Smile 1951You may need graph paper.

When x is ?

What is the value of y ... when x = 10

... when x = 2

. .. when x = 1

. .. when x = 0. 5

. .. whenx = 0.1

. .. when x = 0. 01 .. ?

Draw the graph of the equation y = JL

What happens to the graph

... when x is negative?

. . . when x is very large ?

. . . when x is very close to zero ?

. . . when x is zero ?

© 1991 RBKC SMILE

Investigate the graphs of equations like these.

In what ways are these graphs similar?In what ways are they different ?

You may find a graph plotter or graphic calculator helpful.

.You will need a copy of the Highway Code.

Smile 1953o

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Turn over

1) Use the Highway Code, to find where possible, a sign for each square of the table.

2) A red triangle usually indicates a warning. Find out what other categories mean.

Circle

Triangle

Rectangle

Pentagon

Red

/4Cross roads

Blue Green Black/White

4© 1991 RBKC SMILE

Smile 19540

Line SymmetryYou will need a copy of the Highway Code and a mirror.

This is a road sign which warns people about a hump bridge.

It has 1 line of symmetry.

Check with a mirror.

This sign has 2 lines of symmetry.

Check them with a mirror.

How many signs can you find which have lines of symmetry?

Choose three of these and show the lines of symmetry.

© 1991 RBKC SMILE

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Smile 1955J

Rotational SymmetryYou will need a copy of the Highway Code.

This traffic sign is for a mini-roundabout.

the sign.

How many ways does your tracing fit exactly on top of the original? (You must not turn the tracing paper over.)

The sign has rotational symmetry of order 3.

Find as many signs as you can which have rotational symmetry.

For each one

Sketch the sign.

Write down the order of rotational symmetry.

© 1991 RRKfi SMII F

Thinking and BrakingYou will need a copy of the Highway Code and graph paper.

Smile 19560

Look at the back cover of the Highway Code. It shows the shortest stopping distances.

The information from the table on page 14 of the Highway Code could be shown on graphs.

Thinking Distance (ft)

70 -

60 -

50 -

40 .

30 .

20 .

10 .

0 10 20 30 40 50 60 70

Speed (rrph)

Braking Distance (ft)

350-

300-

250-

200-

150-

100-

50-

0 10 20 30 40 50 60 70

Speed (mph)

'Use graph paper to plot:

thinking distance against speed

braking distance against speed

If the driver was tired, which of the two graphs would be different?

On the same axes draw another line to show how it might look.

What would happen if the car had worn brakes?Show this on the graph of braking distance against speed.

Baling BroadwayYou will need a map of the London Underground.

Smile 1958

WwtHimpitod* Flnchley Road

Swiss Cottage St. John's Wood

"You can get from Ealing Broadway Underground Station to any other underground station with only one change".

Is this true? Convince someone else that your answer is correct.

Is it true for any other stations?1991 RBKC SMILE

Smile Worksheet 1959

Making OneYou will need copies of cut-out sheets a,b and c, scissors and glue.

• Use the cut-out sheets to find different ways of making one in the squares below.There are 9 different ways. The first one is done for you.

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© RBKC SMILE 2001

Making OneSmile Cut-out sheet 1959a

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© RBKC SMILE 2001

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Equiangular

Spirals

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EquiangularSpirals Angle method

Start with equally spaced radii (we've chosen 18).

Choose an angle (we've chosen 95°) and cut a piece of card to that angle.

Choose a point on one radius and draw a line to the next radius, using the card.'

Do the same again until.

^

Get together with other people to share this work. When you have finished, make a poster. Draw several spirals this way using different angles, and different numbers of radii. Measure the length of the sides of the spirals. Is there a pattern? Keep your measurements for later. To get a true equiangular spiral like the one on the next page you should join up all the points on the radii with a smooth curve, not straight lines. Do this with some of your straight-line spirals.

Draw different spirals using the same method.

Sequence methodThis spiral was drawn using the sequence... 7, i, 1, 2, 4, 8, 16... It has an angle of 125°. Can you see what this means? The sequence above was made by multiplying. Explain how.

Get together with some other people to draw some equiangular spirals from other sequences like this.Try ... I, 1,3,9... (multiplying by 3)...0.90, 1, 1.1, 1.21, 1.33, 1.46... (multiplying by 1.1).If you have a calculator, use it.

Using polar graph paper will help.

Patterns TOIH Spirals

Draw some patterns of your own.Start with squares, triangles, pentagons, octagons or any other shape.Colour them and make a poster.Share the work among several people.Make your own patterns like these.

Zx

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Measuring upMeasure up some of your spirals.

measure the radii measure how far apart the turns of the spiral are draw tangents (straight lines that just touch the spiral) and measure angles measure the length of each turn (use string to help)

What things are special about equiangular spirals?Take one spiral you drew using the sequence method. Check it has equal angles.Take one spiral you drew using the angle method. Check the radii increase in a multiplication sequence. (It may help to use a calculator). Some natural things are spirals. Is the inside of this shell an equiangular spiral?

fUse copies of Equiangular Spirals (Smile Worksheet 1999A) to draw the same spirals to different scales.

Here are copies of the same equiangularspiral.Trace the small one and try to fit it on thetwo larger ones.Does it fit?Does this happen with all equiangularspirals?What about other spirals?

Equiangular SpiralsSmile Worksheet 1999a

© RBKC SMILE 2001

Smile 2000OFIBONACCI &

SQUARE ROOTSPIRALS

FIBONACCI SPIRALYOU NEED: SQUARED PAPER, RULER, COMPASS

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Draw thistriangle in themiddle of thepaper.Make your unitlarge.

Make a table and squarethe lengths of thespokes.What do you notice?Why is this called a'square root1 spiral?

Add the next triangle. (Use

.your set square)

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Add on more triangles.

Measure the 'spokes'.

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Smile 2002

This envelope contains ten photographs of real spirals.

What sort of spirals occur in nature and what sort of spirals do people make?

Find out all you can about where spirals and helixes occur.

© 1991 RBKC SMILE

Record grooveThe groove on one side of an LP record is almost 0.5 km (about a quarter of a mile) long. It is an Archimedes' spiral. Why?

You may find it helpful to look at the booklet Archimedes' Spirals (Smile1998)

© 1991 RBKC SMILE

VortexWhen a liquid spins it creates a vortex. Some people believe that in the northern hemisphere vortexes are usually clockwise and in the southern hemisphere, anti-clockwise. Whichever way it turns, the liquid is turning and dropping at the same time so the path of each droplet is a helix. The booklet Helix (Smile 2001) will tell you more about helixes.

© 1991 RBKC SMILE

Nautilus shellShells grow by adding new sections to one end. Surprisingly however the shape of the shell doesn't change as it gets bigger. If this seems strange look at page 7 of the Equiangular Spirals booklet. (Smile 1999)You can see this Nautilus shell cut through on page 6 of the Equiangular Spirals booklet. Many fossil ammonites are similar.

1991 RBKC SMILE

/' " '

RopesRopes are kept coiled on the deck of a ship when they are not being used. Why is this important?

© 1991 RBKC SMILE

BeansMany climbing plants twist round supports so that their leaves can get high enough to catch the sun. This runner bean plant is one.

Can you find out about any others?

© 1991 RBKC SMILE

rfnffranh rniirtesv of NASA/Soace Frontier.

HurricaneThis is a picture taken from an altitude of 180km showing hurricane 'Gladys'. In this hurricane winds of up to 150km per hour were recorded. Hurricanes are formed because hot air rises, cold air flows in to take its place, and the rotation of the earth causes the whole mass of air to rotate. There are a number of related wind movements, for example, tornadoes, typhoons and water­ spouts.

Find out about them.

© 1991 RBKC SMILE

Photograph from the Hale Observatory.

GalaxyGalaxies are huge star systems. Many of them are spirals. Our sun is a star inside the Milky Way galaxy. We are inside one of the spiral arms of the Milky Way, about 35 000 light years from the centre of the system.

Find out all you can about galaxies.

© 1991 RBKC SMILE

Swiss RollWhen it is first rolled up the Swiss roll makes an Archimedes' spiral. Look at the Archimedes' Spirals booklet (Smile 1998) to find out why.

© 1991 RBKC SMILE

PropellerPropellors drive aeroplanes and boats. They can have two, three or more blades. The photograph shows the path (locus) of the tip of an aeroplane propellor as it moves and turns.'

What is the locus of the tip when:- the plane is stationary but the propellor is

turning?- the plane is moving but the propellor is not

turning?

© 1991 RBKC SMILE

SunflowerThe centres of many flowers are arranged in a pattern made up from two sets of equiangular spirals. The numbers of clockwise and anti­ clockwise spirals are not the same but are both usually Fibonacci numbers.

© 1991 RBKC SMILE

Birth dayDotes

You will need the five number strips from the Smile cut-out sheet 20033.

Smile 2003

/ can find the date of your birthday.

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© 1992 RBKC SMILE

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Smile 2004

"54% is a little more than half marks

This envelope should contain ten YELLOW and ten BLUE cards.

Use sensible guesswork to match ill the cards in pairs,

Then check your answers with a calculator.

® 1992 RBKC SMILE

Smile 2004aL These cards and those from 2004b should be cut out and put in envelope 2004.———————————i

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64% of £320

94% of £165

47% of £245

103% of £78

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72% of £250

68% of £210

11% of £900

70% of £80

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£204.80

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r T/^T nSmile 2006

When planning a walk in a mountainous area you need to estimate how long the walk will take.

Modern maps are metric. Invent a new version of the rule which will work if the distances and heights are expressed in metric units.

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