Recently in class, we finished working on a project titled “Scaling Your World”. For this project, we learned about dilation and scaling. I believe that the purpose of this was to give us a deep understanding of how scaling and proportions work, as is it is going to be one of out exhibition topics.
However, that hasn't been the main focus of topics for the last couple of weeks. We first started writing down everything we knew about triangles and similarities and all the topics we started out with. After that, we formed groups, and made posters. Each group was assigned a different topic. There was triangle congruence, dilation, etc… After the posters, we did activities that tied into one another. For example, to find if something is similar, you would have to use a ratio, which tied in to the proportions topic, which could tie into scaling, etc… For our final product, we had to scale any item of our choice. My partner and I chose to do a snowflake. We scaled it larger than the original.
We had 6 different topics to focus on, which I will explain. The first up is congruence and triangle congruency. To me, congruency is when there are two things that are exactly the same. Same width, same height, length, angles, etc… Triangle congruence is when there are two triangles that are exactly the same. I had the privilege of having this as my topic when we split into groups. There are 5 different ways to find out if triangles are congruent. SSS, SAS, ASA, AAS, and HL. S stands for Side, A stands for Angle, H for Hypotenuse, And L for Leg. As you can imagine, SSS means side side side. Which is when you have two triangles that have 3 sides that are the same length. SAS Is when you have two congruent sides, and the angles in between those sides are congruent. All the rest are self explanatory, except for HL. HL is when you have 2 congruent hypotenuses, and two congruent legs. Two congruent triangles would look something like this,
However, that hasn't been the main focus of topics for the last couple of weeks. We first started writing down everything we knew about triangles and similarities and all the topics we started out with. After that, we formed groups, and made posters. Each group was assigned a different topic. There was triangle congruence, dilation, etc… After the posters, we did activities that tied into one another. For example, to find if something is similar, you would have to use a ratio, which tied in to the proportions topic, which could tie into scaling, etc… For our final product, we had to scale any item of our choice. My partner and I chose to do a snowflake. We scaled it larger than the original.
We had 6 different topics to focus on, which I will explain. The first up is congruence and triangle congruency. To me, congruency is when there are two things that are exactly the same. Same width, same height, length, angles, etc… Triangle congruence is when there are two triangles that are exactly the same. I had the privilege of having this as my topic when we split into groups. There are 5 different ways to find out if triangles are congruent. SSS, SAS, ASA, AAS, and HL. S stands for Side, A stands for Angle, H for Hypotenuse, And L for Leg. As you can imagine, SSS means side side side. Which is when you have two triangles that have 3 sides that are the same length. SAS Is when you have two congruent sides, and the angles in between those sides are congruent. All the rest are self explanatory, except for HL. HL is when you have 2 congruent hypotenuses, and two congruent legs. Two congruent triangles would look something like this,
The next topic, is relatively “Similar” to the last. It is the definition of similarity. Similarity is when you have two shapes that are the same, but just different sizes. My partner and I had to use similarity to make sure that our larger version of a snowflake was exactly the same design as a smaller one. To make sure these two shapes are similar, you have to use proportions or angles, which will tie us into our next topic. Here is what two similar triangles look like,
The next topic we are talking about is Ratios and Proportions. Now a ratio is essentially a comparison of two things. When you hear people say, “They won 3 to 1”, 3 to 1 is a ratio. Or a cooking recipe that calls for every cup of water, use a half cup of milk. A proportion on the other hand, is a part of something. It is a piece of a larger item. As you can see in the image above, triangle ABC is a proportion of triangle DEF. These can be shown in equations and used to see if shapes are similar. An example of an equation could be shows using the numbers given above, 10/6 is equal to X/12. To solve this, you would cross multiply 10 to 12, and 6 to X. You would get 120=6X. Divide 120 by 6, which equals 20, and that is your X value. My partner and I had to use proportions to finds out how long each of our legs of the snowflake would be relative to the whole thing.
The next topic is proving similarity. There are multiple ways to prove similarities, but there are two that I will be focusing on today. Those are angles and proportions. Angles have some restrictions when it comes to similarity, they work best with triangles. If you have 2 angles that are the same, then the third will be the same. Therefore, if you have 2 more angles that are the same in two triangles, then they are similar. You can also use proportions. You can measure the sides of your shape, and put them into a proportional format. If they are equal, then your triangles are similar. For example, if you have a triangle with sides of 5, 10, and 15. Then you have another triangle with sides 10, 20, and 30. If you put 5/10 is equal to 10/20, and it is equal, then you have similar triangles.
Now we are going to talk about dilation, the topic used in our scaling your world project. Dilation is basically making something bigger or smaller, and using a scale factor. A scale factor, is the amount you are dilating the object by. For example, if your scale factor is 2, you are doubling the size of the object. My partner and I used a scale factor of about 991.3 in order to scale up a snowflake. Every object that is dilated is similar. It doesn't matter how big or small, every object that is dilated to a different size is still similar to the original. There is another term, called the center of dilation. This is where everythings expands from or shrinks into. This spot does not move. We found out what dilation is by doing an activity where we tied a knot in the center of a rubber band, put a pencil on one end, pinned down the other, and went over an object with the knot it the middle. This would make the pencil draw an object that was dilated bigger than the original.
Last we are talking about how dilation affects area and perimeter. The perimeter of an object will get bigger with multiples the same as the scale factor. For example, if you have a scale factor of 2, then you multiply the perimeter by 2. For the area, you come up with an equation depending on the scale factor and situation. We learned this through and activity we did in class about a bear that we had to redraw with different scale factors.
For our exhibition piece, we had to go through 4 benchmarks. #1 was coming up with the initial idea for what we were going to scale. My partner and I came up with the idea of scaling a snowflake. We planned on scaling it to 3 feet. Benchmark #2 was coming up with all the math to scale it, the scale factor, original lengths, etc… We found the original diameter of a snowflake is about 1mm. With the legs being about 0.5mm. We found that the scale factor was about 991.3. Benchmark #3 was actually making the item and writing a reflection. We made a snowflake that was 3 feet in diameter and used poster board and paper scraps to make it. Benchmark #4 is this writing piece about all the work that we did. Containing what we learned, How we developed, etc… Here is a picture of my partner and I’s snowflake,
The next topic is proving similarity. There are multiple ways to prove similarities, but there are two that I will be focusing on today. Those are angles and proportions. Angles have some restrictions when it comes to similarity, they work best with triangles. If you have 2 angles that are the same, then the third will be the same. Therefore, if you have 2 more angles that are the same in two triangles, then they are similar. You can also use proportions. You can measure the sides of your shape, and put them into a proportional format. If they are equal, then your triangles are similar. For example, if you have a triangle with sides of 5, 10, and 15. Then you have another triangle with sides 10, 20, and 30. If you put 5/10 is equal to 10/20, and it is equal, then you have similar triangles.
Now we are going to talk about dilation, the topic used in our scaling your world project. Dilation is basically making something bigger or smaller, and using a scale factor. A scale factor, is the amount you are dilating the object by. For example, if your scale factor is 2, you are doubling the size of the object. My partner and I used a scale factor of about 991.3 in order to scale up a snowflake. Every object that is dilated is similar. It doesn't matter how big or small, every object that is dilated to a different size is still similar to the original. There is another term, called the center of dilation. This is where everythings expands from or shrinks into. This spot does not move. We found out what dilation is by doing an activity where we tied a knot in the center of a rubber band, put a pencil on one end, pinned down the other, and went over an object with the knot it the middle. This would make the pencil draw an object that was dilated bigger than the original.
Last we are talking about how dilation affects area and perimeter. The perimeter of an object will get bigger with multiples the same as the scale factor. For example, if you have a scale factor of 2, then you multiply the perimeter by 2. For the area, you come up with an equation depending on the scale factor and situation. We learned this through and activity we did in class about a bear that we had to redraw with different scale factors.
For our exhibition piece, we had to go through 4 benchmarks. #1 was coming up with the initial idea for what we were going to scale. My partner and I came up with the idea of scaling a snowflake. We planned on scaling it to 3 feet. Benchmark #2 was coming up with all the math to scale it, the scale factor, original lengths, etc… We found the original diameter of a snowflake is about 1mm. With the legs being about 0.5mm. We found that the scale factor was about 991.3. Benchmark #3 was actually making the item and writing a reflection. We made a snowflake that was 3 feet in diameter and used poster board and paper scraps to make it. Benchmark #4 is this writing piece about all the work that we did. Containing what we learned, How we developed, etc… Here is a picture of my partner and I’s snowflake,
Now for an overall reflection. I definitely had my ups and downs during this project. For the most part, I understood what was going on and how to do it, but there is was a lot of brain growth. First of all, we kind of messed up the symmetrical ness of the snowflake, and is doesn't look incredibly appealing. This was because my partner and I asked for an extension, but we just decided to rush it all into one day instead. If there had to be one habit of a mathematician that really stuck to me during this section, it would be conjecture and test. There was a lot of that going on with the billy bear activity with coming up with equations and equations in general. If I could redo one thing, it would be to make the snowflake with more precision, and have it look nicer.