# Lesson 8

Area of Rectangles Without a Grid

## Warm-up: How Many Do You See: One More, One Less (10 minutes)

### Narrative

The purpose of this How Many Do You See is for students to subitize or use grouping strategies to describe the images they see.

When students use equal groups and a known quantity to find an unknown quantity, they are looking for and making use of structure (MP7).

### Launch

• Groups of 2
• “How many do you see? How do you see them?”
• Flash the image.
• 30 seconds: quiet think time

### Activity

• Display the image.
• 1 minute: partner discussion
• Record responses.
• Repeat for each image.

### Student Facing

How many do you see? How do you see them?

### Student Response

For access, consult one of our IM Certified Partners.

### Activity Synthesis

• “What numbers were easy to see in the images?” (4, 5, 6)
• “How did the first image help you find the number of dots in the next 2 images?” (I know each group in the second image has 1 more dot than each group in the first image, so I figured out 4 groups of 5, then added 4 more dots. For the last image, I subtracted 4 from 20, since one dot was missing in each group.)

## Activity 1: Partially Tiled (15 minutes)

### Narrative

The purpose of this activity is for students to solve an area problem with a partially tiled rectangle. This encourages students to multiply to solve problems involving area, but still provides some visual support to see the arrangement of the rows and columns. This problem includes a product of ten, with which students should be increasingly comfortable. The number of square inches is large in order to discourage one-by-one counting.

MLR8 Discussion Supports. Synthesis: Revoice student ideas to demonstrate and amplify mathematical language use. For example, revoice the student statement, “I saw a complete row, and if I tiled all the rows, then they are all the same” as “I saw a complete row, and if you tiled the rest of the rows, each row would be an equal group.”
Engagement: Develop Effort and Persistence: Differentiate the degree of difficulty or complexity. Some students may benefit from starting with a rectangle with more accessible value. For example, display a partially tiled rectangle with fewer rows.
Supports accessibility for: Conceptual Processing

### Launch

• Groups of 2
• Display images of the painted tiles.
• “What do you notice? What do you wonder?” (Students may notice: There are many square tiles. That is a lot of tiles. The tiles are painted. Students may wonder: How many tiles were used? What else could be tiled? How long did it take to tile that building?)
• 1 minute: quiet think time
• Share and record responses.
• “These are examples of painted tiles called azulejos from Portugal. In Portugal, they have been used for a very long time to decorate walls, floors, and even ceilings. They also show events in Portuguese history.”
• “This problem involves finding the area of an art project that is partially tiled with square tiles. Think about how many tiles are needed to tile the whole rectangle.”
• 1 minute: quiet think time

### Activity

• 3–5 minutes: partner work time
• Monitor for students who use multiplication to find the number of tiles.
• “How did you find how many tiles it would take to cover the rectangle?”
• “Is there a way that’s faster than counting every square one by one?”

### Student Facing

What do you notice? What do you wonder?

After learning about azulejos in Portugal, Elena is making her own tile artwork. This rectangle shows the project Elena is tiling. Each tile has a side length of 1 inch.

How many tiles are needed to tile the whole rectangle? Explain or show your reasoning.

### Student Response

For access, consult one of our IM Certified Partners.

If students count one by one to find the total number of square tiles, consider asking:

• “How many squares would be in each row (or column)?”
• “How could we use this to find the area?”

### Activity Synthesis

• “How did you know how many tiles would be in each row or column?” (The first row had 10 tiles, so I know every other row has 10 tiles because I could put more tiles to fill in the rows. It’s like an array. Each column has to have the same number of tiles, so there is 9 in each column.)
• “How did you find the total number of tiles needed?” (I counted by ten 9 times. I multiplied 9 times 10.)
• “The activity mentions that the tiles are 1 inch on each side. Is the side of each tile actually 1 inch long?” (No). “How can you tell?” (Sample response: We know how long 1 inch is and can see that the sides of the tiles are less than 1 inch. There are 10 tiles across. If they really are 1 inch wide, the image won’t fit on the paper.)
• “Sometimes we will see images labeled with units that are not exactly the size the label says. We can still use these images to represent the situation we are talking about.”

## Activity 2: No More Squares (20 minutes)

### Narrative

In this activity, students find the area of rectangles that are not tiled but whose sides are marked with equally spaced tick marks. The tick marks give students the side lengths of the rectangle, help students visualize a tiled region, and enable them to confirm that multiplying the side lengths give the number of square units in the rectangle. The work here serves to transition students to using only side lengths to find area.

### Required Materials

Materials to Gather

### Launch

• Groups of 2
• Display the first problem and the image.
• Read the first sentence in the first problem.
• “The statement says the tick marks are 1 meter apart. Are they really 1 meter apart?” (No, the spaces between them represent 1 meter each.)
• “We could not draw a rectangle that is actually in meters on a standard piece of paper because it would be much larger than the paper. We can draw this rectangle to represent that larger rectangle.”
• “How is this different from other rectangles whose area we’ve found?” (Before we had squares or tiles to count or we used actual tiles filled into a shape to count to find the area.)
• 30 seconds: quiet think time
• 1 minutes: partner discussion
• Share responses.

### Activity

• “Work with your partner to find the area of this rectangle.”
• 3–5 minutes: partner work time
• “How would you describe the rows and columns if you pictured the squares in the rectangle?”
• “How could you find the total number of square meters?”
• Monitor for students who draw the missing grid lines before they multiply to find the area of the rectangle.
• Invite 2–3 students to share how they found the area of this rectangle.
• Display the second problem.
• “This rectangle is marked off in meters along the top and is labeled with meters on the side. Think about how you might find the area of this rectangle.”
• 1 minute: quiet think time
• 3 minutes: partner work time
• Monitor for students who create the missing grid lines before they multiply to find the area of the rectangle.

### Student Facing

What is the area of the rectangle in square meters?

1. The tick marks on the sides of the rectangle are 1 meter apart.

What is the area of this rectangle in square meters?

2. The top side of this rectangle is marked off in meter lengths. The left side is labeled with the length in meters.

What is the area of this rectangle in square meters?

### Student Response

For access, consult one of our IM Certified Partners.

If students count one by one to find the total number of square units, consider asking:

• “How did you find the area of the rectangle?”
• “How many squares would be in each row (or column)? How could we use this to find the area?”

### Activity Synthesis

• Display samples of student work where students created the missing grid lines.
• “How can creating the grid from the tick marks help us see the missing groups?” (We can see all the squares in a row or in a column. We can see how many rows and columns there are.)
• “Did you have to fill in the grid lines to find the area of these rectangles?” (No, since counting the squares gives the same area as multiplying the side lengths, we can just find the length of each side.)

## Lesson Synthesis

### Lesson Synthesis

“Today we found the area of rectangles where the squares weren’t visible.”

“What did you need to think about to find the area of rectangles where only some of the squares were visible or none of the squares were visible?” (I used the squares that I could see to imagine the rest. I used the tick marks to think about how many squares are in each row and how many rows there are. I multiplied the side lengths to find the area if I couldn’t see all the squares.)

“What do you need to know to find the area of any rectangle?” (The side lengths.)

## Cool-down: Where are the Squares? (5 minutes)

### Cool-Down

For access, consult one of our IM Certified Partners.