Absolute Value Functions
15. Provide an opportunity for each group to share the mind maps with the larger group.
Discuss similarities, differences, and key points brought forth by participants.
16. Distribute the vocabulary organizer template to each participant. Ask participants to
construct a vocabulary model for the term “absolute value functions.”
Maximizing Algebra II Performance
Explore/Explain/Elaborate 3
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Absolute Value Functions
Maximizing Algebra II Performance
Explore/Explain/Elaborate 3
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Absolute Value Functions
Maximizing Algebra II Performance
Explore/Explain/Elaborate 3
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Absolute Value Functions
Maximizing Algebra II Performance
Explore/Explain/Elaborate 3
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Grade level/Course: Algebra II; Regular
Topic: Absolute Value functions
Dates: 11/2, 3
Essential question: What are the critical attributes of absolute value functions? How does changing
the parameters of the function affect the graph? How do you solve absolute value functions?
Focus TEKS:
2A.1A, 2A.1, 2A.2A, 2A.4A,
2A.4B
Prerequisite knowledge:
Graph linear equations
Basic understanding of Absolute Value
Engage:
What factors might affect the cost of homeowner's insurance? Does the distance to a fire station or
fire hydrant matter to your insurance rates? Why? How far do you think your home i from the fire
station?
MTC Fire Station Problem 1-12: Activate Prior Knowledge. Determine the businesses that are on
either side of the Fire station. Apply that to distance and direction. These problems can be worked
independently or in partners in order to formatively assess prior knowledge about linear equations,
domain, range and data.
Explore:
MTC Fire Station Problem 13-23. Create a deeper understanding of limiting domains and how they
effect equations.
Explain:
MTC Fire Station Problem: Discuss questions24-28: Discuss the differences between the Algebra I
concept of absolute value as an expression of distance. how to regress each line from the data
points; graph the equations on the calculator and determine similarities and differences.
Elaborate/Extend:
MTC Fire Station Problem: problem 10 and discussion on questions 11 & 12: Adjusting the window
and the relation to the data points.
Evaluate:
HW: MTC Fire Station Problem: Part 2 question 1-5 (attached after next section)
Knowledge check
Reflection:
Materials/Resources:
MTC Fire Station Problem
houses
Graphing Calculators
“Best Practices”:
x
x
x
x
x
x
x
x
Student choice
Technology
Hands on activity
Differentiated instruction
Small group
Multiple intelligence
Use of visuals
Higher order thinking skills
Literacy strategies
“Real world” connections
Weaves in interventions
Consider ∆
Meaningful assessments
The Fire Station Problem
A fire station is located at Main Street and has buildings at every block to the right and
to the left. You will investigate the relationship between the address number on a building
and its distance from the fire station.
1. Complete the table below that relates the address of a building (x) with its distance
in blocks from the fire station (y).
Address Number (x)
Distance in Blocks
from the Fire Station
(y)
1200
1300
1400
1500
1600
2. Where are these building in relationship to the Fire Station?
3. Draw a scatterplots that represents the data in the table.
Distance in Blocks from the Fire
Station
5
4
3
2
1
400
800
1200
1600
2000
Address Number
4. Make a scatterplot of your data using your graphing calculator, Describe your
viewing window.
5. What function or functions might you use to describe the scatterplot?
6. Find the linear functions that pass through the data points. What is the slope?
What is the y-intercept?
7. Complete the table below that relates the address of a building (x) with its distance
in blocks from the fire station (y).
Address Number (x)
Distance in Blocks
from the Fire Station
(y)
800
900
1000
1100
1200
8. Where are these building in relationship to the Fire Station?
9. Draw a scatterplots that represents the data in the table.
Distance in Blocks from the Fire
Station
5
4
3
2
1
400
800
1200
Address Number
1600
2000
10. Make a scatterplot of your data using your graphing calculator and the given viewing
window.
11. What function or functions might you use to describe the
scatterplot?
12. Find the linear functions that pass through the data points. What is the slope?
What is the y-intercept?
13. Using your data from the two previous tables, complete the following table.
Address Number (x)
Distance in Blocks from the
Fire Station (y)
800
900
1000
1100
1200
1300
1400
1500
1600
14. Draw a scatterplots that represents the data in the table.
Distance in Blocks from the Fire
Station
5
4
3
2
1
400
15. Describe your graph.
800
1200
Address Number
1600
2000
16. Graph the equations on your calculator. How are they similar? How are they
different?
17. Where do the equations fit the graph of the data points? Where do the equations
not fit the graph of the data points?
18. What is the domain and range of each of the linear functions that model the fire
station problem?
Linear Equation
Y1=
Y2=
Domain
Range
19. What is the domain and range of the data set?
20. How do the domains and ranges compare?
21. How do the linear equations compare to each other?
22. At what point do the graphs of the lines intersect?
23. Which parts of the graphs of the lines model our data set? Which parts do not?
Why?
Numerically, when we consider the distance of a number form 0 on a number line, we call
the distance the absolute value of the number.
For example |−7| = 7, since -7 has a distance of 7 from 0. Similarly, |7| = 7 since 7 has a
distance of 7 from 0.
The x-coordinate below represent locations on a number line. The y-coordinates represent
the distance that location is from 0. For the given x-values, use the number line to find
the corresponding y-values.
24. Make a scatterplot of y versus x on your graphing Calculator. Describe your
window. Sketch your scatterplot.
25. What is the shape of the scatterplot?
26. Find two linear functions that model this situation. Graph these functions.
27. Restrict the domain so that the model is an even better fit. Graph the restricted
functions.
28. Now try 3 = ||. How does this function compare to the two linear functions?
Algebraically, the definition of the absolute value function is = −ℎ < 0
ℎ ≥ 0
29. = || is a new parent function to add to your list of parent functions. With
which other parent functions are you familiar?
30. What are some characteristics of the absolute value function?
The Fire Station Problem KEY
A fire station is located at Main Street and has buildings at every block to the right and
to the left. You will investigate the relationship between the address number on a building
and its distance from the fire station.
1. Complete the table below that relates the address of a building (x) with its distance
in blocks from the fire station (y).
Address Number (x)
Distance in Blocks
from the Fire Station
(y)
1200
0
1300
1
1400
2
1500
3
1600
4
2. Where are these building in relationship to the Fire Station? The buildings are to
the right of the fire station.
3. Draw a scatterplots that represents the data in the table.
Distance in Blocks from the Fire
Station
5
4
3
2
1
400
800
1200
1600
2000
Address Number
4. Make a scatterplot of your data using your graphing calculator and the given viewing
window. Check calculators.
5. What function or functions might you use to describe the
scatterplot? A linear equation
6. Find the linear functions that pass through the data points. What is the slope?
What is the y-intercept? = − 12
7. Complete the table below that relates the address of a building (x) with its distance
in blocks from the fire station (y).
Address Number (x)
Distance in Blocks
from the Fire Station
(y)
800
4
900
3
1000
2
1100
1
1200
0
8. Where are these building in relationship to the Fire Station? To the right of the
fire station.
9. Draw a scatterplots that represents the data in the table.
Distance in Blocks from the Fire
Station
5
4
3
2
1
400
800
1200
Address Number
1600
2000
10. Make a scatterplot of your data using your graphing calculator and the given viewing
window.
11. What function or functions might you use to describe the
scatterplot? (Answers may Vary) Parts of two different lines.
12. Find the linear functions that pass through the data points. What is the slope?
What is the y-intercept? = + 12
13. Using your data from the two previous tables, complete the following table.
Address Number (x)
Distance in Blocks from the
Fire Station (y)
800
4
900
3
1000
2
1100
1
1200
0
1300
1
1400
2
1500
3
1600
4
14. Draw a scatterplots that represents the data in the table.
Distance in Blocks from the Fire
Station
5
4
3
2
1
400
800
1200
Address Number
1600
2000
15. Describe your graph. (Answers vary) Looks like a V… one side is the reflection of
the other side.
16. Graph the equations on your calculator. How are they similar? How are they
different? (Answers vary) The lines are the same as the data above the x
axis…there is no data below the axis.
17. Where do the equations fit the graph of the data points? Where do the equations
not fit the graph of the data points? Y1 is the same as the data when ≥ 1200, but
only the equation exists when x<1200. Y2 is the same as the data when x <1200, but
only the equation exists when ≥ 1200.
18. What is the domain and range of each of the linear functions that model the fire
station problem?
Linear Equation
Y1= .01x - 12
Y2= -.01x + 12
Domain
All real numbers
All real numbers
Range
All real numbers
All real numbers
19. What is the domain and range of the data set?
D={ 800, 900, 1000, 1100, 1200, 1300, 1400, 1500, 1600}
R ={0, 1, 2, 3, 4}
20. How do the domains and ranges compare?
(Answers Vary) The domain and range of the linear equations are not restricted,
the data set is restricted, discrete so must be in roster form.
21. How do the linear equations compare to each other?
Y1 = -(Y2)
22. At what point do the graphs of the lines intersect?
(1200, 0)
23. Which parts of the graphs of the lines model our data set? Which parts do not?
Why? Y1 is the same when x ≥ 0. Y2 is the same when x < 0.
Numerically, when we consider the distance of a number form 0 on a number line, we call
the distance the absolute value of the number.
For example |−7| = 7, since -7 has a distance of 7 from 0. Similarly, |7| = 7 since 7 has a
distance of 7 from 0.
The x-coordinate below represent locations on a number line. The y-coordinates represent
the distance that location is from 0. For the given x-values, use the number line to find
the corresponding y-values.
3
2
1
0
1
2
3
24. Make a scatterplot of y versus x on your graphing Calculator. Describe your
window. Sketch your scatterplot.
25. What is the shape of the scatterplot?
It makes a V
26. Find two linear functions that model this situation. Graph these functions.
Y = -x and Y = x
27. Restrict the domain so that the model is an even better fit. Graph the restricted
functions.
Y = -x when x < 0 and Y = x when x > 0 or
= − ∗ < 0
in the calculator
= ∗ ≥ 0
28. Now try 3 = ||. How does this function compare to the two linear functions?
Exactly the same
Algebraically, the definition of the absolute value function is = −ℎ < 0
ℎ ≥ 0
29. = || is a new parent function to add to your list of parent functions. With
which other parent functions are you familiar? Answers Vary
30. What are some characteristics of the absolute value function?
Domain – All Real Numbers, Range y>0,
the fact that two pieces of the equations are linear.