Scaling your world
Project description
During this unit of Math we studied concepts such as Similarity, Ratios/Proportions and Dilation. All these topics revolved around the sizing of different objects and how they can be manipulated to have different dimensions. We also focused on how two objects are similar because of their side lengths, angles, and such. The Scaling Your World Project focused all of these different mathematical terms and combined them to get us to work with objects that we see in our world. The goal was simple, find an object and decide whether you wanted to upscale it or downsize it by a scale factor. This would allow us to employ what we learned and give us an opportunity to see what the results would be.
For the first benchmark of this project, we had to decide what we wanted to scale, and if we wanted to do it in a group, or individually. I decided that I wanted to work in a group, that way I would have people to help me if I didn’t understand something properly. As a group we decide that we wanted to downscale Randy’s donut, which is a 10 meter tall sculpture that is on top of Randy’s Donut Shop. We thought it would be a cool idea, because we would be able to create our versions by making edible donuts. This was our plan and what we would stick with for the entirety of the Project. The second benchmark entailed finding the dimensions of your object and then scaling it according to the size you wanted your project to be. Since our model object was 10 meters tall, we decided that it would be best if we downscaled by 1000%. By doing so, we would be creating 10 cm tall donuts.
For the first benchmark of this project, we had to decide what we wanted to scale, and if we wanted to do it in a group, or individually. I decided that I wanted to work in a group, that way I would have people to help me if I didn’t understand something properly. As a group we decide that we wanted to downscale Randy’s donut, which is a 10 meter tall sculpture that is on top of Randy’s Donut Shop. We thought it would be a cool idea, because we would be able to create our versions by making edible donuts. This was our plan and what we would stick with for the entirety of the Project. The second benchmark entailed finding the dimensions of your object and then scaling it according to the size you wanted your project to be. Since our model object was 10 meters tall, we decided that it would be best if we downscaled by 1000%. By doing so, we would be creating 10 cm tall donuts.
Mathematical concepts
The concepts that we studied while working on this project were:
Congruence and Triangle Congruence This is when we have two shapes and they both have the same size, shape and angles. This means that all off the side lengths must match up and in the corresponding order, in order for them to be congruent. Triangle congruence is defined by having the same angles and having the same side lengths for all three sides. These are measurements that can be acquired by having a few of the dimension in the shape. This was a part of our benchmark number 2, because it focuses on making two congruent shapes and in order for us to be able to create our scale factor and project, we would have to create a replica of the original, and apply our changes to it. |
Definition of Similarity
Similarity is when there are two shapes that have the same sides and angles, but have different side lengths. These side lengths usually tend to be a multiple or factor of the previous shape. These shapes usually look similar and can easily be compared through measuring the angles and their corresponding placement. In our project we used similarity various times because we needed to have a similar product that was a smaller version of the original shape. |
Ratios and Proportions: Congruent Angles + Proportionate Sides
Congruent angles and proportionate sides are best represented with triangles and how they can they can be manipulated and measured easily. If there are two triangles with different side lengths, but congruent angles, then we know that the triangles are similar. Congruent angles can be identified as such by having the same angles on two shapes in the same place. This would make the angles congruent, and the shape would be considered similar. Proportionate sides can also be best represented through triangles because of how two similar triangles will have corresponding sides that have the proportionate sizes. This is the same with all shapes and will always be the same as long as they are similar shapes. In our project since we were making a scaled down version of a round shape, we used the diameter which is also a length and scaled it down to get a proportionate length. |
Proving Similarity: Congruent Angles + Proportional Sides
To prove that the angles are the same on two shapes we would have to use the angles that we have and compare them to the second figure. If the sides are proportional to the original and are in the corresponding place, then we can assume the angles are the same. It is relatively difficult to find the angles and see if the angles are congruent without the side lengths of the shape. With our product, in order to make sure that our calculations were correct, we also scaled down the circumference of the original and saw that it was a proportional change to our creation. |
Dilation, including scale factors and centers of dilation
Dilation is when a shape is shrunken or expanded according to a certain scale factor. This means that a shape can only be changed in size by doubling or reducing the size by half. A scale factor is usually written as a percentage, any value of 99% would be a decrease in size, and anything above 101% would be an increase. By giving a shape a scaling of 100% the shape wouldn’t increase or decrease, it would remain the same. The centers of dilation are usually two different centers. The first is the center of dilation of a shape, which means that the center of dilation is at the center of the shape and the borders are expanding or shrinking from the center. The second center of dilation is on a grid, when a shape is placed on a coordinate plane, the center is on the point (0,0). This was one of the most important processes in our project because it allowed us to be able to create a smaller version of the original. Without a scale factor we would have a shape that was neither congruent or similar to our inspiration, with dilation we found out what the size of our replica would be. |
Dilation: Effect on distance and area
Dilation can affect distance in the same way that it affects shapes, the increases and decreases in distance are proportionate to the scale factor. This is best represented while traveling in a linear fashion, if one is going at 50 mph, they have a scale factor of 50 mph. And this is the same case for area and volume since they can all increase proportionally to a scale factor. Although our project wasn’t about traveling distances, the mathematical concept of dilation was very important because it made up a huge part of our work. |
Exhibition
Our work for the first benchmark was to choose who we wanted to work with and decided what we wanted to scale down/up for our project. I decided to go in a group of three and together we chose that we wanted to scale down a donut sculpture from a donut restaurant. For the second benchmark we had to find the measurements of our object and create a scale factor for us to be able to create our version of the object. The second benchmark was to gather the dimensions of our object and then create new ones, using a scale factor. This would be our culminating product and the centerpiece of our project. This benchmark required the usage of proportions and dilation, in order to be able to create smaller or bigger versions of the original shape. The third benchmark was to create a physical version of your scale replica, and to be able to present it. Within my group since we decided to make a perishable, we made a video instead, displaying our mathematical process, and the actual food that we made. For our project we made donuts, and so we were successful in completing our task.
Video: |
Reflection
During the course of this project I felt that I was able to complete all of our tasks successfully and I put in a lot of effort to get the job done. The first benchmark was relatively simple and didn’t take too long for me to complete. The second benchmarks was slightly more difficult, since we had to do some calculations in order to be able to progress with our work. I decided that the best course of action for us would be to try out various methods and see who could get the calculations correct. We were patiently working on different solutions until we got one that we were happy with and so we completed that benchmark. The third benchmark was my favorite because it was a very hands on experience that allowed me to work in unison with the rest of my partners. Together we worked hard and fast in order to create donuts, which was a huge success. Overall this project allowed for me to learn about mathematical concepts such as dilation and proportions while also applying it to our world.