Problem of the week #3

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**just count the pegs**

For our third Problem of the Week, we focused on the idea of area in relation to geometric parallelograms and quadrilaterals. In the problem we had three students: Freddie Short, Sally Shorter and Frashy Shortest, each of these three students had a formula to find the area of a parallelogram on a Geoboard. Freddie and Sally used specialized formulas that only took into consideration the exterior pegs or the interior pegs, but Frashy used a “superformula” that takes into consideration both the interior and exterior pegs to find the area of the polygon. Your goal in the problem is to find each specialized formula and with the information you found create Frashys “superformula”.

My process began by finding the formulas for Freddie and Sally and then expanding a little on each to see different variations of the formulas. I began with Freddies formula which was specific to a polygon on the Geoboard that contained NO pegs within it. By using an IN & Out table I put in the information I found by creating polygons on a geoboard and found a connection they all had. Using that connection I was able to create a formula that fit all the requirements and was proven by all my experiments.

IN (Pegs on the Exterior) Out (Area)

3 0.5

4 1

5 1.5

6 2

7 2.5

The formula I ultimately created was using this data: X/2 - 1, for X = Exterior Pegs

The first extension I did with this situation was I changed it to finding the formula of a polygon with only one peg on the inside while still creating the formula around the exterior Pegs. I only found one such polygon and the data and formula were as follows:

IN (Exterior Pegs) Out (Area)

8 4

The formula I found for this situation was: X/2, for X = Exterior Pegs

The second extension I did was that I chose a number of pegs larger than one on the inside and found a formula like the last extension that only took into consideration the exterior pegs. I chose that there would be three pegs on the inside, the data and formula were:

IN (Exterior Pegs) Out (Area)

12 8

The formula was: X/2 + 2, for X = Exterior Pegs

The third and final extension was just finding more situations like the second extension.

4 Pegs inside:

IN (Exterior Pegs) Out (Area)

14 10

Formula: X/2 + 3, for X = Exterior Pegs

5 Pegs inside:

IN (Exterior Pegs) Out (Area)

14 11

Formula: X/2 + 4, for X = Exterior Pegs

Then I worked on Sally's formula which was specific to polygons with only FOUR pegs on the exterior AND had at least one peg in the interior. After much searching I only found two polygons that fit all requirements and created an In/Out table to organize the data and created a formula.

IN (Interior Pegs) Out (Area)

1 2

4 5

The formula I ultimately created was: X + 1, for X = Interior Pegs

The first extension for this situation was to pick a number of pegs on the exterior other than four and still create a formula that only took into account the interior pegs.

12 Exterior Pegs:

IN (Interior Pegs) Out (Area)

3 8

Formula: 2x + 2

The second and final extension for this situation was to find more problems like the last extension.

14 Exterior Pegs:

IN (Interior Pegs) Out (Area)

6 12

Formula: 2x, for X = Interior Pegs

14 Exterior Pegs:

IN (Interior Pegs) Out (Area)

5 11

Formula: 2x + 1, for X = Interior Pegs

10 Exterior Pegs:

IN (Interior Pegs) Out (Area)

2 6

Formula: 2x + 2, for X = Interior Pegs

After finding and extending Sally and Freddie's situations and formulas I moved on to finding Frashys “Superformula”. I did this by creating an IN/OUT table that included both interior and exterior pegs. Using this information I used Trial and Error to find the ultimate formula.

X Y

IN.1 (Exterior) IN.2 (Interior) Out (Area)

8 1 4

10 2 6

12 3 8

12 4 9

14 4 10

After going through some variations and different possible equations I found the one equation that proved all the situations true. The Superformula was: y + (X/2) - 1.

Throughout this problem I think the Habits of a Mathematician I used the most were Trial and Error and being Persistent and Patient. I used Trial and Error on paper and in my head while trying to find formulas that fit all requirements and proved all situations I found to be true. Another Habit I used was being Patient and Persistent, in all the situations (especially the “superformula) I went through many different formulas and connections that would work. After it didn’t work I implemented the trial and error to continue my search. Throughout this entire process I had to be persistent and VERY patient to find the formula that would ultimately work.

My process began by finding the formulas for Freddie and Sally and then expanding a little on each to see different variations of the formulas. I began with Freddies formula which was specific to a polygon on the Geoboard that contained NO pegs within it. By using an IN & Out table I put in the information I found by creating polygons on a geoboard and found a connection they all had. Using that connection I was able to create a formula that fit all the requirements and was proven by all my experiments.

IN (Pegs on the Exterior) Out (Area)

3 0.5

4 1

5 1.5

6 2

7 2.5

The formula I ultimately created was using this data: X/2 - 1, for X = Exterior Pegs

The first extension I did with this situation was I changed it to finding the formula of a polygon with only one peg on the inside while still creating the formula around the exterior Pegs. I only found one such polygon and the data and formula were as follows:

IN (Exterior Pegs) Out (Area)

8 4

The formula I found for this situation was: X/2, for X = Exterior Pegs

The second extension I did was that I chose a number of pegs larger than one on the inside and found a formula like the last extension that only took into consideration the exterior pegs. I chose that there would be three pegs on the inside, the data and formula were:

IN (Exterior Pegs) Out (Area)

12 8

The formula was: X/2 + 2, for X = Exterior Pegs

The third and final extension was just finding more situations like the second extension.

4 Pegs inside:

IN (Exterior Pegs) Out (Area)

14 10

Formula: X/2 + 3, for X = Exterior Pegs

5 Pegs inside:

IN (Exterior Pegs) Out (Area)

14 11

Formula: X/2 + 4, for X = Exterior Pegs

Then I worked on Sally's formula which was specific to polygons with only FOUR pegs on the exterior AND had at least one peg in the interior. After much searching I only found two polygons that fit all requirements and created an In/Out table to organize the data and created a formula.

IN (Interior Pegs) Out (Area)

1 2

4 5

The formula I ultimately created was: X + 1, for X = Interior Pegs

The first extension for this situation was to pick a number of pegs on the exterior other than four and still create a formula that only took into account the interior pegs.

12 Exterior Pegs:

IN (Interior Pegs) Out (Area)

3 8

Formula: 2x + 2

The second and final extension for this situation was to find more problems like the last extension.

14 Exterior Pegs:

IN (Interior Pegs) Out (Area)

6 12

Formula: 2x, for X = Interior Pegs

14 Exterior Pegs:

IN (Interior Pegs) Out (Area)

5 11

Formula: 2x + 1, for X = Interior Pegs

10 Exterior Pegs:

IN (Interior Pegs) Out (Area)

2 6

Formula: 2x + 2, for X = Interior Pegs

After finding and extending Sally and Freddie's situations and formulas I moved on to finding Frashys “Superformula”. I did this by creating an IN/OUT table that included both interior and exterior pegs. Using this information I used Trial and Error to find the ultimate formula.

X Y

IN.1 (Exterior) IN.2 (Interior) Out (Area)

8 1 4

10 2 6

12 3 8

12 4 9

14 4 10

After going through some variations and different possible equations I found the one equation that proved all the situations true. The Superformula was: y + (X/2) - 1.

Throughout this problem I think the Habits of a Mathematician I used the most were Trial and Error and being Persistent and Patient. I used Trial and Error on paper and in my head while trying to find formulas that fit all requirements and proved all situations I found to be true. Another Habit I used was being Patient and Persistent, in all the situations (especially the “superformula) I went through many different formulas and connections that would work. After it didn’t work I implemented the trial and error to continue my search. Throughout this entire process I had to be persistent and VERY patient to find the formula that would ultimately work.

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