Create 2 formulas, one that will calculate the last number in terms of the first number and a constant increase in rate as well as the total amount of numbers. The second formula will add ass of the resulting numbers from the first formula together after the last number is calculated. Process: Kevin’s Decisions: In order to put the problem into perspective, I first set up my own possible variables for the first platform height, the difference in height between each platform, and the total number of platforms.
I came up with the numbers for each variable respectively: 6, 3, and 3. The first platform is 6 feet tall. There are 3 platforms. The distance between each platform is 3 feet. The second platform is 6+3 feet tall or 9 feet, the third platform is 9+3 feet or 12 feet. I tried to find a formula for the height of the tallest platform that works. What I had to do, to find the height of the tallest platform, was first to find out how tall the first platform was. Since we don’t know how tall the first platform is, I substituted it for the variable f.
Next we had to determine the difference in height between each platform, which I substituted as d, and multiply that by the total number of platforms because this will show the total increase in height from the first platform to the last platform. However, I had to subtract one from the total number of platforms because I already used the first platform as the starting height in feet for increasing the height from platform to platform. I substituted this for (x-1).
Once I found the total difference in height from the first platform to the last platform, I just added that to the height of the first platform to get the height of the last platform. The formula I came up with was (f+(x-1)d)=1 where f=the height of the first platform, x=number of platforms, and d=the distance between each platform. I checked to see if my formula worked using the numbers from earlier. (f+(x-1)d)=1 (6+(3-1)3)=12 6+(23)=12 6+6=12 12=12 ? Camilla’s Dilemma: In order to find out the total length of fabric that Camilla needs for the platforms, I had to develop a formula that gave us the total height of all of the platforms.
To find out the total height we could add the height of all of the platforms. However, the total height can also be determined by a formula since the increase in height from platform to platform is the same. What I saw was that there are pairs within the heights of the first platform and last platform that when added result to the same number. For example: If there are 5 platforms in total, the first platform is 4 feet and the increase in height is 2 feet, the height of the platforms in order are: 4, 6, 8, 10, 12.
The height of the last platform is 12 and the height of the first platform is 4. When we add 4 and 12 the result is 16 and divided by 2 is 8. The 2nd platform is 6 feet and the 4rth platform is 10 feet When added together and divided by 2 to get the average we get 8. The platform is the average number, which is 8. I saw this and put it into a formula. I took the height of the first platform and added it to the last platform, and substituted it for the variables f+l (where l=the height of the last platform).
Then I took the total number of platforms and divided it by 2 to get the average, and multiplied it by the average of both platforms to get the total height of all of the platforms. This resulting number is the total length required in square feet of fabric to cover the fronts of the platforms. I got the formula (x/2) (f+l)=m from as a result of the process. I used this formula to see if i could get the correct answer using the same situation from Kevin’s Decisions. The first platform is 6 feet tall, there are 3 platforms in total, and the difference in height from each platform is 3 feet.
It would not work is the difference in height from platform from platform was not constant. Camilla’s Dilemma: (x/2)(f=l)=m This formula for Camilla’s Dilemma works because it takes the average of all the platforms. Then it multiplies that average by the total number of platforms there are. The reason this formula works is once again because the rate of increase between each platform is the same. The formula basically takes the average of the first and last platform, and if the rate of increase in height were not the same, the average wouldn’t have been able to be calculated the same way.
I would have had to add all of the numbers individually and divide by the total amount of numbers to get the average if the rate of increase in height were not the same. Also by substituting the last platform for the formula from Kevin’s Decisions we can combine the formula from Kevin’s Decisions in replace of the height of the last platform. The combined formula in Camilla’s dilemma and Kevin’s Decisions will result in the same answer and will look like this.
I learned about creating expressions and equations based on real life scenarios even without any real numbers. I think that this POW was educationally worthwhile and taught me about how real life can be put into equations and expressions. The only thing I might change about the problem was give set variables for each unknown to make it less confusing when comparing papers with others. I did somewhat enjoy working on this because it was worthwhile to my learning but it was difficult and I found myself stuck sometimes.
It was just in the middle between too hard and too easy because I k new some components of the problem but others I was confused on. Overall I thought it was good that we learned this because it will help our understanding of the unit problem. Self-Assessment: I think that I deserve a 3. 8-4 on this because almost all of my work is clear though there is maybe 1 or 2 spots that might be confusing to some but I think I overall did a concise and clear job solving and explain this POW.