Scaling Your World
Overview
The purpose of this project was to remind us of math concepts we’ve seen over the past years, as well as teach us new things that apply to those math concepts. More specifically, we studied four basic math concepts thoroughly: congruence, similarity, proportion, and dilation. (See Mathematical Concepts) These concepts were first planted in our minds when we talked about our prior knowledge through posters and mini presentations. We dove deeper into all of these topics with worksheets and activities. Eventually, we kicked off the bulk of the project by choosing an object and scaling it to be bigger or smaller than what it really is or was. We did all of the calculations and visually represented them. (See Project) Above, you'll see a picture of the product that my project partner and I created.
The purpose of this project was to remind us of math concepts we’ve seen over the past years, as well as teach us new things that apply to those math concepts. More specifically, we studied four basic math concepts thoroughly: congruence, similarity, proportion, and dilation. (See Mathematical Concepts) These concepts were first planted in our minds when we talked about our prior knowledge through posters and mini presentations. We dove deeper into all of these topics with worksheets and activities. Eventually, we kicked off the bulk of the project by choosing an object and scaling it to be bigger or smaller than what it really is or was. We did all of the calculations and visually represented them. (See Project) Above, you'll see a picture of the product that my project partner and I created.
Math Concepts
1. Congruence and Triangle Congruence Two triangles are congruent when they have the same angles and side lengths. Location on the plane and orientation are irrelevant to congruence. Rigid transformations can be performed to test whether or not the two shapes are congruent. Congruence is the root of similarity. 2. Definition of Similarity Two triangles are similar when their side lengths are proportional to each other, and their angles are identical. You should be able to multiply or divide all of the side lengths of one triangle by a common number to get the side lengths of the other triangle. You can do this because they are proportional. Also If two shapes are similar, you should be able to dilate one of the shapes to be congruent to the other. 3. Ratios and Proportions, including solving proportions Ratios and proportions are essentially a relationship between a set of numbers. They can be represented with a colon (a:b = c:d) or with a fraction (a/b = c/d). Similar shapes (or shapes that have been dilated) have proportional side lengths and areas. 4. Proving Similarity: Congruent Angles + Proportional Sides You can prove that two triangles are similar even if you have limited information. You only need to know a certain amount of angle and/or side length values to prove that two triangles are similar. This is because the side lengths are all proportional and the angles are all congruent. 5. Dilation, including scale factors and centers of dilation A dilation is when a shape is shrunken or inflated while preserving its shape and proportions. Important factors to consider when observing the proportions between two similar shapes are the scale factor and the point of dilation. These numbers are also integral to a separate proportion involving “projection lines” that a dilated figure must fall on. 6. Dilation: Effect on distance and area When a figure is dilated, the area and all of the side lengths are increased/decreased in a special way. When the size of an object is increased, the side length values are multiplied by the scale factor. The area is multiplied by the scale factor squared. When an object shrinks in size, the side lengths are multiplied by one over the scale factor. The area is multiplied by one over the scale factor squared. |
Project
Benchmark #1 was all about planning what our artifact was going to be and proposing it. I immediately had an idea in my mind as the teacher was explaining the project. I wanted to scale the heights of Pokemon and visually represent them. I found a partner who was also interested in Pokemon, and together, we got started on Benchmark #2, which involved doing the dilation calculations and turning them in. My partner and I agree that our goal of completing 151 calculations was a bit ambitious, but we were able to finish them all. We multiplied each of the Pokemon's heights by 0.05 to dilate, and then by 100 to convert from meters to centimeters. Here are the 3 largest and the 3 smallest Kanto Pokemon (in meters). Largest: 1.) Onix - 8.8 m (Dilated: 44 cm) 2.) Gyarados - 6.5 m (Dilated: 32.5 cm) 3.) Dragonair - 4.0 m (Dilated: 20 cm)* Smallest: 1.) Diglett - 0.2 m (Dilated: 1 cm) 2.) Caterpie, Weedle, and others - 0.3 m (Dilated: 1.5 cm) 3.) Pikachu, Meowth, and others - 0.4 m (Dilated: 2 cm) Benchmark #3 eventually kicked off. The goal was to create an artifact to display your calculations visually. Some people chose to make sculptures, while others chose to make a two dimensional display. My partner and I were a part of the latter group. Together, we completed 98 out of 151 Pokemon drawings, all representative of their respective heights. Our piece is not void of mistakes, though. Only after completing much of the art piece did my partner and I realize that the snake-like Pokemon, (Onix, Gyarados, Ekans, Dratini) are special in their listed heights. Their listed height most likely referred to length instead. Therefore, those drawings in particular are flawed in logic. Other than the aforementioned snake-like Pokemon, we believe that we properly represented the heights of Pokemon relative to each other. The fourth and final Benchmark involves displaying our work through an exhibition, as well as a page on a digital portfolio. *Not displayed |
Reflection
I learned about more than just mathematics. I learned to consider the ramifications of an idea before executing it, and I also strengthened my ability to stay organized and systematic.
When I proposed my project, I did not consider certain implications of my idea. I didn’t think about the fact that I’d have to do 151 calculations, I’d have to do 151 drawings, I’d have to find a canvas large enough to fit 151 differently sized creatures, among other things. In the end, we weren’t able to complete the latter two tasks, but we were able to form compromise for both of them. Had I considered these things first, I might’ve switched ideas, but what’s done is done, and I’ve learned my lesson.
On a separate note, I’m very proud of my systematic organization throughout this project. The first thing I did when I got the green light for my idea was create a document to store all of the necessary data and calculations. That document is linked here. (Forgive me, I’d include all of the data in the page, but it’s awfully large and dense. Slow computers beware of this link!) I made sure to do the steps accurately and in order for every data point. That, I feel, is something to be happy about.
I think that, despite everything, I have plenty of reasons to be proud of what my partner and I accomplished. The project I chose was more repetitive than it was difficult,and the product itself is a tad sloppier than I wish it was, but It still shows a lot of effort. Overall I’m happy with how I executed this project, and, as always, I look forward to the next.
I learned about more than just mathematics. I learned to consider the ramifications of an idea before executing it, and I also strengthened my ability to stay organized and systematic.
When I proposed my project, I did not consider certain implications of my idea. I didn’t think about the fact that I’d have to do 151 calculations, I’d have to do 151 drawings, I’d have to find a canvas large enough to fit 151 differently sized creatures, among other things. In the end, we weren’t able to complete the latter two tasks, but we were able to form compromise for both of them. Had I considered these things first, I might’ve switched ideas, but what’s done is done, and I’ve learned my lesson.
On a separate note, I’m very proud of my systematic organization throughout this project. The first thing I did when I got the green light for my idea was create a document to store all of the necessary data and calculations. That document is linked here. (Forgive me, I’d include all of the data in the page, but it’s awfully large and dense. Slow computers beware of this link!) I made sure to do the steps accurately and in order for every data point. That, I feel, is something to be happy about.
I think that, despite everything, I have plenty of reasons to be proud of what my partner and I accomplished. The project I chose was more repetitive than it was difficult,and the product itself is a tad sloppier than I wish it was, but It still shows a lot of effort. Overall I’m happy with how I executed this project, and, as always, I look forward to the next.