DP Update
Unit 2 Reflection: Shadows, Similarity and Right Triangle Trigonometry
Q1: What has been the work you are most proud of in this unit?
Studying similarity and right triangles with Khan Academy helped me grasp the concepts quicker and be able to catch up with the class. On my Khan Academy account I have completed 14 modules that have been challenging and interesting, with these concepts. I think Khan academy is a powerful tool that I am proud and willing to use.
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 table initiating/approaching hard problems, e), or learning how to use your graphing calculator to fit equations to data.
The skills I am developing in Geometry are finding the area of shapes. On my worksheets I’m receiving A’s on them which shows that I am grasping the concepts that are being received by the teacher. I think I will be able to use these skills throughout my career in mathematics.
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 (i.e. scaled replicas of sculptures, scaled models for architecture, trigonometry in construction or blood splatter analysis, etc).
I think Trigonometry has been the most beneficial and makes senses. It is fairly difficult skill because of all the options trig can go into, such as inverse SIN, COS, TAN, as well as just SIN, COS, TAN. This makes Trig more challenging. Trig can help me by building houses and finding area of shapes.
Unit 3
Q1: What content/skills have been most interesting to you?
Q2: How have you grown mathematically?
**Be sure to include the questions in the DP Update!
Q1: The topics I have been interested in were visualizing and drawing these shapes out. This interested me because it was more mental and equations that you could draw out, which made it more interesting.
Q2: I believe that I have grown mathematically by having to look at problems differently. In this unit, I had to incorporate many different aspects of math into one project.
POW Update
Eli Cagan
Keegan Hickerson
Caitlyn Kneller p3
3/16/15
POW 4
The problem asked us to solve how many one sided, two sided, three sided, and no sided faces are there in any given cube that has been painted and made up of unit cube.
We began this problem by looking at a 5x5 cube. To find the exact amount of faces painted we counted each individual cube and counted the faces that were painted. Upon seeing the amount of faces painted we saw that not all the cubes were accounted for. There are 125 cubes that make up a 5x5 cube. In other words, the volume of a 5x5 cube is 125 1 cm cubes. Because not all the cubes were painted we found that there were 27 faces not painted. Then while looking at the results from the 5x5 cube we started to see some things that corresponded. For example we saw that all cubes will have 8 cubes with three faces painted. We saw this because all cubes have 8 corners. Then we started to look at how the amount of edges, corners, and faces that a cube had. Every cube has 6 faces, 12 edges and 8 corners. This is how we began to figure out the formula. We first came up with the formula of how to find the amount of cubes that only had one side painted. We saw that the ones that had one side painted were not edges or corners and eventually came up with this formula, 6(n-2)˘2. We proved that it worked by testing it out on a cube that we knew the answers to and it worked. After that we did the same process for the edges and corners and the next formula we came up with was for edges 12(n-2). After that we saw that all cubes have 8 corners that means that all cubes have 8 pieces with three sides painted. Below is an image to represent the faces painted.
The solution to the this problem is there are 54 one sided faces, 36 two sided faces, and 8 three sided faces. For the first one we did 9x6 because there are 9 cubes on one side, and a cube has 6 sides, 9x6. For 36=3x12, 3 two sided blocks and 12 edges. Finally 8x1= 8, one cube with three sides painted and 8 corners.
A rubiks cube is generally a 3 by 3 cube. If the cube was made of all 1cm cubes, how many cubes could fit in it? Now, figure out the volume, how much water can fit in it? This problem made us think about how to formulate an equation out of an object such as the cube. The POW improved our mathematical thinking skills. It helped improve them because it allowed us to think in 3D, instead of just 2D as we are used to. It took a lot of envisioning a cube in order to come up with our formulas.
Problem of the Week Reflection: How have problems of the week helped you grow mathematically? (Communicate ideas, breakdown problems, organize information, develop a growth mindset around mathematics)
Pows have been a challenging part of my geometry career. They have made me try and re-try many different ideas on one problem. Also, they have made me work together with students, and create beautiful work. The vast majority of my POWs were beautiful work. Yes i admit that a few fell a bit short but i think that most of them came out very nicely. I liked working with a partner because we could divide and conquer the project. We got the work done faster and with better quality.
Q1: What has been the work you are most proud of in this unit?
Studying similarity and right triangles with Khan Academy helped me grasp the concepts quicker and be able to catch up with the class. On my Khan Academy account I have completed 14 modules that have been challenging and interesting, with these concepts. I think Khan academy is a powerful tool that I am proud and willing to use.
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 table initiating/approaching hard problems, e), or learning how to use your graphing calculator to fit equations to data.
The skills I am developing in Geometry are finding the area of shapes. On my worksheets I’m receiving A’s on them which shows that I am grasping the concepts that are being received by the teacher. I think I will be able to use these skills throughout my career in mathematics.
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 (i.e. scaled replicas of sculptures, scaled models for architecture, trigonometry in construction or blood splatter analysis, etc).
I think Trigonometry has been the most beneficial and makes senses. It is fairly difficult skill because of all the options trig can go into, such as inverse SIN, COS, TAN, as well as just SIN, COS, TAN. This makes Trig more challenging. Trig can help me by building houses and finding area of shapes.
Unit 3
Q1: What content/skills have been most interesting to you?
Q2: How have you grown mathematically?
**Be sure to include the questions in the DP Update!
Q1: The topics I have been interested in were visualizing and drawing these shapes out. This interested me because it was more mental and equations that you could draw out, which made it more interesting.
Q2: I believe that I have grown mathematically by having to look at problems differently. In this unit, I had to incorporate many different aspects of math into one project.
POW Update
Eli Cagan
Keegan Hickerson
Caitlyn Kneller p3
3/16/15
POW 4
The problem asked us to solve how many one sided, two sided, three sided, and no sided faces are there in any given cube that has been painted and made up of unit cube.
We began this problem by looking at a 5x5 cube. To find the exact amount of faces painted we counted each individual cube and counted the faces that were painted. Upon seeing the amount of faces painted we saw that not all the cubes were accounted for. There are 125 cubes that make up a 5x5 cube. In other words, the volume of a 5x5 cube is 125 1 cm cubes. Because not all the cubes were painted we found that there were 27 faces not painted. Then while looking at the results from the 5x5 cube we started to see some things that corresponded. For example we saw that all cubes will have 8 cubes with three faces painted. We saw this because all cubes have 8 corners. Then we started to look at how the amount of edges, corners, and faces that a cube had. Every cube has 6 faces, 12 edges and 8 corners. This is how we began to figure out the formula. We first came up with the formula of how to find the amount of cubes that only had one side painted. We saw that the ones that had one side painted were not edges or corners and eventually came up with this formula, 6(n-2)˘2. We proved that it worked by testing it out on a cube that we knew the answers to and it worked. After that we did the same process for the edges and corners and the next formula we came up with was for edges 12(n-2). After that we saw that all cubes have 8 corners that means that all cubes have 8 pieces with three sides painted. Below is an image to represent the faces painted.
The solution to the this problem is there are 54 one sided faces, 36 two sided faces, and 8 three sided faces. For the first one we did 9x6 because there are 9 cubes on one side, and a cube has 6 sides, 9x6. For 36=3x12, 3 two sided blocks and 12 edges. Finally 8x1= 8, one cube with three sides painted and 8 corners.
A rubiks cube is generally a 3 by 3 cube. If the cube was made of all 1cm cubes, how many cubes could fit in it? Now, figure out the volume, how much water can fit in it? This problem made us think about how to formulate an equation out of an object such as the cube. The POW improved our mathematical thinking skills. It helped improve them because it allowed us to think in 3D, instead of just 2D as we are used to. It took a lot of envisioning a cube in order to come up with our formulas.
Problem of the Week Reflection: How have problems of the week helped you grow mathematically? (Communicate ideas, breakdown problems, organize information, develop a growth mindset around mathematics)
Pows have been a challenging part of my geometry career. They have made me try and re-try many different ideas on one problem. Also, they have made me work together with students, and create beautiful work. The vast majority of my POWs were beautiful work. Yes i admit that a few fell a bit short but i think that most of them came out very nicely. I liked working with a partner because we could divide and conquer the project. We got the work done faster and with better quality.