# Lesson 11

Arrays and Rectangles

## Warm-up: Which One Doesn’t Belong: All Kinds of Arrays (10 minutes)

### Narrative

This warm-up prompts students to carefully analyze and compare different arrays. The activity also enables the teacher to hear how they talk about arrays, rows, and columns, and how they find the total number of objects in an array.

### Launch

• Groups of 2
• Display the image.
• “Pick one that doesn’t belong. Be ready to share why it doesn’t belong.”
• 1 minute: quiet think time

### Activity

• 2–3 minutes: partner discussion
• Record responses.

### Student Facing

Which one doesn’t belong?

### Activity Synthesis

• “What do B, C, and D have in common?” (They all show arrays with 3 in each row.)
• “D has 4 rows and 3 columns. This rectangle is partitioned into equal-size squares.”

## Activity 1: Use Tiles to Make Arrays (15 minutes)

### Narrative

The purpose of this activity is to make connections between arranging objects to make an array and making a rectangle from equal-size squares. They use the language of arrays to describe the rectangles they create (MP6) and learn that a rectangle composed of equal-size squares is an example of an array.

Representation: Develop Language and Symbols. Activate background knowledge. Have students recall the term “array” and ask, “What makes an array?” and “Tell me more about the rows and columns, are the rows/columns equal/unequal?”
Supports accessibility for: Memory, Language

### Required Materials

Materials to Gather

### Launch

• Groups of 2
• Give students inch tiles and colored pencils.
• Arrange 6 tiles into an array with 2 rows and 3 columns. Pause, then push them together to make a rectangle.
• “What do you notice? What do you wonder?” (There are 6 squares. The squares make a rectangle. You made an array first, then you pushed them together and made a rectangle. Is the rectangle an array of squares? Why did you push them together? Could you make other rectangles?)

### Activity

• “You and your partner can choose 12, 15, 16, 18, or 20 tiles. Arrange all the tiles in an array. Then push them together to make a rectangle.”
• “Shade in the same arrangement of squares on the grid paper to represent the rectangle you’ve made.”
• “Answer the questions about your rectangle. If you have time, choose a different number of tiles and make another rectangle.”
• 10 minutes: partner work time
• Monitor for a variety of different rectangles to display during the synthesis.

### Student Facing

Choose a number of tiles.
• 12
• 15
• 16
• 18
• 20

Arrange all the tiles in an array. Then push them together to make a rectangle.

1. Shade in the same arrangement of squares on the grid paper.
2. How many rows of squares does your rectangle have? ______
3. How many columns does your rectangle have? ______
4. How many tiles are in your rectangle? ______
5. Write 2 equations to represent the number of squares in your rectangle.

### Activity Synthesis

• Display student work.
• “Describe this rectangle.” (It has _____ rows of squares. It has _____ columns of squares. It has _____ squares in all.)
• “How many squares are in each row?”
• “How many squares are in each column?”
• “What equations could we write to represent the number of squares in this rectangle?”

## Activity 2: Make Equal-size Squares (20 minutes)

### Narrative

The purpose of this activity is for students to finish partitioning a rectangle into equal-size squares. This work will prepare students for partitioning rectangles on their own in later lessons. In the synthesis, students are invited to use what they know about the structure of arrays to anticipate how many equal-size squares will fill the rectangle without drawing any lines (MP8).

MLR8 Discussion Supports. Synthesis: At the appropriate time, give students 2–3 minutes to make sure that everyone in their group can explain how they know how many squares will fill the rectangle. Invite groups to rehearse what they will say when they share with the whole class.

### Required Materials

Materials to Gather

### Launch

• Give each student at least 2 different colors of colored pencils or crayons and a ruler.

### Activity

• “In the first activity, you used tiles to make a rectangle and shaded in squares to represent it. In this activity, you will start with a rectangle and decompose it into small equal-size squares by drawing lines with a ruler. For each rectangle, there will be some squares already outlined. Draw lines to continue making equal-size squares inside each rectangle.”
• “After making the squares, use colored pencils or crayons to show patterns in the rows or columns.”
• 10 minutes: independent work time

### Student Facing

1. Draw lines so that the rectangle is completely filled with equal-size squares.

2. Color the rows different colors.
3. How many rows of equal-size squares are there?
4. How many squares are in each row?
5. Write an equation to represent the sum of the squares in each row.
1. Draw lines so that the rectangle is completely filled with equal-size squares.

2. Color the columns different colors.
3. How many columns of equal-size squares are there?
4. How many squares are in each column?
5. Write an equation that represents the sum of squares in each column.
1. Draw lines so that the rectangle is completely filled with equal-size squares.

2. How many columns of equal-size squares are there? How many squares are in each column?
3. How many rows of equal-size squares are there? How many squares are in each row?
4. Write 2 equations to represent the number of equal-size squares in the rectangle.

### Student Response

If students create squares one by one or create squares that are significantly different in size, consider pairing them with a peer who uses a ruler or looks for ways to draw longer lines across rows and columns. Ask students to compare their techniques and try each other’s techniques to complete one row or column of squares.

### Activity Synthesis

• Display the rectangle that has 4 squares in the first row, but is missing squares in each column.
• “How could you find the total number of squares that would fill this rectangle without drawing the lines?” (I can tell there will be 4 columns based on the top row. I can tell there will be 5 rows based on the first column, even though it is missing 1 square.)

## Lesson Synthesis

### Lesson Synthesis

“Today you learned that we can partition a rectangle into equal-sized squares.”

Create or draw a rectangle with inch tiles to show:

“How would you describe this rectangle?” (It has 4 rows of squares. There are 5 squares in each row. It has 5 columns. There are 4 squares in each column.)

“What is the total number of equal-size squares inside the rectangle?” (20)