Scaling your world DP write up
Project description:
Throughout the duration of this project, we have covered many topics surrounding congruence and similarity as well as dilation. It all started with a project sheet. The goal? To create a scaled model of objects in our everyday lives using our understanding of similarities and the relationship it shares with dilation. To start off, we brainstormed what we knew already about similarity. As we progressed in our understanding, we were given a new set of vocabulary including dilation, congruence, and similar shapes. Set into teams we created posters about the concept assigned to us. My group came up with a poster about dilation. Now with a better understanding of our task, I chose to partner with two other students to design a product to scale. To end our project after we created our final products, we dove deeper into dilation and perfected our artifacts what will be shown at a winter exhibition.
Mathematical content:
Throughout this project, we touched on various mathematical concepts, and as a result many helped us through the making of our final product. Here are the six main concepts we learned:
1. Congruence and Triangle Congruence: my understanding of this concept stemmed from our understanding of congruent triangles, where we learned about SSS, SAS, ASA, and AAS. What this means is that to know if triangles have three equal sides, we know they're congruent, hence the three s's. The same goes for the rest, like if we have two sides and one angle, we can figure out if the two triangles are congruent. Congruence means equal in shape as well as size, and when it comes to triangle congruence, that means their corresponding sides are equal.
2. Definition of Similarity: similarity is one of the first concepts we covered, and the definition I go by is that when two shapes are similar, they have corresponding angles and/or sides. For our project, we had to make sure our pans were similar as well as our measured ingredients.
3. Ratios and Proportions, including solving proportions: Ratios and proportions are something that were critical to our final product, for all our ingredients we cut by half, so we had to calculate how much to measure every time the recipe called for a teaspoon. When it comes to the relationship between proportions and ratios, I remember it as proportions are equal ratios. so if the recipe called for 1/2 a cup, we had to create the ratio of 1/2 cup is equal to 1/4, since thats 50% of a half cup. We learned how to solve for proportion in one of our problems of the week, "tiling a patio." In it we had to solve for a 63 by 90 patio, and to solve this I took 63 over 90 and divided by 9 to simplify it which gave me the ratio of 63/90 = 7/10. Therefore, the ratios are proportional.
4. Proving Similarity: Congruent Angles + Proportional Sides: Once we gained definitions about these concepts, it was time to explore why and how to prove these were correct. To prove that two shapes are similar, we completed a worksheet "similar problems". In it, we were given a series problems where we had to solve for two similar triangles. for example, if we were given a triangle with the equation of X/7 = 5/3, then to solve we had to cross multiply which would give us 35 and 3X. The next step would to divide 35 by 3, which in this case equals to 11.6 or rounded up: 12. We got those ratios by looking at the diagram where both the 5 and the 3 were corresponding sides. To fins similar triangles we were given equations that were more complex tot he eye, which we solved to prove the two shapes were similar.
5. Dilation, including scale factors and centers of dilation: For dilation, we ended our project with this concept, and the basic definition of it would be two shapes that are the same, but different sizes. You can "shrink" or "enlarge" when it comes to dilation. Scale factor was part of our first benchmark, and to dilate anything, you have to have a grasp on scale factor. And so to take a hexagon for example, if you applied a scale factor of 50%, then all six sides/ angles would either be scaled up or down by 50%. Now the center of dilation is a point in space, and when the shape either expands or reduces, the center does not move.
6. Dilation: Affect on distance and area (re: Billy Bear): One of the problems we completed in this project was titled "billy bear" where we had to figure out how big a bear would be if every week their scale factor went up by 1 each week. We found out that the perimeter units were equal to 9 multiplied by the week number of the bear. Example: if the bear was at week 6, then the perimeter unit would be 54 (9 X 6). We then found out that the area units were equal to 7 multiplied by the week number squared. For example, with week 6 again the area unit would be 324 (7 X 6 squared). Through this we basically dilated the bear by a scale factor that was equal to the week he was on. We were told that in week one the total amount of area units were 7, so we then scaled him up by a factor of two the next week.
1. Congruence and Triangle Congruence: my understanding of this concept stemmed from our understanding of congruent triangles, where we learned about SSS, SAS, ASA, and AAS. What this means is that to know if triangles have three equal sides, we know they're congruent, hence the three s's. The same goes for the rest, like if we have two sides and one angle, we can figure out if the two triangles are congruent. Congruence means equal in shape as well as size, and when it comes to triangle congruence, that means their corresponding sides are equal.
2. Definition of Similarity: similarity is one of the first concepts we covered, and the definition I go by is that when two shapes are similar, they have corresponding angles and/or sides. For our project, we had to make sure our pans were similar as well as our measured ingredients.
3. Ratios and Proportions, including solving proportions: Ratios and proportions are something that were critical to our final product, for all our ingredients we cut by half, so we had to calculate how much to measure every time the recipe called for a teaspoon. When it comes to the relationship between proportions and ratios, I remember it as proportions are equal ratios. so if the recipe called for 1/2 a cup, we had to create the ratio of 1/2 cup is equal to 1/4, since thats 50% of a half cup. We learned how to solve for proportion in one of our problems of the week, "tiling a patio." In it we had to solve for a 63 by 90 patio, and to solve this I took 63 over 90 and divided by 9 to simplify it which gave me the ratio of 63/90 = 7/10. Therefore, the ratios are proportional.
4. Proving Similarity: Congruent Angles + Proportional Sides: Once we gained definitions about these concepts, it was time to explore why and how to prove these were correct. To prove that two shapes are similar, we completed a worksheet "similar problems". In it, we were given a series problems where we had to solve for two similar triangles. for example, if we were given a triangle with the equation of X/7 = 5/3, then to solve we had to cross multiply which would give us 35 and 3X. The next step would to divide 35 by 3, which in this case equals to 11.6 or rounded up: 12. We got those ratios by looking at the diagram where both the 5 and the 3 were corresponding sides. To fins similar triangles we were given equations that were more complex tot he eye, which we solved to prove the two shapes were similar.
5. Dilation, including scale factors and centers of dilation: For dilation, we ended our project with this concept, and the basic definition of it would be two shapes that are the same, but different sizes. You can "shrink" or "enlarge" when it comes to dilation. Scale factor was part of our first benchmark, and to dilate anything, you have to have a grasp on scale factor. And so to take a hexagon for example, if you applied a scale factor of 50%, then all six sides/ angles would either be scaled up or down by 50%. Now the center of dilation is a point in space, and when the shape either expands or reduces, the center does not move.
6. Dilation: Affect on distance and area (re: Billy Bear): One of the problems we completed in this project was titled "billy bear" where we had to figure out how big a bear would be if every week their scale factor went up by 1 each week. We found out that the perimeter units were equal to 9 multiplied by the week number of the bear. Example: if the bear was at week 6, then the perimeter unit would be 54 (9 X 6). We then found out that the area units were equal to 7 multiplied by the week number squared. For example, with week 6 again the area unit would be 324 (7 X 6 squared). Through this we basically dilated the bear by a scale factor that was equal to the week he was on. We were told that in week one the total amount of area units were 7, so we then scaled him up by a factor of two the next week.
Exhibition:
For our first benchmark in the project, we had to submit a project proposal consisting of what Who else we were working if (unless we were working alone), What item/object we intended to scale, how we would decide on our scale factor, and how it would be constructed and presented. Next came the second benchmark: sketch two similar diagrams of the object we were going to scale. Within the diagram we labeled all the significant dimensions of the original object that we planned on scaling. This was important for my group since we decided to scale down pumpkin pie. We decided on a scale factor of 50, and took a 9" pie pan and scaled it down to a 4.5" pan. To do this we did a simple calculation of 9 divided by 2, and made sure to measure our pans to ensure they were similar before we began the baking process. Since we were baking, we had to dilate the amount of ingredients as well. Since a 9" pan recipe called for an entire can of pumpkin, the scaled pies required half of that, which was half a can, or 7.5 ounces. We did the same with the 12 ounces of evaporated milk, which was reduced to six. And finally, The third benchmark. Since our products would be presented at the December exhibition, we knew we couldn't make a pumpkin pie last that long. Instead we created a video showcasing the process as well as the final product. In the end we decided to create a pattern of a snowflake on the top of our pie and scaled that down as well. At our exhibition, we will showcase our video rather than our actual pie. For the dilation of the whipped cream, we used the height of the whipped cream and scaled it down by half again, creating a 6" height to a 3" height. For this benchmark we shared our video with our teacher and had to complete a self and group assessment, talking of what grade we earned individually as well as what grade our group members should earn with an explanation for both.
Benchmark two and final product:
Reflection:
To me, this project was both challenging as well as exciting and engaging. One big success I think I had throughout this project was the teamwork from my other partners and the crafting of our scaled down pies. Since we couldn't control the rising of the pies in the oven, we decided to control the circumference of each pie tin. Once we decided on that route, it opened up more possibilities to us. It encouraged us to add more than just the actual pie to our project, and we designed a snowflake which we would downsize as well on the face of the pies. However, a challenge I faced along with my group was time. Since we had to buy perishable ingredients and decide on a day to sacrifice for baking over the weekend, it caused problems with scheduling and fitting it into our deadline. A way we overcame this was talking with our teacher to design an extension for our final product, giving us enough time to create the product as well as document and edit our video. Looking back at the experience as a whole, I would've changed how we handled planning our bake day. It took over eight hours to complete not including the time we made to edit our video and make sure we had a beautiful final product. Next time I would take on more of a leadership role when it came to decisions regarding time and make sure we derided on a day to create the product earlier on. However I'm proud of our scaled pies as well as my personal growth, specifically in Habits of Mathematicians. This project really tested me with staying organized not only with each assignment but with the scaling down of all the ingredients and keeping track of the tins we brought in and making sure we had all our measurements correct. Another habit I feel I grew in was the habit of "describe and articulate" because it's one thing to keep track of all the ingredients on paper in a diagram but it was easy to become overwhelmed in the kitchen with all the ingredients and multiple recipes. To cut down all the ingredients by half was a lot to take in, and without the diagrams as a aid I would've been lost. I really had to rely on the diagrams we made in benchmark two and three to represent what the snowflake should look like as well as to have a mental image of our product. But most of all I loved working with my group members and hearing their opinions and suggestions to the project.