Project Description: In this project we focused on similarity and dilation, which is also zooming in or out. We used mathematical modeling and geometry to help us create a model of something scaled up or down. This project was really open ended , we were able to choose anything we wanted, and my group chose to scale a pizza down. We started the project by getting into small groups. Each group had a topic and we made a presentation about what we knew about it. Then, we got handouts everyday that focused on the concepts that would help us understand the scaling project. We also had benchmarks where we presented our learning and our progress making the scale model. Mathematical Concepts The six topics we went over in this project: 1. Congruence and Triangle Congruence Congruence and triangle congruence is when two shapes have equal sides and angles. The relationship between similarity is the shapes are the same shape but they may not be of the same size. For triangles, we can tell when they are congruent if their sides are equal. We didn’t use any triangle shapes in our pizza scale, because the shapes are mostly circular, which is why we didn’t use triangle congruence.
2. Definition of Similarity Similarity is when shapes are similar from the sides or angles. We used similarity in our project by making the pepperoni, crust and slices similar to the original pizza.
3. Ratios and Proportions, including solving proportions I believe the most important concepts that we used to help us get the right measurements for our pizza ingredients were ratios and proportions. Using ratios we measured the diameter of the pepperonis and divided it in half to get the right measurements. For example, if the crust was one inch we would use proportions and ratios to find out how small we needed to scale it down to.
4. Proving Similarity: Congruent Angles + Proportional Sides We proved similarity by working on a worksheet called similar problems. This worksheet had multiple problems that had two similar polygons. We had to make equations that would tell us the lengths of the sides so we can determine how they were similar. We also had to find the length of the variables.
5. Dilation, including scale factors and centers of dilation Dilation is when two shapes are the same but are different sizes. When you are dilating a shape you can shrink or enlarge it. In our first benchmark we used scale factor to show how our scale model would be created. The center of dilation doesn’t move even when you enlarge or shrink the shape, because the center will stay in the middle. One worksheet that helped us practice was “Dilation with Rubber Bands”. For this worksheet we started by putting a dot on a piece of paper. The next step was to take a pen and place it on the center of dilation, and with one side of the band looped around it. With the second pen inside the end part of the other band. Then we stretched the band and traced the knot along the shape while drawing with the other pen too. This worksheet taught me about dilation and the visual hands on part was really helpful. 6. Dilation: Affect on distance and area (re: Billy Bear) For the billy bear problem, he was growing by a scale factor of two then it changed to 3. For each week billy grew, we had to fill in a chart with the perimeter and area of billy. Using dilation, we had to find out what his perimeter and area would be in twenty weeks. Exhibition: For the first benchmark we had to propose a scale model. We wrote down who was in the group, what item we were going to scale, how we were going to decide the scale factor, and how we were going to construct and exhibit it. This benchmark’s purpose was to show that we had an idea of how we were going to make the model. The second benchmark was asking us to do the mathematical calculations to prove that our model would actually be a scale model. This benchmark was important because we did the math to be able to understand the mathematical purpose of the project. Our scale factor was ½ x .5, because we were going to make our pizza scale half of the original. Everything is divided by two. The third benchmark was to make the scale model and present it. We made our pizza out of wood, because we wanted to have a physical form of a pizza not a video. Benchmark two was thinking of the math and benchmark 3 was making it using those calculations. We knew it was important to use the exact measurements to make sure the pizza was to scale and dilated. The last benchmark was the DP update, to show what the project was and how we grew as a mathematician because of it.
This is our final product!
Reflection I think all of the benchmarks and worksheets helped me get more comfortable with the 6 mathematical concepts. My group wasn’t hard to work with, and we all helped each other when we were confused. A challenge I faced in this project was being sick and absent, I was unable to communicate with my partners. However, even though I was gone, we all still got the work done without leaving me to do nothing. Another challenge I faced was the math for the pizza. I didn't really understand scale factor and dilation, which made scaling the pizza down complicated. Something I could have done differently to make the outcome of the first challenge better, is do all of the work when it was assigned, that way we would not be worrying about it later on in the week. A way I could’ve solved the second challenge is by reviewing the worksheets because those covered the topics we needed for that benchmark I was struggling to do. A habit of a mathematician that I used in this project is take apart and put back together. It took us awhile to make our pizza, we had different plans and they didn’t end up working out. This habit of a mathematician really describes benchmark 3 for me. We had to stay calm and slowly come up with ideas to bring the project together. Doing group work is stressful because you have to listen to each other and sometimes your ideas aren’t used in the project. However, in this project the collaborate and listen habit played a role for me personally. I feel like I grew a lot in this skill, group work and collaborating. One moment where I used this habit of a mathematician was the first benchmark. We all had different ideas and eventually came up with one we all liked. Overall, I enjoyed the project and took away new math concepts that I needed to be able to visually represent our pizza.