Arranging Toothpicks

Kim Seay

EMAT6680

The problem begins by asking students to imagine making a rectangular grid using toothpicks where the grid is filled with squares that have one toothpick on each side. I began the exploration by using GSP to construct rectangular grids that fit this description. Each segment represents a toothpick and each circular point represents the space where two or more toothpicks meet.

The problem then asks students to come up with an expression to find the total number of toothpicks in any such rectangular grid when "a" = the length of the grid in toothpicks and "b" = the width of the grid in toothpicks.

After looking at more rectangular grids, it becomes apparent that this answer can be thought of in terms of the rows and columns of the figure. By definition of the rectangular grid, there will always be one more column than there are toothpicks in the length. Likewise, there will always be one more row than there are toothpicks in the width of the grid. This becomes easy to see when I use yellow to represent the rows in the grid.

 

 

 

 

To put this in terms of "a" and "b"the number of columns in each grid is equal to "a+1" and the number of rows in each grid is equal to "b+1". Now, we have a simple multiplication problem. There are (a+1) columns in each grid and "b" toothpicks in each column. So, the total number of toothpicks in the columns of the grid will be (a+1)(b). The total number of rows in each grid is equal to (b+1) and there are "a" toothpicks in each row. The total number of toothpicks in all of the rows can be found by (b+1)(a). Now we know the total number of toothpicks in the rows of any given figure and the total number of toothpicks in the columns of any figure. To find the total number of toothpicks in the grid, we simply add the rows and columns together:

Total number of toothpicks in figure = (a+1)(b) + (b+1)(a)

This can be shown and tested in a spreadsheet.

Toothpicks wide (A) Toothpicks long (B) B(A+1) A (B+1) Total Number of toothpicks B(A+1) + A(B+1)
1 2 4 3 7
2 4 12 10 22
3 6 24 21 45
4 8 40 36 76
5 10 60 55 115
6 12 84 78 162
7 14 112 105 217
8 16 144 136 280
10 20 220 210 430
11 22 264 253 517
12 24 312 300 612
13 26 364 351 715
14 28 420 406 826
15 30 480 465 945
16 32 544 528 1072
17 34 612 595 1207
18 36 684 666 1350
19 38 760 741 1501
20 40 840 820 1660
21 42 924 903 1827
22 44 1012 990 2002
23 46 1104 1081 2185
24 48 1200 1176 2376
25 50 1300 1275 2575

Conclusion:

I thought this experiment was very practical and beneficial. Students often can tell you that "the rows in the figure are one more than the number of toothpicks wide it is," but they have a hard time putting this in terms of variables. This exploration takes a common situation (most of us have played with toothpicks in this manner before) and turns it into a valuable learning experience. In doing this activity with my own students, I think I will use actual toothpicks. once the students think they have a solution, they can design a spread sheet to help test their theory.

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