Mathematics of Circles by Teacher Superstore

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Mathematics Circles © ReadyEdPubl i cat i ons . te

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•f orr evi ew activities pur poses onl y• Mathematics designed to extend and challenge 11 to 13 year olds.

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Written by Susan Levy. Illustrated by Rod Jefferson. © Ready-Ed Publications - 2000. Published by Ready-Ed Publications (2000) P.O. Box 276 Greenwood W.A. 6024 Email: info@readyed.com.au Website: www.readyed.com.au COPYRIGHT NOTICE Permission is granted for the purchaser to photocopy sufficient copies for non-commercial educational purposes. However, this permission is not transferable and applies only to the purchasing individual or institution. ISBN 1 86397 102 5

FOREWORD

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This book is divided into three sections. The first section on circle geometry contains exercises which are of an explanatory nature, and involve some compass and protractor work. The geometry concepts are explored informally, and students are asked to write down their findings.

The second, on circle arithmetic, provides extension exercises for children who have been taught to calculate circumference and area of a circle. The final section contains a variety of activities, all loosely based Re ad yEdconstructions, Publ i cat i ons on circles.© Some are compass which require accurate use of compass and protractor, as well as a steady •f orr evi ew pur posesonl y• hand! There are also some number and logic puzzles.

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There is something for everyone in this book. Enjoy it!

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CIRCLE GEOMETRY 2 FOREWORD 4 Drawing A Circle 5 Parts Of A Circle 6 Chords 7 Angle On The Diameter 8 Angles In A Circle 9 Angles At The Centre And Circumference 10 Hexagon In A Circle 10 Circle In A Hexagon 10 Hexagon Around A Circle 11 Square In A Circle

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CIRCLE ARITHMETIC 12 How Long Is A Circle 13 Parts Of The Circumference 14 Secret Code 15 Secret Code 16 Volume And Capacity 17 Clockwork 18 On The Right Track

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TABLE OF CONTENTS

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CIRCLE PUZZLES AND DIVERSIONS 19 Twenty Fours 20 Twenty Fours 21 Inner Circle 22 Möbius Strip 23 Compass Flower 24 The Horn 25 The Doughnut Twist 26 Round Table 26 Round Track 27 Making A Cone 28 Number Bubbles 29 Cutting The Cake 30 Cutting The Doughnut

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Circle Geometry

Drawing A Circle

EQUIPMENT NEEDED Compass, ruler.

You need a firm compass which will not slip, and which has a sharp pencil and a good point.

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R 1. Draw a circle with a 2 cm radius. Using the same centre, draw another with a 3 cm radius. These circles are concentric.

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How to measure a 5 cm radius

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R 2. Draw a circle with a 3 cm radius below. Now draw another 3 cm circle with its centre on the circumference of the first circle.

What do you notice about the two centres? ...................................................................

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How far apart are the centres? ......................................................................................

R Draw a line through both centres, so that it touches each circle twice. How long is the line between the furthest points on the circles? ..................................... Page 4

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Circle Geometry

Parts Of A Circle

EQUIPMENT NEEDED Ruler, pencil, eraser.

This circle has its centre at O. JO is a radius. KO is another radius.

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LM is a diameter.

LM also contains two radii, LO and OM

Draw a diameter QR.

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Draw a radius PO.

Use a different coloured pencil to trace over the radius OQ.

Arc PAQ is the major (longer) arc.

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Arc PBQ is the minor (shorter) arc. Arc CBD is a semi circle.

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Line KAL is called a tangent. It touches the circle in exactly one place, at A. Draw tangents through B, P, Q, C, D. What do you notice about the tangents through C and D? They are ................................................................................................................................ . Ready - Ed Publications

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Circle Geometry

Chords

EQUIPMENT NEEDED Compass, pencil.

B A

A chord is a line segment whose end points are on a circle. AB is a chord of this circle. R Join A and B to the centre O to form a triangle. R Measure all the angles.

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What do you notice? ..............................................................................................................

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Measure all the angles again. What do you notice? .............................................................. ...............................................................................................................................................

R Describe ∆ AOB. ...............................................................................................................

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Sides AO and OB are ............................................................................................................ R Draw a new circle with centre O, with radius 3·5 cm. R Draw a chord FG, 4 cm long, anywhere in the circle. R Join FO and OG to make a triangle. Measure ∠GOF. R Draw a chord MN 1 cm long in the circle. R Join OM and ON. Measure ∠NOM.

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What do you notice about the size of the angle at the centre of a circle? R Complete this sentence: As the length of the chord of a circle increases, the angle at the centre ...................................

......................................... .

What is the longest chord you can draw in this circle? .......................................................... R Draw it. R Make a triangle with the centre as before, and measure it. It measures .......................... Page 6

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EQUIPMENT NEEDED Protractor, compass.

Circle Geometry

Angle On The Diameter

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R Draw a diameter in the circle. Label it PQ. Now, from any point R on the circle, join RP and RQ to make a triangle.

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R Draw a circle with 5 cm radius here.

It measures .......................... .

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R Use a protractor to measure ∠PRQ.

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R Pick another point on the circle, and again draw a triangle which includes the diameter. Measure the angle on the diameter. It measures .......................... .

R Draw more triangles in the same way. Measure the angle on the diameter. It always measures .............. .

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Circle Geometry

Angles In A Circle

EQUIPMENT NEEDED Compass, protractor.

R Draw a chord (not a diameter) anywhere in this circle. The chord divides the circle into two arcs, the major arc, which is longer, and the minor arc, which is shorter.

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R Pick a point in the minor arc. Form a triangle, with this point and the ends of the chord. Measure the angle on the circle.

R Pick another point in the minor arc, and form another triangle. Again, measure the angle on the circle.

What do you notice? .............................................................................................................. R Repeat this two or three times.

R Complete this sentence: Angles in the minor arc of a circle, which are formed by the

end points of a chord, are ............................................................................................... .

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R Now pick a point in the major arc, and repeat the same steps. Measure the angles in the major arc.

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R Complete this sentence: Angles in the major arc of a circle, which are formed by the

end points of a chord, are ................................................................................................ .

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What type of angle are they? (acute / obtuse)

R Here is another circle. Draw a chord of a different length (not a diameter) and again draw and measure the angles in both arcs. R Describe your results here.

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EQUIPMENT NEEDED Protractor.

Circle Geometry

Angles At The Centre And Circumference

PQ is a chord in this circle. O is the centre. R is a point in the major arc. R

R Measure angles PRQ and POQ.

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What do you notice? ...............................................

................................................................................

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................................................................................ ................................................................................

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R Pick another point S in the major arc. Join PS and QS. R Measure the angle PSQ. Compare angles PSQ and POQ.

© ReadyEdPubl i cat i ons ............................................................................................................................................... •f orr evi ew pur posesonl y• In this circle, AB is a chord and O is the centre. What do you notice about the sizes of the two angles in each case?...................................

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R Measure angle AXB. What type of angle is it? ........................... .

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If the relationship still holds, which you just discovered, what size should the angle at the

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X is a point in the minor arc.

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centre of the circle be? ....................................... What type of angle would this be? .....................

Where do you think you would find this angle? .................................................................... R Measure it and see whether you are right. Fill in the missing word and number of degrees: ................ angle AOB measures ............. . R Mark another point Y in the minor arc. Join and measure angle AYB. Does the above result still hold? ............... . Ready - Ed Publications

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Circle Geometry

Hexagon In A Circle

EQUIPMENT NEEDED Compass, ruler. R Use this circle.

R With your compass point on the perimeter of the circle, and using the same radius as the circle, mark off points on the perimeter. Do this until you have 6 marks evenly spaced around the circle.

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Circle In A Hexagon

R Using the same hexagon, carefully measure to find the midpoint of a side of the hexagon. Now measure from this point to the centre with your compass, and use this as your new radius.

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R Join the points to form a hexagon.

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Hexagon Around A Circle (Use a pencil for this activity)

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R Use this circle. Using your compass, mark off 6 points equally spaced around the circle.

R Join opposite pairs of these to form diameters. At the ends of the diameters, carefully draw tangents to the circle, making sure they are exactly at right angles to the diameters. Join the tangents carefully to form a hexagon. Erase any unwanted parts of the lines.

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EQUIPMENT NEEDED Compass, ruler, pencil.

Circle Geometry

Square In A Circle

There are several ways of constructing a square in a circle, so that all four vertices touch the circle. Here is the easiest way.

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R Draw a circle of radius 5 cm in the space at the foot of the page.

© ReadyEdPubl i cat i ons R With compass point on the end of the diameter and compass radius about one and a f o r e i e wmake pu po s es o y•Now half• times ther radius ofv the circle, twor arcs, one each side ofn thel diameter. R Draw a diameter in the circle.

shift your compass point to the other end of the diameter, and do the same again so that the arcs intersect.

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R Where the arcs cross, join these points with a pencil line. Now, from the two points where this line crosses the circle, (or would cross the circle, if extended), form a square using the ends of the diameter.

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Circle Arithmetic

How Long Is A Circle?

EQUIPMENT NEEDED Thread, string etc., cylindrical objects for measurement, ruler or tape measure.

Here are three ways of estimating the circumference of a circle, if you only need to know its approximate length. 1. Take a piece of string, chain or thread (or hair), and carefully lay it round the circle. Then measure the length of thread you wrapped round the circle.

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Try these.

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2. Find a cylinder of about the same size as your circle, and measure around that. This could be a can or tube or bottle.

3. Measure the diameter of the circle. When you multiply this by 3, you will be just under the actual circumference. R Try it with the circles above. circle 1 = ....................................................

circle 2 = .......................................................

circle 3 = ....................................................

circle 4 = .......................................................

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EQUIPMENT NEEDED Calculator - optional.

Circle Arithmetic

Parts Of The Circumference This circle has a radius of 4 cm. The circumference is

r o e t s Bo r e p ok u S use π = 3·14

Here is half the circle. It is called an arc of the circle. The angle at the centre of a whole circle is 360°, so the angle at the centre of half a circle is 180°. The circumference from P to Q is half of the full circle. Arc PQ = 180 × 25·12 360 = 12·56 cm.

If the radius of a circle is r, If the angle at the centre is A°, Then the length of the arc is

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C =2πr =2×π×4 = 25·12 cm.

Arc = A × 2 π r 360

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= 40 × 2 × 3·14 × 6 360 = .................... cm

= ....................

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Arc = ....................

= ........................

= ....................

= ........................

= ....................

R On the back of this page, draw and calculate these: 5. radius 12·6 m, angle 63°

7. radius 38 m, angle 46°

6. radius 25 cm, angle 155·6°

8. radius 80 cm, angle 212°

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Circle Arithmetic

Secret Code Did you hear the one about the couple who met in a revolving door ...?

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To answer the riddle, find the perimeter of each diagram, and fill in the letter above the number wherever it appears in the table.

5·5 V

3·3

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6.28

18.2

4.11

10.36 21.04 13.82

8.22

10.36 13.19

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H 1·6

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2.83

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13.19 17.27

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8.16

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6.68

21.04 16.01

8.22

4.40

2.83

21.04 10.36

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2.83

4.11

6.28

6.68

6.28

17.27

6.68

12.34 13.82

8.22

1.88

6.28

8.22

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Circle Arithmetic

Secret Code

R Crack the code and discover the title of an amazing true adventure by Joshua Slocum. (Yes he really did it, and then wrote the book!) Find the area for each letter.

2·0

32° 3·3

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5cm

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4·1

8·2

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41°

119° 5·1 D

5·5

2·7

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6·9

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3·9

4·9

18°

56.69 81.67 35.82 30.19 35.82 7.57 3.04 81.67 30.18 18.22 7.57 13.20

81.67 2.86 18.22 69.92 7.57 94.98 7.47 12.56 13.20 12.42 18.22 2.86 30.18 94.98 Ready - Ed Publications

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Circle Arithmetic

Volume And Capacity

EQUIPMENT NEEDED Calculator - optional.

Here is a cylinder. It is 50 centimetres high, and has a radius of 20 centimetres. The volume of the cylinder is calculated like this:

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= (area of base) × height = π r ² × h = 3·14 × 20 cm² × 50 cm = 62 800 cm³

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Notice that the unit of volume is the cubic centimetre. This is what a cubic centimetre looks like.

R Calculate the volumes of these cylinders: 1. radius 12 cm, height 8 cm ....................

2. radius 6·6 cm, height 4·1 cm ................

© R e a d y E d P u b l i c a t i o n s 5. radius 77 mm, height 1·3 cm ................ 6. radius 2 cm, height 7·3 cm ................... •f orr evi ew pur posesonl y• Now, suppose these cylinders were to be used for storing liquid. How much would each 3. radius 3 m, height 1·6 m .......................

4. radius 12·5 m, height 20 m ...................

one hold when full?

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The word used for this is capacity, and the unit of measurement is the litre.

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To work out the capacity of a container, if you know its volume, use the formula 1 litre has a volume of 1000 cubic centimetres. 1 cubic centimetre has a capacity of 1 millilitre.

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To calculate the capacity, divide the volume (must be in cm³) by 1000 to get litres. Therefore, if the above container has a volume of 62 800 cubic centimetres approximately, then its capacity will be 62 800 ÷ 1000 = 62·8 litres. For very large amounts use For very small amounts use

1 kilolitre (kl) = 1000 litres. 1 millilitre (ml) = 1/1000 litre

1 litre = 1000 ml.

R Calculate the capacity of each of the containers in questions 1 - 6, using the best units for each one (ml, l, kl). 1. ................................................................

2. ...............................................................

3. ................................................................

4. ...............................................................

5. ................................................................

6. ...............................................................

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Circle Arithmetic

Clockwork

EQUIPMENT NEEDED Calculator - optional.

This clock has a minute hand 10 cm long, an hour hand 7 cm long and a sweep second hand 10 cm long. 1. How long does it take each hand to go around once?

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Hour hand ..................................................................

second hand ..............................................................

2. How far does the tip of each hand travel as it goes round once? Hour hand

Minute hand

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minute hand ...............................................................

Second hand

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...........................................

.............................................

...........................................

.............................................

...........................................

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go round once. Distance gone in 1 circuit. No. of circuits in 1 hour. Distance gone in 1 hour. Distance gone in 40 minutes. Distance gone in 12 minutes. Distance gone in 2 hours. Distance from 4pm to 5:30pm. Distance from 10 pm to 10:15 pm. Distance from 11:30 pm to 1:10 am. Distance from 6:19am to 6:45 pm.

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Complete this table for the clock. Do your working on another sheet of paper.

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Circle Arithmetic

On The Right T rack Track

EQUIPMENT NEEDED Calculator - optional. The Roundsville Athletic Club has a circular running track.

The track is designed so that the inside lane is exactly 700 metres long on the shorter side. There are eight lanes marked.

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R a) Calculate the radius r1, of the inside (shorter side) of the inside lane to nearest 1 decimal place.

and that

C = 700 m

∴700 = 2 × π × r1

Each lane is 1 metre wide.

r1 =

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We know that C = 2 × π × r1

R b) What is the radius r2 of the inside of the second lane? ...............................................

R c) Calculate the distance c2 that the runner in lane 2 will run, if he stays on the inside of his lane.

R d) Fill in the table. (Do your working on another sheet of paper)

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2 3 4 5 6

Radius

Track Length

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7 8 R e) What is the difference (to the nearest metre) in the lengths of any two neighbouring lanes?

R f) What is the length of the outside fence around the perimeter of the running tracks? R g) On another sheet of paper, draw a diagram of the track, showing how the starting places for the runners must be staggered, so that all runners run the same distance to a straight finishing line. Page 18

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Twenty F ours Fours Here is a number challenge for you. The object of this game is to make 24 by using all of the numbers in the circle, with a single operator (+ – × ÷ ) between each pair. Like this: 8 + 4 + 5 + 7 = 24

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This one is a little harder! (You may need to use brackets here) (4 × 7) – (2 + 2) = 24 or (4 × 7) – 2 – 2 = 24

You try these: (There may be more than one correct way to do them) 1.

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Circle Puzzles

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Circle Puzzles

Twenty F ours Fours 13.

14.

16.

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21.

22.

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24.

25.

20.

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26.

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28.

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31.

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17.

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15.

32.

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EQUIPMENT NEEDED Compass, protractor, pencil, ruler, eraser.

Circle Puzzles

Inner Circle

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R Draw a circle of radius 5 cm. Using a protractor, mark off 15° intervals around the circle. Number the marks lightly from 1 to 24 with pencil so that they can be erased later. Very carefully, using a ruler, join marks 1 and 10, 2 and 11, 3 and 12, and so on, right around the circle.

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R On another page, draw another 5 cm circle. This time join 1 with 8, 2 with 9, 3 with 10,

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and so on. What difference do you notice about the inner circle? ...................................

...............................................................................................................................................

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Variations: try varying * the distance apart of the pairs of numbers; * the number of intervals around the circle.

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R Write notes here about the effect of varying these factors.

............................................................................................................................................... ............................................................................................................................................... ............................................................................................................................................... ............................................................................................................................................... ............................................................................................................................................... ............................................................................................................................................... ............................................................................................................................................... Ready - Ed Publications

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Circle Puzzles

Möbius Strip

EQUIPMENT NEEDED Scissors, ruler, coloured pencil, felt tipped pen, sticky tape. R Follow these instructions:

1. Using scissors, cut off the strip of paper marked on the right hand side of this page.

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2. Colour in one side of the strip lightly in coloured stripes. Rule a line down the middle of the strip, on one side of the paper. 3. Make one twist in the strip, then stick the ends together with tape.

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You now have a loop with one twist in it.

4. With a felt tipped pen or highlighter, colour one cut edge of the paper loop until you get back to your starting point. R What do you notice? ...........................................................................................

..................................................................................................................................

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.................................................................................................................................. 5. With scissors, cut along the line you drew down the middle of the loop, until you have cut right around. You might have expected to get two loops linked together, but what do you actually have?

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A long loop with ............. twists in it.

6. Again, colour along one edge with a felt tipped pen. R What do you notice this time? ............................................................................ .................................................................................................................................. .................................................................................................................................. Page 22

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Compass Flower

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Here’s how you draw one of these: 1. Draw a circle with radius 6 cm in the space below.

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EQUIPMENT NEEDED Compass.

Circle Puzzles

2. Keeping your compass at 6 cm radius, put the point on the circle and draw an arc from one side of the circle to the other.

© ReadyEdPubl i cat i ons 4. Repeat these steps until you return to your starting point (6 arcs). •f orr evi ew pur posesonl y• THERE’S YOUR FLOWER. 3. Now shift the compass point to where the arc meets the circle. Repeat step 2.

R You could:

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What to do next: Cut out the flower. Colour it in. Make more petals, evenly spaced between the ones you already have. Make some flowers out of card or foil for decorations.

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The Hor n Horn

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EQUIPMENT NEEDED Compass, ruler, pencil, protractor.

Circle Puzzles

To make a copy of this drawing, follow these instructions carefully. Work in pencil and have an eraser handy.

1. Take a sheet of paper, and turn it this way. 2. 3.

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Mark the centre.

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4. Use a protractor to divide the semicircle into 5° intervals, and mark each interval lightly on the semicircle. 5. Now, on these marks you measured, draw circles as follows: On the first mark, draw a circle with radius 1 cm (10 mm). On the second, a circle with radius 1.2 cm (12 mm). The third circle will be 14 mm, and so on. Increase the radius by 2 mm each time. Draw 28 circles altogether, all with their centres on the original semicircle. Page 24

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The Doughnut T wist Twist

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EQUIPMENT NEEDED Compass, protractor, eraser.

Circle Puzzles

© ReadyEdPubl i cat i ons To reproduce this drawing, follow the instructions. Work carefully, using a pencil. •f orr evi ew pur posesonl y• R In the middle of a fresh sheet of paper, 1. Draw a circle of radius 6 cm. 2. Measuring with a protractor, lightly mark off 30° intervals around the circle.

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3. With a compass, lightly draw circles of radius 6 cm using these marks as the centres.

4. Copying the drawing above, go over the arcs you want to keep, to make them heavier. Then erase the others.

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5. Colour the doughnut in two or three colours.

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Circle Puzzles

Round T able Table

Seven people, Ali, Ben, Chris, David, Ewen, Frank and Gil, sat around a circular table for breakfast, lunch and dinner one day. After breakfast, there were personality differences, and all decided to sit next to two different people for each of the remaining two meals that day. At breakfast they sat in alphabetical order. For lunch and dinner, Ali wanted to sit as far as possible from Gil, and as near as possible to Ben. A B C D E F G What were the seating arrangements for

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Here’s a hint to get you started - draw some tables!

C D E F

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lunch and dinner? (Remember, no two people could sit together for more than one meal.)

Round T rack Track

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May and June were walking around a circular path. They both started from the same point at the same time, and continued until they both arrived back at the starting point together. If May walks round once in 3 minutes 44 seconds and June takes 6 minutes 4 seconds, how many times has each walked around?

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EQUIPMENT NEEDED Compass, ruler, scissors, tape.

Circle Puzzles

Making A Cone

A cone can be constructed from a piece of paper. Draw a circle with radius 10 cm on a sheet of paper. Cut out the circle.

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Mark the centre of the circle.

Draw two radii which form an angle of 90° at the centre.

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Cut out the wedge bounded by the radii.

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Now, with each piece of circle, join the cut edges and stick together with tape to make two cones. One cone will be tall and thin, the other wider and not very high. The amount you cut out of the circle will determine the height and circumference of the cones. You can regulate this by the angle you have at the centre.

The bigger the angle, the (higher/lower) the cone. ...............................................................

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For both of your cones, estimate by measuring and calculating:

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Height of cone

Diameter of cone

Small angle at centre

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The smaller the angle, the (higher/lower) the cone. ..............................................................

Large angle at centre

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Now cut out another circle. This time cut out a piece with angle 60° between the radii. Form two cones with the pieces, and complete this table. Small angle at centre

Large angle at centre

Height of cone Diameter of cone

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Page 27

Name .................................................................

Number Bubbles

To complete a number bubble, the three numbers in each line should have the same total.

1 4

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See - they all add up to 9!

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Teac he r

Can you complete these number bubbles? (Use consecutive numbers.)

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Did you notice a pattern here, with the number in the middle of each set of bubbles? R Draw yourself a set for the numbers 1 to 15. Now, were these too easy? R Perhaps you would like to try these: 1. The first 5 even numbers. 2. The first 15 even numbers. 3. The first 7 odd numbers. Page 28

4. The first 11 odd numbers. 5. The first 5 multiples of 3. 6. The first 9 multiples of 10. Ready - Ed Publications

Name .................................................................

Cutting The Cake A circular cake is to be cut with six straight cuts of a knife. The cuts must also go from top to bottom of the cake. You may not rearrange the pieces at any time.

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Teac he r

What is the greatest number of pieces you can make? (Of course, they won’t all be the same size.)

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This cake has been cut into 16 pieces with six cuts, but you can do better than that. R Draw some circles and try it.

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Page 29

Name .................................................................

Circle Puzzles

Cutting The Doughnut A doughnut is to be cut with three straight cuts of a knife. The cuts must also go from top to bottom of the doughnut. You may not rearrange the pieces at any time.

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Teac he r

What is the greatest number of pieces you can make? (Of course, they won’t all be the same size.)

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Page 30

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Here is a way which gives six pieces. But it can be cut into more pieces than that. Draw yourself some doughnuts, and try to find it for yourself.

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Answers

Page 14 Secret Code T

2.83

4.11

6.28

18.2

4.11

10.36 21.04 13.82

8.22

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R 6.68

6.28

8.16

6.28

8.22

4.40

21.04 16.01

21.04 10.36

6.28

2.83

4.11

6.28

6.68

8.22

1.88

6.28

17.27

6.28

6.68

12.34 13.82

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13.19 17.27

10.36 13.19

2.83

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Page 15 Secret Code

56.69 81.67 35.82 30.19 35.82 7.57 3.04 81.67 30.18 18.22 7.57 13.20 A

81.67 2.86 18.22 69.92 7.57 94.98 7.47 12.56 13.20 12.42 18.22 2.86 30.18 94.98

4. 9812.5 m³ 6. 91.69 cm³

Capacity: 1. 3.621 3. 45.22 kl 5. 0.242 l or 242.02 ml

2. 0.561 or 560 ml 4. 9812.5 kl 6. 91.69 ml

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3. 45.22 m³ 5. 242.02 cm³

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Page 17 Clockwork

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Time taken to go round once. Distance gone in 1 circuit. No. of circuits in 1 hour. Distance gone in 1 hour. Distance gone in 40 minutes. Distance gone in 12 minutes. Distance gone in 2 hours. Distance from 4pm to 5:30pm. Distance from 10 pm to 10:15 pm. Distance from 11:30 pm to 1:10 am. Distance from 6:19am to 6:45 pm. Ready - Ed Publications

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Hour hand 12 hours 43·96 cm 12 3768 (37·68 cm) 2·44 cm 0·732 cm 7·32 cm 5·5 cm 0·92 cm 6·1 cm 45·50 cm

Minute hand 1 hour 62·8 cm 1 3·66 cm 41·87 cm 12·56 cm 125·6 cm 94·2 cm 15·7 cm 104·67 cm 780·81 cm (7·81 m)

Second hand 1 minute 62·8 cm 60 62·8 cm 25·12 m 754 cm (7·54 m) 75·36 m 56·52 m 9·42 m 62·8 m 468·49 m Page 31

Page 18 On The Right Track a) 111.5 m b) 112.5 m c) 706.5 m d) Lane 1 2 3 4 5 6 7 8

Track Length 700 m 706.5 m 712.8 m 719.1 m 725.3 m 731.6 m 737.9 m 744.2 m

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Teac he r

6m 750.5 m Starting places must be 6 metres apart. Work through circumference of an arc formula backwards. The degree between each starting place is approximately 3°.

Page 19, 20 Twenty Fours

8×2+6+2 6+4+7+7 3×3×2+6 8×2+4×2 6×4×2÷2 (6 × 8) – (6 × 4) 9×4–9–3 (9 – 3) × (9 – 5) 5×6–3–3 7×4–5+1 (9 ÷ 3 + 3) × 4

2. 5. 8. 11. 14. 17. 20. 23. 26. 29. 32.

8+8+7+1 (3 + 9) × (9 – 7) (8 – 5) × 8 × 1 (5 + 3) × 9 ÷ 3 5+6+7+6 (9 – 1) × 6 ÷ 2 (5 + 7) × (5 – 3) 5×2+5+9 (9 × 1) + 6 + 9 (7 – (5 ÷ 5)) × 4 4×7–2–2

3. 6. 9. 12. 15. 18. 21. 24. 27. 30.

3°

(8 × 4) – (8 × 1) 1×8×6÷2 2×5+5+9 (4 + 4 + 4) × 2 (5 – 1) × 6 × 1 8 × (2 + (6 ÷ 6)) (4 – 1) × 6 + 6 8×9×2÷6 5+9+3+7 8×8–8×5

Page 26 Round Table Lunch seating order is Ali, Frank, Ben, David, Gil, Ewen, Chris. Dinner seating is Ali, Ewen, Ben, Gil, Chris, Frank, David.

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1. 4. 7. 10. 13. 16. 19. 22. 25. 28. 31.

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e) f) g)

Radius 111.5 m 112.5 m 113.5 m 114.5 m 115.5 m 116.5 m 117.5 m 118.5 m

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Page 26 Round Track May goes round 13 times in 48 minutes 32 seconds while June goes round 8 times. Page 29 Cutting The Cake The solution is for every line to intersect every other line. The maximum number possible is 22 regions. Page 30 Cutting The Doughnut The solution is to have the three cuts intersect in the doughnut, not in the whole, then there are nine pieces. Page 32

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