# Lesson 17

More Perimeter Problems

## Warm-up: True or False: Fractions and Whole Numbers (10 minutes)

### Narrative

This True or False routine prompts students to recall what they know about addition of fractions with a common denominator and about fractions that are equivalent to whole numbers. The understandings elicited here will be helpful later in the lesson when students find sums or differences of two fractions, or of a whole number and a fraction, to solve problems about the perimeter of rectangles with fractional side lengths.

### Launch

• Display one statement.
• “Give me a signal when you know whether the statement is true and can explain how you know.”
• 1 minute: quiet think time

### Activity

• Share and record answers and strategies.
• Repeat with each statement.

### Student Facing

Decide if each statement is true or false. Be prepared to explain your reasoning.

• $$\frac {8}{12} + \frac{3}{12} + \frac{9}{12} + \frac{4}{12} = 2$$
• $$\frac{20}{4} + \frac{10}{4} + \frac{6}{4} = 8$$
• $$2 = \frac{59}{100} + \frac{41}{100} + \frac{89}{100} + \frac{11}{100}$$
• $$2 = \frac{3}{8} + \frac{3}{8} + \frac{12}{8}$$

### Activity Synthesis

• “In each equation, how do we know that the sum of the fractions can be written as a whole number? For example, how do we know that $$\frac{24}{12}$$ and $$\frac{36}{4}$$ can each be written as a whole number?” (There are 2 groups of $$\frac{12}{12}$$ in $$\frac{24}{12}$$ and 9 groups of $$\frac{4}{4}$$ in $$\frac{36}{4}$$.)
• “How do we know if any fraction is equivalent to a whole number?” (The numerator is a multiple of the denominator.)

## Activity 1: Along the Walls in Tiny Steps (15 minutes)

### Narrative

In this activity, students use their knowledge of feet and inches and the perimeter of a rectangle to solve problems in context. Given the perimeter and one length of a room, they determine its width in feet and the distance along two walls in inches. Students could reason about each problem in a number of ways. As they interpret the quantities in the situation and represent them with expressions or equations, students practice reasoning quantitatively and abstractly (MP2).

MLR7 Compare and Connect. Synthesis: Invite students to prepare a visual display that shows the strategy they used to determine how many inches the ant traveled. Encourage students to include details that will help others interpret their thinking. Give students time to investigate each others’ work. During the whole-class discussion, ask students, “Did anyone solve the problem the same way, but would explain it differently?”, “What did the strategies have in common?”, “How were they different?”

### Launch

• Groups of 2
• “Let’s solve some problems about the side lengths and perimeter of a room.”

### Activity

• 6–8 minutes: independent work time
• 4–5 minutes: partner discussion
• Monitor for the different ways students reason about the number of inches traveled by the ant in the last problem. They may:
• convert the perimeter from feet to inches and divide by 2
• convert the length and the width into inches separately and then add them
• find the sum of the length and width in feet and then convert that sum into inches

### Student Facing

A rectangular room has a perimeter of 39 feet and a length of $$10\frac{1}{2}$$ feet.

1. What is the width of the room in feet? Explain or show your reasoning.
2. An ant walked along two walls of the room, always in a straight line. It started in one corner and ended up in a corner opposite of where it started. How many inches did it travel? Explain or show your reasoning.

### Advancing Student Thinking

To find the width of the room, students may subtract $$10\frac{1}{2}$$ feet from 39 feet and not remember that there are two sides that are $$10\frac{1}{2}$$ feet long and there are two sides with the same width. Encourage them to draw a diagram of the room and to label the sides with known measurements.

### Activity Synthesis

• Invite students to briefly share their response to the first problem and then focus the discussion on the last problem.
• Select students with different strategies to find the distance the ant walked to share their reasoning.

## Activity 2: Missing Measurements (20 minutes)

### Narrative

This activity allows students to consolidate their learning from the past few units to solve problems about length measurements in a mathematical context.

First, students find the perimeter or missing side length of various quadrilaterals. To do so, they apply what they know about adding and subtracting fractions and about rewriting certain fractions as whole numbers. Next, they determine which pairs of figures have a certain multiplicative relationship—for instance, which figure has a perimeter that is 9 times that of another figure. Because the measurements are in different units, students need to attend to the relationship between units and perform conversions accordingly.

The activity can be done in the format of a gallery walk or by giving each group a full set of the diagrams from the blackline master.

Here are the images on the blackline master for reference:

Perimeter =  9 km

Perimeter = __________

Perimeter = __________

Perimeter = 12 in

Perimeter = __________

Perimeter = __________

Representation: Develop Language and Symbols. Invite students to represent each problem as an equation to help them identify strategies for solving and to give them practice interpreting mathematical language.
Supports accessibility for: Conceptual Processing, Language

### Required Materials

Materials to Copy

• Missing Measurements - Small
• Missing Measurements - Large

### Required Preparation

• If the activity is done as a gallery walk, print and cut 1–2 copies of the blackline master with the larger images and post them around the classroom. Otherwise, print and cut 1 copy of the blackline master with the smaller images for each group of 3–4 students.

### Launch

• Groups of 3–4
• As a class, read the opening paragraph and the first two problems. Display the six diagrams. Give students a minute to ask clarifying questions.
• “Recall that quadrilaterals are four-sided shapes. Squares and rectangles are examples of quadrilaterals, but they are not the only ones.”
• If done as a gallery walk, ask each student in a group to visit at least 2 posters, ensuring that, collectively, the members of each group visit all the posters.
• Alternatively, give each group one set of diagrams from the blackline master and ask each member to work on 2 of the diagrams.
• “Coordinate with your group to make sure that all six problems are answered.”

### Activity

• 7–8 minutes: independent work time on the first problem
• 10 minutes: group work time on the remainder of the task
• Monitor for students who can distinguish customary units and standard units and consider them as categories of related units (for instance, yards, feet, and inches in one category, and kilometers, meters, and centimeters as another.)

### Student Facing

Your teacher has posted six quadrilaterals around the room. Each one has a missing side length or a missing perimeter.

1. Choose two diagrams—one with a missing length and another with a missing perimeter. Make sure that all six shapes will be visited by at least one person in your group.

Find the missing values. Show your reasoning and remember to include the units.

2. Discuss your responses with your group until everyone agrees on the missing measurements for all six figures.

3. Answer one of the following questions. Explain or show your reasoning.

1. The perimeter of B is how many times the perimeter of D?
2. The perimeter of one figure is 1,000 times that of another figure. Which are the two figures?
3. The perimeter of F is how many times the perimeter of B?

### Activity Synthesis

• See lesson synthesis.

## Lesson Synthesis

### Lesson Synthesis

Select students to briefly share the 6 missing measurements and their reasoning. Display their calculations or record them for all to see. Then, discuss how students went about reasoning if the comparison statements in the last problem were true.

“How did you know that the perimeter of B is 2 times that of D?” (I know that feet and inches are related, and that 1 foot is 12 inches. So I recognized that 24 inches is twice 1 foot.)

“How did you find a figure with a perimeter 1,000 times that of another figure?” (I know that a kilometer is 1,000 meters, so I started with A and E, converted their perimeters, and then checked if one is indeed 1,000 times as long as the other.)

“How did you know that the perimeter of F is 9 times that of B?” (I converted the yards into feet, and then I could see that 18 feet and 2 feet are related: $$18 = 9 \times 2$$.)

## Cool-down: A Rectangle and a Trapezoid (5 minutes)

### Cool-Down

• One way to solve this problem is by finding $$16 \div 4$$ to find Jada’s throw distance in yards ($$16 \div 4 = 4$$) and then multiplying the result to convert the yards to feet ($$4 \times 3 = 12$$, so 4 yards is 12 feet).
• Another way is to first convert the 16 yards to feet ($$16 \times 3 = 48$$, so 16 yards is 48 feet) and then divide the result by 4 to find Jada’s throw distance ($$48 \div 4 = 12$$).