**Unit 4 - Desmos Drawing and Function Families**

DESMOS Graph Link:

https://www.desmos.com/calculator/ffqezrl1em

(Copy and Paste into your search engine to pull up graph)

https://www.desmos.com/calculator/ffqezrl1em

(Copy and Paste into your search engine to pull up graph)

## Embeded Reflection:

## unit 3 area, volume, measurement

Question 1:

The content and skills that have been the most interesting to me in this unit has been learning about circles and cylinders and the area and volume equations and techniques involved. I really like using pie and using my calculator to find all these things.

Question 2:

These concepts have helped me grow mathematically because I have been challenged and learned to just follow directions and it will be ok. I have problems sometimes with reading directions and the equations for these problems are so specific I've learned that I really need to just follow the directions to succeed. Knowing how to do these things has helped me with the classwork that we have has throughout this unit.

The content and skills that have been the most interesting to me in this unit has been learning about circles and cylinders and the area and volume equations and techniques involved. I really like using pie and using my calculator to find all these things.

Question 2:

These concepts have helped me grow mathematically because I have been challenged and learned to just follow directions and it will be ok. I have problems sometimes with reading directions and the equations for these problems are so specific I've learned that I really need to just follow the directions to succeed. Knowing how to do these things has helped me with the classwork that we have has throughout this unit.

## unit 2 shadows similarity and trigonometry

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

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

The thing I am most proud of was our second to last quiz, I exemplified my newfound trigonomic knowledge through the form of acing my quiz. I was very proud of myself because I had originally thought that I was not good at trig. I grew by finding some confidence in areas that I am not normally confident in.

**Q2: What skills are you developing in geometry/math?**Skills can be applied across mathematics – think graphing, creating tables, creating diagrams or mathematical models, approaching problems in different ways (by testing cases, by testing extreme examples, by setting up a tab initiating/approaching hard problems, e), or learning how to use your graphing calculator to fit equations to data.

I am learning how to do many things including using my TI-84 calculator and the biggest takeaway that I've found is reading directions. Before I used to infer things too much but through this unit I have found that you can't get anything done without reading all of the directions first, that will make your work much better.

**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 that interests you**(i.e. scaled replicas of sculptures, gearing ratios, scaled models for architecture, trigonometry in construction or blood splatter analysis, etc).

Trigonometry is the study of the measures of angles and side of a triangle. It is used to solve for unknown sides or angles of a triangle. This interests me because I think it is cool that you can determine a completely unknown variable by using other sides or angles of a triangle. This can be applied for a contractor finding a roof's pitch or unreachable height.

## how have pow's helped me grow mathematically?

These concepts in POW's this year have helped me grow mathematically because I have been challenged and learned to just follow directions and it will be ok. I have problems sometimes with reading directions and the equations for these problems are so specific. I've learned that I really need to just follow the directions to succeed. Knowing how to do these things has helped me with the classwork that we have has throughout this unit. Now that I have struggled through things like POW, I know that I can succeed in the future by using problem solving and critical thinking to my benefit.

## pow, almost a rubik's cube

TJ Rifkin

March 16th, 2015

Geometry

Caitlyn Kneller 6th period

POW #4 Almost A Rubiks Cube

Problem Statement:

In this pow everyone is given a 5x5x5 cube, what we all know commonly as a rubik’s cube. We are told that each side of the cube has been completely painted and there was no dripping. Each person has to come up with an equation to find the amount of painted faces that will work accurately for different sized cubes.

Proccess:

In the beginning of this POW I was confused so I began by getting some graph paper and sitting down with Orion and he helped me understand. On the graph paper he and I drew the cube diagram to help me better understand the dimensions and how many cubes there were. Once I knew the amount of faces painted looking at the single faces in the diagram I could multiply according to the total amount of sides on the cube. Sadly this diagram could not help us when it came to finding an equation, so we set up tables for each cube to find a pattern. This pattern would eventually help us develop an equation. One thing that was continuous through all the cubes was that as long as the shape you are working with is a cube, then there will always be 8 cubes in the shape that have three of the faces painted. Those are the corner pieces. The next thing that we found was that in the cubes that had 2 faces painted, they were always increasing by 12cm2. This helped us find the equation n-2*12. The last thing we found was that the cubes with one face painted the surface area’s were all multiples of six, this gave us the equation n=6*n-2.

Solution:

5x5x5

Similar Problem:

You are given a pyramid, each face of the pyramid is completely painted and no paint drips. The base of the pyramid is 15 cm long and each unit is 1cm long. The pyramid sides have an angle of elevation of 40 degrees, solve for the height of the pyramid using trigonometry and find the area of the Pyramid. Find an equation for area in a pyramid.

Evaluation:

In this POW I was challenged in many ways, I had a hard time wrapping my mind around the concept for a while until Orion helped me figure it out. I had to push myself to get the work done even when I didn’t want to and I had to reach out to my peers for help. Overall I believe I deserve 24/25 points.

March 16th, 2015

Geometry

Caitlyn Kneller 6th period

POW #4 Almost A Rubiks Cube

Problem Statement:

In this pow everyone is given a 5x5x5 cube, what we all know commonly as a rubik’s cube. We are told that each side of the cube has been completely painted and there was no dripping. Each person has to come up with an equation to find the amount of painted faces that will work accurately for different sized cubes.

Proccess:

In the beginning of this POW I was confused so I began by getting some graph paper and sitting down with Orion and he helped me understand. On the graph paper he and I drew the cube diagram to help me better understand the dimensions and how many cubes there were. Once I knew the amount of faces painted looking at the single faces in the diagram I could multiply according to the total amount of sides on the cube. Sadly this diagram could not help us when it came to finding an equation, so we set up tables for each cube to find a pattern. This pattern would eventually help us develop an equation. One thing that was continuous through all the cubes was that as long as the shape you are working with is a cube, then there will always be 8 cubes in the shape that have three of the faces painted. Those are the corner pieces. The next thing that we found was that in the cubes that had 2 faces painted, they were always increasing by 12cm2. This helped us find the equation n-2*12. The last thing we found was that the cubes with one face painted the surface area’s were all multiples of six, this gave us the equation n=6*n-2.

Solution:

5x5x5

- 54
- 36
- 8

- N=6*n-2
- N=n-2*12
- Always 8

Similar Problem:

You are given a pyramid, each face of the pyramid is completely painted and no paint drips. The base of the pyramid is 15 cm long and each unit is 1cm long. The pyramid sides have an angle of elevation of 40 degrees, solve for the height of the pyramid using trigonometry and find the area of the Pyramid. Find an equation for area in a pyramid.

Evaluation:

In this POW I was challenged in many ways, I had a hard time wrapping my mind around the concept for a while until Orion helped me figure it out. I had to push myself to get the work done even when I didn’t want to and I had to reach out to my peers for help. Overall I believe I deserve 24/25 points.

## semester 1 pow 2 the congruent triangles

POW #2

TJ Rifkin and Grace Wolf-Cartier

11-11-14

Question: For this problem, we were asked to create an original triangle and then re-create that triangle so that 5 of the parts were the same, yet they wouldn’t be congruent. The elements of the triangle that we were able to change were the sides and angles of this problem.

Process: In the process that we used to solve this problem we used scratch paper to draw diagrams of each triangle, we drew the original image, and called it triangle A. We decided that throughout the problem we would not alter triangle A, but rather triangle B. We were given this idea by Caitlyn in order to understand our options more clearly. We used color to understand our triangles better visually, and from there we began altering each of our new triangles. We did this by only changing the sides and not the angles. We moved the sides in relation to the angles, and there were 6 different possible locations for the sides. We used triangle congruence shortcut criteria to rule out corresponding congruent parts. We could see that the other triangles would not work because we rearranged them and they were congruent with the original shape.

Solution: We found that the solution to this problem involved 2 possible answers. Throughout this problem, we also decided to leave the angles in the same place and just move the sides around.

Evaluation: During this POW we struggled a lot because we were unsure of the different criteria of this problem. We worked through most of the problem separately and then joined together to find the final answer and do the write-up. I think we deserve a 27/30 because we thought thoroughly and took our time to make the most sense out of the problem.

TJ Rifkin and Grace Wolf-Cartier

11-11-14

Question: For this problem, we were asked to create an original triangle and then re-create that triangle so that 5 of the parts were the same, yet they wouldn’t be congruent. The elements of the triangle that we were able to change were the sides and angles of this problem.

Process: In the process that we used to solve this problem we used scratch paper to draw diagrams of each triangle, we drew the original image, and called it triangle A. We decided that throughout the problem we would not alter triangle A, but rather triangle B. We were given this idea by Caitlyn in order to understand our options more clearly. We used color to understand our triangles better visually, and from there we began altering each of our new triangles. We did this by only changing the sides and not the angles. We moved the sides in relation to the angles, and there were 6 different possible locations for the sides. We used triangle congruence shortcut criteria to rule out corresponding congruent parts. We could see that the other triangles would not work because we rearranged them and they were congruent with the original shape.

Solution: We found that the solution to this problem involved 2 possible answers. Throughout this problem, we also decided to leave the angles in the same place and just move the sides around.

Evaluation: During this POW we struggled a lot because we were unsure of the different criteria of this problem. We worked through most of the problem separately and then joined together to find the final answer and do the write-up. I think we deserve a 27/30 because we thought thoroughly and took our time to make the most sense out of the problem.

## semester 1 pow 1 the knights

TJ Rifkin

Pow #1 Questions and write up

Problem Statement:

In this problem I was given the task of switching chess pieces, specifically knights, around a 3x3 grid. The black knights began in the bottom two corners of the grid, and the white pieces in the top corners of the grid. In the end I had to make the white pieces switch places with the black pieces The knights can only move in an L shape, meaning they travel two and over one in any direction. However they cannot land on a space that is already occupied by another piece.

Process:

While I attempted to solve this problem I began with a white board, I continually drew the pieces moving around again and again to get the right answer, but I had a hard time. Eventually I did this on a piece of paper. I drew out the moves until they switched. Originally I did it in 20 moves, but I thought I could do better. I tried this process again and came to 16 moves. after trying to do better than 16 moves 3 times in a row I knew that I could not and 16 was the right answer. My evidence would be the way I used the white boards to problem solve in the beginning, and my work on paper eventually to get the right answer. To be honest I made completely random movements, and I could not find an accurate pattern. In fact, every time I came to 16 I moved the pieces differently.

Solution:

My solution was 16 moves, I know my solution is correct and complete because after I came to 16, I tried 3 times to do better and could not. I proved my answer right through rigorous problem solving and trial and error.

Evaluation/Self Assess:

I would say that I deserve 30 points on this POW because I tried my hardest and came to an answer that I know is right, I proved to myself that it’s right through trial and error. I worked my hardest on this pow and worked on it at home as well as school. Therefore my effort and answers should prove that I deserve the score of 30. In this project I learned that even when Problems are challenging, I can work hard and get the right answer.

Pow #1 Questions and write up

- Yes, they can do it.
- 16
- When I sat down with a piece of scratch paper and problem solved by moving the pieces around, I came to a point where I was able to switch them.

Problem Statement:

In this problem I was given the task of switching chess pieces, specifically knights, around a 3x3 grid. The black knights began in the bottom two corners of the grid, and the white pieces in the top corners of the grid. In the end I had to make the white pieces switch places with the black pieces The knights can only move in an L shape, meaning they travel two and over one in any direction. However they cannot land on a space that is already occupied by another piece.

Process:

While I attempted to solve this problem I began with a white board, I continually drew the pieces moving around again and again to get the right answer, but I had a hard time. Eventually I did this on a piece of paper. I drew out the moves until they switched. Originally I did it in 20 moves, but I thought I could do better. I tried this process again and came to 16 moves. after trying to do better than 16 moves 3 times in a row I knew that I could not and 16 was the right answer. My evidence would be the way I used the white boards to problem solve in the beginning, and my work on paper eventually to get the right answer. To be honest I made completely random movements, and I could not find an accurate pattern. In fact, every time I came to 16 I moved the pieces differently.

Solution:

My solution was 16 moves, I know my solution is correct and complete because after I came to 16, I tried 3 times to do better and could not. I proved my answer right through rigorous problem solving and trial and error.

Evaluation/Self Assess:

I would say that I deserve 30 points on this POW because I tried my hardest and came to an answer that I know is right, I proved to myself that it’s right through trial and error. I worked my hardest on this pow and worked on it at home as well as school. Therefore my effort and answers should prove that I deserve the score of 30. In this project I learned that even when Problems are challenging, I can work hard and get the right answer.

## geogebra lab: the burning tent

Geogebra lab questions:

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

They end up becoming the same measure.

Question 2: Why is the path from points Camper to TentFire' the shortest path? Briefly explain. (Think about the shortest distance between two points.)

Because it is the same distance as camper-river-tent fire

Question 3: Where should the point River be located in relation to segment Camper to TentFire' and line AB so that the sum of the distances is minimized?

Move point river to where it intersects the line camper-tentfire

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

They end up becoming the same measure.

Question 2: Why is the path from points Camper to TentFire' the shortest path? Briefly explain. (Think about the shortest distance between two points.)

Because it is the same distance as camper-river-tent fire

Question 3: Where should the point River be located in relation to segment Camper to TentFire' and line AB so that the sum of the distances is minimized?

Move point river to where it intersects the line camper-tentfire

## Snail trail GRAFFITI reflection

I made my snail graffiti by dragging points and mirroring them within the different sections of my circle that I created in geogebra. This was not done entirely easily. Something that I noticed when I turned the trace on was the way that when the points were moving, they all move equally in circles around each other. much like electrons and protons in the outer rings of an atom. It exemplifies the symmetry of a circle, without the color it rotates constantly to make a perfect circle. Overall this lab was fun but challenging and I would probably like to do it again. In the end, this project was very goof practice for me to get better at using geogebra, and I enjoyed it.

## my tessellation

## tessellation questions and response

The theme to my tessellation is "under the sea". I chose this theme because I thought it would look good and I really like turtles. I incorporated turtles and some kind of orange fish to make it look beautiful, I worked very hard to color to the best of my ability to make the poster look well done. I also will add seaweed and bubbles to make it less boring in the end. I started with the polygon that is a square of about 2.5 x 3 inches. I altered it by cutting out the top shape of a turtle and glueing it to the top of the shape from that angle. Then I rotated it 90 degrees and did the same process again but with the shape of a fish. If I had to describe whether a Tessellation is art or math I would say that it is more math. I say this because without the math a free hand art drawing can't tessellate properly. The whole idea has to start with math otherwise it isn't accurate. After you get the math right, sure you can incorporate as much art as you want. But without the math it's not a true tessellation. A professor at Cornell university said" A tessellation all starts with the math, it's that simple, Art is applied post creation." Also Mr. Chad Joyce, a high school math teacher in Wisconsin said, "Math is the base of all things that tesselate. The math is what makes the objects tessellate accurately."

In conclusion, I believe that tessellations are based in math, and that art is applied after you have tessellated correctly. As far as my tessellation goes, I tried my absolute best to make it aesthetically pleasing. I also made it tesselate perfectly. The thing that I did the best on was the coloring, and the only thing I wish I could have done better would be my time usage in class. I wish I could've worked on this because I pushed really hard to finish early and then I had days to just chill. And during that time I got distracted.

In conclusion, I believe that tessellations are based in math, and that art is applied after you have tessellated correctly. As far as my tessellation goes, I tried my absolute best to make it aesthetically pleasing. I also made it tesselate perfectly. The thing that I did the best on was the coloring, and the only thing I wish I could have done better would be my time usage in class. I wish I could've worked on this because I pushed really hard to finish early and then I had days to just chill. And during that time I got distracted.

Sources:

http://www.tessellations.org/

http://www.tessellations.org/essays-escher-or-alhambra-tessellations.shtml

http://www.tessellations.org/

http://www.tessellations.org/essays-escher-or-alhambra-tessellations.shtml

## Cut out and glue to the top in same spot. the repeat when you switch to the other side.

http://www.whatdowedoallday.com/2013/02/math-art-tessellations.html

http://www.mathsisfun.com/geometry/tessellation-artist.html

http://www.tessellations.org/

http://www.mathsisfun.com/geometry/tessellation-artist.html

http://www.tessellations.org/