## Desmos Project

## Desmos Reflection

Desmos has been one of the most confusing websites I have used all year. I think that it was really hard to get started but once I learned what I was doing, I was able to make a very basic face. I used almost all of the equations that I was supposed to, but I didn't get 15 lines like we were supposed to. I started my drawing but creating a circle and staring to put lines into my face. I tried to reflect most of the objects I made creating similarity in my drawing. I have a lot of trouble with desmos so I had to get help from my peers and teachers a lot. Desmos did however help my understand functions. First I learned more about quadratic functions. I created a nose out of a quadratic function. Another function that I learned about was cubic functions. I was going to create hair out of a cubic function because it looks like a squiggly line.

## Desmos Link

**Unit 3 Reflection: Area, Volume, and Measurement **

**Q1: What content/skills have been most interesting to you?**

In unit 3, volume has been the most interesting content to me because this has been the first year that I have felt comfortable with either having to find volume, or even just having a stronger understanding of volume. I really enjoyed developing a deeper understanding of the math content that I am being taught. The reason that I enjoyed volume more than the others is because I was really able to see a real world example of how math can be used in everyday life or just later in my life or career.

**Q2: How have you grown mathematically?**

In unit 3, I have grown mathematically in a couple of ways. First, I gained a deeper understanding of a large math concept, volume. In earlier years I had trouble with area, volume, perimeter, and other measurement equations. I also never really saw the relevance in why someone would deep to know the measurement of a shape, but POW 5 offered a great, real-world example on how optimizing surface area saves has the potential to save companies money. Next, geometry has really tested my temper, and has forced me to persevere through my problems and difficulties. This has been different because in algebra 1, I felt like a lot of the content was super straight forward. For example I really enjoyed how solving equations for a variable is all about using the methods that I had worked on for years to figure out what “a” is. However, geometry was different for me at least because there is usually more than one way to get to the correct answer. Expanding on that, when you have a shape made up of others, you don’t have to start with any specific shape, you have a lot more freedom in how you solve the problem. That freedom was different at first but after I was able to wrap my head around the fact that there isn’t just one formula to solve volume for any shape it was actually cool to be able to use my new learning to solve for the volume of shapes that I have never seen before.

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

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

During this unit, I am most proud of my understanding of trigonometry. All of the trig worksheets and test show that I am really understanding what we are doing in math. I fell very comfortable using any of the trig ratios, and feel able to solve for sides and angles in a triangle using trig. I am excited to explore trig functions further in Algebra 2 next year.

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

In geometry I am developing perseverance most. I am learning that it doesn't matter if I don't understand a math concept right away, and I am learning to deal with the frustration of having to re-visit math concepts in order for me to have a proper understanding. Even though this is frustrating, I think that it is important that I continue to try and work through these problems, because I won't learn unless I put forth the proper effort.

**Q3: Trigonometry. Explain what it is. Provide an example of how it is used in mathematics to solve problems.**Trigonometry is the process of using geometric functions to find the missing side or angle. It is the study of the measures of triangles. I might be interested in becoming an architect later in life and trigonometry is a very large part of architecture. Trigonometry is used to find the angle or slope of an object.

## Problems of the Week

**Problem of the Week Reflection:**These problems actually do help me out a lot even though I do not always like doing the. First, they challenge me and make me think and approach the problem in different ways.This sometimes frustrates me because I have to persevere through the problem when I do not approach the problem correctly. Overall the POWs have helped me develop a growth mind set.

## POW 2 Triangle Problem

POW Pick up Triangles

You have 4 rods that you must use each time, 6, 4 3, and 2. You are given an unlimited amount of whole number rods, but none of the rods are larger than 20. How many possible pairs of similar triangles are you able to make?

First, I tried to make a pair of similar triangles, keeping in mind that I had to use the 4 original rods. I made the first triangle using 4, 3, and 6. I tried to keep it similar by dividing those sides by 2, but that made the triangle’s sides 2, 3, and 1.5, and fractions don’t work so I wouldn't use that pair. Even though the first pair wasn’t correct, it helped me understand one approach I could take. I learned that I could divide the number by 2 and get some similar pairs. With this approach I was able to find 3 similar triangle pairs. On my piece of paper, number 4, number 6, and number 7 are the similar pairs. I thought that I was able to make more pairs than I was, but after I looked closer I realized that a lot of my pairs did not work. The pattern that I used to find most of my triangles was 4 and 6 on the sides, then I would change the base into number that could be divided by 2 be an equal number. The similar triangle would have 2 and 3 on the sides, and then half of the previous base. However, this only worked for a couple of triangles, because if the base is longer than the sides added together, it doesn’t form a triangle. I haven’t found a pattern in the triangles that makes this POW easier.

On the piece of paper I have my examples on, numbers 4, 6, and 7 are the similar pairs. The system I used only made 3 similar triangles, but I know that there are more similar pairs of triangles people can make. I am not sure what other approach I could take to give me more similar pairs.

A similar problem could be, you have 4 original rods that you must use in each time, 14, 12, 8, and 6. You are also given an unlimited supply of rods, but none is larger than 40. You are allowed to use fraction pieces of rods too. How many triangles are you now able to make?

Each week, I have had trouble with the POWs, but with this one I experimented the most difficulty. However I do feel like these POWs help me because it makes me think about the problem differently, and because I don’t usually get the POW right away, it also makes me visit the problem many times. The POWs are not my favorite thing, and they frustrate me because they make me pursue through the difficult parts and because they last all week, I am not able just to forget about this problem like other homework that I have trouble with. I feel like the POWs do help me though because they are teaching me persistence, and also teaching me not to get mad and give up when I don’t understand something. This POW, I didn’t do so well on, I would give myself a 20 out of 25.

You have 4 rods that you must use each time, 6, 4 3, and 2. You are given an unlimited amount of whole number rods, but none of the rods are larger than 20. How many possible pairs of similar triangles are you able to make?

First, I tried to make a pair of similar triangles, keeping in mind that I had to use the 4 original rods. I made the first triangle using 4, 3, and 6. I tried to keep it similar by dividing those sides by 2, but that made the triangle’s sides 2, 3, and 1.5, and fractions don’t work so I wouldn't use that pair. Even though the first pair wasn’t correct, it helped me understand one approach I could take. I learned that I could divide the number by 2 and get some similar pairs. With this approach I was able to find 3 similar triangle pairs. On my piece of paper, number 4, number 6, and number 7 are the similar pairs. I thought that I was able to make more pairs than I was, but after I looked closer I realized that a lot of my pairs did not work. The pattern that I used to find most of my triangles was 4 and 6 on the sides, then I would change the base into number that could be divided by 2 be an equal number. The similar triangle would have 2 and 3 on the sides, and then half of the previous base. However, this only worked for a couple of triangles, because if the base is longer than the sides added together, it doesn’t form a triangle. I haven’t found a pattern in the triangles that makes this POW easier.

On the piece of paper I have my examples on, numbers 4, 6, and 7 are the similar pairs. The system I used only made 3 similar triangles, but I know that there are more similar pairs of triangles people can make. I am not sure what other approach I could take to give me more similar pairs.

A similar problem could be, you have 4 original rods that you must use in each time, 14, 12, 8, and 6. You are also given an unlimited supply of rods, but none is larger than 40. You are allowed to use fraction pieces of rods too. How many triangles are you now able to make?

Each week, I have had trouble with the POWs, but with this one I experimented the most difficulty. However I do feel like these POWs help me because it makes me think about the problem differently, and because I don’t usually get the POW right away, it also makes me visit the problem many times. The POWs are not my favorite thing, and they frustrate me because they make me pursue through the difficult parts and because they last all week, I am not able just to forget about this problem like other homework that I have trouble with. I feel like the POWs do help me though because they are teaching me persistence, and also teaching me not to get mad and give up when I don’t understand something. This POW, I didn’t do so well on, I would give myself a 20 out of 25.

## POW 4 Rubik Cube Problem

POW 4

You have a 5cm by 5cm by 5cm cube made up of 1cm by 1cm cubes. Your friend comes along and paints the six faces of the cube. The paint doesn’t leak through the cube, and the painted cubes stay painted. The way that I started the POW, was by first looking at one side of the cube, and looking at how many of the cubes had only one side painted. I learned that the corners couldn’t be counted because they had more than side painted. Without counting the corners of the cube, I counted that there were 9 cubes with one side painted on one side of the cube. I then multiplied 9 by 6 because there are 9 cubes on each side, and there are 6 sides. There were 54 smaller cubes with one side painted in the whole cube. Next I found how many of the smaller cubes had 2 faces painted. This time, I saw that I wasn’t able to count the corners, or the first cubes that I counted. I saw that there were 3 smaller cubes that had 2 sides painted for every edge of the cube. Once I knew that there were 12 edges on the cube, I then multiplied 12 by 3. I learned that there were 36 smaller cubes with 2 sides painted. Third, I was looking at how many of the smaller cubes have 3 sides painted. I knew that the only cubes with 3 sides painted were the corners, so I knew I was only looking at the corners of the cube. I saw that there were 4 corners on the top of the cube, and because I also remembered to think about the bottom of the cube, I multiplied that by 2, because there were 2 sets of 4 corners. The next problem was asking if there were any cubes with 4, 5, or 6 sides painted. There were not any cubes with more than 3 sides painted because if the smaller cube showed more than three sides, it wouldn't create a 5cm by 5cm by 5cm cube. Lastly, I was asked how many of the cubes aren’t painted. The way I approached this was by trying to find the volume of the cube. The formula for finding volume of a cube is the area of the base multiplied by the height of the object. The area of the base was 125cm³. Next I added all of the cubes with sides painted. I counted 98 smaller cubes were painted. Then I subtracted 98 from 125, or the number of cubes painted, from the sum of all the cubes. I learned that there were 27 cubes that weren’t painted. Finally we were trying to create a generalization that allows you to find the number of smaller cubes when the square is made up of a variable, n. To find the number of cubes with only one side painted, you would do subtract the height of the cube by 2, or (n-2). If the height of the cube was 3, you (3-2). The reason we subtract 2 is because you don’t count the edges. After you have your function (3-2) then you multiply that by 6 because there are 6 sides on a cube and that will never change. To find the number of cubes with 2 sides painted, you would set up your formula like (n-2)* 12. The reason you subtract 2 is because you don’t count the corners, and there are 2 corners per edge. Then you multiply that by 12 because there are 12 edges on a cube and that number will never change. The number of corners will never change and it will always have 8 corners.

You have a master pyramorphix, rubiks made up of smaller triangles. Your friend comes along and paints the outside of your rubiks. How many sides of the triangle have 1 side, 2 sides, and 3 sides painted? How many smaller triangles have 0 faces painted? Can you generalize a formula to be able to find the number of sides painted?

This POW was different than the others, and I actually thought this one was easier than the rest. I felt like this POW was easier for me because I think that I really understand how to find the area of a shape, and I think that a cube is just easy for me to visualize and think about. I enjoy how we learned about how to do almost all of the POW before or while we were working on the POW. Unlike others, this POW was very straightforward and didn’t really require me to approach it multiple times. However, I had to get extra help with the generalization part because it took me a couple of times to fully understand the concept, and it took me a little while to put it into words and for it to make sense. I’m glad that this POW came pretty easy to me because it felt good to not get frustrated with a POW for once. I liked this POW because it allowed me to think about a 3D shape and I learned that it was pretty easy for me to visualize.

You have a 5cm by 5cm by 5cm cube made up of 1cm by 1cm cubes. Your friend comes along and paints the six faces of the cube. The paint doesn’t leak through the cube, and the painted cubes stay painted. The way that I started the POW, was by first looking at one side of the cube, and looking at how many of the cubes had only one side painted. I learned that the corners couldn’t be counted because they had more than side painted. Without counting the corners of the cube, I counted that there were 9 cubes with one side painted on one side of the cube. I then multiplied 9 by 6 because there are 9 cubes on each side, and there are 6 sides. There were 54 smaller cubes with one side painted in the whole cube. Next I found how many of the smaller cubes had 2 faces painted. This time, I saw that I wasn’t able to count the corners, or the first cubes that I counted. I saw that there were 3 smaller cubes that had 2 sides painted for every edge of the cube. Once I knew that there were 12 edges on the cube, I then multiplied 12 by 3. I learned that there were 36 smaller cubes with 2 sides painted. Third, I was looking at how many of the smaller cubes have 3 sides painted. I knew that the only cubes with 3 sides painted were the corners, so I knew I was only looking at the corners of the cube. I saw that there were 4 corners on the top of the cube, and because I also remembered to think about the bottom of the cube, I multiplied that by 2, because there were 2 sets of 4 corners. The next problem was asking if there were any cubes with 4, 5, or 6 sides painted. There were not any cubes with more than 3 sides painted because if the smaller cube showed more than three sides, it wouldn't create a 5cm by 5cm by 5cm cube. Lastly, I was asked how many of the cubes aren’t painted. The way I approached this was by trying to find the volume of the cube. The formula for finding volume of a cube is the area of the base multiplied by the height of the object. The area of the base was 125cm³. Next I added all of the cubes with sides painted. I counted 98 smaller cubes were painted. Then I subtracted 98 from 125, or the number of cubes painted, from the sum of all the cubes. I learned that there were 27 cubes that weren’t painted. Finally we were trying to create a generalization that allows you to find the number of smaller cubes when the square is made up of a variable, n. To find the number of cubes with only one side painted, you would do subtract the height of the cube by 2, or (n-2). If the height of the cube was 3, you (3-2). The reason we subtract 2 is because you don’t count the edges. After you have your function (3-2) then you multiply that by 6 because there are 6 sides on a cube and that will never change. To find the number of cubes with 2 sides painted, you would set up your formula like (n-2)* 12. The reason you subtract 2 is because you don’t count the corners, and there are 2 corners per edge. Then you multiply that by 12 because there are 12 edges on a cube and that number will never change. The number of corners will never change and it will always have 8 corners.

You have a master pyramorphix, rubiks made up of smaller triangles. Your friend comes along and paints the outside of your rubiks. How many sides of the triangle have 1 side, 2 sides, and 3 sides painted? How many smaller triangles have 0 faces painted? Can you generalize a formula to be able to find the number of sides painted?

This POW was different than the others, and I actually thought this one was easier than the rest. I felt like this POW was easier for me because I think that I really understand how to find the area of a shape, and I think that a cube is just easy for me to visualize and think about. I enjoy how we learned about how to do almost all of the POW before or while we were working on the POW. Unlike others, this POW was very straightforward and didn’t really require me to approach it multiple times. However, I had to get extra help with the generalization part because it took me a couple of times to fully understand the concept, and it took me a little while to put it into words and for it to make sense. I’m glad that this POW came pretty easy to me because it felt good to not get frustrated with a POW for once. I liked this POW because it allowed me to think about a 3D shape and I learned that it was pretty easy for me to visualize.

## Burning Tent GeoGebra

## Burning Tent Lab Questions

1. The incoming and outgoing angles are both the answer, because when you move the line they all become equal.

2. The shortest path in the burning tent lab, between two points, is a straight line from the point

3. Move the point "River", until it intersects with the line camper-tent fire.

2. The shortest path in the burning tent lab, between two points, is a straight line from the point

3. Move the point "River", until it intersects with the line camper-tent fire.

## Snail Trail Graffiti

## Reflection

I created my snail trail, by draging points, and then I used geogbra to make them reflect. Because I am not very experienced in geogbra, this was very difficult. One thing that I noticed, when I applied the "trace", was how the points were moving in circles over and over again. Overall, this lab was not my favorite. I really liked how Caitlyn helped us by giving us the instructions on a hard copy of paper, because without them i would have been lost. This lab did how ever help me understand the basics of geogbra, after this lab, I think that the next time I use geogbra, I will be able to use it without as much struggle.