# Lesson 19

Solve Problems Involving Arrays

## Warm-up: Number Talk: One Less Group (10 minutes)

### Narrative

The purpose of this Number Talk is to elicit strategies and understandings students have about equal groups in multiplication expressions and see a pattern as one factor is decreased. These understandings help students develop fluency and will be helpful later in this lesson when students will need to be able to use multiplication to answer questions about array situations.

When students notice that as the number being multiplied by 2 decreases the product decreases by a group of 2, they look for and express regularity in repeated reasoning (MP8).

### Launch

• Display one expression.
• “Give me a signal when you have an answer and can explain how you got it.”
• 1 minute: quiet think time

### Activity

• Keep expressions and work displayed.
• Repeat with each expression.

### Student Facing

Find the value of each expression mentally.

• $$10\times2$$
• $$9\times2$$
• $$8\times2$$
• $$7\times2$$

### Activity Synthesis

• “What do you notice about the products as we worked through this series of problems?” (They decrease by 2.)
• “Why does the product decrease by 2 each time?” (The number of groups decreased by 1.)

## Activity 1: Array of Colors (25 minutes)

### Narrative

The purpose of this activity is for students to use the Co-craft math language routine to write questions that can be asked about array situations and to relate array situations to equations. Students should be encouraged to use whatever strategy or representation feels appropriate to them. Given their prior experiences, they may represent the situation with an array and skip-count or consider equal groups in other ways to find the total.

### Launch

• Groups of 2

MLR5 Co-craft Questions

• Display: “There are 7 rows.”
• “Write a list of mathematical questions that could be asked about this situation.” (What’s in the rows? How many things are in each row? How many things are there altogether?)
• 2 minutes: independent work time
• 2–3 minutes: partner discussion
• Invite several students to share one question with the class. Record responses.
• “What do these questions have in common? How are they different?” (They all have to do with rows. They have to do with more detail about the rows, like what’s in the rows and how many are in each row.)

### Activity

• “Complete the first problem by solving this situation in any way that makes sense to you.”
• 3–4 minutes: independent work time
• 1–2 minutes: partner discussion
• Share a variety of student representations and solution strategies.
• “Think about the situation we have been considering. How could you represent the situation using an equation with a symbol for the unknown?”
• 2 minutes: quiet think time
• “Share your equation with your partner. Together, rewrite the equation with the solution you found in place of the symbol.”
• 2 minutes: partner work

### Student Facing

There are 7 rows. Each row has 5 crayons. How many crayons are there?

1. Solve this problem. Explain or show your reasoning.
2. Represent the situation with an array and an equation with a symbol for the unknown.

### Activity Synthesis

• “What equation(s) did you write?”
• Display an equation with a symbol and one with the solution.
• “How does each part of the equation connect to the situation?" (The 7 is the number of rows. The 5 is the number of crayons in each row. The 35 represents the total number of crayons, but it was a question mark when we didn't know how many there were.)

## Activity 2: Tyler’s Trees (10 minutes)

### Narrative

The purpose of this activity is for students to write an equation with a symbol for the unknown to represent an array situation. Then, they answer the question in the multiplication situation. Encourage students to use whatever strategy or representation feels appropriate to them. Given their prior experiences, they might represent the situation with an array and skip-count or consider equal groups in other ways to find the total.

In the launch of the activity, it may be helpful to ask students to tell their partner a quick story or ask any questions about the context of the first problem. To ensure all students have access, it may also be helpful to display images for students to reference about coconut trees or Mexico.

MLR8 Discussion Supports. Synthesis: If necessary, invite students to repeat their question using mathematical language. For example, “Can you say that again, using the word array?”
Action and Expression: Develop Expression and Communication. Activity: Invite students to show thinking using connecting cubes or counters.
Supports accessibility for: Conceptual Processing

### Launch

• Groups of 2
• Display the image.
• “Coconut trees are grown as a crop in warm climates and have lots of uses. Sometimes they are grown in rows. What are some other crops that are grown in rows?” (Corn. Strawberries. Carrots.)

### Activity

• “Now you are going to practice what we just learned about solving array situations and writing an equation with a symbol for the unknown.”
• 5–7 minutes: independent work time
• As you circulate, consider asking:
• “How does each number or symbol in your equation connect to the situation?”
• “How are you using equal groups to find the solution to the problem?”
• 2 minutes: partner discussion

### Student Facing

For each problem:

• Write an equation with a symbol for the unknown to represent the situation.
• Solve the problem. Show your reasoning.
1. A field of coconut trees in Mexico has 5 rows of trees. Each row has 9 trees. How many trees are there?

2. After learning about growing coconuts in Mexico, Tyler wants to plant coconut trees in his backyard in Florida. His mom will only let him plant 2 rows of 4 trees in his backyard.

How many trees will Tyler plant?

### Activity Synthesis

• “What questions do you still have about solving array problems or writing an equation with a symbol for the unknown?”

## Lesson Synthesis

### Lesson Synthesis

Display the information from the first problem in Activity 2.

A field of coconut trees in Mexico has 5 rows of trees.
Each row has 9 trees. How many trees are there?

$$5\times9=?$$

$$5\times9=45$$

“How do each of these equations represent the array situation?” (The equation with the symbol represents the 5 rows of trees and the 9 trees in each row in the situation before we knew there were 45 trees. The equation with the 45 includes the solution to the problem because there were 45 trees.)

Consider asking: “What does the question mark in the first equation represent in the situation? What does the 45 in the second equation represent in the situation?”