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Arranging Toothpicks

Problem Statement
Toothpicks are used to build a rectangular grid that is 20 toothpicks long and 10 toothpicks wide. The grid is filled with squares that have 1 toothpick on each side. What is the total number of toothpicks used?
If a represents the number of toothpicks in the length of a grid and b represents the number of toothpicks in the width of a grid (again, the grid is filled with squares that have 1 toothpick on each side), write an expression representing the total number of toothpicks in any rectangular grid of this sort.

Problem setup

There will be some sort of an equation (“expression”) to figure out how many toothpicks would be used to make the entire 20 toothpick X 10 toothpick rectangle.

Plans to Solve/Investigate the Problem

My first thought was to draw a grid with half the number of squares as the problem, then double the number of toothpicks.  Then I noticed that I am to write an equation with “a” representing the number of toothpicks in the length, and “b” representing the number of toothpicks in the width.

Investigation/Exploration of the Problem

I started by drawing a grid that was 10 toothpicks X 5 toothpicks.  Unfortunately, I don’t know how to draw this on the computer.  I filled in the entire grid, and discovered that a grid of 10 X 5 toothpicks would use 115 toothpicks.  I had thought it would be less than that.  I came up with an equation that I thought would work: 2*a + 2*b + ((b-1)*(a-1))

(2*10) + (2*5) + (4*9)

20 + 10 + 36 = 66

The only problem was that my equation didn’t work.  So I tried again.  I looked at my grid and saw that the first part of my equation was fine; it measure the perimeter of the grid.  But the inside of my grid was actually better figured by ((b-1)*a) + ((a-1)*b)  Then my equation looked like this:

2*a + 2*b + ((b-1)*a) + ((a-1)*b)

(2*10) + (2*5) +((5-1)*10) + ((10-1)*5)

20 + 10 + (4*10) + (9*5)

30 + 40 + 45 = 115

But this was a grid half the size of the one in the problem.  Because I didn’t want to draw out the entire grid of the problem and count the toothpicks, I decided to test my equation with another smaller grid; 4 X 5:  (2*5) + (2*4) + ((5-1)*4) + ((4-1)*5)

10 + 8 + (4*4) + (3*5)

18 + 16 + 15 = 49

This was the same number of sides I had counted in my smaller grid, so the equation I found works.

To solve for the problem then:

(2*20) + (2*10) + ((20-1)*10) + ((10-1)*20)

40 + 20 + (19*10) + (9*20)

60 + 190 + 180 = 430 toothpicks

Meg Ramsey
mramsey@rockdale.k12.ga.us

Link(s) to resources, references, lesson plans, and/or other materials