# Lesson 8

Count Columns and Objects in Columns

## Warm-up: Estimation Exploration: Rearrange the Dots (10 minutes)

### Narrative

The purpose of this warm-up is to allow students to think about how organizing objects can help them find the total number more quickly. Each image shows 20 counters. Image 1 makes it difficult to count quickly due to a lack of organization, Image 2 makes it more clear and allows for the conversation about even and odd, and Image 3 connects a familiar structure (10-frame) to the new structure being introduced (an array).

The purpose of an Estimation Exploration is to practice the skill of estimating a reasonable answer based on experience and known information. It gives students a low-stakes opportunity to share a mathematical claim and the thinking behind it (MP3). Asking oneself “Does this make sense?” is a component of making sense of problems (MP1). Making an estimate or a range of reasonable answers with incomplete information is a part of modeling with mathematics (MP4).

### Launch

• Groups of 2
• Display the image or arrange 20 counters (10 red and 10 yellow) in a random arrangement:
• “How many counters do you see?”
• “What is an estimate that’s too high?” “Too low?” “About right?”
• 1 minute: quiet think time

### Activity

• 1 minute: partner discussion
• Record responses.
• Arrange counters to show 2 circular arrangements of 10 red and 10 yellow counters:
• “Here are the same counters in a different arrangement. Would you like to revise your thinking?” (I counted the red and there were 10. I think there’s the same amount of yellow, so I said 20.)
• 1 minute: quiet think time
• Share responses.
• After Image 2 ask, “How does this image help you think about whether or not the total is even or odd?” (It helps me see the number of counters as 2 equal groups. So it must be an even number.)
• Arrange the counters into an array of 2 rows of 5 red counters and 2 rows of 5 yellow counters with some space to separate the two colors:
• “This image shows the same number of counters, but in a different arrangement. Would you like to revise your thinking?” (It’s easy now to see that it’s 20. There’s 10 red and 10 yellow.)
• 30 seconds: quiet think time
• Share responses.

### Student Facing

How many counters do you see?

Record an estimate that is:

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

### Activity Synthesis

• “We saw different arrangements of the same number of counters. Which one makes it easier to tell how many there are altogether? Explain.” (In the last way, it is easy to see it is 10 and another 10. It looks like 2 10-frames.)
• Refer to the counters in an array.
• “Organizing the circles into arrays can help us see ways to find the total more quickly. How could I use skip counting to find the total number of counters?” (We could count by 5 or 2.)

## Activity 1: Count by Columns (15 minutes)

### Narrative

The purpose of this activity is for students to describe the structure of an array by identifying the number of columns, the number of objects in each column, and the total number of objects. They learn that the columns of an array go up and down. Students begin to reason about how they can use the structure of the array to find the total number of objects without counting by 1.

This activity uses MLR8 Discussion Supports. Advances: speaking, conversing

Representation: Develop Language and Symbols. Support understanding of the rows and columns by inviting students to act it out. For example, use students to create the first array (3 rows of 2). Discuss the students who create the rows. Now switch to talking about columns (2 columns of 3). Discuss the difference in which students create the columns.
Supports accessibility for: Visual-Spatial Processing, Organization

### Required Materials

Materials to Gather

### Launch

• Groups of 2
• Give students counters.
• “Use your counters to make an array with 3 rows with 4 in each row. Make sure the counters in each row line up with the counters in the other rows.”
• Display counters in an array as shown:
• “Arrays are made up of rows, but they also have columns that go up and down. How many columns are in this array?” (4)
• 30 seconds: quiet think time
• Share responses.
• “How many counters are in each column of the array you made?” (3)
• 30 seconds: quiet think time
• Share responses.

### Activity

• “Now you will look at a few arrays to decide how many columns they have, how many in each column, and find the total number of counters. Then you will make arrays using counters and answer a few questions.”
MLR8 Discussion Supports
• Display sentence frames to support students when they describe the structure of the array:
• “There are _____ columns in the array.”
• “There are _____ counters in each column.”
• “There are _____ counters in all.”
• 8 minutes: partner work time

### Student Facing

1. How many columns are in this array?
2. How many counters are in each column?
3. How many counters are there in all?

1. How many columns are in this array?
2. How many counters are in each column?
3. How many counters are there in all?
1. Use 10 counters to make 2 columns with the same number in each column.

1. How many counters are in each column?
2. How many rows are in the array?
3. How could you count these counters without counting by ones?
2. Use 15 counters to make 3 columns with the same number in each column.

1. How many counters are in each column?
2. How many rows are in the array?
3. How many counters are in each row?
4. How could you count these counters without counting by ones?

### Activity Synthesis

• Display student work for an array with 10 counters.
• “Describe _____’s array.” (It has 10 counters, it has 2 columns and 5 counters in each column. It has 5 rows with 2 counters in each row.)
• “How can we prove there are 10 counters in this array?” (We can skip count by 2. We can add $$2 + 2 + 2 + 2 + 2$$. We can add $$5 + 5$$.)

## Activity 2: Guess My Array (20 minutes)

### Narrative

The purpose of this activity is for students to distinguish between rows and columns of arrays based on given clues. The activity synthesis invites students to use the vocabulary they have learned in this lesson and the previous lesson to describe and create arrays (MP6). When making arrays based on clues, it might be helpful to have partners sit back to back.

### Required Materials

Materials to Gather

### Required Preparation

• Each student needs 25 counters.

### Launch

• Groups of 2
• “4 students made arrays using the same number of counters. Label each array with the student’s name and find the total number of counters.”
• 6 minutes: independent work time
• Share and record responses.
• Give each student 25 counters.

### Activity

• “Now you will work with a partner to see if you can make matching arrays.”
• “Partner A will make an array using up to 25 counters. Don’t let your partner see.”
• “Compare to see if your partner made the same array, and then switch.”
• 10 minutes: partner work time
• Monitor for students who gave clues using odd or even and used the rows and columns language correctly to share in the synthesis.

### Student Facing

Four students talked about their arrays.

• Han said, “My array has an even number of counters. It has 2 rows with 6 counters in each row.”
• Priya said, “My array has more than 10 counters. It has 4 rows with 3 counters in each row.”
• Elena said, “My array is very tall. It has 6 counters in each column.”
• Kiran said, “My array has more columns than rows. It has 3 rows.”

1. Which array belongs to which student? Write the name of each student below their array.

2. Each student used _____ counters to make an array.
3. Make an array using up to 25 counters, but don’t let your partner see.

Be prepared to explain how you knew the total number of counters.

### Student Response

If students give clues that are not related to the rows and columns of their array, consider asking or displaying:
• “How many rows does your array have?”
• “How many columns does your array have?”
• “Does your array have more rows, more columns, or the same number of rows and columns?”

### Activity Synthesis

• Invite previously identified students to share their clues.
• Students can use their counters to see if they make an array that matches the chosen student’s.
• “Which clues helped you most when making your arrays?” (It was easy when they said how many rows and columns.)

## Lesson Synthesis

### Lesson Synthesis

“Today you learned that when arranging counters in an array, the counters in each row should line up with the counters in the other rows. The counters in an array also line up in columns that go up and down. The columns in an array always have the same number of counters in them. Arrays can help us organize objects, so we can find total amounts.”

Arrange counters to show:

“How would you describe this array using rows or columns?” (It has 3 columns with 5 counters in each. It has 5 rows with 3 counters in each. I can count by 5 to find the total. I see a group of 10 and a group of 5.)