3. Circles: Establishing Pi

• various objects with a circular cross-section, some large and some small (for example, paper plates, jar lids, buckets)
• rolls of masking tape
• compasses
• calculator
• protractor
• trundle wheel
• string, rope or tape measure
• Predict, Observe, Explain (POE) template: docx PDF 

This stage of the learning sequence aims to develop students’ understanding that the ratio of circumference to diameter is the same for all circles and use a variety of approaches to find a more accurate value for the ratio.

The following activity is adapted from the ‘How far will this…roll’ task published by the Government of South Australia (2017).

Show students a circular object, such as a roll of masking tape and ask students to estimate how far this roll of tape will roll in one full turn.

Invite students to share and justify their estimations. Take the opportunity to introduce mathematical terms to describe the features of a circle and relate the perimeter of a circle to its circumference. For example, rephrase by saying, “So what you are saying is that how far it will roll depends on the circumference of this circle.”

Provide students with the opportunity to test their estimate. Students may choose to unroll exactly one layer of masking tape and sticking it onto the table.

Then pose the following questions to prompt students to reflect on their estimates:

• How close were you?
• What could you learn from this to help you to make a closer estimate for a different circle?
• If you were able to take measurements, but not the circumference of the circle, what else could you measure?
• Students may suggest measuring the ‘width’ of the circle. Introduce the term ‘diameter’ and ‘radius’ and encourage students to use this language to describe features of a circle.
• Without taking measurements, what connection can you see between the diameter and the circumference of this circle?

Demonstrate the relationship between circumference and diameter of a circle by placing the roll of masking tape along the piece of tape that was unrolled before and marking the diameter length as it is moved along. Students will see that the diameter fits into the circumference a little more than three times.

Support students to test their hypothesis that the circumference of a circle is ‘three and a bit more’ times larger than its diameter. Encourage students to explore how far other circular objects will roll, using circles of different sizes such as bottle caps, wall clock, or a round coffee table.

Ask students, “What do you notice about the relationship between circumference and diameter in each these circles? Do you see a pattern?”

Observe if students notice that the size of the circle has no effect on the relationship between circumference and diameter and that the circumference of any circle is always a little more than three times its diameter.

Enable students by explicitly teaching vocabulary prior to the lesson, and by having the mathematical vocabulary for circles displayed for reference in the room, with labels and description for each term. Demonstrations and activities should include explicit practise with the vocabulary.

Extend students by asking them to try refining their estimate, to the nearest tenth or with fractional parts, the number of times that the diameter of the roll fits into the circumference.

1. Finding Pi

Each of the following activities focuses on refining the ratio of circumference to diameter of a circle through different approaches and it is recommended that students are exposed to all three activities. This can be done by either having students complete each one individually or having the class divided into three groups with students from each team completing a single activity and sharing their results to their peers. The activities are adapted from the ‘A Better Value For Pi’ lesson within the Circumference sequence from the Resolve teaching resources.

Explain to students that now that they have found that the ratio of circumference to diameter of a circle is estimated to be ‘three and a bit’, they will explore how this estimate can be improved.

Show students the symbol to describe this ratio, and discuss its meaning:

Present the ClickView video, Irrational Numbers: Pi and Pies to students to give a brief overview of how pi is derived. Sign into ClickView using your department credentials. Students may be interested in learning about the history of pi through the following articles:

• History of Pi
• A brief history of pi

a. Plates, balls and other round things

Have students carefully measure both the circumference and diameter of a collection of circular objects such as paper plates, jar lid and buckets to find the ratio of circumference to diameter. Also give students the opportunity to measure the circumference of large circles marked on gym floors and playgrounds.

Discuss in the group:

• How do we measure the circumference of these objects?
• How do we know where to measure the diameter?
• How should we organise our data?
• How do we deal with a variety of different ratios from different students? Discuss the importance of finding an average when there are many measurements.
• How can we improve the accuracy of our calculated ratio?

b. Walk around the circumference

This activity will be best completed in a large open space, preferably with a pole.

Attach one end of the rope to a pole. Alternatively, have a student hold one end of the rope and stand in an open space. This will be the centre of a circle. Invite another student to hold the other end of the rope and move away from the centre so that that the rope is taut. Place a marker to indicate the starting point. The student should place the back of their heel against the marker. 

Using the rope as a radius, have students count the number of shoe lengths as they walk around the centre, keeping the rope taut. Remind students to walk back to the start. Ask students, “What length have you measured?” Prompt students that they walked around the circumference of a circle.

Ask students to find the ratio of the number of shoe lengths of the circumference to the number of shoe lengths along the radius.

Discuss in the group:

• Does it matter if we use shoe lengths or more formal units of measurement?
• How do we deal with a variety of different ratios?
• How can we improve the accuracy of our calculated ratio?

As students are working on the activities, encourage them to use the Predict, Observe, Explain (POE) template available in Materials and texts to record what they predict the results of each maths experiment will be; what they found out; and why they think that result was obtained. This can form part of the formative assessment, as outlined in the 'Reflect and consolidate' phase of this sequence.

When students have completed both activities and found a more accurate value for pi, encourage them to explain their findings to their peers in the other groups and to develop a formula to calculate the circumference of a circle.

Challenge students to express the formula for the circumference of a circle in terms of its diameter and in terms of its radius. Students should be able to explain that the formula can be any of the following:

For each activity, enable students requiring further support in finding the ratio by asking them, “How can you calculate how many times bigger is the circumference compared to its diameter?” Suggest that when we want to find the general idea of different values, we can find the average. Discuss how an average of a set of results can be found.

Extend students by asking them, “How can you test your results, ensuring that the ratio of circumference to diameter is the same for circles or different sizes?”

2. Applying the circumference formula

Encourage students to apply their knowledge of circumference and diameter of circles by presenting them with problems in a real-world context. For example:

A go-kart racetrack consists of two straight sections and two semi-circular sections as seen below:

If the distance around the track is 400 m, what can the lengths of the straight section be? What can the radius of the semi-circle be?

Enable students by first asking them, “How can you work out the diameter of a circle if you know the circumference?” Remind students that the circumference of a circle is about three times the diameter and recall the meaning of radius. How can you use this knowledge to find the diameter? How can you find the radius?

Extend students by having them add another three lanes around the outside of the track. If the width of each lane is 1.8 m, what will be the radii of the next few lanes?

Areas for further exploration

1. Measuring Tree Trunks

This lesson follows from the ‘Finding Pi’ activities in the Circumference lesson sequence from Resolve teaching resources. It aims to give students the opportunity to apply their knowledge of the ratio of circumference to diameter. In this lesson, students make a d-tape, which can be used to wrap around a cylindrical object such as a tree trunk, to instantly measure its diameter. Download the resources on the Resolve site to access learning and teaching resources. 

2. Track Design

The Track Design problem from Nrich Maths challenges students to design a running track and to work out where runners should start on the track so that they have all run 200m by the finish.

This stage aims to develop students’ understanding of the relationship between the circumference and the diameter of a circle. Formative assessment of student learning in this stage of the sequence could include observing students’ strategies when conducting the investigations and reviewing their Predict, Observe, Explain tool. Note if students can explain how their investigations help them to discover the value of pi.

To assess students’ application of the circumference formula, ask students to find either of the circumference, radius, or diameter of a circle when one measurement is known. For example:

• What is the diameter of a wheel if one revolution is 1.4 m?
• What is the distance around the famous Stonehenge if its diameter is 100 m?
• If the minute hand of a clock travels 77.2 cm per hour, what is the length of the minute hand?

Enable students requiring further support by suggesting that they calculate the diameter of each donut using the information they already know.

Extend students by presenting them with the following problem:

A donut stall sells donuts in boxes of six. Each donut has a circumference of 28.8 cm. What should the area of the box base be to fit the donuts, so they are not stacked on top of each other?

What if the box has a triangular base? How can the donuts be arranged in the box? What will the area of the base of the box be?

Australian Academy of Science, 2020. Circumference. [Online]
Available at: https://www.resolve.edu.au/circumference
[Accessed 15 March 2022].

ClickView, 2014. Irrational Numbers: Pi and Pies. [Online]
Available at: https://www.clickview.com.au/curriculum-libraries/video-details/?id=3715047&cat=3706391&library=primary
[Accessed 15 March 2022].

Exploratorium, n.d. Pi Day: History of Pi. [Online]
Available at: https://www.exploratorium.edu/pi/history-of-pi
[Accessed 15 March 2022].

Government of South Australia, Department for Education and Child Development, 2017. Using units of measurement: Year 8. [Online]
Available at: https://acleadersresource.sa.edu.au/wp-content/uploads/2018/05/Using_units_of_measurement_Year_8.pdf
[Accessed 15 March 2022].

Purewal, S. J., 2013. A brief history of pi. [Online]
Available at: https://www.pcworld.com/article/511387/a-brief-history-of-pi.html
[Accessed 15 March 2022].

Reys, R. E. et al., 2020. Helping Children Learn Mathematics. Milton: John Wiley & Sons Australia..

Siemon, D. et al., 2015. Teaching Mathematics: Foundations to Middle Years. Melbourne: Oxford University Press.

University of Cambridge, n.d. NRICH: Track Design. [Online]
Available at: https://nrich.maths.org/7359
[Accessed 15 March 2022].

Van de Walle, J., Karp, K., M, B.-W. J. & Brass, A., 2019. Primary and Middle Years Mathematics: Teaching Developmentally. Australia: Pearson.