The Curriculum and Evaluation Standards for School Mathematics (NCTM
1989) calls for the continued study of functions so that all students can-
- model real-world phenomena with a variety of functions;
- represent and analyze relationships using tables, verbal rules, equations,
and graphs;
- translate among tabular, symbolic, and graphical representations of
functions;
- recognize that a variety of problem situations can be modeled by the same
type of function;
- analyze the effects of parameter changes on the graphs of functions. (NCTM
1989, 154)
The curriculum standards also stress the continued study of data analysis
and statistics so that all students can-
- construct and draw inferences from charts, tables, and graphs that
summarize data from real-world situations;
- use curve fitting to predict from data. (NCTM 1989, 167)
The following activities promote these objectives.
Prerequisites: Students should understand the relationships
between various functions and their graphs. Their knowledge should cover
linear, quadratic, higher-degree-polynomial, exponential, and trigonometric
functions. They should also be able to use a graphing calculator with
features that will allow them to edit data, create scatterplots, and draw
functions over those plots.
Directions: The activities will require three to five class
periods to complete with students working in groups of three to four. The
following problem will introduce the activities and allow time for any
necessary instruction on using a graphing calculator. One explanatory
example is usually sufficient for students successfully to use a graphing
calculator for these activities.
Allow time for students to explain why
they chose a particular function
As an example, our class discussed a unit called "How Does Corn Grow?" to
familiarize students with data and graphing. Students planted some
corn. After the seeds sprouted, they measured their growth each
day.
Analyze the collected data.
Day 1 is the first day that sprouts appeared. Each group chose a
sprout and measured its height for five days. The following results from
one group are shown.
| Day |
Height (cm) |
|
1
2
3
4
5 |
1.50
2.59
3.91
6.02
7.11 |
Ask students to enter the data into a graphing calculator, make a
scatterplot of the data on the calculator, and analyze the results.
Before making the scatterplot, students must decide on an
appropriate domain and range so that all their points will appear on the
graph. In this example, we have chosen the day number as the independent
variable (x) and the height of the corn plant as the dependent variable
(y). We will let the x-axis go from -5 to 10 and the
y-axis go from -5 to 15. In so doing, all data will appear on the
screen so that both axes can be seen. After setting the appropriate
domain and range, students should use the calculator to produce a scatterplot
of the data and analyze the results.
Students can discuss how their equations
can predict future function values
Students should decide what type of function the data most
closely represents and write an equation to fit the data. In this
example, the data appear to be linear. The students can make an "eyeball"
fit of the
data by examining the scatterplot and determining a y-intercept and
slope. They might try a y-intercept of 0.05 and a slope of
1.3. The students should use the calculator to draw the equation y
= 1.3x + 0.05 over the scatterplot to check the accuracy of the fit,
which is shown in Figure 1. The
equation appears to
fit the data closely. In many examples, students will try several
equations before they find a close fit.
Figure 1
Allow time for students to explain why they chose a particular
type of function, how they derived their equation, and what restrictions the
problem situation places on the domain and range of the function. In our
example, x and y can be any real number in the algebraic
equation, but in the model, x must be a positive integer and y must be a
positive real number. Also, an upper bound will be necessary for x
and y, since the corn will not continue to grow indefinitely.
The students should also discuss how their equation could be used
to predict future function values. They could use their equation to
predict the height of a corn sprout on future days. If they had actually
grown the corn plants, they could test their predictions by later measuring the
specific sprouts.
The same general procedure that was outlined in the example
problem should be followed for the activity sheets. The student should
collect data, produce a scatterplot, analyze it, choose a function, and write
an
equation. Test the equations by drawing the function over the scatterplot
on the graphing calculator. Class discussion should include why a certain
function type was chosen, whether another function type might also work, how
equations were derived, what restrictions are placed on the domain and range by
the problem situation, and whether the model could be used to predict future
values. Although no one answer is correct for each problem, some
answers may be better than others. Some sample solutions are given.
Answers Sheet
1: Figure 2 graphs the
following data for "EliM&Mination" and sample solution 1.
| Trial Number |
Number Remaining |
|
1
2
3
4
5
6
7
8
9 |
126
72
39
20
10
4
1
1
0 |
Figure
2
Figure 3 gives another view of
data from sample solution 2.
| Trial Number |
Number Remaining |
|
1
2
3
4
5
6
7
8
9 |
125
65
33
15
9
8
6
2
0 |
Figure 3
A possible solution to "The Flaming Function" example graphs the
data presented here (see figure 4).
| Time (sec) |
Height (cm) |
|
0
20
40
60
80
100
120
140
160
180
200
220
240
460
280
300 |
4.6
4.3
4.1
3.9
3.6
3.4
3.3
3.2
3.1
3.0
2.8
2.7
2.5
2.3
2.1
1.9 |
Figure 4
Answers Sheet 2:
Figure 5 graphs these
data for "All Boxed In" sample solution.
| Height (cm) |
Volume (cm 3) |
|
1
2
3
3.5
4
5
6
7
8
9 |
324
512
588
591.4
576
500
384
252
128
36 |
Figure 5
" 'Weather' It's a Function" was calculated
from the normal high
temperature for the Dallas area and was obtained by writing to a local
television weather forecaster; see Figure
6. The average monthly temperature for many cities can be found
in
almanacs. This information reduces the domain to twelve values but still
yields a fairly smooth curve. Be sure the calculator is in the radian
mode to work this problem.
| Day |
Temperature |
|
1
15
32
46
60
74
91
105
121
135
152
196
213
227
244
258
274
288
305
319
335
349 |
55
53
56
59
63
67
72
77
81
84
89
98
99
98
94
90
85
80
72
66
61
58 |
Figure 6
"Water Level" calculations are presented here. See the
resulting graph in Figure 7.
| Time (sec) |
Depth (mm) |
|
0
15
30
45
60
75
90
105
120
135 |
249
219
182
144
110
80
58
36
24
17 |
Figure 7
Discussion and extension activities: The
"EliM&Mination" problem serves as a model of decay. Have students
suggest real-life situations that this problem might model.
The "Flaming Function" is linear. It is interesting to
compare the equation the students obtain from the "eyeball" method with those
from the median-median-line and linear-least-square methods of curve
fitting. The Data Analysis software package from the National Council of
Teachers of Mathematics (1988) is a good source for information about these
methods.
"All Boxed In" presents a cubic model. This problem could
be extended by asking students to find the length plus girth and the surface
area of each box in addition to its volume. The problem would then
include a linear, quadratic, and cubic model. The post office uses the
concept of girth to limit the size of boxes sent in the mail (length + girth
<= 108
inches). Ask students to find the box with the greatest volume for the
least surface area that would meet postal regulations.
The problem " 'Weather' It's a Function"
produces a sine wave. The students can use this model to predict
temperatures and then check to see how close their predictions come to the
actual members.
The "Water Level" problem yields data for a quadratic
model. An interesting activity would be to have different groups use
different-sized jugs and compare the curves obtained from the data.
If computers are available, students may want to print their
graphs rather than draw them. The figures illustrated in this article
were created using The Mathematics Exploration Toolkit (IBM 1988).
Another
source of software is Data Analysis (NCTM 1988), which allows students to print
tables and graphs. The software also includes many algorithms to fit
curves to data sets.
