Pi and the features of circles: a week of ready-to-teach maths

Five lessons for Year 7 Measurement. Print this pack and the week is prepared: each day has a one-page plan and a student worksheet with practice and a stretch section, plus circles to measure, circle cards, a mini-check and every answer with its working.

Start here: five minutes to Monday

1. Skim the week at a glance on the next page.
2. Print the five days. Each day is two A4 sheets: a plan and a worksheet.
3. Print the two cut-out sheets: circles to measure (Day 2), one per pair, and circle cards (Day 4), one set per pair.
4. Open the free interactive unit on the board. Every plan says which picture to show and which buttons to press.
5. Teach straight from the plan. Timings, talk prompts, misconceptions and worked answers are all on the one page.

The idea in plain words

Every circle has a centre. A radius runs from the centre to the edge; a diameter runs right across through the centre, so it is two radii long; and the circumference is the distance all the way around. Measure any circle and divide its circumference by its diameter, and the answer is always the same number, a little over 3. That number is called π (pi). It cannot be written exactly as a fraction or a decimal, and 3.14 is close enough for most work. So C = π × d, and because d = 2r, C = 2 × π × r. Working back, d = C ÷ π.

One day, one lesson

The five lessons fill a week of maths, one lesson of about 60 minutes a day. Run them in order: each day stands on the one before. Every plan marks a split point for two shorter sessions, and every worksheet ends with a stretch section for students who finish early.

How we write circles

Write the rule, then the numbers: C = π × d ≈ 3.14 × 23 = 72.22 ≈ 72.2 cm. Use ≈ once π becomes 3.14, and round a circumference to one decimal place unless the question says otherwise. Estimate first with three diameters. Calculators are welcome from Day 2, where students divide their measurements.

The week at a glance

One lesson a day for a week. Each day stands on the day before, so run them in order.

How the week builds

Day 1 names the parts and links the radius to the diameter; Day 2 finds, by measuring, that C ÷ d is the same for every circle; Day 3 turns that into C = πd; Day 4 brings in the radius and works back from a circumference; and Day 5 uses the circumference in problems about wheels and tracks. It builds on perimeter (AC9M5M02) and decimals, and it leads into the circumference and area of a circle in Year 8 (AC9M8M03).

Materials for the week (one trip)

• From the classroom: pencils, rulers, compasses, scissors, string or wool, small envelopes for the cards, this pack printed.
• Cut-out sheet 1, one per pair: six circles to measure (Day 2).
• Cut-out sheet 2, one per pair: 24 circle cards (Day 4).
• Round things to measure on Day 2: tins, lids, plates, coins, a bin or a wheel; calculators from Day 2.

Words for the word wall

circle, centre, radius, radii, diameter, circumference, pi (π), ratio, compass, semicircle, perimeter, estimate, approximately (≈), turn, revolution

Dear families

This week in maths, Year 7 is learning the parts of a circle and the number π (pi): for every circle, the distance around is a little over three times the distance across.

Try this at home

• Wrap a strip of paper around a tin, mark where it overlaps, and unroll it. Is it a bit more than three times the tin’s width?
• Measure a plate or a lid across the middle, then guess how far it is around before you check with string.
• Count how many turns a bike wheel makes along the driveway or the footpath, then measure how far it went.
• Ask: does a big circle have a bigger π? (No. The distance around grows with the distance across, so π stays the same.)

Circles this week

Fill one row a day with a round thing, its diameter, its circumference, and the circumference divided by the diameter.

Printed from the free seegongsik Pi and the features of circles teaching pack · seegongsik.com/au/y7/measurement/AC9M7M03/pack

Parts of a circle

A circle is every point the same distance from its centre. Students name the centre, radius, diameter and circumference, see that a diameter is two radii, and draw circles with a compass from a radius or a diameter.

We are learning to

• name the centre, radius, diameter and circumference of a circle,
• find the diameter from the radius, and the radius from the diameter,
• set a compass to draw a circle of a given size.

Success criteria

• I can point to each part of a circle and name it.
• I can explain why a diameter is twice as long as a radius.

You need

Compasses, rulers and the worksheet; a board compass or a pin and string for the demonstration.

Lesson flow (about 60 minutes)

Two shorter sessions instead? Split after the compass circles; start the second with Part A.

Watch for these ideas

• Calling any line across a circle a diameter. A diameter must pass through the centre.
• Halving when they should double. A diameter is two radii, so it is the longer of the two.
• Thinking the circumference is a line inside the circle. It is the curved edge itself, the distance all the way around.

Answers

• Part A: 1 diameter, 2 centre, 3 circumference, 4 radius.
• Part B: r = 6 cm, d = 12 cm; d = 19 cm, r = 9.5 cm; r = 1.5 cm, d = 3 cm; d = 11 cm, r = 5.5 cm.
• Part C: (a) the diameter measures 7 cm; (b) set the compass to 4.3 cm, half of 8.6 cm.
• Part D: the big circle is 20 cm across and each small one 4 cm, so 20 ÷ 4 = 5 circles. Using the radii, 10 ÷ 2 gives the same answer.

Parts of a circle

A. Name the parts

B. Radius and diameter

Fill in the empty boxes. All lengths are in centimetres.

C. Compass circles

(a) Draw a circle with a radius of 3.5 cm. Measure its diameter.   ____________
(b) You need a circle with a diameter of 8.6 cm. What do you set the compass to?   ____________  Draw it.

D. Stretch

Small circles with a radius of 2 cm are lined up, touching, along a diameter of a big circle with a radius of 10 cm. How many fit across the big circle? Explain.

Measuring around

Students measure the diameter and circumference of circles and round things, divide, and find that C ÷ d is a little over 3 every time. That number is π.

We are learning to

• measure a diameter and a circumference carefully,
• find C ÷ d for a circle,
• explain why measured values of C ÷ d differ a little.

Success criteria

• I can measure the distance around a circle by rolling it or with string.
• I can say what π is: the circumference divided by the diameter, the same for every circle.

You need

Cut-out sheet 1 (one per pair), scissors, string, rulers, round things, calculators and the worksheet.

Lesson flow (about 60 minutes)

Two shorter sessions instead? Split after measuring; start the second by pooling the data.

Watch for these ideas

• Measuring across but not through the centre. That line is shorter than the diameter, so C ÷ d comes out too big.
• String that stretches or slips, or a circle that skids as it rolls.
• Dividing d by C instead of C by d, which gives about 0.32.

Answers

• Part A: at 100% print, A to F are 45, 25, 65, 35, 75, 55 mm across and about 141, 79, 204, 110, 236, 173 mm around. Accept C ÷ d from about 3.0 to 3.3.
• Part B: 23.2 ÷ 7.4 = 3.14; 81.8 ÷ 26 = 3.15; 7.9 ÷ 2.5 = 3.16; 195 ÷ 62 = 3.15. All close to 3.14; none is exact.
• Part C: 25 ÷ 9.5 = 2.63, far below 3.14. A jar 9.5 cm across should be about 29.8 cm around, so the string slipped or did not go all the way round.
• Part D: every length of the circle is 3 times as long, the diameter and the circumference too. Both grow by the same factor, so C ÷ d does not change.

Measuring around

A. Measure the circles

Use the circles on cut-out sheet 1, then two round things from the room. Divide on a calculator and round to two decimal places.

B. Other people’s measurements

What do you notice about the last column?   ________________________________________

C. Something went wrong

Mia measured a jar: a diameter of 9.5 cm and a circumference of 25 cm. Work out C ÷ d. What might have gone wrong?

D. Stretch

Circle Q is circle P enlarged, so that every length is 3 times as long. What happens to the diameter, to the circumference, and to C ÷ d? Explain.

C = πd

Because C ÷ d is π for every circle, C = π × d. Students estimate a circumference as three diameters, work it with 3.14, and compare 3.14 with the π key on a calculator.

We are learning to

• write C = πd and use it to find a circumference,
• estimate first with three diameters,
• round sensibly and give the unit.

Success criteria

• I can find the circumference from the diameter, with an estimate first.
• I can say why three diameters is always a little too short.

You need

Calculators, with a π key if possible, and the worksheet.

Lesson flow (about 60 minutes)

Two shorter sessions instead? Split after Part A; start the second with Part B.

Watch for these ideas

• Multiplying by 3 and stopping there. Three diameters is the estimate; the circumference is a little more.
• Writing = once π has become 3.14. The answer is approximate, so write ≈.
• Copying every digit from the calculator. Round a circumference to one decimal place.

Answers

• Part A: 15 cm: estimate 45 cm, then 3.14 × 15 = 47.1 cm; 2.5 m: estimate 7.5 m, then 3.14 × 2.5 = 7.85 ≈ 7.9 m; 60 mm: estimate 180 mm, then 3.14 × 60 = 188.4 mm; 1.2 m: estimate 3.6 m, then 3.14 × 1.2 = 3.768 ≈ 3.8 m.
• Part B: 1. 3.14 × 4.5 = 14.13 ≈ 14.1 m; 2. 3.14 × 34 = 106.76 ≈ 106.8 cm; 3. 3.14 × 50 = 157 m.
• Part C: the π key gives 157.08 m, so 3.14 is 0.08 m short, about 8 cm in 157 m. Close enough for a Ferris wheel.
• Part D: the hexagon has six sides, each one radius long, so its perimeter is 6 radii, which is 3 diameters. It sits inside the circle, so the circumference is longer: more than 3d. The square has four sides, each one diameter long, and sits outside the circle, so the circumference is shorter than 4d.

C = πd

C = π × d · estimate first: C is a little more than 3 × d · then use 3.14 and round to one decimal place

A. Estimate, then work it

B. Around the edge

1. A round pool is 4.5 m across. How long is a fence that runs right around its edge?   ____________
2. A pizza has a diameter of 34 cm. How long is its crust, all the way around?   ____________
3. A Ferris wheel has a diameter of 50 m. How far does a seat travel in one full turn?   ____________

C. The π key

Work problem 3 again with the π key on a calculator. How much does using 3.14 change the answer? Does the difference matter for a Ferris wheel?

D. Stretch

Radius and working back

A diameter is two radii, so C = π × 2r = 2πr, and working back, d = C ÷ π. Students sort the circle cards, go from a radius to a circumference and back, and find the perimeter of a semicircle.

We are learning to

• use C = 2πr when we know the radius,
• work back from a circumference to the diameter and the radius,
• find the perimeter of a semicircle.

Success criteria

• I can find a circumference from a radius.
• I can find the diameter of a circle from its circumference.

You need

Circle cards from cut-out sheet 2 (one set per pair, cut out before the lesson), calculators and the worksheet.

Lesson flow (about 60 minutes)

Two shorter sessions instead? Split after the card sort; start the second with Part A.

Watch for these ideas

• Putting the radius into C = πd without doubling it. The answer is then half the circumference.
• Stopping at the diameter when the question asks for the radius, after dividing by 3.14.
• Leaving out the straight edge of a semicircle, or using the whole circumference for the curve.

Answers

• Cards, as radius, diameter, circumference, for r = 2.5, 4.5, 6.5, 8.5, 9, 11, 16, 20 cm: F Q L; V C I; D N S; M T G; A J W; U H O; K X B; P E R. A and C are different circles.
• Part A: r = 12 cm: d = 24 cm, C = 3.14 × 24 = 75.36 ≈ 75.4 cm; r = 0.5 m: d = 1 m, C = 3.14 × 1 = 3.14 ≈ 3.1 m; r = 35 mm: d = 70 mm, C = 3.14 × 70 = 219.8 mm.
• Part C: d = 78.5 ÷ 3.14 = 25 cm, r = 12.5 cm; d = 6.28 ÷ 3.14 = 2 m, r = 1 m; d = 251.2 ÷ 3.14 = 80 mm, r = 40 mm; (d) half of 3.14 × 16 is 25.12 ≈ 25.1, plus the straight edge: 25.12 + 16 = 41.12 ≈ 41.1 cm.
• Part D: (a) C doubles: C = 2πr, so twice the radius gives twice the circumference. (b) d = 100 ÷ 3.14 ≈ 31.85 cm, so r ≈ 15.9 cm.

Radius and working back

C = π × d = 2 × π × r · working back: d = C ÷ π, and r = d ÷ 2 · use 3.14

A. From a radius

B. Card sort

Write the letters of each set of three cards: radius, diameter, circumference.

C. Back from the circumference

D. Stretch

(a) The radius of a circle doubles. What happens to its circumference? Explain with C = 2πr.
(b) A circle has a circumference of 100 cm. Find its radius, to one decimal place.

Wheels and turns

One full turn of a wheel moves it one circumference along the ground. Students use C = πd, forwards and backwards, in problems about wheels, a clock, a track, a garden and a tree, then play the quiz.

We are learning to

• use the circumference to solve a problem, including wheel turns,
• decide whether a problem gives the radius, the diameter or the circumference,
• check an answer against a quick estimate.

Success criteria

• I can find how far a wheel goes in a number of turns.
• I can find a diameter from a measured circumference in a real problem.

You need

The worksheet, calculators, and the unit’s self-check quiz on the board.

Lesson flow (about 60 minutes)

Two shorter sessions instead? Split after problem 3; start the second with problem 4.

Watch for these ideas

• Taking the minute hand as the diameter. It runs from the centre of the clock, so it is a radius.
• Dividing metres by centimetres in problem 2. Change 100 m to centimetres first.
• Counting one straight of the track, or two whole circles for the ends. Two half circles make one circle.

Answers with working

1. 3.14 × 66 = 207.24 ≈ 207.2 cm in one turn.
2. 100 m = 10 000 cm, and 10 000 ÷ 207.24 ≈ 48.3, so about 48 turns.
3. The tip goes round a circle of radius 8 cm, so d = 16 cm and C = 3.14 × 16 = 50.24 ≈ 50.2 cm.
4. 2 × 85 = 170 m of straights; the half circles make one circle, 3.14 × 64 = 200.96 m; 370.96 m, about 371 m.
5. 3.14 × 3 = 9.42 m of edging, and 9.42 × $12 = $113.04.
6. d = 2.2 ÷ 3.14 ≈ 0.7 m, about 70 cm across.
7. d = 40 000 ÷ 3.14 ≈ 12 739 km, about 12 700 km.
8. The diameter grows by 2 × 10 = 20 cm, so the rope by 3.14 × 20 = 62.8 cm, the same for a pond or the Earth.

Wheels and turns

C = π × d, with π ≈ 3.14 · one full turn of a wheel = one circumference along the ground

Problems (7 and 8 are the stretch)

1. A bike wheel has a diameter of 66 cm. How far does the bike go in one full turn of the wheel?
2. About how many turns does the wheel make while the bike rides 100 m?
3. The minute hand of a clock is 8 cm long. How far does its tip travel in one hour?
4. A running track has two straights, each 85 m long, joined by two half circles with a diameter of 64 m. How far is one lap of the inside edge?
5. A round garden bed has a diameter of 3 m. Edging costs $12 a metre. How much does edging for the whole bed cost?
6. A string tied once around a tree trunk is 2.2 m long. About how wide is the trunk?
7. Stretch. The equator, around the middle of the Earth, is about 40 000 km long. About how far is it through the Earth, from one side to the other?
8. Stretch. A rope fits tightly around a round pond. It is made longer so it sits 10 cm out from the edge all the way round. How much longer is the rope? Would the answer change for a rope around the whole Earth?

Circles to measure

Day 2. Cut out each circle along its line. Measure the diameter through the centre dot, in millimetres. Then stand the circle on its edge with the gold tick at 0 on your ruler and roll it one full turn, until the tick touches the ruler again: that distance is the circumference. No time to cut? Lay a piece of string along the edge instead, mark where it meets its start, and straighten it along the ruler.

Print at 100% if you can. A different print size changes every length, but not C ÷ d.

Circle cards

Day 4. Cut out the 24 cards. There are eight circles, and each has three cards: its radius, its diameter and its circumference. Sort the cards into eight sets of three. The circumferences are worked with π as 3.14, to one decimal place.

What we know: Pi and the features of circles

Work on your own. Show your working and write each answer with its unit. Use 3.14 for π and round to one decimal place.

1. A circle has a radius of 21 cm. What is its diameter?   ____________
2. A circle has a diameter of 46 mm. What is its radius?   ____________
3. Find the circumference of a circle with a diameter of 28 cm.   ____________
4. Find the circumference of a circle with a radius of 24 mm.   ____________

5. A circle has a circumference of 314 m. Find its diameter.   ____________
6. Kai measured a lid: a diameter of 5.8 cm and a circumference of 18.4 cm. Work out C ÷ d to two decimal places. Why is it not exactly 3.14? ________________________________________
7. Ana says a bigger circle has a bigger π. Is Ana right? Explain. ________________________________________
8. A wheel with a diameter of 58 cm turns 10 times. How far does it go, in metres?   ____________

Answers and marking guide

Answers with working

1. 2 × 21 = 42 cm.
2. 46 ÷ 2 = 23 mm.
3. 3.14 × 28 = 87.92 ≈ 87.9 cm.
4. d = 2 × 24 = 48 mm, and 3.14 × 48 = 150.72 ≈ 150.7 mm.
5. 314 ÷ 3.14 = 100 m.
6. 18.4 ÷ 5.8 = 3.17. Close to 3.14 but not exact: every measurement of a real object has a small error.
7. No. A bigger circle has a bigger diameter and a bigger circumference, in the same proportion, so C ÷ d, which is π, stays the same.
8. One turn: 3.14 × 58 = 182.12 cm; 10 turns: 1821.2 cm = 18.21 m, about 18.2 m.

A quick three-level guide

Eight questions, four ideas. A student at standard moves between the radius and the diameter, finds a circumference with 3.14 and works back to a diameter, and describes π as C ÷ d, the same for every circle.

Weekly class record

Jot a tick as you move around the room; the mini-check fills any gaps. A tick a day is plenty.

The five columns are the five days: the parts of a circle, measuring around, C = πd, the radius and working back, and wheels.