Problem Of The Week
How have POWs helped you grow mathematically?
They have helped me by making me realize how I can use the math I have learned an connected it to a real world situation. We took almost each and every unit topic and turned it into a POW, a situation where we would be able to contribute this newly learned math to a question needed to be solved. I defiantly think I have grown mathematically because now I can look at a light pole and know it is possible to find the angle of inclination from the point your are at to the top of the pole.
They have helped me by making me realize how I can use the math I have learned an connected it to a real world situation. We took almost each and every unit topic and turned it into a POW, a situation where we would be able to contribute this newly learned math to a question needed to be solved. I defiantly think I have grown mathematically because now I can look at a light pole and know it is possible to find the angle of inclination from the point your are at to the top of the pole.
Semester 2
POW #1
Problem Statement:
You are given a circle. You can cut this circle with only straight line that goes through the whole circle. What is the largest number of pieces that you can make with the given number of cuts?
Process:
At first I was very confused on how to begin to solve this POW or how I could break it down so I could understand it better. I talked to many of my peers about how they were were going to solve this POW. I decided to take good notes from the class discussion about POW #1 and write down what was written on the board that could help me solve this POW. After the class discussion and from the notes I had written It was a lot clearer to me on the different options and ways I could come up with the solution. Through one of the options, the table, I saw a pattern of the number of cuts from the specific amount of cuts, it goes up by one every time. To organize my information I spaced out my notes so that it was clear to me and anyone looking at these notes where the equations are and what's important for me to know.
Solution:
There are three solutions to this POW, one of them equation where y = previous max + number of cuts and the other two were y = f(x) = f(x-1) + x or f(x) = 1 + ∑. I plugged x (the number of cuts) into one of these equations and found what the largest number of pieces could be for certain numbers. The equation I used was y = previous max + the number of cuts to figure out what the maximum number of pieces would be for the cuts.
Extension:
Another way we could test this problem is by changing the shape of the object we are cutting by straight lines. We could use a square or a hexagon to see what would change or if the equation would see be valid.
Evaluate Self:
I think this POW helped me use pervious skills I learned in Algebra. We used an equation that helped us solve multiple problems or variables. I think it was very important for me to continue using those skills I previously learned in Algebra and bring it to my current class, geometry. Out of 30 pts I believe I deserve 30 pts because I worked on this POW to the best of my ability and good amount of time working on it. I used my note taking skills and worked on the problem in and outside of class.
You are given a circle. You can cut this circle with only straight line that goes through the whole circle. What is the largest number of pieces that you can make with the given number of cuts?
Process:
At first I was very confused on how to begin to solve this POW or how I could break it down so I could understand it better. I talked to many of my peers about how they were were going to solve this POW. I decided to take good notes from the class discussion about POW #1 and write down what was written on the board that could help me solve this POW. After the class discussion and from the notes I had written It was a lot clearer to me on the different options and ways I could come up with the solution. Through one of the options, the table, I saw a pattern of the number of cuts from the specific amount of cuts, it goes up by one every time. To organize my information I spaced out my notes so that it was clear to me and anyone looking at these notes where the equations are and what's important for me to know.
Solution:
There are three solutions to this POW, one of them equation where y = previous max + number of cuts and the other two were y = f(x) = f(x-1) + x or f(x) = 1 + ∑. I plugged x (the number of cuts) into one of these equations and found what the largest number of pieces could be for certain numbers. The equation I used was y = previous max + the number of cuts to figure out what the maximum number of pieces would be for the cuts.
Extension:
Another way we could test this problem is by changing the shape of the object we are cutting by straight lines. We could use a square or a hexagon to see what would change or if the equation would see be valid.
Evaluate Self:
I think this POW helped me use pervious skills I learned in Algebra. We used an equation that helped us solve multiple problems or variables. I think it was very important for me to continue using those skills I previously learned in Algebra and bring it to my current class, geometry. Out of 30 pts I believe I deserve 30 pts because I worked on this POW to the best of my ability and good amount of time working on it. I used my note taking skills and worked on the problem in and outside of class.
POW #2
Ben Black, Mira Joyner, Garrett Hagen
Caitlyn Kneller
POW #2
Pick Up Triangles
Problem Statement:
You have four rods they are 2, 3, 4, and 6 inches long. You are also given an unlimited supply of additional rods that can be 1-20 inches long. You have to find as many pairs of similar triangles using all four given lengths.
Process:
Our grouped used the guess and check method. On one of the pairs we came up with 5 different combinations and none of them worked so we had to start over. The way we approached the problem was to see the different combinations that worked in the least amount of time. We organized our information by a list of the triangles that work and don’t work. A lot of the triangles we had made weren’t able to be used because 2 of the sides didn't add up to greater than the other side. We tried many times asking Caitlyn for help but we were only able to come up with a few solutions to the problem.
Solution:
Our solution to this is on the following page. We were only able to find 2 solutions. We were told about there being more but we were not able to find these out. ON the next page also, you can see all the methods that we tried and that didn't work. If you look closely on some of the numbers you can see erase marks where we had tried multiple times to come up with an answer.
One extension that we think would have been interesting to see was if you changed up the side lengths. If you were to change up the side lengths non proportional to each other and totally random between 1-20 inches long. Once you had these lengths set out it would have been fascinating if you had a certain amount of sticks you could use total, instead of an unlimited amount of sticks.
Evaluate/ Self-Assess: Our group thought this was a challenging POW. We didn't find that it helped us educationally wise that much. There were very few hints and help given out, so it made solving it and trying to solve it very difficult. We thought we deserve a 21/25. We didn't find a great solution but we also tried multiple times to get it.
Caitlyn Kneller
POW #2
Pick Up Triangles
Problem Statement:
You have four rods they are 2, 3, 4, and 6 inches long. You are also given an unlimited supply of additional rods that can be 1-20 inches long. You have to find as many pairs of similar triangles using all four given lengths.
Process:
Our grouped used the guess and check method. On one of the pairs we came up with 5 different combinations and none of them worked so we had to start over. The way we approached the problem was to see the different combinations that worked in the least amount of time. We organized our information by a list of the triangles that work and don’t work. A lot of the triangles we had made weren’t able to be used because 2 of the sides didn't add up to greater than the other side. We tried many times asking Caitlyn for help but we were only able to come up with a few solutions to the problem.
Solution:
Our solution to this is on the following page. We were only able to find 2 solutions. We were told about there being more but we were not able to find these out. ON the next page also, you can see all the methods that we tried and that didn't work. If you look closely on some of the numbers you can see erase marks where we had tried multiple times to come up with an answer.
One extension that we think would have been interesting to see was if you changed up the side lengths. If you were to change up the side lengths non proportional to each other and totally random between 1-20 inches long. Once you had these lengths set out it would have been fascinating if you had a certain amount of sticks you could use total, instead of an unlimited amount of sticks.
Evaluate/ Self-Assess: Our group thought this was a challenging POW. We didn't find that it helped us educationally wise that much. There were very few hints and help given out, so it made solving it and trying to solve it very difficult. We thought we deserve a 21/25. We didn't find a great solution but we also tried multiple times to get it.
Semester 1
POW #1
- You’re given a 3x3 grid with two black knight chess pieces and two white. The two black pieces need to switch places with two white pieces. We need to do this by moving the least amount of moves across the chessboard. You can only move a knight two over and one up or down or one over and two up or down.
- To solve this problem of the week I needed to know how the knight chess piece could move across the board. Which is two up one over, one up two over, two down one over, or one down two over. Because this problem is only on a 3x3 board I needed to either create a board where I could physically move the pieces or I needed to draw one out and just use point for the pieces. I decided to create a 3x3 grid on my paper and draw out the two white knights and two black on either side of the board. I plotted point on the board where the possible places the knights could move to. Since I knew prior to finding the least amount of moves that it would talk one if the pieces 4 moves until it reached the other side of the board. I used this knowledge to find the least amount of moves until everyone of the pieces were on the other side, switched places with the opposite piece.
- The solution the this weeks POW is the least amount of spaces you can move your piece across the board to switch places with the opposite color piece is 16. Each knight would have to complete 4 moves across the board to the other side. Which means 4x4 is 16, which is the total amount of moves.
- Another set of problems you could try to solve is a larger board. Such as 4x4, 6x6, or even 8x8. With these larger boards you will aswell try to find the minimal amount of spaces you have to move across the board to switch places with the other pieces.
- During this POW I honestly didn’t learn anything, only the answer to the question. Even though the answer didn't come to me right away I still came to a conclusion. I believe I deserve a good grade on this first POW because I tried my hardest to come up witht the answer, talked to make of my classmates about finding the solution.
POW #2
Problem Statement
You have two triangles made up of three sides and three angles, six parts. Can you find and example of two triangles that are not congruent but with five congruent parts?
Process
To solve this problem we first re-read the question multiple times until we both clearly understood what this POW was asking. At the beginning of this problem we were really confused on how this would even be possible. We knew for sure the there couldn’t be three side in that five parts because having three sides the same makes the triangles congruent. So we knew that there would be three angles and two sides that would make this problem possible. On the Friday my teacher (Kettle Keller) gave the whole class a hint on the problem. She stated that the sides that are congruent don’t have to be in the same place on the triangle. After that we drew all the possible way you could rearrange the two set sides and left all the angles were they were. We found that there were two possibilities to solve this problem and both of them had each side in a different spot in triangle B than it was in Triangle A. You can see the sheet on the back of this paper.
Evaluation
I feel like during this project I didn't learn a whole lot. A lot of this problem was just to think about it and not as much learning. What I did learn was not to give up on a problem. This is a very useful skill for me so I can figure out the problem ahead. I believe that we diseve a 25/30 to a 28/30 I feel this way because We could of used or time better but we still put a lot of work into this POW.
You have two triangles made up of three sides and three angles, six parts. Can you find and example of two triangles that are not congruent but with five congruent parts?
Process
To solve this problem we first re-read the question multiple times until we both clearly understood what this POW was asking. At the beginning of this problem we were really confused on how this would even be possible. We knew for sure the there couldn’t be three side in that five parts because having three sides the same makes the triangles congruent. So we knew that there would be three angles and two sides that would make this problem possible. On the Friday my teacher (Kettle Keller) gave the whole class a hint on the problem. She stated that the sides that are congruent don’t have to be in the same place on the triangle. After that we drew all the possible way you could rearrange the two set sides and left all the angles were they were. We found that there were two possibilities to solve this problem and both of them had each side in a different spot in triangle B than it was in Triangle A. You can see the sheet on the back of this paper.
Evaluation
I feel like during this project I didn't learn a whole lot. A lot of this problem was just to think about it and not as much learning. What I did learn was not to give up on a problem. This is a very useful skill for me so I can figure out the problem ahead. I believe that we diseve a 25/30 to a 28/30 I feel this way because We could of used or time better but we still put a lot of work into this POW.