Consider a rectangle. If the side lengths are whole numbers, then so is the area.

Do an example, i.e., draw a 5 x 7 rectangle and find its area.

We can multiply, or actually note that we are just counting up whole squares.

But is the opposite true? In other words, can an rectangle have integer area but non-integer side lengths?

It seems impossible: imagine filling the rectangle with squares: there will be a bunch of cut off pieces.

For example, consider a 3.5 by by 4.5 rectangle.

It is possible to multiply to find the area, of course, but you can also count up the squares.

There are 12 whole squares in the picture, 7 half squares, and 1 quarter square.

That gives a total of 15.75 squares—certainly not an integer.

But if all those fractional parts fit together just right, is it possible there could be no squares leftover? The answer is yes. But students need to find out how.

**Problem 1**

Find a rectangle with non-integer side length and whole number area.

Side lengths are non-integers (decimals or fractions). Area is a whole number.

If students are stuck and start losing their energy, consider hints to help them tackle an easier version of the problem.

For instance:

**Problem 1a.** Find a rectangle with at least one non-integer side length and whole number area.
- Is it easier to use fractions or decimals in these problems? Have you tried both?

When students have a solution to Problem 1, they should tackle the following extensions.

**Problem 2**.

Find a rectangle with whole number area, with non-integer side lengths greater than 1.

**Problem 3**.

Find as many solutions to problem 2 as you can.

What are the possibilities for side length? For area? Is there some pattern here?