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.
Section A: Size and Location of Fractions
Problem 1
Preunit
Practicing Standards: 3.NF.A.1
What fraction of each figure is shaded?
Solution
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Problem 2
Preunit
Practicing Standards: 3.NF.A.1
Explain why the shaded portion represents \(\frac18\) of the full rectangle.
Solution
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Problem 3
Preunit
Practicing Standards: 3.NF.A.2.a
Label each tick mark with the number it represents. Explain your reasoning.
Solution
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Problem 4
Preunit
Practicing Standards: 3.NF.A.3.b
Explain or show why \(\frac12\) and \(\frac24\) are equivalent fractions.
Solution
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Problem 5

The entire diagram represents 1 whole. Shade the diagram to represent \(\frac14\).
 To represent \(\frac16\) on the tape diagram, would we shade more or less than what we did for \(\frac14\)? Explain your reasoning.
Solution
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Problem 6

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

Shade this diagram to represent \(\frac{2}{10}\).
Solution
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Problem 7
For each pair of fractions, decide which is greater. Explain or show your reasoning.
 \(\frac18\) or \(\frac{1}{10}\)
 \(\frac{4}{10}\) or \(\frac{7}{10}\)
 \(\frac45\) or \(\frac54\)
Solution
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Problem 8
Use the fraction strips to name three pairs of equivalent fractions. Explain how you know the fractions are equivalent.
Solution
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Problem 9

Explain or show why the point on the number line describes both \(\frac35\) and \(\frac{6}{10}\).
 Explain why \(\frac{6}{10}\) and \(\frac35\) are equivalent fractions.
Solution
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Problem 10
For each question, explain your reasoning. Use a number line if you find it helpful.

Is \(\frac45\) more or less than \(\frac12\)?

Is \(\frac45\) more or less than 1?
Solution
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Problem 11
Exploration
Make fraction strips for each of these fractions. How did you fold the paper to make sure you have the rightsize parts?

\(\frac13\)s

\(\frac15\)s

\(\frac{1}{10}\)s
Solution
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Problem 12
Exploration
 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.
 What relationship do you see between \(\frac16\) and \(\frac14\)? Explain your reasoning.
 Can you find a relationship between \(\frac{1}{6}\) and \(\frac{1}{8}\) using fraction strips?
Solution
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Section B: Equivalent Fractions
Problem 1
Name three fractions that are equivalent to \(\frac25\). Explain or show your reasoning.
Solution
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Problem 2
Which of these could be the fraction that the point represents? Explain your reasoning.
\(\frac{86}{100}\)
\(\frac{90}{100}\)
\(\frac{94}{100}\)
\(\frac{101}{100}\)
Solution
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Problem 3
Explain why the fractions \(\frac{10}{3}\) and \(\frac{40}{12}\) are equivalent.
Solution
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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.
Solution
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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 \(\frac75\).
 Explain why Jada’s reasoning is correct.
 Use Jada’s method to find another fraction equivalent to \(\frac75\).
Solution
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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.
 My fraction is equivalent to \(\frac23\).
 The numerator of my fraction is greater than 10.
 8 is a factor of my numerator.
 8 and 5 are a factor pair of my numerator.
Solution
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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.

My fraction is equivalent to \(\underline{\hspace{1.5cm}}\).

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

One multiple of my numerator is \(\underline{\hspace{1.5cm}}\).
 A factor pair of my denominator is \(\underline{\hspace{1.5cm}}\) and \(\underline{\hspace{1.5cm}}\).
Solution
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Problem 8
Exploration

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

Shade \(\frac{18}{24}\) of the diagram. Explain how you know \(\frac{18}{24}\) is shaded.
Solution
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Section C: Fraction Comparison
Problem 1
For each pair of fractions, decide which fraction is greater. Explain or show your reasoning.
 \(\frac25\) or \(\frac26\)
 \(\frac58\) or \(\frac78\)
 \(\frac{9}{10}\) or \(\frac{103}{100}\)
Solution
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Problem 2
Use a \(<\), \(=\), or \(>\) to make each statement true. Explain or show your reasoning.
 \(\frac{2}{3} \> \underline{\phantom{ \hspace{0.7cm} }} \> \frac{10}{15}\)
 \(\frac15 \> \underline{\phantom{ \hspace{0.7cm} }} \> \frac{22}{100}\)
 \(\frac{10}{4} \> \underline{\phantom{ \hspace{0.7cm} }} \> \frac{45}{20}\)
Solution
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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.
Solution
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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 \(\frac78\).
Solution
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Problem 5
List these fractions from least to greatest. Explain or show your reasoning.
 \(\frac13\)
 \(\frac{5}{12}\)
 \(\frac{2}{10}\)
Solution
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Problem 6
List these fractions from least to greatest. Explain or show your reasoning.
 \(\frac{15}{8}\)
 \(\frac{215}{100}\)
 \(\frac{7}{4}\)
 \(\frac{21}{10}\)
Solution
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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
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Problem 8
Exploration
Find a fraction that is between \(\frac25\) and \(\frac38\). Explain or show your reasoning.
Solution
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