Teaching Proofs & Formulas Part 2: Middle School

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The Circle & Pi

What is a circle? What is Pi? What is the relationship between a circle and Pi? Students use Pi everyday with little-to-no understanding of it. To a student, Pi is simply a little symbol that pops up every now and again. Sometime they get to shorten it and write it as 3.14, sometimes they are asked to use it as a fraction 22/7. They apply it when asked, but it has little to know value to them. My philosophy is to generate value by taking it's value away from them. Let them operate in a world without the availability of Pi and see if they begin to value it.

Lets start with the formula for the circumference of a circle

C = (D)(pi)

or

C=(2)(Pi)(r)

An educator can state the relationship of Pi as the quotient of a circles circumference and its diameter. over and over again but the student will tend to neglect the relationship. To remedy this we are going to have students calculate, by hand, the quotient of 10 circles' circumferences to their diameter. In other words, a student is going recalculate an approximation of Pi over and over and over again ten-fold until they start to pick up on a trend.

All Aboard The Pi Train!

What You Will Need:

 

1) A Printout of 8-10 Circles of various sizes and label them by number

2) A ruler

3) A tailor style tape measure (or a bendable ruler to measure circumference)

4) A recorder sheet for answers

For Example: Circle 1 Circumference_______in. Diameter________in. C/D=______

Circle 2 Circumference_______in. Diameter________in. C/D=______

Etc.

Steps:

Step 1: Separate students into pairs (individually works as well)

Step 2: Hand out materials

Step 3: Have Students begin to measure each circles circumference and diameter and record them on the answer sheet

Step 4: Have students calculate, by hand, the ratio of each circles circumference to its diameter.

Step 5: Ask the students to write what trends they have noticed.

Classroom Discussion:

After everyone is done with the exercise, discus what each student just did.

Examples of Questions

“What did you get for the C/D of circle 1? circle 2?”

“Did anyone get anything less than 3 or bigger than 4?”

“Which circles C/D was the closest to Pi?”

“Why do you think this is”

“Does anyone believe they measured their circle perfectly?”

“What would you need to measure it perfectly?”

“If it was measured perfectly and calculated perfectly what would you get?”

Recap Of Lesson:

The biggest idea that you want your students to walk away with is that Pi is something special. It is defined as a circles circumference divided by its diameter (See note below regarding this). Just knowing this bit of information would have allowed your students to have written “Pi” as their answer in the above 10 problems and they would have been right each time (without all the extra work). Now we need to think of a thought experiment to solidify the value to your students.

Thought Experiment:

Try this,

“You are a captain within a 9th century medieval military. Your primary role as head of the archery units is to protect your kings small castle from outside invasion. Your castle has become surrounded by an army of barbarians. You know that your archers can accurately shoot clusters of enemy targets from 100 yards, but you do not know the distance from your men to the targets standing along the entrenched moat. However, you do know the distance all the way around the outside of the circular moat is 600 yards. Can your archers hit their target?

C = (2)(Pi)(r)

We don't know (r) but we know C

600 = (2)(Pi)(r)
dividing 600 by 2 yields

300 = (Pi)(r)

dividing 300 by Pi roughly yields

95.5 yards = radius

Thus, if your archers are standing at the center of the circle, can they hit their target at a radius of 95.5 yards away?

Based on the assessment made earlier, yes!

Homework Question:

Do you think the majority of your archers are located at the center of the circle, closer to the enemy or further away? Why?

User Note To Appease The Math Gods: Although obvious, given the title of this post, to not offend anyone, let me point out that our definition of Pi is limited to Euclidean Space. More formal definitions of Pi are not applicable.