Lesson 11

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 a part of modeling with mathematics (MP4).

The first few times using this routine, ask students to explain the difference between a guess and an estimate. The goal is to understand that a guess is an answer without evidence, whereas an estimate is based on reasoning using the available information.

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. Tell students their answers should be in relation to the person’s height. 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. The person’s height is 6 feet or 72 inches, so the stack of books is about 48 inches or 4 feet. Ask students how accurate their estimates were, as a class. Was the actual value inside their range of values? Was it toward the middle? How variable were their estimates? What were the sources of error? 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 introduced to the idea that variability is important when analyzing and comparing data sets. Students will compare data sets, and make a decision as to which data set represents a better scenario. This activity prepares students to calculate variability in order to compare data sets in a later lesson. Monitor for students who calculate any statistics to inform their decisions. Make note of the students who only calculate a measure of center vs. students who calculate both a measure of center and a measure of variability.

Making statistical technology available gives students an opportunity to choose appropriate tools strategically (MP5).

Students will work individually. Tell them to use statistics to inform their decision.

The purpose of this discussion is to understand that the variability is worth paying attention to. If any students chose to compute the mean, median, IQR, MAD, or range for the two stations, ask them to share their results and how those values help them decide. Interestingly, the mean is the same for the gas stations. But any measure of variability shows that the prices at station B are more stable and predictable than the prices at station A. If no students suggest computing measures of center and variability and comparing them, be prepared to demonstrate.

Here are the summary statistics for each set of data:

• For Station A:

  • Mean: 2.94
  • Five-number summary: 2.38, 2.68, 2.86, 3.26, 3.59
  • Range: 1.21
  • IQR: 0.58
  • MAD: 0.29

• For Station B:

  • Mean: 2.94
  • Five-number summary: 2.84, 2.85, 2.95, 3.03, 3.05
  • Range: 0.21
  • IQR: 0.18
  • MAD: 0.07

If time permits, discuss which measures are helpful. Here are sample questions:

• “Was the mean helpful in deciding the gas station from which you would purchase gas?”(No, because both means were the same.)
• “How did the range (or IQR, MAD, five-number summary) help you decide which gas station you would go to?” (Whichever gas station had the highest range [or IQR, MAD] is the gas station I chose.)
• “In the real world, how would you use the gas prices at each gas station to make a decision?” (I would think about which one generally had lower prices and choose that one, or I would see which one had more predictably cheaper prices and choose that one.)

The purpose of this activity is for students to practice using measures of center and measures of variability to compare data sets. Students look at each pair of data sets and choose the better scenario based on the variability of each. In comparing the data sets, students will apply statistics to real-world scenarios. This activity prepares students to make comparisons of data in a later lesson in which they must order data sets from least variable to most variable. Students should record their answers for each comparison, and they will be used in a subsequent extra support lesson.

Present each problem one at a time. Tell students that higher scores are better in both cases. Give students about 3–5 minutes to choose between the two scenarios and poll the class about which option they think is better. Display the results of the poll for all to see.

The goal of this activity is for students to know that measures of variability are very useful when analyzing and comparing data and ultimately making decisions in the real world. Here are sample questions to promote class discussion:

• “How might the median help you make decisions about situations like these?” (If there is a big difference between the medians, I might choose the one with a more favorable value.)
• “How might the interquartile range help you make decisions about situations like these?” (If the medians are about the same, the IQR could help in deciding which options are more consistent and which are more variable.)
• “What are some other situations for which you might use statistics like these to compare two options in your life?” (I could think about statistics to help make decisions between the best garden materials to help grow plants or reduce weeds, and choosing between things based on their ratings from other people.)