Project Description//
The purpose of this project was for us students to be able to go deeper in similarity, geometric transformation, and to apply functions and algebraic skill. Since our learning of similarity is mostly displayed with triangles we took a leap and chose objects, food, and places to show off our skills. When the project was first introduced we got into a group of three that would then be the group for the rest of the project. In the groups we had a chance to brainstorm ideas of what we would scale and if it was challenging enough for the three of us. There were different benchmarks. Benchmark number one was answering the following questions, (1. Who else is on your team? 2. What item/object are going to scale? 3. How are you going to decide on the scale factor? 4. How will your scale model be constructed and exhibited?) which we then turned in onto edmodo. Benchmark number two involved each group member to draw what we were going to scale and to calculate how much the object/food/place was going to either volumise or shrink. Benchmark number three was to turn in our actual artifact, either it was a video or an object. Most people asked for extensions for the assignment so they can make the final piece look professional and ready to be displayed during exhibition. This is the forth bench mark which is displaying all we learned on this very page.
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Mathematical Concepts//
Congruence and Triangle Congruence//
Congruence is when there are two shapes that are exactly the same in everything. You will know that two shapes are congruent if you can successfully place the two on top of one another without one of them "spilling" out from the top. No matter how you should turn them they are going to be congruent. Triangle congruence is when two triangles have the same three sides, and the same exact three angles. Congruence was used in my project when we were getting the materials, for example we bought two Styrofoam "tomatoes" and they needed to be congruent, as well as the onions and the buns.
Definition of Similarity//
Similarity is when there are two shapes with the same angles and sides, but their side lengths are different. Similar shapes have corresponding angles that are equal, and the lines are in proportion. Sometimes you need to flip the shape around in order to see that the two are similar. Similarity was used while looking at the original product and making the scale model. They needed to be similar but doubled in size.
Ratios and Proportions, including solving proportions//
A proportion is an equation that expresses the equality between two ratios. The proportions are not only numbers but also include variables. Proportions help solve for the missing side of a shape. For example, if there are two triangles that are similar we can use proportions to find the missing side because the similar triangles have corresponding side lengths.
Proving Similarity: Congruent Angles + Proportional Sides//
To prove similarity you need to look at the shape, if the angles are equal and the corresponding sides are in the same ratio then you have a similar shape. Congruent angles means that the angles are the same. The same that goes for triangle congruence goes for angle congruence, the angle does not need to be the same direction. For Proportional sides, If the measures of the corresponding sides of the two triangles are proportional then the triangles must be similar.
Dilation, including scale factors and centers of dilation//
Dilation: Affect on distance and area// (re: Billy Bear)
(we also looked at transversals, vertical angles and corresponding angles)
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Exhibition//
Explain your Benchmarks #1, #2 and #3. There are two things you must emphasize:
1. The process (a clear description of each benchmark and its purpose)
As I explained earlier, during the project we had to go through benchmarks, 1, 2, and 3. Benchmark number one was answering the following questions, (1. Who else is on your team? 2. What item/object are going to scale? 3. How are you going to decide on the scale factor? 4. How will your scale model be constructed and exhibited?) which we then turned in onto edmodo. The purpose of the first benchmark was to propose the scale model we were leaning towards. Benchmark number two involved each group member to draw what we were going to scale and to calculate how much the object/food/place was going to be either volumized or shrunk. The purpose of the second benchmark was to prove that the scale model was going to be true to size by contributing math as the evidence. Benchmark number three was to turn in our actual artifact, either it was a video or an object. My group and I decided to double our double double. We collected the measurements of the ingredients used to make the burger and doubled it. We made sure to make the scale model as similar to the original as possible.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Reflection//
I believe this project was successful because I got to work with two of my friends and got it done without fighting over a small detail. It was easy to communicate with each other when there was a change that was going to be happening. It was also fun when we were trying to brainstorm ideas of what we were going to scale. Some challenges we faced were minor and not that big of a deal because it was just trying to find the materials that we needed. Once we found them it all came together nicely. We had the opportunity to ask for an extension which is absolutely what we did, we took it so our piece could look nicely put together instead of thrown together last minute. This project really was involved in so many habits of a mathematician and the ones that really played a part were, Looking for Patterns, Starting Small, and Staying Organized. The "Starting Small" habit didn't really take place until after we came to our senses, and I say this because at first we wanted to create a diagram of a continent, then of the deepest part of the Grand Canyon. We finally chose to create a Double Double burger from In-N-Out. The "Looking for Patterns" habit took part in putting the hamburger together in a way that it looked balanced. Out of all of this, we also stayed organized, which as a matter of fact is another Habit of a Mathematician that was used during the project. We kept all of our information and calculations together and written neatly, there were no scattered papers that had gotten lost, everything was in place and used our time wisely. We were all very good at contributing in this project which is also why I believe this project was successful. If non of us contributed or spoke up we would have not gotten to he point of actually coming up with an idea, let alone creating the scale model itself.
The purpose of this project was for us students to be able to go deeper in similarity, geometric transformation, and to apply functions and algebraic skill. Since our learning of similarity is mostly displayed with triangles we took a leap and chose objects, food, and places to show off our skills. When the project was first introduced we got into a group of three that would then be the group for the rest of the project. In the groups we had a chance to brainstorm ideas of what we would scale and if it was challenging enough for the three of us. There were different benchmarks. Benchmark number one was answering the following questions, (1. Who else is on your team? 2. What item/object are going to scale? 3. How are you going to decide on the scale factor? 4. How will your scale model be constructed and exhibited?) which we then turned in onto edmodo. Benchmark number two involved each group member to draw what we were going to scale and to calculate how much the object/food/place was going to either volumise or shrink. Benchmark number three was to turn in our actual artifact, either it was a video or an object. Most people asked for extensions for the assignment so they can make the final piece look professional and ready to be displayed during exhibition. This is the forth bench mark which is displaying all we learned on this very page.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Mathematical Concepts//
Congruence and Triangle Congruence//
Congruence is when there are two shapes that are exactly the same in everything. You will know that two shapes are congruent if you can successfully place the two on top of one another without one of them "spilling" out from the top. No matter how you should turn them they are going to be congruent. Triangle congruence is when two triangles have the same three sides, and the same exact three angles. Congruence was used in my project when we were getting the materials, for example we bought two Styrofoam "tomatoes" and they needed to be congruent, as well as the onions and the buns.
Definition of Similarity//
Similarity is when there are two shapes with the same angles and sides, but their side lengths are different. Similar shapes have corresponding angles that are equal, and the lines are in proportion. Sometimes you need to flip the shape around in order to see that the two are similar. Similarity was used while looking at the original product and making the scale model. They needed to be similar but doubled in size.
Ratios and Proportions, including solving proportions//
A proportion is an equation that expresses the equality between two ratios. The proportions are not only numbers but also include variables. Proportions help solve for the missing side of a shape. For example, if there are two triangles that are similar we can use proportions to find the missing side because the similar triangles have corresponding side lengths.
Proving Similarity: Congruent Angles + Proportional Sides//
To prove similarity you need to look at the shape, if the angles are equal and the corresponding sides are in the same ratio then you have a similar shape. Congruent angles means that the angles are the same. The same that goes for triangle congruence goes for angle congruence, the angle does not need to be the same direction. For Proportional sides, If the measures of the corresponding sides of the two triangles are proportional then the triangles must be similar.
Dilation, including scale factors and centers of dilation//
Dilation: Affect on distance and area// (re: Billy Bear)
(we also looked at transversals, vertical angles and corresponding angles)
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Exhibition//
Explain your Benchmarks #1, #2 and #3. There are two things you must emphasize:
1. The process (a clear description of each benchmark and its purpose)
As I explained earlier, during the project we had to go through benchmarks, 1, 2, and 3. Benchmark number one was answering the following questions, (1. Who else is on your team? 2. What item/object are going to scale? 3. How are you going to decide on the scale factor? 4. How will your scale model be constructed and exhibited?) which we then turned in onto edmodo. The purpose of the first benchmark was to propose the scale model we were leaning towards. Benchmark number two involved each group member to draw what we were going to scale and to calculate how much the object/food/place was going to be either volumized or shrunk. The purpose of the second benchmark was to prove that the scale model was going to be true to size by contributing math as the evidence. Benchmark number three was to turn in our actual artifact, either it was a video or an object. My group and I decided to double our double double. We collected the measurements of the ingredients used to make the burger and doubled it. We made sure to make the scale model as similar to the original as possible.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Reflection//
I believe this project was successful because I got to work with two of my friends and got it done without fighting over a small detail. It was easy to communicate with each other when there was a change that was going to be happening. It was also fun when we were trying to brainstorm ideas of what we were going to scale. Some challenges we faced were minor and not that big of a deal because it was just trying to find the materials that we needed. Once we found them it all came together nicely. We had the opportunity to ask for an extension which is absolutely what we did, we took it so our piece could look nicely put together instead of thrown together last minute. This project really was involved in so many habits of a mathematician and the ones that really played a part were, Looking for Patterns, Starting Small, and Staying Organized. The "Starting Small" habit didn't really take place until after we came to our senses, and I say this because at first we wanted to create a diagram of a continent, then of the deepest part of the Grand Canyon. We finally chose to create a Double Double burger from In-N-Out. The "Looking for Patterns" habit took part in putting the hamburger together in a way that it looked balanced. Out of all of this, we also stayed organized, which as a matter of fact is another Habit of a Mathematician that was used during the project. We kept all of our information and calculations together and written neatly, there were no scattered papers that had gotten lost, everything was in place and used our time wisely. We were all very good at contributing in this project which is also why I believe this project was successful. If non of us contributed or spoke up we would have not gotten to he point of actually coming up with an idea, let alone creating the scale model itself.