Section B: Practice Problems Addition and Subtraction of Fractions

Section Summary

Details

In this section, we added and subtracted fractions with the same denominator, using number lines to help with our reasoning.

First, we learned that a fraction can be decomposed into a sum of smaller fractions. For example, here are a few ways to write $\frac{6}{10}$ :

$\frac{6}{10} = \frac{4}{10} + \frac{2}{10}$
$\frac{6}{10} = \frac{5}{10} + \frac{1}{10}$
$\frac{6}{10} = \frac{2}{10} + \frac{2}{10} + \frac{2}{10}$

If the fraction is greater than 1, it can be decomposed into a whole number and a fraction less than 1. For instance, we can decompose $\frac{17}{10}$ and rewrite it as $1 \frac{7}{10}$ . A number such as $1 \frac{7}{10}$ is called a mixed number.

$\frac{10}{10} + \frac{7}{10}$
$1 + \frac{7}{10}$
$1 \frac{7}{10}$

Later, we decomposed fractions into sums and wrote equivalent fractions to help us add and subtract fractions. For example, to find the value of $1 \frac{2}{5} - \frac{3}{5}$ , we can:

Decompose $1 \frac{2}{5}$ into $1 + \frac{2}{5}$ or $\frac{5}{5} + \frac{2}{5}$ , which is $\frac{7}{5}$ .
Find the value of $\frac{7}{5} - \frac{3}{5}$ , which is $\frac{4}{5}$ .

Finally, we organized and analyzed measurement data on line plots. The data were lengths measured to the nearest inch, $\frac{1}{2}$ inch, $\frac{1}{4}$ inch, and $\frac{1}{8}$ inch.

Dot plot titled Colored Pencil Data from 0 to 6 by 1’s.

Because the measurements have different denominators, we used equivalent fractions to plot them. Then, we used the line plots and what we know about addition and subtraction of fractions to solve problems about the data.

Problem 1 (Lesson 7)

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.

Problem 2 (Lesson 8)

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.

Problem 3 (Lesson 9)

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

Problem 4 (Lesson 10)

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

Problem 5 (Lesson 11)

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.

Problem 6 (Lesson 12)

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.

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

Problem 7 (Lesson 12)

The lengths of the shoes of a dad and his two daughters are shown.

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.

For each question, show your reasoning.

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

Problem 8 (Exploration)

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

What are different combinations of the measuring cups 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.

Problem 9 (Exploration)

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

U.S. shoe size	insole length
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?
What will the insole length increase be from size 7 to 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.