Section B Practice Problems

Problem 1

Write $\frac{4}{3}$ in as many ways as you can as a sum of fractions.
Write $\frac{9}{8}$ in at least 3 different ways as a sum of fractions.

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Solution

$\frac{3}{3} + \frac{1}{3}$ , $\frac{2}{3} + \frac{2}{3}$ , $\frac{2}{3} + \frac{1}{3} + \frac{1}{3}$ , $\frac{1}{3} + \frac{1}{3} + \frac{1}{3} + \frac{1}{3}$
Sample responses: $\frac{3}{8} + \frac{3}{8} + \frac{3}{8}$ , $\frac{7}{8} + \frac{2}{8}$ , $\frac{1}{8} + \frac{1}{8} + \frac{2}{8} + \frac{5}{8}$

Problem 2

Draw jumps on the number lines to show two ways to use fourths to make a sum of $\frac{7}{4}$ .
Represent each combination of jumps as an equation.

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Solution

Sample responses:
$\frac{3}{4} + \frac{4}{4} = \frac{7}{4}$ and $\frac{2}{4} + \frac{5}{4} = \frac{7}{4}$

Problem 3

Explain how the diagram represents $\frac{13}{5} - \frac{4}{5}$ .

Number line. 16 evenly spaced tick marks. First tick mark, 0. Sixth, 1. Eleventh, 2. Sixteenth, 3. Arrow begins at the point on the fourteenth tick mark, points to the left, and ends at the point on the tenth tick mark.

Use the diagram to find the value of $\frac{13}{5}- \frac{4}{5}$ .
Use a number line to represent and find the difference $\frac{9}{4} - \frac{3}{4}$ .

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Solution

Sample response: The point to& the far right represents $\frac{13}{5}$ , and a jump of 4 spaces to the left represents subtraction of $\frac{4}{5}$ . $\frac{13}{5} - \frac{4}{5}=\frac{9}{5}$
$\frac{9}{4} - \frac{3}{4} = \frac{6}{4}$

Problem 4

Show two different ways to find the difference: $2 - \frac{3}{4}$

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Solution

Sample responses:

2 is equivalent to $\frac{8}{4}$ and $\frac{8}{4} - \frac{3}{4} = \frac{5}{4}$ .
2 is equivalent to $1 + \frac{4}{4}$ and $1 + \frac{4}{4} - \frac{3}{4} = 1\frac{1}{4}$ .

Problem 5

Elena is making friendship necklaces and wants the chain and clasp to be a total of

18\frac{1}{4}

inches long. She is going to use a clasp that is

2\frac{3}{4}

inches long. How long does her chain need to be? Explain or show your reasoning.

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Solution

15\frac{2}{4}

inches or equivalent. Sample response: 18 – 2 = 16 and

16 - \frac{3}{4} = 15\frac{1}{4}

, and then I need to add back the extra

\frac{1}{4}

inch to get a total of

15\frac{2}{4}

Problem 6

For each of the expressions, explain whether you think it would be helpful to decompose one or more numbers to find the value of the expression.

$\frac{4}{3} + \frac{5}{3}$
$5\frac{1}{5} - 2\frac{2}{5}$
$9\frac{5}{6} - 6\frac{1}{6}$

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Solution

No, because you can add $\frac{4}{3} + \frac{5}{3}$ since both values are written as fractions and not as mixed numbers
Yes, because there are not enough fractional parts to subtract $\frac{2}{5}$ from $5\frac{1}{5}$ .
No, because you can find $9 - 6$ and there are enough fractional parts to find $\frac{5}{6} - \frac{1}{6}$ .

Problem 7

This shows the shoe lengths for a dad and his two daughters.

For each question, show your reasoning.

Image of 3 pairs of shoes and their lengths. Pink shoes, 8 and 5 eighths inches. Cat shoes, 3 and 6 eighths inches. Dad's shoes, 12 and 1 eighth inches.

How much longer is the older daughter’s shoes than her sister’s shoes?
Which is longer, the dad’s shoes or the combined lengths of his daughters’ shoes?

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Solution

$8\frac{5}{8} - 3\frac{6}{8} = 4\frac{7}{8}$ inches. Sample response:
The combined length of the daughters’ shoes is longer. Sample response: $8\frac{5}{8} + 3\frac{6}{8} = 11\frac{11}{8}$ , which is equivalent to $12\frac{3}{8}$ inches.

Problem 8

A chocolate chip cookie recipe calls for $2\frac{3}{4}$ cups of flour. You have only a $\frac{1}{4}$ -cup measure and a $\frac{3}{4}$ -cup measure that you can use.

What are different combinations of the cup measures that you can use to get a total of $2\frac{3}{4}$ cups of flour?
Write each of the combinations as an addition equation.

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Solution

Eleven $\frac{1}{4}$ -cup measures
One $\frac{3}{4}$ -cup measure and eight $\frac{1}{4}$ -cup measures
Two $\frac{3}{4}$ -cup measures and five $\frac{1}{4}$ -cup measures
Three $\frac{3}{4}$ -cup measures and two $\frac{1}{4}$ -cup measures
$\frac{1}{4} + \frac{1}{4} + \frac{1}{4} + \frac{1}{4} + \frac{1}{4} + \frac{1}{4} + \frac{1}{4} + \frac{1}{4} + \frac{1}{4} + \frac{1}{4} + \frac{1}{4} = \frac{11}{4}$
$\frac{3}{4} + \frac{1}{4} + \frac{1}{4} + \frac{1}{4} + \frac{1}{4} + \frac{1}{4} + \frac{1}{4} + \frac{1}{4} + \frac{1}{4}= \frac{11}{4}$
$\frac{3}{4} + \frac{3}{4} + \frac{1}{4} + \frac{1}{4} + \frac{1}{4} + \frac{1}{4} + \frac{1}{4}=\frac{11}{4}$
$\frac{3}{4} + \frac{3}{4} + \frac{3}{4} + \frac{1}{4} + \frac{1}{4}=\frac{11}{4}$

Problem 9

The table shows some lengths of different shoe sizes in inches.

U.S. shoe size	insole length (inches)
1	$7\frac{6}{8}$
1.5	8
2	$8\frac{1}{8}$
2.5	$8\frac{2}{8}$
3	$8\frac{4}{8}$
3.5	$8\frac{5}{8}$
4	$8\frac{6}{8}$
4.5	9
5	$9\frac{1}{8}$
5.5	$9\frac{2}{8}$
6	$9\frac{4}{8}$
6.5	$9\frac{5}{8}$
7	$9\frac{6}{8}$

What do you notice about the insole lengths as the size increases?
Based on the data in the table, what is the insole length increase from size 7 to size 7.5? What is the insole length of a size 7.5 shoe?
Predict the insole length for sizes 9, 10, and 12. Explain your prediction. Then solve to find out if your prediction is true.

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Solution

The insole lengths do not grow the same amount, but there is a pattern in these growth amounts: $\frac{2}{8}$ , $\frac{1}{8}$ , $\frac{1}{8}$ .
The increase will be $\frac{2}{8}$ , so the insole length will be 10 inches since $9\frac{6}{8} + \frac{2}{8} = 10$ .
Sample response: Size 9 will be $10\frac{4}{8}$ inches long since every three half-sizes the insole increases by $\frac{4}{8}$ inches. Size 9 is three half-sizes away from size 7.5 and $10 + \frac{4}{8} = 10\frac{4}{8}$ .
Size 10 will be $10\frac{6}{8}$ inches long since there are only two half-size increases from size 9 to size 10 and the $\frac{2}{8}$ increase happened from size 8.5 to size 9, $10\frac{4}{8} + \frac{1}{8} + \frac{1}{8} = 10\frac{6}{8}$ .
Size 12 will be $11\frac{4}{8}$ inches long because I noticed that for every three whole-sizes on the chart, the insole length increases by 1 inch. Since size 9 is $10\frac{4}{8}$ inches and size 12 is three whole-sizes away, I have $10\frac{4}{8} + 1 = 11\frac{4}{8}$ .