Lesson 1

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).

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.

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.

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

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.)

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.

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.)