Lesson 9

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 the students that 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. Tell students that the person’s height is 6 feet or 72 inches, so the stack of books is about 24 inches or 2 feet. (For estimation activities in following lessons, the same person will appear, so it is important that students know he is about 6 feet tall.)

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 the error? Consider developing a method to record a snapshot of the estimates and the actual value so students can track their progress as estimators over time.

In this activity, students use their understanding of spreadsheets to compute useful statistics from data sets. In the associated Algebra 1 lesson, students will learn alternative methods for computing these values.

If students need a reminder about how to use a spreadsheet as a calculator, demonstrate how to do basic calculations such as =A1+A2, =SUM(A1:A10), =(A1+A2)/A3, and =COUNT(A1:A50) or =LENGTH(A1:A50) in GeoGebra.

Select students to share their responses and methods for finding the mean from a list of values in a spreadsheet. The purpose of the discussion is to point out the advantage of using a spreadsheet to calculate statistics, especially for very large sets of data. For further discussion, consider asking these questions:

• "Which part of the process for finding the mean for these values took the most time?" (Typing the values into the spreadsheet and making sure they are correct.)
• "Often large data sets are collected electronically and automatically added to a spreadsheet. When there are thousands of data values in a spreadsheet, what other advantages do you think might be helpful when using technology to compute the mean?" (It may be difficult to understand the data when there are so many values, so using the spreadsheet to calculate the mean can help in understanding what is typical. It would definitely be faster to use the spreadsheet to calculate the mean than doing it by hand.)
• "What other statistics would be useful and easier to calculate using a spreadsheet?" (If the data set is large or the numbers are difficult to understand at a glance, the median, quartiles, interquartile range, mean absolute deviation, maximum, and minimum should all be easy for a computer to find from a list.)

In the associated Algebra 1 lesson, students will learn how to use technology to create data displays. In this lesson, students examine data displays to select an appropriate one for the data given.

Select students to share their responses and reasonings. The purpose of the discussion is to examine the strengths and weaknesses of each of the types of data display as well as thinking about the interval size for histograms and axis marks for ease of reading.

The goal of the lesson is for students to have an introduction to navigating technology to calculate statistics and arrange a data display to be informative. Students should understand how to input data into the spreadsheet, create graphical representations, and calculate some statistics. To discuss what they learned, ask,

• “What are some things that technology allowed you to do that will make computing statistics easier in the future?” (After putting the data into the spreadsheet, it is relatively simple to compute the appropriate value, and it can update automatically with new data or after changing a value.)
• "What are some considerations when arranging a data display?" (The scale of the axes, which display type best shows the distribution, and the interval size for histograms are all important to consider when displaying data to be informative.)