Introduction
During this week of Inspirational maths we learned about the brain, did 4 insightful problems, and collaborated with our peers. To start off each day we watched a video to inspire us/help us for our problem of the day. The video series basically consisted of helping understand that math is someone anyone can learn and that nobody is born a math person, at least that is what I got out of it. The video that really spoke to me the most was that we learn the most when we make mistakes in math. This spoke to me because in maths you can make mistakes over and over again and you can feel discouraged. But I learned from this video that you actually learn from making mistakes. It is really interesting because your brain actually grows, It fires a synapse. That same week in the ‘Stairs to Squares’ problem I actually thought of this because my equation was not working. I later found out why and this made me excited because while I was struggling, I was learning. Another Video we watches was about how nobody is born to be better at math or to not be good at math. This was encouraging to me because it shows that no matter what problem you have you can work through it. Throughout the week we also did four problems. Tiling a 11x13 rectangle, Squares to Stairs, Hailstone Sequences, and the Painted Cube. These problems were meant to make us think, not just process math. We also talked about these problems with a group, and with our class, this was interesting to me because you could hear other methods of solving this problem to get the same answer.
Squares to Stairs
The Squares to Stairs problem was the most interesting to me but also the most challenging. The problem consisted of a pattern that seemed to be growing. We were given 4 figures and questions to answer about them. (1) What does figure 10 look like and how many squares does it have? (2) What does figure 55 look like and how many squares does it have? (3) Can you use 190 squares to make a stair like structure? Justify your thinking with different representations visually, numerically, algebraically. These problems seemed overwhelming at first. The 4 Figures are shown below.
At first I saw the pattern growing diagonally and I still see it this way. See example below. Others saw it like it was Tetris, or adding some horizontally and vertically.
The first question states “How does figure 10 look like and how many squares does it have?” I could just draw up to figure 10 but I wanted to find a pattern mathematically. I put the number of squares into an X,Y table to try to find a pattern through math.
After looking at these numbers to try to find a relationship I found a pattern but I was not sure how to describe it in math. Here is the first thing that I found.
I found a pattern. The previous number of squares plus the next figure number will give you the how many squares. For example Figure 2 has 3 squares, and we want to find how many squares figure three will have so we add 3+3=6. There are 6 squares in the 3rd figure. This is a pattern but it will not let me just give it any number, it will take a lot of work. After looking at the figures I found that the base of any staircase is the figure number and every one afterwards is just one less. So figure 5 would be 1+2+3+4+5 which equals 15. Figure 6 would be 1+2+3+4+5+6 so it would be 21. This also lines up with the first pattern that I found. I used this to answer question number 1, I added 1+2+3+4… till I got to 10 and got 55. This is a great approach to solve small staircases with small bases but doing this for problem number 2 ‘What does figure 55 look like and how many squares does it have?’ would take a lot of work. I wanted to find an equation to try to solve this for me. After looking at the figures for the 3rd time I realized that they almost look like half of a square. So my first Idea was to square the base number(the figure number) and divide it by 2 to get the total number of squares. I did this with countless figures and found that they would usually get close to the number of squares but never the same amount. I further released by drawing it out that they are not exactly half of a square, the figure previous would make it a full square. So I added the figure number to account/make it exactly half a square. My final equation is as seen below.
The final question, number 3 states ‘Can you use 190 squares to make a stair like structure? Justify your thinking with different representations visually, numerically, algebraically.’ I knew that I could just plug numbers into my equation until I hit 190 or go over it but I wanted to challenge myself and do it algebraically. To do this I plugged in 190 for blocks and solved my equation. I simplified the equation down to figure#squared+figure#=380. I got stuck at this point because I did not know how to surpass this point using algebra. I started to plug in numbers for figure# and I got 19. I checked my answer using my formula and the other patterns I stated before.
Reflection
During this week I think I put all of my effort into the work but not collaborating and helping others. By helping others I can learn even further. I need to share out with the class more and further throughout the year.