Don't Be Square

Materials: calculator, spreadsheet software (optional), large sheet of paper greater than 15 inches on a side, yard stick, protractor

Work in a group of two or three.

In this activity, you will discover an interesting relationship when you construct figures inside of squares. Then you will see how you can use rational numbers to find the area of these figures.

Procedure for the Activity

Step 1 Carefully draw a square that measures 16 inches on a side. This will be Figure 1. Be sure that the angles measure 90° and that the sides are congruent, or have equal measure. Find the midpoint of each side of the square. Carefully connect the midpoints with line segments. This is Figure 2. Repeat this process four more times until you have Figures 3 through 6. The first two figures are shown in the diagram below. What type of quadrilateral is each figure?

Step 2 Copy or print out the table below. Fill in Columns 2 and 3. Measure each side of a figure to the nearest sixteenth of an inch. Then use your calculator to convert the fraction to a decimal rounded to the nearest hundredth. Use the fact that the area of a rectangle is length times width to find the area of each figure. Find the area of each figure to the nearest hundredth.

Step 3 Fill in Column 4 of the table by finding the ratio of the areas. You cannot fill in the cell for Figure 1. For the row containing Figure 2, find the ratio of the area of Figure 2 to Figure 1. Use your calculator to express the ratio as a decimal rounded to the nearest hundredth. For the row containing Figure 3, find the ratio of the area of Figure 3 to Figure 2. Continue the pattern for Figures 4 through 6. Compare the ratios. If you were to draw more and more figures, what do you think the ratio of the area of any two consecutive figures would be?
Step 4 Suppose you were to draw Figure 7. What would be the area of that figure?
Step 5 Describe how to use multiplication of rational numbers to find the area of the tenth figure.
Step 6 Write a general rule or a description of how you could find the area of any figure number drawn inside of the original square with side of measure 16 inches.

Wrapping Up the Activity

Make a poster to display your figures and the table of areas. Be sure to include a description of how to find the area of any figure number. If you prefer, repeat this activity using an equilateral triangle with sides of length 16 inches. Connect the midpoints of each side to form the inner figures.