## Paper Pool Game

The paper pool game provides an opportunity for students to develop their understanding of ratio, proportion, greatest common divisor, and least common multiple.

This is the first part of a four-part project. We recommend that students work on this project with a partner. 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.

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.

To ensure that all students understand how the game is played, launch the Paper Pool game. (Start with a 5 × 3 table.) Alternatively, you can use a transparency of the Introduction to Paper Pool to discuss the rules with your students.

### Try It!

Allow students to play the Paper Pool game with 4 × 2 and 4 × 4 tables. Ask students, "In which pocket does the ball land on each table?"

On each of these tables, allow students to explore the applet by adjusting the controls to the right. 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 when exploring later in this unit. In particular, they will be able to perform more investigations if they increase the speed.

Set up the rest of this unit by saying the following to students:

After designingPaper Pooland playing several games on tables of various sizes, Marisa wondered about two things.In the rest of this unit, you will perform investigations and try to answer her questions.

- 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?

Students can continue their investigations of other Paper Pool tables by proceeding to the next lesson, Explore More Tables, or by completing the Paper Pool Tables Activity Sheet.

Paper Pool Tables Activity Sheet

### Reference

This lesson plan on Paper Pool was adapted with permission and guidance from:

Lappan, Glenda, et al. "Comparing and Scaling: Ratio, Proportion, and Percent," Connected Mathematics Project, Prentice Hall, 2004.

- Computers with internet connection
- Centimeter grid paper
- Handouts
- Colored pencils or markers

**Assessment Option**

After students have explored several Paper Pool tables, be sure to discuss with students the form of assessment using Paper Pool Project assignment sheet or online in the fourth lesson of this unit, Going the Distance.

**Extension**

Move on to the next lesson,

Explore More Tables.

**Teacher Reflection**

- Think
about what you would like students to do and learn during this lesson.
Here are two questions in particular that you should answer:
- Do you want students to be responsible for organizing the data on their own? Or do you want to provide them with help in organizing the data so that they can focus solely on looking for patterns?
- Do you want students to explore the path length of the ball?

- Tailor the project to meet your instructional goals. If you wish to have your students organize data, do not hand out the Paper Pool Record Sheet. Decide if you want students to answer the question of how far the ball travels in terms of diagonal units. Students can use the Extension Record Sheet to explore this question effectively, or they can organize data on their own.

### Explore More Tables

### Look for Patterns

### Going the Distance

### Learning Objectives

Students will:

- Gather and organize data.
- Search for patterns.
- Recognize rectangles with sides in the same ratio (similar rectangles).
- Use the simplest ratio to predict the stopping pocket and the number of hits.

### NCTM Standards and Expectations

- Represent, analyze, and generalize a variety of patterns with tables, graphs, words, and, when possible, symbolic rules.

- Model and solve contextualized problems using various representations, such as graphs, tables, and equations.

- Formulate questions, design studies, and collect data about a characteristic shared by two populations or different characteristics within one population.

- Select, create, and use appropriate graphical representations of data, including histograms, box plots, and scatterplots.

### Common Core State Standards – Mathematics

Grade 6, Ratio & Proportion

- CCSS.Math.Content.6.RP.A.1

Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities. For example, ''The ratio of wings to beaks in the bird house at the zoo was 2:1, because for every 2 wings there was 1 beak.'' ''For every vote candidate A received, candidate C received nearly three votes.''

Grade 8, Stats & Probability

- CCSS.Math.Content.8.SP.A.1

Construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities. Describe patterns such as clustering, outliers, positive or negative association, linear association, and nonlinear association.

Grade 6, The Number System

- CCSS.Math.Content.6.NS.B.4

Find the greatest common factor of two whole numbers less than or equal to 100 and the least common multiple of two whole numbers less than or equal to 12. Use the distributive property to express a sum of two whole numbers 1-100 with a common factor as a multiple of a sum of two whole numbers with no common factor. For example, express 36 + 8 as 4 (9 + 2).

### Common Core State Standards – Practice

- CCSS.Math.Practice.MP8

Look for and express regularity in repeated reasoning.