Section A Practice Problems

Problem 1

What fraction of each figure is shaded?

square partitioned into 4 equal parts, 1 part shaded

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Solution

$\frac{1}{3}$ of the circle and $\frac{1}{4}$ of the square

Problem 2

Explain why the shaded part represents $\frac18$ of the whole rectangle.

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Solution

There are 8 equal parts in the rectangle, and 1 part is shaded.

Problem 3

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

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Solution

Some students may label $\frac24$ as $\frac12$ .

There are 4 equal parts, so each part is $\frac14$ .

Problem 4

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

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Solution

Sample response:

The shaded portion represents

\frac24

because 2 of 4 equal parts of the diagram are shaded. It also represents

\frac12

because the shaded portion is 1 of 2 equal parts of the diagram.

Problem 5

The whole diagram represents 1. Shade the diagram to represent $\frac14$ .
To represent $\frac16$ on the diagram, would you shade more or less than what you shaded for $\frac14$ ? Explain your reasoning.

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Solution

Sample response:
Less. Sample response: When there are 6 equal parts in 1 whole, each part is smaller than when there are 4 parts in the same size whole.

Problem 6

The whole diagram represents 1. What fraction does the shaded part represent? Explain your reasoning.
Shade this diagram to represent $\frac{2}{10}$ .

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Solution

$\frac{7}{10}$ . Sample response: The rectangle is broken into 10 equal parts, and 7 of them are shaded.
Sample response:

Diagram. 10 equal parts, 7 parts shaded.

Problem 7

Circle the greater fraction in each pair. Explain or show your reasoning.

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

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Solution

Sample reasoning: (Students may create a diagram to show their reasoning for each question.)

$\frac18$ . There are fewer eighths in a whole than tenths in a whole.
$\frac{7}{10}$ . I know that 7 tenths is more than 4 tenths.
$\frac54$ . Since $\frac44$ is 1 whole, 5 fourths is greater than 1. Since $\frac55$ is 1 whole, and 4 fifths is less than 1.

Problem 8

Use the fraction strips to name 3 pairs of equivalent fractions. Explain how you know the fractions are equivalent.

Two diagrams of equal length. Top diagram, 12 equal parts, each labeled 1 twelfth. Bottom diagram, 6 equal parts, each labeled 1 sixth.

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Solution

Sample responses:

\frac{2}{12}

and

\frac16

\frac{4}{12}

and

\frac26

\frac{6}{12}

and

\frac36

. The fractions in each pair represent the same size, but they are formed by d with different sized parts.

Problem 9

Explain or show why the point on the number line represents both $\frac35$ and $\frac{6}{10}$ .
Explain why $\frac{6}{10}$ and $\frac35$ are equivalent fractions.

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Solution

Sample response: There are ticks marking 10 equal-size spaces between 0 and 1 and the point is on the 6th tick mark from 0, so that’s $\frac{6}{10}$ . The 10 spaces can be seen as 5 groups of 2 spaces, or fifths, and the point is on the third tick that marks fifths, so that’s $\frac35$ .
$\frac{6}{10}$ and $\frac35$ are equivalent fractions because they represent the same point on the number line or the same distance from 0.

Problem 10

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

Is $\frac45$ greater than or less than $\frac12$ ?
Is $\frac45$ greater than or less than 1?

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Solution

$\frac45$ is greater than $\frac12$ because it is more than half way from 0 to 1.
$\frac45$ is less than 1 because one whole is $\frac55$ and $\frac45$ is less than $\frac55$ (or because the point for $\frac45$ is to the left of 1 on the number line).

Problem 11

Fold a strip of paper to represent each fraction. How did you fold the paper to make sure the parts are the correct size for each fraction? Use the blank diagrams to show how you folded.

$\frac13$ s
$\frac15$ s
$\frac{1}{10}$ s

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Solution

Sample response: For thirds, I folded the two ends so that they met in the middle.
Sample response: For fifths, I tried to fold the paper 4 times, to the left, right, left and right. Then I tried to get the folds to be close to equal.
Sample response: For tenths, I just folded my fifths back up and then folded that in half since there are two tenths in each fifth.

Problem 12

Andre looks at these fraction strips and says, “Each $\frac12$ is the same size as $\frac13$ 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? Explain your reasoning.

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Solution

Sample responses:

Andre is correct. The fraction strips show that there are three $\frac13$ s in 1 and two $\frac12$ s in 1, so that means there is one $\frac13$ and another half of $\frac13$ in each $\frac12$ .
$\frac36$ is the same as $\frac24$ , and that means that there is one $\frac16$ and another half of $\frac16$ in each $\frac14$ .
It looks like $\frac16$ is $\frac18$ and maybe another third of $\frac18$ . I can see from the fraction strips that $\frac36$ is $\frac38$ and another $\frac18$ . This means that $\frac16$ is equal to $\frac18$ and another third of $\frac18$ .