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Activity 1

Standards Alignment

Building On

Addressing

Building Toward

Instructional Routines

Materials

To Copy (from Blackline Masters)

Card Sort Area Cards

To Gather

Grid paper Inch tiles

Activity Narrative

The purpose of this activity is for students to learn about prime numbers and composite numbers. Students are given a set of cards with rectangles on them. They sort the rectangles by area and then attempt to draw an additional rectangle for each category. They notice that some areas can be represented by more than one rectangle and some areas can only be represented by one rectangle.

During the Activity Synthesis, highlight that the side lengths of each rectangle represent one factor pair (each pair of side lengths should be used only once), and that the area of each rectangle represents a multiple of each side length. Students learn that a number with only one factor pair—1 and the number itself—is a prime number, and a number with more than one factor pair is a composite number.

MLR8 Discussion Supports. Invite students to take turns selecting a rectangle, and explaining how they should sort it to their partner. Display the following sentence frames: “This rectangle belongs with _____ , because . . .” Encourage students to challenge each other when they disagree.
Advances: Conversing, Representing

Engagement: Develop Effort and Persistence. Chunk this task into manageable parts to support organizational skills in problem solving. Some students may benefit from explicit guidance for how to begin. For example, before sorting, students can find the area of each rectangle.
Supports accessibility for: Conceptual Processing, Visual-Spatial Processing, Organization

Launch

Groups of 2
Give each group a set of cards.

Activity

“Work with your partner to sort the cards into categories in a way that makes sense to you.”
2 minutes: partner work time
Monitor for the different ways students sort the cards, including grouping the cards into rectangles with the same area.
Invite 2–3 previously selected students to share their categories.
“If you did not already, group the cards into rectangles that have the same area.”
3–5 minutes: partner work time
Ask students to check their work with another group to make sure the cards in each category match.
“Now, create at least one rectangle to add to each category in your card sort.”
3–5 minutes: partner work time
Observe the rectangles students add to each category. Monitor for students who notice that no new rectangles could be drawn for the area of 7 square units.

Your teacher will give you a set of cards that show rectangles.

Sort the cards into categories in a way that makes sense to you. Be ready to explain the meaning of your categories.
Group the cards into rectangles that have the same area. Be ready to explain your reasoning.
For each group of cards that have the same area, think of at least one more rectangle. Record its length and width. Be prepared to explain your reasoning.

Student Response

Activity Synthesis

Invite 2–3 students to share the rectangles they added to each category.
“Why were you able to create more rectangles for some areas and not others?” (Some of the numbers had more factor pairs. For some numbers, there was only one possible factor pair.)
Repeat student reasoning. “Only one rectangle can be made for the area of 7. Numbers like 7 are called prime numbers. Prime numbers have only one factor pair: 1 and itself.”
“Numbers like 15 that have more than one factor pair are called composite numbers.”
“What other composite numbers did you work with? How do you know they are composite?” (Twenty-four is a composite number because I can make 2 rows of 12 or 4 rows of 6. Eighteen is composite because it has factor pairs of 2 and 9 and 3 and 6.)

Activity 2

Standards Alignment

Building On

Addressing

Building Toward

Instructional Routines

None

Materials

To Gather

Grid paper Inch tiles

Activity Narrative

In this activity, students use the area of rectangles to find all factor pairs of a given whole number and decide if the number is prime or composite. The Activity Synthesis focuses on finding all possible rectangles for a given area as a strategy to find all the factor pairs of a number. Students may notice that they do not need to find all possible rectangles to determine whether a number is prime or composite.

Launch

Groups of 2
Give each group access to inch tiles and grid paper.
“If you were given a number that is the area of rectangle, how could you find out how many rectangles with that area can be made?” (Test it out with tiles. Think about factor pairs for the number.)
1 minute: partner discussion
Share and record responses.

Activity

“Work with your partner to complete this table. Inch tiles and grid paper are available if you’d like them.”
10 minutes: partner work time
Monitor for different ways students find the number of rectangles, such as:
- Building the rectangles from inch tiles.
- Drawing rectangles on grid paper.
- Drawing rectangles freehand.
- Listing the factor pairs of the number and knowing that one rectangle corresponds to each pair.

The table shows different areas. How many rectangles with whole-number side lengths can be made for each area?

Complete the table.

Rectangles with the same pair of side lengths should be counted only once. For example, if you count a rectangle with 4 units across and 6 units down, you don’t need to also count a rectangle with 6 units across and 4 units down.

area (square units)	how many rectangles?	prime or composite?
2
10
48
11
21
23
60
32
42
31
56

Student Response

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Advancing Student Thinking

Activity Synthesis

Invite 3–4 groups to share their strategy for finding the number of rectangles for a given area.
“How does the number of factor pairs relate to the number of rectangles?” (The side lengths of each rectangle is a factor pair. So, finding all the rectangles would give us all the factor pairs. Finding all the factor pairs of the number would tell us how many rectangles have that number for their area.)
“What are all of the prime numbers in our list? How do we know they are prime?” (2, 11, 23, 31. They each only have one factor pair, 1 and the number itself.)
“What do you notice about the prime numbers?” (They are odd numbers except the number 2.)
“What is the smallest prime number in our set? Is it the smallest prime number?” (2. I don’t know. Is 1 a prime number?)
Display a rectangle with an area of 1 square unit.
“What are the side lengths of a rectangle with an area of 1 square unit?” (1 and 1)
“Since 1 only has 1 factor, it doesn’t have any factor pairs, so it is neither prime nor composite.”
“What are all the composite numbers in our set? How do we know they are not prime?” (10, 48, 21, 60, 32, 42, 56. They each have more than 1 factor pair.)