Cutting a rectangle
Tools and materials.
Index cards, rulers, scissors, calculators.
Task.
Each student gets an index card. The task it to cut it into 5 slices having equal areas. A slice must be cut by two straight cuts starting at the center and extending to the border.
(You may tell students a story about dividing a rectangular pizza or a rectangular cake.)
A solution.
Divide each of the four sides into 5 equal parts and draw 20 narrow slices toward the center.
All of the slices have the same area. Cut along every fourth line, forming 5 bigger slices.
Proof of correctness.
Let the rectangle’s sides have length a and b. Each small slice has either a base measuring a/5 and height b/2, or a base measuring b/5 and height a/2. In each case the area is a*b/20.
Generalization.
The same method works for dividing a rectangle into any whole number n > 1 slices of equal area. Divide each side into n equal parts, and form 4*n narrow slices. Let 4 narrow slices form one bigger slice to be cut out.