The purpose of this Warm-up is to practice the skill of estimating a reasonable answer based on experience and known information, and also to help students develop a deeper understanding of the meaning of standard units of measurement. It gives students a low-stakes opportunity to share a mathematical claim and the thinking behind it (MP3). Asking, “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).

The goal of this discussion is for students to analyze the reasonableness and accuracy of their estimates.

Ask a few students to share their estimates and their reasoning. If a student is reluctant to commit to an estimate, ask for a range of values. Display these responses 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 ? Is anyone’s estimate greater than ?” If time allows, ask students, “After this discussion, does anyone want to revise their estimate?”

Then, reveal the actual value, and add it to the display.

Encourage students to think about and discuss the accuracy of their estimates and the estimates that are displayed, by asking questions, such as:

• “Is the actual value inside their range of values?”
• “Is it toward 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 the 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. 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.)

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. 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.)