Description
The purpose of the project went beyond just learning about congruence, similarity, and scaling. The point was to show how to use these concepts to better your understanding of the outside world. To better our concept of how truly big or truly small something is like. How if you make a scale model of the moon and earth, how the distance between the two is large enough to fit every planet in our solar system and still have room, scales like that. Below is my example shown visually.
What We Did
We started simple. We were asked what we knew about similarity and congruence. We were then put into groups and given different mathematical concepts similar triangles, definition of congruence. However the one that most didn't understand was dilation., which was the concept given to my group. All of the groups shared out. The proceeding work we did was all about finding sides and angles of similar triangle. I chose the challenge which was an exceptionally difficult and in the end after hours and hours spent trying to solve it I couldn’t, I don’t think anyone did. Then we got to the worksheet that really kicked off the project for me. It was called Billy Bear. You would take a drawing of a bear and see as it scaled up how both its perimeter and area changed. After that worksheet we were given the prompt of the project. Take something or things and either scale it up or down and create a physical representation of that scale. Me and my partner both avid Pokemon fans Decided that we would create a portrait of 150 or so pokemon to scale with each other. The result was 150 calculations, 150 drawings, 150 traces, and two sets of very tired hands. And all that work will be explained below in the Benchmarks section.
Mathematical ConceptsBelow is a list of every mathematical concepts we covered, what they mean.
1. Congruence and Triangle Congruence: Congruency is when to shapes or segments are the exact same size. Triangle Congruence is how you can prove that two triangles are similar. If two sides and an angle are equal, if two angles are equal it is similar and combinations of those. 2. Definition of Similarity: If two shapes have proportional sides and the same angles they are similar. 3. Ratios and Proportions, including solving proportions: Ratios are a relationship between two or more values, such as a fraction. Like 3/5 4. Proving Similarity: Congruent Angles + Proportional Sides: If you have an unknown such as this situation 1/2 = 4/x. This means that the ratio of 1 and 2 is equal to 4 and x. So finding out what multiplied to 1 to get 2 gives you what multiplied to 4 gives you x. For the example that number is 2. 4 times 2 equals x 5. Dilation, including scale factors and centers of dilation: Dilation could be summed up as the act of scaling an object up or down. An image for showing center of dilation can be found below. The the scale factor is how much the original either increased or decreased. For example if the Factor is 2 than the new image is twice as big. And if it's 0.5 it's twice as small. For negative numbers the image is mirrored and then the factor is applied.
6.Dilation:effects on distance and area: Dilation does two different things to area and distance. For distance it is multiple by the scale factor so it creates a linear relationship. For area however it is the area squared time the area, so the relationship is non-linear because graphed it doesn't make a straight line. 7. Vertical angles and corresponding angles: Vertical angles are a pair of opposing angles made by two intersecting lines. Vertical angles are always equal to each other. Corresponding angles are two angles on the same side of one line with another line intersecting them. Both added together are always equal to 180 degrees. |
BenchmarksFor this project we had four major benchmarks set to keep us on track. Here are them all.
Benchmark 1: This benchmark was quite simple in the simple and required little to no math. For this benchmark we had to first choose a partner, if we wanted. Then because I did choose a partner we both had to come up with an object to scale up or down. We thought a little out side the box and came up with scaling Pokemon to each other. Both fans of the game we thought it would be cool seeing how they all accurately scaled to each other. We then had to decide how we would make it. Because Pokemon exist in a virtual space we decided that because of the large amount of Pokemon we were scaling it would be easiest to make them on a canvas. And that was the end of the first benchmark. Benchmark 2: This benchmark required the most math. This required us to do most if not all of our calculations of the Pokemon, so that meant about 150 calculations. We started by getting all 150 heights in meters. Because we for practical reasons couldn't draw out 150 I decided to make 1 meter equal 5 cm. So to apply this ratio to all of the other measurements I needed to scale of this ratio. Because I needed the same unit I converted 1 meter into 100 centimeters. I then proceeded to divide 5cm by 100 cm because I was going from 100 to 5 so the scale was shrinking. 5/100=0.05 so that was my scale factor. I then applied that to all 150 heights and got their scale heights. We turned in all our work and moved on to the next benchmark. Benchmark 3: This was the big one that took the longest. In this benchmark we had to make our final product. So me and my partner took two canvas and had to use the scaled heights to mostly freehand all 150 pokemon. While she did most of the more complex pokemon I helped fill in gaps with smaller ones. we then finished off them all by outlining them in sharpie. While it wasn't very hard or required complex materials It was very time consuming. You can find a picture of the final product below. Benchmark 4:This benchmark required the most thought. In this benchmark we had to upload everything we did onto our digital portfolio. This entire project page is my benchmark 4. |
Reflection
I quite enjoyed this project, very few projects have let me draw Pokemon for the final product. But in a seriousness hitting benchmark 3 was a challenge for both me and my group member. even though I didn't do even half of 150 Pokemon drawings it's a lot to do in the allotted time. We really had to work effectively to hit the dead line, which because of the amount of work our teach let use push back a couple of days so we could finish all of the work. We learned to keep on track, how to communicate what we were doing, and how to manage a lot of work. I felt like during benchmark 2 when calculating all the data I sub-consciously formed a pattern in my head, I did thing to quickly figure out the answers in a short amount of time. Normally you would have to convert the meters into centimeters then multiply that by 0.05. Instead I found that if you took the decimal point and moved it once to the left then divided that number in half I would get the answer I was looking for. So that let me breeze through all the math. For example: 1m would be 10 half of that is 5cm, or 0.7m would be 7 half of that is 3.5cm. if not for that method I developed I don't know how long it would have taken. I believe given more time my partner and I could have still made a better product, we could have maybe added details to the Pokemon or completely shaded them in. For my self I wished I had helped more with the drawing of the Pokemon so my partner wouldn't have to have done so many. However, I was very content with how this project turned out.