Scaling Your World
Project Description
We first started off this project by learning about congruence and similarity. We were split into groups and created posters about the different concepts of congruent and similar shapes and figures. When we began to start the project, we were given 4 benchmarks:
Benchmark 1: On a document, fill out what we were going to scale and what the scale factor was going to be between our scale model and the original figure.
Benchmark 2: Figure out all the measurements for your scale model and the original figure and draw a blueprint.
Benchmark 3: Grade yourself and your group members.
Benchmark 4: DP Write- up.
For this project, when we figured out what item/figure we were going to scale, and figured out the scale factor and measurements, we created a 2D or 3D scale model of the figure we chose to build.
What my group and I decided to scale was the tallest building in the world, the Burj Khalifa. It's a building in Dubai. My group and I decided to scale this because it was something we were interested in and we had a plan on how to build the scale model and what materials we would use to build it.
Mathematical Concepts
Congruence and Triangle Congruence
Congruency is when two shapes are exactly the same, and equal in size and shape. The sides of the two shapes must be the same lengths and widths, and the angles must have the same measure.
Congruent triangles must have the same sides and the interior angles must have the same measure.
Similarity
Similarity is when two shapes have proportionate sides from each other and the same angles.
Ratios and Proportions
A ratio is the relationship between two numbers. And proportion means that two ratios are the same or equal.
Congruent Angles and Proportional Sides / Proving Similarity
When proving similarity, the sides of the two shapes would have to be proportionate form each other, and equal. But the angles have to be congruent/exactly the same.
Dilation
Dilation is when a shape becomes smaller by the shape shrinking towards the center point, or when a shape grows larger by the shape growing farther away from the center point.
Dilation: Affect on Distance and Area
The affect dilation has on distance and area is, when a shape dilates larger, the distance of the shape from the center point goes farther away, and the area increases. But when a shape dilates smaller, the distance of the shape from the center point becomes shorter, and the area decreases.
Relationship Between Similarity and Proportion
The way similarity and proportion are connected is similar shapes have proportionate sides.
Relationship Between Dilation and Similarity
Dilation is related to similarity because similar shapes are a dilation of one another, whether the shape is a larger or smaller version of itself.
Connections to Benchmark 2 & 3
In my benchmark 2, I used ratios & proportions and similarity because when my group and I found the scale factor of our original figure to the scale model, we basically dilated the original figure into a smaller version. So the original figure, which in our case would be the Burj Khalifa, was similar to our scale model of the building. And the measurements were proportionate to each other. This is what helped us build our scale model, which was benchmark 3.
Calculations
The scale factor of our original figure to the scale model was 1,361. We came up with the scale factor by taking the height of the Burj Khalifa building, 2722 ft, and dividing it by 2 because we wanted our scale model to be 2 ft.
Reflection
I felt my group's project was successful because we got it done. We did face some challenges though. One of the challenges we experienced was actually building the scale model. Because ours was a building, it was hard to figure out what to make it out of. But my group and I collaborated and figured out what materials we should use and how we were going to actually build the scale model. It was also a challenge finding time to finish it because majority people in my group had extracurriculars after school, so we had to fit doing the project during the weekend. Also, there was a bit of lack of communication within my group. But my group members and I dealt with the challenges we faced and got the project done. A habit of a mathematician I felt I grew in was starting small. As I mentioned earlier, it was hard figuring out how to build our scale model so my group members and I came up with small ideas, and transformed them into one big idea on how we were going to build it. Also I had growth in the habit of a mathematician of collaborate and listen because I had to collaborate a lot with my group to deal with the problems we faced. What I would have done differently in this project is managing my time more and communicating with my group a little better. I should've managed my time between sports and school so my group and I didn't have to do the whole project in a weekend, and we should've communicated better so each group member had an equal amount of participation in the project.
We first started off this project by learning about congruence and similarity. We were split into groups and created posters about the different concepts of congruent and similar shapes and figures. When we began to start the project, we were given 4 benchmarks:
Benchmark 1: On a document, fill out what we were going to scale and what the scale factor was going to be between our scale model and the original figure.
Benchmark 2: Figure out all the measurements for your scale model and the original figure and draw a blueprint.
Benchmark 3: Grade yourself and your group members.
Benchmark 4: DP Write- up.
For this project, when we figured out what item/figure we were going to scale, and figured out the scale factor and measurements, we created a 2D or 3D scale model of the figure we chose to build.
What my group and I decided to scale was the tallest building in the world, the Burj Khalifa. It's a building in Dubai. My group and I decided to scale this because it was something we were interested in and we had a plan on how to build the scale model and what materials we would use to build it.
Mathematical Concepts
Congruence and Triangle Congruence
Congruency is when two shapes are exactly the same, and equal in size and shape. The sides of the two shapes must be the same lengths and widths, and the angles must have the same measure.
Congruent triangles must have the same sides and the interior angles must have the same measure.
Similarity
Similarity is when two shapes have proportionate sides from each other and the same angles.
Ratios and Proportions
A ratio is the relationship between two numbers. And proportion means that two ratios are the same or equal.
Congruent Angles and Proportional Sides / Proving Similarity
When proving similarity, the sides of the two shapes would have to be proportionate form each other, and equal. But the angles have to be congruent/exactly the same.
Dilation
Dilation is when a shape becomes smaller by the shape shrinking towards the center point, or when a shape grows larger by the shape growing farther away from the center point.
Dilation: Affect on Distance and Area
The affect dilation has on distance and area is, when a shape dilates larger, the distance of the shape from the center point goes farther away, and the area increases. But when a shape dilates smaller, the distance of the shape from the center point becomes shorter, and the area decreases.
Relationship Between Similarity and Proportion
The way similarity and proportion are connected is similar shapes have proportionate sides.
Relationship Between Dilation and Similarity
Dilation is related to similarity because similar shapes are a dilation of one another, whether the shape is a larger or smaller version of itself.
Connections to Benchmark 2 & 3
In my benchmark 2, I used ratios & proportions and similarity because when my group and I found the scale factor of our original figure to the scale model, we basically dilated the original figure into a smaller version. So the original figure, which in our case would be the Burj Khalifa, was similar to our scale model of the building. And the measurements were proportionate to each other. This is what helped us build our scale model, which was benchmark 3.
Calculations
The scale factor of our original figure to the scale model was 1,361. We came up with the scale factor by taking the height of the Burj Khalifa building, 2722 ft, and dividing it by 2 because we wanted our scale model to be 2 ft.
Reflection
I felt my group's project was successful because we got it done. We did face some challenges though. One of the challenges we experienced was actually building the scale model. Because ours was a building, it was hard to figure out what to make it out of. But my group and I collaborated and figured out what materials we should use and how we were going to actually build the scale model. It was also a challenge finding time to finish it because majority people in my group had extracurriculars after school, so we had to fit doing the project during the weekend. Also, there was a bit of lack of communication within my group. But my group members and I dealt with the challenges we faced and got the project done. A habit of a mathematician I felt I grew in was starting small. As I mentioned earlier, it was hard figuring out how to build our scale model so my group members and I came up with small ideas, and transformed them into one big idea on how we were going to build it. Also I had growth in the habit of a mathematician of collaborate and listen because I had to collaborate a lot with my group to deal with the problems we faced. What I would have done differently in this project is managing my time more and communicating with my group a little better. I should've managed my time between sports and school so my group and I didn't have to do the whole project in a weekend, and we should've communicated better so each group member had an equal amount of participation in the project.