Engage:

 What can you tell me about pi? Listen to students as they tell you what they know. Try to emphasize answers that refer to pi as a ratio instead of a given value.

Time: 3-5 minutes

Where does this value come from?

Students are expected to participate in the discussion. They should offer up any explanations or answers they can. They are also expected to question other students or the teacher during any unclear explanations.

Explore:

The students will be divided into groups of 2-3. Each student will be given an Archimedes Recipe for Pi worksheet and the other needed materials.

Briefly demonstrate how to inscribe an octagon in a circle. This can be done by drawing a circle and then dividing it into 8 equal "pie slices" with lines. Every point on one of the lines that intersects with the circle is a vertex of the octagon.

The teacher should then instruct the students to work in their groups to complete the worksheet. During work, the teacher shall be available to answer questions.

Time: 25-30 minutes

Students will either divide themselves into groups or work with their assigned partners, depending on teacher preference.

Students will observe the procedure for inscribing a polygon in a circle.

The students will work in their groups toward the completion of the worksheet. They are expected to help, check, and question each other during this process. If the other students do not offer sufficient help, they are expected to ask the teacher for clarification.

Explain:

Call attention back to the center of the class. Using the final two questions from the worksheet, initiate a class discussion about the value pi and the process of discovering it. Thoroughly explore both questions on the worksheet. Be sure to check for misconceptions in student answers and seek to alleviate them if they arise.

After discussion has begun to wind down on this topic, begin to ask why pi could not be directly measured by Archimedes.

Time: 10-15 min

What do you think characterizes a more accurate approximation?

What technological limitations did they face?

Are measurements with instruments absolutely "true" numbers?

Students stay with their groups and work together briefly to review the questions if they have not answered them. Every group is expected to contribute at least something to the discussion.

Extend / Elaborate:

Ask the students to explain why the approximation becomes more accurate as the number of sides of the polygon increases. Try to guide them towards its approximation of a circle.

Time: 5-7 minutes

What would a regular polygon with 1000 sides look like? 100000 sides?

Students need to offer their opinions and theories as to the effect of the sides of the polygon. All students are asked to briefly discuss in their groups and then one person from each group can address the class.

  Evaluate:

This lesson is to reinforce the idea of discovery in math. The teacher needs to actively observe students as they go about the task of discovering the value of pi. The teacher will use their responses in groups as well as their questions to gauge their progress. This lesson really serves to deepen the understanding of the use of pi.

Perhaps at a later time, so it is not immediately fresh in their minds, ask them to approximate pi showing work.

Time: Ongoing through lesson

Could you derive pi on your own?

How would you define pi in terms of a circle?

Students should be actively engaged in the lesson and working within their groups to discuss and evaluate their data.

If students are later asked to perform such an operation, they will recreate Archimedes method.