Illuminations: Paper Pool: Analyzing Numeric and Geometric Patterns

Paper Pool: Analyzing Numeric and Geometric Patterns

Going the Distance

The interactive paper pool game in this i-Math investigation provides an opportunity for students to further develop their understanding of ratio, proportion, and least common multiple.

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

Materials
 
  • Computer and Internet connection
  • Handouts

Instructional Plan

Below are additional questions that students can investigate for the Paper Pool game.

  1. What kind of symmetry does the path of the ball have for different pool tables?

    2. For what dimensions of pool tables, does the path have a vertical line of reflection?

    For what dimensions of pool tables, does the path have a horizontal line of reflection?

    For what dimensions of pool tables, does the path have a rotational symmetry?

  2. Can you predict the length of the ball's path on any size Paper Pool table? Each time the ball crosses a square, the distance it travels is 1 diagonal unit. For instance, the ball will travel 15 diagonal units on a 5 × 3 table, as shown below. How many diagonal units will a ball travel on its entire journey?

    3. Path Length = 15 diagonal units

Allow students to examine the path of the ball for several pool tables using the Paper Pool Tool or by hand using grid paper.

Paper Pool Tool

Assessment Options
 

After students have thoroughly explored several sizes of Paper Pool tables and have reached some conclusions, write a report on your work. Include the following in your report:

  1. A summary of the rules you found, why you think your rules are correct, and anything else you discovered. You might discuss what you noticed as you examined the paths for different tables and what helped you to arrive at your rules.
     
  2. A drawing of one Paper Pool table for each rule that demonstrates that the rule accurately predicts what will happen.
     
  3. Any tables, charts, drawings or other tools you used to organize your information and look for patterns.
     
  4. An explanation of any other patterns you found or ideas you have about Paper Pool tables and the path of the ball.

NCTM Standards and Expectations
 
Data Analysis & Probability 6-8
  1. Compute probabilities for simple compound events, using such methods as organized lists, tree diagrams, and area models.
  2. Use conjectures to formulate new questions and plan new studies to answer them.
  3. Use observations about differences between two or more samples to make conjectures about the populations from which the samples were taken.

This Paper Pool Table Investigation adapted with permission and guidance from:

Comparing and Scaling: Ratio, Proportion, and Percent, Connected Mathematics Project, G. Lappan, J. Fey, W Fitzgerald, S. Friel and E. Phillips, Dale Seymour Publications, (1998), Paper Pool Project, pp.106-111.

  
1 period   

NCTM Resources

Principles and Standards for School Mathematics
National Council of Teachers of Mathematics Thinkfinity Verizon Foundation
More and Better Mathematics for All Students
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