We all think we know what size means, but there are many different words in the English language for size: big, small, huge, tiny, vast, length, width, height, diameter, perimeter, area, volume, height ... these are just a few of the words we use to describe the size of of something. In my writing-intensive geometry class, my students explored the concept of size using the following worksheet, with 3 questions about perimeter and area:
Question 1) Which triangle is larger, a right triangle with one leg 1 cm and the other leg 40 cm or an equilateral triangle with sides 8 cm? Lauren drew the following diagram, and calculated the perimeter and area for both triangles:
In answering the question, Lauren wrote: "The perimeter of the right triangle is 41 + the square root of 1601 cm, while the perimeter of the equilateral triangle is 24 cm. But the area of the right triangle is 20 square centimeters while the area of the equilateral triangle is 16 times the square root of 3. So it's hard to say which triangle is larger, because larger is actually a rather vague word. The right triangle with legs 1 cm and 40 cm has a larger perimeter, but the equilateral triangle with sides 8 cm has a larger area. So it depends on what you mean by larger. "
She went on to say: "When we ask a question about the relative size of figures, we need to define what we mean by size: area or perimeter. Without a clear definition, there is no single answer to the question."
In her reflections on this project, Lauren wrote: "I really had a lot of fun with this project because I found out some very interesting things in the process. One thing I learned was that my intuition cannot always be relied on because it is not always right. Overall, I really liked doing this project because not only did I discover new things, but I also had fun doing it!"
She went on to say, in her reflections on the process: "I chose this piece for my portfolio because I worked hard on it. The assignment was challenging, but interesting. It made me really think about how area and perimeter are related. I thought this was an interesting question: what does size really mean? Is it area, or perimeter? Actually, sometimes we mean volume when we talk about size, like a big box, but we might also call a wheel big because of its circumference. Mathematically, both area and volume each have a more precise meaning."
In his reflections on this same project, Deane wrote "Size is a combination of both length, width, height, area and perimeter, in geometry. Actually, when you think about 3 dimensional figures, like spheres and cubes, there is surface area and volume, too. So size is a pretty complicated thing! There's weight, too, which is another way of measuring things but I don't think we study that in geometry except maybe in word problems. So the simple question 'which is bigger...?' is not so simple, is it!"
Question 2: If you double each side of a triangle, what would happen to the perimeter? What would happen to the area?
John said: "If you double each side of a triangle, you get a similar triangle. Since each side of the new triangle is twice the length of the corresponding side of the old triangle, the new triangle would have double the perimeter. But the area of the new triangle would be quadruple the area of the old. This is because your area is 1/2 base times height, and you are doubling the base and doubling the height so you are multiplying the area by 4."
Jane responded to the same question with the following observation: "Measurements can be a bit tricky, like the difference between inches and square inches. When the length of each side of a triangle is doubled, the perimeter is doubled, but the area of the triangle is quadrupled.The reason the perimeter is doubled is that it is a linear ratio, inches. But when you double the area you are actually doubling both the length and the width, and so the are is in square inches (like inches x inches), so the area of the new triangle is four times the area that it was before."
You can see her diagrams below:
Question 3) What would happen if you doubled the length a pair of opposite sides of a rectangle but left the height the same? How would the perimeter be affected? The area?
After exploring this concept, David constructed the diagram below, and wrote this answer to the question: "If you double the length of each side (the top and bottom) of a rectangle but leave the height the same, the perimeter is doubled. This is easily proven using algebra:
Original rectangle: perimeter = x + y + x + y
Original area: xy
So here's the new perimeter: 2x + 2y + y + y
And this would make the area of the new rectangle 4xy"
He also included his mathematical calculations, which I will leave as a challenge to the reader.
In his reflections on this and the related projects on area and perimeter, David wrote the following essay:
"Size, perimeter, and area were the focus of this project. In it, I investigated how different polygons are affected by increases of perimeter, area, and other measurements. I learned a lot about the ways in which size can be judged. Also. I learned how increases in dimension affect quadrilaterals and triangles. It was pretty interesting, particularly because it wasn't in the textbook! This made it much more challenging, but definitely more interesting!!"
With regard to the same assignment, Janine had this to say: "Who would ever have thought you could get in a debate about a math problem! Our group started out with some guesses about what the answers would be, and then we decided to experiment with triangles and quadrilaterals using Sketchpad. It was interesting because some people had different ideas about how to go about things. We found out we were right about some of our guesses but wrong about others. After lots of experimentation and discussion, we came up with a pretty nice project, and got every answer correct, hooray!"
There are so many interesting questions that can be asked on this topic. Here are a few of them; I will leave the answers for the reader to discover on his or her own.
If you increase the length at width of a rectangle by 50 per cent, how is the perimeter changed? What if you increase the length of a rectangle but do not change the width? How is the perimeter affected? What if you double the radius of a circle? How is the circumference affected? The area? What if the side of a square is tripled? What would happen to the length of a diagonal? What if you triple each side of an equilateral triangle - what would happen the altitude?
When asked interesting questions, students are given the opportunity to explore mathematics themselves, and become more involved in their learning than if they spend their days memorizing facts from a textbook. I also found that the students had interesting comments to make about what they were learning. Zenny wrote the following essay on our Area and Perimeter Explorations: "Size, area, and perimeter were the focus of this writing assignment. In it, I investigated how different polygons are affected by the increase of various elements. I learned a lot about the ways in which size can be judged. Who would have thought how much there is to learn on this one little topic!"
Go To Homepage Go To Introduction
1) Constructions 2) Clock Problem 3) Test Corrections 4) ASN Explain 5) Thoughts About Slope 6) What is Proof?
7) Similar Triangles 8) Homework Corrections 9) Quads Midpoints 10) Quads Congruence 11) Polygons
12) Polygons Into Circles 13) Area and Perimeter 14) Writing About Grading 15) Locus 16) Extra Credit Projects
17) Homework Reflections 18) Students' Overall Reflections 19) Parents' Evaluate Method 20) In Conclusion