To the Teacher:
Many students tend to memorize formulas that we use in geometry or other mathematics areas without much understanding. For example, most students know the formula for the circumference and area of a circle for a long, long time, but they never really understand why Pi exists in the formulas for the area and circumference of a circle. This particular lesson allows students to discover why pi works in solving problems dealing with finding circumference.

Materials:
Various size circles or circular items, meter sticks, metric tape measures, rulers, scissors, Scotch tape, drywall tape, calculators, and table worksheet.

Geometer's SketchPad.

1. Consider the follow situation:

• How long a piece of paper must be to cover a round container for an art project?
• How long a piece of wire is needed to make a round perch for a bird feeder?

• For helmets we need the circumference of your head at the temple, the distance from your eyebrows to the base of your neck in back, and the distance from ear to ear over the head.
• For bracers the length and circumference of your forearm at the elbow and wrist.
• For breastplates the circumference and length of your chest.
• For warrior's belts and kidney belts, the circumference of your waist.

Part 2. The  relationship between the diameter and circumference of a circle

1. Students get into groups of 2-3.

2. Each group has some round objects (coins, cylinders, jars, lids).

3. Discuss: how can we measure the diameter and circumference of a round object?

4. With the measuring method found, measure the diameter of each of the round objects to at least two decimal places. Each person should do their own measurements. Fill the data into the following table.

5. Measure the circumference, distance around each circle, to at least two decimal places. Each person should do their own measurements. Fill the data into the following table.

6. For each object, calculate the ratio of circumference/diameter and round to the nearest two decimals. Each person should do their own calculations.

7. Within your group calculate the (group) average ratio.

8. Fill in the table.
 

9. Examine the table. Do you see anything? Discuss with your group members. If you find anything, state your conjecture(s).

10. For each object, assume that diameter is the X coordinate and circumference is the Y coordinate, then you can use a point (X, Y) on the coordinate plan to represent the object. Plot the eight points on the coordinate plan. What do you find? State your conjecture(s).

11. We will use GSP to test our conjecture(s) by exploring the relationship between the diameter and circumference of a circle.

12. Use GSP to draw a circle. Measure the diameter and circumference. Compute the ratio of circumference to diameter.

13. Drag the construction point on the circle to shrink or enlarge the circle. Notice the change of the dragging to the measure of the diameter and circumference and their ratio. What do you find?

14. Go back to your table of the measurements. Compare the average of the ratios on your table to the ratio on the GSP. Are they close?

15. As a group decide which of the following is the correct choice.

As a circle gets larger, the ratio of the circumference to the diameter
a) becomes large
b) stays about the same
c) becomes smaller

16. If the ratio becomes larger or smaller how much larger or smaller does it become?
______________________________________

17. If the ratio stays the same, about what number does it equal?
___________________________________________

18. What can you say about the relationship between the diameter and circumference of a circle? Can you come up with a formula to find the circumference of an object knowing only the diameter of that object?

19. Share what you have learned with other members of the class.

20. Write down your conclusions for the activities you have just done.

21. (As an extension, the teacher can provide more information about Pi, such as the history of Pi.)