# Lesson 19

Fraction Games

## Warm-up: Estimation Exploration: Multiply Fractions (10 minutes)

### Narrative

The purpose of this Estimation Exploration is for students to develop strategies for finding the product of a fraction and a mixed number. Since $$2 \frac{8}{9}$$ is so close to 3, a good estimate is $$3 \times 28$$ or 84. Students may refine this estimate using the distributive property

$$\begin{array} 28 \times 2\frac{8}{9} &=& 28 \times \left(3 - \frac{1}{9}\right)\\ &=& 28 \times 3 - 28 \times \frac{1}{9} \end{array}$$

Since $$\frac{28}{9}$$ is about 3, $$84 - 3$$ or 81 is a very good estimate. Students will use these ideas in the lesson when they find products of fractions, whole numbers, and mixed numbers.

### Launch

• Groups of 2
• Display the image.
• “What is an estimate that’s too high? Too low? About right?”

### Activity

• 1 minute: quiet think time
• 1 minute: partner discussion
• Record responses.

### Student Facing

$$28 \times 2 \frac{8}{9}$$

Record an estimate that is:
too low about right too high
$$\phantom{\hspace{2.5cm} \\ \hspace{2.5cm}}$$ $$\phantom{\hspace{2.5cm} \\ \hspace{2.5cm}}$$ $$\phantom{\hspace{2.5cm} \\ \hspace{2.5cm}}$$

### Student Response

For access, consult one of our IM Certified Partners.

### Activity Synthesis

• “How does $$28 \times 2\frac{8}{9}$$ compare to $$28 \times 2$$? How do you know?” (It’s larger because $$2 \frac{8}{9}$$ is greater than 2.)
• “Why is $$28 \times 3$$ a good estimate?” (Because $$2\frac{8}{9}$$ is really close to 3.)
• “Is $$28 \times 2\frac{8}{9}$$ greater or less than $$28 \times 3$$? How do you know?” (Less because $$2\frac{8}{9}$$ is less than 3.)
Optional: Reveal the actual value, $$80\frac{8}{9}$$, and add it to the display.

## Activity 1: Largest Product or Quotient (20 minutes)

### Narrative

The purpose of this activity is for students to apply what they have learned about multiplication and division of fractions to strategically write expressions with the greatest value. Students notice and explain patterns (MP7) such as:
• To make a product as large as possible, the two factors should be chosen as large as possible.
• To make a quotient or fraction as large as possible, the dividend should be as large as possible and the divisor as small as possible.
MLR8 Discussion Supports. Synthesis: Before inviting students to share their strategies for making the expression as large as possible, give groups time to rehearse what they might say if selected.
Engagement: Provide Access by Recruiting Interest. Synthesis: Invite students to generate a list of additional examples of problems that can be solved using multiplication or division that connect to their personal backgrounds and interests.
Supports accessibility for: Attention, Conceptual Processing

• Groups of 2

### Activity

• 15 minutes: partner work time
• Monitor for students who reason about the size of the product or quotient based on the location of the digits.

### Student Facing

For each expression, work with your partner to decide what is the greatest product or quotient you can make with the numbers 1, 2, 3, 4, 5, and 6. For each expression, you can only use each number once. Explain or show your reasoning.

1. $$\frac{\boxed{\phantom{\frac{000}{00}}}}{\boxed{\phantom{\frac{000}{000}}}}\times\frac{\boxed{\phantom{\frac{000}{00}}}}{\boxed{\phantom{\frac{00}{000}}}}$$
2. $$\boxed{\phantom{\frac{000}{0}}} \div\frac{1}{\boxed{\phantom{\frac{0000}{0}}}}$$
3. $$\frac{1}{\boxed{\phantom{\frac{0000}{0}}}}\div\boxed{\phantom{\frac{0000}{0}}}$$

### Student Response

For access, consult one of our IM Certified Partners.

### Activity Synthesis

• Invite previously selected students to share their responses.
• Display the second expression.
• Invite students to share their strategies for making the expression as large as possible.
• “Why is $$6 \div \frac{1}{5}$$ a good choice for making this expression as large as possible?” (When I find the value I am multiplying 6 and 5. Those are the two biggest numbers so I know that will give me the biggest value.)
• “Is there another choice for filling in the blanks that gives the same value?” (Yes, I can also do $$5 \div \frac{1}{6}$$.)

## Activity 2: Smallest Product or Quotient (15 minutes)

### Narrative

In the previous activity, students chose digits to create expressions whose value was as large as possible. The purpose of this activity is for students to create expressions with the smallest possible value. The expressions and digits that students use are the same so the patterns that they identified in the previous activity will apply here as well but they lead to a different choice of expressions.

• Groups of 2

### Activity

• 7–8 minutes: independent work time
• 2–3 minutes: partner work time
• Monitor for students who reason about the size of the product or quotient based on the location of the digits.

### Student Facing

For each expression, work with your partner to decide what is the smallest product or quotient you can make with the numbers 1, 2, 3, 4, 5, and 6. You can only use each number once for reach expression. Explain or show your reasoning.

1. $$\frac{\boxed{\phantom{\frac{000}{00}}}}{\boxed{\phantom{\frac{000}{000}}}}\times\frac{\boxed{\phantom{\frac{000}{00}}}}{\boxed{\phantom{\frac{00}{000}}}}$$
2. $$\boxed{\phantom{\frac{000}{0}}} \div\frac{1}{\boxed{\phantom{\frac{0000}{0}}}}$$
3. $$\frac{1}{\boxed{\phantom{\frac{0000}{0}}}}\div\boxed{\phantom{\frac{0000}{0}}}$$

### Student Response

For access, consult one of our IM Certified Partners.

### Activity Synthesis

• Display the last expression.
• “Why is $$\frac{1}{6}\div 5$$ a good choice for making the expression as small as possible?” (The numerator is 1 so I want the denominator to be as large as possible. That’s why putting in the 6 and 5 is a good strategy.)
• “Is there another choice for filling in the blanks that gives the same value?” (Yes, I can also do $$\frac{1}{5} \div 6$$.)
• “What is the value of $$\frac{1}{6} \div 5$$ and $$\frac{1}{5} \div 6$$ ? How do you know?” ($$\frac{1}{30}$$ because I am either cutting $$\frac{1}{6}$$ into 5 equal pieces or $$\frac {1}{5}$$ into 6 equal pieces. Either way there are $$6 \times 5$$ or 30 of those pieces in a whole.)

## Lesson Synthesis

### Lesson Synthesis

“Today we looked at the value of different multiplication and division expressions involving unit fractions.”

Display the first expressions from the two activities.

“What numbers will make the value of this expression as large as possible?” (I use the 5 and 6 for the numerators and the 1 and 2 for the denominators.)

“What numbers will make it as small as possible?” (I use the 1 and 2 for the numerators and the 5 and 6 for denominators.)

“How are the expressions we wrote for the largest and smallest values the same? How are they different?” (They use the same numbers but they are in the numerator in one expression and in the denominator in the other.)

## Cool-down: Fill in the Blanks (5 minutes)

### Cool-Down

For access, consult one of our IM Certified Partners.

## Student Section Summary

### Student Facing

We used the relationship between multiplication and division to write both multiplication and division equations to represent the same situation. For example, there are 2 pounds of beef in the package. Each burger uses $$\frac{1}{4}$$ pound. How many burgers will the package make? We can write $$2\div\frac{1}{4}=8$$ and $$8\times\frac{1}{4}=2$$ to represent the situation.

We also wrote multiplication and division equations to represent the same diagram. For example:

We can write $$6 \times \frac{1}{3} = 2$$ because the diagram shows 6 groups of $$\frac{1}{3}$$ and the total value is 2. We can also write $$2 \div \frac{1}{3} = 6$$ because the diagram shows that the number of groups of $$\frac{1}{3}$$ in 2 is 6.