This project was called 'Scaling Your World.' In this project we picked something important to us to scale. I started out wanting to do the U.S.S. Enterprise, then realized that, "Hey, maybe let's not do something super complicated." We started off by brainstorming what we wanted to do and then spent a little time working on our projects. We were allowed to work in a pair or alone, and mine was a very weird situation. Started out working in a team, then alone, then in an odd pair were we created two similar scaled drawings of whales. (She drew an Orca, I drew a Beluga). We began this by talking about congruency, similarity, and dilation. We made posters to elaborate on each subject. We spent several days on dilation and it became the central feature of this project, which ended, as said above, with a scaled artifact.
So, my understanding on each individual subject was fairly limited to what I'd learned in ninth grade. So I knew a little about all of them beforehand, just not enough, I guess. Congruency In Geometry, congruence is defined as "Identical in form, coinciding when superimposed." Triangles are congruent when they have three sides exactly the same, and the exact same three angles. They are essentially the same thing. if you can flip a triangle on it's side, it will still be congruent to the one not flipped on its side. They have corresponding sides and angles. The flippers on my whale were congruent because they are the same shapes on opposite sides of the whale. The two sides of the fluke were congruent, as well. If you fold up the fluke, the shapes would be the same and so would the angles. If they weren't, the whale wouldn't be able to swim straight. Similarity: In geometry, Similarity occurs when the only difference is size and possibly the need to move, turn, or flip the object. Congruence and similarity are very similar (no pun intended). They both require the shapes to have the same angles. The only major difference is congruence requires both objects to be the same size. This whole project is basically about similarity. The original item or drawing is always going to be similar to the scaled up or down version because the only difference between them is size. Ratios and Proportions A proportion is the name we give to two equal ratios. Two equal fractions, A over B, is equal to C over D. Proportions say that two ratios (or fractions) are equal. If the ratios are the same, they are in proportion. Between the scaled down or scaled up whales, they are in proportion because their ratios are the same. We scaled down the whale using a a 1ft equals 14ft ratio. This kept my whale in proportion. Proving Similarity, Congruent Angles, and Proportionate Sides Two pairs of proportionate sides and a pair of equal included angles mean a triangle is similar. There are three major techniques for proving similarity and congruence, "Side angle side," "Side side side," and "Angle Angle." My whale doesn't have a dorsal fin, so this made proving congruence and similarity more difficult. Dorsal fins are typically triangular. Dilation (Including Scale Factors and Centers Of Dilation) Dilation is were the polygon grows or shrinks, but keeps the same overall shape. The scale factor is the amount it grows or shrinks, and the center of dilation can be anywhere but it's the point it grows from or shrinks to. It directly affects a shape. When you grow or shrink a whale and dilation occurs, instead of staying congruent, the object becomes similar instead. Dilation: Affect on distance and area The area of a dilation is going to grow or shrink depending on an object. Distance is going to be a product of the scale factor instead. If the scale factor is two, the point on the object is twice the distance from the center of dilation. We used the center of the body mass of the whale as the center for our dilation, so the ratios, dilation and scale factor are all the same.
Benchmarks Benchmark 1 was essentially planning our individual project. I wanted to work with Hannah, but our ideas just ended up in different places until the end. We had to pick a partner or alone, and then explain our scale factor and turn it in on Edmodo. Benchmark 2 was when I decided to do my own thing and draw a beluga whale. It's 1' to 14' scaling factor which can be deduced to 6'' to 7', 3'' to 3.5', or 1.5'' to 1 3/4.' All of these are equivalent scaling factors. This was my final art piece:
It's still a 1' to 14'.
Reflection: I found this project pretty much in the middle of my skills. I enjoyed the idea of scaling something, sure, and I enjoyed getting to pick it, but I was bored with the math seeing as it was kind of extended review for me. I think that I wish I would have tried to find a more interesting thing to scale, as much as I love whales, I don't think it was the right choice for me. If anything, this did help with my starting small skills. I had to start small, just to make this project work the right way.