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 <, and justify the conclusions.
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Section A: Introduction to Fractions

Problem 1

Pre-unit

Practicing Standards:  2.G.A.2

Partition the rectangle into 10 equal squares.

Diagram. Rectangle.

Solution

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Problem 2

Pre-unit

Practicing Standards:  2.G.A.3

Here are two equal-size squares. A part of each square is shaded.

Is the same amount of each square shaded? Explain or show your reasoning.

Diagram. Square partitioned into 4 equal parts, 1 part shaded.
A square, partly shaded.

Solution

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Problem 3

Pre-unit

Practicing Standards:  2.MD.B.6

Number line. First tick mark 0, last tick mark 100. 9 unlabeled tick marks between 0 and 100.

  1. Label the tick marks on the number line.
  2. Locate and label 45 and 62 on the number line.

Solution

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Problem 4

Pre-unit

Practicing Standards:  2.NBT.A.4

Fill in each blank with \(<\) or \(>\) to compare the numbers.

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

  2. \(106\, \underline{\hspace{1cm}} \,89\)

  3. \(806\, \underline{\hspace{1cm}} \,809\)

Solution

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Problem 5

Partition the rectangle into 6 equal parts.

Diagram. Rectangle.

Solution

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Problem 6

  1. What fraction of the rectangle is shaded?

    Diagram. Rectangle partitioned into 6 equal parts, one of them shaded.

  2. Partition the rectangle into 8 equal parts.

    What fraction of the whole rectangle does each part represent?

    Diagram. Rectangle.

Solution

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Problem 7

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

    Diagram. Rectangle partitioned into 8 equal parts, 5 parts shaded.

  2. Shade \(\frac{4}{6}\) of the rectangle.

    Diagram. 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.

Diagram. Rectangle.

Solution

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Problem 9

Exploration

Write a situation represented by the diagram. Explain why the diagram represents your situation.

Diagram. Rectangle partitioned into 8 equal parts, 6 parts shaded.

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.

  1.  
    Diagram. Rectangle partitioned into 2 parts, 1 part shaded.
  2.  
    Diagram. Rectangle split into 2 unequal parts, one part shaded.
  3.  
    Diagram. Rectangle partitioned into 2 unequal parts, 1 part shaded.

Solution

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Section B: Fractions on the Number Line

Problem 1

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

    Number line. Five evenly spaced tick marks, 0 and 1 labeled. First tick mark, 0. Last tick mark, 1.

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

    Number line. Tick marks labeled zero and one with unlabeled tick marks between.

Solution

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Problem 2

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

    Number line. Two tick marks, labeled 0 and 1.

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

    Number line. Evenly spaced tick marks labeled 0, 1, and 2.

Solution

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Problem 3

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

    Number line. Evenly spaced tick marks labeled 0, 1, and 2.

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

    Number line. Evenly spaced tick marks labeled 0, 1, and 2.

  3. Diego marks and labels fourths on the number line like this:

    Number line. Tick marks labeled 0, 1 fourth, 2 fourths, 3 fourths, 1, 4 fourths, and 2. Larger space between 4 fourths and 2.

    Do you agree with Diego? Explain your reasoning.

Solution

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Problem 4

  1. Label the tick marks on the number line.

    Number line. Tick marks labeled zero and one fourth. The rest of the tick marks unlabeled.
  2. 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.

Number line. Tick marks labeled 0 and 5 thirds.

Solution

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Problem 6

Exploration

How are the fraction strip and number line the same? How are they different?

Diagram. Rectangle partitioned into 8 equal parts, each labeled one eighth.
 
Number line. Scale 0 to 1 by eighths. Evenly spaced tick marks. First tick mark, 0. Last tick mark, 1.

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?

Number line. Two tick marks labeled 0 and 15 eighths.

Solution

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Section C: Equivalent Fractions

Problem 1

Select all correct statements.

Diagram. Rectangle partitioned into 2 equal parts, each labeled one half.
Diagram. Rectangle partitioned into 3 equal parts, each labeled one third.
Diagram. Rectangle partitioned into 4 equal parts, each part labeled one fourth.
Diagram. Rectangle partitioned into 6 equal parts, each labeled one sixth.

A:

\(\frac{1}{2}\) is equivalent to \(\frac{3}{6}\)

B:

\(\frac{1}{2}\) is equivalent to \(\frac{1}{3}\)

C:

\(\frac{2}{2}\) is equivalent to \(\frac{4}{4}\)

D:

\(\frac{2}{2}\) is equivalent to \(\frac{6}{6}\)

E:

\(\frac{2}{3}\) is equivalent to \(\frac{4}{6}\)

F:

\(\frac{2}{3}\) 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.

a
b

Solution

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Problem 3

  1. Tyler draws this picture and says that \(\frac{3}{4}\) is equivalent to \(\frac{2}{3}\). Explain why Tyler is not correct.

    2 number lines. First, 0 to 2 thirds by thirds, unevenly spaced tick marks. Second, 0 to 3 fourths by fourths, unevenly spaced tick marks.
  2. Find a fraction equivalent to \(\frac{2}{3}\).
  3. Find a fraction equivalent to \(\frac{3}{4}\).

Solution

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Problem 4

  1. Write 10 as a fraction in 2 different ways.
  2. 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.

  1. \(\frac{100}{2}\)
  2. \(\frac{100}{3}\)
  3. \(\frac{100}{4}\)
  4. \(\frac{100}{6}\)
  5. \(\frac{100}{8}\)

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

  1. Are \(\frac{2}{3}\) and \(\frac{4}{6}\) equivalent? Show your thinking using diagrams, symbols, or other representations.
  2. 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.

  1. \(\frac{5}{3} \, \underline{\phantom{\frac{1}{1}\hspace{1.05cm}}} \, \frac{5}{2}\)
  2. \(\frac{3}{4} \, \underline{\phantom{\frac{1}{1}\hspace{1.05cm}}} \, \frac{5}{4}\)

Solution

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Problem 4

  1. 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.
  2. 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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