Insights Into Algebra 1: Teaching for Learning
Linear Functions and Inequalities Lesson Plan 1: The Phone Bill Problem – Linear Functions
In this lesson, students will learn how to write an equation of a linear function when given a set of data. They will interpret the meaning of the slope and y-intercept and then use the equation to find other values of x and y.
One 50-minute class period
Students will be able to:
- Make sense of a set of data and plot it on a graph.
- Find the equation of the line that contains the data points.
- Understand the meaning of the slope and y-intercept.
- Use the equation to predict other x- and y-values.
Principles and Standards for School Mathematics, National Council of Teachers of Mathematics (NCTM), 2000:
NCTM Algebra Standard for Grades 6-8
NCTM Algebra Standard for Grades 9-12
Students will need the following:
- Graphing calculators
- Copies of The Phone Bill Problem handout
Teachers Activities and Assignment
1. Explain to the class that the handout contains a copy of an actual phone bill. Have students look at the phone bill data – code, minutes, and cost – and determine whether to include all of the calls in the data set.
2. Students should notice that all of the calls have the code “EC” except for the second call, which is coded “NC.” They might hypothesize that they should eliminate the call coded NC from the data set.
3. Prompt students to examine the cost and length of the first two calls.
4. Students should notice that the EC call is more expensive than the NC call, even though it’s shorter. They should conclude that the class should eliminate the call coded NC from the data set. (You may want to tell them that NC is a lower billing rate applied to nighttime calls.)
1. Ask the students to arrange the EC calls in a table, from the shortest in length to the longest.
2. Make a scatterplot of the data using a graphing calculator. Ask students to discuss the meaning of the x- and y-values and to decide the appropriate values to use in the window setting. In the video, the students proposed the following: x = minutes, y = cost of the call, xmin = 0, xmax = 50, xscl = 10, ymin = 0, ymax = 20, yscl = 5. Make sure students can justify their choices, and remind them that these values represent the domain and range in this problem.
3. Elicit from the class that if the data points were connected, they would form a line.
4. Allow students time to devise a plan for finding the equation of the line formed by connecting the data points.
5. Ask students to report on their plan. Students should mention that they would find the slope and y-intercept of the line. Focus on the slope first and make sure students understand how to use the data to find the slope.
6. Give the students time to choose two data points and find the slope (m = 0.24).
7. Ask students to discuss how to find the value of the y-intercept. They should conclude that the y-intercept, b, is 0.85.
8. Have the class verify the values for both the slope and the y-intercept and determine that the equation of the line is y = 0.24x + 0.85.
9. Check the equation by graphing it and decide how well the equation fits the data. The students should notice that the equation falls on all of the data points, and is therefore a perfect fit.
10. Ask students to consider whether or not it matters which two data points they choose to find the equation of the line. Students should determine that because the data points all lie on the line, they could choose any two and arrive at the same equation.
11. Trace to x = 8 and y = 2.77 and ask students to discuss the meaning of this point. They should understand that this point means that if you talk for eight minutes, the cost of the call would be $2.77.
12. Ask students to choose another x-value and find the corresponding y-value, and explain how they found it. Students may choose to find the value by using a table or a graph, or by substituting into the equation.
13. Ask students to look at the table values for x = 0 to x = 5 and describe any patterns they see. Students should notice that for each increase of x, the y-values increase by $0.24.
14. Give the students some time to determine the meaning of the slope and the y-intercept in this problem context.
15. Make sure they understand that the slope represents the fact that the increase in cost for the call is 24 cents per minute and the y-intercept of 0.85 represents the initial fee for the call.
16. Ask the class to determine the cost of a one-minute phone call. Students should report that this call would cost $1.09 ($0.85 as the initial fee plus $0.24 for up to one minute). Ask students how much it would cost if the call lasted only 15 seconds. Students should realize that any call of one minute or less would cost $1.09.
17. Write the equations using different variables and using function notation:
y = 0.24x + 0.85
C = 0.24t + 0.85
C(t) = 0.24t + 0.85
Explain to the class that these are different ways to express the same equation. To demonstrate that fact, ask students to substitute the same values into each equation.
18. Ask the class to find the cost of a one-hour phone call. Discuss the method they chose to solve the problem – table, graph, or equation – and their reason for choosing that method.
19. Ask students how much it would cost to talk on the phone for 24 hours. Students will first need to determine the number of minutes in 24 hours. For example:
20. Students should then substitute the number of minutes, 1440, into one of the equations above to find the cost. (Answer: $346.45)
21. Ask the students to determine how long a call lasted if it cost $6.13. (Answer: 22 minutes)
1. Ask students to write a paragraph that describes what they learned today, what they re-learned today, and what they liked about today’s lesson.
2. Assign practice problems for homework.
Related Standardized Test Questions
The questions below dealing with linear functions have been selected from various state and national assessments. Although the lesson above may not fully equip students to answer all such test questions successfully, students who participate in active lessons like this one will eventually develop the conceptual understanding needed to succeed on these and other state assessment questions.
- Taken from the North Carolina End-of-Course Test, Algebra 1 (October 2003):
The cost of a large pizza is given by the formula C(t) = 1.5t + 7.5, where C(t) is the cost of the pizza and t is the number of toppings. What does the slope represent?
A. Number of toppings
B. Cost per slice
C. Cost of each topping (correct answer)
D. Cost of pizza with no toppings
- Taken from the North Carolina End-of-Course Test, Algebra 1 (October 2003):
Denisha bought a car for $15,000. The value depreciates linearly. After three years, the value is $11,250. What is the amount of yearly depreciation?
C. $1,250 (correct answer)
- Taken from the Virginia Standards of Learning, Algebra 1 (2002):
The graph of is shown below. If the line in the graph is shifted up 2 units, what is the equation of the new line?
D. (correct answer)
- Taken from the North Carolina End-of-Course Test, Algebra 1 (October 2003):Which of the following equations describes the data in the table below?
A. 2x + y = -27
B. x – y = -3
C. x + y = -21
D. 2x – y = 3 (correct answer)
Student Work: The Phone Bill Problem
Matthew took good notes. He had a difficult time with the spelling of the word “intercept,” spelling it three different ways in a matter of minutes. He also didn’t realize at first that m is for slope and b is for the y-intercept.
It was not always clear to me how he decided upon the answers to some of the questions we asked. He would just write “graph” or “table”.
I do see evidence that he understands what he is doing. I also see some unsure areas. I appreciated Matt’s comments at the end of the assignment and how he referred to this problem as a “day-to-day” problem.
I would not change this [assignment]. I have been using it for several years and it has served my students well at all levels of mathematical maturity. I just need to adapt the class to the [assignment].
Workshop 1 Variables and Patterns of Change
In Part I, Janel Green introduces a swimming pool problem as a context to help her students understand and make connections between words and symbols as used in algebraic situations. In Part II, Jenny Novak's students work with manipulatives and algebra to develop an understanding of the equivalence transformations used to solve linear equations.
Workshop 2 Linear Functions and Inequalities
In Part I, Tom Reardon uses a phone bill to help his students deepen their understanding of linear functions and how to apply them. In Part II, Janel Green's hot dog vending scheme is a vehicle to help her students learn how to solve linear equations and inequalities using three methods: tables, graphs, and algebra.
Workshop 3 Systems of Equations and Inequalities
In Part I, Jenny Novak's students compare the speed at which they write with their right hands with the speed at which they write with their left hands. This activity enables them to explore the different types of solutions possible in systems of linear equations, and the meaning of the solutions. In Part II, Patricia Valdez's students model a real-world business situation using systems of linear inequalities.
Workshop 5 Properties
In Part I, Tom Reardon's students come to understand the process of factoring quadratic expressions by using algebra tiles, graphing, and symbolic manipulation. In Part II, Sarah Wallick's students conduct coin-tossing and die-rolling experiments and use the data to write basic recursive equations and compare them to explicit equations.
Workshop 6 Exponential Functions
In Part I, Orlando Pajon uses a population growth simulation to introduce students to exponential growth and develop the conceptual understanding underlying the principles of exponential functions. In Part II, a scenario from Alice in Wonderland helps Mike Melville's students develop a definition of a negative exponent and understand the reasoning behind the division property of exponents with like bases.
Workshop 7 Direct and Inverse Variation
In Part I, Peggy Lynn's students simulate oil spills on land and investigate the relationship between the volume and the area of the spill to develop an understanding of direct variation. In Part II, they develop the concept of inverse variation by examining the relationship of the depth and surface area of a constant volume of water that is transferred to cylinders of different sizes.
Workshop 8 Mathematical Modeling
This workshop presents two capstone lessons that demonstrate mathematical modeling activities in Algebra 1. In both lessons, the students first build a physical model and use it to collect data and then generate a mathematical model of the situation they've explored. In Part I, Sarah Wallick's students use a pulley system to explore the effects of one rotating object on another and develop the concept of transmission factor. In Part II, Orlando Pajon's students conduct a series of experiments, determine the pattern by which each set of data changes over time, and model each set of data with a linear function or an exponential function.