Circles: a week of ready-to-teach maths

Five lessons for Year 8 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 every circle drawn, a circle to cut into wedges, twelve job cards to sort, 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. Every circle is drawn on the worksheets.
3. Print the two cut-out sheets, one copy per pair: the wedge circle is for Day 3, the job cards for Day 4.
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

For every circle, the distance round (the circumference) divided by the distance across (the diameter) is the same number, π = 3.14159..., which never ends or repeats. So C = πd, and since the diameter is two radii, C = 2πr. Cut a circle into thin wedges and lay them head to tail: they make a shape close to a rectangle r tall and half the circumference, πr, wide, so the area is A = πr². Circumference is a length, in cm or m; area is in square units. The hardest part is choosing: edging, fencing and crust need the circumference; seed, paint and cloth need the area.

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.

Exact and rounded answers

Answers are given exactly, in terms of π (a circle of radius 5 cm has C = 10π cm), and rounded to two decimal places with the π key on a calculator (31.42 cm). Using 3.14 for π, as the interactive unit does, gives answers a little smaller (31.4 cm); accept either when the working is right, and ask which is closer.

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 finds π by measuring; Day 2 uses it for the distance round; Day 3 sees why the area is πr²; Day 4 learns to choose between the two; and Day 5 puts them to work in longer problems. It builds on the parts of a circle and π in Year 7 and on composite areas earlier in Year 8, and prepares the volume and surface area of cylinders in Year 9.

Materials for the week (one trip)

• From the classroom: rulers, sharp pencils, calculators, scissors, glue sticks, this pack printed.
• For Day 1: string (about 1 m per pair) and five round things to measure, such as a drink can, a mug, a roll of tape, a plate and a bin.
• Cut-out sheet 1, one per pair: the wedge circle (Day 3).
• Cut-out sheet 2, one per pair: 12 job cards (Day 4).

Words for the word wall

circle, centre, radius, radii, diameter, circumference, π (pi), exact, rounded, semicircle, quarter circle, ring, sector, wedge, area, square centimetre, perimeter, composite shape

Dear families

This week in maths, Year 8 is working with circles: the distance round (circumference, C = π × diameter) and the space inside (area, A = π × radius × radius), where π is about 3.14. They measure real round things, see why the area formula works by cutting a circle into wedges, and decide which formula a job needs.

Try this at home

• Wrap a string round a can, then lay it across the top: the string is a bit more than three times as long as the width.
• Ask: a plate is 26 cm across. About how far is it round the edge? (About 3 × 26 = 78 cm; more exactly 81.7 cm.)
• Ask: which needs the area, the lace round a round tablecloth or the cloth itself? (The cloth.)
• Compare pizza sizes on a menu: is one large pizza more than two small ones? Use A = πr² to check.

My circles this week

Fill one row a day. Find a round thing, measure its diameter, and work out its circumference or area.

Printed from the free seegongsik Circles teaching pack · seegongsik.com/au/y8/measurement/AC9M8M03/pack

Radius, diameter and π

Every circle has the same ratio of the distance round to the distance across. Students name the parts of a circle, measure round things to find that C ÷ d is always a little more than 3, and compare values people use for π.

We are learning to

• name the centre, radius, diameter and circumference, and use d = 2r,
• measure a circumference and a diameter and find C ÷ d,
• explain why π is the same for every circle and why no fraction is exactly π.

Success criteria

• I can give the diameter of a circle of radius 3.5 cm.
• I can explain what the number π measures.

You need

The worksheet, a ruler and a calculator each; per pair, about 1 m of string and five round things.

Lesson flow (about 60 minutes)

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

Watch for these ideas

• A diameter that misses the centre: it is a chord, and too short.
• Mixing up the radius and the diameter.
• Thinking π is exactly 3.14, or exactly 22/7.

Answers

• Part A: r = 4 gives d = 8 cm; d = 15 gives r = 7.5 cm; r = 2.5 gives d = 5 m; d = 9 gives r = 4.5 mm.
• Part B: every C ÷ d should come out between about 3.0 and 3.3; the class average is close to 3.14.
• Part C: closest first, 3.1416 (out by 0.000007), 22/7 (out by 0.0013), 3.14 (out by 0.0016), 3.1 (out by 0.0416), 3 (out by 0.1416); none of them is π exactly.
• Part D: a circle twice as wide is the same circle scaled by 2, so every length doubles, the circumference too: C ÷ d stays the same.

Radius, diameter and π

A. Radius and diameter

B. Measure and divide

Wrap string once round the object and straighten it along a ruler. Measure the diameter through the centre. Give C ÷ d to two decimal places.

The class average of C ÷ d: __________   What do you notice?

C. Values for π

People use these values for π = 3.14159265...:   3.14,   3,   22/7,   3.1416,   3.1. Put them in order, the closest to π first. Which one is π exactly?

D. Stretch: twice as wide

A plate is twice as wide as a saucer. Explain why its circumference is twice as long, so that C ÷ d is the same for both.

Circumference

C = πd = 2πr turns a diameter into a distance round. Students give circumferences exactly and rounded, find how far a wheel rolls, and work back from a circumference to a diameter.

We are learning to

• find a circumference with C = πd or C = 2πr,
• give the answer exactly, in terms of π, and rounded to two decimal places,
• find a diameter from a circumference by dividing by π.

Success criteria

• I can find the circumference of a circle of radius 9 cm, exactly and to two decimal places.
• I can find the diameter of a circle from its circumference.

You need

The worksheet and a calculator with a π key each.

Lesson flow (about 60 minutes)

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

Watch for these ideas

• Putting the radius into C = πd, so the answer is half what it should be.
• Rounding in the middle of a long working instead of at the end.
• Giving a circumference in square units: it is a length.

Answers

• Part A: r = 5 cm: C = 10π = 31.42 cm; d = 12 cm: C = 12π = 37.70 cm; r = 3.5 m: C = 7π = 21.99 m; d = 70 cm: C = 70π = 219.91 cm.
• Part B: one turn = 66π = 207.35 cm, about 2.07 m; 1 km = 100 000 cm, and 100 000 ÷ 207.35 = 482.3, so about 482 turns.
• Part C: d = 75 ÷ π = 23.87 cm; d = 2.2 ÷ π = 0.70 m.
• Part D: semicircle ½ × 10π + 10 = 25.71 m (the straight edge counts too); quarter circle ¼ × 8π + 4 + 4 = 14.28 cm.

Circumference

C = πd = 2πr  ·  Use the π key and round to two decimal places.

A. Exact and rounded

B. How far does a wheel roll?

A bike wheel is 66 cm across. How far does the bike go in one turn of the wheel? About how many turns does the wheel make in 1 km?

C. Working back

Find each diameter to two decimal places.

D. Stretch: half and quarter circles

Find the perimeter of each shape: the curved part and the straight edges.

The area of a circle

Cut a circle into wedges and lay them head to tail: they make a shape close to a rectangle r tall and πr wide. Students see why A = πr², find areas of circles, rings and semicircles, and settle a pizza argument.

We are learning to

• explain A = πr² from a circle cut into wedges,
• find the area of a circle from its radius or diameter, exactly and rounded, in square units,
• find the areas of rings and semicircles.

Success criteria

• I can find the area of a circle 14 cm across, halving the diameter first.
• I can explain why the wedges make a shape close to a rectangle πr wide.

You need

The worksheet and a calculator each; per pair, the wedge circle (cut-out sheet 1), scissors and a glue stick.

Lesson flow (about 60 minutes)

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

Watch for these ideas

• Squaring the diameter instead of the radius: the answer is four times too big.
• Doubling the radius instead of squaring it: r² means r × r, not 2r.
• Writing cm for an area: it is in square units.

Answers

• Wedges: about 5 cm tall and 15.7 cm wide, so about 78.5 cm². Part A: r = 5 cm: A = 25π = 78.54 cm²; d = 8 cm: A = 16π = 50.27 cm²; r = 1.2 m: A = 1.44π = 4.52 m²; d = 30 cm: A = 225π = 706.86 cm².
• Part B: ring π × 6² − π × 4² = 20π = 62.83 cm²; semicircle ½ × π × 5² = 12.5π = 39.27 m².
• Part C: one big pizza 225π = 706.86 cm²; 2 small ones 2 × 100π = 628.32 cm². The big one is more, by 78.54 cm².
• Part D: π × 5² = 78.54 and π × 6² = 113.10, so a circle of 100 cm² has a radius between 5 and 6 cm (it is 5.64 cm).

The area of a circle

A = πr²  ·  Use the π key and round to two decimal places.

A. Wedges, then areas

B. A ring and a semicircle

C. The pizza argument

Ava says one pizza 30 cm across is more pizza than 2 pizzas 20 cm across. Jo says the two small ones are more, because 2 × 20 is more than 30. Who is right? Show the areas.

D. Stretch: estimate first

A circle has an area of 100 cm². Without finding it exactly, show that its radius is between 5 cm and 6 cm.

Circumference or area?

Real jobs do not say which formula to use. Students sort job cards into circumference and area, work four of them out, then find the perimeter and area of a running track and a window with a round top.

We are learning to

• decide whether a job needs the circumference (the edge) or the area (the inside),
• find the perimeter and area of shapes made of rectangles and part circles,
• work back from an area to a radius.

Success criteria

• I can say whether edging or seed for a round garden needs the circumference or the area, and why.
• I can find the perimeter of a shape with a semicircle on one side.

You need

The worksheet and a calculator each; the job cards from cut-out sheet 2, one set per pair, cut out.

Lesson flow (about 60 minutes)

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

Watch for these ideas

• Leaving out the straight edges of a perimeter, or counting a dashed join line.
• Adding a whole circle where only a semicircle is drawn.
• Forgetting to halve the area of a semicircle.

Answers

• Part A: circumference 1, 3, 4, 7, 9, 10, 12; area 2, 5, 6, 8, 11. Card 1 (d = 3 m): C = 3π = 9.42 m; 2 (r = 6 m): A = 36π = 113.10 m²; 5 (r = 0.8 m): A = 0.64π = 2.01 m²; 9 (d = 30 cm): C = 30π = 94.25 cm.
• Part B: perimeter 2 × 100 + 60π = 388.50 m; area 100 × 60 + π × 30² = 8827.43 m².
• Part C: glass 80 × 100 + ½ × π × 40² = 10 513.27 cm²; frame 80 + 2 × 100 + ½ × 80π = 405.66 cm (the dashed line is not part of the frame).
• Part D: r² = 50 ÷ π = 15.92, so r = 3.99 m, about 4 m.

Circumference or area?

A. Job cards

Write each card number in a pile. Then work out the four cards below, exactly and to two decimal places, with units.

B. A running track

C. A window with a round top

D. Stretch: from area to radius

A round pond has a surface area of 50 m². Find its radius to two decimal places.

Problems with circles

Circle problems take more than one step and need rounding that suits the job. Students solve problems from sprinklers to tables, find a radius-or-diameter mistake, and play the self-check quiz as a class game.

We are learning to

• plan a circle problem in steps and write each step with its unit,
• round an answer up or down to suit the job,
• check an answer by estimating with π ≈ 3.

Success criteria

• I can estimate the area of a circle with π ≈ 3 to check my answer.
• I can explain why using the diameter in A = πr² gives an answer four times too big.

You need

The worksheet and a calculator each; the self-check quiz at the bottom of the unit page for the game.

Lesson flow (about 60 minutes)

Two shorter sessions instead? Split after Part B; start the second with Part C and the quiz.

Watch for these ideas

• Rounding up when the job needs rounding down, as with people round a table.
• Putting the diameter into A = πr².
• Forgetting to take away the hole in the middle of a disc.

Answers

• Part A: 1. 16π = 50.27 m²; 2. π(6² − 0.75²) = 35.4375π = 111.33 cm²; 3. 25π = 78.54 m².
• Part B: 1. 1.2π = 3.77 m, ÷ 0.6 = 6.28, so 6 people (round down); 2. 9π = 28.27 m², ÷ 10 = 2.83, so 3 tins (round up); 3. 5π = 15.71 m, so buy 16 m (round up).
• Part C: Lia used the diameter as the radius: π × 10² = 314.16. With r = 5: 25π = 78.54 cm², a quarter of Lia’s answer.
• Part D: the square has sides 10 cm and area 100 cm²; the circle has r = 40 ÷ 2π = 6.37 cm and area 127.32 cm². The circle holds more.

Problems with circles

Estimate each answer with π ≈ 3 first. Then work it out with the π key and round to two decimal places.

A. Problems in steps

1. A sprinkler waters the grass in a circle of radius 4 m. What area does it water?
2. A disc is 12 cm across, with a hole 1.5 cm across cut from its centre. What is the area of the disc that is left?
3. A goat is tied to a post by a rope 5 m long, in a big flat paddock. What area of grass can it reach?

B. Round for the job

C. Find the mistake

Lia found the area of a circle 10 cm across and wrote π × 10² = 314.16 cm². What went wrong? Find the right area, and say how many times too big Lia’s answer is.

D. Stretch: same string, more space

A string 40 cm long is made into a square, then into a circle. Which shape holds more area, and by how much?

A circle to cut into wedges

For Day 3, one per pair, printed at full size

Cut round the circle, then along every dashed line to make 16 wedges. Lay them in a row, point up, point down, point up, so the shaded and plain wedges take turns. Cut the last wedge in half and put one half at each end. Glue the shape in your book and measure its height and its width.

Teacher note: the radius is 5 cm at full size. The row of wedges is about 5 cm tall and 15.7 cm wide (half the circumference), so about 78.5 cm², which is π × 5². If the sheet printed smaller, the height still matches the radius you measure.

Circumference or area? Job cards

For Day 4, one set per pair: twelve jobs

Sort the cards: does the job need the distance round the edge, or the space inside?

Teacher note: circumference: 1, 3, 4, 7, 9, 10, 12; area: 2, 5, 6, 8, 11.

What we know: Circles

Work on your own with a calculator. Give each answer exactly and to two decimal places where you can, with units.

1. A circle has a diameter of 18 cm. What is its radius? ______________________
2. Find the circumference of a circle of radius 7 cm. ______________________
3. Find the area of a circle 16 cm across. ______________________
4. A wheel is 50 cm across. How far does it roll in one turn? ______________________
5. How much cloth is needed for a round tablecloth of radius 0.9 m? Say first whether you need the circumference or the area. ______________________
6. Find the perimeter of this semicircle. ______________________
7. Find the area of the shaded ring. ______________________
8. Ben says, “If you double the radius of a circle, you double its area.” Is Ben right? Show an example. ______________________

Answers and marking guide

Answers with working

1. r = 18 ÷ 2 = 9 cm.
2. C = 2 × π × 7 = 14π = 43.98 cm.
3. r = 8, so A = π × 8² = 64π = 201.06 cm².
4. C = 50π = 157.08 cm, about 1.57 m.
5. The area: A = π × 0.9² = 0.81π = 2.54 m².
6. ½ × 12π + 12 = 6π + 12 = 30.85 cm.
7. π × 5² − π × 3² = 16π = 50.27 cm².
8. No: radius 1 gives π, radius 2 gives 4π. Doubling the radius makes the area 2² = 4 times as big. Any correct example earns the mark.

A quick three-level guide

Eight questions, four ideas. A student at standard answers most questions with working, gives exact and rounded answers, and keeps lengths and areas in their own units. Answers with 3.14 for π are a little smaller; accept them when the working is right.

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 and π, circumference, area, choosing between them, and problems.