Tri-Square Rug Games
Just Count the Pegs P.O.W. – I.M.P. II
All that this Problem of the Week is about is formulas for finding area of a shape on a geometry board depending on the number of boundary pegs (pegs that make the perimeter of the shape) and inside pegs (any pegs inside the shape). The main, and pretty much only goal was to find different formulas to figure out the area of a shape with certain characteristics. The characteristics were depending on the number of outside pegs, the number of inside pegs, and both of them together. I came up with many formulas for many different characteristics of the shape(s). First, I tried to figure out Justin’s Formula and the variations to his formula. Justin’s original formula was that he could find the area of any shape on a geometry board that has no pegs on its interior, but and number of pegs on the boundary. To figure this formula out (and all the others) I made different shapes on the geometry board that fit the criteria, found the area by using the measure button, and recorded my findings down on a table. After finding five examples, I would stop and look at the relationship between the outside pegs, inside pegs, or both, depending of the instructions, and try to find a relationship between those and the area. Once I thought I found the pattern, I would make the formula I thought was correct, plug in all the numbers, or at least two or three of them, and make sure they all came out with the correct area. If they did, I decided that was the correct formula, and if they didn’t, I would repeat the last two steps until I found a/the formula that worked for all of the data I had collected.
Justin’s First Formula (For any shape without interior pegs) Formula
Here’s the table with the data I made and the shapes I made.
Outside Pegs
Area
4
1
6
2
8
3
10
4
12
5
The first variation of Justin’s Formula (exactly one peg on the interior)
Formula
All that this Problem of the Week is about is formulas for finding area of a shape on a geometry board depending on the number of boundary pegs (pegs that make the perimeter of the shape) and inside pegs (any pegs inside the shape). The main, and pretty much only goal was to find different formulas to figure out the area of a shape with certain characteristics. The characteristics were depending on the number of outside pegs, the number of inside pegs, and both of them together. I came up with many formulas for many different characteristics of the shape(s). First, I tried to figure out Justin’s Formula and the variations to his formula. Justin’s original formula was that he could find the area of any shape on a geometry board that has no pegs on its interior, but and number of pegs on the boundary. To figure this formula out (and all the others) I made different shapes on the geometry board that fit the criteria, found the area by using the measure button, and recorded my findings down on a table. After finding five examples, I would stop and look at the relationship between the outside pegs, inside pegs, or both, depending of the instructions, and try to find a relationship between those and the area. Once I thought I found the pattern, I would make the formula I thought was correct, plug in all the numbers, or at least two or three of them, and make sure they all came out with the correct area. If they did, I decided that was the correct formula, and if they didn’t, I would repeat the last two steps until I found a/the formula that worked for all of the data I had collected.
Justin’s First Formula (For any shape without interior pegs) Formula
Here’s the table with the data I made and the shapes I made.
Outside Pegs
Area
4
1
6
2
8
3
10
4
12
5
The first variation of Justin’s Formula (exactly one peg on the interior)
Formula