22.2: How Many Groups? (Part 2)

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Lesson

Let's use blocks and diagrams to understand more about division with fractions.

Exercise $\PageIndex{1}$: Reasoning with Fraction Strips

Write a fraction or whole number as an answer for each question. If you get stuck, use the fraction strips. Be prepared to share your reasoning.

How many $\frac{1}{2}$s are in $2$?
How many $\frac{1}{5}$s are in $3$?
How many $\frac{1}{8}$s are in $1\frac{1}{4}$?
$1\div\frac{2}{6}=?$
$2\div\frac{2}{9}=?$
$4\div\frac{2}{10}=?$

Figure $\PageIndex{1}$: Fraction strips depicting 2 in 8 different ways, by rows. First row, two 1s. Second row, 4 of the fraction one over two. Third row, 6 of the fraction one over three. Fourth row, 8 of the fraction one over four. Fifth row, 10 of the fraction one over five. Sixth row, 12 of the fraction one over six. Seventh row, 16 of the fraction one over eight. Eighth row, 18 of the fraction one over nine.

Exercise $\PageIndex{2}$: More Reasoning with Pattern Blocks

Use the pattern blocks in the applet to answer the questions. (If you need help aligning the pieces, you can turn on the grid.)

If the trapezoid represents 1 whole, what do each of these other shapes represent? Be prepared to explain or show your reasoning.
1. 1 triangle
2. 1 rhombus
3. 1 hexagon
Use pattern blocks to represent each multiplication equation. Use the trapezoid to represent 1 whole.
1. $3\cdot\frac{1}{3}=1$
2. $3\cdot\frac{2}{3}=2$
Diego and Jada were asked “How many rhombuses are in a trapezoid?”
- Diego says, “$1\frac{1}{3}$. If I put 1 rhombus on a trapezoid, the leftover shape is a triangle, which is $\frac{1}{3}$ of the trapezoid.”
- Jada says, “I think it’s $1\frac{1}{2}$. Since we want to find out ‘how many rhombuses,’ we should compare the leftover triangle to a rhombus. A triangle is $\frac{1}{2}$ of a rhombus.”
  Do you agree with either of them? Explain or show your reasoning.
Select all the equations that can be used to answer the question: “How many rhombuses are in a trapezoid?”
- $\frac{2}{3}\div ?=1$
- $?\cdot\frac{2}{3}=1$
- $1\div\frac{2}{3}=?$
- $1\cdot\frac{2}{3}=?$
- $?\div\frac{2}{3}=1$

Exercise $\PageIndex{3}$: Drawing Diagrams to Show Equal-sized Groups

For each situation, draw a diagram for the relationship of the quantities to help you answer the question. Then write a multiplication equation or a division equation for the relationship. Be prepared to share your reasoning.

The distance around a park is $\frac{3}{2}$ miles. Noah rode his bicycle around the park for a total of 3 miles. How many times around the park did he ride?
You need $\frac{3}{4}$ yard of ribbon for one gift box. You have 3 yards of ribbon. How many gift boxes do you have ribbon for?
The water hose fills a bucket at $\frac{1}{3}$ gallon per minute. How many minutes does it take to fill a 2-gallon bucket?

Are you ready for more?

How many heaping teaspoons are in a heaping tablespoon? How would the answer depend on the shape of the spoons?

Summary

Suppose one batch of cookies requires $\frac{2}{3}$ cup flour. How many batches can be made with 4 cups of flour?

We can think of the question as being: “How many $\frac{2}{3}$ are in 4?” and represent it using multiplication and division equations.

$?\cdot\frac{2}{3}=4$

$4\div\frac{2}{3}=?$

Let’s use pattern blocks to visualize the situation and say that a hexagon is 1 whole.

Figure $\PageIndex{2}$: Diagram of 4 hexagons. Each hexagon is made up of 3 rhombuses using pattern blocks. 6 pairs of rhombuses are shown by pairs of arrows and labeled 1, 2, 3, 4, 5, and 6.

Since 3 rhombuses make a hexagon, 1 rhombus represents $\frac{1}{3}$ and 2 rhombuses represent $\frac{2}{3}$. We can see that 6 pairs of rhombuses make 4 hexagons, so there are 6 groups of $\frac{2}{3}$ in 4.

Other kinds of diagrams can also help us reason about equal-sized groups involving fractions. This example shows how we might reason about the same question from above: “How many $\frac{2}{3}$-cups are in 4 cups?”

Figure $\PageIndex{3}$: Diagram with 4 rectangles, partitioned into thirds. In the first rectangle two parts are each labeled one third. Across all 4 rectangles, 6 pairs of one third pieces are circled.

We can see each “cup” partitioned into thirds, and that there are 6 groups of $\frac{2}{3}$-cup in 4 cups. In both diagrams, we see that the unknown value (or the “?” in the equations) is 6. So we can now write:

$6\cdot\frac{2}{3}=4$

$4\div\frac{2}{3}=6$

Practice

Exercise $\PageIndex{4}$

Use the tape diagram to find the value of $\frac{1}{2}\div\frac{1}{3}$. Show your reasoning.

Figure $\PageIndex{4}$: A tape diagram on a square grid, composed of 6 squares and is partitioned by a solid line into two equal parts. The tape diagram is also partitioned by two vertical dashed lines resulting in three equal parts. A brace extends over the first 3 squares and is labeled one half.

Exercise $\PageIndex{5}$

What is the value of $\frac{1}{2}\div\frac{1}{3}$? Use pattern blocks to represent and find this value. The yellow hexagon represents 1 whole. Explain or show your reasoning.

Exercise $\PageIndex{6}$

Use a standard inch ruler to answer each question. Then, write a multiplication equation and a division equation that answer the question.

How many $\frac{1}{2}$s are in $7$?
How many $\frac{3}{8}$s are in $6$?
How many $\frac{5}{16}$s are in $1\frac{7}{8}$?

Exercise $\PageIndex{7}$

Use the tape diagram to answer the question: How many $\frac{2}{5}$s are in $1\frac{1}{2}$? Show your reasoning.

Exercise $\PageIndex{8}$

Write a multiplication equation and a division equation to represent each sentence or diagram.

There are $12$ fourths in $3$.

How many $\frac{2}{3}$s are in $6$?

(From Unit 4.2.1)

Exercise $\PageIndex{9}$

At a farmer’s market, two vendors sell fresh milk. One vendor sells 2 liters for $3.80, and another vendor sells 1.5 liters for $2.70. Which is the better deal? Explain your reasoning.

(From Unit 3.3.1)

Exercise $\PageIndex{10}$

A recipe uses 5 cups of flour for every 2 cups of sugar.

How much sugar is used for 1 cup of flour?
How much flour is used for 1 cup of sugar?
How much flour is used with 7 cups of sugar?
How much sugar is used with 6 cups of flour?

(From Unit 3.3.2)

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