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.

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.