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Unit 8, Lesson 13

Absolute Value Functions (Part 1)

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Integrated Math 1

13.1

Warm-up

Standards Alignment

Building On

Addressing

Building Toward

Instructional Routines

None

Materials

To Copy (from Blackline Masters)

How Good Are Your Guesses Handout

Activity Narrative

In this Warm-up, students compute absolute guessing errors using class data. 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 (MP5). 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

Tell students to guess the number of objects in the collection. If no collection is available, display the image and ask students to guess the number of snap cubes in the jar.

Collect all of the guesses from the students, and display them for all to see.

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 can split up the calculations, if desired. Provide access to calculators and spreadsheet technology.

Give each student a copy of the blackline master to record their calculations, then reveal the actual number of items in the collection. 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.

Activity

None

Use the actual number of items to calculate the absolute guessing error of each guess, or how far the guess is from the actual number. For example, suppose the actual number of objects is 100.

If a guess is 75, then the absolute guessing error is 25.
If a 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 (or elsewhere, as directed by your teacher).

Student Response

to access Student Response.

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 about 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

Activity

Instructional Routines

Notice and Wonder

Materials

None

Activity Narrative

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.

13.3

Activity

Instructional Routines

MLR2: Collect and Display

Materials

None

Activity Narrative

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.

Engagement: Develop Effort and Persistence. Encourage and support opportunities for peer interactions. Invite students to talk about their ideas with a partner before writing them down. Display sentence frames to support students when they explain their strategy. For example, “One thing that is the same is . . . .” and “One thing that is different is . . . .”
Supports accessibility for: Language, Social-Emotional Functioning

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. Collect the language students use to describe similarities and differences between the scatter plots. Display words and phrases, such as “form a V,” “no negative numbers,” and “shift to the right.” During the Activity Synthesis, invite students to suggest ways to update the display: “What are some other words or phrases we should include?” Invite students to borrow language from the display as needed.
Advances: Conversing, Reading

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|.)

to access Cool-down.

Student Lesson Summary

Have you played a number guessing game in which 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.

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.

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

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 nonnegative. If we plot these points on a coordinate plane, they are also on or above the horizontal line and form a V shape.

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

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Lesson 13

Absolute Value Functions (Part 1)

Warm-up

Standards Alignment

Building On

Addressing

Building Toward

HSF-IF.C.7.b

Instructional Routines

Materials

To Copy (from Blackline Masters)

Activity Narrative

Launch

Activity

Student Response

Activity Synthesis

Activity

Instructional Routines

Materials

Activity Narrative

Launch

Activity

Instructional Routines

Materials

Activity Narrative

Launch

Lesson Synthesis

Student Lesson Summary

Building on Student Thinking

Activity

Student Response

Building on Student Thinking

Activity Synthesis

Activity

Student Response

Building on Student Thinking

Activity Synthesis

Standards Alignment

Building On

Addressing

Building Toward

HSA-CED.A.1

HSF-IF.C.7.b

Standards Alignment

Building On

Addressing

HSA-CED.A.1

HSF-IF.C

HSF-IF.C.7.b

Building Toward