THE TRIANGLE PROBLEM
WHAT MY CLASS TOOK AWAY FROM THE VIDEO
- How many toothpicks were used in total?
- How many triangles are in the base of the overall shape?
- What is the minimum number of triangles you could possibly make? What is the maximum?
- How many total rows?
- Is there some kind of formula that we can use to figure out a) the total number of triangles or b) the total number of triangles per row?
- Is there a pattern or sequence to this problem?
THE PROBLEM
After we watched the video and went through a short Q & A session as a class, we were told that there were a total of 250 toothpicks that could be used to make a large triangle structure that consisted of rows made of triangles. As a group, we decided to pursue the following questions: How many perfect rows can we make with 250 toothpicks? How many triangles are in the whole structure? How many triangles are in each row? Can we find a pattern that we can use to make a formula?
MY PROCESS
At first, we all took about 15 to 20 minutes to work on our own individual approach to the problem. As the rest of my group worked away at trying to immediately find a formula, I took my time to draw out part of the structure and start to make a chart.
Once I finished the structure, I started to count out how many triangles, toothpicks, and rows were in the first three layers. I used that information to start my chart. In the chart, I collected information that would help me determine a pattern that would eventually make it a lot easier to know how many toothpicks and triangles would be in the following rows. We used this data to plug into multiple equations and formulas that the other members in my group were working on.
Once I noticed that there was a definite pattern, I made a separate chart that clearly showed how many triangles were added to each new row. It helped my group find numbers to plug into a formula that we were working on online. This formula made us believe that the pattern was a parabolic equation, but we were quickly proven wrong when the graph did not give us the exact coordinates that we had already come up with.
In the end, we were not able to come up with a definite equation that could give how many triangles were added to the shape as we added each row. However, we did find a pattern, and all of the data needed, to do so. The only problem was that we did not know how to translate all of our information into a sequence or equation. Other than that, I think that we did a really good job at combining all of our ideas and feeding off of each other's opinions to come up with as close of an answer as we could. We really had a lot of fun experimenting with all of the different ways to solve this problem, and surprisingly, were all fascinated by the way that the problem would leave us stumped or at a dead end. It was especially hard to find new ways to come up with ways to communicate our thoughts and transfer them into a formula, but that was all part of the learning process. Perhaps further into the year, if we are given a problem similar to this one again, we will be able to get through it with ease because of what we learned while attempting to do this problem.