it helpful to know the common factors for the numerators and denominators?
• In #4, you were asked to use the largest common factor to rename the fraction: $\frac{4}{12}=\frac{1}{3}$. By doing so, you renamed $\frac{4}{12}$ using larger units. How is renaming fractions useful?
• Do fractions always need to be renamed to the largest unit? Explain.
• How can you tell that a fraction is composed of the largest possible units?
• What equivalent fractions did you generate in #5? Is there more than one correct answer?

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### Problem 1

Find two fractions that are equivalent to ${{{12}\over18}}$ and whose numerators are less than 12.

#### References

Achieve the Core Fraction Concepts Mini-AssessmentQuestion #6

Fraction Concepts Mini-Assessment by Student Achievement Partners is made available by Achieve the Core under the CC0 1.0 license. Accessed March 23, 2018, 9:16 a.m..

Modified by The Match Foundation, Inc.

### Problem 2

Explain how you know $\frac{21}{28}$ is equivalent to $\frac{21\div 7}{28\div7}$. Use a model to support your reasoning.

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