Don’t Be Square

Teacher Notes:

• If students have a good geometry background, they could find the side length of each square by using the Pythagorean Theorem. When students have finished the activity, you could show them how to verify that the ratio of the areas of the consecutive figures is always exactly one-half by computing areas using the exact measures found by using the Pythagorean Theorem. For example, the side of Figure 2 is , so the area is or 128. Then the ratio of Figure 2 to Figure 1 is 128 ÷ 256 or 0.5.
• If you prefer, students could complete this activity using metric measure. Select a metric measure close to 16 inches for them to use as the original square. However, using inches is a good way for them to practice measuring with fractions. They also will get the opportunity to convert fractions to decimals.
• It is suggested in Wrapping up the Activity that students could repeat this activity using an equilateral triangle. You may want to have part of the class use the square and the other part use an equilateral triangle. Then you can compare the students’ results with the two different types of figures.
• For Step 6, most students will only be able to describe a method for finding the area of any figure. More advanced students may be able to write a formula, but only if they understand exponents.

Procedure for the Activity

Answers:

Step 1  Each figure is a square.

Steps 2-3 Answers may vary slightly due to rounding. Sample answers:

Step 3  It appears that the ratio of the smaller figure to the next larger figure is 0.5.

Step 4  4 in²

Step 5  Start with the area of the first figure, 256. Multiply 256 by 9 factors of .

Step 6  Multiply the area of the first square, 16, by n – 1 factors of , where n is the figure number. A possible formula where n is the figure number is .

Wrapping Up the Activity

See students’ work.