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 ° Internal angles 180°  3 = 60° 120° 90° External angles 180° - 90° = 90° External angles 180° - 60° = 120° Internal angles 360°  4 = 90°

LJR March °72° Internal angles 540°  5 = 108° External angles 180° - 108° = 72° 120°60° Internal angles 720°  6 = 120° External angles 180° - 120° = 60°

LJR March ° Sides of Polygon Angle total Internal angles External angles 60° 120° 3 360° 90° 4 540° 108° 72° 5 720° 120° 60° 6 (n-2)180° n n 180°- (n-2)180° n

LJR March 2004 Draw a regular hexagon of side 4cm. Sketch to identify angles 60º 120º 60º 4cm Use this information to accurately draw the hexagon.

LJR March 2004 Draw a regular pentagon of side 6cm. Sketch to identify angles 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 3cm 4 cm 8 cm 3cm now draw

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LJR March 2004 a b c Hypotenuse 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. c 2 = a 2 + b 2 6m 8m x 9cm x 7cm to 1 dp

LJR March 2004 Pythagoras theorem can be rearranged so that a shorter side can be calculated. also 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. c 2 = a 2 + b 2 5m x 13m 27cm x 24cm to 2 dp

LJR March 2004 Find the distance between A and B. A B to 1 dp

LJR March 2004 Click here to return to the main index. Click here to try some Pythagoras problems. Click here to repeat this section.

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 ST D = S = D T T = D S S x T D ST

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 ST

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 ST

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 ST

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 ST

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 ST

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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 =  r 2

LJR March cm = 3·14  8 = 25·12cm An approximation for  is 3·14 C =  d Calculate the circumference of this circle.

LJR March 2004 A =  r 2 = 3·14  5 2 = 3·14  25 = 78·5cm 2 10cm Calculate the area of this circle. Diameter is 10cm  Radius is 5cm

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 6280cm 2. A =  r = 3·14  r 2 r 2 = 6280 ÷ 3·14 = 44· = 2000 r =  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· = 47·13m

LJR March 2004 Composite Shapes Calculate the shaded Area. 28cm A =  r 2 = 3·14  14 2 = 3·14  196 = 615·44cm 2 Area of square = 28  28 = 784cm 2 Shaded area = 784  615·44 = 168·56cm 2

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LJR March 2004

base height Example: Calculate the area of this triangle. 7cm 4cm 9cm

LJR March 2004

Example: Calculate the area of these shapes. 6m 9m 4m 8m 3m

LJR March 2004 base height base This shows that the area of a parallelogram is similar to the rectangle.

LJR March 2004 Example: Calculate the area of this parallelogram. 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 90cm 2 A B 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 165m 2 Area B = × 6 × 11 = 33 Area A = 12 × 11 = 132

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LJR March 2004 Organise this data in a Stem & Leaf chart |2 means 42n = 25 Stem Leaf

LJR March 2004 Click here to return to the main index. Click here to try some more Stem & Leaf. Click here to repeat this section.

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. rackets f ivide ultiply dd ubtract eg: ¼ of 20

LJR March 2004 Examples Evaluate top and bottom separately first then divide.

LJR March 2004 Given a = 4, b = 5 and c = 3, find the values of:

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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 10 2, so 300 can be written as 3 x = 3 x 10 2 This is scientific notation. In general a x 10 n where 1  a  10 and n is an integer. Positive or negative whole number.

LJR March decimal point moves 9 places left decimal point moves 8 places left decimal point moves 6 places left Normal numbers to scientific notation Examples

LJR March decimal point moves 10 places right Examples decimal point moves 7 places right You do not need to remember this but it is the reason why we can write small numbers as follows.

LJR March decimal point moves 9 places right 4700 decimal point moves 3 places right decimal point moves 5 places right Scientific notation to normal numbers. Examples

LJR March decimal point moves 7 places left decimal point moves 4 places left decimal point moves 5 places left Scientific notation to normal numbers. Examples

LJR March 2004 You must be able to enter and understand scientific notation on a calculator. To enter 4 EXP 7 To enter 3 EXP 4 1 +/- On a calculator display 4  ·1 

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LJR March 2004 Everything in the bracket is multiplied by what is outside the bracket.

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.

LJR March 2004 Examples:Factorise Try to work out the answer to each question before pressing the space-bar.

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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 = = 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 = = 14 = 15 – = 44 Find an expression for the number of matches in design x An expression for the no. of matches in design x is 4x + 1

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LJR March 2004 Example:Solve

LJR March 2004 Example:Solve

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LJR March 2004 Here is some information (or data) – imagine it is a set of test marks belonging to a group of children 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 First of all rearrange the data in order 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 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 The sum of the values is 105. The number of values is 7. The mean is = 15

LJR March This set of data shows shoe sizes. Find the mean, median, mode and range The mode is 4 as it occurs most often. The median is the middle value 4. The range is 7 – 3 = 4 The sum of the values is 41. The number of values is 9. The mean is

LJR March Here is another set of data. Find the mean, median, mode and range. There is no mode as each value occurs just once 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 205 The range is 28 – 17 = 11. The value of the range is 11. The sum of the values is 172. The number of values is 8. The mean is

LJR March 2004 Number of goals scored Frequency The sum of the values is: 13 X 0 goals = 0 21 X 1 goal = X 2 goals = 22 8 X 3 goals = 24 6 X 4 goals = 24 So the sum of the values is: = 91 The number of values is the total frequency: = 59 The mean of the goals scored is = 154 to 2dp

LJR March 2004 Marks out of 10 Frequency of marks The mode is 8 because this test mark has the highest frequency. The total frequency is 34. The range is 9 – 5 = 4. The value of the range is 4. 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) = 175 The 17 th value is 7 and the 18 th value is 8. The median is 75

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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 Does this show a connection between height and shoe size?

LJR March 2004 Height (cm) Shoe size This graph shows a strong positive correlation. Height and shoe size in S3

LJR March 2004 Absence (in days) Exam mark This graph shows a strong negative correlation. Exam Marks & Attendance

LJR March 2004 Exam mark This graph shows no correlation. Exam marks & Height Height (cm)

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

LJR March 2004 Find the prime factors of = 2 x 36 = 2 x 3 x 12 = 2 x 3 x 3 x 4 = 2 3 x x 3 = 3 2 = 2 x 3 x 3 x 2 x 2 2 x 2 x 2 = 2 3

LJR March 2004 Click here to return to the main index. Click here to investigate further. Click here to repeat this section.

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 32:56 16:28 8:14 4:7 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 : 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 : 12 3 x 4 = 12

LJR March 2004 Share £800 between 2 partners in a business in the ratio 3: = 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: = 9 shares  45  9 = 5 sweets 5 x 5 = 25 sweets 4 x 5 = 20 sweets = 45 The friends receive 25 and 20 sweets each.

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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 R Q P N M K S B G F E D C 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° 117° 63° 117° 180° - 63° = 117°

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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

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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 Rings160 Calendar Girls150 Pirates of the Caribbean190 The Last Samurai170 Touching the Void130 Use these results to draw a pie chart.

LJR March 2004 Lord of the Rings160 Calendar Girls144 Pirates of the Caribbean192 The Last Samurai176 Touching the Void128 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 Lord of the Rings160 Calendar Girls144 Pirates of the Caribbean192 The Last Samurai176 Touching the Void128 72° 65° 86° 79° 58° Lord of the Rings Calendar Girls Pirates of the Caribbean The Last Samurai Touching the Void 360°

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LJR March 2004 Impossible UnlikelyEven chanceMost likelyCertain 0 1 ½ Probability is a measure of chance between 0 and 1. Probability of an impossible event is 0. Probability of a certain event is 1.

LJR March 2004 The probability of throwing a 3 is 1 out of 6 The probability of throwing an even number is 3 out of The probability of choosing a 5 of diamonds from a pack of cards is 1 out of 52

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