In the first lesson of this unit, students gathered some preliminary data about the game of Paper Pool. In this lesson, students will gather more data using the
Paper Pool Tool, organize the data using a table or spreadsheet, and discover a rule for the number of hits that occur. (Although students will collect data regarding the pocket in which the ball lands, discovering a formula to predict the pocket will be covered in the fourth lesson of the unit,
Which Pocket?.)
Remind students of the Paper Pool game that they began to investigate in the previous lesson. Set up the rest of the unit by saying the following to students:
After designing Paper Pool and playing several games on tables of various sizes, Marisa wondered about two things.
- Is there a way to predict the pocket at which the ball will stop?
- Is there a way to figure out how many hits will occur?
In the rest of this unit, you will perform investigations so that you are able to answer her questions.
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Ask students to pull out the More Tables activity sheet that they completed in the Learn The Game lesson.
Ask students to identify any patterns that they notice on the More Tables activity sheet. Allow students to point out a few findings, then ask, "How could you organize your results to make it easier to identify patterns?" Allow students to suggest ideas, such as making a chart or using a spreadsheet.
You can distribute the Paper Pool Record Sheet for students to use to organize the results. A few tables have already been listed.
Alternatively, students can use a spreadsheet program to organize the data. One possibility for using a spreadsheet is shown in the Paper Pool Excel Spreadsheet below. After a number of rows have been filled in, students can sort the data by Length, Height, Hits, or Pocket, which may make it easier to identify patterns. (To save this file to your computer, right click and choose "Save Target As....")
The data that students collect from the Paper Pool Introduction and the More Tables activity sheets will be a good start, but students may wish to investigate tables of other sizes. To do so, allow students to play with the Paper Pool Tool, an interactive version of the Paper Pool game.
Allow students to explore the applet by adjusting the controls along the right side. Have students change the speed and check or uncheck some of the boxes. Although you do not want to devote a large amount of time to teaching students how to use this tool, it is important that they understand the features. The better students understand this tool, the more efficient they will be while exploring. Most notably, they will be able to investigate more tables in less time if they increase the speed at which the ball travels.
Once students are familiar with the tool, allow them to invetigate as many tables as they like. The Paper Pool Tool will allow them to investigate tables up to 21 × 21. They should continue to record the results of these investigations on the Paper Pool Record Sheet or in the Paper Pool Excel Spreadsheet.
The table below, though not exhaustive, shows the results for many games of Paper Pool in which the number of hits is 11 or fewer:
| Pocket Where The Ball Stops |
Number of Hits (including Start and Finish)
|
| 2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
10 |
11 |
| B |
|
2 × 1
4 × 2
6 × 3
8 × 4 |
|
2 × 3
4 × 1
4 × 6 |
|
2 × 5
4 × 3
6 × 1 |
|
2 × 7
4 × 5
8 × 10 |
|
2 × 9
4 × 7
6 × 5
8 × 3
10 × 1
|
| C |
1 × 1
2 × 2
3 × 3
4 × 4
|
|
1 × 3
3 × 1
2 × 6 |
|
1 × 5
5 × 1
10 × 2 |
|
1 × 7
5 × 3
3 × 5
7 × 1
6 × 10 |
|
1 × 9
3 × 7
7 × 3
9 × 1 |
|
| D |
|
1 × 2
2 × 4
3 × 6
4 × 8 |
|
1 × 4
3 × 2
6 × 4 |
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1 × 6
3 × 4
5 × 2 |
|
5 × 4
7 × 2
10 × 8 |
|
1 × 10
3 × 8
5 × 6
7 × 4
9 × 2
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After giving students time to explore, gather the class for a group discussion. Ask each group to offer one observation that they made during their exploration. (Be sure to solicit at least one observation from every group before getting a second observation from any group, to ensure that all students have a chance to participate.) On the overhead projector or chalkboard, record student observations. Based on the results in the table above, some of the things that students may notice include:
- If the number of hits is even, the ball lands in pocket C.
- If the number of hits is odd, the ball lands in either pocker B or D.
- The entries in row B are same as the entries in row D, but the dimensions are reversed.
- Tables with proportional dimensions have the same number of hits. For example, a 4 × 2 table has dimensions that are exactly double a 2 × 1 table, and the number of hits on both tables is 3.
- If the dimensions of two tables are proportional, the ball will land in the same pocket.
- If both dimensions are odd (but not equal), the number of hits is equal to the sum of the dimensions. For instance, on a 3 × 5 table, a total of 3 + 5 = 8 hits occur.
Ask students, "Based on the observations you made, can you determine a rule that will allow you to predict the number of hits if you know the dimensions of the table?" Students should realize that each dimension needs to be divided by the greatest common factor before predicting the number of hits. They may state that the dimensions of the table should be reduced to lowest terms, and then the number of hits is equal to the sum of the reduced length and reduced height. For instance, both dimensions of a 4 × 10 table should be reduced by the greatest common factor of 2 to give dimensions of 2 × 5; then, the number of hits is 2 + 5 = 7.
Symbolically, students may state the rule for an l × h table as follows:
Number of Hits = (l + h) ÷ GCF(l, h)
Discovering this rule on their own is very important. It is the essence of problem solving. Allow students to struggle as they search for a rule, and avoid offering too many hints. If necessary, allow students to ponder the question overnight. In fact, students who are unable to find a rule quickly should continue to reflect on the question throughout the unit. They should be pushed to record all of their observations in their final report, even if they never identify a rule for the number of hits.
