Setting the Stage
Consider the following situation:
A ball rolls down an incline. If the mass of the ball and the length and
height of the incline are changed, which of these variables has an effect on
the time it takes the ball to reach the bottom of the incline?
Students may respond with the following variables. You may need to guide the discussion toward these four variables:
 the time it takes the ball to reach the bottom of the incline
 variations in the mass of the ball;
 the length of the incline plane; and
 the height of the incline;
Once these variables have been identified, a sample "experiment"
could be conducted. Roll a ball down a plane. Ask students to observe
what happens. Ask students how they could determine the effect of each
variable.
The Class Activity
The class discussion should now focus on a method for measuring
each variable the class has agreed to use. For example, the students
might suggest that three different values that can be directly
controlled, that is, three different masses, three different lengths,
and three different heights. Next the students can be encouraged to
collect all the data. This process is greatly facilitated by organizing
the class in small cooperative groups of three or four students.
The figure below depicts setting up an incline to vary the height while
holding the mass and length constant. Students might simply stack textbooks to
produce various heights. What is more important is not the precision of the
measures of the heights, lengths, and masses but that at least three distinct
values exist for each of them. Often students do not realize that
three different masses, lengths, and heights mean that twentyseven separate
repetitions of the experiment are needed. Many believe that three experiments
are enough. Frequently the data are collected in a haphazard manner. But at
this
point, it is important that they proceed to collect some data so that they
begin to understand the experiment.
Setting up an experiment to vary only the height of the incline
Ask students how they will collect and organize their data. One
possible method is organizing the data into tables. By asking students
to focus on the quantities that they are varying, the teacher can
negotiate a threebythree twoway table of length against height Then
if the teacher asks for which of the masses this table will be used,
students may suggest a separate table for each of the three masses. If
not, the teacher should assign the first table to one of the masses and
ask what might be done for the second mass.
The figure below illustrates such a set of three twoway tables. Students should
then be able to visualize the extent of the experiment because these tables
contain twentyseven empty cells.
 Length (cm)  Height (cm)  50  75  100  10     20     30     Mass 1
  Length (cm)  Height (cm)  50  75  100  10     20     30     Mass 2
  Length (cm)  Height (cm)  50  75  100  10     20     30     Mass 3

A twoway table for balls with three
different masses
Distribute the Data Tables Activity Sheet to the students.
Data Tables Activity Sheet
You may need to record the above sample tables on the overhead or chalkboard to give students a model.
At this point, the data must be collected, which can usually be accomplished
in one or two additional class periods. Notice that, as yet, the teacher has
not
imposed a question on this situation. If teachers listen to students and give
them time to think of appropriate responses, the questions will often be
supplied by students. One possible approach is to assign for homework only one
question, "Once
your three tables are completely filled with data, and all twentyseven cells are
filled in, can you think of a question that you could use your tables to
answer?" You will return to this question at the end of the lesson.
For example, students might wonder whether the texture of the surface would
affect the time. Others may suggest that patterns might appear in the
data.
The figure below shows the threebythree tables completed using
sample classroom data. Students should be asked to look for patterns in the data
for a specified length for each of the masses. Repeating this process for each
of the three lengths should allow students to conjecture that time and height
are inversely related: as the height increases, the time for each given length
for
each of the masses decreases. Likewise, it is not difficult to elicit a
relationship between length and time by looking at rows rather than at columns.
Students do have difficulty with the question of a possible relationship
between
mass and time. The structure of the tables requires looking at the same cell
entry across all three tables to make an inference about such a relationship.
 Length (cm)  Height (cm)  50  75  100  10  0.98  1.40  1.82  20  0.61  0.96  1.29  30  0.40  0.64  1.06  Mass 1
  Length (cm)  Height (cm)  50  75  100  10  1.03  1.58  1.82  20  0.60  0.92  1.22  30  0.41  0.66  1.04  Mass 2
  Length (cm)  Height (cm)  50  75  100  10  0.93  1.62  1.67  20  0.57  0.96  1.19  30  0.39  0.63  1.07  Mass 3

Sample classroom data presented in a twoway
table for each mass
The tables below highlight one such set of entries. 0.96. 0.92, and 0.96,
respectively. Then, the fact that no such relationship is apparent is somewhat
counterintuitive and difficult for some middle school students to accept. It is
difficult, but not impossible. To quote one sixth grader, "We need to look to
see if they [the corresponding cell entries] are the same or almost the
same. If they are, then we can say that the weight doesn't matter"
[emphasis added].
 Length (cm)  Height (cm)  50  75  100  10  0.98  1.40  1.82  20  0.61  0.96  1.29  30  0.40  0.64  1.06  Mass 1: 14g
  Length (cm)  Height (cm)  50  75  100  10  1.03  1.58  1.82  20  0.60  0.92  1.22  30  0.41  0.66  1.04  Mass 2: 28g
  Length (cm)  Height (cm)  50  75  100  10  0.93  1.62  1.67  20  0.57  0.96  1.19  30  0.39  0.63  1.07  Mass 3: 57g

Highlighting the time for each of the
balls on an
incline of the same length and
height
Note that the highlighted values represent an incline of the
same length (75 cm) and the same height (20 cm), but with varying
masses. You could use this sample data, or actual data collected by the
students, to lead into a discussion of whether or not the mass of the
ball affected the time it took the ball to roll down the incline.
Closing the Activity
Prior to the end of the lesson, return to the question posed earlier in the lesson:
Once your three tables are completely filled with data, and all twentyseven cells are
filled in, can you think of a question that you could use your tables to
answer?
Sample student questions may include:
 Does the mass of the ball affect the speed at which it rolls?
[Students should observe that mass generally does not have an effect on the speed of the ball.]
 Does the height of the incline affect the speed at which the ball rolls?
[Students should observe that the higher the incline, the faster the ball tends to roll.]
 Does the length of the incline affect the speed at which the ball rolls?
[Students should observe that a shorter incline generally produces a faster time.]