Linear Relationships
Linear Relationships
Objectives
In this unit, students will develop an understanding of linear relationships and situations that are represented by these relationships. Students will:
- use patterns to predict missing values in a table or other representation of a linear relationship.
- use and meaningfully represent the concept of slope as a relationship between the change in one variable compared to the change in another variable.
- learn what defines a linear function.
- represent and understand proportional relationships and direct proportion.
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 can data be organized and represented to provide insight into the relationship between quantities?
- 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.
- http://www.learner.org/teacherslab/math/patterns/
- http://www.learner.org/teacherslab/math/patterns/number.html
- http://math.rice.edu/~lanius/Lessons/Patterns/rect.html
- http://nlvm.usu.edu/en/nav/frames_asid_185_g_2_t_1.html
- http://id.mind.net/~zona/mmts/functionInstitute/linearFunctions/linearFunctions.html
- http://id.mind.net/~zona/mmts/functionInstitute/linearFunctions/lsif.html
- http://www.zweigmedia.com/RealWorld/tutorialsf0/frames1_3.html
- http://en.wikipedia.org/wiki/Linear_function
- http://www.freemathhelp.com/point-slope.html
- http://id.mind.net/~zona/mmts/functionInstitute/linearFunctions/lpsf.html
- http://www.sparknotes.com/math/algebra1/writingequations/section2.rhtml
- http://www.algebralab.org/studyaids/studyaid.aspx?file=algebra1_5-5.xml
- http://courses.wccnet.edu/~palay/precalc/22mt01.htm
- http://mathforum.org/library/drmath/view/60365.html
- http://edhelper.com/LinearEquations4.htm
- http://mathforum.org/library/drmath/view/54471.html
Formative Assessment
-
View
Multiple-Choice Items:
- Which is the next number in the pattern 55, 44, 33, 22, 11, …?
A
−1
B
0
C
1
D
11
- Which rule describes the pattern −5, −3, −1, 1, 3, . . .?
A
−2x +1 = y
B
−2x −1 = y
C
2x − 5 = y
D
2x − 7 = y
- The rule for a pattern is y = 3x − 5. What is the 45th term in this pattern?
A
40
B
130
C
222
D
340
- Which of the following is not a linear equation?
A
y = 0.5x + 5
B
y = −3x
C
D
y = 165x + 1245
- What is the mathematical term for the quantity that measures the steepness of a line?
A
x-intercept
B
y-intercept
C
domain
D
slope
- Which equation represents a line with a slope of 3 and a y-intercept of 4?
A
y = 3x + 4
B
x = 3y + 4
C
y = 4x + 3
D
x = 4y + 3
- Which linear relationship is most likely to have a negative slope when the first variable is plotted against the second variable?
A
hours shopping; total spent
B
hours exercising; calories burned
C
hours driving, fuel remaining
D
hours studying; grade point average
- An attorney charges an initial fee of $300 and $150 per each hour of work. Which equation most accurately represents the total cost, c, for h hours of work?
A
c = 150h + 300
B
c = 150 + 300h
C
c = 150 + 300
D
c = 450h
- The y-intercept of a certain linear equation is −7. Which statement is always true?
A
The graph of the equation crosses the x-axis at (−7, 0).
B
The graph of the equation crosses the y-axis at (0, −7).
C
The slope of the graph is positive.
D
The slope of the graph is negative.
Multiple-Choice Answer Key:
1. B
2. D
3. B
4. C
5. D
6. A
7. C
8. A
9. B
Short-Answer Items:
- Consider the following pattern: −1, 3, 7, 11, 15, …
Identify the pattern as linear or nonlinear. Create a graph of the pattern, using the numbers listed as y-values in the ordered pairs.
_______________________________
- What are two properties of linear functions? Your answers should use some of the following terms: domain, range, exponent, slope.
- Provide a brief description of a real-world linear function, and identify the rate of change.
Short-Answer Key and Scoring Rubrics:
- Consider the following pattern: −1, 3, 7, 11, 15, …
Identify the pattern as linear or nonlinear. Create a graph of the pattern, using the numbers listed as y-values in the ordered pairs.
The pattern is linear. The graph includes the points (1, −1), (2, 3), (3, 7), (4, 11), and (5, 15).
Points
Description
2
The student correctly identifies the pattern as linear and draws the graph with points (1, −1), (2, 3), (3, 7), (4, 11), and (5, 15).
1
The student either correctly identifies the pattern as linear OR correctly draws the graph.
0
The student does not correctly identify the pattern as linear AND does not correctly draw the graph.
- What are two properties of linear functions? Your answers should use some of the following terms: domain, range, exponent, slope.
Answers will vary.
Points
Description
2
The student’s answer identifies two of the following properties of linear functions:
- The graph of a linear function is a line.
- The highest exponent for the independent variable in a linear function is 1.
- The domain of a linear function is all the real numbers.
- The range of a linear function is all the real numbers.
- A linear function has a constant slope.
- A linear function can be written in slope-intercept form (y = mx + b, where m is the slope and b is the y-intercept).
1
The student’s answer identifies one of the properties of linear functions.
0
The student’s answer does not identify any of the properties of linear functions.
- Provide a brief description of a real-world linear function, and identify the rate of change.
Answers will vary.
Points
Description
2
The student correctly writes a brief description of a real-world linear function AND correctly identifies the rate of change.
1
The student correctly writes a brief description of a real-world linear function, but does not correctly identify the rate of change.
0
The student does not provide a brief description of a real-world linear function AND does not identify the rate of change.
Performance Assessment:
Find a food product in your house and look at the nutritional information, which is always given per serving. Choose one of the nutritional values (calories, sodium, cholesterol, etc.).
- Make an x/y chart, where x represents the number of servings of the food product you have chosen, and y represents the amount of calories, sodium, etc., that you have chosen. Complete 5 rows in your x/y chart. For instance, if you chose something that has
200 calories per serving, your x/y chart would look like:
x
y
1
200
2
400
3
600
4
800
5
1000
- Write a rule that relates the x and y values in your chart. For the chart above, a rule might be y = 200x.
- Describe the slope within the context of the problem.
- Describe the y-intercept within the context of the problem.
- Finally, graph the line. Remember that your graph should only exist in the first quadrant, because you cannot have a negative number of servings or a negative number of calories, sodium, etc.
Performance Assessment Scoring Rubric:
Points
Description
4
The student:
- makes an x/y chart with no errors.
- writes a correct equation for the rule.
- correctly describes the slope.
- correctly describes the y-intercept.
- correctly graphs the line with axes labeled with x and y.
3
The student:
- makes an x/y chart with one error.
- writes a correct equation or expression for the rule.
- correctly describes the slope but not in context.
- correctly describes the y-intercept but not in context.
- correctly graphs the line.
2
The student:
- makes an x/y chart with two errors.
- writes an equation for the rule that has some errors.
- describes the slope with some errors.
- describes the y-intercept with some errors.
- graphs the line with minimal errors.
1
The student:
- makes an x/y chart with three or more errors.
- writes an equation for the rule that is mostly incorrect.
- incorrectly describes the slope.
- incorrectly describes the y-intercept.
- graphs the line with multiple errors.
0
The student:
- fails to make an x/y chart or has multiple errors.
- fails to write an equation for the rule or writes an equation that is completely incorrect.
- fails to describe the slope.
- fails to describe the y-intercept.
- fails to graph the line.
Final 06/28/2013
