Discovering the meaning of π Lesson Plan

Set-up for the activity

You would have asked students to bring in objects with circular surfaces and made a collection of these. Sort these through to make sure that it is easy to measure the diameter and circumference of the circular surface. You will need about `12-15` objects; one per pair. I have used mugs, tins, water bottles, and tape rolls, among other things.

Demonstrate how to measure

Do not assume that your `7`th graders know how to measure using a ruler. Demonstrate using an object - emphasize that they have to start at `0`, that they should count beyond the whole numbers. Make them familiar with the different units of measurement on the ruler, including millimeters and the sections of an inch on the ruler.

Introducing the activity

Tell students that today you will measure the around and across in a circle. Students might already know that the across is called diameter. 

• When measuring the around, they should use the string to wrap around the object and then measure it on the ruler. 
• To measure the across, they should use a ruler - in most objects the center might not be visible, so they to visually estimate where the center is.
• They should use the same unit for both the measurements. Emphasize they should be as accurate as they can be. For centimeters, they could use decimals. For inches, they should write the measurement as a mixed number or with decimals.

Each pair gets one object with a circular surface, a ruler, and a string.

Recording the data

Students will first record the data in their notebook. I then make a large table on the whiteboard and call on students to write down the name of the object and the measurements for the around and the across. Once they have recorded the measurements, I add a fourth column for Around `\div` Across. Sometimes, after we have done the Around `\div` Across for every object, I might ask a pair to do the measurements again, knowing that the ratio looks off; sometimes I do it as I walk around if it seems like any measurements don’t make sense.

When recording the Around `\div` Across, students wonder what they should do with the units. Hopefully, some students can explain that when you divide, the units are canceled out and the ratio does not have any units... as long as they did both measurements with the same unit.

What do you notice?

Ask students to write down `3` things that they notice about the chart. Students are likely to point out that the different objects have different measurements, that students have used different units of measurement, that the same object might have been measured by two different groups using different units. They will also notice that the ratio of Around `\div` Across hovers around `3`. 

Introducing `\pi`

Students are astonished that when you take any circle and divide the Around by the Across, you always get the same number and that is it called `\pi` and that the value is close to `3.14`.  Some students would have heard of `\pi` and would know a few digits of `\pi`. Show the first hundred digits of `\pi`. Sometimes we have done a competition for memorizing the most number of digits - nothing conceptual about it; it is just fun way for students to remember `\pi`!

This might be a good time to talk about a type of irrational numbers - that their decimal expansion goes on and on without any predictability and that `\pi` is one such number.

Formula for circumference of a circle

In the next lesson, you would formally establish the formula for finding the circumference of a circle. You will first take them through the definition of Circumference and Diameter. 

Then ask them what they learned about the relationship between Around and Across.

Once you have reviewed yesterday’s lesson and talked about Circumference and Diameter, you can show them this slide and ask them to find the formula for circumference. After some algebra, students will realize that to find the circumference, they simply have to multiply the diameter by `\pi`. You could also introduce the formula for circumference given radius `(2\pi r)` but I believe that at this stage it is unnecessary. If they can remember using words that the Circumference is `\pi` times the diameter, it is better than remembering formulas.

Finding circumference using diameter

Now that students are familiar with the formula for circumference using diameter, have them attempt the next problems. Make sure they understand they need to write their answers in terms of `\pi` (exact answer) and rounded to the nearest hundredth. They can check their work with a partner.

Students may be confused by “in terms of `\pi`”, so it can be helpful to have a discussion on why their answers in terms of `exactlyare exact compared to the rounded answers. Students may also have rounding errors, so it can be helpful to have students explain how they rounded.

Finding circumference using radius

If your class is capable, let them try these next problems independently. If your students seem to be struggling though, you can ask them what is different about the images. This should help students recognize that the radius only goes halfway across the circle. From there, they should be able to use their reasoning skills to determine the length of each diameter.

Have students explain what they did to find the circumference for each circle. If you haven’t already, you can show students the formula for circumference with radius `(2\pi r)` and relate it to the steps they took to find each circumference.