Lesson 2

Factor Pairs

Warm-up: Number Talk: Multiplication (10 minutes)

Narrative

The purpose of this Number Talk is to elicit strategies and understandings students have for multiplying single-digit numbers. These understandings help students develop fluency and will be helpful later in this lesson when students find factor pairs of numbers.

As students use earlier problems to find the new products, they look for and make use of structure (MP7) and use repeated reasoning (MP8).

Launch

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

Activity

• Keep expressions and work displayed.
• Repeat with each expression.

Student Facing

Find the value of each expression mentally.

• $$2\times7$$
• $$4\times7$$
• $$3\times7$$
• $$7\times7$$

Activity Synthesis

• “How did the first three expressions help you find $$7 \times 7$$?” (The 7 breaks apart into 3 and 4, so I could multiply in parts and add them.)
• “Who can restate _____’s reasoning in a different way?”
• “Did anyone have the same strategy but would explain it differently?”
• “Did anyone approach the expression in a different way?”
• “Does anyone want to add on to _____’s strategy?”

Activity 1: How Many Rectangles? (20 minutes)

Narrative

The purpose of this activity is for students to find all the possible pairs of whole-number side lengths given the area of a rectangle. Each group is assigned 2 areas for which they find all the possible rectangles. They draw and cut out the possible rectangles with that area. In the next activity, they will display the rectangles in a gallery walk. To find all possible rectangles with a given area, students may use tiles but they may also start to observe patterns such as if there are an even number of rows then the number of tiles in the rectangle is an even number (MP7).

Areas to assign (in square units):

Group A: 11, 27

Group B: 25, 5

Group C: 16, 8

Group D: 9, 18

Group E: 24, 12

Group F: 14, 28

Group G: 15, 30

Group H: 19, 20

This activity uses MLR7 Compare and Connect. Advances: representing, conversing.

Engagement: Provide Access by Recruiting Interest. Students may benefit from 2–3 minutes of independent work time to make sense of and begin the task before joining a group.
Supports accessibility for: Conceptual Processing, Visual-Spatial Processing

Required Materials

Materials to Gather

Materials to Copy

• Centimeter Grid Paper - Standard

Required Preparation

• Each of the 8 groups needs tools for creating a visual display.

Launch

• 8 groups
• Give each group access to inch tiles, grid paper, poster paper, scissors, and glue.

Activity

MLR7 Compare and Connect

• “You are going to be given 2 numbers. Each number represents the area of a rectangle. With your group, draw all the possible rectangles with that area and create a poster for each area. Inch tiles are available if you find them helpful.”
• “Your poster should show the rectangles with each of your assigned areas. Include details such as area and side lengths to help others understand your thinking.”
• Assign each group 2 area values.
• 15 minutes: small-group work time

Student Facing

Your teacher will assign 2 numbers to your group. Each number represents the area of a rectangle.

1. On grid paper:

• Draw all the possible rectangles that have the given area.
• Label the area and the side lengths.
• Use each pair of side lengths only once.

(For example, if you draw a rectangle with 4 units across and 6 units down, you don’t need to also draw a rectangle with 6 units across and 4 units down because they have the same pair of side lengths.)

2. When you think you've drawn all the possible rectangles for both areas, cut out your rectangles and put them on a poster for each area you were assigned.

3. Display your poster for all to see.

Student Response

Students may list only some of the factor pairs for a given number. Consider asking: “Are there any other rectangles you can draw?” or “How can you be sure that you have drawn all of the possible rectangles?”

Activity Synthesis

• “Before we walk around and look at all the posters, take a minute to reflect on the numbers you worked with. What did you notice and wonder as you worked on this activity?”
• 1–2 minutes: quiet think time

Activity 2: How Many Rectangles: Gallery Walk (15 minutes)

Narrative

In this activity, students examine the rectangles drawn by their classmates and learn the term factor pairs. Students recognize the side lengths of each rectangle as a factor pair of its area.

This activity uses MLR7 Compare and Connect. Advances: representing, conversing

• Groups of 2

Activity

• 5–7 minutes: gallery walk
• Monitor for different explanations students offer for how they know whether all possible rectangles with a given area have been found.

Student Facing

As you visit each poster, discuss with your partner:

1. What do you notice? Use the following sentence frames when you share:

1. “I notice that some of the posters . . . .”
2. “I notice the posters for numbers _____ and _____ are alike because . . . .”
2. How do you know that all possible rectangles were found for the given area?

Activity Synthesis

MLR7 Compare and Connect

• “What is the same and what is different between the rectangles on the posters?”
• 30 seconds: quiet think time
• 1 minute: partner discussion
• “How do you know that all possible rectangles have been found for the given area?” (We could not find any other numbers that multiply together to make the area.)
• Display the rectangles for 21: 1 by 21 and 3 by 7.
• “Are there any more rectangles we can draw? Why or why not?” (No, there are no more whole-number factors of 21. Or, no, because to get 21 we can multiply only 1 and 21, 3 and 7, 7 and 3, and 21 and 1.)
• “We call 1 and 21 a factor pair of 21 because each of them is a factor of 21 and multiplying them gives 21. Another factor pair of 21 is 3 and 7.”
• “Work with your partner to write down the factor pairs for the areas you were assigned.”
• 2 minutes: partner work time

Lesson Synthesis

Lesson Synthesis

“Today we learned that a factor pair of a whole number is a pair of whole numbers that multiply to result in that number. For example, 5 and 4 are a factor pair of 20.”

“What are the factors pairs of 24?” (1 and 24, 2 and 12, 3 and 8, and 4 and 6)

“How do we know if we have found all of the factor pairs of 24?” (We went in order. When we reached 4 and 6, there are no more pairs between 4 and 6, so we can stop there. Or, we used multiplication to see how many facts we could pair to make 24. Or, we used division, and these were all of the numbers that we could divide equally.)

“Can you use the same strategies to find all of the factor pairs of 45?” (Yes, 1 and 45, 3 and 15, 5 and 9. There are no more factors between 5 and 9, so I have found all of the factor pairs.)

“Can you use these strategies to find the factor pairs of any whole number?”

Math Community

After the cool-down, give students 2–3 minutes to discuss any revisions to the “Doing Math” actions in small groups. Share ideas as a whole group and record any revisions.