Smile 1670

Find the fakes!Six students were asked to toss a coin 50 times each and record the results. Some K them couldn't be bothered tossing a coin, and simply invented their results. They all filled in their tables row by row.

Here are the six sets of results (T=tail, H = head)

Artie

Sue

lana

Simon

HTTTTHHTHH TTHTTTTHHH TTHHTTTHTT TTHTHHHTHH .HHHHTHHHHH

TTHTTTTHHH TTHTHTHTTT THTHHTHHHH THHHTHHHTT THHTHTHTHH

HTTHHTTHTH THTHTHTTHT HTHHTHTTHH

T HT H T H T H H T H

T H T H T H T H H

THHHHHHHTH HTTTTHHTHT THTTTTTHTH TTTHTHTTTT HTHHHTTTTT

Debbie TTHHTHTTTT HHTTHHHTTH HHHHTHHHTT HTHHTHHTTH HTHHTHHHHH

Bob HTTHTHTTHT HHTTHTHHTH TTHHTH'TTHT THTHHTHTHT HTHTTHTHTT

Which sets of results were faked? Justify your conclusions.

Smile 1672o

SOA1A SOLIDSThe Soma Cube Puzzle has 7 pieces.

1. Sketch each piece on isometric paper.

Pieces A and D can be placed together to make this solid.

2. Copy and shade these diagrams to show how the solids can be built from ....

. . . pieces B and C

. . . pieces E and F

. . . pieces B and G

3. Which two pieces can be put together to make this solid? Sketch your answer.

4. Use pieces D, E and F to make cuboid.

5. Can you make this cuboid? Which pieces did you use?

Turn over

All seven pieces of the Soma Puzzle fit together to build a cube.

6. Can you build a Soma Cube?Sketch your solution to show how the pieces fit together.

There are, in fact, hundreds of different ways to build the Soma Cube. There are also many other shapes that can be made with the Soma set. You can find out more about these in ....

More Mathematical Puzzles and Diversions Martin Gardner (Penguin) ISBN 0 14020748 1

Creative Puzzles of the World Pieter Van Delf & Jack Botermans (Cassell) ISBN 0 30430300 3

Smile 1673

HCF & LCM

HCFHCF stands for highest common factor . . .. . . The HCF of 6 and 9 is 3 because 3 is the largestnumber which divides exactly into both 6 and 9.

LCM stands for lowest common multiple . . . . . . The LCM of 6 and 9 is 18 because 18 is the smallest number which 6 and 9 both divide into.

(1) Work out (a) the HCF of 6 and 8.(b) the LCM of 6 and 8.

The answers are at the foot of the back page. If you have got them wrong, make sure you understand HCF and LCM before you continue.

Draw a table to show the HCF of some pairs of numbers.

HCF 1 8 10

10

Describe any patterns in your table.

Explain them if you can.

All numbers can be expressed as the product of prime numbers:

15 / <3' "5

20.

/\2 5

15=3x5 20=2x2x5

These prime factors can be put into a diagram:

Prime factors of 15

Prime factors of 20

)raw diagrams like this for some other pairs of numbers.

How do the diagrams help to find the HCF and LCM easily?

turn over

Tabulate your results from the previous page:

FIRSTNUMBER

1566

"VVv,'^-^/^- W*^-^''^-

SECONDNUMBER

2098

XNXSXWX, ^-^^

HCF532

^^^^^

LCM601824

What is the relationship between any two numbersand their HCF and LCM?(You may need to add some more results to yourtable.)

Explain the rule, if you can - the work on the previous page might help.

HOJ SH3MSNV

Board Order 5m(lel675- a game for 2 to 4 playersYou will need counters and a board for each player.

1 . Get someone to choose 15 counters for you.

2. Each player must use at least 8 counters to make a pattern with symmetry.

3. Scoring

»One line of symmetry - score 3 points

E

Two lines of symmetry - score 5 points

E>

Point symmetry - no lines of symmetry but the board looks the same when turned half-way round -

flfe score 1 0 points

fv ^ ^

^ counters, eacn *^i^ " / bonus point. * /K/^Jp "^

- 1

Oo

O 01 1

ioo

oxamp/e; One line of symmetry

i

O.__-

oo

0 0

ooo

:amp/e: Two lirtes of symmetry

OC

0o

v_.-'

v^ ;

o

C

Example: Point symmetry

4. Calculate the score for your own pattern.

5. Play 5 rounds. The player with the highest total wins.

ipoa 'sudiunoo g anyi a/oiu asn norf // sjuiOfi sniiog

S]UlOd Q I

dJioos -

Board Order- a game for 2 to 4 players

One line of symmetry - score 3 points

Two lines of symmetry - score 5 points

Point symmetry - score 10 points

Bonus Points//you use more than 8 counters, each

Co>.vs\j|

<

'( ^r'lv l;i

Smile 1676

24

12

Pythagorean Triples

Here are the first few in a famous "family" of right angled triangles. Continue the sequence.

(3, 4, 5)(5, 12, 13)(7, 24, 25)(9, 40, 41)

(n, )

There are other right angled triangles which do not fit this pattern. Some are multiples of ones in the list above, like (6, 8, 1.0) but others, like (8, 15, 17) aren't multiples of any other triples. A triple that is not a multiple of any other whole number triple is called primitive.

This card will lead you through a proof of a method of finding all primitive Pythagorean triples. The proof is basically the one given by Diophantus (see SMILE 1460) but in modern notation. What follows is a sketch of Diophantus' proof. There are sixteen lettered statements and you are asked to give reasons why they are true -in other words, to prove them. One technical word is used in the proof and that word is parity. Two numbers have the same parity if they are both odd or both even but they have different parity if one is odd and the other is even.

Turn over

Remember: every lettered statement needs to be proved.

A Finding all primitive Pythagorean triples gives us all Pythagorean triples.

=» We only need to consider primitive triples (x, y, z)

B x, y, z cannot all be odd

C x, y, z cannot be two even numbers and one odd number

P x, y, z cannot all be even

=> x, y, z consists of one even number and two odd numbers.

E z is not the even number (Hint: work in modulo 4)

=> either x or y is even

F We can assume x is the even number

G We can write x2 = (z + y)(z - y) instead of x2 + y2 = 72

H x, z + y, z-y are all evenSo we can write x = 2u, z + y = 2v, z-y = 2w, where u, v and w are positive whole numbers

J v and w have no common factor

=» in particular v and w have no prime factor in common

K v x w = u 2 tells us that v and w are both square numbers

=> We can find positive whole numbers p and q so that p2 = v and q 2 = w

L p and q have no common factor

M z=p2 + q 2 y = p2 -q2

N x = 2pq

O p>q

P p and q have different parity

All primitive Pythagorean triples are of the form (2pq, p2 -q2 p2 + q 2) where(j) P>q(ii) p and q have different parity.Copy and continue this table of primitive Pythagorean triples up to p = 8

P2 3 4 4 5 5

Q1 2 1 3 2 4

x4

12 8

24 20 40

y3 5

15 7

21 9

z5

13 17 25 29 41

One last question:To what values of p and q does the triple(4961, 6480, 8161) correspond?

Smile 1677

The ancient Greeks believed for a long time that any two lengths were "commensurable". By this they meant that the ratio of their lengths could be written as p : q where p and q were whole numbers. The whole foundation of Greek mathematics was shaken by the discovery, some time before 410 BC, that some ratios in geometrical

*es were incommensurable. The proof given here is ably not the one that led to the initial discovery but

it was well known to Aristotle. The proof is an example of proof by contradiction. The idea of such a proof is as follows:

(i) We assume the reverse of what we want to prove, (ii) We try to determine the consequences.(iii) If the consequences are nonsensical, then our original

assumption must have been wrong.

We shall show that the ratio of the diagonal of a square to its> side cannot be written in the form -2- where p and q have no common factor.

SMILE CENTRE MIDDLE ROW SCHOOL

KEIMSAL ROADLONDON W10 5DB

Tel: 01-960 7330

Turn over

7. By using Pythagoras' theorem, show that the ratio of the diagonal of a square to its side is ~

2. Assume V2~= -jp where p and q have no factors in common (apart, of course, from 1)

Pi - q2=> p2 =

=> p2 is even (why?)

=> p is even (why?)

3. If p is even then p can be written as double another whole number so we could say p = 2k => p 2 =

4. Now write down an equation involving q and k but not p.What does this tell you about q? You should now have found that p and q are both even (why?)

5. Can you see why this leads to a contradiction?

Use the method of 'proof by contradiction' to show that the planet Earth is not flat.

Smile 1679

SpheresThis pack is designed for a group of 2 to 4 students. Start with the Straight Lines booklet, It introduces you to straight lines on a sphere which are needed for the

second booklet Triangles. Here you will investigate triangles on a sphere, anddiscover that they are not always the same as triangles on flat paper. Finish with

the Maps booklet. Here you will explore the problems of representing the3-dimensional world on a flat sheet of paper.

Contents1679A Straight Lines

1679B Triangles1679C Maps

You will need Smile Worksheets 1679D, 1679E and 1679F. You will also need 1679G which contains arubber ball, map pins and rubber bands

Straight lines Smile 1679A

Straight lines

What is a straight line?

A straight line is the shortest path between On a piece of paper you can draw straight two points. lines with a ruler.

But you can't use an ordinary ruler on a sphere.

Straight lines on a sphere

Put two pins into the ball.Take a piece of string or a rubber bandand pull it tight between them.

This is the shortest path between the two pins.

We call this a straight line on the sphere.

If you continue the straight line round the ball, you find that the ends meet. Notice that the straight line goes right round the middle of the ball.

Take some thin rubber bands. Put the bands round the ball to show straight lines.

Remember: The bands should not wiggle and the bands should go right round the middle of the ball.

Like this

. . . or like this

Parallel lines

These two lines are parallel and straight.

These two lines are parallel, but not straight.

Use rubber bands to try to find parallel straight lines on the sphere.

But take care!The rails of this track aren't parallelstraight lines.

Why not?

Dividing a flat surface into regions

Draw a line across a sheet of paper. It divides the sheet into 2 regions.

Draw a second line. Make sure it cuts thefirst line.Now there are 4 regions.

Draw a third line to make as many new regions as possible. Make sure only 2 lines meet at any point and the new line crosses both the others.

You should have made 7 regions.

Now you draw a fourth line on your drawing to make as many new regions as possible.

Make sure only 2 lines meet at any one point.

Like this

Not like this

Make sure the new line crosses all the others.

Like this

Not like this

Copy this table. Fill in the number of regions four lines make.

Lines

Regions

Look at the pattern of numbers. Predict how many regions 5 lines can make.Draw a fifth line and count the regions to check.

Do the same for more lines. Write down what you find out.

Dividing a sphere into regions

Now take the ball and several rubber bands.

Use a rubber band to put one straight lineon the ball.It divides the surface into 2 regions.

Put on a second straight line. Now there are 4 regions.

Hint: To count regions, put one pin in each and then count the pins as you take them out.

Put on a third straight line to make as many regions as possible. Make sure it doesn't pass through the points where the first and second lines met.

Now there are 8 regions.

Copy this table.

Lines

Regions

1

2

2

4

3

8

4 5 6 7 i5 9

Predict the number of regions for 4 andmore straight lines.Use rubber bands to check yourpredictions.Remember that only 2 lines should meet atany point.

Write down what you find out.

Comparing regions for paper and ballLook at the table of numbers for the ball. Compare it with the table of numbers for the paper.

What do you notice?

Can you explain why?

How long is a line?You can leave the work on this page until you have done the other booklets if you wish.

How long is a curved line?

You can measure lengths on a flat surface with a ruler. But if you try to use a ruler on a curved surface you have to bend it.

When the ruler is flat the top and bottom edges are the same length.

But when you bend the ruler, the top becomes longer than the bottom.

We can't tell if the top has been stretched, or if the bottom has been squashed, or both.

Why not?

If the ruler has changed length it will not measure accurately.

How could you make an improved ruler for curved surfaces?

Would it be absolutely accurate?

The only lines we can measure accurately are straight lines on flat surfaces. So for curves we have to try to approximate them with straight lines. This may mean leaving the surface if the surface is curved.

Start with two end points and one in the middle.

First we approximate the curve with chords (which are shorter than the curve).

9 -yy

Length of chords = 3'9 + 3-15 = 7'05cm

Then we approximate the curve with tangents (which are longer than the curve). , 3-1

Length of tangents =2-65 + 3-1 + l-7=7'4Scm

Put new points between the existing three points.

Length of tangents =1-3 + 1-75 + 1-9 + 1-5 + 0-8 = 7.25cm

We can add new points as often as we wish.

Length of chords =2-05 + 2-1 + 1-55 + 1-5 =7.2cm

Try finding the length of a curve this way yourself.Draw a curve on a sheet of paper. Mark the two ends and a point in the middle. Draw the tangents and the chords. Measure their total lengths. Add more points until you have sandwiched the length of the curve to the nearest millimetre.

Hint: Use the edge of a piece of paper to mark off the lengths of tangents or chords. Then you will only need to measure the total lengths with your ruler.

8

tefe

How to use this booklet

This booklet is about triangles: triangles drawn on paper and triangles on the sphere.

Pages 3 and 5 tell you how to measure your triangles.Pages 6 and 7 suggest how you can investigate the differences between the two kinds of triangles.

There is a lot of work to do so, if you are working in a group, share it out. But remember to talk over your results with everyone and to check any results that seem odd.

When you've finished, write up a groupreport.Talk about what you've found out andthen decide which member of the group isgoing to write each section of the report.

What is unexpected about the sum of the angles of this curved triangle?

It is difficult to use an angle indicator to measure the angles (why?) Instead look at the sphere to determine the size of each angle.

Measuring triangles on a flat surface

You are probably used to measuring triangles drawn on paper. Measure the triangles below and record your measurements on worksheet 1679D like this:

Then draw two triangles of your own and record their measurements.

Making triangles on a sphere

Check the three things below.

1 The band is not too loose.

Not like this

If it is, use a smaller rubber Jjarixl or move the pins further apart.

2 The band is round the shaft of the pin.

If they are not, pull each one out from ball and let it snap back.

You need:Rubber ball3 map pinsThin rubber bands.

Stick the 3 pins into the ball. Put a rubber band round them to make a triangle.

If it is not, use a thinner rubber band or pull the pin out a little.

3 All the lines are straight.

4

Measuring triangles on a sphere

You need:3 angle measurers and a paper ruler cut from worksheet 1679E.

Now you are going to measure triangles on the sphere.

Put an angle measurer onto each pin.

Stick the pins in the ball and put on a rubber band.

Check that the band is not loose, the band is tight round the pins and all the lines are straight.

Choose one of the pins. Turn the angle measurer until the 0° line is under one of the rubber-band lines.

Check that the sides are still straight.Measure the angle.Record it on worksheet 1679D.Measure the other two angles in the sameway.

Now use the paper ruler to measure the sides.

Nine statements about triangles

1 The sum of the angles of a triangle is always 180°.

58+67°+55°= 180°

4 The largest angle in a triangle is opposite its longest side.

2 The sum of the lengths of the sides of a triangle is always less than 50cm.

3-4+3-l+3-8=10-3

5 The sum of the two shorter sides of a triangle is always more than the length of the longest side.

2- 2cm

2+2-2=4-2 4-2>3-9

3 If three sides of a triangle are equal then all its angles are the same.

2 . 7 crn

6 The sum of the two smallest angles of a triangle is always bigger than the largest angle.

7 If two angles of a triangle are equal, so are two of its sides.

3-3cm

8 The sum of the angles of a triangle is bigger in bigger triangles.

60

9 The angles of a triangle don't change if you double the length of its sides.

48°,

3cm30°

Checking statement 8

It isn't easy to measure area on the sphere. So, when you are checking statement 8 on the ball, start with a small triangle. Measure its angles. Then move just one pin so that you make a slightly larger triangle. Measure its angles. Then move another pin to make a slightly larger triangle still.

Your triangles will be getting bigger and bigger.

What is happening to the sum of their angles?

Compare your results for triangles on paper and for triangles on the ball. Write up a report. If you're working in a group, share out the work.

6

True or false?

On the opposite page are nine statements about triangles. Some are true for triangles on a flat surface; some are true for triangles on a sphere; some are true for both.

Use the measurements you recorded on worksheet 1679D to help you check each statement.

Copy and complete the table. Try to put a reason each time.

But be careful. Even if you check a statement on all your triangles and find that it's true, there could be others for which it is false.

Always try to think of triangles which don't work, and if you think there aren't any write down the reasons why.

What are the differences?

Triaiigtes on paper Triangles

4

Areas and angles of curved trianglesYou can leave the work on this page until you have done the other booklets if you wish.

Here are pictures of 5 spherical triangles.

For each triangle work out:

• the sum of the angles

• the difference between that and 180°

• the number of each triangle you would need to completely cover the sphere's surface

Copy and complete this table. B D

Sum of angles 195°

Difference between sum of angles and 180° 15 C

Number of triangles needed to cover sphere's surface 48

Is there a connection between the sum of the angles of a spherical triangle and its area?

8

Maps Smile 1679C

Making solids from flat paper

Take a sheet of paper.

Can you make it into a cylinder? Do you have to:

bend itfold itcut itstretch it?

Can you make it into a cone? Do you have to:

bend itfold itcut itstretch it?

Can you make it into a cube? Do you have to:

bend itfold itcut itstretch it?

Can you make it into a sphere? Do you have to:

bend itfold itcut itstretch it?

Fixing positions on the sphere

LatitudeNorth Pole

South PoleLatitude tells you how far north or south a place is. It is measured from the equator.

The North Pole is 90°N. The equator is 0°. The South Pole is 90°S.

You need a globe (if your teacher has one).

If you don't have a globe you can use this picture to answer these questions.

1 Which countries are these places in?a) 50°E20°Nb) 100°E40°Nc) 35°E55°Nd) 35°E 5°Se) 10°W20°N (Answers on back page)

2 Are the circles of latitude all the same size?

3 Are the circles of longitude all the same size?

4 Are all the 'squares' on the grid the same size?

You can fix the positions of any point on the sphere with two numbers:

latitude and longitude. North PoleLongitude

120°E

Longitude tells you how far east or west a place is.It is measured from the Greenwich meridian, which passesthrough London.

90°NSON

r10S

£o°s

Making Maps

Here are three ways of cutting or stretching the surface of a sphere to make a map.

Look at the maps and compare them with a globe if you have one. Write down what you notice.

Map 1: Cut down lines of longitude and open out the surface.

Map 2: Open out the surface by stretching out the circles of latitude.

Map 3: Open out the surface by stretching out the circles of latitude and squashing the circles of longitude.

Some other maps

Map 4: This map was designed by a Belgian, Geradus Mercator. He first published his new map of the world in

1569, while he was working in Germany. It was a real breakthrough in maps for sailors. When you've finished this booklet

you'll probably see why it is so useful. Mercator was also the first person to use the word 'atlas' for a collection of maps.

Map 5: This map is made by squashing the globe onto a piece of paper at the pole.

How good are the maps?

We had to cut and stretch the surface of the globe to make the maps. So the map is different from the surface of the globe. On this page are several statements which are true.

If you have a globe you can check them.

But because the maps cut and stretch the earth's surface some of the statements are not true on the maps.

Use the maps on worksheet 1679 F to check the statements.

If you are working in a group, share out the work, but don't forget to discuss the work together at the end.

Write two paragraphs for each map saying what its advantages and disadvantages are.

DistanceThe shortest routes:

from New York to London from London to Accra and from Accra to Mogadishu

are the same length.(If you have a globe use a piece of string tocheck.)

Measure the three distances on each of the 5 maps. What do you notice?

Which of these maps can be used to measure distances? Can you explain why?

Shortest RouteThe shortest route from London to Tokyo goes over Murmansk. (If you have a globe use a piece of string to check.)

Draw the straight line from London to Tokyo on each of the five maps.

Do any of them go through Murmansk?

Which of these maps can be used to find the shortest route between two places? Can you explain why?

Area and ShapeIf you have a globe, trace Greenland and India. Otherwise use the outlines below.

BearingsIf you set off in a plane from Georgetown, Guyana keeping a compass bearing of 045° (north east) you will eventually arrive over London.

Copy them onto graph paper and find their areas by counting squares.

Which is bigger?

Now find the areas of Greenland and India on each of the five maps. How well does each map show area?

Compare the shape of Greenland and India on the globe with the shape on each of the five maps. How well does each map show shape?

Which of these maps can be used tomeasure area?Which of these maps can be used to findshape?Can you explain why?

Draw the line from Georgetown to London on each of the maps.

Measure the angle it makes with the north line.

Which of the maps tells you the bearing is45°?

Other mapsHowever you draw a map it will distort thesurface of the earth.Even the same type of cutting andstretching will produce different maps ifthe cuts and stretches are in differentplaces. Both the maps below are made likeMap 2.Compare Maps 2, 6 and 7.

Map 6

Map?

8

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BUTIO fa BI°iBJV IPnBS (B• i uoijsanb oj SJSAVSUy

3J3qdS 31^1 UO SUOTJTSOd SUTXIJ

Spheres Smile Worksheet 1679d

Use the 12 triangles below to record your results from page 3 (1679b).

Area = Area = Area = Area =

Area = Area = Area = Area =

Area = Area = Area = Area =

Use the 12 triangles below to record your results from page 5 (1679b).

© RBKC SMILE 2001

SpheresMap 1

Smile Worksheet 1679f

Map 2

turn over

Map 4

Map 5

© RBKC SMILE 2000

Smile Worksheet 1679e

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© RBKC SMILE 2000

Smile 1680

In this pack you will find a mirror card and ten bug pictures.

Can you put the mirror on the bug onthe back of this card to make the

other bug pictures?

Which of the bug pictures can you make? Which ones can't you make?

There are some more cards like these in apack called "Materials for Mirror Cards"(ISBN 07-018418-6) which should be

in your school library.

D

1

4

8

10

Smile 1681

FoldingWhich shape was folded twice to make this triangle?

original shape was folded along a line each time.

Try to find all possible solutions.

Do the same for the shapes below:

Number JumbleEach of the following groups of statements describes a number less than 25. In fact, all the letters represent an integer less than 25.

SMILE 1682

g is a two-digit number whose square root is a power of 2

Identify the integers represented by a, b, c, d, e, f and g. Write a description for h, i and j. Test them on a friend.

0

ROBLEflOF

POWER

ie work on this card is divided into 3 parts: n page 2 there is a table to copy and complete.ou can use the results from this table to solve tl "oblem on page 3.this sort of work interests you, page 4 shows h< >u could investigate the patterns in some simila bles.

A Useful TableBelow is a table of powers in mod 10 - all the numbers are remainders after division by 10.

Powers1 2

149656941

The first column has the numbers 1 to 9

The second column shows 1 2,22,32,...,92 in mod 10

The third column shows 1 3, 23, 33,.. .,93 in mod 10

... and so on.

Copy and continue the table.What do you notice about the 5th column?What happens after that?

A problemThe table from page 2 should help you to solve this problem without a calculator. In any case you certainly shouldn't use the square root function.

1601613

One of these numbers is a fourth power. Q£ of these numbers is a cube. One of them is only a square number. Which is which?

Turn over.

... and an investigationBelow is the start of the table for powers of numbers in mod 7.

Powers1 21 12 43 24 25 46 1

3 11 7|1 fl6 I1 m6 M6 1|

Copy and continue the,table. Describe any patterns in the table. When do the columns start to repeat? Investigate other tables.

Milk CrateSMILE 1685 Q

Can you arrange 18 bottles in the crate so that each row and column has an even number of bottles?

You may want to use counters to help you.

SquaresSmile 1686

The pattern has been developed from the MicroSMILE program TAKEHALF.mmmmm

EBfflfflB

w\mmDescribe the rule used to create one row of the pattern.How can your rule be adapted so that it will describe the whole pattern?

Which squares have more black than red? Which squares have more red than black?

Justify your answers.Turn over

Describe the rule used to create one row of the pattern.How can your rule be adapted so that it will describe the whole pattern?

How many lines of symmetry does the pattern have?Which lines of symmetry reflect black on to red and red on to black?

Does the pattern have rotational symmetry?

You can see this pattern as a 'bird's eye view* of square based pyramids. Which of these pyramids are identical?

You may like to create your own poster.

©RBKC SMILE 1994.

Change

Smile 1687

= 9p

• P

• P

• P

Use 5p and 2p coins only.Try to make all the amounts of money up to 30p.

Which amounts cannot be made?

Colombia.: ,

Many national flags are designed using simple fractions. The flag of Colombia is l/2 yellow, ¥4 blue and V4 red.

Jse fractions to describe these flags:

1. Poland

4. Belgium

2. Holland

5. Benin

3. Mauritius

6. United Arab Emirates

Some flags have more than one section the same colour. Austria has 2/3 red.

Austria

Use fractions to describe the flags below:

7. Nigeria 8. Uganda

For some flags, like Abu Dhabi, we need to draw in a few extra lines to recognise what fractions have been used. The flag of Abu Dhabi is 3/4 red and % white.

Abu Dhabi

9- Czechoslovakia

11. Thailand

i r i i

i ii ii i

10. Sharjah

i it-Hj. -J I •

i iI I i i

12. Switzerland

Ships can use the signal flags below to send messages.

ALPHA

G MlGOLF HOTEL

M NMIKE NOVEMBER

T

DDELTA ECHO FOXTROT

KJULIETT KILO LIMA

Q RPAPA QUEBEC ROMEO

w fflTANGO UNIFORM VICTOR WHISKEY XRAY

YYANKEE ZULU

What message is the ship signalling?

These outlines will help you to answer the next four questions about the signal flags:

14. What fraction of the flag is white on F?75. What fraction of the flag is red on U?16. What fraction of the flag is blue on N ?17. What fraction of the flag is blue on A?18. What fractions can you see on other flags ?

Finally, you may like to look at the national flags of Kuwait, Guyana and Brunei. Their designs are more unusual.

Kuwait

Guyana

Brunei

Hints:

LOGICAL KITTY Smile 1690

ar trial results

Car A CarB CarC Car D Car A Car B Car C Car D

Car A CarB CarC Car D Car A CarB CarC Car D

Motorcycle RatiosG

Smile 1697

\

This activity is about the various speeds of a motorbike engine.

An engine speed (in revolutions per minute - rpm) does not always produce the same vehicle speed. It depends upon which gear is being used. The bottom gear is engagedwhen the motorbike is starting off, and the top gear is engaged when travelling on the open road. A bike may have as many as six different gears.

Speedometer Rev counter(measures engine speed)

Complete these two lists:

In sixth gear (top gear) :

6000 rpm corresponds to 60 mph 4000 rpm corresponds to 40 mph 5000 rpm corresponds to 50 mph 8000 rpm corresponds to • mph•Bl rpm corresponds to 35 mph 6500 rpm corresponds to • mph

In first gear (bottom gear) the engine has to work harder so:

6000 rpm corresponds to 20 mph•B rpm corresponds to 10 mph 1500 rpm corresponds to • mph BBI rpm corresponds to 12V2 mph

Running the engine at more than 9500 rpm damages it.

What is the top speed of the bike in sixth gear?

What is the top speed in first gear?

In second gear.

The diagram above shows the correspondence of mph to rpm in second gear.

What is the top speed in second gear?

3.

In third gear

MPH60 •

50

30

20

10

1000 2000 3000 4000 5000 6000 7000 8000 9000 10000

The graph above shows the correspondence of mph to rpm in third gear.

What is the top speed in third gear?

RPM

In fourth gear:

In this gear 37% mph corresponds to 6000 rpm. Draw a graph for speed against revs in fourth gear. Use the graph to find' top speed in this gear.

On the same graph:

Add a new line to your graph for each of the other gears.

At 30 miles per hour:

Work out the engine speeds at 30 mph:

(a) In first gear(b) In second gear(c) In third gear(d) In fourth gear(e) In sixth gear

Estimate the engine speed at 30 mph in fifth gear.

Smile 1698

IdentikitYou might find it

helpful to work witha partner.

1describe a circle without using the words circle or round.When you are satisfied with your description, write it down.

2Choose at least 3 other items from the list below. Write out a description for each item

« a separate piece of paper. , cube (b) square

(c) sphere (d) parallel (e) tessellation

3Give your descriptions to someone else. Can she identify what you have described?

Smile 1699

Fifteen GameYou will need cards from 'Smile 2226, Sum Number Cards.

A game for two players.

• Lay the cards 1 to 9 face up on the table.

• Take turns to choose a card. The chosen card must be kept face-up where it can be seen by the other player.

• The winner is the first person to make a total of 15 with any of her cards.

©RBKC SMILE 1997.

FittingYou will need triangles, kites, and

arrowheads (from Generators Gl by Leapfrogs)

Smile 1700

Fit 6 arrowheads on the star

Fit 3 kites on this triangle

Try to make each of these with 2 kites and 2 triangles

Make a shape of your own. Try it on a friend

2 arrowheads and 2 triangles:

'Kites and 2 arrowheads:

3 arrowheads and 2 kites

If you are ready for a challenge, turn over!

This one is more difficult!

What other star-shapes can you make?

HINT

This can be done using: 6 arrowheads 6 kites and 12 triangles

but there is more than one way to do it!!!