# Lesson 9

## Warm-up: Estimation Exploration: How Many Waffles? (10 minutes)

### Narrative

The purpose of this Estimation Exploration is to practice the skill of making a reasonable estimate. Students consider how the arrangement of the objects helps them estimate the total number of objects (MP7). For the first image, students should keep their books closed and discuss estimates as a group. They will record their estimates after seeing the second image. In the synthesis, students discuss their confidence in their estimates of the total number of waffles on the tray even though they cannot see all the waffles. It is an opportunity to highlight how they look for and use the structure of rows and columns to justify their estimates.

### Launch

• Groups of 2
• Have students keep their books closed.
• Display the first image:
• “How many waffles are on one tray?”
• “What is an estimate that’s too high?” “Too low?” “About right?”
• 1 minute: quiet think time

### Activity

• 1 minute: partner discussion
• Share responses.
• Display the second image:
• “Would you like to revise your thinking?”
• 1 minute: quiet think time
• Record responses.
• Share responses.

### Student Facing

How many waffles are on the tray?

Record an estimate that is:

too low about right too high
$$\phantom{\hspace{2.5cm} \\ \hspace{2.5cm}}$$ $$\phantom{\hspace{2.5cm} \\ \hspace{2.5cm}}$$ $$\phantom{\hspace{2.5cm} \\ \hspace{2.5cm}}$$

### Student Response

For access, consult one of our IM Certified Partners.

### Activity Synthesis

• “Even though we can’t see the whole tray, some students are very confident about their estimates. Why do you think that is?” (If the tray is full, it has 5 rows with 4 in each row. Even though we can’t see all the waffles, we know how many fit.)

## Activity 1: Sums of Rows and Sums of Columns (10 minutes)

### Narrative

In the first section of the unit, students determined whether a group of objects had an even or odd number of members. They found the total number of objects in even groups by finding the sum of two equal addends or by skip counting by 2. In this activity, they analyze expressions that have more than two equal addends and connect these expressions to the structure of an array. The purpose of this activity is for students to recognize that an expression with equal addends can represent the sum of the number of objects in each row or the sum of the number of objects in each column. When students analyze Diego and Mai’s equations, they reason abstractly and quantitatively by relating the expressions to the array of dots (MP2).

### Required Materials

Materials to Gather

• Groups of 2

### Activity

• “Mai and Diego represented the number of objects in the same array with different expressions. Diego wrote $$2+ 2 + 2 + 2 + 2 +2$$. Mai wrote $$6 + 6$$.”
• “Who do you agree with? Work with a partner to decide who you agree with and be prepared to explain your reasoning.”
• 1 minute: quiet think time
• 4 minutes: partner work time

### Student Facing

Mai and Diego represented the same array with different expressions.

Diego’s expression

$$2 + 2 + 2 + 2 + 2 +2$$

Mai’s expression

$$6 + 6$$

Who do you agree with?

Show your thinking using diagrams, symbols, or other representations.

### Student Response

For access, consult one of our IM Certified Partners.

### Activity Synthesis

• “Some students agreed with Mai, some students agreed with Diego, and some students agreed with both. How can both expressions represent the array?” (One shows adding up the numbers in each row and one shows adding up the number in each column)
• “Which expression represents the sum of the number in each row? Explain.” ($$2 + 2 + 2 + 2 + 2 + 2$$ because there are 2 in each row.)
• “Mai’s expression is $$6 + 6$$. What are the different ways Mai may see $$6 + 6$$ in this array?” (She may see 6 red and 6 yellow counters. She may see 2 rows with 6 counters in each row.)
• Consider displaying and circling the counters to illustrate student thinking:
• “We can find the total number of objects in an array by adding up the number of objects in each row, or by adding up the number of objects in each column. We can represent these sums with expressions like the ones Mai and Diego wrote.”

## Activity 2: Card Sort: Arrays and Expressions (15 minutes)

### Narrative

The purpose of this activity is for students to connect expressions to the array structure. Students match arrays, 2 expressions, and the total number of objects represented. As they do so, they look for different ways of representing the number of objects in an array in terms of the columns or the rows (MP2, MP7)In the synthesis, student share how they found their matches and discuss why there was only 1 expression that represented the array that had the same number of rows as columns.

MLR2 Collect and Display. Collect the language students use as they complete the card sort. Display words and phrases such as: “array,” “rows,” “columns,” “groups,” “equation,” “total,” “repeated addition,” and “skip count.” During the synthesis, invite students to suggest ways to update the display: “What are some other words or phrases we should include?” and so on. Invite students to borrow language from the display as needed.

### Required Materials

Materials to Copy

• Match Arrays to Expressions Card Sort

### Required Preparation

• Create a set of cards for each group of 2–3.

### Launch

• Groups of 2–3
• Give each group a set of cards.

### Activity

• “Sort the cards into 8 groups.”
• “Each group should have an array, 2 expressions, and a total number.”
• “Each group has 4 cards, but one will only have 3.”
• 12 minutes: partner work time

### Student Facing

Your teacher will give you a set of cards that show arrays, expressions and a total number. Find the cards that match.

### Student Response

For access, consult one of our IM Certified Partners.

### Activity Synthesis

• Display Card A and Card F.
• Select students to share the expressions that match Card A and Card F ($$4 + 4 + 4 + 4 + 4$$ and $$5 + 5 + 5 + 5$$)
• “How is it possible that these 2 arrays have the same expressions?” (Sample response: $$4 + 4 + 4 + 4 + 4$$ shows the sum of the columns for the basketballs, but it shows the sum of the rows for notebooks.)
• Display Card C (popcorn in 3 rows of 3 bags).
• “What was different about this array?” (There was only 1 expression because the number of rows and columns are the same.)

## Activity 3: Add It All Up (10 minutes)

### Narrative

The purpose of this activity is for students to determine the total number of objects in an array and match expressions to arrays by paying attention to the number of objects in each row and the number of objects in each column. For example, students recognize that 3 rows with 4 in each row would be $$4 + 4 + 4$$. The arrays in this task provide students opportunities to compare different ways an array could be decomposed to find the total number of objects. In the synthesis, students compare the different ways they find the total number of objects in the array to expressions that use equal addends to represent the sums of rows or sums of columns.

Representation: Develop Language and Symbols. Support understanding of the problem by inviting students to use a blank piece of paper to cover counters and show each row in one expression, and then each column in the other expression. For example, in task 1, students will use a piece of paper to cover all counters except the first row of 4 counters, slide it down to reveal the second row of 4 counters, and slide it down one last time to show the last row of 4 counters ($$4 + 4 + 4$$). Then, in a similar fashion, cover all counters but the first column of three counters, slide right to reveal the second column of three columns, and so on. ($$3 + 3 + 3 + 3$$)
Supports accessibility for: Visual-Spatial Processing, Organization

### Required Materials

Materials to Gather

### Activity

• “Now you will find the total number of counters in arrays using a method that makes sense to you. Then match each array to expressions.”
• 6 minutes: independent work time
• Monitor for the array that generates the largest variety of different ways students find the total number of counters in each array, including skip-counting or adding on based on the number in each row or column.

### Student Facing

1. How many counters are there in all?

2. Explain how you found the total number of counters.
3. Circle 2 expressions that represent the array.

$$3 + 3 + 3 + 3$$

$$3 + 3 + 3$$

$$4 + 3$$

$$4 + 4 + 4$$

$$4 + 4 + 4 + 4$$

1. How many counters are there in all?

2. Explain how you found the total number of counters.
3. Circle 2 expressions that represent the array.

$$2 + 2 + 2 + 2 + 2 + 2 + 2$$

$$6 + 6$$

$$7 + 7$$

$$2 + 2 + 2 + 2 + 2$$

### Student Response

For access, consult one of our IM Certified Partners.

### Activity Synthesis

• Invite previously identified students to share how they found the total for the same array. Record each method using an expression.
• If no student counted by adding the sum of each row or sum of each column, select a student to share the expressions they circled to match the array.
• “How are these expressions the same? How are they different?” (All the expressions show the total number of counters. They have different numbers of addends. They use different numbers. Some show a way to find the total, but they don’t match the number of counters in each row or each column.)

## Lesson Synthesis

### Lesson Synthesis

“Today you learned that the number of objects in an array can be represented using expressions that show the sum of the number of objects in each row or the sum of the number of objects in each column.”

Display counters to show:

“What can you tell me about this array?” (It has 5 rows with 4 in each. It has 4 columns with 5 in each. An expression that represents the sum of the rows is $$4 + 4 + 4 + 4 + 4$$. An expression that represents the columns is $$5 + 5 + 5 + 5$$. There are 20 counters in all.)

Share and record responses.

## Cool-down: Match Expressions with Arrays (5 minutes)

### Cool-Down

For access, consult one of our IM Certified Partners.