# Lesson 15

Find Missing Side Lengths

## Warm-up: Estimation Exploration: The Garden (10 minutes)

### Narrative

The purpose of this Estimation Exploration is to recall the concept of area. Students need to think strategically because the one point of reference for the size of the grassy area in the image is the car and the road. In order to facilitate mental calculation, expect students to choose multiples of ten for the length and width of the rectangle.

### 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 is the area of one of the large rectangles in the garden? 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

• “What did you use in the image to make an estimate for the area of the garden?” (The car and the person gave me an idea of how big it is.)

## Activity 1: Find the Missing Side Length, Part 1 (15 minutes)

### Narrative

The purpose of this activity is to use multiplication and division to solve area problems. In most cases the area and one side length are given and students can use division to find the missing side length. In one case the two side lengths are given and students find their product which is the area.

### Launch

• Display garden from warm-up.
• “The area of one of the large rectangular pieces is 9,175 square feet and the length is 75 feet.”
• Display:
Area: 9,175 square feet
One side length: 75 feet
• “What is a reasonable estimate for the width?” (100 because and $$100 \times 75 = 7,\!500$$ and $$7,\!500 \div 75 = 100$$.)
• 2 minutes: partner discussion
• “How can we find the exact width of the garden?” (Divide the area by the length.)

### Activity

• 1–2 minutes: quiet think time
• 5–8 minutes: partner work time
• Monitor for students who use a partial quotients algorithm to divide to share during the synthesis.
• If students try to divide to find the missing area, consider asking, “How do we find the area of a rectangle given the length and width?”

### Student Facing

Complete the table.
area
(square feet)
length
(feet)
width
(feet)
816 24
1,248 48
23 253
5,796 36

### Student Response

For access, consult one of our IM Certified Partners.

### Activity Synthesis

• Ask selected students to share their work and explain their steps.
• “How can we make sure that we found the correct missing side lengths?” (Multiply the length and width together to make sure it gives the area or divide the area by one of the side lengths and make sure it gives the other side length.)
• “How was finding the area of the rectangle using the length and width different than finding one of the side lengths using the area and the other side length?” (I had to multiply to find the area from the length and width. I used division to get one side length from the area and the other side length.)

## Activity 2: Find the Missing Side Length, Part 2 (20 minutes)

### Narrative

The purpose of this activity is for students to apply what they learned about dividing multi-digit numbers to find the missing side length(s) of rectangular prisms given the volume and at least one other side length. As in the previous activity, both multiplication and division are important to solve the problems. This is true both because finding partial quotients uses multiplication and because they have different choices how to find a missing side length when two side lengths and the volume are given. Monitor for students who
• find the product of the two given side lengths and then use division to find the third side length.
• divide successively by the two given side lengths.
MLR8 Discussion Supports. Prior to solving the problems, invite students to make sense of the situations and take turns sharing their understanding with their partner. Listen for and clarify any questions about the context.
Engagement: Provide Access by Recruiting Interest. Provide choice. Invite students to decide in which order to complete the task and choose what strategy they want to use.
Supports accessibility for: Organization, Social-Emotional Functioning

• Groups of 2

### Activity

• 5-8 minutes: independent work time
• 5 minutes: partner work time
• Monitor for students who:
• use a partial quotients algorithm to divide.
• find the missing side length in the third table by dividing twice, similar to Clare’s steps in the next problem.
• find the missing length in the third table by first multiplying the given side lengths and then dividing the volume by the product.
• notice there are multiple possible lengths and widths for the last rectangular prism in the third table

### Student Facing

1. Complete the table.
volume
(cubic feet)
base
(square feet)
height
(feet)
375 15
1,176 28
2. Clare wants to find the height of a rectangular prism with the following measurements:
volume
(cubic feet)
length
(feet)
width
(feet)
height
(feet)
882 6 7
1. First, Clare finds the quotient $$882 \div 6$$. What could she do next to find the height?
2. Find the missing height to finish the problem for Clare.
3. Complete the table.
volume
(cubic feet)
length
(feet)
width
(feet)
height
(feet)
936 8 9
1,536 48 2
1,008 36

### Student Response

For access, consult one of our IM Certified Partners.

### Activity Synthesis

• Ask students to share their work and reasoning.
• Highlight multiple approaches to find the missing side length in the second and third tables.
• Display second row of third table.
• “What are some different ways you can find the length of this rectangular prism?” (I can first divide the volume by 2 and then divide by 48 or I can multiply 2 and 48 and then divide the volume by that product.)
• “Which method did you prefer? Why?” (I divided the volume by 2 and then by 48 as that was quick and kept the numbers smaller.)

## Lesson Synthesis

### Lesson Synthesis

“Today we found missing side lengths of rectangles and rectangular prisms using division.”

Display last row of the table from the last problem of the last activity.

Invite students to share different responses for the width and height.

"What is the value of $$1,\!008 \div 36$$?" (28)

"Why is there more than one solution for the width and height of this rectangular prism?" (I only know that the product of the width and the height is 28. But there are different factors whose product is 28.)

## Cool-down: The Area of the Garden (5 minutes)

### Cool-Down

For access, consult one of our IM Certified Partners.

## Student Section Summary

### Student Facing

In this section, we learned how to divide multi-digit whole numbers. To find a quotient like $$448 \div 16$$ we broke 448 down into multiples of 16 and then added these partial quotients.

\begin{align} 320\div 16&= 20\\ 80\div 16 &= \phantom{0} 5\\ 48 \div 16 &= \phantom{0} 3\\ \overline {\hspace{5mm}448 \div 16} &\overline{\hspace{1mm}= 28 \phantom{000}}\end{align}

Then, we worked with a way to record these calculations that we first saw in an earlier course.