The purpose of this project was to obtain information about dilation and how to apply it to real scenarios. Within the project, we were taught different mathematical skills such as: congruence and triangle congruence, the definition of similarity, ratios and proportions, proving similarity, and scale factors of dilation. To kick off the project we were told to get into groups of 4. Dr. Drew put up a list of geometry terms and each group picked one to research. We had two class periods to design a poster that could be presented to the class. Each of the 7 topics were related to congruent, proportional, and similar shapes. The topic my group got was rigid motion. Within the topic of rigid motion there are 3 types of motion: rotation, reflection, and translation. After presentations, we got a series of worksheets that went more in depth of topics we were learning such as ratios and proportions and scale factors of dilation. We would get usually a half hour to an hour to work on each worksheet. After completing each sheet, we would go over it as a class so everyone would be on the same page about approaches to take to get to the same answer. After finishing around 12 worksheets over the course of about 2 weeks, Dr. Drew gave us our final project for this series called “Scaling Your World”. For this project you would take an object and scale it either larger or smaller. For my product, I scaled a mural that is painted in Carlsbad called “Catnap”. I would scale this painting onto a canvas painting it with acrylic paints. I went to the mural and measured all of the dimensions I needed such as the height, width, and size of a tiger. I then used those measurements and scaled the rest of the mural. I used the scale factor of every 1 foot would be scaled down to one inch. The actual mathematical scaling part was taking me a lot longer than I anticipated, so I emailed Dr. Drew for an extension of a few days. After scaling each raindrop and the size of the tiger, I was ready to paint. It took me around 3 1/2 hours to paint the whole mural. After I was done painting, I was happy with the way it came out and how taking extra time was very beneficial in order to produce a product I felt comfortable turning in. The products from the whole class will be exhibited at our Winter Exhibition.Mathematical Concepts 1. Congruence and Triangle Congruence Both of the triangles on a grid have identical sides lengths and angles but are not always in the same position. The definition of congruence is “Exactly equal in size and shape”.
2. Definition of Similarity 2 triangles that are the same scale but have a size difference.
3. Ratios and Proportions In the sense of triangles, equal ratios = proportional 4. Proving Similarity: Proportional Sides and Congruent Angles To prove that triangles are similar find equal sides and if the corresponding sides are the same ratio. AA, SAS, SSS are ways to figure out if they’re similar. AA is stands for angle: if the angles are the same. SAS means side, angles, side: if the ratios of 2 sides are the same and the angles will be the same. SSS is side, side, side: the 3 sides have the same ratio.
5. Dilation: Scale Factors and Centers of Dilation Transforming a shape that is constructed the same shape but a different size.
6. Dilation: Effect on Distance and Area The shape stays in it's original place but the size can vary.
All of these concepts can connect to triangles. Dilation and similarity connect because both triangles are different sizes and their shapes can be in 2 different forms. Similarity and proportion have identical ratios. Similar triangles are proportional meaning they have equal ratios. Benchmark #2 can connect because trying to figure out a scale factor that would be reasonable was challenging. Benchmark #3 can connect because I was actually making the product while applying terms such as dilation and proportions.
Exhibition
Benchmark #1 This benchmark was a short edmodo assignment that was an intro to the project. For this, we had to answer 4 questions: 1. Who else is on your team? 2. What item/object are going to scale? 3. How are you going to decide on the scale factor? 4. How will your scale model be constructed and exhibited?
To which I answered: 1. I am working by myself 2. I am going to scare the "Catnip" mural that is located in Carlsbad 3. Using inches, I will convert to a fraction of the actual size as the mural is 18 feet 4. I will paint it on a canvas with acrylics and scale it down
Benchmark #2 This benchmark was all the math behind the product. I went to Carlsbad and measured out the size of the mural. I then wrote out each rain drop, it's length, and the length after scaling it. On grid paper, I drew out the replica of the mural scaling it 1 cm = 1 ft. I then drew it out again scaling it with 1 cm = 1/2 ft, which made it a bit challenging not to get confused with how many feet I had drawn out. I wanted to see the difference between the two using a skill I had been previously taught: dilation. With doing this, I was able to know I wanted my final product to have a scale factor of 1 in = 1 foot on the canvas.
Benchmark #3 This benchmark was actually creating the final product. I knew I wanted to use acrylic paints because they’re easy to work with. I first sketched out the whole mural onto my canvas using the scale factor 1 in = 1 foot. After sketching, I started using the paint. Each raindrop was a different color and trying to match all the colors was hard. Eventually my palette looked like a rainbow. After about 3 hours of painting, I let whatever I had done dry in order to prevent smudging when I put a top coat on. I was very happy with the way the final product came out as I am artistically challenged. I chose to do a painting to try and challenge myself, and it was definitely a challenge.
Reflection This project was definitely a challenge for me, but I put that on myself. I chose to create something that I knew would be difficult. I liked the challenge. I chose to work independently because I work best alone. I was able to create a product that I felt was successful by myself. I think I used the habit of a mathematician “stay organized”. There was a massive amount of calculations that went into this project and I knew I had to keep my work organized in order to not miscalculate or get confused. I kept my notes clean and spaced to not have a bunch of work smashed together. I was very happy with the outcome of my final product and knew that asking for an extension could benefit “beautiful work”. If I could go back, I would have chosen to scale it onto a bigger canvas in order to have a bigger final product. Overall, I enjoyed this project because I got to pick something that would challenge me.