# Lesson 1

Multiples of a Number

## Warm-up: Which One Doesn’t Belong: All Kinds of Area (10 minutes)

### Narrative

This warm-up prompts students to carefully analyze and compare the area of different figures. In making comparisons, students have a reason to use language precisely (MP6) as they describe the area of different figures. It also enables the teacher to hear the terminologies students know and how they talk about characteristics of shapes that help them find different areas.

For all warm-up routines, consider establishing a small, discreet hand signal that students can display to indicate they have an answer they can support with reasoning. This signal could be a thumbs-up, a certain number of fingers that tells the number of responses they have, or a different subtle signal. This is a quick way to see if students have had enough time to think about the problem. It also keeps students from being distracted or rushed by hands being raised around the class.

Math Community

• After the warm-up, ask students to reflect on both individual and group actions while considering the questions, “What does it look and sound like to do math together as a mathematical community? What am I doing? What are you doing?”
• Record and display their responses under the “Doing Math” header. Students might mention things such as: we talked to each other and to the teacher, we had quiet time to think, we shared our ideas, we thought about the math ideas and words we knew, you were writing down our answers, you were waiting until we gave the answers.

### Launch

• Groups of 2
• Display the image.
• “Pick one that doesn’t belong. Be ready to share why it doesn’t belong.”
• 1 minute: quiet think time

### Activity

• 23 minutes: partner discussion
• Share and record responses.

### Student Facing

Which one doesn’t belong?

### Activity Synthesis

• “How could we determine the area of each figure?” (We can use multiplication for most of them or count the units in one of them.)
• Consider saying: “Let’s find at least one reason why each one doesn’t belong.”

## Activity 1: Build Rectangles and Find Area (20 minutes)

### Narrative

The purpose of this activity is for students to find the area of a rectangle by tiling and to recall that the area can also be found by multiplying the side lengths. Students use inch tiles to build rectangles with a given side length and find the area of those rectangles. They work together to compare and explain the strategies used to find the area of rectangles and make connections between strategies. Students observe how the area of rectangles with a given width varies as the length changes and make predictions about what areas are possible with the given widths (MP7).

MLR2 Collect and Display. Circulate, listen for and collect the language students use as they build rectangles. On a visible display, record words and phrases such as: “row,” “column,” “area,” “length,” “width,” “wide.” Invite students to borrow language from the display as needed, and update it throughout the lesson.

### Required Materials

Materials to Gather

### Required Preparation

• Each group of 2 needs at least 36 tiles.

### Launch

• Display the rectangle in the book.
• “Look at the rectangle on your page and describe it to a partner.” (It has 6 units. There are 2 rows and 3 columns.)
• Give each group 10 tiles.
• “Build all the rectangles you can using all 10 tiles. Describe them to a partner.”
• 2 minutes: partner discussion
• “Who has a rectangle that is 2 tiles wide, 5 tiles wide, 10 tiles wide?”
• Draw each rectangle as students share their responses.

### Activity

• Give students more inch tiles.
• “Now build five different rectangles that are each 2 tiles wide. Record the area of each rectangle in the table.”
• “Repeat with rectangles that are each 3 tiles and 4 tiles wide.”
• 5–7 minutes: partner work time
• Monitor for students who:
• build one row or column and repeat the same number of tiles over again to build the area
• skip-count or multiply to determine the area of each rectangle
• combine skip-counting with another counting strategy

### Student Facing

1. Build 5 different rectangles with each of the given widths. Record the area of each rectangle in the table.

area of rectangle
2 tiles wide
3 tiles wide
4 tiles wide
2. Discuss with a partner what you notice about the areas in each row of the table.

3. Predict the area of another rectangle that has each width. Explain your reasoning.

• 2 tiles:

• 3 tiles:

• 4 tiles:

### Student Response

Students may count tiles by 1 to determine the area of the rectangles they build. Consider asking, “Do you see groups of tiles that could help you count?”

### Activity Synthesis

• Collect predictions for areas of rectangles with a width of 2. (18, 14, 20, 30)
• “For rectangles that are 2 tiles wide, how can we tell if our area predictions are true without building each rectangle?” (Each area is an even number. It is what we say when counting by 2 or multiplying by 2.)
• “How can we check our predictions for rectangles that are 3 or 4 tiles wide?” (The predictions are numbers we get when we multiply a number by 3 or 4.)

## Activity 2: What Areas Can You Build? (15 minutes)

### Narrative

The purpose of this activity is for students to explore the idea of multiples through an area context. Students learn that a multiple of a number is the result of multiplying any whole number by another whole number. As students build and find the area of rectangles given one side length, they see that every area is a multiple of each of the side lengths of a rectangle.

Representation: Develop Language and Symbols. Synthesis: Maintain a visible display to record new vocabulary. Invite students to suggest details (words, pictures, or equations) that will help them remember the meaning of the terms. In this lesson, include the terms “multiple,” “even,” and “odd.” Throughout the unit, add the terms “factor,” “factor pair,” “composite,” and “prime.”
Supports accessibility for: Language, Memory

### Required Materials

Materials to Gather

Materials to Copy

• Centimeter Grid Paper - Standard

### Required Preparation

• Each group of 2 needs at least 36 tiles from the previous activity.

### Launch

• Groups of 2
• “I am thinking of a rectangle that is 2 tiles wide. What could be the area of my rectangle?”
• 1 minute: partner discussion
• Share and record responses.
• “How do we know all of these are possible areas?” (We can multiply another number by 2 to get those numbers.)

### Activity

• “In this activity, we are going to think about what the area of a rectangle could be if we only knew one side length. Work with your partner to answer these questions.”
• 5–7 minutes: partner work time
• Monitor for students who notice that areas you can build are a result of multiplying 3 by another possible side length.

### Student Facing

1. Elena is building rectangles with a width of 3 units and an area of 30 square units or less.

1. Build the rectangles Elena could make and draw the rectangles on grid paper. Label the area and the side lengths of each rectangle.
2. What is the area of each rectangle you built?

3. What do you notice about the areas?

2. Why is 28 square units not a possible area for a rectangle with a width of 3 units?

3. If the area of the rectangle can be more than 30 square units, find 2 other areas it could have. Explain or show your reasoning.

4. What is an area that is not possible for a rectangle with a width of 3 units? Explain or show your reasoning.

### Activity Synthesis

• Display a set of student-generated areas that are less than 30 square units.
• “What did you notice about the areas you found?”
• Ask students to explain why the rectangle couldn’t have an area of 28 square units.
• “For the last two questions, how did you know whether a rectangle with a width of 3 units could have that area?”
• “We can have an area of 12 square units when the width of the rectangle is 3 units. That is because 12 is a multiple of 3.”
• “A multiple of a number is the result of multiplying a number by a whole number.”
• “Look back at your work and discuss with your partner: Which numbers are multiples of 3?” (3, 6, 9, 12, 15, 18, 21, 24, 27, and 30)
• “Which numbers are not multiples of 3?” (29, 28, 26, 25, 23, 22, 20, 19, 17, 16, 14, 13, 11, 10, 8, 7, 5, 4, 2, 1)
• 2 minutes: partner discussion

## Lesson Synthesis

### Lesson Synthesis

“Today we built rectangles and learned about multiples of a number. A multiple of a number is the result of multiplying that number by a whole number.”

“How would you decide whether 28 is a multiple of 4?” (I think about whether there is a number I can multiply 4 by to get 28.)

“What is a number that would not be a multiple of 4? How do you know?” (Twenty-five is not a multiple of 4 because I cannot multiply 4 by any whole number to get 25.)

Math Community

After the cool-down, revisit the “Doing Math” list of actions. Ask students to discuss with a partner where they saw evidence of the actions during the rest of the lesson. As a whole group, add any missing actions and revise earlier ideas.