Project Description:
I believe that the main purpose of executing this project was to fully understand ratios and proportions and how they may directly correlate to our surroundings. I also think that problem solving was greatly incorporated into this project as well as dilation. Before the process of this project took place, I was aware of the word dilation and some of its definitions, but hadn’t fully gone into the mathematical uses/definition of it so it was helpful to learn more about that in the course of this project.
In the beginning of this project, we started with a review on the topic of congruence/similarity by creating posters based on a component of similarity and then presenting them. My group’s poster went over the principles of reflection, rotation, and translation. We also reviewed ratios and how to prove that two triangles are similar for certain by using the rules of angle-angle (AA), proportional sides (SSS), and two sides and angle (SAS). After that (and the Shortest Path problem), we launched into dilation and learned more about dilation’s mathematical meaning and scale factors/centers of dilation. To learn about the effect on distance and area that dilation has, we did a worksheet called Billy Bear Grows Up, where we used isometric dot paper to dilate a shape representing Billy bear. We also estimated/calculated Billy bear’s size if he continued growing in the same dilation pattern. To further our investigations on dilation, we watched a video where a group of men decided to scale down the solar system and create a to-scale model of every planet/star and place them the correct distance from each other. This video lead to a series of benchmarks where we first had to, either working in a group or alone, come up with an object, building, or place that we would be scaling. Then, we did the calculations necessary to figure out the size of our scaled model and drew diagrams so we could label them with our calculations. Finally, we created our scaled model and did a final peer/self assesment.
Mathematical Concepts:
The main mathematical concepts we went over during this project were congruence, similarity, ratios, proportions, solving proportions, and dilation. The deeper lessons we went into throughout the project were: Congruence and Triangle Congruence, the Definition of Similarity, Ratios and Proportions, including solving proportions, Proving Similarity: Congruent Angles + Proportional Sides, Dilation, including scale factors and centers of dilation, and Dilation: effect on distance and area.
As far as congruence, my understanding is that it means that two sides of something, especially two shapes, are compatible with each other, or completely balanced. When two triangles are congruent, they share all three same angles and same lengthed sides. My definition of similarity is when one shape can become another shape through dilation, or when one shape shares all the same angles and is proportional to another shape. To me, ratios and proportions are another way to represent dilation. A ratio is a relationship between two numbers or amounts showing how many times one value is contained within another. A proportion is essentially a part of a whole, or another way to say that two things are balanced/proportional. Solving proportions is basically setting the ratios up as fractions and either cross-multiplying or using another method to solve the resulting equation. In order to prove similarity, you determine if the angles are all the same (they must be, no exceptions), and if the side lengths of a shape are proportional. It isn't essential for them to be the same, but it is essential that they are divisible by each other and proportional to one another. As for dilation, it's a transformation producing an image/shape that is similar to the original because it shares the same shape but a different size. A scale factor of dilation is a number determining the reduction or expansion of the object/shape. If a scale factor is larger than 1, the object will expand, and if the factor is between 0 and 1, it will contract. The center of dilation is a fixed point where either all points expand or contract. It is the only point that always remains the same when something is dilated.
I am able to connect similarity and proportion because two figures that have the same proportional shape are said to be similar, and a proportion is an equation stating that two ratios are equal, so the two have much in common in the sense that they both share a common meaning.
As far as the relationship between dilation and similarity, the two concepts also have much in common because when something is dilated, say Mount Everest (my group’s example), it contains the similar form of it’s original size because it hasn’t been changed in height or width. They are directly correlated because similarity is a result of dilation. Dilating something results in the similar version of the original.
In benchmark #2, we made every initial plan for how our final product would appear as far as height and width in relation to the original size of our object. We drew two diagrams of our object where one represented the original size and one the size we were going to scale it to. We demonstrated all of our calculations on the paper so we could lay it all out before starting our product. This lead to benchmark #3, where we finally constructed all of our calculations into a final product. These two benchmarks directly connect because benchmark #2 was basically the planning process of benchmark #3.
Exhibition:
For benchmark #1, when we started off the Scaling Your World project, we were to find out who we were to work with, or if we were working independently, find an item/object to scale, decide on a scale factor, and think about how our model would be constructed/exhibited. I think the purpose of this initial benchmark was to get ourselves situated and with clear ideas in mind. For benchmark #2, we were to essentially get all of our scaling factors and calculations onto paper. We first completed all necessary scaling calculations according to what size we wanted our model. We then sketched two diagrams of our object (in our case, Mount Everest and the Mariana Trench), using one as a representation of the original size (not to scale, of course) of the object and the other as a representation of what we would scale our objects down/up to. For benchmark #3, we of course were to complete our scaled model and turn it in along with a self/peer evaluation, giving ourselves and peers a grade and evidence as to why we deserved that grade. The purpose of this benchmark was to have a final product prepared for exhibition and to clearly give our classmates and selves an evaluation so we could reflect on our participation and involvement in the project.
During the process of benchmark #2, we got our scale factor in place, we came up with all of our calculations according to our scale factor, and we showed our work. We first calculated the height of mount Everest if our scale factor was 2/29,029. mount Everest is already 29,029 feet tall, so our scaled down model would be 2 feet tall. We also determined that the width of mount Everest was about the same, so our model would ultimately be 2 feet wide as well. For the Mariana trench, we calculated that if our scale factor for mount Everest was 2, Mariana trench's would have to be 2.46 because it was slightly taller (35,814 ft.) and it needed to be changed accordingly. We got this by dividing 2 by 29,029 (.0000688), then multiplying that by 35,814 (Mariana’s depth). For the width of the Mariana trench, we found that the average width was 69 kilometers, so we converted it to feet and got 226,378. From there, we divided the scale factor, 2.46 ft., by 226,378 ft and got .00107 ft. This would be the width of our scaled model for the ultimate product, benchmark #3.
Final scaled down models of mount Everest and the Mariana trench below:
Reflection:A main success I believe I had during the course of this project was working by myself on the various worksheets we had involving proportion/dilation, such as Similar Problems, Inventing Rules, and the Billy Bear Grows Up problem. I in general had a good understanding on how dilation is used mathematically as well as ratios and proportions throughout the project, and I would consider that a strength. A main habit of mathematicians I feel I exercised during the project, especially in benchmark #3, was being systematic. When you take on something with a systematic approach, you are more likely to notice patterns and use those patterns to your advantage. It was helpful to use this approach during the calculating part of the project because it kept us at a steady pace. Another habit that I feel like I improved on was Collaborate and Listen because I obviously had to do more of that than usual because of my project group. A lot of the project was weighing all of our individual ideas and making compromises based on what was realistic/efficient for us to do and what our desired outcome was. Sometimes, a struggle was agreeing on one thing, such as the material we would use for our product, but we generally found ways to work together in a productive manner. Looking back, I would do some things differently if I could, like buying materials myself in order to produce a better final product, but I'm glad that I was able to improve on certain habits as well.