Project Description
The purpose of this project was to study about similarity and dilation of many shapes. both are very similar but still different so it was also to see the difference between the two. and then to take what we just learned about the two and make it into a real life model of scaling to prove we know what we are talking about. We did this project in four main benchmarks. Benchmark #1 was the scale model proposal where we decided if we wanted to work in a group or just work by our selves. once we got that sorted out we then proposed a scale model we would make. Benchmark #2 was The Mathematical Calculations. If we were to get our scale model proposal approved by Dr.Drew then we could start doing the mathematical calculations to show that it can be scaled up or down and how much of a scale it's going to be. Benchmark #3 was The Scale Model Exhibition where we would take the drawing and math and actually build the models and present it compared to the original object that is being scaled. And finally the benchmark #4 is what you are reading right now, a documentation of the process and work that was done on this project and to put it on our student DP. while we were doing these benchmarks in between then we worked on 12 activities that helped us learn more about the subject the project was about.
#1 Similar Problems
#2 Inverting Equations
#3 Polygon Equations
#4 More Similar Triangles
#5 Ladder Crossing
#6 The Shortest Path
#7 Now You See It
#8 Dilation with Bands
#9 Copy Machine
#10 Repeated Dilations
#11 Billy Bear
#12 Isometric Dot Paper
There was 6 main content areas that we learned about throughout the whole project
#1 Similar Problems
#2 Inverting Equations
#3 Polygon Equations
#4 More Similar Triangles
#5 Ladder Crossing
#6 The Shortest Path
#7 Now You See It
#8 Dilation with Bands
#9 Copy Machine
#10 Repeated Dilations
#11 Billy Bear
#12 Isometric Dot Paper
There was 6 main content areas that we learned about throughout the whole project
- Congruence and Triangle Congruence
- Definition of Similarity
- Ratios and Proportions, including solving proportions
- Proving Similarity: Congruent Angles + Proportional Sides
- Dilation, including scale factors and centers of dilation
- Dilation: Effect on distance and area
Mathematical Concepts
Congruence and triangle congruence is where there are two same shapes have a few common similarities. Even if the shape, sides, and angles is labeled different. A few things have to be true about the two shapes to make it truly congruence. All sides compared to the other similar shape has to be the exact length, and all angles must be the same compared to the other similar shape. Both of the congruence shapes can in both a different rotation and/or different location compared to the other congruent shape and still be considered congruent shapes. If the two shapes are triangles there is a few extra ways to figure out if the two are congruence. If two of the sides are congruent and one angle that is being used by the two sides are congruent then it can be said that its congruent even if you don't have the last side length or the other two angles. The same can be said with the other way if there is only side shown but there is two given angles that are the same then its congruent.
The Definition of similarity is when there are two same shapes but one of the shapes is smaller or bigger than the other one. But for the two shapes to be considered similar both must have evenly scaled on on all dimensions of the shape. So for two similar squares one of them must have all four sides evenly scaled up or down. It can be in any rotation and location and still be similar. But if any of the dimensions of the shape are not scaled the same with the other dimensions then it is not considered a similarity.
Ratios and Proportions, including solving proportions. First ratios is a factor where there is two numbers of anything and the larger of the two is compared to how many times it's bigger to the smaller one. For example if there was a space battle against two space fleets and one of the fleets has three times the space ships than the other smaller fleet than it is a ratio of 3 to 1. It doesn't mean that its three ships against just one it means for every one ship the other side has three.
“Shipmaster, they outnumber us three to one.” “Then it is an even fight, all cruisers fire at will! Burn their mongrel hides!” Shipmaster- Halo 3, 2007
Proportions are two different ratios used as fractions but when they are crossed multiplied they are equal to each other. An example of this is 6/12 and 3/6, they may look different but if you cross multiply the two (6 x 6 and 12 x 3) both will be 36 making both ratios proportions.
“Shipmaster, they outnumber us three to one.” “Then it is an even fight, all cruisers fire at will! Burn their mongrel hides!” Shipmaster- Halo 3, 2007
Proportions are two different ratios used as fractions but when they are crossed multiplied they are equal to each other. An example of this is 6/12 and 3/6, they may look different but if you cross multiply the two (6 x 6 and 12 x 3) both will be 36 making both ratios proportions.
Proving Similarity: Congruent Angles + Proportional Sides. To prove similarity with congruent angles and proportional sides is very easy. First with congruent angles if the size of a similar shape changes the angles will never change, so for a triangle if all three sides are the same with the similar shape then it is similar. You even don't need all three angles you can only need two because you can find the 3rd angle easily. For other shapes it depends, for a square all angles are always 90 degrees so that doesn't tell much so you need to still figure out if all the sides are proportional to really find out if a square is similar. For circles they don't really have a angle so that's impossible. And for everything else if all the angles match up then it's a similarly. Now for proportional sides this is a little more tricky, if two different size but are similar shapes then no matter what it will have different length sides. Remember how above we learned how to prove proportional sides, well we will be using that. Let's say you have two rectangle and you are trying to prove if they are similar. The height is 6 and 24, and the length is 10 and 40. You first setup all the numbers for cross multiply. Sense 6 and 24 are heights of both rectangles they go in a ratio, same goes for both lengths. You then cross multiply the two ratios and if both outcomes are the same (240) then the sides are proportional which makes them similar
Dilation, including scale factors and centers of dilation. Dilation is very similar to similarity in that the size can change with all measurements begin scaled equally but it does not move around or rotate. When dilations are shown they are on a graph and each corner of the shape has a point of the graph. And so when the shape is scaled it is scaled by all the points it has and not by the length and angles. So if a shape is two times bigger than all the points of the original are multiplied by two. For each dilation there is a center of dilation. The center of dilation is a point somewhere in the shape or is even one of the points of the shape does not change when there is a scaling. Most of the time it is one of the corner points or the direct center of the shape. This is so that since the dilation can't move or rotate like similar shapes it need a kinda of a anchor to make it self not move. If it did not have this center/anchor then it could move around while being scaled and mess all the proportions around.
Dilation: Effect on distance and area. When a dilations happens the the distance and area of the shape changes a lot. When a shape is being scaled up by a factor of 2 the distance/perimeter is just doubled, but for the area is multiplied by four of the original number. But that is just for a scale of 2. If we go to a scale factor of 3 then the distance/perimeter is multiplied by 3 and the area is multiplied by 9. The list goes on forever with each different scale factor.
The relationship between similarity and proportion. When there is two similar shapes then they must have a proportion. Each shape has its own ratio and since there is two shapes that means you can make a proportion to figure out and prove that the two shapes are similar.
The relationship between dilation and similarity. Both dilation and similarity are very similar but still have a few differences that make them stand apart. Similarity is when you take a shape and scale is up or down from the original shape. It is scaled by the lengths of its sides and angles. Then after that it is moved around and/or rotated. Dilation starts off with the same fact that it's a shape that is scaled up or down from the original shape, but that is when the similarities end. After a dilation has happened it can't move around or rotate. This is because there is a center point of dilation. This point never moves when the scaling is happening to prevent it from moving. And dilations are not measured by the length or angels they are measured by where very cromer point is on a graph.
Connections i made with benchmark #2 and #3 is that the measurements are the same. I wrote the scale measurements for 1x 2x 3x 4x 5x 6x 7x 8x 9x and 10x. And with my final product i made five scaled bricks each one with a different scale factor starting from 1x to all the way to 5x. All of which had very accurate measurements to the measurements i did on paper.
The relationship between dilation and similarity. Both dilation and similarity are very similar but still have a few differences that make them stand apart. Similarity is when you take a shape and scale is up or down from the original shape. It is scaled by the lengths of its sides and angles. Then after that it is moved around and/or rotated. Dilation starts off with the same fact that it's a shape that is scaled up or down from the original shape, but that is when the similarities end. After a dilation has happened it can't move around or rotate. This is because there is a center point of dilation. This point never moves when the scaling is happening to prevent it from moving. And dilations are not measured by the length or angels they are measured by where very cromer point is on a graph.
Connections i made with benchmark #2 and #3 is that the measurements are the same. I wrote the scale measurements for 1x 2x 3x 4x 5x 6x 7x 8x 9x and 10x. And with my final product i made five scaled bricks each one with a different scale factor starting from 1x to all the way to 5x. All of which had very accurate measurements to the measurements i did on paper.
Exhibition
Benchmark #1 was the scale model proposal where we decided if we wanted to work in a group or just work by our selves. once we got that sorted out we then proposed a scale model we would make. We had to get this purpose idea past Dr.Drew to check with any problems that could happen when building the model
Benchmark #2 was The Mathematical Calculations. If we were to get our scale model proposal approved by Dr.Drew then we could start doing the mathematical calculations to show that it can be scaled up or down and how much of a scale it's going to be. These calculations would then be used exactly on the model so the model had to be exact or the model was not a proper scale.
Benchmark #3 was The Scale Model Exhibition where we would take the drawing and math and actually build the models and present it compared to the original object that is being scaled. We took the calculations we did before and made sure the models were good. This was to prove that an accurate scale model of our object could be scaled up
Benchmark #2 was The Mathematical Calculations. If we were to get our scale model proposal approved by Dr.Drew then we could start doing the mathematical calculations to show that it can be scaled up or down and how much of a scale it's going to be. These calculations would then be used exactly on the model so the model had to be exact or the model was not a proper scale.
Benchmark #3 was The Scale Model Exhibition where we would take the drawing and math and actually build the models and present it compared to the original object that is being scaled. We took the calculations we did before and made sure the models were good. This was to prove that an accurate scale model of our object could be scaled up
Reflection
Overall i really liked this project it was simple yet very challenging. I felt very successful with coming up with a good concept that took a few different tries. I started off about to make a scale model of a surfboard but went through so many different changes i ended up making super big lego brick. I also found it easy to find the measurements of the the lego brick because apparently there is some people who are really into the engineering of a lego brick
http://www.robertcailliau.eu/Lego/Dimensions/zMeasurements-en.xhtml
So scaling up the measurements was very easy. When i got to the point of actually building the lego brick i ran into many problems. I wanted to make the whole lego brick out of smaller lego bricks to make the scaling perfect to then realize rounding up all the lego bricks with the same color in my house was very hard. And then the biggest problem is that making the four circles that are on top of the lego brick were impossible to make to scale with lego bricks. So if there is one thing i would change is finding a better way at making the circles on top to scale.
http://www.robertcailliau.eu/Lego/Dimensions/zMeasurements-en.xhtml
So scaling up the measurements was very easy. When i got to the point of actually building the lego brick i ran into many problems. I wanted to make the whole lego brick out of smaller lego bricks to make the scaling perfect to then realize rounding up all the lego bricks with the same color in my house was very hard. And then the biggest problem is that making the four circles that are on top of the lego brick were impossible to make to scale with lego bricks. So if there is one thing i would change is finding a better way at making the circles on top to scale.