Lesson 3

Tile Rectangles

Warm-up: Which One Doesn’t Belong: Tiles (10 minutes)

Narrative

The purpose of this warm-up is to draw students' attention to different ways of covering a plane figure with squares and reinforce the idea that tiling involves covering a region without gaps and overlaps. It gives students a reason to use language precisely (MP6). It gives the teacher an opportunity to hear how students use terminology and talk about characteristics of the items in comparison to one another. During the synthesis, ask students to explain the meaning of any terminology they use, such as rows, columns, area, gaps, overlap, and tiling.

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

• “Discuss your thinking with your partner.”
• 2–3 minutes: partner discussion
• Share and record responses.

Student Facing

Which one doesn't belong?

Student Response

For access, consult one of our IM Certified Partners.

Activity Synthesis

• “How could you use the squares in each of these rectangles to find the area of each rectangle?” (In C, I can just count the tiles. In B, I could finish tiling the rectangle and count the tiles. In D I would need to straighten out the tiles so they cover all of the rectangle. In A, I could count the blue tiles and double the number since in each row there are the same number of white tiles as there are blue tiles.
• Consider saying:
• “Let’s find at least one reason why each one doesn’t belong.”

Activity 1: Time to Tile (15 minutes)

Narrative

The purpose of this activity is for students to use square tiles to find the area of rectangles. They use their knowledge of tiling to complete the tiling that is started in each rectangle. Students may use physical tiles on copies of the blackline master or reason directly on the images in the student book, which may not be the right size for physical tiles. The synthesis focuses on solidifying the idea that area is the number of square units that cover a flat figure with no gaps or overlaps.

This activity uses MLR1 Stronger and Clearer Each Time. Advances: reading, writing

Required Materials

Materials to Gather

Materials to Copy

• Time to Tile

Required Preparation

• Each group of 2 needs 24 square tiles.

Launch

• Groups of 2
• Give each student 1 copy of the blackline master.
• Give students access to inch tiles.
• “Take a minute to think about how you would finish measuring the area of these rectangles that are partially tiled.”
• 1 minute: quiet think time

Activity

• “Work with your partner to describe how to use square tiles to find the area of each rectangle. You can use the square tiles and rearrange what’s shown in each rectangle, if needed. Then complete the last problem independently.”
• 5–7 minutes: partner work time

Student Facing

Your teacher will give you square tiles and a handout showing rectangles and squares.

1. Describe or show how to use the square tiles to measure the area of each rectangle. You can place square tiles on the handout where squares are already shown. You can also move the tiles, if needed.

1.
2.
3.
4.

2. Describe how to use square tiles to find the area of any rectangle.

Student Response

For access, consult one of our IM Certified Partners.

Activity Synthesis

• “Why did the square tiles in the first rectangle, the third rectangle, and the fourth rectangle need to be adjusted before we could finish finding the area of the rectangle?” (In the first rectangle, the square tiles had to be moved over because they weren’t going to fill the whole rectangle if we left them in the center. In the third rectangle, the squares in the second row needed to be lined up with the first row so there would be the same number of squares in each row. In the fourth rectangle, the squares need to be adjusted so they are not crooked or overlapping and one square needs to be removed from the first row.)
• “If someone told you four squares would fit across the top of the rectangle, but only three squares would fit across the bottom of the rectangle, how would you know this didn’t make sense?” (The top and bottom have the same length so they should fit the same number of squares.)

MLR1 Stronger and Clearer Each Time

• “Share your response to the last problem with your partner. Take turns being the speaker and the listener. If you are the speaker, share your ideas and writing so far. If you are the listener, ask questions and give feedback to help your partner improve their work.”
• 2 minutes: structured partner discussion.
• Repeat with 1–2 different partners.
• “Revise your initial draft based on the feedback you got from your partners.”
• 2–3 minutes: independent work time
• “We just found the area of rectangles, and learned that when we cover the rectangle with square tiles, the tiles can’t have gaps or overlaps. The same ideas are important with any flat figure. Area is the number of square units that it takes to cover a flat figure without gaps or overlaps.”

Activity 2: Card Sort: Rectangles (20 minutes)

Narrative

The purpose of this activity is for students to recognize that different shapes can have the same area. Students first sort the cards in any way that makes sense to them and then by area. After the cards are sorted by area, students create another rectangle that would fit into one of the categories (by having a particular area). A sorting task prompts students to look for structure and make connections across the representations and statements being analyzed (MP7).

Students may start to notice that the organization of the squares in rectangles makes it efficient to count: The squares can be grouped by row, column, or in other ways. As students sort and create rectangles with certain areas, monitor for students who leverage the structure of a rectangle to find area. Invite them to share in the synthesis.

In this activity, the squares on the gridded rectangles are not the same size as the square tiles, but students could still use tiles as a support. Provide students access to square tiles if they would like to use them, but encourage them to draw what they create on the grid provided.

MLR8 Discussion Supports. Students should take turns finding a match and explaining their reasoning to their partner. Display the following sentence frames for all to see: “I noticed _____ , so I matched . . . .” Encourage students to challenge each other when they disagree.
Advances: Conversing
Representation: Internalize Comprehension. Synthesis: On chart paper, record students’ rectangles with justifications in each category. Record students’ efficient ways for counting to find the area of rectangles.
Supports accessibility for: Memory

Required Materials

Materials to Gather

Materials to Copy

• Card Sort: Rectangles

Required Preparation

• Create a set of cards from the blackline master for each group of 2.

Launch

• Groups of 2
• Display the image.
• “What do you notice? What do you wonder?” (Students may notice: There are 3 rectangles. One of the rectangles is made up of square tiles. One of the rectangles is shaded and the other rectangle isn’t. They all have 12 squares. They all have an area of 12 square units. Students may wonder: Why are there 3 rectangles? Why is one rectangle shaded and the other one isn’t? Do the blue squares show tiles?)
• 1 minute: quiet think time
• 1 minute: partner discussion
• Record responses.
• “These are ways that we can represent a rectangle with 12 square units. When the squares are shaded in the image they look like square tiles, but we can also make a rectangle on a grid and say that it has an area of 12 square units, because it contains 12 squares.”
• “Draw a rectangle with an area of 8 square units on the grid.”
• 30 seconds: independent work time
• Share responses.
• Distribute one set of pre-cut cards to each group of students.
• Give students access to inch tiles.

Activity

• “Work with your partner to sort the cards into categories. Be prepared to explain how you sorted your cards.”
• 5 minutes: partner work time
• Select groups to share their categories and how they sorted their cards.
• If no groups sort their rectangles by area, give students 2-3 minutes to do so and then ask them to share their new categories.
• “Take a minute to think about what other rectangles might fit into these categories.”
• 1 minute: quiet think time
• “Now, work with your partner to create at least one different rectangle that has the same area as the rectangles in each group. Be prepared to share how you know your rectangles belong in each group.”
• 5 minutes: partner work time
• Monitor for the strategies students use to find the area of rectangles.

Student Facing

What do you notice? What do you wonder?

Draw a rectangle with an area of 8 square units on the grid.

Your teacher will give you a set of cards that show rectangles. Sort the cards into categories of your choosing. Be prepared to explain your categories.

1. Create a rectangle that would fit in each group.

Student Response

For access, consult one of our IM Certified Partners.

Activity Synthesis

• Invite students to share the rectangles they created for each category.
• Consider asking:
• “How did you know that your rectangle belongs here?”
• “How did you know that the rectangle you created had the same area as the other rectangles in that category?”
• Discuss some of the efficient ways that students counted to find the area of rectangles.

Lesson Synthesis

Lesson Synthesis

“Today we learned we can draw squares in rectangles to represent tiling. We can count the squares to find the area of a rectangle just like we would count tiles.”

“What helpful features do rectangles have that help us find their area?” (The rows and columns in a rectangle show equal groups of squares, so we can just count one row or column and then skip-count to find the area.)

Cool-down: Tile a Rectangle (5 minutes)

Cool-Down

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