PROBLEM STATEMENT: There are 1176 squares. When you use all of them can it create a stair-like structure.
PROCESS: You must make an equation to solve this problem with ease. We found that X over 2(x+1)=1176. We wanted to get rid of the fraction so we multiplied both sides by 2. Now we have x(x+1)=2352. From here you know that you must multiply two numbers to get 2352. From here you can guess and check, you know that the two numbers won’t be the same. So I started with 40 and 41 and that wasnt enough. Eventually I tried 48 and 49 and that was the perfect number. now we know that x=48.
SOLUTION: The solution is X=48. You can check by plugging it in. 48/2(48+1)=1176. 24(48+1)=1176. 1152+24=1176. 1176=1176. It is clear that X=48.
Extension/Reflection: This problem was definitely a challenge for me. I consider it to be a good thing to know how to do. I feel like now I could do this problem over and over. I do think there could be a easier way to do this and I would like to know how. I didn’t that you could get rid of fractions by multiplying the denominator to everything else in the problem.
I think I deserve full 15 points for this problem because I gave it my all and got it right in the end. Also I know how to solve it step by step. I even taught the class to do it on the whiteboard. That is why I deserve the full 15 points.
POW 10
Problem Statement:
You’re checking the prices to fill up your gas tank, and it costs $4.21 per gallon. At what rate is the price increasing by on the price meter?
Process:
This problem was fairly self explanatory. First, you had to read the variables (the first paragraph) which gave all of the information required to solve the problem. Rate, is interpreted as “price as it moves up”. I know that it costs $4.21 for the gas, so I knew that, the price as it went up per gallon was going to be $4.21. There wasn't much solving to do in this problem, except for knowing that rate = price in this situation, in which I knew that the $4.21 was going to increase for every gallon, but the price it increased by will not.
Solution:
The rate in which the price increase is $4.21, and I know this because they’re not asking about the gallons nor how much I need. I know that this is the answer because this is the only way that price is mentioned, and it didn't ask about gallons followed by the price for the gallons.
Extension:
In the beginning of this problem, I honestly over thought what the question was trying to ask, so I didn't think that the answer was as obvious as it was. I usually don’t picture problems to be easier than Roberto makes them seem. I thought it was being much more complicated than it had been, but they pretty much gave you the answer. I got the help from Franky which made me sure that
2. In order to fill up your tank, how much will you have to pay?
Problem Statement:
If you have 1.35 gallons in your tank at the moment, and you can hold a total capacity of 10.8 gallons, how much money will it take to fill up the remainder of the tank?
Process:
We have a tank that can hold 10.8 gallons and your car can give you 40 miles per gallon. Currently, my car only has 1/8 gallons filled up.In order to solve how much I need to pay, I need to consider this information I was given in the beginning of the problem:
Solution:
The total price for the gasoline required to fill up my tank is $39.78 cents but without rounding its $39.7845. I know that this is my final answer because I did the procedures mentioned above and double checked with Lizzy to make sure that I didn't do the problem incorrectly. I don’t think that there was much more I could do in order to figure out this was correct, because almost everyone in my class got this answer or rounded up several cents.
Extension:
I found this problem to be very quick and easy, but I don't think that I would have been able to solve problem 4 without it, so it was important. I didn't need any help with solving this problem, so I think that it might have been a little bit too easy.
3. Graph the action of filling up your gas tank in the provided graph. Make sure you label your axis correctly and increase by the correct rate.
Problem Statement:
Given the information from the previous problems, how can you graph the rate of the price for every gallon. Make sure that you are labeling the x and y axis correctly.
Process:
Originally, I thought I knew what I was doing, but I labeled the price of the gallons on my x axis, and the gallons on the y axis. I noticed that in the beginning, my graph looked like it was going up straight, and I knew that I had to be doing something wrong. I asked Lizzy for help, and she told me that I was labeling my axis’ wrong, so I went back and fixed it to have:
After I had fixed my graph, and correctly plotted my points, I got the graph as shown below: I made a hand-drawn graph to show a bit more clearly how I labeled and wrote my axis, and to show the increase in price as I go by gallons. Towards the end, you can see that from 9-10 on the gallons there's a slight curve, and the reason being is because the price for the remaining gallons I need is $39.78
Extension:
This problem was definitely one that I am glad we covered. I'm not the best at graphing, especially when it comes to labeling what goes one what axis. I had to redo this graph about 5 times, until I got it to what I think made the most sense. I think that this helped me really understand how to label a graph, because I learned this last year but I forgot how to properly do it.
Grade for problems 1-3:
I added all three of these problems together for a complete grade, mainly because I felt like they were fairly easy and quick to solve, and they each deserve to be graded as a whole. However, I think that doing all of theses problems, I should get a 15/15 points because I worked on them somewhat independently. I also tried to help out my peers while working on problems 1-2, the first couple of days being in class. I don't think that there was enough work to do in order to show that I deserve the 15 points, but I still feel like I tried my best to be as thorough as I could be.
4. Million dollar question: Many People Drive to fill up at the cheapest gas station in their neighborhood. Given the same car specified above, how far can the gas station be from you if the gas cost $4.11? How about $4.01?
Problem Statement:
You found a cheaper gas station that you normally go to, and you want to see how far you would be able to drive in order to still be saving money. Using the same car mentioned above, if the gas cost $4.11, how far could you drive. What if the gas costs $4.01?
Process:
This problem was by far the most confusing problem I solved. I tried to do this problem with equations, and very complicated math, which left me being confused and I had no idea where I was going with the problem. I had a pattern going, but I didn't know what I was trying to do. The math was right, and I was getting somewhere, but I still personally did not understand what I was doing. To the right I have the work in which I originally started to tackle the problem However it didn't work so I then got help from Jacob. The way she solved it was more visually clear than what I had done. I find that drawing out patterns helps me understand a problem much more easier than trying to do an equation.
Solution:
Like I had mentioned, I got help from Jacob, and came to the conclusion that the gas for $4.11, would be a 7.5 mi distance max which would cost .94 cents which is the same as I would have saved. For the $4.01 priced gas, the furthest I could drive to get there is 17.5 mi which would cost $1.84, giving me 5 cents leftover. I made the line graph which helped me a lot, and below I have the work to show how I got what I did. I compared my answers to Jacob afterwards, and I think that her's are a bit different because she rounded her cents, and I didn't, but the answers are still fairly close to each other.
Extension:
I found this by far the most difficult problem, and I think that I would not have been able to solve this without the help of my tablemates. Although I was very confused in the beginning, I liked working on this because it helped me find a new way of solving a problem. I did consider this problem to be fairly challenging, especially because I was trying to solve it, without fully understanding what the problem was asking for.
Grade:
For this problem, I think that I would deserve a 15/15 because I tried to solve it multiple times, regardless if I was doing it wrong or not. Now, I think that I am fairly close to the answers, and I have my work to prove that I tried solving it to the best of my ability. I put in the most work in this problem, and I didn't give up at all, even though I was getting frustrated and confused.
You’re checking the prices to fill up your gas tank, and it costs $4.21 per gallon. At what rate is the price increasing by on the price meter?
Process:
This problem was fairly self explanatory. First, you had to read the variables (the first paragraph) which gave all of the information required to solve the problem. Rate, is interpreted as “price as it moves up”. I know that it costs $4.21 for the gas, so I knew that, the price as it went up per gallon was going to be $4.21. There wasn't much solving to do in this problem, except for knowing that rate = price in this situation, in which I knew that the $4.21 was going to increase for every gallon, but the price it increased by will not.
Solution:
The rate in which the price increase is $4.21, and I know this because they’re not asking about the gallons nor how much I need. I know that this is the answer because this is the only way that price is mentioned, and it didn't ask about gallons followed by the price for the gallons.
Extension:
In the beginning of this problem, I honestly over thought what the question was trying to ask, so I didn't think that the answer was as obvious as it was. I usually don’t picture problems to be easier than Roberto makes them seem. I thought it was being much more complicated than it had been, but they pretty much gave you the answer. I got the help from Franky which made me sure that
2. In order to fill up your tank, how much will you have to pay?
Problem Statement:
If you have 1.35 gallons in your tank at the moment, and you can hold a total capacity of 10.8 gallons, how much money will it take to fill up the remainder of the tank?
Process:
We have a tank that can hold 10.8 gallons and your car can give you 40 miles per gallon. Currently, my car only has 1/8 gallons filled up.In order to solve how much I need to pay, I need to consider this information I was given in the beginning of the problem:
- 1/8 of the 10.8 gallons is 1.35 gallons which I currently have in my tank.
- For each gallon, the cost is $4.21
- I need 9.45 in order to fill up my tank completely. I figured that in order to find out my total cost, I would have to multiply the $4.21 (price per gallon) by 9.45 gallons (what I need to fill up my entire tank) in order to get the total that I would have to pay in order to have a full tank. The price of gas x the gallons I need in order to fill up my tank = the total cost
Solution:
The total price for the gasoline required to fill up my tank is $39.78 cents but without rounding its $39.7845. I know that this is my final answer because I did the procedures mentioned above and double checked with Lizzy to make sure that I didn't do the problem incorrectly. I don’t think that there was much more I could do in order to figure out this was correct, because almost everyone in my class got this answer or rounded up several cents.
Extension:
I found this problem to be very quick and easy, but I don't think that I would have been able to solve problem 4 without it, so it was important. I didn't need any help with solving this problem, so I think that it might have been a little bit too easy.
3. Graph the action of filling up your gas tank in the provided graph. Make sure you label your axis correctly and increase by the correct rate.
Problem Statement:
Given the information from the previous problems, how can you graph the rate of the price for every gallon. Make sure that you are labeling the x and y axis correctly.
Process:
Originally, I thought I knew what I was doing, but I labeled the price of the gallons on my x axis, and the gallons on the y axis. I noticed that in the beginning, my graph looked like it was going up straight, and I knew that I had to be doing something wrong. I asked Lizzy for help, and she told me that I was labeling my axis’ wrong, so I went back and fixed it to have:
- x axis = gallons
- y axis= money
After I had fixed my graph, and correctly plotted my points, I got the graph as shown below: I made a hand-drawn graph to show a bit more clearly how I labeled and wrote my axis, and to show the increase in price as I go by gallons. Towards the end, you can see that from 9-10 on the gallons there's a slight curve, and the reason being is because the price for the remaining gallons I need is $39.78
Extension:
This problem was definitely one that I am glad we covered. I'm not the best at graphing, especially when it comes to labeling what goes one what axis. I had to redo this graph about 5 times, until I got it to what I think made the most sense. I think that this helped me really understand how to label a graph, because I learned this last year but I forgot how to properly do it.
Grade for problems 1-3:
I added all three of these problems together for a complete grade, mainly because I felt like they were fairly easy and quick to solve, and they each deserve to be graded as a whole. However, I think that doing all of theses problems, I should get a 15/15 points because I worked on them somewhat independently. I also tried to help out my peers while working on problems 1-2, the first couple of days being in class. I don't think that there was enough work to do in order to show that I deserve the 15 points, but I still feel like I tried my best to be as thorough as I could be.
4. Million dollar question: Many People Drive to fill up at the cheapest gas station in their neighborhood. Given the same car specified above, how far can the gas station be from you if the gas cost $4.11? How about $4.01?
Problem Statement:
You found a cheaper gas station that you normally go to, and you want to see how far you would be able to drive in order to still be saving money. Using the same car mentioned above, if the gas cost $4.11, how far could you drive. What if the gas costs $4.01?
Process:
This problem was by far the most confusing problem I solved. I tried to do this problem with equations, and very complicated math, which left me being confused and I had no idea where I was going with the problem. I had a pattern going, but I didn't know what I was trying to do. The math was right, and I was getting somewhere, but I still personally did not understand what I was doing. To the right I have the work in which I originally started to tackle the problem However it didn't work so I then got help from Jacob. The way she solved it was more visually clear than what I had done. I find that drawing out patterns helps me understand a problem much more easier than trying to do an equation.
Solution:
Like I had mentioned, I got help from Jacob, and came to the conclusion that the gas for $4.11, would be a 7.5 mi distance max which would cost .94 cents which is the same as I would have saved. For the $4.01 priced gas, the furthest I could drive to get there is 17.5 mi which would cost $1.84, giving me 5 cents leftover. I made the line graph which helped me a lot, and below I have the work to show how I got what I did. I compared my answers to Jacob afterwards, and I think that her's are a bit different because she rounded her cents, and I didn't, but the answers are still fairly close to each other.
Extension:
I found this by far the most difficult problem, and I think that I would not have been able to solve this without the help of my tablemates. Although I was very confused in the beginning, I liked working on this because it helped me find a new way of solving a problem. I did consider this problem to be fairly challenging, especially because I was trying to solve it, without fully understanding what the problem was asking for.
Grade:
For this problem, I think that I would deserve a 15/15 because I tried to solve it multiple times, regardless if I was doing it wrong or not. Now, I think that I am fairly close to the answers, and I have my work to prove that I tried solving it to the best of my ability. I put in the most work in this problem, and I didn't give up at all, even though I was getting frustrated and confused.