Linear Functions
Linear Functions
Objectives
In this unit, students will explore the concept of linear functions. They will compare function descriptions, rules, tables, and graphs to make connections. Students will:
- identify relations and functions.
- determine whether a function is linear, and why or why not.
- describe the similarities and differences between representing a function in descriptive, rule, table, or graph form and understand the advantages of each form.
- determine a function rule given a list of possible values that satisfy that rule.
- calculate missing values for a stated function or function pattern.
- use multiple methods to define/describe a real-life linear function situation.
- recognize and move between multiple representations of linear functions.
- match multiple descriptions (words, rule, table, or graph) which represent the same linear function.
- use technology to create and analyze linear functions.
Essential Questions
- How are relationships represented mathematically?
- How can expressions, equations, and inequalities be used to quantify, solve, model, and/or analyze mathematical situations?
- How is mathematics used to quantify, compare, represent, and model numbers?
Related Unit and Lesson Plans
Related Materials & Resources
The possible inclusion of commercial websites below is not an implied endorsement of their products, which are not free, and are not required for this lesson plan.
Formative Assessment
-
View
Multiple-Choice Items:
1. Which relation is not a function?
A
y = −2x + 7
B
x
y
3
8
5
6
1
−4
2
8
C
D
(3, 4), (6, 1), (10, 3), (3, 2), (7, 4)
2. Which of the following does not represent a linear function?
A
cents paid per word
B
y = x2 − 8
C
(−1, 7), (1, −5), (4, −23), (−5, 31)
D
5, 2, −1, −4, …
3. Identify the constant rate of change in the table below.
x
y
−2
−11
0
−3
2
5
4
13
A
4
B
8
C
−3
D
3
4. Given the function
, which is a point on the line with an x-value of −6?A
(–6, −89)
B
(–6, 87)
C
(–6, −105)
D
(–6, 71)
5. For an x-value of −16, which function has the largest y-value?
6. Which linear function is steepest?
7. Which representation is most likely not linear?
A
y = −x + 9
B
the number of umbrellas purchased
C
D
5 trees planted per hour
8. Which representation is linear?
A
y = x2 – 9
B
(2, 4), (6, 36), (−3, 9)
C
−8, −16, −32, −64,…
D
x
y
−10
−183
−6
−107
3
64
7
140
9. Which situation is most likely to have a linear relationship?
A
Number of candy bars bought per hour at the vending machine
B
Amount owed in cab fare per miles driven
C
Amount of money spent on purchases each week
D
Number of donations given per hour at a fundraiser
Multiple-Choice Answer Key:
1. D
2. B
3. A
4. C
5. B
6. D
7. B
8. D
9. B
Short-Answer Items:
10. Write the function rule for the relationship between x and y in the following table. Then graph the function on a coordinate grid.
x
−6
−3
0
4
9
y
96
51
6
−54
−129
11. Given
, state the slope in the equation, make a table with at least three input values, and graph the line.12. Write a linear function using the representation of your choice. Provide a justification for the linearity and provide at least one alternative representation.
Short-Answer Key and Scoring Rubrics:
10. Write the function rule for the relationship between x and y in the following table. Then graph the function on a coordinate grid.
x
−6
−3
0
4
9
y
96
51
6
−54
−129
Points
Description
3
The student writes
AND- shows the correct process for finding the rule.
- correctly graphs the function.
2
The student correctly provides two of the three answers.
1
The student correctly provides one of the three answers.
0
The student does not provide an answer or incorrectly answers all three parts.
11. Given
, state the slope in the equation, make a table with at least three input values, and graph the line.Points
Description
3
The student correctly
- states a slope of −3.
- creates a table showing at least three input values and their correct corresponding output values.
- graphs the line of the equation.
2
The student correctly writes two of the three answers.
1
The student correctly writes one of the three answers.
0
The student does not provide an answer or incorrectly answers all three parts.
- Write a linear function using the representation of your choice. Provide a justification for the linearity and provide at least one alternative representation.
Points
Description
3
The student provides all of the following:
- an acceptable linear function.
- a justification.
- an acceptable alternative representation.
2
The student meets two of the requirements.
1
The student meets one requirement.
0
The student does not provide an answer or incorrectly provides all three pieces.
Performance Assessment:
Write a letter to future students in this class explaining what they will learn about linear functions. Clearly address and provide the following:
- important ideas learned, with supporting detail.
- at least five examples of linear functions with illustrations and discussion of the characteristics, including at least two real-world examples.
- related topic(s) you wish to explore.
Performance Assessment Scoring Rubric:
Points
Description
4
- A complete, accurate, and insightful description of linear functions is included. It may include comparisons or definitions of relations, functions, linear functions, and nonlinear functions.
- More than five accurate examples of linear functions are included with sketches; three or more are real-world examples.
- A detailed explanation of linear characteristics including constant rate of change (slope) is included for each example, as is how slope can be found in each example.
- Two or more topics for further study are discussed, including reasons why.
- The student demonstrates advanced understanding of the mathematical concept of linear functions.
- The student worked beyond problem requirements by including additional examples and depth of explanation, collecting real-world data, or using technology to make sketches or report.
3
- An accurate description of linear functions is included. It may lack some detail.
- Four accurate examples of linear functions are included with sketches; one to two are real-world examples.
- Linear characteristics are addressed to some extent but not in detail; constant rate of change (slope) is mentioned but not discussed.
- One topic for further study is described.
- The student demonstrates a solid understanding of the mathematical concept of linear functions.
- The student meets or partially meets all of the problem requirements.
2
- A description of linear functions is attempted. It lacks significant detail or is detailed but with several inaccuracies.
- Three to four examples of linear functions are included but are partially inaccurate, more than one sketch is missing, or no or only one real-world example is included.
- Discussion of linear characteristics lacks significant detail or is difficult to follow.
- One topic for further study is mentioned.
- The student demonstrates a limited to fair understanding of the mathematical concept of linear functions.
- The student meets approximately half of the problem requirements.
1
- A description of linear functions is attempted but is difficult to understand, or inaccurate.
- One to two examples of linear functions are included but are partially inaccurate, more than one sketch is missing, or no real-world examples are included.
- Discussion of linear characteristics lacks significant detail, is almost completely inaccurate, or is not addressed at all.
- No topic for further study is mentioned.
- The student demonstrates very limited understanding of the mathematical concept of linear functions.
- The student does not fully meet most of the problem requirements, but partially meets some.
0
- The description of linear functions is completely inaccurate or not attempted.
- Two or fewer (or no) examples of linear functions are attempted, all are inaccurate, no sketches are provided, or no real-world examples are included.
- Discussion of linear characteristics is completely inaccurate or missing.
- No topic for further study is mentioned.
- The student demonstrates no understanding of the mathematical concept of linear functions.
- The student did not meet any problem requirements.
Final 04/12/13
