Lesson 13

In this warm-up, students compute absolute guessing errors using the class data collected in advance. The absolute errors they find here will be plotted in the next activity.

The absolute errors could be calculated and compiled in different ways, by hand or using technology, including a spreadsheet tool. For example, students could:

• Complete a table of values individually.
• Complete a table in groups of 2–4, with the calculations done by hand but divided among group members.
• Complete a table as a class, with each student calculating only one absolute guessing error and sharing it with the class.

Read the task statement with the class. Make sure students understand what they are asked to compute. Consider arranging students in groups of 2–4 so students could split up the calculations, if desired. Provide access to calculators and spreadsheet technology.

Before revealing the actual number of items in the jar used to collect students' guesses, give each student a copy of the blackline master. If using the image given in this lesson, the actual number of snap cubes in the jar is 47.

Students should compute at least 12 absolute guessing errors, and more if time permits.

Some students may record negative values for the absolute guessing errors of guesses that are lower than the actual number, not realizing that the term absolute error refers to "how far away," and therefore cannot be negative. Suggest that they revisit the examples in the activity statement and clarify the term as needed.

If desired, display a completed table for all to see, or simply invite students to share some observations about the absolute guessing errors they found. If no one mentioned that all the values are positive, ask them about it and solicit some ideas as to why this is the case.

Also ask students if they could tell from the data how good the guesses were. (Were the guesses close? Were there a lot of overestimates or underestimates?)

Tell students that they will plot the data next.

In this activity, students create a scatter plot of the data they compiled earlier and examine the scatter plot. Unless the data they collected consist mostly of overestimates or mostly of underestimates, students are likely to notice the data points forming a V shape. They then consider whether the relationship between the guesses and absolute guessing errors form a function.

Though a blank coordinate plane is provided (in the blackline master) so that students could individually plot the values by hand, there are other alternatives to consider, if desired. For instance, students could:

• Create individual scatter plots using graphing or statistical technology.
• Create a class scatter plot by hand. Display a large coordinate plane for the entire class to use. Give each student a dot sticker and assign them one data point to plot, by sticking the dot sticker on the large coordinate plane.
• Create a class scatter plot using technology, by accessing a shared spreadsheet or table online and generating a scatter plot. Display the scatter plot for all to see.

If plotting the data by hand, give students a few minutes to plot at least 12 points from the data set (or more if possible) on the given coordinate plane and time to think about the last two questions. Be sure to leave time for a whole-class discussion.

Display a scatter plot for all to see. Ask students to share something they notice and something they wonder. If no students commented on the shape of the data points, bring it up. Discuss questions such as:

• “If someone’s guess matches the actual number exactly, where would the point representing that guess be on the scatter plot?” (\((47,0)\))
• “Where do the worst guesses appear on the scatter plot? What about the best guesses?” (The worst guesses are far away from 47. The best guesses are close to \((47,0)\).)
• (If time permits:) “Did more people overestimate or underestimate? Explain how you know.” (In the sample response, more people underestimated. The scatter plot shows more points with \(x\)-value greater than 47 or to the right of 47 on the horizontal axis. The table shows more numbers that are greater than 47.)

Then, ask students to share their response to the last question and their reasoning. Highlight that the absolute guessing error is a function of the guess because there is only one possible absolute guessing error for each guess.

This activity allows students to see how changing the target number in a guessing game changes the absolute guessing errors and changes the scatter plot of the data.

Here, students see that they are still finding the difference between the guesses and a number, that the absolute guessing errors are still all positive, and that the data points still seem to form a V above the horizontal axis. What is different is that the position of the V shape has now shifted horizontally and that the number of points forming the two pieces of the V may have changed.

The work here prepares students to later see the absolute value function in terms of finding the distance between input values and 0.

The activity is written so that students could perform the calculations and graphing individually and by hand. To make more time for reasoning and analysis, however, use of statistical and graphing technology for computation and plotting is recommended, as is splitting up the work among students.

Tell students that suppose there had been a mistake in the reported number of items in the jar, and that their job is to find out how the absolute guessing errors and the scatter plot would change once they find out the corrected number of items.

Consider giving one half of the class one value for the actual number of objects, and giving the other half of the class another value so that they could observe the general behavior of the function. (For example, give “50” to half the class and “40” to the other half.)

Students could follow the same process as earlier: calculating the absolute errors using the new "actual" number, recording them in a table, and plotting the data points on a coordinate plane. If doing so by hand, ask students to use Table B and the second coordinate plane on the handout given earlier. Keep students in groups of 2–4 so they could split up the calculations. Provide access to calculators.

Alternatively, give students the option of using technology (statistical or graphing tool) to calculate the absolute errors and to create the scatter plot. Another option is to arrange for the calculations to be split up among students and collected in a shared table or spreadsheet, and then create a class scatter plot, either by hand or using technology, displayed for all to see.

Invite students to share their observations of the new data set (or data sets, if two new "actual" numbers were given to students).

If time is limited, focus the discussion on the features of the new scatter plot, and on whether the relationship between the guesses and the absolute guessing errors form a function.

Display a completed scatter plot for all to see. Ask questions such as:

• "How is this scatter plot like the first scatter plot we saw earlier?" (The points form two straight lines that seem to converge at a point, forming a V.)
• "How is this scatter plot different than the first one?" (The points seem to have been shifted horizontally. Earlier, the two sets of points—each forming a line—seemed to meet at \((47,0)\). Now they see to meet at \((50,0)\).)

Students may also note that there were more points (or fewer points, depending on the data) that represent underestimates (or overestimates) in the new scatter plot than in the first one.