Paper Pool

The Navigations book argues that, "...problems that can be solved by using tables, graphs, verbal descriptions, concrete or pictorial representations, or algebraic symbols offer opportunities for students to build their understanding of mathematical functions (NCTM, 2001)."

Students have already considered four of these five representations for the Paper Pool game. In the first lesson of this unit, Learn The Game, students learned about the game of Paper Pool by considering a verbal description of the game as well as a picture that explained it. In the second lesson, How Many Hits?, students organized the data in a table and discovered an algebraic rule for the number of hits.

In this lesson, students will consider graphical representations for the number of hits.

Ask students, "How could you graph the information you collected about the Paper Pool game and the number of hits on each table?" Students may have a difficult time answering this question, because there are three variables: the Length of the table, the Height of the table, and the Number of Hits. However, it's possible to graph the data if one of the variables is held constant. For instance, what does the graph look like if you only consider tables with a length of 5?

If the Excel chart is sorted by Length, then by Height, the following set of data might result for tables with length 5:

When considering this table of values, students may notice a partial pattern in the Number of Hits column. For the most part, the number of hits increases by 1 each row, but there is an anomaly in each row where the height is a multiple of 5. The pattern is even more obvious when students consider this information represented in a graph:

Discuss this graph with students. What title would they give this graph, and what labels would they use? [The title could be, "Paper Pool Tables of Length 5." The horizontal axis should be labeled, "Height of Table," and the vertical axis should be labeled, "Number of Hits."] Allow students to make some observations. They may notice that it appears to have two sets of "lines." The first, very obvious set of linear points passes through (1,6), (2,7), (3,8), ..., and has a slope of 1. A less obvious set of linear points passes through (5,2), (10,3), (15,4), ..., and has a slope of 1/5.

Ask students to explain each "line" that occurs in the graph. [The lower line represents the tables whose length and height have a common factor. The upper line represents all other tables, for which the length and height are relatively prime.]

Ask students to consider all tables with length of 18. Have them use the Paper Pool Tool to investigate. They should record the results in the Excel Spreadsheet or on the Paper Pool Record Sheet. By looking only at a table of values, ask them to predict what a graph of the data would look like. Then, have them graph and interpret the results.

You might want to display the following PowerPoint slide for students to consider (save this file to your computer by right-clicking and choosing "Save Target As..."):

Tables of Length 18 [PowerPoint]

(If you do not have access to PowerPoint, click here to save or view a printable image of this file.)

Discuss this graph as a class, and allow students to suggest what each line represents. [The bottom line passes through the points for heights of 18, 36, 54, ..., and it has a slope of 1/18. This represents the tables whose heights are multiples of 18. The line just above the bottom line passes through the points for heights of 9, 27, 36, ..., and its slope is 1/9. This line represents the tables whose heights are multiples of 9 but not multiples of 18. The other lines correspond to the other factors of 18: 6, 3, 2, and 1.]