Lesson 12
Partition Rectangles into Squares
Warmup: Estimation Exploration: Fill it Up (10 minutes)
Narrative
The purpose of this Estimation Exploration is to practice the skill of making a reasonable estimate. Students consider how the placement of the first 2 squares can help them think about the total number of squares needed to fill the rectangle (MP7). These understandings will be helpful later when students will need to partition rectangles into equalsize squares.
Launch
 Groups of 2
 Display the image.
 “How many little squares would fill the rectangle?”
 “What is an estimate that’s too high?” “Too low?” “About right?”
 1 minute: quiet think time
Activity
 “Discuss your thinking with your partner.”
 1 minute: partner discussion
 Record responses.
 Draw equalsize squares to show:
 “Would you like to revise your thinking?”
 1 minute: quiet think time
 Share responses.
Student Facing
How many little squares would fill the rectangle?
Record an estimate that is:
too low  about right  too high 

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Student Response
For access, consult one of our IM Certified Partners.
Activity Synthesis
 “How did the second image help you revise your thinking?”
 Consider asking, “How could you tell that 11 would not be a reasonable estimate?” (The bottom row already has 4, so the 2 bottom rows will have 10. It has to be more than 11 because there is another row to fill.)
Activity 1: How Many Squares? (15 minutes)
Narrative
The purpose of this activity is for students to partition rectangles to create rows and columns of equalsize squares. In the launch, students build an array with tiles and then represent it on a rectangle with tick marks as guidance. They will partition rectangles without tick marks in the next activity.
Advances: Conversing, Reading
Required Materials
Materials to Gather
Launch
 Groups of 2
 Give students rulers and inch tiles.
Activity
 “Use 8 tiles to make a rectangle. Your tiles should be in 2 rows.”
 1 minute: independent work time
 “Now, draw lines in the rectangle to show the squares. It should have the same number of equalsize squares as the rectangle you made out of tiles. You may use a ruler if it helps you.”
 1 minute: independent work time
 “You will draw lines to partition 2 more rectangles into equalsize squares. You can also use the tiles to build it first, if that helps.”
 10 minutes: independent work time
Student Facing

Build a rectangle with 8 tiles arranged in 2 rows. Use a ruler to partition the rectangle to match the rectangle you made.

Use a ruler to partition the rectangle using the tick marks as a guide.
 How many rows of equalsize squares did you make?
 How many columns did you make?
 Write 2 equations to represent the total number of equalsize squares.

Use a ruler to partition the rectangle using the tick marks as a guide.
 How many rows of equalsize squares did you make?
 How many columns did you make?
 Write 2 equations to represent the total number of equalsize squares.
Student Response
For access, consult one of our IM Certified Partners.
Advancing Student Thinking
If students create equations that match the total number of squares, but do not represent the sum of squares in each row or the sum of squares in each column, consider asking:
 “How does your equation match the array?”
 “Can you write an equation with equal addends that shows the sum of each row or column?”
Activity Synthesis
 Invite 1–2 students to share their rectangles and equations for their rectangle with 3 rows and 5 columns. (\(3 + 3 + 3 + 3 + 3 = 15\) or \(5 + 5 + 5 = 15\))
 Draw a rectangle and partition to show:
 “How is this partition different from what we did in this activity?” (The squares aren’t the same size and the last column has 4 not 3.)
Activity 2: Partition Rectangles (20 minutes)
Narrative
The purpose of this activity is for students to partition rectangles into rows and columns of equalsize squares. They use tiles to help them see how to draw lines to partition the rectangles. Although it is important for students to use what they know about the structure of arrays and composing rectangles from squares to partition the rectangles (MP7), it is not necessary that students’ drawings are perfect. As long as it is clear that the student intended for the squares to be equal in size and they can articulate their reasoning. If their squares are significantly different or result in more squares in one column or row than another, offer a ruler and consider having students trace tiles for practice or guidance.
Supports accessibility for: VisualSpatial Processing, Organization
Required Materials
Materials to Gather
Launch
 Groups of 2
 Give students rulers and inch tiles.
Activity
 “You will be partitioning rectangles. For the first one, work with a partner to make a rectangle using tiles to help you plan, and then partition one of the rectangles to match.”
 5 minutes: partner work time
 Monitor for a group who used each of the rectangles.
 Selected groups share.
 “How did you decide which drawn rectangle to use to represent your tile rectangle?” (Our rectangle had 2 tiles in each row so we used the taller rectangle.)
 “Could you have used the other rectangle?” (No, because the squares wouldn’t be the same size.)
 30 seconds: quiet think time
 1 minute: partner discussion
 Share responses.
 “For the next 2 rectangles, work on your own to partition them. Remember that all the squares should be the same size. After you are done, compare with your partner.”
 6 minutes: independent work time
 Monitor for students who have a solid strategy for making equalsize squares. Choose at least 1 student who thought about the number of squares first and 1 who drew the lines first and then counted the squares.
Student Facing

Use 12 tiles to make a rectangle. Split one of the rectangles into equalsize squares to match your rectangle made of tiles.
 Write 2 equations to represent the total number of squares.

Split this rectangle into equalsize squares.

Write 2 equations to represent the total number of squares.


Split this rectangle into equalsize squares.
 Write 2 equations to represent the total number of squares.
Student Response
For access, consult one of our IM Certified Partners.
Activity Synthesis
 Invite previously identified students to share how they partitioned their rectangles.
 “What strategies did you use for making your squares all the same size?” (I knew I wanted _____ squares, so I did _____ lines across and down. I started by making my rows, then made the lines for the columns, and then saw how many squares it made.)
Lesson Synthesis
Lesson Synthesis
“In this unit you learned about even and odd numbers and different types of arrays. Looking at these images, think about 1 thing you could say about each one. What are some things that are the same or different?”
Share and record responses.
Cooldown: How Many Squares? (5 minutes)
CoolDown
For access, consult one of our IM Certified Partners.
Student Section Summary
Student Facing
In this section, we learned that arrays are groups of objects that are organized into rows and columns. Arrays have the same number of objects in each row and the same number of objects in each column. We practiced different ways to count the objects in an array and used expressions with equal addends to show that you can find the total objects in an array by adding the sum of each row or the sum of each column. We also learned that rectangles can be composed of an array of equalsize squares. We practiced partitioning rectangles into rows and columns of equalsize squares.
\(4 + 4 + 4 + 4 + 4 = 20\) or \(5 + 5 + 5 + 5 = 20\)