Project Description
In this project we were asked to scale up/down a object that we were interested in, through this process we learned about, Congruence and Triangle Congruence, Definition of Similarity, Ratios and Proportions, including solving proportions, Congruent Angles + Proportional Sides, Dilation, including scale factors and centers of dilation, and much more. Not only did we just scale our object we had side “problems’’ to help us on our journey. We also created a scale model of object. By doing this it further implemented the math and geometry into our brain.
We started this project by brainstorming what we knew about congruence and similarity, than researching things about them in teams and presenting to the class. We then dove into more topics over the following weeks and finally, made a scale model. I created a snowflake. For our final scale model we had benchmarks we had to make, 4 to be exact.
We started this project by brainstorming what we knew about congruence and similarity, than researching things about them in teams and presenting to the class. We then dove into more topics over the following weeks and finally, made a scale model. I created a snowflake. For our final scale model we had benchmarks we had to make, 4 to be exact.
- Scale Model Proposal
- Thinking about what you will make and how you will scale it. In my case, a snowflake.
- Mathematical Calculations
- Coming up with the calculations for your scale model
- The Scale Model Exhibition
- Finishing your scale model
- Finishing your scale model
Mathematical Concepts
Congruence and Triangle Congruence
Congruence basically means the exact same size and shape. 2 shapes can be in different places and still be congruent. They can also be rotated and reflected, as long as they have the same side lengths and the same angles they are congruent, because that is what makes a shape a shape, that is what defines it. There are 5 main ways to tell if a tell if a triangle is congruent: SSS, SAS, ASA, AAS, and HL. S stands for side and A stands for angle.
SSS: If all sides of the triangle have the same length the triangle must be congruent.
SAS: IF you know 2 sides of the triangle and the corresponding angle that touches them both they must be congruent.
ASA: If you know 2 angels and the length of the side that shares the angles the triangle is congruent.
AAS: If you know 2 of the side lengths and a non included side of the triangle they are congruent.
HL (Hypotenuse, Leg): If you know the length of the hypotenuse and the length of a leg they are congruent. Note: This only works for 90 degree triangles.
Definition of Similarity
The most basic definition of similarity is “same shape, different size”. In other words, shapes that share the same angles but not the same side lengths. This also means that their corresponding side lengths are proportional to each other. The shape can be rotated and reflected too.
Ratios and Proportions
Ratios and Proportions are extremely helpful when it comes to shapes and scaling things up/down. Similar shapes corresponding side lengths are proportional to each other. Taking advantage of this you can find missing side lengths very easily. For example you can use 2 ratios. ½= 2/x. By looking at this you can easily find using algebra that x is equal to 4. Using this mechanic you can not only scale things up and down but find side lengths.
Proving Similarity
Any 2 shapes with the same angles are similar. Also the ratios of side lengths will all match if the shapes are similar. You could also dilate the shape to see if it is the same.
Dilation
Dilation is a type of rigid motion that essentially makes a shape bigger or smaller while remaining similar. If you dilate something by a scale factor of 2 it will be twice as big or 200% the size. While applying dilation angles do not change, only the side length. Therefor the side lengths are still proportional and similar. The point of dilation is where you start the simulation from (0). Once you pass this point the object is reflected in both dimensions.
Exhibition
Benchmark #1
For this benchmark we just had to come up with what we were going to scale and how we were going to do it. I choose a snowflake. Benchmark #2 For this benchmark, we needed a sketch, a scale factor (913.7) and all of our measurements. We scaled a snowflake from 1mm to 3 feet. Benchmark #3 For this benchmark we created our final products. |
Reflection
The most challenging part of this project was creating the scale model itself. Everything seams easy when you plan it on paper but when you jump into it and try to construct it, it does not always go as planned. My biggest success was understanding the mathematical process behind everything, but creating the snowflake really helped me embed it in my head. During this project I differently grew in looking for patterns. Without this, It would of been lot harder to scale everything because you cold not apply previous knowage.