Menu
Scaling your world
The Project
An Exploration of Proportion and Dilation
October 2016 -November 2016
In this project, we delved further into our understanding of similarity and dilation.
We used mathematical reasoning and understanding to create scale models that would give our audience a new perspective.
The Concepts
CongruenceObjects that are the same size and shape are congruent.
Proving SimilarityDue to the properties of triangles, similarity between two can be proved with two given angles and one given side or two given sides and one given angle.
|
SimilarityObjects whose angles are the same are similar. The lengths of their sides are proportional.
DilationDilation is a geometric transformation where all points are moved proportionally towards or away from a central point, the center of dilation.
|
Ratios and ProportionsA proportion is seen when objects are similar. They can be written as either a ratio or a proportional equation (shown above).
Dilation - Distance & AreaWhen a shape is dilated, the perimeter changes linearly and the area changes exponentially. The linear change in lengths results in similarity.
|
The Lessons
These activities helped our class learn about those concepts
Board-StormingTo start the project, we shared our prior knowledge of the essential concepts by writing on the board and having a class discussion. |
Poster PresentingStudents were put into groups where we were each asked to research and present on one of the concepts. This helped us to gain a deeper understanding. |
Problem SolvingLastly, we used what we had learned to solve and discuss problems that involved similarity, dilation, and their effects on the properties of a shape. |
The Benchmarks
Benchmark #1 - Brainstorm Our first task was to come up with an idea for our final product. We looked for objects that could be represented to our audience at a different size. My group chose to scale a neuron and it's axon we brainstormed criteria that would help us decide on a scale factor a means of construction. |
Benchmark #2 - Calculate the ScaleOur next step was to use our criteria to decide on a scale. Our understanding of similarity, proportions, and dilation was essential as we tested potential scale factors and drew two dimensional models. My group worked with metric unit conversions to turn a microscopic neuron into an interactive piece. We decided to scale a 4 cubic micron neuron and a 100 by 1 square micron axon by a 100,000 scale factor. This gave us the final dimensions of a 0.4 cubic meter neuron and a 10 meter by 0.1 cubic meter axon.
|
Benchmark #3 - Create Final Product
We implemented our design ideas and our scale factor to create our exhibition product. My final product was a two dimensional neuron drawing on a cardboard stand with a metallic border. The axon was represented as a slinky that could be extended to the full size of ten meters.
The Takeaway
"It's not the destination, it's the journey"
Content Understanding
Through the LessonsThe class lessons and worksheets gave me a deeper understanding of dilation and similarity. I was able to apply these concepts to scaling problems as well as SAT practice problems where I was asked to find a missing side or angle. I now feel more confident in solving complex problems of this sort. |
Through the ProductThe product we made helped me to see an application of dilation. With my group, I was able to turn qualitative goals into materials and dimensions for our piece. Looking back at how we executed the project, I would have liked to consult the mentors who made themselves available to us earlier in the project to make the highest quality product possible. |
Habits of a Mathematician
Be SystematicIn order to meet our specific needs for creating the final products, I created sub-requirements within each benchmark. This was particularly true for benchmark two. The range of sizes and relationships between neuron types and axons gave us flexibility, but it also gave the added challenge of deciding which would best fit our ideas for the product. I worked within the second benchmark to resolve this issue in and organized manner. Having worked with this challenge will help me to be more systematic in future projects as well. |
Be confident, Patent, and PersistentThe original idea for our project, I soon realized, neglected to depict the thickness of the axon. We had to quickly redesign and decided upon a two dimensional depiction that would be adhered to the floor. However, the material we were planning to use would damage the floor. We had to redesign once again. Instead of abandoning our goal, we came up with new solutions. I will continue to staying flexible and persevere in both projects and math problems. |