Lesson 13

Absolute Value Functions (Part 1)

13.1: How Good Were the Guesses? (5 minutes)

Warm-up

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.

Launch

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.

Student Facing

Before this lesson, you were asked to guess the number of objects in the jar. The guesses of all students have been collected. Your teacher will share the data and reveal the actual number of objects in the jar.

Use that number to calculate the absolute guessing error of each guess, or how far the guess is from the actual number. Suppose the actual number of objects is 100.

  • If your guess is 75, then the absolute guessing error is 25.
  • If your guess is 110, then the absolute guessing error is 10.

Record the absolute guessing error of at least 12 guesses in Table A of the handout from your teacher (or elsewhere as directed).

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Anticipated Misconceptions

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.

Activity Synthesis

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.

13.2: Plotting the Guesses (15 minutes)

Activity

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.

Launch

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.

Student Facing

Refer to the table you completed in the warm-up, which shows your class' guesses and absolute guessing errors.

  1. Plot at least 12 pairs of values from your table on the coordinate plane on the handout (or elsewhere as directed by your teacher).
  2. Write down 1–2 other observations about the completed scatter plot.
  3. Is the absolute guessing error a function of the guess? Explain how you know.

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Student Facing

Are you ready for more?

Suppose there's another guessing contest that comes with a prize. Each class can submit one guess. It is up to the students to decide on the number to be submitted. Here are some ideas that have been proposed on how to decide on that number:

  • Option A: Ask the person or persons who did really well in the previous guessing game to make a guess.
  • Option B: Ask everyone to make a guess and have a discussion to narrow the list and then choose a number.
  • Option C: Ask everyone to make a guess and find the mean of all the guesses.
  • Option D: Ask everyone to make a guess and find the middle point between the largest number and the smallest number.

Which approach do you think would give your class the best chance of winning? Explain your reasoning.

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Activity Synthesis

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.

Speaking: MLR8 Discussion Supports. Use this routine to support whole-class discussion of the question “Is the absolute guessing error a function of the guess? Explain how you know.” For each response that is shared, ask students to restate what they heard using precise mathematical language. Give students time to restate what they hear to a partner before selecting one or two students to share with the class. Ask the original speaker if their peer was able to accurately restate their thinking. Call students’ attention to the phrase “exactly one output” and any other words or phrases that helped to clarify which variable depends on the other. This routine provides more students with an opportunity to produce language as they interpret the reasoning of others.
Design Principle(s): Support sense-making
Engagement: Develop Effort and Persistence. Encourage and support opportunities for peer interactions. Invite students to share something that they notice and something that they wonder about the scatter plot with their partner before they record their observations.  
Supports accessibility for: Language; Social-emotional skills

13.3: Oops, Try Again! (15 minutes)

Activity

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.

Launch

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.

Student Facing

Earlier, you guessed the number of objects in a container and then your teacher told you the actual number.

Suppose your teacher made a mistake about the number of objects in the jar and would like to correct it. The actual number of objects in the jar is \(\underline{\hspace{0.5in}}\).

  1. Find the new absolute guessing errors based on this new information. Record the errors in Table B of the handout (or elsewhere as directed by your teacher).
  2. Make 1–2 observations about the new set of absolute guessing errors.
    1. Predict how the scatter plot would change given the new actual number of objects. (Would it have the same shape as in the first scatter plot? If so, what would be different about it? If not, what would it look like?)
    2. Use technology to plot the points and test your prediction.
  3. Can you write a rule for finding the output (absolute guessing error) given the input (a guess)?

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Activity Synthesis

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.

Conversing, Writing: MLR2 Collect and Display. Before the whole-class discussion, invite students to share with their partner their observations of the new data set and compare them with what they saw in the previous activity. Listen for and collect vocabulary and phrases students use to describe similarities and differences between the scatter plots on a visual display. Include words and phrases such as “form a V,” “no negative numbers,” and “shift to the right” on the display for all to see. Remind students to borrow language from the display as needed. This will help students discuss the features of a scatter plot using appropriate mathematical language.
Design Principle(s): Support sense-making 
Engagement: Develop Effort and Persistence. Encourage and support opportunities for peer interactions. Invite students to talk with their partner about their observations of the new data set before writing them down. Display sentence frames to support students when they explain their strategy. For example “One thing that is the same is . . .”, “One thing that is different is . . . .”
Supports accessibility for: Language; Social-emotional skills

Lesson Synthesis

Lesson Synthesis

One key thread across the activities in this lesson is the idea of distance from a certain value. To help students see this thread, discuss questions such as:

  • "In either set of data, how is the absolute guessing error related to the actual number?" (The absolute guessing error is the distance of a guess to the actual number.)
  • "What affects the size of the absolute guessing error? What makes it larger or smaller?" (The farther a guess is from the actual number, the greater the absolute guessing error. The closer it is to the actual number, the smaller the absolute guessing error.)
  • "Does it make a difference to the absolute guessing error if a guess overestimates by 5 or underestimates by 5? Why or why not?" (No, what matters is the distance, not whether the guess is above or below the actual number.)
  • "Is the absolute guessing error a function of the guess? Why or why not?" (Yes. For every guess, there is one absolute guessing error.)
  • "Is there a rule we can write to define the relationship between the input and output of this function?" (The output is the distance of a guess from the actual number of objects, which we can express as a difference: “output = input - actual number”.)
  • "If we subtract the actual number from the guess, wouldn't we end up with a negative number if the guess is less than the actual number?" (Because the error is about ‘how far away’, what we want is the absolute value of that difference: “output = |input - actual number|”.)

13.4: Cool-down - Almond Bags (5 minutes)

Cool-Down

Cool-downs for this lesson are available at one of our IM Certified Partners

Student Lesson Summary

Student Facing

Have you played a number guessing game where the guess that is closest to a target number wins?

In such a game, it doesn’t matter if the guess is above or below the target number. What matters is how far off the guess is from the target number, or the absolute guessing error. The smaller the absolute guessing error, or the closer it is to 0, the better.

Suppose eight people made these guesses for the number of pretzels in a jar: 14, 15, 19, 21, 23, 24, 26, and 28. If the actual number of pretzels is 22, the absolute guessing error of each number is as shown in the table.

guess 14 15 19 21 23 24 26 28
absolute guessing error 8 7 3 1 1 2 4 6

In this case, 21 and 23 are both winning guesses. Even though one number is an underestimate and the other an overestimate, 21 and 23 are both 1 away from 22. Of all the absolute guessing errors, 1 is the smallest one.

If we plot the guesses and the guessing errors on a coordinate plane, the points would form a V shape. Notice that the V shape is above the horizontal axis, suggesting that all the vertical values are positive.

horizontal axis, guess. scale 0 to 32, by 4's. vertical axis, absolute guessing error. scale 0 to 12, by 2's. 

Suppose the actual number of pretzels is 19. The absolute guessing errors of the same eight guesses are shown in this table.

guess 14 15 19 21 23 24 26 28
absolute guessing error 5 4 0 2 4 5 7 9

Notice that all the errors are still positive. If we plot these points on a coordinate plane, they are also above the horizontal line and form a V shape.

horizontal axis, guess. scale 0 to 32, by 4's. vertical axis, absolute guessing error. scale 0 to 12, by 2's. 

Why does the relationship between guesses and absolute guessing errors always have this kind of graph? We will explore more in the next lesson!