LJR March 2004 The internal angles in a triangle add to 180° The angles at a point on a straight...

LJR March 2004

LJR March 2004

The internal angles in a triangle add to 180°

The angles at a point on a straight line add to

180°

LJR March 2004

Since any quadrilateral can be split into two triangles its internal angles add to 360°

LJR March 2004

A pentagon can be split into three triangles so its

internal angles add to 540°

A hexagon can be split into four triangles so its internal angles add to

720°

LJR March 2004

60°

Internal angles 180° 3 =

60°

120°

90° 90°External angles

180° - 90° = 90°

External angles 180° - 60° =

120°


90°

LJR March 2004

108° 72°


108°External angles 180° - 108° =

72°

120° 60°


120°External angles 180° - 120° =

60°

LJR March 2004

180°

Sides of

Polygon

Angle total

Internal angles

External angles

60° 120°3

360° 90° 90°4

540° 108° 72°5

720° 120° 60°6

(n-2)180°n (n-2)180°n

180°-(n-

2)180°n

LJR March 2004

Draw a regular hexagon of side 4cm.

Sketch to identify angles

60º

120º

60º4cm

Use this information to accurately draw the hexagon.

LJR March 2004

Draw a regular pentagon of side 6cm.

Sketch to identify angles

72º

72º

6cm

Use this information to accurately draw the pentagon.

108º

LJR March 2004

Draw a rhombus with diagonals 8cm and 6cm.

Sketch first

8cm

6cm

3cm3cm

4 cm

4 cm

8 cm

3cm

3cm

now draw

LJR March 2004

Click here to return to the main

index.

Click here to practice this further.

Click here to repeat this section.

LJR March 2004

a

b c

Hyp

otenu

se

222 bac

In any right angled triangle the square on the hypotenuse is equal to the sum of the squares on the two shorter sides.

LJR March 2004

Examples: Calculate x in these two triangles.

Calculating the hypotenuse. c2 = a2 +

b2

6m

8m

x

222 68 x

3664 100

100x

9cm

x7cm

222 79 x

4981

130130x

m10

cm411 to 1 dp

LJR March 2004

222 bac

Pythagoras theorem can be rearranged so that a shorter side can be calculated.

222 cba 222 bca 222 acb als

o

Write the biggest number first

Add to find the hypotenuse

Subtract to find a shorter side

LJR March 2004

Examples: Calculate x in these two triangles.

Calculating a shorter side. c2 = a2 +

b2

5m

x

13m

222 513 x

25169

144

144x

27cm

x

24cm222 2427 x

576729

153153x

m12

cm3712 to 2 dp

LJR March 2004

Find the distance between A and B.

A

B

222 57AB 2549

74

74x cm411 to 1

dp

LJR March 2004


index.

Click here to try some Pythagoras

problems.


LJR March 2004

When we talk about the speed of an object we usually mean the average speed. A car may speed up and slow down during a journey but if the distance covered in one hour is 50 miles, we would say its average speed was 50mph.

When we are doing calculations using speed, distance and time, it is important to keep the units consistent.

If distance is measured in kilometres and time is measured in hours, then the speed is in kilometres per hour (km/h).

LJR March 2004

D

S T

D =

S =

D

TT =

D

S

S x T

D

S T

LJR March 2004

Problem:

Stewart walks 15km in 3 hours.

Calculate his average speed.

Stewart covers 15km in 3hours

So his average speed is 15 3 = 5 km/h

Speed = 5km/h

distance covered

Average speed =

time taken

D

S T

LJR March 2004

Problem

Claire cycled at a steady speed of 11 kilometres per hour.

How far did she cycle in 3 hours?

In 1 hour she covers 11 km

So in 3 hours she covers 11 X 3 = 33km

Distance = 33km

Distance = average speed X time taken

D = S X T

D

S T

LJR March 2004

Problem

Paul drives 144 kilometres at an average speed of 48km/h.

How long will the journey take?

He drives 48 km in 1 hour.

144 48 = 3 (there are three 48s in 144)

So the journey takes 3 hours.

Time = 3 hours.

distance covered

Time taken =

average speed

D

T =

S

D

S T

LJR March 2004

Problem

A car travelled for 2 hours at an average speed of 90km/h.

How far did it travel?

D = S X T D = 90 km X 2 hours

= 180 km

The car travelled 180 km

D

S T

LJR March 2004

Problem

A car on a 240km journey can travel at 60km/h.

How long will the journey take?

D

T =

S

T = 240 km 60

= 4 hours

The journey will take 4 hours

D

S T

LJR March 2004


index.

Click here to try some harder SDT.


LJR March 2004

LJR March 2004

Diameter

Radius

LJR March 2004

• The formula for the circumference of a circle is:

where C is circumference

and d is diameter

C = d

Circumference

LJR March 2004

Area

• The formula for the area of a circle is:

where A is area

and r is radius

A = r2

LJR March 2004

8cm

= 3·14 8

= 25·12cm

An approximatio

n

for is 3·14

C = d

Calculate the circumference of this

circle.

LJR March 2004

A = r2

= 3·14 52

= 3·14 25

= 78·5cm2

10cm

Calculate the area of this

circle.

Diameter is 10cm

Radius is 5cm

LJR March 2004

LJR March 2004

• Calculate the diameter and radius of a circle with a circumference of 157m.

C = d

157 = 3·14 d

d = 50m

d = 157 ÷ 3·14

r = 50 ÷ 2 = 25m

LJR March 2004

• Calculate the radius and diameter

of a circular slab with an area of

6280cm2.A = r2

6280 = 3·14 r2

r2 = 6280 ÷ 3·14

= 44·72135955

= 2000r = 2000

44·7cm

d = 89·4cm

LJR March 2004

Composite Shapes• Calculate the Perimeter of this shape

12m

9m

C = d

= 3·14 9

= 28·26

28·26 2 = 14·13m

Perimeter = 14·13 + 12 + 12 + 9 = 47·13m

LJR March 2004

Composite Shapes

• Calculate the shaded Area.

28cm

28cmA = r2

= 3·14 142

= 3·14 196

= 615·44cm2

Area of square = 28 28 =

784cm2

Shaded area = 784 615·44

= 168·56cm2

LJR March 2004


index.

Click here to try some more Circle.


LJR March 2004

LJR March 2004

base

heightheight base

21

Area

bh 21

A

Example: Calculate the area of this triangle.

7cm

4cm

9cm bh 21

A

4 7 21

214cm

LJR March 2004

diagonal2 diagonal1 21

Area

21dd

21

A

1d

2d

1d

2d

LJR March 2004

Example: Calculate the area of these shapes.

21dd

21

A

8 11 21

244m

6m

9m

21dd

21

A

6 9 21

227m

4m

8m

3m

LJR March 2004

base

height

base

This shows that the area of a parallelogram is similar to the rectangle.

height base Area

LJR March 2004

height base Area

bh A

Example: Calculate the area of this parallelogram.

bh A

5 8 240cm

base

height

8cm

5cm

LJR March 2004

A composite shape can be split into parts so that the area can be calculated.

Examples: Calculate the area of the following shapes.

Area A = 10 × 6 = 60

90cm2

AB

6cm

6cm

11cm

10cm Area B = 6 × 5 = 30

LJR March 2004

A shape can be split into as many parts as necessary.

A

B

18m

12m

11m 165m2

Area B = × 6 × 11 = 33

2

1Area A = 12 × 11 = 132

LJR March 2004


index.

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this.


LJR March 2004

Organise this data in a Stem & Leaf chart

27 24 31 28 33 42 50 29 30

26 32 45 48 51 45 34 26 51

33 41 44 37 22 52 35

2

3

4

5

7 6

3

4

2

1

1

5 4

8

8

7 3

1

2

2 5

2 0

4 5

9 6

0

1

2

3

4

5

2 4 6 6 7 8 90 1 2 3 3 4 5 71 2 4 5 5 8

0 1 1 2

4|2 means 42n = 25

Stem Leaf

LJR March 2004


index.

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Leaf.


LJR March 2004

Calculations must be carried out in a certain order.

Brackets first and any ‘of’ questions,

then multiply and divide before add and subtract.

racketsf

ivide

ultiply

dd

ubtract

eg: ¼ of 20

LJR March 2004

Examples

2753 14 15

29

312-57

4 - 35 31

7)(8 of 31

15 of 31

5

28-1)(27

28 - 37 4 - 21

17 3-8

427 5

8 7 3

515

Evaluate top and bottom separately first then

divide.

LJR March 2004

Given a = 4, b = 5 and c = 3, find the values of:

3b 2a 5342

23

2abc 3542 120

2ac)-7(b 423)-7(5

8 2 7

22

bca6cab

5-34

3654

21820

15 8

8 14

238

19

LJR March 2004


index.

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LJR March 2004

Very large or very small numbers can be written in scientific notation (also known as standard form) to ease calculations and allow the use of a calculator.

300 can be written as 3 x 100 and we know that

100 can be written as 102 , so 300 can be written as

3 x 102

300 = 3 x 102 This is scientific

notation.In general a x 10n where 1 a 10 and n is an

integer.Positive or

negative whole number.

LJR March 2004

100000000077000000000 9107

7000000000

decimal point moves 9 places

left

10000000035530000000 81035

530000000


left

10000007144710000 610714

4710000


left

Normal numbers to scientific notationExamples

LJR March 2004

-22

10310

31003

1003030

0 0000000008


right

Examples -1010800000000080

0 000000692


right

-7109260000006920

You do not need to remember this but it is the reason why we can write small numbers as follows.

LJR March 2004

10000000005105 9 5000000000

5000000000


right

1000741074 3 4700

4700


right

10000089310893 5

389000389000


right

Scientific notation to normal numbers.Examples

LJR March 2004

00000040104 -7 0 0000004


left

0001601061 -4 0 00016


left

0000254010542 -5 0 0000254


left

Scientific notation to normal numbers.Examples

LJR March 2004

You must be able to enter and understand scientific notation on a calculator.

7104To enter 4 EXP 7

-41013 To enter 3 EXP 4• 1 +/-

On a calculator display

41007 7104 represents

3·110-04-41013 represents

LJR March 2004


index.



LJR March 2004

)( 34 x 344 x

)( qp 523 qp 156

)( 723 tt tt 216 2

Everything in the bracket is multiplied by what is outside the bracket.

124 x

)( aba 52 2102 aab

LJR March 2004

The reverse process is called factorising.

To factorise

• look for factors which are common to all terms.

• identify the highest common factor.

)( 34124 xx

124 x Factorise

. and and and are of factors xxxx 422414 ,,

. and and and are of factors 436212112 ,,

LJR March 2004

Examples: Factorise

68 y )( 342 y pp 84 2 )( 24 pp

21418 xx )( xx 792 tt 216 3 )( 723 2 tt

21824 gg )( gg 346 21025 xxy )( xyx 255

Try to work out the answer to each question before pressing the space-bar.

LJR March 2004


index.



LJR March 2004

Letters are used to represent missing numbers.Expressions

An expression contains letters and numbers.

x + 3, 2t – 5, 7 + 4y etc are all expressions.

The value of an expression depends on the value given to the letters in the expression.

If x = 4, give the value of (i) x + 3 (ii) 5x – 7

(i) x + 3 = 4 + 3 = 7

(ii) 5x – 7 = 20 – 7 = 13

LJR March 2004

If a = 3, b = 0, c = 5 and d = 7 find the value of the

following expressions (i) 4b + 2d (ii) 3c – 2a +

5d (i) 4b + 2d (ii) 3c – 2a + 5d

= 0 + 14

= 14

= 15 – 6 + 35

= 44

Find an expression for the number of matches in design x.

5 9 134

An expression for the no. of matches in design x is 4x + 1

LJR March 2004


index.



LJR March 2004

Example: Solve 135342 )( x13568 x13118 x248 x3x

24387 xxExample: Solve

2484 x324 x8x

LJR March 2004

Example: Solve 178523 )( x178156 x17236 x66 x1x

22345 xxExample: Solve

2242 x222 x11x

LJR March 2004


index.

Click here to try more

Equations/Inequations.


LJR March 2004

Here is some information (or data) – imagine it is a set of test marks belonging to a group of children

19 21 20 17 18

24 20 16 20

This data can be organised and used in different ways.

LJR March 2004

The mode (or modal value) is the value in the data that occurs most frequently.

19 21 20 17 18

24 20 16 20

First of all rearrange the data in order -

16 17 18 19 20 20 20 21 24

The mode is 20 as it occurs most often.

LJR March 2004

The median is the value in the middle of the data when it is arranged in order.

16 17 18 19 20 20 20 21 24

19 21 20 17 18

24 20 16 20

The median is 20 as this is the value which is in the middle.The range is a measure of spread: it tells us how the data is spread out. The range = the highest value – lowest value.

The range is 24 – 16 = 8. The value of the range is 8.

LJR March 2004

The mean of a set of data is the

sum of all the values divided by

the number of values.Unlike the median and mode, the

mean uses every piece of data. It

gives us an idea of what would

happen if there were equal shares.

Temperatures in ºC

13 13 11 14 17 19 1811 13 13 14 17 18 19

The sum of the values is 105.The number of values is 7.

The mean is 105

7

= 15

LJR March 2004

3 4 6 3 7 5 4 5 4

This set of data shows shoe sizes. Find the mean, median, mode and range.

3 3 4 4 4 5 5 6 7

The mode is 4 as it occurs most often.

The median is the middle value 4. The range is 7 – 3 = 4


The mean is 95

4941

LJR March 2004

19 21 20 17 22 18 28 27

Here is another set of data. Find the mean, median, mode and range.

There is no mode as each value occurs just once.

17 18 19 20 21 22 27 28

The median is the middle value. As there is an even number of data, the median is half way between 20 and 21.

The median is 20•5



The mean is 5218

172

LJR March 2004

Number of goals scored

Frequency

0 13

1 21

2 11

3 8

4 6

The sum of the values is:

13 X 0 goals = 0

21 X 1 goal = 21

11 X 2 goals = 22

8 X 3 goals = 24

6 X 4 goals = 24

So the sum of the values is: 0 +21 + 22 + 24 + 24 = 91

The number of values is the total frequency:

13 + 21 + 11 + 8 + 6 = 59

The mean of the goals scored is 91

59

= 1•54 to 2dp

LJR March 2004

Marks out of 10

Frequency of marks

5 2

6 6

7 9

8 10

9 7

The mode is 8 because this test mark has the highest frequency.

The total frequency is 34.


In general, when there are n pieces of data, the median is the value

of the ½(n +1) term.

The median is ½(n +1) value so ½(34 +1) = ½(35) = 17•5

The 17th value is 7 and the 18th value is 8.

The median is 7•5

LJR March 2004


index.



LJR March 2004

Scatter graphs are used to identify any correlation between two measures.

Graph the following data taken from a class of S3 students.

Height (cm)Shoe size

2

12513

013

514

014

014

515

015

015

516

016

517

54 3 5 6 7 6 7 8 1

09 1

1

Does this show a connection between height and shoe size?

LJR March 2004

Height (cm)

Sh

oe s

ize

13

12

11

10

9

8

7

6

5

4

3

2

1

120 130 140 150 160 170 180

This graph shows a strong

positive correlation.

Height and shoe size in S3

LJR March 2004

Absence (in days)

Exam

mark

28

26

24

22

20

18

16

14

12

10

8

6

4

2

0 2 4 6 8 10 12 14

This graph shows a strong

negative correlation.

Exam Marks & Attendance

LJR March 2004

Exam

mark

28

26

24

22

20

18

16

14

12

10

8

6

4

2

0

This graph shows no

correlation.

Exam marks & Height

Height (cm)

120 130 140 150 160 170 180

LJR March 2004


index.

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this.


LJR March 2004

A factor is a number that divides another number exactly.

Find all the factors of 24

1 x 24

2 x 12

3 x 8

4 x 6

Factors of 24 are 1, 2, 3, 4, 6, 8, 12, 24

LJR March 2004

Find the prime factors of 24

24 = 2 x 12

= 2 x 3 x 4

= 2 x 3 x 2 x 2

= 23 x 3

2 x 2 x 2 = 23

LJR March 2004

Find the prime factors of 72

72 = 2 x 36

= 2 x 3 x 12= 2 x 3 x 3 x 4

= 23 x 32 3 x 3 = 32

= 2 x 3 x 3 x 2 x 2 2 x 2 x 2 = 23

LJR March 2004


index.

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LJR March 2004

A ratio compares quantities

You must be able to

• Simplify ratios

• Find one quantity given the other

• Share a quantity in a given ratio

LJR March 2004

The ratio of cars to buses is 4 to 3

also written as 4:3

It is essential to write ratios in the correct order

cars to buses is 4:3

LJR March 2004

The ratio of eggs to bunnies is 3 to 5

also written as 3:5

eggs to bunnies is 3:5

Remember order is important

LJR March 2004

Simplify the ratio 28:21

both numbers divide by 7

28:21

4:3

Simplify the ratio 32:56

both numbers divide by 8

32:56

4:7

32:5616:288:144:7

You can take as many steps as you need

LJR March 2004

The ratio of boys to girls in a class is 5:4

If there are 12 girls how many boys are there?

Boys : Girls

5 : 4

:1233

155 x 3 = 15

The ratio of oranges to apples in a fruit bowl is 2:3

If there are 8 oranges how many apples are there? Oranges : Apple

2 : 3

8 : 44

12 3 x 4 = 12

LJR March 2004

Share £800 between 2 partners in a business in the ratio 3:5

3 + 5 = 8 shares

£800 8 = £100

3 x £100 = £300

5 x £100 = £500

£300 + £500 = £800

The partners receive £300 and £500 respectively.

LJR March 2004

Share 45 sweets between 2 friends in the ratio 5:4

5 + 4 = 9 shares

45 9 = 5 sweets

5 x 5 = 25 sweets

4 x 5 = 20 sweets25 + 20 =

45

The friends receive 25 and 20 sweets each.

LJR March 2004


index.



LJR March 2004

Remember : A right angle is 90°

A straight angle is 180°

There are 360° round a point

An acute angle is less than 90°

An obtuse angle is more than 90° and less than 180°

A reflex angle is more than 180° and less than

360°

LJR March 2004

More Angle terms

Reflex ABC = 280°

A reflex angle is greater than 180° but less than 360°

A

B

C

280°

80°

LJR March 2004

A line parallel to the earth’s horizon is horizontal.

A line perpendicular to a horizontal is called vertical.

Two lines are perpendicular if they intersect at right angles.

LJR March 2004

ABD and DBC are supplementary

If two angles make a right angle they are said to be complementary.

A

B C

D

ABD and DBC are complementary

If two angles make a straight angle they are said to be supplementary.

A B C

D

LJR March 2004

ABD = 90° – 60° = 30°

A

B C

D

60°

DBC = 180° – 40° = 140° A B C

D

40°

••

A

D C

B

E*

*AED = BEC

Vertically opposite angles are equal.

AEB = DEC

LJR March 2004

L

RQP

NM

K

S

B

GFE

DC

A

H

Angles and parallel

Lines

Parallel lines (F shape) so HFG =

80°

When two parallel lines are involved F and Z shapes can be used to calculate angles.

80°

Parallel lines (Z shape)

so PQM = 65°

65°

LJR March 2004

Fill in all the missing angles.

63° 63°117°117°

63°63°117°117°

180° - 63° = 117°

LJR March 2004


index.



LJR March 2004

21

factor Scale

21

factor Scale 2 factor Scale

2 factor Scale

41

factor Scale 4 factor Scale

LJR March 2004

Identify the scale factor

scale factor 3

scale factor 2

scale factor

½

scale factor 6

scale factor

¼

What other scale factors can you identify?

LJR March 2004

scale factor 3

scale factor 4 scale

factor 2

scale factor 1½

scale factor

¾

scale factor

½

LJR March 2004


index.



LJR March 2004

To draw a pie chart we need to work out the different fractions for each group.

One evening the first 800 people to enter a Cinema complex are asked which film they plan to see.

The results are as follows:

Lord of the Rings 160

Calendar Girls 150

Pirates of the Caribbean 190

The Last Samurai 170

Touching the Void 130

Use these results to draw a pie chart.

LJR March 2004


Calendar Girls 144

Pirates of the Caribbean

192



20%100800160

18%100800144

24%100800192

22%100800176

16%100800128

20%

18%

24%

22%

16%

Lord of the

Rings

Calendar Girls

Pirates of the

Caribbean

The Last Samurai

Touching the Void

100%

LJR March 2004


Calendar Girls 144

Pirates of the Caribbean

192



72360800160

65864360800144

86486360800192

79279360800176

58657360800128

72°

65°

86°

79°

58°

Lord of the

Rings

Calendar Girls

Pirates of the

Caribbean

The Last Samurai

Touching the Void

360°

LJR March 2004


index.



LJR March 2004

Impossible Unlikely Even chance

Most likely Certain

0 1½

Probability is a measure of chance between 0 and 1.

Probability of an impossible event is 0.

Probability of a certain event is 1.

outcomes of number Totaloutcome favourable of Number

yProbabilit

LJR March 2004

The probability of throwing a 3

is 1 out of 6 61

Pr(3)

The probability of throwing an even number is 3 out of 6

21

63

Pr(3)

5

5

The probability of choosing a 5 of diamonds from a pack of cards is 1 out

of 52521

diamonds) of Pr(5

41

5213

)Pr(diamond

LJR March 2004


index.



LJR March 2004 The internal angles in a triangle add to 180° The angles at a point on a straight...

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