# 4.2 Fraction Equivalence and Comparison

## Unit Goals

• Students generate and reason about equivalent fractions and compare and order fractions with the following denominators: 2, 3, 4, 5, 6, 8, 10, 12, and 100.

### Section A Goals

• Make sense of fractions with denominators 2, 3, 4, 5, 6, 8, 10, and 12 through physical representations and diagrams.
• Reason about the location of fractions on the number line.

### Section B Goals

• Generate equivalent fractions with the following denominators: 2, 3, 4, 5, 6, 8, 10, 12, and 100.
• Use visual representations to reason about fraction equivalence, including using benchmarks such as $\frac{1}{2}$ and 1.

### Section C Goals

• Use visual representations or a numerical process to reason about fraction comparison.

### Problem 1

#### Pre-unit

Practicing Standards:  3.NF.A.1

What fraction of each figure is shaded?

### Problem 2

#### Pre-unit

Practicing Standards:  3.NF.A.1

Explain why the shaded portion represents $$\frac18$$ of the full rectangle.

### Problem 3

#### Pre-unit

Practicing Standards:  3.NF.A.2.a

Label each tick mark with the number it represents. Explain your reasoning.

### Problem 4

#### Pre-unit

Practicing Standards:  3.NF.A.3.b

Explain or show why $$\frac12$$ and $$\frac24$$ are equivalent fractions.

### Problem 5

1. The entire diagram represents 1 whole. Shade the diagram to represent $$\frac14$$.

2. To represent $$\frac16$$ on the tape diagram, would we shade more or less than what we did for $$\frac14$$? Explain your reasoning.

### Problem 6

1. The entire diagram represents 1 whole. What fraction does the shaded portion represent? Explain your reasoning.

2. Shade this diagram to represent $$\frac{2}{10}$$.

### Problem 7

For each pair of fractions, decide which is greater. Explain or show your reasoning.

1. $$\frac18$$ or $$\frac{1}{10}$$
2. $$\frac{4}{10}$$ or $$\frac{7}{10}$$
3. $$\frac45$$ or $$\frac54$$

### Problem 8

Use the fraction strips to name three pairs of equivalent fractions. Explain how you know the fractions are equivalent.

### Problem 9

1. Explain or show why the point on the number line describes both $$\frac35$$ and $$\frac{6}{10}$$.

2. Explain why $$\frac{6}{10}$$ and $$\frac35$$ are equivalent fractions.

### Problem 10

For each question, explain your reasoning. Use a number line if you find it helpful.

1. Is $$\frac45$$ more or less than $$\frac12$$?

2. Is $$\frac45$$ more or less than 1?

### Problem 11

#### Exploration

Make fraction strips for each of these fractions. How did you fold the paper to make sure you have the right-size parts?

1. $$\frac13$$s

2. $$\frac15$$s

3. $$\frac{1}{10}$$s

### Problem 12

#### Exploration

1. Andre looks at these fraction strips and says “Each $$\frac12$$ is $$\frac13$$ and another half of $$\frac13$$”. Do you agree with Andre? Explain your reasoning.

2. What relationship do you see between $$\frac16$$ and $$\frac14$$? Explain your reasoning.

3. Can you find a relationship between $$\frac{1}{6}$$ and $$\frac{1}{8}$$ using fraction strips?

### Problem 1

Name three fractions that are equivalent to $$\frac25$$. Explain or show your reasoning.

### Problem 2

Which of these could be the fraction that the point represents? Explain your reasoning.

$$\frac{86}{100}$$

$$\frac{90}{100}$$

$$\frac{94}{100}$$

$$\frac{101}{100}$$

### Problem 3

Explain why the fractions $$\frac{10}{3}$$ and $$\frac{40}{12}$$ are equivalent.

### Problem 4

Find two fractions equivalent to $$\frac{10}{6}$$. Explain or show why they are equivalent to $$\frac{10}{6}$$. Use the number line if you think it is helpful.

### Problem 5

Jada says that $$\frac75$$ is equivalent to $$\frac{14}{10}$$ because the numerator and denominator of $$\frac{14}{10}$$ are each 2 times the numerator and denominator of $$\frac75$$.

1. Explain why Jada’s reasoning is correct.

2. Use Jada’s method to find another fraction equivalent to $$\frac75$$.

### Problem 6

#### Exploration

Jada is thinking of a fraction. She gives several clues to help you guess her fraction. Try to guess Jada’s fraction after each clue.

1. My fraction is equivalent to $$\frac23$$.
2. The numerator of my fraction is greater than 10.
3. 8 is a factor of my numerator.
4. 8 and 5 are a factor pair of my numerator.

### Problem 7

#### Exploration

Think of a fraction: $$\underline{\hspace{1.5cm}}$$

Write several clues so a friend or family member can guess your fraction. Then, present the clues one at a time and ask them to make a guess after each one.

1. My fraction is equivalent to $$\underline{\hspace{1.5cm}}$$.

2. The numerator of my fraction is less than $$\underline{\hspace{1.5cm}}$$.

3. One multiple of my numerator is $$\underline{\hspace{1.5cm}}$$.

4. A factor pair of my denominator is $$\underline{\hspace{1.5cm}}$$ and $$\underline{\hspace{1.5cm}}$$.

### Problem 8

#### Exploration

1. Diego says he shaded $$\frac{10}{20}$$ of the diagram. Do you agree with Diego? Explain your reasoning.

2. Shade $$\frac{18}{24}$$ of the diagram. Explain how you know $$\frac{18}{24}$$ is shaded.

### Problem 1

For each pair of fractions, decide which fraction is greater. Explain or show your reasoning.

1. $$\frac25$$ or $$\frac26$$
2. $$\frac58$$ or $$\frac78$$
3. $$\frac{9}{10}$$ or $$\frac{103}{100}$$

### Problem 2

Use a $$<$$, $$=$$, or $$>$$ to make each statement true. Explain or show your reasoning.

1. $$\frac{2}{3} \> \underline{\phantom{ \hspace{0.7cm} }} \> \frac{10}{15}$$
2. $$\frac15 \> \underline{\phantom{ \hspace{0.7cm} }} \> \frac{22}{100}$$
3. $$\frac{10}{4} \> \underline{\phantom{ \hspace{0.7cm} }} \> \frac{45}{20}$$

### Problem 3

There is a water fountain $$\frac{7}{10}$$ mile from the start of a hiking trail. There is a pond $$\frac{3}{5}$$ mile from the start of the trail. If a hiker begins walking at the start of the trail, which will they come across first, the water fountain or the pond? Explain your reasoning.

### Problem 4

Tyler said he grew $$\frac32$$ centimeters since his height was measured six months ago.

Diego said, “Oh, you grew more than I did! My height went up only by $$\frac78$$ inch in the past six months.”

Explain why Tyler may not have grown more than Diego did, even though $$\frac32$$ is greater than $$\frac78$$.

### Problem 5

List these fractions from least to greatest. Explain or show your reasoning.

• $$\frac13$$
• $$\frac{5}{12}$$
• $$\frac{2}{10}$$

### Problem 6

List these fractions from least to greatest. Explain or show your reasoning.

• $$\frac{15}{8}$$
• $$\frac{215}{100}$$
• $$\frac{7}{4}$$
• $$\frac{21}{10}$$

### Problem 7

#### Exploration

Jada lists these fractions that are all equivalent to $$\frac12$$: $$\quad \frac24, \frac{3}{6}, \frac{4}{8}, \frac{5}{10}$$

She notices that each time the numerator increases by 1 and the denominator increases by 2. Will the pattern Jada notices continue? Explain your reasoning.

### Solution

Find a fraction that is between $$\frac25$$ and $$\frac38$$. Explain or show your reasoning.