## Free-Think Friday Problems:

## Fibonacci Fun

Math Lab Write Up: Fibonacci Fun

When we first looked upon this problem we had no idea what we were looking at. Their was four formations that all started from one right triangle, and then each formation it looked as if more and more of the original triangle was being taken out by smaller triangles and above each formation it looked as if their was a different equation above all of them, the first had n= 0, the second n= 1, the third had n= 2, and the last was n= 3. Starting this problem we looked at each transition of every triangle and it just seemed so confusing because each triangle had a triangle of the same shape taken out of the bigger triangle, and these triangles got smaller with more as the problem went on, it looked like random pieces of smaller right triangles were being taken out of a bigger right triangle. We observed it for a little and we began to try and identify what's happening, we presumed that “n” was the number of triangles being taken out, and then we started building on theories that each triangle being taken out was half of one other half of a bigger triangle, and this was consistently happening every formation. But we looked closer into it and then we thought what if their were triangles actually being added, so we looked at it closely and we thought, what if the first triangle was getting smaller and then are adding more triangle, and we started to realize that each formation that was being made was being rotated 60 degrees and then the previous formation that was made was being added to the next, and it was getting smaller, while at the same time more formations were being added, and this gave the illusion that these triangles were actually being taken out, but weren't completely sure if this was what it was actually doing, but we soon started to think that maybe it wasn't so much that these triangles were being added in and the original formation was just getting smaller, rather that sections were just being divided out. We looked at each formation and it appeared that the section of the triangle being divided out was from the smallest triangle or triangles that looked like they were added in, and from this we had determined that a fourth of the sections were being divided out as the formation progressed, and our final equation we had finalized was n/4.

Math Lab Write Up: Fibonacci Fun

When we first looked upon this problem we had no idea what we were looking at. Their was four formations that all started from one right triangle, and then each formation it looked as if more and more of the original triangle was being taken out by smaller triangles and above each formation it looked as if their was a different equation above all of them, the first had n= 0, the second n= 1, the third had n= 2, and the last was n= 3. Starting this problem we looked at each transition of every triangle and it just seemed so confusing because each triangle had a triangle of the same shape taken out of the bigger triangle, and these triangles got smaller with more as the problem went on, it looked like random pieces of smaller right triangles were being taken out of a bigger right triangle. We observed it for a little and we began to try and identify what's happening, we presumed that “n” was the number of triangles being taken out, and then we started building on theories that each triangle being taken out was half of one other half of a bigger triangle, and this was consistently happening every formation. But we looked closer into it and then we thought what if their were triangles actually being added, so we looked at it closely and we thought, what if the first triangle was getting smaller and then are adding more triangle, and we started to realize that each formation that was being made was being rotated 60 degrees and then the previous formation that was made was being added to the next, and it was getting smaller, while at the same time more formations were being added, and this gave the illusion that these triangles were actually being taken out, but weren't completely sure if this was what it was actually doing, but we soon started to think that maybe it wasn't so much that these triangles were being added in and the original formation was just getting smaller, rather that sections were just being divided out. We looked at each formation and it appeared that the section of the triangle being divided out was from the smallest triangle or triangles that looked like they were added in, and from this we had determined that a fourth of the sections were being divided out as the formation progressed, and our final equation we had finalized was n/4.