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 Task Statement, and clarify the term as needed.

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.

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?” ()
• “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 .)
• “It couldn’t have been a correct guess, but how would you graph a guess of 48.3?” (It would be a point at .)
• (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 -values 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.

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 from 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 . Now they see to meet at .)

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.