Rectangles with Whole Area and Fractional Sides

Math concepts: Fraction/decimal multiplication, area model of multiplication
Materials: Paper and pencil
Tags: 5.NBT.B.7, 5.NF.4, 5.NF.6,

I love the question at the heart of this lesson—it’s simple but counter intuitive. This is a perfect way to let kids look for something without

The Launch

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?

The Wrap

Once students have worked through the problems, bring them together to wrap up the lesson with a discussion. Let students share their solutions. In particular, explore Problem 3. What patterns did they notice? What observations and conjectures do they have? For problem 1, any fractional reciprocals work, i.e., 3/4 and 4/3. Problems 2 and 3 are much more interesting. For reference, here are some solutions to the problem: So my first conjectures might be that one of the sides is a multiple of 2.5 and the other is a multiple of 1.6, or that the area is always a multiple of 4. However, the much more surprising solution given below destroys the first conjecture (not the second!). Fractions and decimals each suggest a different tactic in approaching this problem. Is there some perfect conjecture about what areas work? Leave that as an open question for your students to explore.

Here is a final problem for any students who are really look for a challenge:
Challenge Problem

Is there a square with non-whole side lengths and integer area? This is, in fact, a famous and important problem in the history of mathematics. It takes a somewhat deeper argument to show that such a square cannot exist. But don’t tell your students this! The search for the square that answers this problem can be fascinating, even if the students will come up empty in the end. Side lengths are equal non-integers (decimals or fractions) Area is a whole number

Volume Lessons

Now that the students have solidified their understanding of the two-dimensional cases, we will extend to solid geometry and volume. Some of these lessons also involve surface area as well; while surface area isn’t in the fifth grade standards, exploring it can actually strengthen the comprehension of volume, and make clear how it is different than area. A note about materials: you must have some kind of cubes for the beginning of these lessons. It is practically impossible to understand volume without actually building out of cubes when you begin. I like Unifix cubes or snap cubes if you have them, though the centimeter cubes in a set of base ten blocks will also work. As students become more comfortable with the physical objects, they may transition to recording their work using perspective paper, aka isometric paper.

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