3.5 Fractions as Numbers
Unit Goals
 Students develop an understanding of fractions as numbers and of fraction equivalence by representing fractions on diagrams and number lines, generating equivalent fractions, and comparing fractions.
Section A Goals
 Understand that fractions are built from unit fractions such that a fraction $\frac{a}{b}$ is the quantity formed by $a$ parts of size $\frac{1}{b}$.
 Understand that unit fractions are formed by partitioning shapes into equal parts.
Section B Goals
 Understand a fraction as a number and represent fractions on the number line.
Section C Goals
 Explain equivalence of fractions in special cases and express whole numbers as fractions and fractions as whole numbers.
Section D Goals
 Compare two fractions with the same numerator or denominator, record the results with the symbols >, =, or
Section A: Introduction to Fractions
Problem 1
Preunit
Practicing Standards: 2.G.A.2
Partition the rectangle into 10 equal squares.
Solution
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Problem 2
Preunit
Practicing Standards: 2.G.A.3
Here are two equalsize squares. A part of each square is shaded.
Is the same amount of each square shaded? Explain or show your reasoning.
Solution
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Problem 3
Preunit
Practicing Standards: 2.MD.B.6
 Label the tick marks on the number line.
 Locate and label 45 and 62 on the number line.
Solution
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Problem 4
Preunit
Practicing Standards: 2.NBT.A.4
Fill in each blank with \(<\) or \(>\) to compare the numbers.

\(718\, \underline{\hspace{1cm}}\, 817\)

\(106\, \underline{\hspace{1cm}} \,89\)

\(806\, \underline{\hspace{1cm}} \,809\)
Solution
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Problem 5
Partition the rectangle into 6 equal parts.
Solution
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Problem 6

What fraction of the rectangle is shaded?

Partition the rectangle into 8 equal parts.
What fraction of the whole rectangle does each part represent?
Solution
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Problem 7

What fraction of the rectangle is shaded? Explain how you know.

Shade \(\frac{4}{6}\) of the rectangle.
Solution
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Problem 8
Jada walks across the street at a stoplight \(\frac{5}{6}\) of her way from home to school. Represent the situation on the fraction strip. Explain your reasoning.
Solution
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Problem 9
Exploration
Write a situation represented by the diagram. Explain why the diagram represents your situation.
Solution
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Problem 10
Exploration
Lin shaded part of some fraction strips. What fraction did she shade in each one? Explain how you know.
Solution
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Section B: Fractions on the Number Line
Problem 1

Locate and label \(\frac{1}{4}\) on the number line. Explain your reasoning.

Locate and label \(\frac{1}{6}\) on the number line. Explain your reasoning.
Solution
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Problem 2

Locate and label \(\frac{1}{8}\) on the number line.

Locate and label \(\frac{1}{3}\) on the number line.
Solution
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Problem 3

Locate and label \(\frac{4}{8}\) on the number line.

Locate and label \(\frac{7}{6}\) on the number line.

Diego marks and labels fourths on the number line like this:
Do you agree with Diego? Explain your reasoning.
Solution
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Problem 4

Label the tick marks on the number line.
 Which numbers on the number line are whole numbers? Explain how you know.
Solution
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Problem 5
Locate and label 1 on the number line. Explain your reasoning.
Solution
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Problem 6
Exploration
How are the fraction strip and number line the same? How are they different?
Solution
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Problem 7
Exploration
Han says that he can find 1 on the number line without finding \(\frac{1}{8}\). What might Han’s method be?
Solution
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Section C: Equivalent Fractions
Problem 1
Select all correct statements.
\(\frac{1}{2}\) is equivalent to \(\frac{3}{6}\)
\(\frac{1}{2}\) is equivalent to \(\frac{1}{3}\)
\(\frac{2}{2}\) is equivalent to \(\frac{4}{4}\)
\(\frac{2}{2}\) is equivalent to \(\frac{6}{6}\)
\(\frac{2}{3}\) is equivalent to \(\frac{4}{6}\)
\(\frac{2}{3}\) is equivalent to \(\frac{3}{4}\)
Solution
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Problem 2
Write as many fractions as you can that represent the shaded part of each diagram.
Solution
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Problem 3

Tyler draws this picture and says that \(\frac{3}{4}\) is equivalent to \(\frac{2}{3}\). Explain why Tyler is not correct.
 Find a fraction equivalent to \(\frac{2}{3}\).
 Find a fraction equivalent to \(\frac{3}{4}\).
Solution
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Problem 4
 Write 10 as a fraction in 2 different ways.
 Is \(\frac{88}{8}\) equivalent to a whole number?
Solution
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Problem 5
Exploration
Decide if each fraction is a whole number. Explain or show your reasoning.
 \(\frac{100}{2}\)
 \(\frac{100}{3}\)
 \(\frac{100}{4}\)
 \(\frac{100}{6}\)
 \(\frac{100}{8}\)
Solution
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Problem 6
Exploration
If you continue to fold fraction strips, how many parts can you fold them into? Can you fold them into 100 equal parts?
Solution
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Section D: Fraction Comparisons
Problem 1
 Are \(\frac{2}{3}\) and \(\frac{4}{6}\) equivalent? Show your thinking using diagrams, symbols, or other representations.
 Are \(\frac{6}{8}\) and \(\frac{7}{8}\) equivalent? Show your thinking using diagrams, symbols, or other representations.
Solution
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Problem 2
Han says there is no fraction with denominator 8 that's greater than \(\frac{8}{8}\) because \(\frac{8}{8}\) is a whole. Do you agree with Han? Explain your reasoning.
Solution
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Problem 3
Use the symbols \(>\) or \(<\) to make each statement true. Explain your reasoning.
 \(\frac{5}{3} \, \underline{\phantom{\frac{1}{1}\hspace{1.05cm}}} \, \frac{5}{2}\)
 \(\frac{3}{4} \, \underline{\phantom{\frac{1}{1}\hspace{1.05cm}}} \, \frac{5}{4}\)
Solution
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Problem 4
 Jada threw the ball \(\frac{3}{4}\) of the length of the gym. Clare threw the ball \(\frac{6}{8}\) of the length of the gym. Clare says she threw the ball farther. Do you agree? Show your thinking.
 Tyler kicked the ball \(\frac{7}{8}\) the length of the playground. Andre kicked the ball \(\frac{7}{6}\) the length of the playground. Andre says he kicked the ball farther. Do you agree? Show your thinking.
Solution
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Problem 5
Exploration
Clare walked \(\frac{3}{4}\) of the way around a park. Tyler walked \(\frac{3}{6}\) of the way around a different park. Who walked farther? Explain your reasoning.
Solution
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Problem 6
Exploration
Choose a fraction that you can compare with both \(\frac{3}{8}\) and \(\frac{5}{6}\) by looking at the numerators and denominators.
Solution
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