Lesson 12

The purpose of an Estimation warm-up is to practice the skill of estimating a reasonable answer based on experience and known information, and also help students develop a deeper understanding of the meaning of standard units of measure. It gives students a low-stakes 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 part of modeling with mathematics (MP4).

Display the image 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.

Select 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. The books are 830, 276, and 32 pages long, so the average number of pages in the books is about 379 pages. Ask students how accurate their estimates were, as a class.

• “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 actual value so 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 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.

Allow students to work individually or in pairs.

Discuss how students made their decisions. Here are sample questions to promote class discussion:

• “How did the variability influence your choices?" (When there is greater variability, the values are more spread out, so that helped me think about whether I wanted consistent values or a wider range.)
• "What are some situations in which less variability might be preferred?" (When you want the results to be consistent and reliable. Examples include the number of points scored by a good player, the miles a car can go on a single gallon of gas, or the price of milk.)
• “How would the process of choosing the best situation be different if you were given a box plot?” (With a histogram or box plot, I wouldn’t have known exact numbers, so I would probably have to use a measure like IQR or the median to help make my decision.)

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. 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 in order 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.

Display the situation for all to see. Ask students to think of at least one thing they notice and at least one thing they wonder. Give students 1 minute of quiet think time, and then 1 minute to discuss the things they notice and wonder with their partner, followed by a whole-class discussion.

Ask students to share the things they noticed and wondered. Record and display their responses for all to see. If possible, record the relevant reasoning on or near the situation. After all responses have been recorded without commentary or editing, ask students, “Is there anything on this list that you are wondering about now?” Encourage students to respectfully disagree, ask for clarification, or point out contradicting information.

If the affect of the extreme value on the measures of center and measures of variation does not come up during the conversation, ask students to discuss this idea.