In this project, similarity and the geometric transformation dilation (also known as scaling, and colloquially referred to as “zooming in” or “zooming out”) are studied through modeling, exploration, and reasoning. Students will explore important ideas in geometry, and apply skills with functions and algebra. The primary context for this exploration is mathematical modeling and how geometry (specifically similarity and dilation) can be used to create physical models of real-world phenomena.
Purpose of the project
This project had many steps to it, but there were four assignments that mattered for our final draft. The project started by learning about different math concepts, but it ended with these four Benchmarks. Benchmark #1; Deciding what to scale down or up. In this benchmark, we had to decide what to scale up or down and what scale factor that will be. We had to write a paragraph to our teacher explaining who we will be working with, our object we are choosing to scale, the scale factor to which we are scaling, and how we are presenting it. Benchmark #2; Math behind the work. In this benchmark, we had to complete the math behind the work. We had to do all the math to scale up or down our object. At least three of the measurements had to be correct, but we were expected to try and get them all correct. Benchmark #3; Final Draft. In this benchmark, we had to create and turn in our finial draft. Some people made models of their exhibition piece, others made videos. It had to show what you chose to scale and the new version of that; bigger or smaller. Benchmark #4; DP Update. In this benchmark, we had to upload this DP update informing everyone of what we did for this project.
Mathematical Concepts
During this project we learned many different mathematical topics, the whole project was based on dilation and similarity. Definition of Similarity; Geometry. (of figures) having the same shape; having corresponding sides proportional and corresponding angles equal: similar triangles. Definition of Proportion; A proportion is a name we give to a statement that two ratios are equal. I Definition of Dilation; A dilation is a transformation (notation ) that produces an image that is the same shape as the original, but is a different size. A dilation stretches or shrinks the original figure. The description of a dilation includes the scale factor (or ratio) and the center of the dilation. The relationship between similarity and proportion tie together nicely. For two shapes to be similar, all the angles have to be in proportion. The side lengths do not need to be the same, because they would not be the same, not similar. For something to be similar, the proportions all have to line up. Also, you use proportions to find if two shapes are similar. If you are doing a math problem, that involves finding a missing side in two shapes, looking for proportions will help you find out if the shapes are similar. Basically, when you use one, you always use the other. The relationship between similarity and dilation are fairly simple. Similarity is the process of dilation. If you have one image that is bigger than the other, but with all mathematically similar sides, it is considered similar. One image is a dilation of the other image. They play together very nicely. If there are two image under dilation, they are similar. Benchmark #2 and benchmark #3 also had a relationship that flowed together nicely. In benchmark 2 we did all the math that we needed for benchmark 3. We took the twenty foot giraffe and scaled it down to two feet. We had to then scale all the body parts down to match. To do this, we took each number, from the original image, and multiplied it by 1/10. Once we completed that section (benchmark 2) we used those exact measurements to make our giraffe in benchmark 3.
Exhibition
For each benchmark my group did our own work around what it required.
Benchmark #1; Deciding we will scale down a giraffe. In this benchmark, we decided that we will scale down a twenty food giraffe to two feet. We told our teacher that we will scale it down by one tenth. We also had to chose how we will present this. We chose to present it on a black poster board and we would make the to scale giraffe with push-pins and connect them with yarn.
Benchmark #2; Figuring out the measurements. In this benchmark, we had to do the math and figure out how to accurately scale down a giraffe from twenty feet to two. First, we had to figure out how long each part of the body was. The whole giraffe is two hundred forty inches. The legs are 96 inches, the body is 36 inches, the neck is 96 inches, and the head is 12 inches. We multiplied each number by 1/10 to get the size of the smaller giraffe. After we did that we got all our measurements; the whole giraffe is now 24 inches, legs are now 9.6 inches, body is 3.6 inches, neck is 9.6 inches, and the head is now 1.2 inches. As reference, the picture to the left is our actual math that we turned in. Benchmark #3; Final Draft. In this benchmark, we had to create our actual two foot giraffe. We started by drawing it out onto the poster board. After we drew all the body parts to scale, we used almost one hundred push-pins to outline the drawing. Then, we used yarn to connect them and finish our final exhibition draft. The picture down below is our final draft.
Reflection
During this project I faced many successes and challenges. I faced challenges in areas such as deciding what to scale, what factor to scale it to, how to make it look professional and more. One challenge that really stuck out to was figuring out how to create our final draft. To not run into this challenge, I think I could have thought the idea thoroughly at the very beginning of the project. First, we wanted to just draw our giraffe onto a poster board, but we didn't think it was professional enough. It took us multiple drafts to come up with our final idea, but we got there and I am proud for over coming that challenge. One success that I had was the math behind it. I felt very proud when we accomplished the math and it was all correct. The Habits of a Mathematician came in handy for this project. I feel like I used "Collaborate and Listen" and "Be Confident, Patient, and Persistent". My partner and I had to collaborate and consider each others ideas to make many final decisions on our piece. I feel like I was confident, patient, and persistent during this project for a few reasons. I believe this because we had many drafts of our project, and it took lots of time to feel confident with the piece that we turned in. This project taught me math skills and skills that I will use in next projects. My number one take away skill is to never give up.I realized that no matter how difficult a project gets you can always keep trying and make it better. I thought that we could not make our final draft look professional enough, but with endless amounts of effort it was possible. I will definitely use this skill in my next project because I now know that my project can always be improved if I do not love it.