It’s
late in the school year, but I hope you had a chance to try out the perimeter and area comparison problem with your
students. If not, you might try it early next year.

Recall
that students were going to start with a rectangle and then make another one
with half the area. The task was to see what fraction of the old perimeter the
new perimeter could be.

I tried
this problem in a sixth-grade class and had a really interesting experience.
The regular classroom teacher had four students who normally did not stay with
the others for math; they usually went to a special education classroom. We
asked if, for this problem, the students could stay and, in fact, each of them
was successful in creating the second rectangle and in realizing that the new
perimeter was not only not half the old perimeter but also that it was greater
than half.

Some
students figured out that if the cutting in half was on a line of symmetry, the
new perimeter had to be more than half of, but less than all of the old perimeter.
That’s because even if you cut the length (or width) in half, you retain the
full width (or length) of the original.

For
example, in the two rectangles below, note that the length of the top edge and
bottom edge do not change when the shape is cut on a line of symmetry; however,
the lengths of the left and right sides (shown in blue on the original figure)
are half the length of the new figure (shown in green on the halved figure).

**fig. 1** Perimeter
= 2 full red + 2 full blue Perimeter
= 2 full red + 2 half-blues

Some
students realized that if you have a long rectangle or a tall and skinny oneand
cut it in half by making it even skinnier, you end up keeping most of the
perimeter, so the fraction of the old perimeter and that of the new perimeter
turn out to be really close to 1.

Compare
the change in perimeter on the left versus on the right. It is much less of a
change on the left.

**fig. 2**

Keeping the
same perimeter or even increasing the perimeter actually is possible, but only
if the original shape is not cut on a line of symmetry. For example, a 1 ´ 6 rectangle has half the
area of a 3 ´ 4
rectangle but the same perimeter. And a rectangle that is 6 ´ 5 with a perimeter of 22
can be halved in area by using a rectangle that is 1 ´ 15. But the new perimeter
is 32, even bigger than the original perimeter.

How did
your students respond to the task?

Marian Small is the former dean
of education at the University of New Brunswick, where she taught mathematics
and math education courses to elementary and secondary school teachers. She has
been involved as an NCTM writer on the Navigations series, has served on the
editorial panel of a recent NCTM yearbook, and has served as the NCTM
representative on the MathCounts writing team. She has written many
professional resources including *Good
Questions: Great Ways to Differentiate Mathematics Instruction *(2012)*, Eyes on Math *(2012)*, *and *Uncomplicating Fractions to Meet Common Core Standards in Math, K–7 *(2013)*,* all co-published by NCTM.