Absolute Value Functions (Part 2)

10 min

Narrative

In this Warm-up, students begin the transition from thinking about absolute guessing error to thinking about the absolute value function. Using a target value of 0, students see that the function in this activity is equivalent to the distance function.

Student Task

Toronto is a city at the border of the United States and Canada, just north of Buffalo, New York. Here are twelve guesses of the average temperature of Toronto, in degrees Celsius, in February 2017. 

  • 5
  • 2
  • -5
  • 3
  • 0
  • -1
  • 1.5
  • 4
  • -2.5
  • 6
  • 4
  • -0.5
  1. The actual average temperature of Toronto in February 2017 is 0 degrees Celsius.

    Use this information to sketch a scatter plot representing the guesses, xx, and the corresponding absolute guessing errors, yy.

    <p>horizontal axis, guess. scale -8 to 8, by 1's. vertical axis, absolute guessing error. scale 0 to 8, by 1's. </p>

  2. What rule can you write to find the output given the input?

Sample Response

  1. See graph.
  2. Sample responses:
    • To find the output, subtract 0 from the input and take the absolute value. 
    • To find the output, take the absolute value of the input.

<p>scatterplot with v shape</p>

Synthesis

Select a student to display the completed scatter plot. Ask students:

  • "How is the scatter plot for this data like the scatter plot for the absolute guessing errors from an earlier lesson?" (The points still form a V shape. There are still no negative yy-values.)
  • "How are they different?" (The graphs are shifted horizontally toward the vertical axis. The two parts of the V now meet at (0,0)(0,0).)

To help students see that the "actual average temperature" is like the "actual number of items in a jar" they saw earlier, highlight that:

  • Earlier, we saw that when the actual number of items is aa, the absolute guessing error is "the distance of guess from aa," which can be expressed as "the absolute value of (guess - aa)," or guessa|\text{guess} - a|
  • Here, likewise, when the actual average temperature is aa and the guess is xx, the absolute guessing error, yy, is "the distance of xx from aa," which can be written as y=xay = |x-a|.
  • When the actual temperature in Toronto is 0 degrees Celsius, yy is "the distance of xx from 0," which can be written as y=x0y=|x-0|, or simply y=xy=|x|.
Anticipated Misconceptions

Some students may struggle to plot the data without explicitly computing the absolute guessing errors first or creating a table of values. Encourage them to take those intermediate steps if they are helpful.

Standards
Addressing
  • F-BF.1.a·Determine an explicit expression, a recursive process, or steps for calculation from a context.
  • F-BF.1.a·Determine an explicit expression, a recursive process, or steps for calculation from a context.
  • F-BF.1.a·Determine an explicit expression, a recursive process, or steps for calculation from a context.
  • F-BF.1.a·Determine an explicit expression, a recursive process, or steps for calculation from a context.
  • F-IF.7.b·Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions.
  • F-IF.7.b·Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions.
  • F-IF.7.b·Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions.
  • F-IF.7.b·Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions.
  • F-IF.7.b·Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions.
  • HSF-BF.A.1.a·Determine an explicit expression, a recursive process, or steps for calculation from a context.
  • HSF-IF.C.7.b·Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions.
Building Toward
  • A-CED.1·Create equations and inequalities in one variable including ones with absolute value and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions.
  • A-CED.1·Create equations and inequalities in one variable including ones with absolute value and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions.
  • A-CED.1·Create equations and inequalities in one variable including ones with absolute value and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions.
  • A-CED.1·Create equations and inequalities in one variable including ones with absolute value and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions.
  • A-CED.1·Create equations and inequalities in one variable including ones with absolute value and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions.
  • A-CED.1·Create equations and inequalities in one variable including ones with absolute value and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions.
  • HSA-CED.A.1·Create equations and inequalities in one variable and use them to solve problems. <span>Include equations arising from linear and quadratic functions, and simple rational and exponential functions.</span>

15 min

10 min

15 min