Lesson 17

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.

• 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

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

• “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.)

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).

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

• 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

• 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.

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:

A

Perimeter =  9 km

B

Perimeter = __________

C

Perimeter = __________

D

Perimeter = 12 in

E

Perimeter = __________

F

Perimeter = __________

• 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.

• 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.”

• 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.)

• See lesson synthesis.