## Unit 4-Desmos Drawing and Function Families

In this project we had to make a drawing completely out of equations in an online graphing calculator called Desmos. When I started to make this, I came up with an idea fairly quickly. I knew what basic shapes I woulld need right from the beginning. I knew I would use several linear equations and a couple parabolas. The difficult part was incorperationg the rest of the required equations in. Eventually I made a circle into a G in the title, and surrounded the title with two cubic functions. I made an elipse around the Batman symbol and it was finished.
Mostly in this project, I graphed the parent function of the equation I wanted and modified it until I got something I liked. This served me well for most of the project. At one point I deviated from this. When I got to about halfway through the lettering, I found out that for straight lines, it was easier to actually figure out the equation first and then plug it in. This was harder if not impossible for equations such as the elipse, but for the straight lines in the title it worked just fine. |

## Unit 3 Reflection

1. What content/skills have been the most interesting to you?

I liked some of the review of area because I hadn't quite memorized all the formulas for calculating it. It was nice to go over that so I could learn them well.

2. How have you grown mathematically?

Much the same as my first answer, it has been a huge help to review and thus memorize the basic area formulas. Since I wasn't quite sure of a lot of them, it helped immensely in many assignments I did this unit.

I liked some of the review of area because I hadn't quite memorized all the formulas for calculating it. It was nice to go over that so I could learn them well.

2. How have you grown mathematically?

Much the same as my first answer, it has been a huge help to review and thus memorize the basic area formulas. Since I wasn't quite sure of a lot of them, it helped immensely in many assignments I did this unit.

## Unit 2 Reflection

**Unit 2 Reflection: Shadows, Similarity and Right Triangle Trigonometry**

**Q1: What has been the work you are most proud of in this unit?**

I have been most proud of my POW #1. I worked well on it and found the answer quickly and stated my reasoning well. I worked hard and got it done quickly and early. It turned out really well and I even had time to make some Geogebra diagrams.

**Q2: What skills are you developing in geometry/math?**

Some skills I have been developing have been graphing trigonometry problems, testing ideas, and creating useful diagrams. We leaarned what basic sine, cosine, and tangent functions look like about halfway through the unit and it actually made sense. I was able to eliminate ideas that didn't work to solve problems quicker. And I applied my Geogebra skills to make simple diagrams that conveyed lots of information.

**Q3: Choose one topic: similarity (ratios) or trigonometry. Explain what it is. Provide an example of how it is used in mathematics to solve problems. State an application of the topic in the adult world.**

Similarity is when you have two objects of different side lengths which have angles that remain the same. This is very useful for many buisnesses. It is far easier to make a scale model of a building with similar sides than make a practice building first. That's both much harder and just plain impractical.

## Problems of the Week

Will Klumpenhower

POW Write-up 1

The Four Knights

In this week’s POW, we were given a problem. Four knights in Chess, two black and two white, were on the corners of a 3 square by 3 square chessboard. The white ones were on the bottom side, and the black ones were on the top side. Is it possible, using only real chess moves, to have the black knights and the white knights switch places? If so, do it in the least number of moves.

I immediately assumed it was possible, because it just didn’t seem right for there not to be a way. In chess, knights are very versatile pieces and I thought that it was almost obvious that the problem could be solved. Having determined this, I set to work on the problem.

When I looked at the board, I saw that there were four vacant spaces that a knight could occupy. There was one vacant spot that no matter what, a knight could not ever be there, or move from it once it was: the center square. according to the rules of chess, A knight has to move in a pattern of two squares either up, down, and to the side, and then one square up down or to the side, depending on the original move. A knight can jump over other pieces. So since it is impossible to have a starting point that is two spaces away from the middle, it was impossible for a knight to be placed there. So I knew I had to use the edges for this puzzle.

Since it is difficult to display how I came to my answer, I will try to explain it as best I can. I solved the problem the day I got it. I moved all the knights around in a circle until they came to a point where they had switched places. I moved each piece four times to come to a total of 16 moves. I know that this is correct because I cross-referenced my answer with my classmates. Several of them were in their spots at only about six moves, but the way they were placed made it impossible to put their counterparts in their spots. If this is still confusing, see the paper attached.

An interesting extension to this problem would be to make the board 4 by 4 and see if it was still possible. I think it still would be, but it would be much more difficult, since there are no borders. I think the borders helped a bit in the original, since they made it less confusing.

I learned from this project that the problem was slightly more complicated than I thought, but still not too difficult. I learned also that while I figured out the problem quickly, it was much harder for me to describe how I solved it than I thought. This made it difficult for me to get it done on time. I think I deserve somewhere from 20 to 25 points out of 30 on this problem, since, while I understood and solved the problem well and easily, I was not the most punctual on getting it written down.

POWs or Problems of the Week have helped me develop several math skills, including trigonometry and similarity. They help improve problem soliving abilities by giving us a large problem that takes a while to find a solution. Sometimes it is difficult, but it is good to struggle to build math skills. Breaking down a problem and picking it apart are important skills for real-life problems we will face in the future.

POW #1 Write-up

Will Klumpenhower

Geometry

1-7-15

Slices of Pie

You are given a pie that you have to cut with straight lines that go all the way through it,but they don’t have to have the same shape. How many pieces can you make with a given number of cuts?

When we received this POW, I started working on it immediately. I figured out within the class period that the maximum for 4 cuts was 16. I already knew that 1 cut was 2, 2 cuts was 4, and 3 cuts was 7. Then the next day, I figured out that 5 cuts made 16 at the most. I started to make a diagram to show these findings simply and had cranked out a pretty good diagram when Caitlyn showed us on the board what the answer was, which sort of surprised me. I was also really surprised that I hadn’t realized it before. The equation used to find the maximum pieces of pie was the number of cuts plus the former number of pieces. 1+1=2, 2+2=4, 3+4=7, etc. So I put my findings in a Geogebra document.

We also had to figure out what the maximum for 10 cuts would be, using our formula,

f(n) =f(n-1)+n. Plug in 10 for n and you get 56. It’s pretty basic algebra.

An extension to the problem could be having all the cuts form a regular triangle, in sets of three. This would pose some interesting questions, but would probably result in a less complicated formula.

I thought the problem was more interesting than the last POW we had, and thus I think I did better on it and it kept me working on it more. It reminded me of the chess problem, my favorite so far. It was maybe a little too easy, but I didn’t mind. I think I should receive a 25 out of 25 for the problem. I think I did a good job and I feel I expressed my ideas well.

POW Write-up 1

The Four Knights

In this week’s POW, we were given a problem. Four knights in Chess, two black and two white, were on the corners of a 3 square by 3 square chessboard. The white ones were on the bottom side, and the black ones were on the top side. Is it possible, using only real chess moves, to have the black knights and the white knights switch places? If so, do it in the least number of moves.

I immediately assumed it was possible, because it just didn’t seem right for there not to be a way. In chess, knights are very versatile pieces and I thought that it was almost obvious that the problem could be solved. Having determined this, I set to work on the problem.

When I looked at the board, I saw that there were four vacant spaces that a knight could occupy. There was one vacant spot that no matter what, a knight could not ever be there, or move from it once it was: the center square. according to the rules of chess, A knight has to move in a pattern of two squares either up, down, and to the side, and then one square up down or to the side, depending on the original move. A knight can jump over other pieces. So since it is impossible to have a starting point that is two spaces away from the middle, it was impossible for a knight to be placed there. So I knew I had to use the edges for this puzzle.

Since it is difficult to display how I came to my answer, I will try to explain it as best I can. I solved the problem the day I got it. I moved all the knights around in a circle until they came to a point where they had switched places. I moved each piece four times to come to a total of 16 moves. I know that this is correct because I cross-referenced my answer with my classmates. Several of them were in their spots at only about six moves, but the way they were placed made it impossible to put their counterparts in their spots. If this is still confusing, see the paper attached.

An interesting extension to this problem would be to make the board 4 by 4 and see if it was still possible. I think it still would be, but it would be much more difficult, since there are no borders. I think the borders helped a bit in the original, since they made it less confusing.

I learned from this project that the problem was slightly more complicated than I thought, but still not too difficult. I learned also that while I figured out the problem quickly, it was much harder for me to describe how I solved it than I thought. This made it difficult for me to get it done on time. I think I deserve somewhere from 20 to 25 points out of 30 on this problem, since, while I understood and solved the problem well and easily, I was not the most punctual on getting it written down.

**PROBLEMS OF THE WEEK:****Problem of the Week Reflection:**POWs or Problems of the Week have helped me develop several math skills, including trigonometry and similarity. They help improve problem soliving abilities by giving us a large problem that takes a while to find a solution. Sometimes it is difficult, but it is good to struggle to build math skills. Breaking down a problem and picking it apart are important skills for real-life problems we will face in the future.

POW #1 Write-up

Will Klumpenhower

Geometry

1-7-15

Slices of Pie

You are given a pie that you have to cut with straight lines that go all the way through it,but they don’t have to have the same shape. How many pieces can you make with a given number of cuts?

When we received this POW, I started working on it immediately. I figured out within the class period that the maximum for 4 cuts was 16. I already knew that 1 cut was 2, 2 cuts was 4, and 3 cuts was 7. Then the next day, I figured out that 5 cuts made 16 at the most. I started to make a diagram to show these findings simply and had cranked out a pretty good diagram when Caitlyn showed us on the board what the answer was, which sort of surprised me. I was also really surprised that I hadn’t realized it before. The equation used to find the maximum pieces of pie was the number of cuts plus the former number of pieces. 1+1=2, 2+2=4, 3+4=7, etc. So I put my findings in a Geogebra document.

We also had to figure out what the maximum for 10 cuts would be, using our formula,

f(n) =f(n-1)+n. Plug in 10 for n and you get 56. It’s pretty basic algebra.

An extension to the problem could be having all the cuts form a regular triangle, in sets of three. This would pose some interesting questions, but would probably result in a less complicated formula.

I thought the problem was more interesting than the last POW we had, and thus I think I did better on it and it kept me working on it more. It reminded me of the chess problem, my favorite so far. It was maybe a little too easy, but I didn’t mind. I think I should receive a 25 out of 25 for the problem. I think I did a good job and I feel I expressed my ideas well.

## Burning Tent Lab

Question 1: Once you have a minimal path, what appears to be true about the incoming angle and the outgoing angle?

Answer: They stay the same when you are moving them on the minimal distaance.

Question 2: Why are the path from points Camper to Tentfire the shortest path? Briefly explain.

Answer: The direct path is slightly shorter because it has no turn in it, it goes straight to the point.

Question 3: Where should the point River be located in relation to segment Camper to Tentfiire and line AB so that the sum of the distance is minimized? Answer: Point River is intersecting segment Camper Tentfire'.

Answer: They stay the same when you are moving them on the minimal distaance.

Question 2: Why are the path from points Camper to Tentfire the shortest path? Briefly explain.

Answer: The direct path is slightly shorter because it has no turn in it, it goes straight to the point.

Question 3: Where should the point River be located in relation to segment Camper to Tentfiire and line AB so that the sum of the distance is minimized? Answer: Point River is intersecting segment Camper Tentfire'.