This lesson provides students with the description of an authentic
situation and data points that fit that situation. Students graph the
line that contains those points; determine its y‑intercept, slope, and equation; and interpret the slope and y‑intercept
in the context of the problem. Finally, students will extrapolate to
determine the a functional value that is out of the domain they have
To begin, present students with the following situation:
While surfing the Internet, you find a site that claims to offer
"the most popular and the cheapest DVDs anywhere." Unfortunately, the
website isn’t clear about the how much they charge for each DVD, but it
does give you the following information:
Number of DVDs Ordered
Total Cost (includes S&H)
Although you can present the problem to students verbally, it
will likely be necessary to project the table of costs on the overhead
projector or draw it on the chalkboard. Students will need to refer to
it throughout the first part of the lesson.
Ask students to plot the data shown in the table and to connect
the points. Discuss what the graph of these points looks like [A line.]. When
students justify that the points form a line, ask them to write an equation for the line.
Then, ask them to answer the following questions:
- Assume that total cost is a linear function of number of DVDs ordered.
- What is the slope of the line that contains the data points? [9.]
- What does that slope represent in the context of this problem? [You will be charged $9.00 for each additional movie purchased.]
- What is the y‑intercept of the line that contains the data points? [6.]
- What does that y‑intercept represent in the context of this problem? [It would cost you $6.00 for the store to send you an empty box.]
- Your friend says that he can get a dozen DVDs from this web site for $90. Is he correct? Explain. [No. The equation for this line y=9x+6. If you buy a dozen DVDs, then y=9(12)+6, or y=114. It would cost you $114, not $90, to buy a dozen DVDs.]
- How much would it cost to order 50 DVDs from this web site? [y=9(50)+6=456. It would cost $456.00 to buy 50 DVDs.]
Students may work individually at first, but then they should share
their answers with a partner. Each pair should then compare their
results with another pair. A whole class review of the solutions should
Then, tell students that there is another web site offering
movies at a cost of $10 per DVD (including S&H). Allow students time to follow the same
steps in analyzing the cost for DVDs: plot data points (if necessary),
draw the line, and write an equation. The graph for this situation
should be drawn on the same set of axes. The equation of the second web site offering movies is: y=10x.
When students graph both lines, the results will look like this:
Students can then compare the results from both situations to answer the following questions:
- How do the graphs differ? [The have different slopes; thus, at first, the original website is more expensive, but over time, the second website is more cost efficient. They also have different y-intercepts.]
- How many movies would you have to rent for the price to be exactly the same at both sites? [6 movies (solve 9x+6=10x for x).]
- What would be the price difference if you bought a dozen movies at each site? [The first website, as stated before, would cost you $114. The second website would cost you: y=10(12), or $120.]
Conclude the lesson by having a whole-class discussion about how these two situations are similar and how they are different.
1. Give students other linear equations with m and b greater than 0 and ask them to assume the context of the problem is the same as in the DVD problems above.
- For each line, ask students to state the y‑intercept and
slope and interpret both in the context of the problem. From these
values, have them generate an equation for the line.
- Ask students to explain the meaning of a y‑intercept of 0 in the context of these problems.
- Ask students to explain the meaning of a slope of 0 in the context of these problems.
- Ask students to find the number of DVDs that they would need to purchase for both websites to be equal in cost.
2. Ask students to come up with two linear equations, and a real world context, where one had a positive slope, while the other had a negative slope. Have students repeat finding the answers to the four bullet points in Assessment Question #1.
- Allow students to write authentic problems to fit given linear equations and to interpret slope and y‑intercept in the context of each problems.
- Allow students to search for similar data on the web,
in catalogs, and elsewhere. Then, have them provide the graphs,
equations, and tables to represent those situations.
Questions for Students
1. How did you find the slope of each equation? What does the slope mean in this problem?
[From the graph of each situation, the slope can be identified by determining the ratio of the vertical change to the horizontal change (that is, rise to run). Additionally, students should realize that the rate of change for this situation is equivalent to the cost per movie. In the first situation, the total cost increases by $9 each time the number of DVD’s increases by 1, so the slope is 9/1 = 9. Similarly for the second situation, the slope is 10/1 = 10.]
2. How did you find the y‑intercept? What does the y‑intercept mean in this problem?
[The y‑intercept is the point at which the line crosses the y‑axis. For the first situation, the y‑intercept is 6, and for the second situation, the y‑intercept is 0. In the context of these problems, the y‑intercept represents the surcharge applied to any order; for the first web site, there is a $6 surcharge, but the second web site has no surcharge.]
- Do students need more practice in interpreting slope and y‑intercept in the context of an authentic problem?
- Do students understand the power of algebra in checking "their friend’s calculation"? Did they rely on the graph?
- Were clear expectations discussed so that students knew
what was expected of them? If not, how can you make expectations more
clear in the future?
- This lesson is very open‑ended. Once given the
scenario, students are expected to make progress by themselves, with
very little guidance. How might this lesson be structured for
low‑ability students to minimize frustration?