Exploring the relationship (or
lack of relationship) between perimeter and area is interesting for students—even
for simple shapes like rectangles. For example, if you cut a rectangle’s area
in half, do you also cut the perimeter in half?

Using the rectangles shown below,
it is easy to see that the figure on the left was cut in half to create the
figure on the right. When we measure the area, the rectangle on the left is 16
units and the area of the smaller rectangle is 8 units—exactly half of the
original rectangle. However, it turns
out that the perimeter of the figure on the left was not cut in half when the
new rectangle was created. In fact, the new perimeter is a full 3/4 of the old
perimeter.

Is the new perimeter always 3/4
of the old one? Let’s try a different rectangle. This time, let’s cut it
vertically instead of horizontally.

Once again, the area is halved, but
the perimeter changes from 16 units to only 10 units. This time, the ratio of
new perimeter: old perimeter, is not 1/2 and also not 3/4. Instead, it is 5/8.

You could provide students with
square tiles with which to build rectangles, or they could explore the
challenge using geoboards and geobands. Alternatively, students might digitally
access squares they can put together to make rectangles using the Patch Tool at
http://illuminations.nctm.org/Activity.aspx?id=3577, the Shape Tool at http://illuminations.nctm.org/Activity.aspx?id=3587,
or the virtual geoboard available at the National Library of Virtual Manipulatives website.

Encourage students to then
explore exactly what fractions of the old perimeter the new perimeter could be
if a rectangle’s area is cut in half.

Could it be 2/3?

Could it be 1/3?

Could it be 5/6?

Could it be really close to 1?

Could it be really close to 0?

Is it ever 1/2?

Alternatively, if time is
limited, ask students to determine the dimensions of rectangles with specific new
perimeter: old perimeter ratios, such as 5/6 or 2/3.

Have your students try the
problem and see how it goes for them. If you are already on summer break, you
could challenge you own children or neighborhood children to explore the task. I
welcome you to share your students’ experiences with us.

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.