## Analyzing Numeric and Geometric Patterns of Paper Pool

- Lesson

This mathematics excursion is based on the Paper Pool Project from the *Comparing and Scaling* unit of the Connected Mathematics Project, G. Lappan, J. Fey, W Fitzgerald, S. Friel and E. Phillips, Dale Seymour Publications, (1998), Paper Pool Project, pp.106-111.

### Introduction

Paper Pool is an application involving many of the concepts encountered throughout grades 6 and 7: factors, multiples, rectangles, the relation of being relatively prime. Before seeing how to apply these concepts, students must gather and organize data, then search for patterns.

Paper Pool is played with an imaginary ball being hit from the lower left-hand corner marked A, at a 45° angle. A ball hit in this way will bounce off the sides at a 45° angle. Also, if a grid is placed on the table, the ball always traverses on diagonals of the squares of the grid. For example, the illustration below shows the path of a ball on a 5 x 3 table. The ball ends up in pocket C; there are 8 hits and 15 squares are crossed.

Students first learn how to predict the pocket into which a ball will fall and the number of hits as the ball crosses the table. Students further develop their analytical skills by investigating the number of squares crossed. All three relations depend upon the lengths of the sides of the pool table.

**Using the Excursion**

This Internet Mathematics Excursion is a brief mathematics activity. To maximize student learning, certain prerequisites are necessary to use this activity. Thus, it would be appropriate to include this activity as part of a more fully developed Standards-based lesson, but it should not be used as a complete stand-alone lesson.

Another series of 5 lesson plans exist, which explore this interactive in much more detail.

**Conducting the Excursion**

We recommend that students work on this project with a partner. Each student or pair will need graph paper, handouts, and colored pencils or markers.

- One class period will be needed for pairs to collect their data. They can continue to investigate the task and draft their reports outside of class.
- Part of a second class period could be used for comparing results and finalizing reports. You may want to have pairs or individuals share their results in a class summary of the project. If the extension question was given as an extra challenge, be sure to ask any pair who attempted it to share their answers.

Use a following activity sheet or the Interactive Pool Table to introduce the game.

Have students investigate other Paper Pool tables using either the on-line tools or the Paper Pool Table handouts.

Record Sheet (optional)

After students have explored a few more Paper Pool tables, be sure to discuss with students the form of assessment using Project Assignment handout.

Tailor the project to meet your needs: Decide the extent of the project you wish students to investigate. If you wish to see how your students organize data, do not hand out the record sheet which helps students to organize their data. Decide if you want students to answer the question of how far the ball travels in diagonal units. Students will need to create a new record sheet to explore this question effectively.

**Resources for Teachers**

- Graph Paper
- Introduction to Paper Pool
- Paper Pool Tables
- Project Assignment
- Sample Rubric
- Colored pencils or markers
- Record Sheet (optional)
- Computers with internet access (optional)
- Advanced Paper Pool (extension option)
- How Many Squares are Crossed? (extension option)

**Assessment Options**

Use students' activity sheets to assess whether or not students have mastered the objectives and standards listed in the lesson plan.

**Extension Options**

Use the following activity sheets to continue exploring the game.

**Questions for Students**

Refer to the activity sheets for questions for students.

### Paper Pool

Develop students' understanding of ratio, proportion, greatest common factor and least common multiple.

### Learning Objectives

Students will be able to:

- Recognize rectangles whose sides have the same ratio (similar rectangles).
- Use the concept of common factor to find the rectangle with the smallest area having a given ratio of sides.
- Practice gathering and organizing data and looking for patterns.
- Use the concept of common factor (simplest ratio) and least common multiple to predict the behavior of a ball on a Paper Pool table: final corner, number of hits, length of path.

### NCTM Standards and Expectations

- Understand relationships among the angles, side lengths, perimeters, areas, and volumes of similar objects.

- Recognize and apply geometric ideas and relationships in areas outside the mathematics classroom, such as art, science, and everyday life.