## Inclined Plane

• Lesson
6-8
4

Students time balls rolling down inclines of varying lengths and heights, then try to make inferences about the relationship among the variables involved. Students decide which variables are important (and can be measured), how best to collect the data, and how to interpret the data. This lesson plan is adapated from an article by Thomas Edwards, which appeared in the November - December 1995 edition of Mathematics Teaching in the Middle School.

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 twenty-seven 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 three-by-three two-way 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 two-way tables. Students should then be able to visualize the extent of the experiment because these tables contain twenty-seven empty cells.

 Length (cm) Height (cm) 50 75 100 10 20 30
 Length (cm) Height (cm) 50 75 100 10 20 30
 Length (cm) Height (cm) 50 75 100 10 20 30
A two-way table for balls with three different masses

Distribute the Data Tables activity sheet to the students.

 Data Tables
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 twenty-seven 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 three-by-three 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
 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
 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
Sample classroom data presented in a two-way 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
 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
 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
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 twenty-seven 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.]

For each group of students:

Extensions

A discussion of the relationship between the mass of the ball and gravity will most likely follow. Students may wish to research Newton and his work with gravity. One possible website for research is located at the Mathematician Biography Index.

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### Learning Objectives

Students will:

• Apply appropriate techniques, tools, and formulas to determine measurements
• Organize and consolidate their mathematical thinking through communication
• Develop and evaluate inferences and predictions that are based on data
• Recognize and apply mathematics in contexts outside of mathematics