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!*"

Halmos, Paul R.

**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**