What’s the biggest rectangle?

Math concepts: Area, perimeter, number patterns
Materials: Graph paper, pencil
Tags: 3.MD.7.a-b, 4.OA2, 4.OA.3, 4.OA.5, 4.OA.5, 5.OA.3, MP1, MP2, MP3, MP5, MP6, MP7.

Of all the rectangles with a given perimeter, which one has the largest area? This simple question launches a fascinating exploration.

Remind students of the definition of area and perimeter.
Then pose the following question: if you know the perimeter of a rectangle, can you find the area? For example, what if we have a rectangle with perimeter 20 units. Can you find the area? Let students work on this for a bit, and then discuss.

Ideally, different students will have found different possibilities for what the area could be, (i.e., if sides are 3 and 7, area would be 21 square units). This motivates a return to the question: if we can’t find one single area, we could still ask: what is the largest area a rectangle of perimeter 20 could have? Again, let the students work on the problem, then bring them back to discuss.

If any student has written down all the possibilities, share this work. Otherwise, model how you might solve this problem by organizing your data.

There’s a lot to notice in the numbers you get for the area. For example, the numbers are symmetric around 25. Not only that, look at the differences between the numbers. They’re odd numbers! And the square numbers are hiding in there too (25 – 21 = 4). You could also extend this table if you want by saying that a side length could be 0. This would be a “flat” rectangle—a degenerate case, in mathematical parlance.

But the patterns keep working! Note that the largest area rectangle in this case is actually a square. Is a square a rectangle? Yes—rectangles need to have four equal (hence right) angles. Squares have four equal sides as well, but they still count as rectangles.

Let the students try to solve some more similar problems. Use the handout on the following page, or let the students come up with their own numbers to try. They can work till the end of math, and do the problems they don’t solve for homework.

Students may have noticed that the largest-area rectangles on the first two problems are squares. What about questions 3 and 4? If you stick with whole numbered sides, you can’t construct a square, since 70 and 75 aren’t evenly divisible by 4. But if you use fraction or decimal side lengths, you can. These questions can provide a nice motivation or review for how to multiply decimals or fractions. Question 5 is truly fascinating.

Once again, if we allow fractions, we can have tiny areas. For example, a 0.0001 by 49.9999 mile rectangle has a perimeter of 100 miles, but an area of just 0.00499999 square miles. Could we make it smaller? Easily—just add more zeroes after the decimal place in 0.0001. We are force to conclude, bizarrely and wonderfully, that there is no smallest rectangle with perimeter 100 miles.

Whatever rectangle we choose, we can find a smaller one. If we insist on whole side lengths, then the minimum area would be 49 square miles. Many students may have this as their answer, and that’s fine. But dropping the whole side length stipulation leads to some more interesting places.

• MFL Retangle Worksheet 1
• MFL Retangle Worksheet 2Q