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 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 estimates 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. (The books are 830, 276, and 32 pages long, so the average number of pages in the books is about 379 pages.)

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 the range of values?”
• “Is 379 near the middle of the estimates?”
• “How variable are the estimates?”
• “What are the sources of errors?”

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.

The goal of this activity is for students to be reminded of the importance of measures of variability. Students reason abstractly and quantitatively (MP2) when they compare data sets and decide which is the better option. Students must explain their reasoning for choosing one of the options and use statistics to justify their answers. This activity helps students gain more practice working with using variability to compare data sets.

The purpose of this activity is to elicit the idea that extreme values tend to have little effect on the median and interquartile range, which will be useful when students explore the effects of outliers in a later activity.  The structure of Notice and Wonder gives space for students to notice and be curious about the extreme value in Tyler’s results. While students may notice and wonder many things about these data displays, the presence of extreme values and the stability of the median and quartiles are the important discussion points.

When students articulate what they notice and wonder, they have an opportunity to attend to precision in the language they use to describe what they see (MP6). They might first propose less formal or imprecise language, and then restate their observation with more precise language to communicate more clearly.

This prompt gives students opportunities to see and make use of structure (MP7). Students should use the structure of the data displays to notice that extreme values affect the mean and mean absolute deviation much more than median and interquartile range.