Project Description
The purpose of the "Scaling Your World" project was to understand dilation, and how shapes can transform. We learned about similarity, dilation, and congruence. To start the project, we got into groups of 4, and were assigned different geometry topics to create a poster on. These topics included congruent, proportional, and similar shapes. I got the topic of similar shapes. After we presented our posters, we used the topics we learned to scale an object. We started by drawing a scale model of the object, and measuring every line and angle in the shapes. We then scaled the object larger or smaller. To do this, I multiplied all the lines of the original by 2, to create a 2x scale model. The product for benchmark 2 was a piece of paper with the original drawing of the iPhone at normal size, and the 2x larger iPhone. After, we made a final product to be displayed at exhibition. My partner and I drew a detailed 2x scale drawing of our object. We did this in pen, on graph paper. To make the curves of the iPhone look more smooth, we used a technique where we made a rough outline of a curve, and loosely traced it over and over again with short strokes to make the line look smooth. We used a ruler for the rest of our lines.
Mathematical Concepts
Here are some mathematical concepts we learned in this project:
1.) Congruence and Triangle Congruence
Two triangles that have the same size, side lengths, angles, and proportions but are not neccesarily in the same position. The real iPhone was congruent to the first scale of the iPhone we drew.
2.) Definition of Similarity
Triangles that have the same angles, and have lines that are proportionate to each other, but are not the same size. Our drawing for benchmarks 2 and 3 were similar.
3.) Ratios and Proportions
The word proportional means having the same ratios. If one triangle has the side lengths 1,2,3, and another triangle has the side lengths 2,4,6, they are proportional. This is because 1:2 is equal to 2:4 and 3:6 when simplified. We used this in benchmarks 2 and 3, because we multiplied the sides lengths of the iPhone by 2. Because they were all multiplied by the same amount, we knew they were still proportionate.
4.) Proving Similarity
There are a few methods to prove that triangles are similar. These are identified by the acronyms AA, SAS, and SSS. AA stands for angle angle. If two angles are the same, the third angle must be the same, and the triangle are similar. SAS stands for side angle side. If two sides are proportionate, and one angle is the same, the triangles are similar. SSS stands for side side side. If all three sides are proportionate, the shapes are similar. Our 2x scale iPhone was similar to the real iPhone.
5.) Dialation
This is when a shape grows or shrinks in size, but remains in the same place with the same proportions. This means that all the sides are still proportional to each other, as they were in the original shape. Dialation a shape affects its area; if a shape is dilated by the scale factor X, the shape's area is changed by X^2 (X squared). We could find the area of the 2x iPhone using this equation.
All of these concepts conect to Triangles, and the scales we drew in our Benchmarks. Similarity is a key element in dialtion. If the shape is made larger but not kept proportionate, it is not similar. Congruency requires similarity, but also requires the exact same side lengths.
Exhibition
We had three benchmarks in this project.
The first one required us to decide on what item we were going to scale, and how we were going to do it. This helped me brainstorm and come up with different ideas on what to create. This benchmark helped me think realistically about this project. Originally, I wanted to do something involving scaling food, like a giant brownie. This bench mark made me realize that wasn't realistic.
The second benchmark required us to submit our first drawing of the dilated object. My partner and I decided to scale an iPhone playing the game Flappy Bird. We took the original screenshot, measured every part of it, and multiplied the whole thing by two. We had to be very meticulous about measuring every line on the screenshot we took, and on the actual phone. At one point, our drawing stopped looking proportionate to the original. We erased the drawing and tried again. It looked much more like the original the second time. We used some of the math concepts we learned in this benchmark. For example, if the iPhone had a height of 50 cm, 50x2=100. So, the 2x scale model had a height of 100 cm. We used the concepts of similarity and dialation.
Benchmark three had us create a model of our scaled product and submit it. This caused us to bring our measurements to life. My partner and I drew a 2x scale model of the phone playing Flappy Bird. We decided not to color in our drawing, because we wanted it to look more like a blueprint. In Hindsight, I think it would've looked better had we colored it in carefully, and payed attention to detail. It could've looked like an actual iPhone actually playing Flappy Bird. We used the same mathematical concepts that we had in the second benchmark.
Reflection
I think that this project was fun and exciting. A success I had in this project was working with my partner in scaling our phone for benchmark 2. We agreed on how to do things and got it done very quickly. A challenge I encountered in this project was miscommunication. My partner and I thought we only had to draw a model of the iPhone for the second benchmark. In actuality, we had to make a scale of it. We didn't realize this until we turned it in though, so we ended up having to revise the assignment. I used the being systematic habit of mathematicians in this project. I was very systematic in how I measured every possible line and distance in the screenshot of the game. This project made me think about scaling things in a whole new way. I think that if I were to do this project again, I would try to pick something more original to scale. Lots of people ended up scaling iPhones. I also would have come up with a better product than just a drawing. I also would have tried coloring in the final drawing. As I explained before, my partner and I were going for a blueprint-type look with this drawing, as evidenced by the lack of color and blue pen writing. I think if we had colored it in really well, it could have looked very nice.
The purpose of the "Scaling Your World" project was to understand dilation, and how shapes can transform. We learned about similarity, dilation, and congruence. To start the project, we got into groups of 4, and were assigned different geometry topics to create a poster on. These topics included congruent, proportional, and similar shapes. I got the topic of similar shapes. After we presented our posters, we used the topics we learned to scale an object. We started by drawing a scale model of the object, and measuring every line and angle in the shapes. We then scaled the object larger or smaller. To do this, I multiplied all the lines of the original by 2, to create a 2x scale model. The product for benchmark 2 was a piece of paper with the original drawing of the iPhone at normal size, and the 2x larger iPhone. After, we made a final product to be displayed at exhibition. My partner and I drew a detailed 2x scale drawing of our object. We did this in pen, on graph paper. To make the curves of the iPhone look more smooth, we used a technique where we made a rough outline of a curve, and loosely traced it over and over again with short strokes to make the line look smooth. We used a ruler for the rest of our lines.
Mathematical Concepts
Here are some mathematical concepts we learned in this project:
1.) Congruence and Triangle Congruence
Two triangles that have the same size, side lengths, angles, and proportions but are not neccesarily in the same position. The real iPhone was congruent to the first scale of the iPhone we drew.
2.) Definition of Similarity
Triangles that have the same angles, and have lines that are proportionate to each other, but are not the same size. Our drawing for benchmarks 2 and 3 were similar.
3.) Ratios and Proportions
The word proportional means having the same ratios. If one triangle has the side lengths 1,2,3, and another triangle has the side lengths 2,4,6, they are proportional. This is because 1:2 is equal to 2:4 and 3:6 when simplified. We used this in benchmarks 2 and 3, because we multiplied the sides lengths of the iPhone by 2. Because they were all multiplied by the same amount, we knew they were still proportionate.
4.) Proving Similarity
There are a few methods to prove that triangles are similar. These are identified by the acronyms AA, SAS, and SSS. AA stands for angle angle. If two angles are the same, the third angle must be the same, and the triangle are similar. SAS stands for side angle side. If two sides are proportionate, and one angle is the same, the triangles are similar. SSS stands for side side side. If all three sides are proportionate, the shapes are similar. Our 2x scale iPhone was similar to the real iPhone.
5.) Dialation
This is when a shape grows or shrinks in size, but remains in the same place with the same proportions. This means that all the sides are still proportional to each other, as they were in the original shape. Dialation a shape affects its area; if a shape is dilated by the scale factor X, the shape's area is changed by X^2 (X squared). We could find the area of the 2x iPhone using this equation.
All of these concepts conect to Triangles, and the scales we drew in our Benchmarks. Similarity is a key element in dialtion. If the shape is made larger but not kept proportionate, it is not similar. Congruency requires similarity, but also requires the exact same side lengths.
Exhibition
We had three benchmarks in this project.
The first one required us to decide on what item we were going to scale, and how we were going to do it. This helped me brainstorm and come up with different ideas on what to create. This benchmark helped me think realistically about this project. Originally, I wanted to do something involving scaling food, like a giant brownie. This bench mark made me realize that wasn't realistic.
The second benchmark required us to submit our first drawing of the dilated object. My partner and I decided to scale an iPhone playing the game Flappy Bird. We took the original screenshot, measured every part of it, and multiplied the whole thing by two. We had to be very meticulous about measuring every line on the screenshot we took, and on the actual phone. At one point, our drawing stopped looking proportionate to the original. We erased the drawing and tried again. It looked much more like the original the second time. We used some of the math concepts we learned in this benchmark. For example, if the iPhone had a height of 50 cm, 50x2=100. So, the 2x scale model had a height of 100 cm. We used the concepts of similarity and dialation.
Benchmark three had us create a model of our scaled product and submit it. This caused us to bring our measurements to life. My partner and I drew a 2x scale model of the phone playing Flappy Bird. We decided not to color in our drawing, because we wanted it to look more like a blueprint. In Hindsight, I think it would've looked better had we colored it in carefully, and payed attention to detail. It could've looked like an actual iPhone actually playing Flappy Bird. We used the same mathematical concepts that we had in the second benchmark.
Reflection
I think that this project was fun and exciting. A success I had in this project was working with my partner in scaling our phone for benchmark 2. We agreed on how to do things and got it done very quickly. A challenge I encountered in this project was miscommunication. My partner and I thought we only had to draw a model of the iPhone for the second benchmark. In actuality, we had to make a scale of it. We didn't realize this until we turned it in though, so we ended up having to revise the assignment. I used the being systematic habit of mathematicians in this project. I was very systematic in how I measured every possible line and distance in the screenshot of the game. This project made me think about scaling things in a whole new way. I think that if I were to do this project again, I would try to pick something more original to scale. Lots of people ended up scaling iPhones. I also would have come up with a better product than just a drawing. I also would have tried coloring in the final drawing. As I explained before, my partner and I were going for a blueprint-type look with this drawing, as evidenced by the lack of color and blue pen writing. I think if we had colored it in really well, it could have looked very nice.