This is the first part of a multi-part project. Students will investigate several questions related to the Paper Pool game, and at the end of the project, they will be required to write a report that explains their findings.
In this lesson, students will learn the rules of the Paper Pool game and collect data. In subsequent lessons, they will continue the investigation, and they can draft their reports outside of class. (Note: Depending on the level of your students and the technology available to your class, you may wish combine this lesson with the second lesson of this unit, How Many Hits?.)
Introduce the Paper Pool game to your class with the following description:
| Marisa created a game called Paper Pool. Her pool tables were rectangles drawn on grid paper. The pockets at each corner were labeled A (lower left), B (lower right), C (upper right), and D (upper left). Marisa described each table by its size, giving the horizontal length first and the vertical height second. The figure below shows a 6 × 4 table.
How to Play Paper Pool
- The lower-left corner is always corner A, and the labeling continues counterclockwise with B, C, and D.
- The ball always starts in corner A.
- The ball is hit with an imaginary cue (a stick for hitting a pool ball) so that it travels at a 45° diagonal across the grid.
- If the ball hits a side of the table, it bounces off at a 45° angle and continues its travel.
- The ball continues to travel until it hits a pocket.
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To ensure that all students understand how the game is played, launch the interactive version of the Paper Pool game. When the tool first opens, it will show a 5 × 3 table. (Alternatively, you can show a demonstration on the overhead projector using a transparency with grid lines. Mark off a 5 × 3 rectangle, label the corners, and show the path of the ball with a marker.) This is a good first table to show students as an example—it uses small numbers yet gives a result that is not obvious.
After showing a demonstration with a 5 × 3 table, ask the class, "How many hits occurred before the ball reached a pocket?" Students should count the number of hits in the demonstration. (Note that some students may count only six hits, while other students may count seven or eight hits. There may be some discussion as to whether or not to count the first hit (when the ball is struck at corner A) or the last hit (when the ball reaches a pocket). Explain that both of those hits should be counted. As students continue their investigation, some patterns will be more obvious if the number of hits is counted in this way.)
Adjust the sliders on the table, and demonstrate 5 × 2 and 5 × 4 tables. Based on these three examples, students may begin to suspect that the total number of hits is equal to the sum of the horizontal length and the vertical height (for instance, the number of hits on a 5 × 2 table is 5 + 2 = 7, and the number of hits on a 5 × 4 table is 5 + 4 = 9.)
Therefore, ask students, "How many hits do you think will occur on a 5 × 5 table?" Following the pattern, students will likely guess, "10 hits." When you show them a demonstration with a 5 × 5 table, they will be surprised that only 2 hits occur.
Rather than just seeing demonstrations, it is important for students to play a few games on their own. Distribute the Paper Pool Introduction activity sheet, and allow students to work in pairs or groups of three to explore the tables given on the sheet.
When students complete the two tables on this sheet, conduct a brief discussion about what they've observed. Ask students:
- What observations have you made?
- Does there seem to be a pattern for the number of hits? Do you think that there is a rule that you could use to predict the number of hits?
- How does the size of a table affect the pocket in which the ball lands, the number of hits, and the path that the ball travels?
Explain to students that they will attempt to answer these questions and many others as they continue to explore the Paper Pool game.
Allow students to continue their investigations of other Paper Pool tables by completing the More Tables activity sheet. Students will want to save their work, so that they can refer to it in future lessons. (At this point, you may wish to proceed directly to the second lesson, How Many Hits?, and allow students to use technology to explore tables with the Paper Pool Tool instead of completing the More Tables activity sheet by hand.)
As students work on the More Tables activity sheet, circulate and talk with them. Be sure to visit each group to make sure that they understand how the game is played and that they are drawing the paths accurately. Offer assistance as necessary.
