Lesson 16

This Number Talk elicits the strategies and understandings students have for multiplying a whole number and a fraction mentally. The reasoning here prepares students to perform multiplication to solve problems about the perimeter of rectangles with fractional side lengths later in the lesson.

• Display one expression.
• “Give me a signal when you have an answer and can explain how you got it.”
• 1 minute: quiet think time

• Record answers and strategy.
• Keep expressions and work displayed.
• Repeat with each expression.

• “How do the first three expressions help you find the value of the last expression?”

In this activity, students consider possible side lengths for a rectangle with a perimeter of 12 inches and visualize each rectangle. Students may notice many patterns as they find different rectangles (MP7) including

• the sum of the length and width is 6 inches
• the length and width can be exchanged to give a length and width pair
• when the length is a fraction so is the width and vice versa

• Groups of 2
• Display a 12-inch pipe cleaner.
• “This pipe cleaner is 12 inches long. If we use the entire length to form a rectangle, what might be one possible pair of length and width?” (Sample responses: 4 inches and 2 inches)
• “Think of some other pairs of length and width and record them in the table.”
• “Each pair should be unique. If you have listed 4 inches and 2 inches as a pair, do not list 2 inches and 4 inches as another pair.”

• 5 minutes: partner work time

Students may list only whole-number side lengths. consider asking: “Have you tried using a fraction as a side length?”

• Invite students to share a set of side lengths.
• Record responses. This table will be used in the next activity.
• If students do not offer fractional side lengths as an option consider asking students to consider a pair of side lengths that includes fractions.

In this activity, students build rectangles with a perimeter of 12 inches and varied side lengths. Then, they reason about the side lengths of rectangles whose perimeters are multiples of 12.

• Each group of 2 needs a 12-inch pipe cleaner, an inch ruler, and tape.

• Groups of 2
• Assign one pair of side lengths from the table created in the previous lesson for each group to build.

• “After you have built your rectangle, find its perimeter. You may remember that the perimeter is the total distance all the way around a shape.”
• 2 minutes: partner work time
• When the rectangles are built, ask: “What is the perimeter of your rectangle?” (12 inches) “How do you know?” (The length of the pipe cleaner is 12 inches and we used the entire length.)
• Display two pipe cleaners that are joined without overlaps. “What is the combined length of these pipe cleaners?” (24 inches) “How do you know?” (It’s 2 times 12 inches.)
• “Work with your partner to complete the rest of the activity.”
• 5–7 minutes: partner work time

Students may confuse the idea of perimeter with that of area. For instance, when asked to find possible side lengths for squares with perimeters 24 inches and 60 inches, they may think that they are to find numbers that multiply to 24 and 60 inches, respectively. Urge them to visualize a 24-inch long or 60-inch long pipe cleaner being bent to form a square. “How long would each side be?”

• Invite students to share their responses to the last two problems and their reasoning.
• “How did you know that 5 pipe cleaners were used in the last problem?” (60 is 5 times 12.)
• “How did you know that the side length of the last square is 15 inches?” (The perimeter of a square is 4 times its side length. \(4 \times 15 = 60\))
• Consider displaying a table as shown here to highlight the relationships between the number of pipe cleaners used, the perimeter of the square, and the side length of the square.

  number of pipe cleaners	perimeter (inches)	side length of square (inches)
  1	12	3
  2	24	6
  5	60	15

In this activity, students continue to think about the relationship between side lengths and perimeter by drawing (on grid paper) rectangles when given the perimeter, one or both side lengths, or the relationship between two rectangles. They apply what they learned in an earlier unit about comparing quantities multiplicatively.

• Groups of 2
• Give each student a sheet of centimeter grid paper and a straightedge or ruler.
• “Each square on the grid is 1 centimeter by 1 centimeter.”

• “Take a few quiet minutes to draw some rectangles based on the requirements given to you. Be sure to label each rectangle with its side lengths and perimeter.”
• 7–8 minutes: independent work time
• 2–3 minutes: partner discussion
• Monitor for students who choose different pairs of side lengths for rectangle A.

• See lesson synthesis.