Project Description
This project focused on similarity of shapes and dilation, a geometric transformation. Dilation is also known as scaling, or zooming in/out. We combined these topic with functions and algebra to get a better understanding of why the geometry works. Our first activity in the project was to create posters explaining similarity, congruence, and dilation. Then, combined with related SAT practice problems and Problems of the Week, we started our final product. We were tasked to create our own scale model of anything we wanted. There were three benchmarks in this project: The first was simply to write up a proposal of our idea and how we plan to create it, the second was doing and writing down all the mathematical calculation involved with our idea, and the third was actually building our final product.
Mathematical Concepts
Congruence and Triangle Congruence: In order for two shapes to be congruent, they have to have two things: equal measurements and equal angles. (Measurements could be side lengths, diameter, radius, distances across a shape, etc.)
Definition of Similarity: The most basic definition of similarity is same shape, different size. We had to use similarity in our second benchmark, to make sure that all measurements of our pie tins were in proportion.
Ratios and Proportions: A proportion is an equation that expresses an equality between two ratios. The ratios are generally set up like fractions. Ratios can be set up multiple ways, depended on the complexity of the problem, and they can involve variables. The easiest way to solve a proportion problem is with algebra. The connection between proportions and similarity is that when two triangles are similar, you can use proportions to find missing side lengths.
Proving Similarity: Two shapes that are similar will share all the same angles, but have different side lengths. The two shapes are similar to each if it can be proved that corresponding sides are proportional.
Dilation: This was the last concept that we learned about in this unit, and probably the most complex. Dilation is a transformation, applied onto a shape, that produces a new shape that is similar to the first. The center of dilation is a fixed point in space, or on a plane, around which all the points of the shape will expand or contract. It is the only point involved in the dilation process that never changes. Based on the scale factor used in the dilation, the shape will either stretch or shrink. A scale factor (generally defined with the variable k) that is less than one, but more than zero, will shrink the shape. A scale factor that is more than one stretches it. If the scale factor equals one, the shape will stay the same. A scale factor that is less than zero, however, gets complicated. Any scale factor less than zero can result in either a stretched OR shrunk shape, as well as a reflection through the center of dilation. Scale factor was part of our first benchmark, where we had to specify by how much our object would stretch or shrink.
Dilation - Affect on Distance and Area: In a dilated image, distances, such as perimeter, scale proportionally to the scale factor. The equation (d'=k*d) represents this relationship. Areas will scale proportionally to the square of the scale factor (a'=k^2*a). Volumes scale proportionally to the cube of the scale factor.
Definition of Similarity: The most basic definition of similarity is same shape, different size. We had to use similarity in our second benchmark, to make sure that all measurements of our pie tins were in proportion.
Ratios and Proportions: A proportion is an equation that expresses an equality between two ratios. The ratios are generally set up like fractions. Ratios can be set up multiple ways, depended on the complexity of the problem, and they can involve variables. The easiest way to solve a proportion problem is with algebra. The connection between proportions and similarity is that when two triangles are similar, you can use proportions to find missing side lengths.
Proving Similarity: Two shapes that are similar will share all the same angles, but have different side lengths. The two shapes are similar to each if it can be proved that corresponding sides are proportional.
Dilation: This was the last concept that we learned about in this unit, and probably the most complex. Dilation is a transformation, applied onto a shape, that produces a new shape that is similar to the first. The center of dilation is a fixed point in space, or on a plane, around which all the points of the shape will expand or contract. It is the only point involved in the dilation process that never changes. Based on the scale factor used in the dilation, the shape will either stretch or shrink. A scale factor (generally defined with the variable k) that is less than one, but more than zero, will shrink the shape. A scale factor that is more than one stretches it. If the scale factor equals one, the shape will stay the same. A scale factor that is less than zero, however, gets complicated. Any scale factor less than zero can result in either a stretched OR shrunk shape, as well as a reflection through the center of dilation. Scale factor was part of our first benchmark, where we had to specify by how much our object would stretch or shrink.
Dilation - Affect on Distance and Area: In a dilated image, distances, such as perimeter, scale proportionally to the scale factor. The equation (d'=k*d) represents this relationship. Areas will scale proportionally to the square of the scale factor (a'=k^2*a). Volumes scale proportionally to the cube of the scale factor.
Exhibition
Benchmark #1: For this benchmark, we created a project proposal that described who we were working with, what we planned to scale, and how we would create our final product.
Benchmark #2: For this benchmark, we used a sketch to show what our scale factor would be and what our final product would look like. We labeled the diagram with all our mathematical calculations regarding the dimensions that we planned to scale. My group decided to scale a pumpkin pie using a scale factor of 50%. The circumference of the original pie was 9", so our scaled version ended up being 4.5". We made sure to measure our pie tins before we started the baking process for accuracy. The amount of filling used in each pie also ended up being scaled proportionally, so the smaller pies contained half the filling that the larger one did. We also created a design, piped in whipped cream, on top of each pie that was also scaled according to pie size so that we had more than one calculation being used.
Benchmark #3: For this benchmark, we created our actual final products. This project was supposed to be showcased a couple months later, at an exhibition, and we knew that our pies wouldn't last that long. So that we would have something to present at exhibition, we filmed the whole baking process.
(Watch our video here: youtu.be/LJLeGHkCbZ8)
Benchmark #2: For this benchmark, we used a sketch to show what our scale factor would be and what our final product would look like. We labeled the diagram with all our mathematical calculations regarding the dimensions that we planned to scale. My group decided to scale a pumpkin pie using a scale factor of 50%. The circumference of the original pie was 9", so our scaled version ended up being 4.5". We made sure to measure our pie tins before we started the baking process for accuracy. The amount of filling used in each pie also ended up being scaled proportionally, so the smaller pies contained half the filling that the larger one did. We also created a design, piped in whipped cream, on top of each pie that was also scaled according to pie size so that we had more than one calculation being used.
Benchmark #3: For this benchmark, we created our actual final products. This project was supposed to be showcased a couple months later, at an exhibition, and we knew that our pies wouldn't last that long. So that we would have something to present at exhibition, we filmed the whole baking process.
(Watch our video here: youtu.be/LJLeGHkCbZ8)
Reflection
The actual execution of this project was relatively simple for me and my group. Our biggest success was working together while planning and creating the pies, and in the end we had a great final product. However, I wish we would have planned out our video in advance, since it didn't turn out as well as it could have. The video was thrown together kind of last minute, and I feel like it shows. We spent almost a full day working on the pies, and failed to leave enough time to make a really engaging and impressive video. If we had made more decisions sooner regarding the video, it might have turned out better and reflected our excitement for the project. Another major challenge for our group throughout this project was scheduling. We eventually had to request an extension, since it was so difficult for us to find a time where we could all get together. If I could do this project again, I would probably challenge myself more, mainly with our scale factor. Since we decided on pies, we didn't have much leeway since there were only a few sizes of pie tins available to us, but I would've liked to have chosen something more complicated and interesting. I think that the Habit of a Mathematician that I utilized the most in this project was Staying Organized. While we were baking the pies, it was very easy to get all our recipes and ingredients mixed up, since there were three of us working at once. We had to make sure all our measurements were correct and that all our ingredients made it in.