The Gum Ball Dilemma POW
Problem Statement:
For this problem, we had to find a formula to fit a specific situation. We had to find out how much money a parent might have to spend (for her children) at a gum ball machine, with a certain number of colors of the gum balls, and with however many kids she might have, who theoretically, all want the same color.
For this problem, we had to find a formula to fit a specific situation. We had to find out how much money a parent might have to spend (for her children) at a gum ball machine, with a certain number of colors of the gum balls, and with however many kids she might have, who theoretically, all want the same color.
Process Description:
To figure out this problem, we were given three examples/scenarios of a situation like this. The first one was about Ms. Hernandez and her twins. The gum ball machine that they were at had two colors, white and red. The twins wanted the same color gum ball, and each gum ball cost one cent. The conclusion that the problem asked was why (theoretically), Ms, Hernandez only needed to spend three cents, maximum, to get the same color of gum balls for her twins.
The second scenario that would help us figure out the overall problem was that Ms. Hernandez and her twins passed a gum ball machine with three different colors of gum balls, red, white, and blue, and both of her kids want the same color again. Assuming that these gum balls also cost one penny, what is the maximum amount Ms. Hernandez would have to spend, in order to get the same color gum balls for her children?
The third and final practice problem, was with Mr. Hodges and his triplets. They passed a gum ball machine that had three different colors, and each of the triplets want the same color, like the others. This question asked how much Mr. Hodges might have to spend in total, for all of his kids to be happy with the same color gum balls, from the three-color gum ball machine.
These three 'warm-up questions' helped me figure out the overall problem because they gave me an idea of what I'm going to do in order to find the answer. For this problem, we had to find a formula that would fit any given situation, that would satisfy any number of kids that all want the same color gum ball, from any quantity of gum ball colors a gum ball machine might have. To solve this problem, I played around with a few numbers and variables. After a while, I found out that if you keep a certain variable constant, and then change around some other variables, you could potentially answer this problem. When I tried this with a few different scenarios, and figured out that they all worked, I found my answer. The arranged formulas are presented below.
To figure out this problem, we were given three examples/scenarios of a situation like this. The first one was about Ms. Hernandez and her twins. The gum ball machine that they were at had two colors, white and red. The twins wanted the same color gum ball, and each gum ball cost one cent. The conclusion that the problem asked was why (theoretically), Ms, Hernandez only needed to spend three cents, maximum, to get the same color of gum balls for her twins.
The second scenario that would help us figure out the overall problem was that Ms. Hernandez and her twins passed a gum ball machine with three different colors of gum balls, red, white, and blue, and both of her kids want the same color again. Assuming that these gum balls also cost one penny, what is the maximum amount Ms. Hernandez would have to spend, in order to get the same color gum balls for her children?
The third and final practice problem, was with Mr. Hodges and his triplets. They passed a gum ball machine that had three different colors, and each of the triplets want the same color, like the others. This question asked how much Mr. Hodges might have to spend in total, for all of his kids to be happy with the same color gum balls, from the three-color gum ball machine.
These three 'warm-up questions' helped me figure out the overall problem because they gave me an idea of what I'm going to do in order to find the answer. For this problem, we had to find a formula that would fit any given situation, that would satisfy any number of kids that all want the same color gum ball, from any quantity of gum ball colors a gum ball machine might have. To solve this problem, I played around with a few numbers and variables. After a while, I found out that if you keep a certain variable constant, and then change around some other variables, you could potentially answer this problem. When I tried this with a few different scenarios, and figured out that they all worked, I found my answer. The arranged formulas are presented below.
Solutions:
If you keep the kids constant, and just change the amount of colors, you get the same results as you would keeping the colors constant and changing the number of kids. I got the following formulas:
C= the amount of colors
P= pennies
For 2 kids:
1C + 1 = P
For 3 kids:
2C + 1 = P
For 4 kids:
3C + 1 = P
For 5 kids:
4C + 1 = P
This is all I've gotten so far, but I predict if you go further into this problem, with this same format, it would all work out to get the same answer. Here is an example of how to go through the equations: for 4 kids, you would first multiply 3 times however many colors of gum balls there are, add one to the product, and that should sum up to the amount of pennies needed for each equation.
Self-Assessment and Reflection:
While working through this problem, I found it really interesting and challenging at the same time. I learned that if you really look deep into a problem, and try as many possible solutions that could work, you can eventually get your answer. I found this out during this problem. I tried as many solutions that I could find, and I was about to lose faith and give up, but I managed to motivate myself to keep trying, and I finally got the answer. This was a problem where I used three habits of a mathematician. They were 'be persistent and patient,' 'look for patterns,' and 'conjecture and test.' I used the of 'being persistent and patient' when I was about to lose hope and, finally motivated myself to keep trying. I used the 'look for patterns' when I was trying to solve this problem, because this problem was all about finding a pattern. Once I finally conquered a pattern that worked, I knew that I could say that I used this habit of a mathematician. Finally, I used the 'conjecture and test' while I was solving this problem, because in order to find the right equations that would fit this problem, I had to test my thoughts on this multiple times. For this problem, I would give myself a 10 out of 10, because I continuously tried many, many ways of completing this problem, and I hurt my brain doing it. I had a lot of fun throughout this problem, even though it was challenging at times, as well.
If you keep the kids constant, and just change the amount of colors, you get the same results as you would keeping the colors constant and changing the number of kids. I got the following formulas:
C= the amount of colors
P= pennies
For 2 kids:
1C + 1 = P
For 3 kids:
2C + 1 = P
For 4 kids:
3C + 1 = P
For 5 kids:
4C + 1 = P
This is all I've gotten so far, but I predict if you go further into this problem, with this same format, it would all work out to get the same answer. Here is an example of how to go through the equations: for 4 kids, you would first multiply 3 times however many colors of gum balls there are, add one to the product, and that should sum up to the amount of pennies needed for each equation.
Self-Assessment and Reflection:
While working through this problem, I found it really interesting and challenging at the same time. I learned that if you really look deep into a problem, and try as many possible solutions that could work, you can eventually get your answer. I found this out during this problem. I tried as many solutions that I could find, and I was about to lose faith and give up, but I managed to motivate myself to keep trying, and I finally got the answer. This was a problem where I used three habits of a mathematician. They were 'be persistent and patient,' 'look for patterns,' and 'conjecture and test.' I used the of 'being persistent and patient' when I was about to lose hope and, finally motivated myself to keep trying. I used the 'look for patterns' when I was trying to solve this problem, because this problem was all about finding a pattern. Once I finally conquered a pattern that worked, I knew that I could say that I used this habit of a mathematician. Finally, I used the 'conjecture and test' while I was solving this problem, because in order to find the right equations that would fit this problem, I had to test my thoughts on this multiple times. For this problem, I would give myself a 10 out of 10, because I continuously tried many, many ways of completing this problem, and I hurt my brain doing it. I had a lot of fun throughout this problem, even though it was challenging at times, as well.