# Lesson 1

Areas Around Areas

These materials, when encountered before Algebra 1, Unit 7, Lesson 1 support success in that lesson.

## 1.1: Estimation: Tennis Area (5 minutes)

### Warm-up

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

### Launch

Display the images for all to see. Ask students to silently think of a number they are sure is too low, a number they are sure is too high, and a number that is about right, and write these down. Then, write a short explanation for the reasoning behind their estimate.

### Student Facing

Estimation:

Estimate the area of the flat, gray ground surface that is not in the tennis court.

1. Record an estimate that is:

too low about right too high

### Student Response

For access, consult one of our IM Certified Partners.

### Activity Synthesis

Ask a few students to share their estimate and their reasoning. If a student is reluctant to commit to an estimate, ask for a range of values. Display these for all to see in an ordered list or on a number line. Add the least and greatest estimate to the display by asking, “Is anyone’s estimate less than $$\underline{\hspace{.5in}}$$? Is anyone’s estimate greater than $$\underline{\hspace{.5in}}$$?” If time allows, ask students, “Based on this discussion does anyone want to revise their estimate?” Then, reveal the actual value and add it to the display. Ask students how accurate their estimates are, as a class. Is the actual value inside their range of values? Is it towards the middle? How variable are their estimates? What are the sources of the error? Consider developing a method to record a snapshot of the estimates and actual value so that students can track their progress as estimators over time.

## 1.2: Pool Walkway (15 minutes)

### Activity

In this activity, students begin writing quadratic expressions based on areas of rectangles. Students practice working with different units to find the area of rectangles and the difference between the rectangles, then write an equation to compute the areas when some of the dimensions are variable.

Monitor for students who find the area by:

1. counting all the squares
2. multiply the length and the width of the rectangles

### Student Facing

The pool is represented by the shaded region, and the walkway around a pool is tiled as shown in the diagram.

1. What’s the area of the (shaded) pool if each tile is 1 square foot? 1 square meter? 4 square feet?
2. What’s the area of the walkway and the pool when each tile is 1 square foot? 1 square meter? 4 square feet?
3. What is the area of the walkway only when each tile is 1 square foot? 1 square meter? 4 square feet?
4. Write an expression to find the total area of the walkway and pool if the length of the pool is $$\ell$$ feet, the width of the pool is $$w$$ feet, and the walkway is 5 feet wide on each side of the pool.
5. What is the area of just the walkway?

### Student Response

For access, consult one of our IM Certified Partners.

### Activity Synthesis

The purpose of the discussion is to recall how to calculate areas of regions inside one shape, but outside another. Select students to share their solutions including previously identified students to share their methods for finding area. Ask students:

• “How are the solutions for the areas for which tiles are 1 square foot and for which the tiles are 4 square feet related?” (When the tiles are 4 square feet, the areas are 16 times greater than when the tiles are 1 square feet.)
• “If the pool is shifted 1 tile to the side, does that change any of the answers? Explain your reasoning.” (No, the number of tiles inside and outside of the pool would remain the same even if the pool is not centered in the walkway.)

## 1.3: Painting the Walls (20 minutes)

### Activity

In this activity, students practice finding areas of areas between rectangles again, this time they examine a square wall with a window. Students have the opportunity to reason abstractly and quantitatively (MP2) when they make sense of the situation to find the desired area.

### Student Facing

Clare wants to paint the square wall in her bedroom that has a rectangular window. She needs the area of the wall space not including the window in order to determine how much paint is needed.

1. Does it matter if the window is centered on the wall or not when trying to find the wall area that will be painted? Explain your reasoning.
2. The total area of the window is 34 square feet. Find the height of the window if it has a width of 6 feet. Explain or show your reasoning.
3. Find the area of wall space that will be painted if the window has an area of 34 square feet and the wall is square shaped and 10 feet wide? 13 feet wide? 15.5 feet wide? Explain or show your reasoning.
4. Write an expression to find the total area of the wall space that will be painted if the height of the window is $$h$$ feet, the width is $$w$$ feet, and the entire wall is a square with length 20 feet.

### Student Response

For access, consult one of our IM Certified Partners.

### Activity Synthesis

The purpose of the discussion is to solidify student learning about the area of a region between two rectangles when one is inside the other. Select students to share their solutions. Ask students,

• “Does it help to draw a picture of the situation when there is not one provided?” (Sometimes it is helpful. In some cases, I can figure out the answer without a picture, but sometimes it can help me think about what information I have and what information I need.)
• “How does your answer change to the final question if the window is a square?” (The height and width would be the same, so the painted area would be $$20^2 - h^2$$ or equivalent.)