Unit 4 - Desmos Drawing and Function Families
Unit 3: Area, Volume, Measurement
Reflection:
1.) What content/skills have been the most interesting to you?
During class when we went over Surface Area I thought this was the most interesting content to me. I thought this was interesting because you can get the general SA from one equation (SA= 2 x area of base + lateral area) This stood out to me because when you look for area in any other shape you have to remember a specific equation for each shape. Being able to use a general equation for the SA is helpful and convenient. This is why I thought SA was the most interesting content to learn and have.
2.) And how has this content/skill helped you grow mathematically?
Learning Surface Area has helped me grow mathematically. By being able to "exercise" my brain really appeals to me because it's fascinating to be able to have the skills of knowing how much space a three-dimensional object has. Also knowing the area of a two-dimensional shape is good knowledge. Being able to have these skills is going to benefit me in the long run of my future careers and hopefully in college.
1.) What content/skills have been the most interesting to you?
During class when we went over Surface Area I thought this was the most interesting content to me. I thought this was interesting because you can get the general SA from one equation (SA= 2 x area of base + lateral area) This stood out to me because when you look for area in any other shape you have to remember a specific equation for each shape. Being able to use a general equation for the SA is helpful and convenient. This is why I thought SA was the most interesting content to learn and have.
2.) And how has this content/skill helped you grow mathematically?
Learning Surface Area has helped me grow mathematically. By being able to "exercise" my brain really appeals to me because it's fascinating to be able to have the skills of knowing how much space a three-dimensional object has. Also knowing the area of a two-dimensional shape is good knowledge. Being able to have these skills is going to benefit me in the long run of my future careers and hopefully in college.
Unit 2 Reflection: Shadows, Similarity and Right Triangle Trigonometry
Reflection:
Q1: What has been the work you are most proud of in this unit?
I am most proud of everything that I have done this semester. I feel like everything has had a significant importance this semester. Math is numbers and rules, based on a constant medium, to describe the world around us. I feel that everything we have right now in the 1st world county is all possible through math. Most of what we take for granted today is possible because of math.
Q2: What skills are you developing in geometry/math?
Besides giving me the natural analytic skills that math does I feel that Geometry has helped me grow as a student because it will give me skills that I will be using later in my life and career. I also feel that the main reason we are exposed to all this math is because we don't know what career we are going into and what math is required for it. Even if you do know what you are going to do you may change your mind later on and you won't have to start from the beginning. So not only have I been set with skills in geometry but I feel that it will benefit us all later in life.
Q3: similarity (ratios) or trigonometry.
Trigonometry is all about triangles. Trig is needed for calculus, which is needed for physics. Without trig, we could not calculate how much weight the floor could hold, buildings would collapse, and lots of people would be killed. Airplanes could not fly. etc.
Q1: What has been the work you are most proud of in this unit?
I am most proud of everything that I have done this semester. I feel like everything has had a significant importance this semester. Math is numbers and rules, based on a constant medium, to describe the world around us. I feel that everything we have right now in the 1st world county is all possible through math. Most of what we take for granted today is possible because of math.
Q2: What skills are you developing in geometry/math?
Besides giving me the natural analytic skills that math does I feel that Geometry has helped me grow as a student because it will give me skills that I will be using later in my life and career. I also feel that the main reason we are exposed to all this math is because we don't know what career we are going into and what math is required for it. Even if you do know what you are going to do you may change your mind later on and you won't have to start from the beginning. So not only have I been set with skills in geometry but I feel that it will benefit us all later in life.
Q3: similarity (ratios) or trigonometry.
Trigonometry is all about triangles. Trig is needed for calculus, which is needed for physics. Without trig, we could not calculate how much weight the floor could hold, buildings would collapse, and lots of people would be killed. Airplanes could not fly. etc.
POW #1: Knight Boredom No Longer
Problem Statement
Four knights, 2 white, and 2 black are sitting on a 3x3 chessboard. The knights were really bored, since they spent all of their time sitting on the chessboard doing nothing, so they decided to try switching places so that the white knights would end up where the black knights started our and the black knights would end up where the white knights started out. To do this, the knights had to follow the following rules:
- Two chess pieces can not occupy the same square at the same time
- Knights can jump or pass over each other on the way to an empty square
- The knights can only move 2 squares up (or down, or left, or right) and 1 square to the left (or right, or up, or down.) The moving combinations must be 2 up or down and 1 to the left or right. Or, 2 to the left or right and 1 up or down. Example: They can't move 2 to the left and 1 to the left. They must always move in an "L" shape.
- Two pieces can not switch spots at the same time
- The knights can only move one at a time
- They must stay within the 9 squares of their 3x3 chessboard
Four knights, 2 white, and 2 black are sitting on a 3x3 chessboard. The knights were really bored, since they spent all of their time sitting on the chessboard doing nothing, so they decided to try switching places so that the white knights would end up where the black knights started our and the black knights would end up where the white knights started out. To do this, the knights had to follow the following rules:
- Two chess pieces can not occupy the same square at the same time
- Knights can jump or pass over each other on the way to an empty square
- The knights can only move 2 squares up (or down, or left, or right) and 1 square to the left (or right, or up, or down.) The moving combinations must be 2 up or down and 1 to the left or right. Or, 2 to the left or right and 1 up or down. Example: They can't move 2 to the left and 1 to the left. They must always move in an "L" shape.
- Two pieces can not switch spots at the same time
- The knights can only move one at a time
- They must stay within the 9 squares of their 3x3 chessboard
Process
To solve this POW I read over exactly what it was asking and what the rules to solving it were. The easiest method to solve it was by Drawing different diagrams of the problem. I started out with drawing a 3x3 square and marking the positions on the board where the 2 white and 2 black knights were. To keep track of which knight was which, I named the uppermost left black knight Jim, or "J", the uppermost right black knight Bob, or "B", the bottom left white knight Timmy, or "T", and the bottom right white knight Alex, or "A." To start out, I drew a 3x3 square and drew the initials of the place where each knight was supposed to be. When I moved one Knight, I drew an arrow to the spot where I wanted the knight to go, and then showed the moved knight in the 3x3 square, along with another arrow showing a different knight moving. To keep track of the various moves that the knights took, for every new move the knights took, I drew a new 3x3 square and I drew where the knights from the previous square moved (drawn by arrows) ended up. I also “double-checked” my answer by doing a simple guess and check type method. Using a simple 3x3 grid I was able to check my answer again by: using a 3x3 grid with 4 green squares and 5 white squares. The left side of the tiles is labeled “1,2,3” and the bottom of the grid the tiles are labeled “K,A,Y”. To show my work I Found 16 ways to get the knights on the bottom of the grid to the top of the grid and vice versa with the knights on top of the grid. The way I did this was by: (w)= white squares and (g)= green squares.
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Solution
The "Big" Questions
1. It is possible for the black and white knights to switch places on a 3x3 chessboard.
2. For the solution to this POW I counted 16 moves for the black and white knights to switch places. I believe that my answer is the least number it would take for the knights to switch places, because I followed all of the rules and guidelines correctly, (No two chess pieces occupied any spot at the same given time, no two chess pieces switched into each others spot at the same time, the knights only moved one at a time, and they stayed within the 9 squares of the chessboard.) So, following all of the rules to this POW correctly, 16 moves would be the least number of moves it would take for the black and white knights to switch places.
3. It is possible.
The "Big" Questions
1. It is possible for the black and white knights to switch places on a 3x3 chessboard.
2. For the solution to this POW I counted 16 moves for the black and white knights to switch places. I believe that my answer is the least number it would take for the knights to switch places, because I followed all of the rules and guidelines correctly, (No two chess pieces occupied any spot at the same given time, no two chess pieces switched into each others spot at the same time, the knights only moved one at a time, and they stayed within the 9 squares of the chessboard.) So, following all of the rules to this POW correctly, 16 moves would be the least number of moves it would take for the black and white knights to switch places.
3. It is possible.
Evaluation
I definitely considered this POW to be educationally worthwhile, because it made me think critically, and it taught me that the answer to a problem won't always be obvious, and that in order to figure it out, you need to look at the problem from different perspectives, and in different ways. POW : Knight Boredom No Longer was challenging, because even if one move was incorrect, then it threw off all of the other moves that were made after it, and therefore I had to do this problem several times in order to get the solution.
I definitely considered this POW to be educationally worthwhile, because it made me think critically, and it taught me that the answer to a problem won't always be obvious, and that in order to figure it out, you need to look at the problem from different perspectives, and in different ways. POW : Knight Boredom No Longer was challenging, because even if one move was incorrect, then it threw off all of the other moves that were made after it, and therefore I had to do this problem several times in order to get the solution.
Self-Assessment
For the POW: The Big Knight Switch, I believe I deserve an B, because I am positive I found the correct solution by following all of the rules and guidelines to the problem correctly, and I understand the problem thoroughly. Although I did forget to turn this in on time and I possibly left out a few ideas.
For the POW: The Big Knight Switch, I believe I deserve an B, because I am positive I found the correct solution by following all of the rules and guidelines to the problem correctly, and I understand the problem thoroughly. Although I did forget to turn this in on time and I possibly left out a few ideas.
GGB Lab: The Burning Tent
TESSELLATION Project
Project Reflection
Question 1:
“ What is the idea /theme behind your tessellation?”
When I first started the project I had no idea that there were so many ways to create a different tessellations. After researching what tessellations are, I found out that they are “tiles” used to create an image on a flat surface without gaps or any overlapping. While creating my tessellation I used a post it note and made a translation tile. My tile ended up looking like a lobster. When I colored the end product, I really liked it so the title of my tessellation is “Sea Scavengers”. This is also a nickname for lobsters. That is basically the “theme” of my tesselation.
Question 2:
“What polygon(s) did you start with and how did you alter it (what transformations did you use)?”
I started with a simple 3inx3in Post It Note. You can do the same or construct a 3inx3in square. Then you draw a design on the square. After this you translate the design to the top, to the side, or onto both. Once you've translated them. Look at the “square” ask yourself “what do I see”? In my tessellation I saw a lobster. I added a smile and eyes to make it look realistic. This is the polygon that I used and how I altered it.
Question 4:
“In your opinion, are tessellation math or art? Justify your answer.”
In my opinion, tessellations are art. Tessellations are used in quilts, paintings, and architecture. They can be used in math as well but not as regularly. Tessellations can be used in quilts, the most common patterns that are tessellated are the “honeycombs, brick walls, and turtle shells”. Maurits Cornelis Escher was an artist who used graphics in his art. He was not a mathematician but still he was mainly the person who “created” tessellations. In architecture there is use of tessellations all around us. In ancient egypt and rome, they used tessellations to create mosaics using small pieces of stone embedded in walls and floors. This is an my opinion why tessellations are art.
Bibliography
http://nrich.maths.org/2577
http://en.wikipedia.org/wiki/Tessellation
http://www.math.com/students/wonders/tessellations.html
http://www.csun.edu/~lmp99402/Math_Art/Tesselations/tesselations.html
Question 1:
“ What is the idea /theme behind your tessellation?”
When I first started the project I had no idea that there were so many ways to create a different tessellations. After researching what tessellations are, I found out that they are “tiles” used to create an image on a flat surface without gaps or any overlapping. While creating my tessellation I used a post it note and made a translation tile. My tile ended up looking like a lobster. When I colored the end product, I really liked it so the title of my tessellation is “Sea Scavengers”. This is also a nickname for lobsters. That is basically the “theme” of my tesselation.
Question 2:
“What polygon(s) did you start with and how did you alter it (what transformations did you use)?”
I started with a simple 3inx3in Post It Note. You can do the same or construct a 3inx3in square. Then you draw a design on the square. After this you translate the design to the top, to the side, or onto both. Once you've translated them. Look at the “square” ask yourself “what do I see”? In my tessellation I saw a lobster. I added a smile and eyes to make it look realistic. This is the polygon that I used and how I altered it.
Question 4:
“In your opinion, are tessellation math or art? Justify your answer.”
In my opinion, tessellations are art. Tessellations are used in quilts, paintings, and architecture. They can be used in math as well but not as regularly. Tessellations can be used in quilts, the most common patterns that are tessellated are the “honeycombs, brick walls, and turtle shells”. Maurits Cornelis Escher was an artist who used graphics in his art. He was not a mathematician but still he was mainly the person who “created” tessellations. In architecture there is use of tessellations all around us. In ancient egypt and rome, they used tessellations to create mosaics using small pieces of stone embedded in walls and floors. This is an my opinion why tessellations are art.
Bibliography
http://nrich.maths.org/2577
http://en.wikipedia.org/wiki/Tessellation
http://www.math.com/students/wonders/tessellations.html
http://www.csun.edu/~lmp99402/Math_Art/Tesselations/tesselations.html