Unit 4: Desmos Drawing and Function Families- |
-How did you go about drawing this image? (Did you plan? Did you use patterns, reflections of curves? Did you experiment? Did you consult with peers/teacher?)
For this drawing I first had no idea what I was wanting to do but then I walked around the class and talked to my peers. They all had mentioned that they did something they enjoyed. So then I decided to do an flower. The way I started this was by finding a mage of an flower on the internet and decided I would use it as an tracing tool. From there I had put it into desmos and started tracing by making all the functions to outline the flower petals. Once I was finished I feel it looks somewhat like a flower but not exactly.
-How did using Desmos and creating this drawing help you understand function families and their transformations? (Be specific and discuss two types of functions and what you learned about them - for example, discuss how you understand linear and quadratic functions.)
Desmos was slightly confusing but in the end did help me better understand the functions. The ones that I best understood by the end of the project were the quadratic functions and the circle functions. I best understood these ones because I used them the most throughout my desmos drawing.
For this drawing I first had no idea what I was wanting to do but then I walked around the class and talked to my peers. They all had mentioned that they did something they enjoyed. So then I decided to do an flower. The way I started this was by finding a mage of an flower on the internet and decided I would use it as an tracing tool. From there I had put it into desmos and started tracing by making all the functions to outline the flower petals. Once I was finished I feel it looks somewhat like a flower but not exactly.
-How did using Desmos and creating this drawing help you understand function families and their transformations? (Be specific and discuss two types of functions and what you learned about them - for example, discuss how you understand linear and quadratic functions.)
Desmos was slightly confusing but in the end did help me better understand the functions. The ones that I best understood by the end of the project were the quadratic functions and the circle functions. I best understood these ones because I used them the most throughout my desmos drawing.
Unit 3: Area, Volume and Measurement-
1. What content/skills have been the most interesting to you?
- Surface area has been fairly interesting to me. It has been interesting for multiple reasons. One reason was because it taught me how to find the areas surface of any object.
2. And how has this content/skill helped you grow mathematically?
- This has helped me for many reasons. One reason is that it can help me in a future career. Also for math tests and other classroom uses.
- Surface area has been fairly interesting to me. It has been interesting for multiple reasons. One reason was because it taught me how to find the areas surface of any object.
2. And how has this content/skill helped you grow mathematically?
- This has helped me for many reasons. One reason is that it can help me in a future career. Also for math tests and other classroom uses.
Problem of the Week:
Problem of the Week Reflection:
How have problems of the week helped you grow mathematically? They have helped me think outside of the box as well as make sure my work is organized so it is more understandable.
How have problems of the week helped you grow mathematically? They have helped me think outside of the box as well as make sure my work is organized so it is more understandable.
Semester 2, POW 1 – Slices of Pie
Skylar Barr
January 7th, 2015 POW #1-Cutting The Pie Problem Statement: For this problem we were given a circular pie, and a figurative pie cutter. We were told from the beginning that our cutter can only slice straight lines through the pie. The task was to find the maximum amount of pieces of pie, according to a given maximum amount of cuts. For example, if you only cut a pie once, the maximum amount of pieces creatable it two. We were given the maximum amount of pieces for cuts one through 3, and we had to find the answer for ourselves on slices four and five. Question: Given the number of straight cuts, what’s the maximum number of pieces you can get? Process: Our process involved a bit of trial and error, and also we used a formula that we figured out as a class. At first we tried making a circle and drawing in lines, seeing how we could get the most pieces. The way in which we got the most pieces was if we crossed all the lines we had previously made. That began to get tedious so with Caitlyn’s guidance we made a table (as seen in the solution). We had an input and an output variable. The number of cuts was our input and the maximum number of pieces was our output. Then using this table we began to try and find a pattern but looking at the both the relationship between the output number and the relationship between the input and output numbers. With those relationships we came up with a pattern. The pattern was to add the number of cuts to the previous maximum number of pieces, we would get the next maximum number of pieces using this pattern. Again with Caitlyn’s help we came up with a formula for this pattern that we could use in future problems and to finish the POW. The formula was [f(n)=f(n-1)+n]. We used this to figure out the solution to our problem Solution: Using our formula we came up with the solution, [f(n)=f(n-1)+n]. This means add the number of cuts to the previous maximum number of pieces.The way in which we found our formula was first we said n= the number of cuts. The number of cuts (n)- 1 plus previous maximum number of pieces + n. That was our formula and we plugged in the numbers to test it, [f(3)=4+3=7], we tested multiple sets of numbers from our table. Our solution was that four cuts of the pie would create eleven pieces. Then five cuts of the pie would create sixteen pieces. The pattern and what the formula was telling us, was to add the number of cuts to the previous maximum number of pieces. Extension: An extension of this problem would be to find the max number of pieces, given the number of straight cuts in an octagon. I also feel that an extension could be to find the least number of pieces in a pizza, from a given number of straight cuts, although that would not be that hard. Also a question we could ask is: can this same question apply to a 3D object? Can I use a given number of straight cuts to make the maximum number of pieces? An example of this could be in a cylinder or a cone. Self-Evaluation: As a group we really enjoyed this pow more than any of the other pows that we have been assigned this year. We thought that it was educationally worthwhile because it challenged us mentally and we really had to think deeply. We learned that to get the most pieces, you have to cross each line that has already been made. We think that this problem was well laid out and couldn’t have been more specific. It was very clear in that regard. We used our time efficiently and had fun while doing it. It was confusing at first, and challenged us. Therefore it was the perfect level of challenge for us. We would say we deserve 25/25 points on this pow. |
Semester 2, POW 2 – Pick Up Triangles
POW #2: Pick up Triangles
Lillian and Skylar- 2/3/15 Problem Statement: How many similar triangles can you make using four rods, their lengths being 6,4,3 and 2 and taking two more rods from a pile with an unlimited supply of rods, their lengths being all whole numbers up to 20 inches? You must use the original four rods in all pairs of similar triangles you find. Process: To start we took our four rods and laid them out in a proportion. If the rods were in proportion to each other we knew that they would be similar. For example the rods 2,3 were part of a triangle and the rods 4,6 were part of triangle. That was my first proportion 2:4 and 3:6. This proportion is 1:2. we then looked at the last side. The proportion still had to be 1:2 so starting at 3 we added the other sides, keeping in mind the sum of the shorter sides had to be greater than the longest or else it wouldn’t form a triangle. With that in mind we knew we couldn’t have 5 and 10 as a side. Between 3:6 and 10:20 with the exception of 5:10, we made similar triangles where all sides were proportionate. After we couldn’t come up with anymore we set up a new proportion, 3:2 and 6:4. The proportion of 2:3. Using the same method as we used for the last proportion we found 2 more triangles within that proportion of 2:3. We knew there were no more similar triangles because there were no more proportions we could set up. Some of our “key” insights on this POW were probably when we actually figured out we were only using 2 sticks from the pile with unlimited supply and also when we figured out how to match the proportion we had set up. Solution: The solution we came up with was 10 total similar triangles. The triangles we came up with were: 2,3,2 and 4,6,4 2,3,3 and 4,6,6 2,3,4 and 4,6,8 2,3,6 and 4,6,12 2,3,7 and 4,6,14 2,3,8 and 4,6,16 2,3,9 and 4,6,18 2,3,10 and 4,6,20 3,6,6 and 2,4,4 3,6,3 and 2,4,2. Extension: An extension of this project could be to have different length rods, both to start off with and in the pile of unlimited supply. Using this same problem method we could use the same principles of proportion and the extra rods, but for a trapezoid. Evaluation: So far we think that this has been the best POW and the most related to what we are learning. At first we found it confusing and unclear but when we sat down and truly looked at the problem we felt it was more clear and not as difficult as we originally anticipated. We learned even more about similar triangles and proportions and we feel much more confident. We definitely thought about this problem and we feel we deserve a 25/25 on this POW. |
Unit 2 reflection: Shadows, Similarity and Right Triangle Trigonometry.
Q1: What has been the work you are most proud of in this unit?
The work I have been most proud of in this unit would be the shadow work. I am most proud of it because I feel like I really tried hard and put a lot of effort into it to try and understand it. I also paid closer attention to the subject then I have in the past. Also I had liked learning it more than the previous topics which made it a more understandable topic to learn. In the end I had grasped the concept of how to find the length of an shadow.
Q2: What skills are you developing in geometry/math?
In geometry I am learning many skills but one particular which would be the graphing calculator. I feel this skill will help me in future math classes because you can do so many different things on the calculator that other calculators are unable to do. They can also help me when I am unsure of how an graph should look. So then I can put in all my data and it would create a graph for me then I would know how the graph should look. I will have this skill throughout my entire life and could become very helpful depending on where I end up working or majoring in.
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.
One of the topics we are learning is Trigonometry. Trigonometry is used to define relations between elements in a triangle. It can also be defined as the study of triangles. In a triangle there are 3 sides and 3 angles. Measuring angles is the most fundamental skill of trigonometry. One example would be how to find the sin of 35 degrees, which would be .57. An application of trigonometry in the real world that interests me would be in scaled models for architecture. This interests me because I like architectures that are more different from other regular buildings with different shapes.
The work I have been most proud of in this unit would be the shadow work. I am most proud of it because I feel like I really tried hard and put a lot of effort into it to try and understand it. I also paid closer attention to the subject then I have in the past. Also I had liked learning it more than the previous topics which made it a more understandable topic to learn. In the end I had grasped the concept of how to find the length of an shadow.
Q2: What skills are you developing in geometry/math?
In geometry I am learning many skills but one particular which would be the graphing calculator. I feel this skill will help me in future math classes because you can do so many different things on the calculator that other calculators are unable to do. They can also help me when I am unsure of how an graph should look. So then I can put in all my data and it would create a graph for me then I would know how the graph should look. I will have this skill throughout my entire life and could become very helpful depending on where I end up working or majoring in.
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.
One of the topics we are learning is Trigonometry. Trigonometry is used to define relations between elements in a triangle. It can also be defined as the study of triangles. In a triangle there are 3 sides and 3 angles. Measuring angles is the most fundamental skill of trigonometry. One example would be how to find the sin of 35 degrees, which would be .57. An application of trigonometry in the real world that interests me would be in scaled models for architecture. This interests me because I like architectures that are more different from other regular buildings with different shapes.
Geo-Gebra labs:
GGB: Snail Trail Lab-Questions:Question 1: What appears to happen to the other five points when you drag point D around the screen? How does every other point move (i.e. D, D', D", D''')?
- When I dragged point D around, all the other points started to follow and they made the same pattern as the one I was dragging. Every other point moves like a reflection from the original one. |
GGB Burning Tent Lab-Questions:Question1: Once you have a minimal path, what appears to be true about the incoming angle and outgoing angle?
-The incoming angle is not that much different from the outgoing angle, they both had a similar degree. Question 2: Why is the path from points Camper to Tentfire' the shortest path? Briefly Explain. - The path is short because to get the the tent you have to go by the river to even get the tent. But to get to point tentfire you have to make a detour to go get the water. 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? - It should be in exact relation to both points, because it will make the distance the shortest. If the River point were in a different place the distance would be longer and not much of an efficient way to get the camper to the river to the tentfire'. |