Repeating digitsSmile 0752

You will need a calculator.

Enter any 3 digits on yourcalculator.e.g.

Repeat them to make a 6 figure numbere -g -

jSiSiSiS&sSSssSiiS^tSf&K&i^^t

Divide by 13

Enter 3 different digits on your calculator.

Write down what you notice.

Haveyou tried this at least

5 times?

Can youexplain your

results?Look at the hints overleaf.

Explain them.

Try the challenge on the back of the card.

Turn over

Hints

• Multiply any 3-digit number by 7, then 11, then 13. What do you get?

• What is 7x11 x13?

Challenge

Make up another flow chart based on the same idea which uses:

10001 = 73x137

©RBKC SMILE 1994.

Smile 0755

Rectangles to RegionsYou will need graph paper.

Each of these rectangles has an area of 36 square units.They are not drawn to scale.

Rectangle Bbase 6 units height 6 units area 36 square units

Rectangle Abase 4 units height 9 units area 36 square units

Rectangle Cbase 14.4 unitsheight 2.5 unitsarea 36 square units

1. Find as many rectangles as you can which have an area of 36 square units.

2. Copy this graph and plot the points for each of the rectangles you have found.

40

JS 35

30

25

20

15

10

10 IS 20 25 30 35 40

base(b)

B

Join the points with a smooth curve.

ft*)

-H-

3. Choose 4 new points on the curve.Read off the value of the base (b) and the height (/?). Find the value of base x height (bh) for each point. What do you notice?

4. Choose 4 points in the region R (below the curve). Read off the value of b and h. Calculate bh for each point.p| Points in region R represent rectangles with HI areas less than 36 square units.

5. Choose 4 points in the region S (above the curve). Calculate bh for each point. What do points in region S represent?

Draw a new graph to represent rectangles with an area of 24 square units, bh = 24

6. Find 4 points on your graph where bh < 24. \ Find 4 more points where bh > 24.

7. If a rectangle has b = 8.5 units, give a value of h for which:a) /tf? = 24b) bh>24c) b/?<24

©RBKC SMILE 1995.

C753smila

11111% II OTEKSEBTIDl

When two lines intersect we can find the point of intersection.

One method is' to draw the two lines on a graph.

(1) Plot the two lines y = x + 2 and y = 2x on a graph.

What is the point of intersection?

(see card O744 if this is difficult)

(2) Plot the two lines y = x + 5 and y = 4o; + 1 on a new graph.

What is the point of intersection?

Can you give an exact answer?

-2-

Your answer to question (2) was probably not very accurate because the values of x and y at the point of intersection are not integers.

There are several ways to get a more accurate answer.One of these is the ITERATIVE METHOD.

"* 3 •—

o PROBLEM

Find the point of intersection of the two ] your answer correct to 2 decimal places. ,

o METHOD

Rewrite the equations as

y = x + 5 and x = y - 14

Make a guess at the x

co-ordinate of the answer

Let y = x + 5

Let x = y - 1 4

last two x values

ies y = x + 5 and y = 4rc + 1 and give

SOLUTION

(3) Copy and complete Y

First Guess

1.5

lis is the co-ordinate

: your answer

Your answer for x to 2

decimal places shouldbe 1.34 and so y mustbe 6.34 (y=x+5)

So the point of intersectionis (1.34, 6.34)

- 5 -

Use the iterative method to find the points of intersection of the following

(4) y = x + 3 and y = 5x

Rewrite the second equation as

w ^ • • •

and start with x = 1

(5) y = 2x + 1 and y = Ix + 3

Rewrite the second equation asT* rrtAs • • •

and make your own guess for a

starting value of x

(6) For both questions (4) and (5), sketch the lines on a graph and trace your path

- 6 -

(7) For both questions (4) and (5) rewrite the first equation as

T* ^~w ~" • • *

and use the iterative method.

What happens? Why? ( see question (6) )

(8) Which of the following equations will you have to rewrite and why, in order to use the iterative method successfully?

y = 5x + 1 and y = 2x

CENTIGRADEAND

FAHRENHEIT

An activity for 2 or 3 people.

This envelope should contain:

work-card 0757 additional cards 0757 A-F

SMILE 0757

SMILE 0757

CENTIGRADE

FAHRENHEITCONVERSION FORMULAScentigrade to fahrenheit •.

fahrenheit to centigrade •.

= 5(F-32) 9

Do these two formulas agree?

Try substitution, plotting F against C, inverse flags and re-arranging the formulas

For Discussion:

Do the two formulas agree? Do any of the activities you have carried out actually prove your answer?

PLOTTING SMILE POINTS 0757 C

F = + 32

Choose some values of C and calculate F in each case.

Plot a graph of C and F with C along the horizontal axis.

5(F-32 u 9

Choose some values of F and find the corresponding values of C.

Plot these results on the same graph.REMEMBER C is on the horizontal axis.

SUBSTITUTING FUR C SMILE0757A

Try substituting a number for C, say 40,9C in F = —- + 32

What value do you obtain for F?

Substitute this value of F in C = 5(F" 32)y

Do you get C = 4O?

Try four or five other values for C.

SUBSTITUTING FOR FC = 5(F-32) F = + 32

Put F = 68

c = 5(68-32)

Put C =

F =5

?

+ 32

Choose four more values for F and repeat the process for each value.

REARRANGEMENT Z SMILE0757F

r __ 5(F-32) c 9

Multiply both sides by 9

9C = 5(F-32)

Divide both sides by

9C = F - 32

Add H to both sides

= F

REARRANGEMENT 1 SMILE0757E

F = ^p + 32

Subtract 32 from both sides

F - 32 = ~i

Multiply both sides by 5

= 9C

Divide both sides by 9

= C

INVERSE FLAGS SMILE 0757D

Odd one outSmile 0758

Look at these five numbers: 137 158

Which is the 'odd one out'?

is the 'odd one out' with this ruleDivide by 3 and the remainder is 2

53 + 3 123 -»- 3 137 1- 3 143 H- 3 158 -»- 3

17 remainder 24145 remainder 247 remainder 252 remainder 2

is the 'odd one out' with this ruleDivide by 5 and the remainder is 3

53+5 123 + 5 137 -«- 5 143 H- 5 158 •*- 5

10 remainder 3 24 remainder 327 remainder 228 remainder 3 311 remainder 3

Each of the remaining three numbers can be made 'odd one out' with one of these rules.

Oddnumbers

Divide by 7and theremainderis 4

3 digitnumbers

1. Which rule makes which number 'odd one out'?

Here is another set of numbers

2. Find four rules to make each number in turn 'odd one out'.

3. Make up a set of numbers and rules yourself.

©RBKC SMILE 1994.

Smile 0760

QUICKLY TO ZERO|Start with a 4 digit number. Reduce it to zero using 2 digit numbers only.

The number 5617 needs 3 steps to reduce it to zero.

Try this number. — — — Find the least number of steps to reduce it to zero.

Investigate other 4 digit numbers... 5 digit numbers... 6 digit numbers...©RBKC SMILE 1994.

OrbitsSmile 0761

One quarter of the circumference of the earth, measured through Paris is 10 000km.Use the n button on your calculator or take K as 3.14.

1. What is the circumference of the earth?

2. What is the diameter of the earth?

3. What is the radius of the earth?

4. A satellite is put into orbit 8km (5 miles) above the earth.

Assuming that the orbit is circular, what distance does the satellite travel in one orbit.

5. If the satellite travels at 30 000km per hour approximately, how long will it take to circle the earth?

6. A rope fits exactly around the equator.

A large group of people then lift the rope one metre above the earth.

The rope is now too short.

How much more rope is needed?

©RBKC SMILE 1994.

Smile 0772You will need an angle indicator.

Angle estimationAn activity for two people.

'Measure the angle.

The player with the best estimate scores 1 point. Finish when one of you reaches 20 points.

You may like to moke up your own rules on scoring.

Smile 0775

Measuring anglesYou will need: An angle indicator.

Several thousand years ago, the Babylonians needed an accurate measure of angles. Because there are roughly 360 days in a year, it is said that they decided to split a turn into 360 divisions called degrees.

This activity will teach you how to measure angles using an angle indicator.

The arrow is pointing to zero.

TRUEFITT ANGLE INDICATOR

Rotate the arrow one full turn. 7. How many degrees have you turned it?

Now rotate the arrow through a right angle. 2. How many degrees have you turned it?

3. Copy and complete this table.It shows three ways to measure angles.

Angle

d)•D

a

T£>

k

size (in turns)

1 turn

H turn

H turn

Hturn

-J-tum

size (in degrees)

360 degrees (360°)

180°

•

•

•

size (in right angles)

4 right angles

H right angles

H right angle

• right angles

H right angle

To measure the size of this angle in degrees.

Vertex

SteplMake sure the arrow on your angle indicator is pointing to zero.

Step 2Place the angle indicator on the angle matching the zero line with one arm.Make sure that the centre_o_f the angle indicator is on the vertex.

StepSTurn the arrow to point along the other arm of the angle.

Step 4Record the size of the angle. This angle is 57°

4. Estimate the sizes of the following angles and then use your angle indicator to measure them accurately.

Before going any further check your answers to a and b on page 8.

Answers

a is 66° bis 120°

8

Smile 0776

r An activity for two people.

How to draw an anjpe of 60

j Draw a line.\ Choose one end to be\;

I the vertex.

Place the zero line exactly on your line...

the centre of the indicator on the vertex.

Move the arrow through 60° and mark this point.

TBUEFITT ANGLE INDICATOR

! Remove the angle \ "\ indicator and join the |

vertex to the point. f^J

Do these excercises on your own and check each other's answers by measuring:

1. Draw angles of the following sizes:

(a) 30° (c) 115° (e) 160°

(b) 330° (d) 245° (f) 200°

2. On the same diagram, using the same zero line draw angles of:

310° clockwise and

50° anti clockwise

What do you notice?

3. In question (1) what is the connection between:

(a) and (b)

(c) and (d)

(e) and (0

© RBKC SMILE 2001

Smile Worksheet 0777

You will need an angle indicator.

The radio telescope receives signals from satellite A then turnsto each of the other satellites in order.Estimate the angle, clockwise or anticlockwise, the telescope mustbe turned to aim at the next satellite.Record your estimates in this table:

:- : --i"' : Turn -•.,.-;"'••'

A to BB to CC to DD to EE to F.F to GG to H

EstimateClockwise or anticlockwise..-..':';-'.••••. '- •

•''••'"'" V •'•''•'' - • •• "

Measure

Now use your angle indicator to measure these angles. Record them in the table.You pick up the signal if you are within 5° of the position. How many satellites did you pick up signals from?

© RBKC SMILE 2001

I

SMILE0780Long Multiplication Revision

264 x 37

264 x 37

0

2 6 f x 37

264 x 37

264- x 377,9,20

9 768

First of all examine what 264 is multiplied by .... 37 .... that's thirty and seven

FIRST STEP Multiply by 30. The easiest way is to multiply by 10 ....

.... and then multiply by 3

SECOND STEP Multiply by seven. NO PROBLEM IF YOU KNOW YOUR TABLES!

THIRD STEP — Add your answers to steps one and two

WHY DO YOU WRITE. A NOUGHT?

THE SAME AS X30?

NOWTK/THE. ONE OVERLEftF

MULTIPLY423x26

4-2 3 26

423

4-2x

423 x 2 6

42 3 x 2

x1————^r First of all examine what 423 is^ multiplied by .... 26 .... that's

>. twenty and six

N

<FIRST STEP - Multiply by :wenty. The easiest way is to multiply by 10 . . . ————————

. . . and then multiply by 2

SECOND STEP - Multiply by six. NO PROBLEM IF YOU KNOW YOUR TABLES!

;THIRD STEP - Add your answers to steps one and two

NOW TKV THESE. FOUR

64-1 X 83 374x29 544 x 47 20S X 36

©RBKC SMILE 1995.

The Inverse Smile 0781

The inverse of on operation takes you back to where you started.

1 "Going to the sea 'Borrowing a £1.'

What is the inverse?

Breaking an egg.

Is there an inverse? What is it? Is there an inverse?

Do the following hove inverses? If so what are they?A Pumping up a tyre

C Adding 6 to a number

Pouring out a cup of tea

"7 Turning clockwise through an angle of 7 60°

Q Lighting a match

Q Dividing a number by 2

IQ Turning an empty mug upside down

1 I Turning a full mug upside down

"JO Multiplying a number by 0

13 Moke up your own list of operations. Which ones have inverses ? What are the inverses?

Smile 0782

Number Pattern Proof

1 _X 2~

X 3~

4~

1--12

~3

4

Are these equations true? Work out both sides of each one to find out.

2. Use the pattern of numbers to write the next three equations. Are they true?

3. Is the 20th equation in the pattern true?

4. To find out if the equations in thispattern are a/ways true, write down

_ the nth one.

n x = = n - =•

Prove that the right-hand side is equal to the left-hand side.

5. Verify that the equation in question 4 is true when:

a) n is an integer

b) n is a fraction

c) n is negative.

©RBKC SMILE 1995.

Smile 0783

trianglesThere is only 1 triangle here,

How many trianglesme?(There are more than 3.)

How many in this one?

Look at your answers so far.

Do they help you to find the answer to this one?

Check your answer by counting triangles,

Use your results to find the genera! relationship.

Smile 0784

142 857 times table1 . Use a calculator or a spreadsheet to

generate the first seven rows of the 142 857 times table.

142 857

142857

285714

2. What do you notice about your answers so far?

3. Use a calculator or spreadsheet to extend your table for the next two rows.

4. Look for number patterns in your table.Fill in the next five rows of your table without using a calculator or spreadsheet.

5. Check your results if you wish.

6. Now use a calculator or spreadsheet to work out

2,7

1, ...7

Any comments on your results?

©RBKC SMILE 1994.

Free hand anglesYou will need paper circles and an angle indicator.

Smile 0788

Fold a paper circle in half, and again... and again.. and once again.

Open out the paper circle.

Draw lines on the folds.

If you have folded carefully, angle a should be 90° (a right angle).

1. Copy and complete Zb (angle b) = 22-5°

Z c = BJ

^ d = •

Z e = B

2. An angle of 20° is slightly smaller than angle b. (Zb = 22-5°) Draw a free-hand drawing (ruler and pencil only) for each of the following angles: 20°, 30°, 40°... 350'

Check all your angles with an angle indicator.

For further practice in estimating angles use the MicroSMILE programs Angle 90 and Angle 360.(ft ORkY-. SMII F 1001

rSMILE 0789

GRADIENT

Perhaps you know what these signs mean?

In mathematics the word 'gradient' has a precise mean

Gradient = Jl=

•| In the diagram above, the gradient was measured in two places. The result was the same. IS THIS ALWAYS TRUE? Use tracing paper and the diagrams opposite.

g ........GRADENT=-sdistance up distance across

Gradient = 2 Gradient = .3 10

Gradients? Gradients?

2 On squared paper, draw lines with gradients:

a) I b)1 c)3 d)-2

An investigation— how can you find the gradient of a line from a mapping?

2x+3^V"

-10 12 7

—— >

—— -

—— -

——

——

—— >-

z_^v

1

7

Gradients?

Try some:

Keep the co-efficient of x the same

X—»2;x: t 1 x—>2xt4- x—*2xt 3

Keep the same constant

x—> x -t-3

O791YOU Mrt'U. nggrf s today's fAftt f

//f <?^i/j newspaper you should be abU, to a foreign currency tabU tvhiik gives && mfe of

-for £

/. /J a HtiUionaire m fete United Jtates a,* A, HniUionaire m

as

. /fetv AT wortk3.

d,money would a FtexcA *Ht'(lio*aire,

a n*iUioH.aire, i*.

4v // an oii *kij>meH.t is wottk, 3 warte, How much, will it Cost a British, buyer ?

Ami££ionaire

,*»«• /

j^^r* ' >«~ > %.Ul

w^

IV V

2. THEOFF/C6

t>o

OUT HOW MUCHIF

our ffowBY

3, WHICH ivxjy IVOULP you voreIF YOU W€*€ (a) TH€

to A SKILL?*

3>o youMOST PeofLe voret> &>* T

0793 SMILE

Materials; circle shapes, scissors.

APPROXIMATION & TTFold a circle shape into quarters and cut along the folds.

Cut one of the quarters into two equal parts.

Stick them into your book.

The rectangle ABCD has approximately the same area as the circle.

(1) Calculate this area by measuring AB and BC.

The more sectors you start with, the better the approximation will be. O

(2) Fold a circle of the same size into 8 sectors and repeat. Calculate the approximate area.

(3) Repeat this again using 16 sectors.

(4) Explain why the statement is true

(5) Follow through the working below to understand the statement'

Circumference of circle = 2irr

AB is approximately half of the circumference (why?)

So AB = Trr (approx.)

BC = r

(why?)

Area of rectangle, ABCD

= AB x BC

= irr x r (approx)

= Trr 2 (approx)

(6) Work through the following to find a value for IT .

\

Draw a circle, radius 10cm , on graph paper.

Divide the circle into strips, width 1cm.

Each strip is approximately a trapezium. Find the area of each one to calculate the approximate area of the circle.

Copy and complete:-

Area of circle = TT r :So H TT|

SO TT = •

(approx.) (approx.)

(7) Repeat with a circle of different size to check your answer,

The TrapeziumYou will need a pinboard, rubber band and dotty paper.

A trapezium is a quadrilateral with 2 sides which are parallel.

Smile 0794

1. Make this trapezium on a pinboard.

2. Call the lengths of the parallel sides a and b, and the distance between them (perpendicular height) h.

Write a = • unit b = • units h = • units.

3. Find the area of the trapezium.

4. Make at least 5 more trapezia on your pinboard.

Draw them on dotty paper, find their areas and record your results.

a1

^s^SSfc*^ _

b

4

a + b

5

h

2

area

5

5. Can you see a pattern in your table? If not, record some more results and look again.

6. Explain in words how you could calculate the area of a trapezium.

Turn over.

The formula for the area of a trapezium

can be written as (a + b) h

a

7. Find the areas of trapezia with the following lengths:

a) a = 4 b = 2 h = 2

b) a = 3 b = 1 h = 4

c) a = 3 b = 4 h = 3

8. Draw 2 different trapezia for each of a), b) and c) above - it may be helpful to use a pinboard first.

9. Draw a trapezium in which a is very small.

If a = 0 what shape does the trapezium become?

Put a = 0 in the formula for the area. Does this agree with what you already know?

10. Describe the special cases when

a) b = 0

b) h = 0

c) a = b

d) a = b = h

Find the formula for the area in each case.

©RBKC SMILE 1994.

0796 SMILE

DARTS PROBABILITYJohn is a very poor player. Although he always hits the board, he has no skill and is equally likely to hit any part of the board.

(1) What is the probability that he scores a double with his first throw? (You will need to calculate the area of the ring)

(2) What is the probability of his scoring 50 with one dart?

(3) About how often can he be expected to miss the scoring area altogether?

(4) What is the probability of scoring 6O with one dart?

(5) John is throwing one dart to see which team starts first in a competition. His opponent scores 45.

(a) In how many different ways can John beat this score?

(b) What is the probability that John's team starts first?

The board has radius 30cm The 50 circle has radius lets The 25,double and treble rings ^£ all 1cm wide.

Matrices and Transformations

Smile 0797

/

3_

—— 2-

i

/jj

\I

•5-4-3-2-10 1 2345

This machine changes the point (2, 3) to (2, -3).

.(2,3)

Change the signof the (2, -3)

y co-ordinate. f\

1. Copy and complete, for the corners of the house:

(2, 3) ——> (2, -3) (3, 4) ——> (m, •) (4, 4) ——> (m, m)

2. On squared paper draw both the original house, and the house after it has been through the machine.

Describe the transformation of the house.

3. Repeat questions 1 and 2 for each of the following machines, drawing just the new house in each case.

a) b) c),(2,3) (2, 3)

Swap the X andy (3, 2)

co-ordinates.

(2, 3)

Change the signof the (-2, 3)

X co-ordinate. N

Double the y co-ordinate. (2, 6)

4. Feed the corners of your original house through the two machines below and draw the house you obtain. Describe the transformation.

(2,3)

Change the signof the (2, -3)

y co-ordinate. N

Double the(4 -3) X co-ordinate. ^ ' '

5. Repeat question 4 for these machines:

. (2, 3)

Change the signof the (-;

X co-ordinate. [\

Swap the two co-ordinates. ^ ' ~ '

6. What machines are needed to make the change below? You will find it easier if you break the change into two steps.

y> k

-5 -4 -3 -2 -1

7. This machine can be replaced by multiplication by a matrix.

J2, 3)

Change the signof the (2 , .3)

y co-ordinate. K

To do this the co-ordinates have to be written as a column vector:

'* *)(i* */\

a) Find what numbers must replace the stars.

Two of the stars will be replaced by O's.

b) Check that your matrix changes to 31 -41

8. Copy and complete the following to show the effect of

0 12 0

on the house.

0 1,2 0

0 12 0

0 1 i2 0

0 12 0

0 12 0

0 12 0

Describe the transformation.

9. Find a matrix corresponding to each of the following machines:

a) b)

Double the y co-ordinate.

Change the signof the

X co-ordinate. N

10. a) Find what happens to the house when it is fed through this machine.

Add the two co-ordinates together to make the newX co-ordinate.

Leave the y co-ordinate as it is.

(5,3)

b) Draw the new house.

c) The matrix corresponding to this machine is made up of three 1 's and one 0.

Find it, and check that *i* * 3

11. a) Find what happens to the house when it is fed through this machine, and draw the new house.

linateis iinate. I'

The new y co-ordinate is the old X co-ordinate. The new X co-ordinate (6, 2)

is double the old y co-ordinate.

b) Which of the following matrices corresponds to this machine?

0I 2 ;) 2 ° 1° 1 I /° i I 1 °J (i 0 2 )

©RBKC SMILE 1994.

Polygons: Interior Angles Smile 0800

a + b + c = 360d + e +/+ g = 360C

= 360C

We know that the exterior angles of any polygon add up to 360C ... but what about the interior angles... ?

-Starting with a triangleYou probably already know that the interior angles of any triangle add up to 180°.

By cutting and sticking

By measuring

108°+ 32°+ 40° = 180° z= 180°

Mathematical proof——————————————————————————————>ere is a mathematical proof which shows why the interior angles of any triangle add up to 180C

Read it through carefully. Copy and complete it in your book.

k.

Known: The exterior angles of any polygon add up to 360°.So a + b + c =

Known: Angles on a straight line add up to 180°.So x + a =

Adding all the angles x + a+y + b + z + c = Rewriting gives x+y + z + a + b + c = Subtracting (a + b + c) from x+y + z + a + b + c

x+ya + b + c =

x+y + z

is will help you to find the sum of the interior angles of any polygon.—Copy and complete the following to find the sum of the interior angles of a quadrilateral, (i) exterior angles a + b + c + d =

a+b+c+d+w+x+y+z=

w + x+y + z =

(iv) The sum of the interior angles of a quadrilateral =

Work out the sum of the interior angles of: (i) a pentagon (ii) an octagon

Copy and complete the table. Polygontriangle

pentagon

•j "i.- •'; '••^-:~-.i::f ,/••',

decagon--

Number of sides34568

'••^'^^^'^^^•^&f-223

Sum of interior angles(3x1 80°) -360° = 180°(3x1 80°) -360° = 180°(3x1 80°) -360° = 180°

'^' : .^:':-^i^^^^B^l^^^i'^iyi^'^i^sBli'Plipiill^% ;4i^:MflW^1SG^^isi:J'^•'f ~L'"& : ^f :;^S$:S»-»::- :'?x : , ;;';.'«?: {? '^f '. '":?'&}

•i' £/, :-y < X?. S ^': ~ W:, - ;:: ; ry ;":::^ir f ;2ili'

RBKC SMILE Mathematics 2005

Polygons: Interior AnglesSmile 0800

We know that the exterior angles of any polygon add up to 360°...

but what about the interior angles ...?

\

Starting with a triangle

You probably already know that the interior angles of any triangle add up to 180°.

By cutting and sticking

By measuring

108 +32°+40° =180 x + y+ 2 = 180°

Mathematical proofHere is a mathematical proof which shows why the interior angles of any triangle add up to 180°.

Read it through carefully.Copy and complete it in your book.

Known:The exterior angles of any polygonadd up to 360°, ___

SO a + b + c = •^H

Known:Angles on a straight line add upto180°,

so x + a = U y + b = •2 + C= ••

Adding all the angles x + a + y + b + z + c

Rewriting gives x + y + z + a + b + c

Subtracting (a + b + c) from x + y + z + a + b

x + y + z + a + b + ca + b + c

x+y+z

This will help you to find the sum of the interior angles of any polygon

1. Copy and complete the following to findthe sum of the interior angles of a quadrilateral.

exterior angles: a + b + c+ d =

a + w- b + x =c + y = d+ z -

a+b+c+d+w+x+y+z=

(iv) The sum of the interior angles of a quadrilateral

2. Work out the sum of the interior angles of: (i) a pentagon (ii) an octagon

3. Copy and complete

Polygon

triangle

pentagon

decagon

Number of sides

22

n

Sum of interior angles

(3x180°)-360° = 180°

(4x180°)-360° = 360(5x180°)-360°=

©RBKC SMILE 1997.

InflationSmile 0804

Inflation is a measure of the rate at which prices increase.

Inflation is measured by taking the price of a standard "basket" of food and other services.

Look at this table for 1991-1993. The inflation rate in 1992 was 3.7% so that the standard "basket" costing £100 in 1991 would cost £103.70 in 1992. In 1993 the inflation rate decreased to 1.6%. Actual prices still increased but by less. When the rate of inflation decreases, prices still rise. The rate of increase in the prices is slower.

Year

1991

1992

1993

Inflation rate

3.7%

1.6%

Actual price

£100

£103.70

£105.36:i.SM.»-HW-!.H«.!.SS'-'-w.vH-W'

This table gives the prices of seven foods in six successive years during the 1970s.

Food

1 1b sausages

4oz coffee

1 1b potatoes

12 eggs

2lbs sugar

Iptmilk

1 1b carrots

1972

21p

29p

2p

20p

10p

6p

3p

1973

24p

30p

2p

20p

9p

6p

4p

1974

29p

32p

2p

47p

10p

6p

5p

1975

32p

40p

3p

31p

29p

5p

7p

1976

37p

41p

7p

39p

23p

9p

7p

1977

44p

72p

12p

48p

21p

10p

14p

This graph shows the annual change in the prices of 11b sausages.

The points are joined by straight lines because there is no information about how the prices vary during the course of a year.

oIIT EL

50

40

30

20

10

72 73 74 Year

75 76 77

1. Draw graphs to illustrate the annual changes in price of the six other foods listed.

2. Use your graphs to answer these questions about prices from 1972 to 1977.

a) Which food did not increase in price between 1975 and 1976?

b) Which foods did not record their highest prices in 1977? Which foods did not record their lowest prices in 1972? Suggest reasons for your answers.

3. a) The price increase for 11b of sausages between 1972 and 1977 is 23p; for 2lbs sugar it is only 11 p. Any comment?

b) Find the price increases for the other foods.

4 Which food do you think increased most in price between 1972 and 1977? Which food price do you think increased least?

Sausages and sugar increased by different amounts but in each case the price increase is approximately the same as the original 1972 price.

Food

1 Ib sausages

4 oz coffee

1 Ib potatoes

12 eggs

2 Ibs sugar

1 pt milk

1 Ib carrots

1972 price

21p

29p

2p

20p

10p

6p

3p

1977 price

44p

72p

12p

48p

21 p

10p

14p

price increase

23p

11p

price increase -*- 1972 price

oq|f =1.095

11 -1110 - 1 ' 1

percentage increase

1.095 x 100% . 109.5%

1.1 x 100% = 110%

5. Copy and complete this table which compares the increase with the original 1972 price.

6. Which food do you now think has increased most in price?

7. Compare your answers to questions 4 and 6. Comment.

8. This family shopping list is the same, year afte/ year.

/ Ib sausages

8 oz. coffee

5 Ibs potatoes

6 eggs

3 pts milk

1 Ib carrots

a) What was the bill for this list in 1972?

b) What was the bill in 1977?

c) What is the percentage increase in the cost of the shopping?

d) Does your answer to c) give an accurate value for the rate of inflation between1972 and 1977? Give a reason.

9. Find out the current price of some of the foods.

Find out the current rate of inflation.

Calculate what price the foods are likely to to be in one years time.

©RBKC SMILE 1994.

Smile 0805

Anaverage pack of work-cards

This envelope contains one booklet (0805A) and eight cards (0805B to 08051).

Work through the booklet. It will explain how to use the cards.

The back of the booklet provides a summaryof the three main different mathematical averages.

X '

You will need to discuss this work.©RBKC SMILE 1994

Smile 0805A

The word 'average' is used in many different ways .. .

1. Work through cards B to I and look out for different meanings of 'average'.

2. Copy out the definitions of the three averages most often used in mathematics (see page 8).

3. Look back at cards B to I and decide which of these three averages (if any) is appropriate.

W] Average shoe size

You probably chose the mode, possibly the median.

4. Explain why the mean is not very useful here.

5. Calculate the mode and the median.

6. Six more join the class. Their shoe sizes are 6, 8, 9, 6, 8 and 10. Have the mode and the median changed?

How? Comment.

Average ability and fp] Average mark

For both of these you could have chosen the median or the mean.

7. Explain why the mode is gseless in instances such as these.

~E\ Average breakfast cereal

If the data is not in figures the mode is the most appropriate.

8. Explain why.

Average speed and |G| Average weekend

Mean, median and mode are all unsuitable here.

9. Which average did you use for each these?

Explain in each case, what the word average does mean.

Hi Average weight

10. Can you use the mean, median or mode to enable Asma to be more precise?

Average wage

The foreman goes to see the director to demand a pay rise:

"We must have a rise, our average wage is only £3500 a year."

"I'm sorry" said the director "you've calculated it wrongly, your average wage is £5000. You don't need a rise."

11. Which average suits the director best?

12. Which average suits the workers best?

"That isn't right" replied the foreman. "Your wage is in a different class than ours. If you don't count your wage, you can use any average you like."

13. How does it affect the different averages if you do not count the director's salary?

Mathematical averages

In a set of data, the mode is the item which is most common.

If a set of figures is listed in order, the median is the middle one (or halfway between the middle two).

To find the mean of a set of figures add them up and divide by the number of figures.

Average shoe size B

In Asma's class there are 25 pupils. Their shoe sizes are:

3, 7, 7, 5, 6, 8, 5, 7, 9,

7, 4, 4, 6, 9, 7, 6, 6, 6,

5,8,7,3, 10,7,6.

Asma's shoe size is average. What do you think it is?

Average ability

Asma's geography teacher said her mark was average.

The marks for the whole class were:

How many marks do you think Asma scored?

Average mark

These are Asma's exam marks:

Work out Asma's average mark.

Average breakfast cereal

Asma drew a pie-chart of the cereals her classmates had for breakfast and it looked like the one on the right.

What is the average breakfast in Asma's class?

en CD"o tooUlm

Average speed

The distance from London to Leicester is 100 miles.

7. A car does the journey in 2 hours. What is the average speed?

2. The car takes 3 hours to come back. What is the average speed for the return journey?

3. What is the average speed for the 200 mile round trip?

An average weekend

Write down briefly what you do during an average weekend.

CDoOBOat 0

Average weight H

Weight (KG) 38-9

40—1 42—3 44—5 46—7 48—9 50-1 52-3 54—5 56-7 58—9 60-1 62-3 64—5

66-7 68-9

The bar-chart shows the weights (to the nearest kg) of Asma's classmates.

THEPUPIL IN MY CLRSS

60 Kb AMD 65 Kb

1 2345678 Frequency

C/D

cb~ o oa o en

Average wageAsma's mother is chief clerk in a factory employing 31 people. The ncomes of the director and his 30 staff are below. What is the average wage?Position in firm

Director

Manager

Chief clerk

Clerk

Secretary

Number of persons

1

1

1

1

1

Salary

£26,000

£7,000

£5,600

£5,300

£5,000

Position in firm

Foreman

Skilled Workers

Unskilled Workers

Apprentice

Number of persons

1

11

13

1

Salary

£5,800

£4,800

£3,500

£2,000

SMILE 0806You will need:- squared paper, scissors

Trapezium to Parallelogram

4cm

Cut along the middle.... so that youcan tip the top over,

(3)

to make a parallelogram

base = (7 + 4) cm

height = 1 cm

area of

parallelogram = base x height So area =

Use this method to find the areas of some more trapezia

(5)

a

Can you use this

method to find a

formula for the area

of a trapezium?

6)

Can you use this method

to find the area of a

triangle?......and the formula for the area or a triangle?

,_ . Smile 0808Code breakingYou will need a copy of worksheet 0808a.

• Here is a coded message:

L.HDLJ AOJ Did V >DJJAL

AOJ

EUDAII rjvvu L.HVL. JVAH<U L.HJ VnAHV<JL

nvu <JJY JJAHVAJ> <n v >DUUJJJVL

Use worksheet 0808a to find which symbol represents which letter of the alphabet.

Decode the message.

SMILE MATHEMATICSIsaac Newton Centre for

Professional Development108A Lancaster Road

London wn IQS Turn over for hints__Tel o 171 -221 :8966^._._. and a challenge.

Hints• The V symbol must be A or I. Why ?

E is the most common letter in the English language. You will need to use worksheet 0808a to find which is the most frequently used symbol.

The most frequent letters in the message are E, T, A, H, I and S, O - in that order.

ChallengeDecode this message:

ZJDIIL. CL. < ACUVALIU JCLLOEim

The original code was developed over 200 years ago. Can you work out how the symbols were chosen ?

Hint

©RBKC SMILE 1997.

Code BreakingThis is the same secret message, written out for you to work from. Cross out each symbol as you record it in the tally column opposite. A space has been left between each line so that you can write in letters under the symbols.

LUDld AOJ Did V >DjJJALL

Smile Worksheet 0808a

AOJ

EUDAH FJVVLd b.HVLi. JVAH

HVI±] <JJV JJAHVAJ> <R

V

Frequency Table

©RBKC SMILE 2001

Fold It Smile 0809

1 . Fold a sheet of paper.

A(/V

2. Open it out.

/

3. Fold the bottom edge on to the upper edge.

i \

Fold 2

4. Fold the fold on to the upper edge.

\FoldS

Open it out.

5. Use a ruler to draw along the folds. Label the angles.

&$-g/fi-

W

Use a rotagram to find sets of equal angles.

b =

6. Comment on these sets of equal angles and the folds.

7. Fold the paper again and unfold.

8. Label some angles and note which ones you expect to be equal.

Check with a rotagram.

a/b

e1g/h i/i

9. Angles on a straight line add up to 180°.

Write down all the pairs of angles which add up to 180°.

a + b =180° a+ ... = 180°

Use this information to explain why a = d and b = c. The horizontal folds are parallel.

Write down what parallel means and check it in a Mathematical dictionary or Smile 2163

Use this information to explain why a = e and b = f.

RBKC SMILE 2005

Smile 0809

FOLD ITYou will need a rotagram.

1. Fold a sheet of paper.

2. Open it out.

3. Fold the bottom edge on to the upper edge.

Fold 2

4. Now fold the fold on to the upper edge.

FoldS... and open it out.

5. Use a ruler to draw along the folds. Label the angles.

c/'d

•g/1r-

Use a rotagram to find sets of equal angles.

b = . ..

6. Comment on these sets of equal angles and the folds.

7. Fold the paper again and unfold to get...

8. Label some angles and note which ones you expect to be equal.

Check with a rotagram.

9. Angles on a straight line add up to 180C

Write down all the pairs of angles which add up to 180°.

a + b = 180°

Use this information to explain why a = d and b = c.

The horizontal folds are parallel.

Write down what parallel means and check it in a Mathematical dictionary or Smile 2163 Geometry Facts.

Use this information to explain why a = e and b = f.