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. It takes a somewhat deeper argument to show that such a square cannot exist. This is, in fact, a famous and important problem in the history of mathematics. Is there a square with non-whole side lengths and integer area? Here is a final problem for any students who are really looking for a challenge: Leave that as an open question for your students to explore. Ask students to use their knowledge of fractions to. Is there some perfect conjecture about what areas work? Draw two rectangles, one with a side length of 1/2 and the other with a side length of 2/3, on the board. However, the much more surprising solution given below destroys the first conjecture (not the second!).įractions and decimals each suggest a different tactic in approaching this problem. 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. Problems 2 and 3 are much more interesting. 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. Once students have worked through the problems, bring them together to wrap up the lesson with a discussion. What are the possibilities for side length? For area? Is there some pattern here? When students have a solution to Problem 1, they should tackle the following extensions.įind a rectangle with whole number area, with non-integer side lengths greater than 1.įind as many solutions to problem 2 as you can. Is it easier to use fractions or decimals in these problems? Have you tried both?.Find a rectangle with at least one non-integer side length and whole number area. If students are stuck and start losing their energy, consider hints to help them tackle an easier version of the problem. Side lengths are non-integers (decimals or fractions). But students need to find out how.įind a rectangle with non-integer side length and whole number area. That gives a total of 15.75 squares-certainly not an integer.īut if all those fractional parts fit together just right, is it possible there could be no squares leftover? The answer is yes. There are 12 whole squares in the picture, 7 half squares, and 1 quarter square. It is possible to multiply to find the area, of course, but you can also count up the squares. It seems impossible: imagine filling the rectangle with squares: there will be a bunch of cut off pieces.įor example, consider a 3.5 by by 4.5 rectangle. We can multiply, or actually note that we are just counting up whole squares.īut is the opposite true? In other words, can an rectangle have integer area but non-integer side lengths? For example, a garden shaped as a rectangle with a length of 10 yards and. If the side lengths are whole numbers, then so is the area.ĭo an example, i.e., draw a 5 x 7 rectangle and find its area. The area of any rectangular place is or surface is its length multiplied by its width.
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