The first step in exploring linear data is understanding the
data. For some relation there is clearly an independent, or operating,
variable and a dependent, or response, variable — for example, time and
distance. The choice when fitting lines does not always depend on the
physical relation between the operating and response variables. It is
often more important to know which of the two is to be predicted. If,
for example, students investigate the relationship between the
temperature and the number of times a cricket chirps in a given period
of time, there is a linear relationship. Clearly, the temperature
determines to a large extent the rapidity with which crickets chirp,
not the other way around. If the students want to predict the
temperature by counting chirps, however, the better prediction would
come from using the number of chirps as the independent variable and
erature as the dependent variable.
Oil Changes and Engine Repairs
The table below displays data that relate the number or oil
changes per year and the cost of engine repairs. The activity which
follows uses these data to introduce students to modeling with a linear
function. To predict the cost of repairs from the number of oil
changes, use the number of oil changes as the x variable and engine-repair cost as the y variable.
When graphing the data, it is important to ask students how the
axes should be labeled. After a class discussion, it will be decided
that the x‑axis should be labeled "Number of Oil Changes per Year" (with a scale of 0–10), and the y‑axis should be labeled "Engine Repair Cost (in dollars)" (with a scale of 0–700).
Oil Changes and Engine Repair
Line of Best Fit Tool
Use the Line of Best Fit Tool
to graph the data for the class. Project the graph onto the overhead
projector or computer/television monitor. (Alternatively, students may
graph the data using grid paper and pencil.)
The figure below displays the data from the above table
graphically. The students are asked to visualize a straight line as a
representation of the data. Each student should draw a line that seems
to "fit" the plotted points. A line that "fits" the points should have
the same characteristics as the set of points; it should actually
summarize the data. A line drawn this way is called an eyeball-fit line.Oil Changes vs. Engine Repair
Once the graph has been drawn, ask students, "What is the
significance of the line's downward (or negative) slope?" Students will
say it means that the more oil changes made per year, the less money
that needs to be spent on engine repairs. Note that the correct wording
should be, "There appears to be a relation between the number
of oil changes and the money spent on repairs." Although the line
represents the data, there is no indication that the number of oil
changes causes the need for engine repairs. There are many other
variables that affect the cost of engine repairs. More oil changes may
just mean those cars have more careful drivers.
Students should examine the slope, which can be found by counting
units. First they must define what a unit represents for each variable.
In the graph above, one horizontal unit represents one oil change. One
vertical unit represents $100 in engine repairs. Since students have
drawn different lines, the slopes will vary. Note that whereas a
fraction or ratio is easily understood as the slope or rate of change, a decimal representation is more useful to compare slopes. After
listing all the slopes reported by the class (perhaps in a
stem-and-leaf plot), the class should determine a "consensus" slope.
The number should be simple to use. For example, ‑70 is much more
useful than ‑73.07 (the result obtained using the Illuminations Line of Best Fit Tool)
or ‑68.42 (the result obtained by choosing the two points (0,650)
and (9.5,0) from the yellow line on the graph above). Since utility is
more highly valued than precision in this instance, a slope of ‑70
seems to be a reasonable value.
Students should be able to interpret slope as rate of change. Ask
students, "What is the change in the cost of repairs for each oil
change?" A rate of change (or slope) of ‑70 indicates that for each
additional oil change per year, the cost of engine repairs will tend to
decrease by $70. (Students should note that a slope of ‑70 actually
represents ‑70/1, which indicates a decrease of $70 per 1 oil change.)
Associating measurement units with the slope is more important to give
students a concrete basis for understanding. Students should recognize
that changing the units on an axis will affect the slope. If the
vertical axis were in cents rather than dollars, the slope would
Next examine the intercepts. The y-intercept is
about 650. This means that if there are zero oil changes, engine
repairs will cost about $750. From the graph, the x-intercept
is about 9.5, which means that a car owner would expect to spend
nothing on engine repairs if she changed the oil 9.5 times a year. Is
this a sensible number of oil changes per year?
The slope and the y-intercept can be used to write the equation of the line:
Writing y = mx + b for m = ‑70 and b = 650, we get y = ‑70x + 650.
If the y‑intercept is not accessible because of the scale,
the equation of the line can be found by using any two points (not
necessarily data points) on the line. Students can then use the
equation to predict the cost of engine repairs expected for a specific
number of oil changes.
For example, if you change your oil four times a year, how much can you expect to pay in engine repairs?
Let x = 4, then
y = ‑70(4) + 650 = $370.
With four oil changes per year, you can expect to pay about $350 in engine repairs.
Additional points on the graph can be discussed as needed.
Bike Weights and Jump Heights
Distribute the Bike Weights and Jump Heights activity sheet to the students.
Bike Weights and Jump Heights Activity Sheet
Students may work individually or in pairs to complete the activity sheet. Students may plot the graph by hand, on the grid paper provided, or they may use the Line of Best Fit Tool for their graphs.
Answers to the activity sheet are provided below.
Bike Weights and Jump Heights Overhead
- Check student graphs. Alternatively, you may display the Bike Weights and Jump Heights Overhead which has another graph of the same data.
Bike Weight vs. Jump Height
- -0.150; for every 1‑pound increase in weight, the height decreases slightly less than 2/10 of an inch.
- A 21.5‑pound bike would be able to jump about 10 inches.
Weights and Drug Doses
Distribute the Weights And Drug Doses activity sheet. Once again, students may work individually or in groups.
Weights and Drug Doses Activity Sheet
Answers to the activity sheet are provided below.
This problem about prescription medicine illustrates the
importance of slope and reinforces the notion of rate of change. You
may instruct students to draw median-fit lines, eyeball lines, or
regression lines depending on the background of the class.
- Check student graphs.
Check student graphs.
Weights vs. Drug Doses
Weights and Drug Doses Overhead
Alternatively, you may display the Weights and Drug Doses Overhead which has another graph of the same data.
- The slope for usual dosage is about 0.45, and for maximum
dosage is about 0.76. For every pound of increase in weight, you can
increase the usually dosage by 45%, compared to a 76% increase per
pound for maximum dosage.
- The (weight, usual dosage) equation is approximately y = 0.46x. The (weight, maximum dosage) equations is approximately y = 0.76x - 0.05.
- The lines are not parallel because they have different slopes.
- y = 1.67x - 0.46. The ratio of the slopes, the
change in maximum dosage to weight to the change in usual dosage to
weight (0.76 / 0.46), is the slope of the new line. (Weight factors out
of the ratio.)