Lesson 20

How Much in the Group? (optional)

Warm-up: Estimation Exploration: What Number Goes in the Blank? (10 minutes)

Narrative

The purpose of this Estimation Exploration is to estimate a whole given the value of a fraction of the whole. This prepares students for the type of problem they will solve in this lesson.

Launch

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

Activity

• 1 minute: partner discussion
• Record responses.

Student Facing

What number goes in the blank? Record an estimate that is:
too low about right too high
$$\phantom{\hspace{2.5cm} \\ \hspace{2.5cm}}$$ $$\phantom{\hspace{2.5cm} \\ \hspace{2.5cm}}$$ $$\phantom{\hspace{2.5cm} \\ \hspace{2.5cm}}$$

Activity Synthesis

• “What strategies did you use to determine the number that goes in the blank?” (I tried to see how many 15's fit in the whole rectangle.)

Activity 1: Different Equations (20 minutes)

Narrative

The purpose of this activity is to solve problems about how many students are in the whole fifth grade. Students should use whatever strategy makes sense to them. As students consider which equations represent the problem, they may use the context, the relationship between multiplication and division, or computations to make sense of the equations. When students interpret the meaning of their answer in a context, they are reasoning abstractly and quantitatively (MP2).

MLR8 Discussion Supports. Invite students to begin partner interactions by repeating the question, “How many students are in the class?” This gives both students an opportunity to produce language.

• Groups of 2

Activity

• 1–2 minutes: independent think time
• 6–8 minutes: partner work time
• Monitor for students who:
• describe the total number of students as 27 because there are 3 groups of 9 students
• describe the total number of students as 27 because when 27 is divided into 3 equal groups, there will be 9 in each group
• describe the total number of students in the class as 27 because $$\frac {1}{3} \times 27$$ is 9

Student Facing

1. If $$\frac {1}{3}$$ of the class is 9 students, how many students are in the class?

2. Explain how each of these equations represents this situation.

1. $$\frac {1}{3} \times \underline{\hspace{1 cm}} = 9$$
2. $$\underline{\hspace{1 cm}} \div 3 = 9$$
3. $$3 \times 9 = \underline{\hspace{1 cm}}$$

Activity Synthesis

• Ask previously identified students to explain how each equation can represent the situation.
• Display:
$$27 \div 3 = 9$$
$$3 \times 9 = 27$$
• “What is the relationship between these two equations? Discuss with a partner.” (They are like opposites. 27 divided into 3 equal groups is 9 and 3 groups of 9 is 27.)
• Display:
$$\frac {1}{3} \times 27 = 9$$
• “How does this equation represent the situation?” (We know that $$\frac {1}{3}$$ of the class is 9, so the class must have 27 kids in it, because $$\frac {1}{3}$$ of 27 is 9.)

Activity 2: How Big is the Class? (10 minutes)

Narrative

The purpose of this activity is for students to reason about which equations represent a situation. They use their understanding of the relationship between multiplication and division to make their selections.

Engagement: Provide Access by Recruiting Interest. Synthesis: Optimize meaning and value. Invite students to share equations with fractional values that would represent the total number of students in their classroom.
Supports accessibility for: Conceptual Processing, Attention

• Groups of 2

Activity

• 5–8 minutes: partner work time
• Monitor for students who use the relationship between multiplication and division to make their selections.

Student Facing

1. Jada’s class has 24 students in it. That is $$\frac {1}{4}$$ of the total students in the 5th grade. How many students are in the 5th grade? Explain or show your reasoning.
2. Select all the equations that represent this situation.

1. $$\frac {1}{4} \times 24 = \underline{\hspace{1 cm}}$$
2. $$\underline{\hspace{1 cm}} \div 4 = 24$$
3. $$\frac {1}{4} \div 24 = \underline{\hspace{1 cm}}$$
4. $$24 = \frac {1}{4} \times \underline{\hspace{1 cm}}$$

Student Response

If students do not select appropriate equations, show these equations and ask them to explain how they are the same and different.
• $$24 = \frac {1}{4} \times 96$$
• $$\underline{\hspace{1 cm}} \div 4 = 24$$

Activity Synthesis

• Ask previously selected students to share their reasoning and solutions.
• Display:
$$\frac {1}{4} \div 24 = \underline{\hspace{1 cm}}$$
• “How do we know this equation does not represent the situation?” ($$\frac {1}{4}$$ divided into 24 groups is going to be really small. There will be $$\frac{1}{96}$$ in each group.)
• Display:
$$24 = \frac {1}{4} \times \underline{\hspace{1 cm}}$$
• “How does this equation represent the situation?” (We know there are 24 students in Jada’s class and we know that is $$\frac {1}{4}$$ of the whole grade, but we don’t know how many students are in the whole grade.)
• Display:
$$4 \times 24 = \underline{\hspace{1 cm}}$$
• “How does this equation help us figure out how many students are in the whole grade?” (If 24 is $$\frac {1}{4}$$ of the grade, we can multiply 24 by 4 to figure out how many are in the whole grade.)

Activity 3: How Many in One Group? (10 minutes)

Narrative

The purpose of this activity is for students to solve more “how many in one group” division problems in which the dividend is a whole number and the divisor is a unit fraction. The numbers are larger but still well-suited for a tape diagram representation. No method of solution is suggested or requested so students may draw a picture or a tape diagram or write an equation. For the second problem, the distance context may encourage students to use a number line representation to solve the problem.

Launch

• Groups of 2
• 2 minutes: quiet think time

Activity

• 8 minutes: partner work time
• Monitor for students who use a tape diagram or number line to represent and solve each problem.

Student Facing

Solve each problem. Show or explain your reasoning.
1. 250 mg of calcium is $$\frac {1}{4}$$ of the daily recommended allowance. What is the daily recommended allowance of calcium? Show or explain your reasoning.
2. A rocket took 60 days to get $$\frac {1}{5}$$ of the way to Mars. How many days did it take the rocket to get to Mars? Show or explain your reasoning.

Student Response

Students may not be familiar with the contexts in this task. Consider incorporating a launch into the activity that supports students’ understanding of the context, for example, an image of daily recommended nutritional values, or a video of a rocket launching into space.

Activity Synthesis

• Invite previously selected students to share their solutions.
• Display:
$$60 = \frac {1}{5} \times \underline{\hspace{1 cm}}$$
$$60 \div \frac {1}{5}$$ = _______
• “These equations represent the rocket problem. We can solve both of these equations by multiplying 60 by 5. Why do we multiply by 5?” (60 days is $$\frac{1}{5}$$ of the trip, so $$5 \times 60$$ is the whole trip.)

Lesson Synthesis

Lesson Synthesis

“Today we solved problems using the relationship between multiplication and division.”

Display: Jada’s class has 24 students in it. That is $$\frac {1}{4}$$ the total students in the 5th grade. How many students are in the whole grade?

“How did we use multiplication to solve this problem?” (We multiplied 24 by 4.)

Display equations:

• $$\frac {1}{4} \times 96 = 96 \div 4$$
• $$24 \div \frac{1}{4}= 4 \times 24$$

“Why are these equations true?” (If I divide 96 into 4 equal parts then each part is $$\frac{1}{4}$$ of 96. To find out how many $$\frac{1}{4}$$s are in 24 I can multiply 24 by 4 since there are four $$\frac{1}{4}$$s in each whole.)