# Lesson 10

Write Expressions and Equations to Represent Arrays

## Warm-up: True or False: Expressions that Represent Arrays (10 minutes)

### Narrative

The purpose of this True or False is to elicit strategies and understandings students have for expressions with equal addends. Arrays are displayed for the first 2 equations as a support for students to explain how they know the equations are true.

### Launch

• Display one statement.
• “Give me a signal when you know whether the statement is true and can explain how you know.”
• 1 minute: quiet think time

### Activity

• Share and record answers and strategies.
• Repeat with each statement.

### Student Facing

Decide if each statement is true or false. Be prepared to explain your reasoning.

• $$2 + 2 + 2 = 3 + 3$$
• $$4 + 4 + 4 = 3 + 3 + 3 + 3$$
• $$5 + 5 + 5 = 3 + 3 + 3$$

### Activity Synthesis

• Arrange counters to show 5 rows of 3 counters:
• “How could we make the last statement true based on this array? Explain.” (To make this true we need $$3 + 3 + 3 + 3 + 3$$. I know that if $$5 + 5 + 5$$ means 3 columns of 5, there would be 5 rows of 3.)

## Activity 1: Build Arrays and Write Equations (15 minutes)

### Narrative

The purpose of this activity is for students to write equations that represent the number of objects in the rows or columns of an array. In a previous lesson, students matched arrays to expressions. In this activity, they write their own equations and describe how each equal addend represents the number of counters in each row or each column.

MLR2 Collect and Display. Synthesis: Direct attention to words collected and displayed from the previous lesson. Invite students to borrow language from the display as needed, and update it throughout the lesson.
Action and Expression: Develop Expression and Communication. Invite students to show thinking using red and yellow counters. For example, show the rows using red in the first row, yellow in the second row, red in the third row, yellow in the 4th row, and so on. Then have students represent the columns using the two colors as well. This shows a concrete representation for each number in an expression.
Supports accessibility for: Conceptual Processing, Organization

### Required Materials

Materials to Gather

### Launch

• Groups of 2
• Give students counters.

### Activity

• “First, you will arrange counters to make an array. Then you will write an equation that has equal addends. There are 2 equations that match each array. To find the total number of counters, you can use any method that makes sense to you.”
• 6 minutes: independent work time
• “For number 3, work with a partner. You should each make your own array and follow the steps. Then show your partner your array and let them answer the same questions. Check to see if you agree.”
• 6 minutes: partner work time
• Monitor for students to share different equations for an array they create that uses between 6 and 24 counters.

### Student Facing

1. Use 20 counters to make an array with 4 rows.

1. How many columns does your array have?
2. Fill in the blanks to create equations with equal addends that represent the array.

$$\underline{\hspace{1 cm}}+\underline{\hspace{1 cm}}+\underline{\hspace{1 cm}}+\underline{\hspace{1 cm}}=\underline{\hspace{1 cm}}$$

$$\underline{\hspace{1 cm}}+\underline{\hspace{1 cm}}+\underline{\hspace{1 cm}}+\underline{\hspace{1 cm}}+\underline{\hspace{1 cm}}=\underline{\hspace{1 cm}}$$

2. Use 15 counters to make an array with 3 columns.

1. How many rows does your array have?
2. Fill in the blanks to create equations with equal addends that represent the array.

$$\underline{\hspace{1 cm}}+\underline{\hspace{1 cm}}+\underline{\hspace{1 cm}}+\underline{\hspace{1 cm}}+\underline{\hspace{1 cm}}=\underline{\hspace{1 cm}}$$

$$\underline{\hspace{1 cm}}+\underline{\hspace{1 cm}}+\underline{\hspace{1 cm}}=\underline{\hspace{1 cm}}$$

3. Choose an even number of counters between 6 and 24. Make an array.

1. How many rows does your array have?
2. How many columns does your array have?
3. Write equal addends equations that represent the array.

### Student Response

If students create equations that do not match their array, consider asking:
• “Does the sum of your equation match the total number of counters?”

### Activity Synthesis

• Invite previously identified students to share their arrays and ask others to decide how many rows and columns the student’s array could have.
• “_____ wrote this equation. How many rows and columns could _____’s array have?”

## Activity 2: Arrange Veggies to Make Arrays (20 minutes)

### Narrative

The purpose of this activity is for students to practice writing equations with equal addends to represent the number of objects in each row or column of an array. They create arrays based on the context of a vegetable garden and compare different arrangements of their crops (MP2).

### Required Materials

Materials to Gather

### Launch

• Groups of 2
• “What do you know about growing plants?” (Plants need soil, water, and sunlight. You can grow plants to eat them. You can grow flowers. Plants can grow inside or outside.)
• 1 minute: quiet think time
• 1–2 minutes: partner discussion
• Share responses.
• “How are plants arranged in gardens or in fields?” (Sometimes the same plants are together. Flowers might be mixed up to make a design. On a farm, some plants are arranged in rows.)

### Activity

• “The local farmer needs help arranging crops in the vegetable garden. Based on the number of veggies planted, draw arrays to show how each crop could be arranged. Use counters if it helps.”
• 10 minutes: independent work time
• Monitor for a variety of ways students arrange the 16 carrot seeds, including 4 rows of 4.

### Student Facing

1. Make an array that shows how to plant 9 potatoes. Draw it.
Write an equation to represent your array.

2. Make an array that shows how to plant 16 carrot seeds. Draw it.
Write an equation to represent your array.

3. Make an array that shows how to plant 15 potatoes. Draw it.
Write an equation to represent your array.

4. Make an array that shows how to plant 12 carrot seeds. Draw it.
Write an equation to represent your array.

### Student Response

If students represent the crops using one row or multiple rows with 1 in each, consider asking:

• “Is there another way to arrange the crops so there are at least 2 rows?”

### Activity Synthesis

• Invite 2–3 previously identified students to share how they arranged 16 carrot seeds. Select the student who arranged in 4 rows of 4 carrots seeds last.
• If no student arranged the carrots seeds in 4 rows, arrange counters to show 4 rows of 4 counters.
• “What equation would represent the array using equal addends?” ($$4 + 4 + 4 + 4 = 16$$)
• “Do the addends represent the number of carrot seeds in each row or in each column?” (It could represent both.)
• “Can you find another equation that shows the number in each row or each column?” (No, since there are the same number of rows and columns, there is only one equation.)

## Lesson Synthesis

### Lesson Synthesis

“Today you learned that you can write equations to show the sum of the number of objects in rows or columns of arrays.”

Arrange counters to show:

“Elena wrote $$6 + 6 = 12$$ as an equation to represent the number of objects in this array. Do you agree? Explain.” (Yes, but it doesn’t show the number in the rows or the number in the columns. $$6 + 6$$ can help us find the total, but the equations $$4 + 4 + 4 = 12$$ or $$3 + 3 + 3 + 3 = 12$$ show the sum of the rows or columns.)