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. When students try using a variety of measurements and generalize with an equation, they look for and express regularity in repeated reasoning (MP8).

Monitor for students who find the area by:

1. Counting all the squares.
2. Multiplying the length and the width of the rectangles.
3. Subtracting the area they are not including from the larger area.

The purpose of the discussion is to recall how to calculate areas of regions inside one shape, but outside another.

Direct students’ attention to the reference created using Collect and Display. Ask students to share their solutions, including previously identified students to share their methods for finding area. Invite students to borrow language from the display as needed. As they respond, update the reference to include additional phrases.

Here are some questions for discussion:

• “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 foot.)
• “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.)

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. Here are some questions for discussion:

• “Does it help to draw a picture of the situation when there is not one given?” (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 , or equivalent.)