22.2: How Many Groups? (Part 2)
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 \(1\frac{1}{4}\)?
- \(1\div\frac{2}{6}=?\)
- \(2\div\frac{2}{9}=?\)
- \(4\div\frac{2}{10}=?\)
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.)
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If the trapezoid represents 1 whole, what do each of these other shapes represent? Be prepared to explain or show your reasoning.
- 1 triangle
- 1 rhombus
- 1 hexagon
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Use pattern blocks to represent each multiplication equation. Use the trapezoid to represent 1 whole.
- \(3\cdot\frac{1}{3}=1\)
- \(3\cdot\frac{2}{3}=2\)
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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.”
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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.
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?”
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.
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)