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Discovering Pi

Discussion, Suggestions, Possible Solutions

Make sure that you provide students with circles of varying sizes, including some that are relatively small and some that are
relatively large. Suggestions include jewelry (rings, bracelets), spools of thread, flower pots, tires, hula hoops, round tables,
etc.

You may want to begin this task with a mini-lesson on rounding since students are asked to round to the nearest hundredth.
Allow students to decide on the units they should use to measure the lengths on each circle. After the groups begin, check to
be sure that students understand that they must use the same units of measure to measure the circumference and the
diameter of a given circle.

In a closing discussion, be sure to clear up any issues or misconceptions related to measurement that still exist. Guide
students to the idea that the ratio C/d is around 3.14. End the lesson by telling students some form of the following. “You
have discovered one of the most important ratios in mathematics. It is so important that it was given a special name by the
Greeks. That name is Pi and is represented by the symbol , π. Later on, when you study irrational numbers, you will learn
that the actual value of Pi is a decimal value that is non- terminating and non -repeating. It goes on forever! For our work
this year, we will use the approximation 3.14 for π.”

Using the Equation C/d = π

a. In the equation C/d = π, C and d are called variables. π is referred to as a constant. Explain why.

Use this circle to help you answer parts b-d.
b. Look at the given circle. What is its radius? What is its diameter? Explain how you know.
c. Use the equation you have discovered, C/d = π and what you have learned about solving equations to find the
circumference of this circle.
d. We can write the equation C/d =π as C = πd. Explain why.
e. We say the equation C =πd is “solved for C in terms of d”. In other words the equation gives us the length of the
circumference of a circle if we know its diameter. We can also find the diameter if we know the circumference.
Solve the equation for d. What operation did you use? Why?
f. Suppose the circumference of a circle is 24 inches. What is its diameter to the nearest tenth of an inch?
g. Use what you know about the relationship between the diameter and the radius of a circle to write an equation for the
circumference in terms of the radius. Explain your thinking.
h. Practice what you have learned. Find the indicated length for each circle below. Round each of your answers to the
nearest tenth of a unit.
A circle has a radius of 5 inches. Find its circumference.
A circle has a circumference of 21 feet. Find its diameter.
A circle has a radius of ¾ yards. Find its circumference.
A circle has a diameter of 12.8 inches. Find its circumference.

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Using the Equation C/d = π

Discussion, Suggestions, Possible Solutions

An excellent mini-lesson for this task could involve the conventions for multiplying and dividing numbers and letters . Be sure
that students know that 4n means 4 times n and could also be written as 4·n. Ask students why they think we might not want
to write 4 xn in algebra . Conventions that should be mentioned include:

• 1·n, 1xn, 1n are usually written as n. Ask students why they think this might be true.
• mn means m times n
• n/m means n divided by m and can also be written n÷m.
• n/n = 1. Ask students why they think this might be true.

At some point students will also need to discuss the difference between calculating values using the approximation 3.14 for π
and calculating values using the π button on their calculators.