# Lesson 18

Represent Arrays with Expressions

## Warm-up: How Many Do You See: An Array of Shapes (10 minutes)

### Narrative

The purpose of this How Many Do You See is for students to subitize or use grouping strategies to describe the images they see. When students use the structure of the array to figure out how many objects are shown, they look for and make use of structure (MP7).

### Launch

- Groups of 2
- “How many do you see? How do you see them?”
- Flash the image.
- 30 seconds: quiet think time

### Activity

- Display the image.
- “Discuss your thinking with your partner.”
- 1 minute: partner discussion
- Record responses.

### Student Facing

How many do you see? How do you see them?

### Student Response

For access, consult one of our IM Certified Partners.

### Activity Synthesis

- “How did seeing equal groups help you know how many triangles there were in the array?”
- Consider asking:
- “Who can restate the way _____ saw the triangles in different words?”
- “Did anyone see the triangles the same way but would explain it differently?”

## Activity 1: Represent Array Situations (20 minutes)

### Narrative

The purpose of this activity is for students to represent multiplication situations with arrays and multiplication expressions. Students should have the option to use math tools, such as counters or connecting cubes, to create the arrays before they draw them. Connecting situations, arrays, and expressions reinforces the idea that multiplication can be used to express the total number of objects in equal groups (MP2).

*MLR2 Collect and Display.*Amplify language used to describe arrays. On a visible display, record words, phrases and expressions such as: row, column, each, for every, 3 by 5, 5 by 3, \(5\times3\), and \(3\times5\). Include diagrams and annotations. Invite students to borrow language from the display as needed, and update it throughout the lesson.

*Advances: Representing, Reading, Speaking*

*Representation: Internalize Comprehension.*Make connections between representations visible. Invite students to use gestures or drawings as they verbally describe correspondences between their arrays and expressions. For example, “The 4 in my expression \(4\times5\), shows the number of rows, and the 5 shows that there are 5 cars in each row.”

*Supports accessibility for: Conceptual Processing, Memory*

### Required Materials

Materials to Gather

### Required Preparation

- Each group of 2 will need 20 connecting cubes or counters.

### Launch

- Groups of 2
- Give students access to connecting cubes or counters.
- Display the first situation.
- “Take a minute to represent this situation with an array. You can use drawings or objects.”
- 2 minutes: independent work time
- “Discuss your ideas with your partner.”
- 2 minutes: partner discussion
- Share responses. Emphasize ways students used equal groups to create their arrays.

### Activity

- “Work with your partner to represent the next three situations with an array. Be prepared to share how you see equal groups in your array.”
- 5 minutes: partner work time
- Have students share an array for problems 2–4. Try to show both drawings and arrays made of objects.
- “How do you see equal groups in your arrays?” (The rows can show the number of groups, and then you put however many are in each group across the row.)
- Display the first situation again.
- “Let’s revisit the first situation. What multiplication expression would represent this situation?”
- 30 seconds: quiet think time
- Share responses. Emphasize how students use equal groups to write the expression.
- “Work with your partner to write multiplication expressions for the other three situations.”
- 2 minutes: partner work time

### Student Facing

Use objects or drawings to represent each situation with an array.

- There are 3 rows of chairs. Each row has 5 chairs.
- There are 4 rows of cars. Each row has 5 cars in it.
- There are 2 rows of eggs. Each row has 6 eggs.
- There are 2 teams of students lined up. Each team has 10 students.

Write a multiplication expression to represent each situation.

### Student Response

For access, consult one of our IM Certified Partners.

### Activity Synthesis

- “How did you use equal groups to write your multiplication expression?” (I thought about how many groups were in the situation and then how many things were in each group. For example, I know that 2 teams of 10 students is 2 groups of 10, so I can write \(2\times10\).)

## Activity 2: Connect Arrays to Expressions (15 minutes)

### Narrative

The purpose of this activity is for students to apply their knowledge from previous activities to draw arrays to match multiplication expressions. Have connecting cubes or counters available for students who need them. In the launch, students use their bodies to make an array for the expression \(4\times6\). Feel free to adjust this expression to better fit the number of students in your class.

### Launch

- Groups of 2
- Display \(4\times6\)
- “Work together as a class to use your bodies to create an array for this multiplication expression.”
- If the number of students does not exactly match the product, ask extra students to monitor the array and be prepared to explain where they see parts of the expression in the array.
- 3–5 minutes: whole-class work time

### Activity

- “Work with your partner to draw an array for each multiplication expression. Be ready to share connections you notice between the multiplication expressions and arrays.”
- 2–3 minutes: partner work time

### Student Facing

Draw an array for each multiplication expression. Be prepared to share your reasoning.

- \(2\times3\)
- \(5\times2\)
- \(4\times4\)

### Student Response

For access, consult one of our IM Certified Partners.

### Advancing Student Thinking

If students draw equal groups that are not arrays, consider asking:

- “How did you use the expression to create your drawing?”
- “How could we rearrange your drawing into an array?”

### Activity Synthesis

- Share an array or two for each expression.
- “What connections did you and your partner see between the multiplication expressions and arrays?” (The factors tell us how many things are in each row and column. For \(2\times3\), we drew an array with 2 columns that have 3 things in each column. For \(2\times3\), we drew an array with 2 rows that have 3 things in each row.)

## Lesson Synthesis

### Lesson Synthesis

Display a situation from the first activity, and an array and an expression that represents the situation.

“We learned that multiplication is how we express the total number of objects in equal groups.”

“How did your knowledge of equal groups help you create arrays and write expressions for multiplication situations?” (I thought about how many groups there were and drew the groups as each row [or column]. Then the number of groups tells me how many rows [or columns] there are. The array and the expression represent the total number of objects in the problem.)

## Cool-down: Array Situation (5 minutes)

### Cool-Down

For access, consult one of our IM Certified Partners.