Smile 0812

Irregular Areas

High Field is bounded by two roads at right-angles and a river.

How would you find the area of High Field?

Using squares

1) Trace the map. Place the tracing over centimetre squared paper and find an estimate for the area by counting squares.

Using rectangles

2) Trace the map. Draw rectangles with width 1cm on the tracing paper.

a) Calculate the area of each rectangle and add the areas together.

b) Why is the total area using this method inaccurate?

Using trapezia

This is a more accurate method. It uses trapezia with width 1cm.

Area of Trapezium A

Area of Trapezium B

1 /2 x 100 (600+ 580)

50x1180

59000m2

V2 x 100 (580+ 530)

3) What is the total area of High Field?

( Turn over for a quick method.}

The trapezium method is the most accurate, but calculating each area separately takes a long time. If each strip has the same width, calculation can be done more quickly using this method:

1/2 X 100 (600 + 580) + 1 /2 x 100 (580 + 530) + 1/2 x 100 (530 +

Find the total of the brackets first.

1 /2 x 100 (1180) + 1/2 x 100 (1110)

\———Then . . . 1 /2 x 100 (1180+ 1110+ . . . . .)

4) Find the area of Low Field by drawing trapezia 1cm wide on each side of the footpath.

©RBKC SMILE 1996.

Sectors of Circles SMILE 0813

COPY AND COMPLETE.If you cut a portion of| apple tart, and the angle at the centre is

you take x 16O of

the apple in the middle

.... and also x360

the pie crust around the edge.

ofCHECK

Use the answer book to check that you have the correct formula for the area of a sector of a circle.

SOLVE.(1) Does this formula remind

you of any other area formula? Is there a connection?

(2) Find the area of the metal sheet in this medallion - try it as a mental arithmetic exercise (you don't need to multiply 3.5 by 3.7 - there's a short cut)

(3) Design another medallion made up of sectors and find the area of metal sheet needed.

Smile 0817

Straight line graphsYou may like to use a graphic calculator or graph plotting package

1. Copy and complete the mapping diagram and graph for

2. Do the same for x ——> ^x + 3 and x ——> \ Use the same axes.

Is. What do you notice about the point where each line cuts the y axis?

4. What else do you notice about the three lines?

5. Draw the graphs of these three mappings on another set of axes:

x ——» 2x + 1x——> 2x-3x ——> 2x + 3

Write down what you notice - how are these lines different from the first set of lines? How are they the same?

6. x ——* x + 1x ——* 2x + 1x ——> 3* + 1

What will these graphs look like?Will they be parallel? Where will they cross the vertical axis?

Draw the graphs to check your answers.

What will the graph of x ——> -f x + 1 look like?

7. What will the graph of this mapping look like?

x ——* 5* + 3

©RBKC SMILE 1994.

0818You may want to use pegboard and pegs SMILE

Differencesbetween

squares

-2-

Make a 4x4 square

o o o o o o o o o oo o oooo

Take away 1

Move the bottom row to make a rectangle.

oooo oooo oooo

oooo oooo oooo

4 2 - 1 Are they the same?

(4+1) (4-1)

They're both 15

Try other

Record your

size squares,

results:4 2 -1 =5 2 -1 =6 2 -I -

taking away 1 each time.

(4+1) (4-1) (5+1)(5-1) (6-m (6-D

-3-

This is a summary of what you have done on page 2:

Make any size square

a 2

Take away 1 Rearrange into a rectangle (a+1)(a-1)

- 1 Are they the same ?

(a+1)(a-1)

Substitute a = 10 in a 2 - 1 = (a+1)(a-1) to check that the identity works.

Use the identity to evaluate 21 x 19= (20+1)(20-1)

Take away a 2x2 square

o o o o o oo o o o o oo o o o o oo o o o o o

o o oo o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o

Move the bottom 2 rows to make a rectangle.

oooooo** oooooo** oooooo** oooooo**

6 2 - 2 Are they (6+2)(6-2) £ the same ?

Try other size squares, taking away 2 2 each time.Record your results 6 2 - 2 2 = (6+2) (6-2)

3 2 - 2 2 =

-o -

This is the summary of your work on page 4

Make any size square

a 2

Take away2 2

a 2- 2 2

Rearrange into a rectangle (a+2)(a-2)

a 2 - 2 2 Are they the same?

(a+2) (a-2)

Substitute a = 2^ina 2 -2 2 = (a+2)(a-2) to check if the identity works.

Use the identity to evaluate 32 x 28

= (30+2)(...)

-6-

Make a 5x5 square

Take away a 3x3 square

O O O O O O O O O O

O O O O OO O O O OO O O O OO O O O Oo o o o o

Move the botrom 3 rows to make a rectangle

O O O O O • O O O O O •

5 2 - 3 2 Are they the same ?

(5+3) (5-3)

Try other size squares, taking away 3 2 each time.Record your results (5+3)(5-3)

Here is the summary of the work on page 6

Make any size square a 2

Take away3 2 a 2 - 3 2

Rearrange into a rectangle (a+3)(a-3>

a 2 - 3 2 Are they the same?

(a+3) (a-3)

What identity can you write? Check that the identity works .. for integers

.. for fractions

Does it work for a = 3 ?

-8-

This is a summary of all the work on pages 2, 4, 6:

Make any size square a 2

Take away a smaller square a 2 - b 2

Rearrange into a rectangle (a+b)(a-b)

(a 2 - b 2 ) Are they (a+b)(a-b) the same?

Choose any pair of numbers for a and b (choose b<a) to see whether

a 2 - b 2 = (a+b)(a-b)

If you are satisfied that the identity works, write the identity in your own words.

O8I9You may want to use pegboard and pegs_________

SMILE

PROVE YOUR

IDENTITY

(1) Copy and complete:

-*• I 2 = O 2 + 1

- 2 2 = I 2 + 3

-*• 3 2 = 2 2

+ 7

39

In general,

- 2 -

The general equation from page 2 is n 2 = (n-1)'2 + (2n-l)

We know that this equation is true for some values of n (the ones on page 2) . We could easily find out if the equation is true for

her particular values of n by substitution. But is the equation an identity? (That means it is always true?)

To find out, start with the right-hand side of the equation,

i.e. (n-1) 2 + (2n-l)

Simplify this expression - start by rewriting (n-1) 2 as n 2 -2n+l

You should get n 2 . This proves that the equation is true for every value of n. It is an identity.

N.B. The proof might be set out like this :i)

RES = (n-1) 2 + (2n-l) = (n 2 -2n+l)

= n 2- 2n + 1 + 2n -

:. n 2 = (n-1) 2 + (2n-l)

-3-

(2) Copy and complete:

O 2 _

• O XX O I 2 + 2 + 1

O O O

O O O O

10'

+ 3 +2

+ 3

+ 2O

In general, n 2 = • + • + Can you prove that this is an identity?

- 4 -

(3) Copy and complete:

AO

AGO

AOOO

AOOOO

2 _ + 4x1

3 2 = X 2 + 4x2

4x3

10

+ 4x2O

In general n 2 = • + •

you prove that this is an identity?

- 5 -

(4) Copy and complete

Xx Xx 2 z = 0 z + 2x2 + 2xO

xox xoxXOX

XOOX

xoox

XOOOX

xOOOx

3 2 = I 2 + 2x3 + 2x1

+2x4+2x2

3 2 +

In general, n'

Can you prove that this is an identity?

- 6 -

(5) Construct another series according to a rule.

(6) Try to write an equation for the general case.

(7) Try to prove your equation is an identity.

- 7 -

Make a square of pegs

Rearrange the pegs according to a rule

Write a number equation

Check the equation is true

aveyou done this

at least A time

Doe different

number equations follow a attern

Write an algebraic equation using n to stand for any number

Check the algebraic equation with some different values of n, e.g. fractions, negative numbers.

Canyou prove

that the equationis true?

Yes

SMILE 0820

You may want to use pegboard and p

Equations from squaresTake a quick look at this flow-chart, work through the booklet and then study the flow chart again.

It would be helpful for two people to work together on this task.

SMILE MATHEMATICSIsaac Newton Centre for

Professional Development108A Lancaster Road

London W11 1QSTel 0207 598 4841

No

No

© RBKC SMILE 2001

ooo ooo ooo ooo <f

ooo ooo ooo ooo

Make a square with pegs on a pegboard

Write down the pattern you have made 4x4

Move a column of pegs to the bottom

Write down the pattern you have made (3x5) + 1

Write down an equation => 4x4= (3x5) -f 1

Check that it is true 16 = 15 + 1

- 2 -

(1) Follow the flow-chart opposite for these squares:

(a) 3 x 3 (b) 8 x 8 (c) 6 x 6

(2) Complete the equations for each one:

(a) 3x3= (BxH) +1 (b) 8 x 8 = (•>:•) + 1 (c) 6x6=

(3) Complete the equation for a 20 x 20 square:

20 x 20 = (HxH ) +1

(4) Complete the equation for an n x n square

n x n =

(5) If n = 7 then,

n x n (n+1) (n-1) + 1 =7x7 =(8x6) +1 = 49 =48 +1

49So the equation is true when n = 7 Check the equation for two other positive integers.

(6) Check the equation when (a) n = 1%(b) n = O(c) n = 8.7(d) n = -4

- 3 -

Make a square with pegs on a pegboard =>

o o o o

Add another column of the pegs

Write down the rectangle pattern you have made 4x5

Write down an equation 4 2 + 4 = 4 x 5

Check that it is true 16 + 4 = 2O

- 4 -

(1) Follow the flow-chart opposite for these squares

(a) 5 x 5 (b) 3 x 3 (c) 7 x 7

(2) Complete the equations for each one:

(fc (a) 5 Z + 5 =|x|j (b) 3 2 + 3 =|x| (c) 7 2 + 7 =

(3) Complete the equation for a 20 x 20 square:

20 2 + 20 = MxM

(4) Complete the equation for an n x n square:

n 2 + n =

(5) Is the equation true for any value of n ?

Check the equation .... (a) when n is an integer(b) when n is a fraction(c) when n is negative

and then discuss your answer.

(6) Look back at the front cover and study the flow-chart,

_ 5 _

• ••00• ••00• ••00

o o o o o o

Make a square with pegs on a pegboard

Remove a square of pegs

Write down the pattern of pegs

Rearrange the pegs to make a rectangle

Write down the rectangle pattern

Write down an equation

Check that it is true

5 2 - 2'

7x3

5 2 - 2 2 = 7 x 3

25 - 4 = 21

- 6 -

(1) Follow the flow-chart opposite starting with a 7 x 7 square and removing a 3 x 3 square:

Complete the equation 7 2 - 3 2 =HxH

(2) Using the pegboard if necessary, copy and complete :

(a) 5 2 - 3 2 =Hx| (b) 7 2 - 2 2 = |x|

(c) 6 2 - 2 2 =HxH (d) 8 2 - 5 2 = HxH

(3) Complete the equation for starting with a 2O x 20 square and removing a 3 x 3 square:

20 :

(4) Complete the equation for starting with an x x x square and removing a y x y square

(5) The equationx 2 - y 2 = (x + y)(x - y)

is true for all values of x and yty

Can you follow this working!L.H.S. = x 2 - y 2

= x 2 + xy - xy - y 2

= (x 2 + xy) - (xy + y 2 )

= x(x + y) - y(x + y)

= (x - y)(x + y)

= R.H.S. The proof works both ways. Copy and complete:

R.H.S. = (x + y)(x - y)

= x (x - y) + Jjj^(x - y) = x 2 - xy + | - | = * 2 -•

= L.H.S.

(6) Use the equation to evaluate(a) 51 2 - 49 2(b) 77 2 - 67 2

- 7 -

Start with 1 peg Write I 2 = 1

Add 3 pegs

Add 5 pegs

Add 7 pegs

• o oo

• o00

Write 2 2 =1+3

Write 3 2 =1+3+5

• 00O00*0 • ••o oooo

Write 4 2 =• + •+• +

(1) Complete the equations:

i) 5 2 =• + • + •+• +

ii) 10 2 =

(2) Complete the equation for a 2O x 2O square:

20 2 = 1 + 3 +

(3) Write the equation for an n x n square:

n 2 = 1 + 3 + • • • • ••••+•

(4) Check if the equation is true for the following values of n:

i) n = 6 ii) n = 15 iii) n = h iv) n = -2

The equation is not true for all values of n.For which values of n do you think it is true. Discuss this and try to give reasons.

- 8 -

Smile0824

THE GOLDEN RECTANGLEContentsWorkcards 0824 a, b, c, d Booklets 0824 e, f, g Reading list 0824k

iYou will need: Worksheet 0824h (3 copies) Worksheet 0824J

This work should be shared between a group of 2 or 3.You must all start with card 0824A. When you have worked through this and understood it, you may choose what else to do....

.....remember it is much more important to do two or three of the other cards thoroughly than it is to do everything in a hurry.

Smile 08241

A Dissection Problem

1 Draw this square, cut along the lines, and rearrange the 4 shapes to make a rectangle.

You probably made this rectangle?.

..... but something is wrong.

2 What are the areas of the original square and the rectangle above? So what is wrong?

3 Repeat the dissection for a 21 x 21 square.What is the area of the rectangle?

2-1

By how much do the areas of square and rectangle differ?

4 Repeat again for an 8 by 8 square.

There is a connection between these dissection problems and the Fibonacci Sequence

1, 1, 2, 3, 5, 8, 13, 21........

5 Can you find this connection?

6 In each case the area of the square is 1 square unit more or less than the area of the rectangle.

Give the sizes of 3 more squares where this happens and show the dimensions of each dissection on a diagram.

A Fibonacci-type sequence can be constructed from any two starting numbers.

eg. 1, 3, 4, 7, 11, 18, 29.....

A dissection, similar to the previous ones, can again be designed using 3 consecutive numbers from the sequence.

The area of this square is 121.....

but the area of the rectangle will not be 120 or 122.

u

1 Investigate this for different Fibonacci-type sequences.

Smile 0824g

The Pentagram

You will need 3 copies of Worksheet 0824 h , calculator and angle indicator.

Take a copy of worksheet 0824 H and cut out the ten triangles and the pentagon.

Can you rearrange the pieces so that the small pentagon is in a corner of the large one?(You can use another worksheet to put the pieces on).

We want to find the connections between the angles x, y arid z.

1 Take a copy of worksheet 0824, and label all the angles x, y or z (You can use the pieces from the dissection puzzle to help you).

2 Copy and complete using x, y or z.

x + y =•

2x =•

3x =•

Angles

Look at the triangle in the diagram and then copy and complete:

3 The angles marked add up to

4 The angles of a triangle add uptorn5 Write an equation in terms of

so x =

6 y = (See question (2))z =

Check your answers for x, y and z.

7 Find another triangle and use it to find another equation. Check the equation using your answers to (5) and (6).

Lengths

There are many connections between the lengths a, b, c and d.

8 Complete this list (your dissection pieces will help):

a

KPMN

b

EL

Phi

C

THMH

d

FH

9 Copy and complete using a, b, c or d.a + b =| d - c =|

a + 2b =1 2c - d =l

10 There are many more equations like these. How many can you find?

Golden Ratio1 1 Very carefully, measure the lengths of a, b, c and d, correct to two significant figures. Copy and complete: a = Mem

b = Bern c = •cm d = •cm

12 Use a calculator to work out the following ratios correct to two significant figures.!=•C

13 Why do you think the answers to (12) are all approximately the same?

14 Why do you think the answers are all approximately equal to the golden ratio?

These last two questions are difficult but the work on the last two pages will help.

Similar Triangles

15 Compare the angles of the three triangles shown above and write down anything you notice.

16 Why are the triangles similar?

17 Why are these equations true?d_ = £ = _b c b a

18 Copy and complete:From the work on the previous two pages we know that:_c_=liL b •

From the work on page 5 we know thatc = a +•

Substituting this value of c in the first equation gives

19 Look at card 0824 aCan you see why b^d are

a b call equal to the golden ratio?

Pythagoras was a member of a society which used the pentagram as its secret sign. Find out more about this.

Smiie0824j

Rectangle Worksheet

Rectangle A

Rectangle B

Smile0824h

PENTAGRAM WORKSHEET

You will need squared paper and a calculator. Smile0824e

FIBONACCI SEQUENCE

Start near the centre of your paper with 2 squares the same size.

Add on squares to fit the longer side, one at a time.......

....until no more will fit on the paperMake sure your turn anti-clockwise every time.

1 If the first two squares are 1 x 1 squares, the next is 2 x 2 and the next is....How does the sequence continue? Give the sizes of the first eight squares.

2 Measure the sides of your final rectangle and calculate their ratio.Any comments!

1> 1> 2, 3, 5, 8, 13, -, -, -, -

This is called the Fibonacci Sequence.

3 How is the Fibonacci Sequence made?

4 What are the next three terms (after 13)?

5 Look at the ratio of successive terms.....

1,2,3,1,^, _,_,_^1 12 3 5 ;

Use a calculator to find the first eight terms in this sequence.• ! ' -•• • ' '

6 What is the connection between questions (3) and (4) and question (1)?

7 What is the connection between question (5) and question (2)?

What happens if you start with 2 squares which are not the same size?

1, 3, 4, 7, 11,8 How does this Fibonacci-type sequence continue?

Calculate the ratios of successive terms.

9 If you kept on drawing squares like this, what would the shape of the rectangle become?

10 Investigate the questions above for other Fibonacci-type sequences

e.g. 2, 7, 9, 16, 25,.

You should have noticed that in any Fibonacci-type sequence, the ratio of successive terms gets closer and closer to the golden ratio.,

..... but why does this happen?

......... a, b, c, ..'.........

Suppose a, b and c are three successive terms.

If successive ratios get closer and closer, it is approximately true that:

If the sequence is a Fibonacci j- type sequencec = • + •

Substituting this value of c in the first equation gives

a =

11 Look at card 0824 aCan you explain the connection between the Fibonacci Sequence and the golden ratio?

Leonardo FibonacciLeonardo Fibonacci was born about 1175 in Pisa, which was a commercial centre of Italy. His father was a merchant, which probably accounted for Leonardo's early interest in arithmetic. Trips to Egypt, Sicily, Greece and Syria brought him in contact with Eastern and Arabic mathematics and Fibonacci became thoroughly convinced of the practical superiority of the Hindu- Arabic methods of calculation. In 1202 he published his famous work Liber abaci. This book strongly advocated the Hindu-Arabic notation and did much to encourage the introduction of these numerals into Europe.Fibonacci published two other important books: Practica geometricae in 1220 and Liber quadratorum in 1225.

Is it Golden?You will need worksheet 0824 J and a calculator.

Smile 0824d

On worksheet 0824J, one of the rectangles is a golden rectangle — the other isnot.Can you find out which one is the golden rectangle?

1 For each of the rectangles A and B:a) find the ratio, longer side, for the complete rectangle.

shorter side

b) cut off the square.c) find the ratio, longer side, for the remaining rectangle.

shorter side

2 In which rectangle does the ratio, longer side, remain the same when the square is cut off? shorter side

3 You now have 2 squares and 2 smaller rectangles.What will happen if you repeat (1) and (2) for these rectangles? If you are not sure, do it and find out.

COTMHMA game for 2 players.

QwetuYou will need an 8 x 8 board and a counter.

7

6

5

4

3

2

1

00 1

RulesThe first player puts the counter on any shaded square.The players then take turns to move the counter any number ofsquares west, south, or south-west.The player who moves onto the black square (0, 0) is the winner.

1) Play a few times.2) If a player lands on (2, 1) or

(1, 2) he must win. Why?3) Where are the other winning

squares?4) Where are the safe starting

squares?5) If the same game were played

on a very large board, .......?

Investigate the safe squares.

l)or 3hv?11 y • nning o

ting -|

played9 t)•

0

I::::::::::::::::::

iililllllliil:i::::::::::::HH:

1 2 3

o

00oL E

CO

Two Rectangle Surveys

It is said that the golden rectangle has a more attractive shape than all other rectangles because it has the most pleasing proportions.

What do you think?

Survey 1If the golden rectangle is the most pleasing one it will probably be used frequently to aid the sale of packets of washing powder, cereals and so on.

Measure the sides of as many rectangles as you can find and calculate the ratio, longer side

shorter side shorter side for each one. of Rectangle

Plot your results on a scattergram.

Longer Side of Rectangle

Survey 2Devise your own survey to find out the proportions of the rectangle which most people find the most pleasing.

ResultsPresent the results of your surveys as reports using diagrams/short paragraphs/conclusions.

CM 00 O-=

CO

THE GOLDEN RECTANGLEYou will need a calculator

The large rectangle on this card is a golden rectangle. It is not the size which makes it golden, but the shape. The long side (b) and the short side (a) are in the right proportion - the golden proportion. The ratio is called the golden ratio.

1 . Measure a and b and calculate the golden ratio, b/a. In a golden rectangle the following equation is always true:

_ _ .- a b

(There is an explanation to follow through on the back of the card)

2. Check that this equation is true for the values of a and b which you measured.

3. Use a calculator and a method of trial and error to find a more accurate value for the golden ratio, b/a. Look at the note-book below if you have not met this method before.

4. Use your results to draw an accurate golden rectangle in your book.

03^tCM 00 O-=

O)

Smile 0824k

Smile Worksheet 0824h

Pentagram Worksheet

© RBKC SMILE 2001

Smile Worksheet 0824J

Rectangle B

© RBKC SMILE 2001

Clover Leaf

77/5 DIAGRAM A CWV£R L&I RT TH£ JUNCTION Of WOMOTOR ways.

fit X and similar locations the radios of the bend is 60m. f\t Y %nd similar locations the radius of the c/ouer leaf is 75m and the radius of the adjacent bend is lOOm. The motor way is 60m wide. ignore width of feeder roads. .A

Smile0827

/) Find the distances: <?) A to £ b) F toC c) A to& J) e> to £ e) DtoC

2) A motorist Joins the motorway at '/\ ' and finds he is going in the wrong direction. How far does he have to trauel to reach the point opposite, 'f\ l on the other carriageway ?

Re-groupingYou will need small triangles, squares and circle logiblocks.

Smile 0830

Take out 3 triangles,9 squares and 6 circles.

This can be written in short form as 3T + 98 + 6C

Split your shapes into three smaller identical groups.

(T + 3S + 2C) + (T + 3S + 2C) + (T + 3S + 2C)

Copy and complete the identity.

3(T + 3S + 2C)

3T + 9S + 6C = 3(T + 3S + 2C)

• How many triangles, squares and circles are needed for 4(T + 3S + 2C)?

1. Take out 4 squares and 8 circles.

Split the group of shapes into 4 smaller identical groups.

Copy and complete

2. Split 12 squares and 18 circles into

• 3 smaller identical groups.Complete the identity. 12S + 18C = 3 (•+•)

• 6 smaller identical groups.Complete the identity. 12S + 18C = 6 (•+•)

• 2 smaller identical groups.Complete the identity. 12S + 18C = 2 (•+•)

There are three different identities for 12S + 18C.

3. Find as many different identities for 8T + 12C as you can. Write them in your book.

How many different identities did you find?

4. Split 32C + 8T + 24S into identical groups and write the identities which describe them. What is the largest number of identical groups you can make from 32C + 8T + 24S? Complete the identity

32C + 8T + 24S = | (•+• + •)^This number needs to be large as possible.

5. Here are some other groups.For each one, find the largest number of identical groups and write the identity.

a. 4C + 12T b. 6S + 12T + 6C c. 7F+14L d. 10x+15v e. 16a + 4b + 8c + 8d

6. Write about how you decided how many groups can be made.©RBKC SMILE 1997.

V _rjd

[fMake

sure you

have

:wo

rk-c

ards

A,

B, C

information

card D

additional hints

E

You

don'

t ha

ve to work through

them in

any

order

and

it

doesn't

really matter if you

don't

attempt

all

the

work,

.....

but

you

MUST

work with a

SMALL

group

of 2

or 3.

You will ne

ed to talk about

the

work

to un

dersta

nd it

proper

ly.

FldditiDial HintsThe work on the back pages of 0831B and 0831C is very difficult. Use this leaflet sensibly for hints if you need them. Look at hint 1 and try again before you look at hint 2, and so on.

Turn to the index on page 2 to find the hint you need.

Index:

A Proof About Primes '(0831B)

1st hint ....... page 3

2nd hint ....... page 5

3rd hint ....... page 7

Primes and Factorials (0831C)

1st hint ....... page 4

2nd hint ....... page 6

-2-

A Proof About PrimesJ083 IB)

Hint 1

p must be an odd number , so (p-1)

and (p+1) must both be .......

Every other even number is also2 a multiple of ...... /2>

so either (p-1) or (p+1) must /^>\o.

be ....... 10©14

-3-

Primes and Factorials (0831C)

Hint 1

Why is 8 not a factor of 71 + 1 ?

4 is not a factor of 71 + 1 (why not?) 3 and if 8 were a factor 4 would be a factor too.

... similarly,

2 is not a factor of7! + 1and if .......

A Proof About Primes (0831B)

Hint 2

(p-l) a PJ (p+1) ewe consecutive numbers3 so one of them must be a multiple of 3

p is not a multiple of 3, so

-5-

Primes and Factorials (0831C)

Hint 2

is 4 not a factor of 7! ?

71 = 7x5x5x4x3x2x1

So 4 is_ a factor of 71

So ......

-6-

A Proof About Primes (0831B)

Hint 3

Either (p-1) or (p+1) is a multiple of 3 and

either (p-1) or (p+1) is a multiple of 4 ................................... and whichever is not a multiple of 4 is still a multiple of 2

So (p-1) (p+1) must be .....

CH.S809IMJS

Shakespeare, Leonardo da Vinci and Beethoven all excelled in their particular subjects. We appreciate English literature by reading the works of Shakespeare; if we study art we look at Leonardo's paintings and how he worked; we learn more about music by studying the life and music of Beethoven.For a fuller understanding of mathematics we study the work of great mathematicians. This booklet looks at two famous Greeks, Euclid and Pythagoras, who worked out principles so clearly that they laid a basis for the mathe­ matics we use today. If you want to know about how these mathematicians lived and worked, how they and others developed mathematics, there is a list of interesting books on the back page.When reading mathematics you should have a pencil and paper handy to make notes or work things out for yourself. You may have to read some parts twice before you fully understand them.

Euclid and Greek Mathemathics

Euclid was a Greek who lived around 3OO BC - more than 20OO years ago. At this time, the centre of culture and learning had moved from Athens to the Egyptian port of Alexandria. The main attraction to scholars and teachers was the vast Alexandrian library of art, philosophy and science. The ideas and theories evolved over the previous centuries were collected together and Euclid not only taught the theories of older mathematicians like Pythagoras, but also wrote several books about Greek mathematics. Very few of these ancient books survive, but we do have copies of the works of Euclid, the most famous being his books about geometry called 'The Elements'. Hardly any of the ideas in The Elements were new, but their historical importance was in the way Euclid insisted on a proof for every point he described - thus laying a basis for mathematics for centuries to come. EUCLID REALISED THAT THERE ARE TWO STAGES IN MATHEMATICAL DEVELOPMENT: THE FIRST IS TO HAVE AN IDEA, MAKING AN HYPOTHESIS ABOUT MATHEMATICS WHICH WORKS IN A FEW CASES. THE SECOND STAGE IS TO PROVE THAT THIS GUESS OR HYPOTHESIS IS CORRECT, TO ESTABLISH THAT THE HYPOTHESIS WILL WORK IN ALL CASES.

The Infinity of PrimesAn example of Euclid's approach is the theoryof the infinity of prime numbers. Mathematicians before Euclid had suspected that there was an infinite number of primes - however large a prime number is, someone can always find a larger one. But Euclid actually proved that this is so, raising the guess to the level of a fact, established beyond doubt.His method of proof is called "REDUCTIO AD ABSURDOM": assume the opposite of what you want to prove and then show that this leads to some­ thing absurb - a contradition.

Euclid's proof of the infinity of primes

(1) Assume that the number of primes -is not infinite.

Suppose there are n different primes

and the largest prime is p .

(2) Multiply all n primes together and then .add. 1. Call the answer P.

P = Cp i x P 2 x P 3 x . ...x pn ) + 1

(3) P is not a multiple of any of the primes

Pi Pa ..... p because whichever one 12 nyou divide by, the remainder is always

one.

So P must be prime.

(4) The largest prime is p but "? is clearly larger than this.

(5) .... so if there is a largest prime number (p ) j then there is an even larger prime (P)

This is obviously nonsense and so the first assumption (1) must be wrong.

THE NUMBER OF PRIMES IS INFINITE

Any mathematician or computer could calculate 1OO prime numbers. Any computer could calculate 10OO prime numbers. BUT IT TOOK THE GENIUS OF EUCLID TO ESTABLISH THAT HOWEVER MANY PRIME NUMBERS WERE CALCULATED IT WOULD BE POSSIBLE TO FIND MORE - THE NUMBER OF PRIMES IS INFINITE.

Pythagoras' Theorem is another example of the great power of mathematical proof. Mathematicians had known for many years before Pythagoras that the triangle shown has a right angle and also that:

0923 2 + 4 = 5 Z

Bern5cm

4cm

They knew that a similar rule worked for other right-angled triangles:

eg. 5 2 2 + 12 = 13But it was Pythagoras who PROVED that for every possible right-angled triangle the rule must work.

HOW DTD PYTHAGORAS PROVE IT?

Consecutive Composites

Are there 5 consecutive numbers which are not prime? .......

are there 10 consecutive numbers which are not prime? .......

are there 50 consecutive numbers which are not prime? .......

Euclid proved that he could find any number of consecutive numbers that do not include a prime.

The proof is on the page opposite. Try to follow it through.

These 5 numbers are consecutive:

6! + 2, 6! + 3, 6! + 4, 6! + 5, 6.' + 6

But 2 is a factor of 6!, so

2 is a factor of 6! + 2 (why?)

3 is a factor of ti! + 3 (why?)

4 is a factor of 6.' + 4 (why?)

5 is a factor of 6! + 5 (why?)

6 is a factor of 6! + 6 (why?)

They all have factors, so none of them is prime.

Here are 10 consecutive numbers, none of which is prime:

11! + 2, 11! + 3, 11! + 4, 111+5, 11! + 6,

11! + 7, 11! + 8, 11! + 9, 11! + 10, 11! + 11

Whatever number n is, none of the numbers in this list is prime.

n! + 2 3 n! + 3, n! +4, ....... n! + n

By making n as large as we like, we can make the list as long as we like.

Do you understand this proof?

Write down 20 consecutive numbers which are not prime.

Explain why none of them is prime.

If you want to find out more about Euclid, Pythagoras and Greek mathematics, the following books are interesting:

1. Life Science Library: Mathematics-D.Bergamini

2. Mathematics, A Human Endeavour-H.Jacobs(W.H.Freeman & Co.)

3. Greek Mathematicians (Exploring Maths.) French(McGraw Hill)

4. Mathematics and Imagination-Kasner & Newman(Bell)

5. A Short Account of the History of Mathematics - W.W. Rouse Ball (Dover)

6. A History of Mathematics - Freebury(Cassell)

SMILE 0831C

Primes and Factorials

Factorial four means 4x3x2x1

4x3x2x1 is written 4! So 4! = 4x3x2x1 = 24

1! = 12! = 231 = 6*j = 2^ Check the first few5. = 120 numbers in this list,6. = 720 to make sure you71 = 5 040 understand it.81 = 40 32091 =362 880

101 = 3 628 800 111 =39 916 800 121 = 479 001 600 131 = 6 227 020 800 141 = 87 178 291 200 15! * 1 307 674 368 000 16! =20 922 789 888 000 17! = 355 687 428 096 000 18! = 6 402 373 705 728 000 19! = 121 645 100 408 832 000 20 I = 2 432 902 008 176 640 000

Start with a whole number,

n

Subtract 1

Find (n-1)!

Add 1

Divide by n

YES Write n in list

Write n in listB

Try n = 7:

n = 7

(n-1) = 6

(n-1) ! = 720

(n-1)! + 1 = 721

Try several integer values for n.

Describe the numbers you get in list A and in list B

Is103 awhole

number? \M v_\ST

TURN OVER

You should have found that:

list A contains only BHHHI numbers list B contains only HUH numbers

Can you see why all composite numbers must be in list B?

Hints: What are the factors of 8? So why is 8 not a factor of 7x6x5x4x3x2x1 +1?

If you need additional hints look at 0831E.

SMILE 0831B

Start with any prime (not 2'or 3)

Square it

Subtract 1

Divide by 24

Isthe answer a whole number?

You've made a mistake Try again

(1) Try 11 (2) Try someother primes

(3) Try p:

yes

1

> -1 24

If the final answer -1 is always a 24 whole number, what can you say about

(p 2 -!) and 24?

Turn over

2(4) Is p -1 always a multiple of 24?

You can't test every prime (why not?) but you might be able to find a proof.

Hints: (1) p 2-! = (p-1) (p+1)

(2) 1,2,3,4,5, ...., (p-1) ,p, (p+1),

(3) What factors do (p-1) and (p+1) have? Remember p is prime and so 2 and 3 cannot be factors of p.

If you need additional hints, look at 0831E

SMILE 0831A

SQUARES & PRIMES

29

31

37

= 5 2

= 6 2 + I

Some primes are the sum of 2 squares .....

Examine all the primes up to 1OO and find whether they are the sum

of 2 squares or not.

(1) What can you find out about the primes which are the sum of 2 squares? (Hint: multiples of 4)

(2) What do you notice about the pairs of square numbers which add up to prime numbers?

4ru-l

11

14

18

22

Q

15

12

16

20

24

You should have noticed that:a) If a prime number is the sum of 2

squares then it is one more than a multiple of 4 (it is of the form 4n + 1)

b) When the sum of 2 square numbers is prime, one of the squares is even and the other is odd.Can you prove these statements? It is easier to take (b) first because this result will help to prove (a).

What sort of numbers are the sum of 2 squares? (odd) 2 + (odd) 2 = odd + odd = (even)2 + (even)2 =(even)2 + (odd) 2 =So if the sum of 2 squares is prime, .......

So we're only interested in (odd) 2 + (even)An odd number is of the form (2p+l)An even number is of the form 2qSo (odd) 2 + (even) 2 is of the form .......

You will need: Base Ten Apparatus. Smile 0832

SHORT DIVISION

ANSWER 69 + 3 = 23

Share into 3 equal piles.

69 . .

units

. . that's 60 + 9 . that's 6 tens and 9

(ShareShare 6 tens 3 waysthat's 2 tens each

fohare 9 units 3 ways I. . . . that's 3 units each

(ANSWER12 tens, 3 units or 23

Now try these:

1) 84 - 2 4) 63 -f- 3

2) 77 -s- 7

3) 84 -s- 4

5) 80 + 8

6} 286 -s- 2

7) 840 - 4

8) 693 - 3

9) 404 + 4

You will need Base Ten Apparatus. Smile 0833

SHORT DIVISION— CARRYING

How many 4 sin 6?

1 remainder 2

Share this 4 ways.

Share the tens .... that's 1 each and 2 left over

ANSWER 68 -s- 4 = 17

Now try these:1) 72 -s- 3

2) 91 -s- 7

3) 56 -*- 4

Change the 2 tens left over into 20 units .... that makes 28 units altogether

Now share out the 28 units .... 7 each

ANSWER1 ten, 7 units or 17

4) 84 + 3

5) 934 + 3

6) 812 * 4

7) 906 -s- 6

8) 253 + 7

9) 486 -*- 4

10) 158 -s- 6

11) 289 * 4

©RBKC SMILE 1999.

You will need: gummed strips Smile 0834

24-4

Cut a strip 24cm

Fold into 4.

How long?

Stick % of 24cm into your book: write on it 24cm -5- 4 = 6cm.

16-8

Cut a strip 16cm

Fold into 8.

How long?

Stick 1/s of 16cm into your book: write 16cm -=- 8 =

18-3

Folding into 3 is more difficult!

Stick 1/s of 18cm into your book;

Use strips of gummed paper to divide:

4) 155)206)327)20

3488

8) 10 - 49) 52 - 4

10) 24 -j- 6 (How will you fold it?)

Inverse MappingsSmile 0837

The machine ( x4 J -1 represents the mapping y —— » 4y - 1

11

The mapping diagram shows what happens to the numbers -1, 0, 2 and 3 as a result of this mapping

4y - 1

1. The mapping k ——> t k + 4 is represented by this machine:

Draw a mapping diagram and show what happens to the numbers -4, 0, 4, 6, 8,10 and 12 as a result of this mapping.

2. Here the numbers are put through this compound machine:

Copy the mapping diagram and show the result of putting some other numbers through this compound machine.

3. What is the central line of the mapping diagram used for?

4. x ——» 2x + 1 describes the left-hand mapping, x ——> x - 4 describes the right-hand mapping. What single mapping could replace the combined mapping?

5.

6.

Draw a mapping diagram with 3 number lines for this compound machine.

Your left-hand mapping should show x——» x + 3Your right-hand mapping should show x ——> x - 3What single mapping could replace the combined mapping?

7. This diagram shows the mappings

x——»x + 7 and x——»2x-7 2

Copy and complete the diagram and put some other numbers through the combined mapping.

What is the result of the combined ) mapping?

x + 7 2x-7

5

4

3

2

1

0

-1

-2

-3

-4

-5

-6

You should have found in question 6 and in question 7 that the arrows are symmetrical - whatever is done by the left-hand mapping is undone by the right-hand mapping.

x ——> x + 3 and x ——> x - 3 are inverse mappings,

x ——* x + 7 and x ——* 2x - 7 are inverse mappings.

Turn over

8. Use mapping diagrams with 3 number lines to find which of the following pairs are inverse mappings:

a) x——»3x and x——*>jc.3

b) x ——»> 7 + x and x ——>• 7 - x

c) x ——* 6 - x and x ——> 6 - x

d) x ——* x. + 3 ancl x ——*•.*. ~ 3 2 2

9. In two of the four examples in question 8 the mappings are not inverses of one another. Find the correct inverse of each of these mappings.

10. If x ——»• 2x - 7 is the inverse of x ——* x + 7,2

what is the inverse of x ——> 2x - 7?

©RBKC SMILE 1994.

IT'6 TOO SMALL '• WflNT THAT

ffS BI&

easy ENOUGH,FOLLOW THIS '

Mark, a dot near the letter

... measure from the dot to one corner... treble the distance... pot a mark. ..

Smile0838

J J

... do the same for the next comer

.. . and the rest.

When you 'i/e done eaoh corner, join thox op. Measure each side of theJetber. What do you notice ?

© RBKC SMILE 2000

is

THRTS ft GOOD

OOES IT MRTTGR WHERE YOU PUT THE POT?

A /) USe THIS MZTHOD TO MRKE R LZTTGR f THREE T/MBS as BIG. 2) /S ZI/6RV SIDE OF THE

5 THERE ARE IN FACT, SBVeiZRi Q06STIONS TO RNSW6R ABOUT THIS METHOD'-0 DOES IT MflTTeK WHERE YOU POT THE DOT? .... TfZY SB\J6RRL POSITIONS flZOUMD f\ SIMPL2

.... TRi INSIDZ THE SHflPE. . ... TRY ON THE EDGE OF THE SHAPB.

ON TH£ CORN6R OF WE

21 DOES IT WORK FOR ANY

3) DOBS IT WOKk FOR QOUBL/NG (2x) ftS W£LL flS TREBLING . . ..7Y?x

10

4) DOES IT WORK FOR. X

5) THE LRST QUESTION IS USUftUV TH£ MOST P/Ff/CUtf! Do£S THE SCftLt fflCTOR METHOP WORK. . . . 7#y '4 x- - - . TRY fy

O HOW DOES THE ftKER CHflNG& ERCH TIME ?

Smile Worksheet 0839

Rotate this wayYou will need:compasses and an angle indicator

To rotate triangle ABC 50° clockwise about the point O:

A1 B'c'is the image of ABC rotated 50° clockwise about O

Use an angle indicator to mark a line 50° clockwise from line OA

Checkthat A1 B1 C'

and ABC arecongruent

Using compasses measure OA and mark off the same distance from O along the 50° line

This gives the point A1 which is the image of A after a rotation of 50° clockwise about 0

Repeat 1, 2, and 3 for f vertices B and C to give f points B1 and C1 I

Rotate these shapes about a point in your book:

A)

B)

C)

60° clockwise

110° anti-clockwise

150° anti-clockwise

© RBKC SMILE 2001

-Smile 0843

VERYLARGENUMBERS

VERY LARGE NUMBERS

Approximate distances from the Sun to its nearest planets are:

MERCURY 60 000 000 km VENUS 100 000 000 kmEI^TH 150 ooo ooo kmMARS 200 000 000 km

Scientists engaged in space research calculate and work with numbers that are extremely large.

For example .... Pluto is approximately 6000000000 km from the Sun .... the star Cygni is approximately 300000000000000 km from Earth ....

The scientists have a shorter way to write these numbers. It's called standard form:

6000000000= 6 x 1000000000 written as 6 x 109

300000000000000= 3 x 100000000000000 which is 3 x 10 14

To use this method you must be able to recognise the powers of 10. Check these:

1) 1000 = 10 x2) 10000000 =3) 106 =

10 x 10 = 10

, 4) Change these numbers to standard form:

a) 300000 = 3 x 100 000 = 3 x 10" \b) 70000 = 7 x 10000 = Hx|c) 80000000 = 8 x B = |- •d) 2000e) 90000000000

5) How would you write 100000 in standard form?

6) Use the information on page 1 of this booklet to write in standard form, thedistances of Mercury, Venus and Mars from the Sun.

7) If you have the Leapfrogs LINKS BOOK, look at pages 36-37. Write in standard form the distance:a) to the edge of our galaxy (Milky Way)b) to the edge of the known Universe

LARGE NUMBER

NUMBER BETWEEN 1 AND 10

POWER OF 10

standard form

The Moon is 382 171 km from the Earth. How do you write a number like that in standard form?

1fc>t too difficult if you know how to multiply decimals by powers of 10.

8) Multiply any decimal by some powers of 10:Try 3-64 x 101

3-64 x 102 3-64 x 103 3-64 x 104 3-64 x 105

9) What is the connection between the power and the change in the number?

10) Now try the reverse problem:a) 256-3 = 2-563 x 100 = 2-563 x 101b) 137-6 = 1-376 ••=•••c) 9200-0 = 9-2 ••=•••d) 72310-0 -•••==•••e) 22-0 • •- • •

So the Moon is 382 171 km away. Thats 3-821 71 x ....

11) Write in standard form the distance of:a) the Moon from Earthb) from Earth to the edge of the Solar

System

Standard form allows very large numbers to be compared more easily.

12) Put the following distances from the Sun in standard form:

EARTH 1-49 x 10

;A|PHA CENTAURI000 000 000

;.'=/': •' '

SATURN 1-43 x 10

VENUS 108 000 000ALPHA CYGNI

15 300 000 000 000 000

MERCURY 58 000 000

13) Arrange these distances in order of size.

14) Which is further from the Sun — Saturn or Earth? About how many times further?

15) Compare Alpha Centauri with Earth in the same way.

16) Which is bigger? ,a) 9-3 xlO8 or 3-8 xlO9?b) 1-9 xlO6 or 9-1 x 10s?

17) If a = 3 x 105 b = 2xl07 c = 5x103

Can you find the value of the following, writing your answer in standard form?i) a2 ii) ab iii) be iv) ac v) a + b

Find out the following sizes and distances and record your answers in standard form.• Speed of light

• How long is a light year?

• Population of London.

• Approximately 3-386 x 106 people buy The Sun newspaper every day.

• Number of heartbeats in a Life-time.

The next booklet about standard form is SMILE 0844 Very Small Numbers. Itshows how scientists write very small numbers in this shorter way.

You will need to do SMILE 0614 Powers of 10 before tackling this.

© RBKC 1998

#::£

>&

y£:::^

Scientists studying microbiology calculate and work with numbers that are very small indeed.

For example .... the smallest living cells are bacteria with a diameter of 0.000 025cm . . . blood cells have a diameter of about 0.000 75cm.

Very small numbers can be written in standard form in a similar way to large numbers.

SMALL NUMBER =

NUMBER BETWEEN X POWER 1 AND 10 OF 10

- --. - —— ̂ -

For example, '-0.000 000 6 = 6 x 0.000 000 1So0.000 000 6 = 6 x 10B

1) What is the power of 10 for 0.000 000 1 ?

// you had difficulty answering question (1) look at POWERS OF TEN worksheet 0614.

If you need a reminder about standard form, look at VERY LARGE NUMBERS 0843.

2) Change these small numbers to standard form:a) 0.6 = 6 x 0.1 = 6 x 10*b) 0.000 6 = 6 x 0.000 1 =•/•c) 0.000 62 = 6.2 x 0.000 1 = BX|d) 0.001 29 = 1.29 X|=BX|e) 0.007 5

3) Try some other very small numbers. What do you notice about the power of 10 and the change in the number?

4) Change the size of the bacteria cells and the blood cells (from above) into standard form

5) Arrange the following in order of size:

Protozoa (paramecium) 0.2 mm Bacteria (pneumonia) 0.000 001 mm Virus (flu) 5 x 10~ 5 mm Virus (mumps) 0.000 225 mm Molecule (egg white protein) 0.000 01 mm Atom (hydrogen) 0.000 000 2 mm Pin-prick 10" 1 mm

PROTOZOA X+00

6) Which is bigger ?a) 9.3 x 108 or 3.8 x 109b) 4.6 x 1(T 3 or 3.6 x 1CT 4

7) Find out the following sizes and distances (and more if you are interested). You will need to use the library.

Record your answers in standard formwhere appropriate.

9.8 x 10" miles from your cornea toretina wall.

*Mass of a neutron.* IWavelength of light. "

*Diameter of an electron.

*Limit of human vision.

*Units for measuring very small distances:

angstrommicrometrenonometrepicometre

*Mass of a carbon atom.

Smile 0845 c

/) Use the SCRIBE fRCTOR. method C see cart 0838) to complete the enlargement on riorksheeb

a factor of

On the same worksheet change the shape using scale factors .-- £x

/fcx

1&x Ox

happens to the shape as the scale factor diminishes ?

Think about the number fine as it passes be/oi/o zero: where do you think the shapeon the worksheet will be if we change ti: by a factor of ~

Turn over

ft negative scale factor like ~2 instructs yov to double the distance from the dot} but in the Opposite direction

/8cm

5) Draw the shape, on the worksheet,Does your answer to question (4-) agree ?

h) What effect does the scale factor -/

7) Does the position of the dot affect the negatiue, scale factor?

ARROWS

$) Draw an arrow ^2 cm long.

the arrow +2 by a scale factor +3

Measure the new arrou). Record your result

9) Now to change an amw ~2 by a scale factor

Can you predict the answer ?

It-, check your result.

IG) Change the arrouj+2 by the negative scale factor "5

Record ywr result

//) NOUJ to change the arrow -2 by the negatiue scale factor "5

Can yew predict the revolt ?, ..~b X ^ *

answer.

7>y several positive and negative scale factors: what a-ffect do they have on (a) positive anouos?

(bl negative arrows?

Smile Worksheet 0845a

Negative Scale Factor4x

© RBKC SMILE 2001

Smile Worksheet 0849

Anywhere on the Number Line

-7 -6'5 -4 -3 -2 -1•4—————•———»———•*—-—»———•—~—«——j—*-

1. If you start at 2, can you continue to rename the number line?

-7 -6 -5 -4 -3 -2-1 01 34567

2-9 2- 2- 2-2 2-1 2+1 2+22+

It is possible to rename the number line from any starting point. 2. Try this one:

3. Choose a starting point of your own:254 255 256 257 258 259 260 261

-8 ~7 -6I — . ——— . _____ . ——

11 H'5-1

SIl -4-3-2-1012345678~~i*-j —— , —— , —— , —— , ——— • —— , ——— , —— . ——— , —— , ——— , —— . —— , — i"5+1"5+2| || II II II II II II It II

262

4. Let n be any position on the number line:

5. If in question 4, n=3 which number is (n+1)?

which number is (n-3)?

which number is (n+3)?

6. If in question 4, n=7 which number is:

(n-1) --_-__- n(n+3)

3(n+1) ______ n(n-4)(n+3)

4 —— . ————— , ————— , ————— , ————I n-4 II n-2

K;.-.••«:»W

n+1 n+2 || n+5

© RBKC SMILE 2001

Multiplication problem?Aysha wants to multiply 17 by 13.

77

77X73 73

Explain how Aysha's method works.

10 100

30

70

21

Smile 0850

77X73 = 27

Copy and complete Aysha's method to work out 23 x 16.

23 10 10

16

60 18

1. Use Aysha's method to work out the following: a) 18x13 b) 21x19 c) 23x34 d) 47x17

2. Is it sensible to use Aysha's method for bigger numbers...e.g. 53 x 47? How could you adapt Aysha's method to help you with these? 103 x 17 121x24 84x93

© RBKC SMILE 2001