Section A: Practice Problems Equal Groups of Fractions

Section Summary

Details

In this section, we learned to multiply a whole number and a fraction by thinking about equal-size groups, just as we did when multiplying two whole numbers.

For instance, we can think of $6 \times 4$ as 6 groups of 4. A diagram like this can help to show that the product is 24:

Likewise, we can think of $6 \times \frac{1}{4}$ as 6 groups of $\frac{1}{4}$ . Diagrams can help us see that the product is $\frac{6}{4}$ :

6 diagrams of equal length. 4 parts. 1 part shaded. Total length, 1.

Diagram. 6 equal parts, each labeled 1 fourth.

After studying patterns, we saw that when we multiply a whole number and a fraction, the whole number is multiplied only by the numerator of the fraction and the denominator stays the same. For example:

$6 \times \frac{1}{2} = \frac{6}{2}$

$2 \times \frac{4}{5} = \frac{8}{5}$

We also learned that:

Every fraction can be written as a product of a whole number and a unit fraction. For example, $\frac{5}{4}$ can be written as $5 \times \frac{1}{4}$ .
We can write different multiplication expressions for the same fraction. For example, $\frac{8}{3}$ can be written as:

$8 \times \frac{1}{3}$

$4 \times 2 \times \frac{1}{3}$

$4 \times \frac{2}{3}$

$2 \times \frac{4}{3}$

Problem 1 (Pre-Unit)

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

Problem 2 (Pre-Unit)

Locate and label $\frac{3}{4}$ and $\frac{6}{4}$ on the number line.
Explain why your points represent $\frac{3}{4}$ and $\frac{6}{4}$ .

Problem 3 (Pre-Unit)

Write a multiplication expression for each image. Explain your reasoning.

Problem 4 (Pre-Unit)

Here are the lengths of some lizards in inches. Use the lengths to complete the line plot.

$2 \frac{1}{4}$
$1 \frac{1}{2}$
$2 \frac{2}{4}$
3
$3 \frac{2}{4}$
2

$2 \frac{1}{4}$
$2 \frac{1}{4}$
$2 \frac{3}{4}$
2
$2 \frac{1}{4}$
3

Problem 5 (Lesson 1)

Write an expression that matches each diagram. Then, find the value of each expression.

Problem 6 (Lesson 2)

Five friends go on a hike. They each bring $\frac{1}{4}$ cup of nuts.

If the shaded parts represent the amount of nuts the friends bring on their hike, which diagram matches the story? Explain your reasoning.

A

B
How many cups of nuts do the friends bring on the hike?

Problem 7 (Lesson 3)

Kiran’s cat eats $\frac{1}{2}$ cup of food each day.

How much food does Kiran’s cat eat in a week?
Draw a diagram to represent the situation.

Problem 8 (Lesson 4)

Draw a diagram to show $3 \times \frac{7}{8}$ .
How does the diagram help you find the value of the expression $3 \times \frac{7}{8}$ ?

Problem 9 (Lesson 5)

Find the number that makes each equation true. Draw a diagram if it is helpful.

$\frac{10}{3} =$ $\times \frac{1}{3}$
$\frac{10}{3} =$ $\times \frac{2}{3}$
$\frac{10}{3} =$ $\times \frac{5}{3}$

Problem 10 (Lesson 6)

Each bead weighs $\frac{5}{8}$ gram. How much do 7 beads weigh? Explain or show your reasoning.

Problem 11 (Exploration)

Measure how thick your workbook is to the nearest $\frac{1}{8}$ inch.
If all of your classmates stacked their workbooks together, how tall would the stack be? Explain or show your reasoning.
Check your answer by measuring, if possible.

Problem 12 (Exploration)

Diego walked the same number of miles to school each day. He says that he walked $\frac{48}{5}$ miles in total, but does not say how many days that distance includes.

What are some possible number of days Diego counted and the distance he walked each of those days?