LJR March 2004 LJR March 2004 LJR March

LJR March 2004

LJR March 2004

LJR March 2004

LJR March 2004

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

Internal angles 180° 3 = 60° 120° External angles 180° - 60° = 120° Internal angles 360° 4 = 90° External angles 180° - 90° = 90° 90° LJR March 2004

Internal angles 540° 5 = 108° External angles 180° - 108° = 72° 108° 72° Internal angles 720° 6 = 120° External angles 180° - 120° = 60° 120° 60° LJR March 2004

Sides of Polygon Angle total Internal angles External angles 3 180° 60° 120° 4 360° 90° 5 540° 108° 72° 6 720° 120° 60° n (n-2)180° n 180°- (n-2)180° n LJR March 2004

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

Draw a regular pentagon of side 6 cm. Sketch to identify angles 6 cm 72º 108º 72º Use this information to accurately draw the pentagon. LJR March 2004

Draw a rhombus with diagonals 8 cm and 6 cm. Sketch first now draw 4 cm 3 cm 3 cm 8 cm 4 cm 6 cm LJR March 2004

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Hy s nu te po b c e a 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

Calculating the hypotenuse. Examples: c 2 = a 2 + b 2 Calculate x in these two triangles. 9 cm x 8 m 7 cm x to 1 dp 6 m 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

Calculating a shorter side. Examples: c 2 = a 2 + b 2 Calculate x in these two triangles. x 5 m 24 cm m c 27 13 m x to 2 dp LJR March 2004

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

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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 50 mph. 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

S = D T = D S S T D = S x T LJR March 2004

Problem: Stewart walks 15 km in 3 hours. D Calculate his average speed. S T Stewart covers 15 km in 3 hours So his average speed is 15 3 = 5 km/h Speed = 5 km/h distance covered Average speed = time taken LJR March 2004

D S T 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 = 33 km Distance = average speed X time taken D = S X T LJR March 2004

D S T Problem Paul drives 144 kilometres at an average speed of 48 km/h. How long will the journey take? He drives 48 km in 1 hour. 144 48 = 3 (there are three 48 s in 144) So the journey takes 3 hours. Time = 3 hours. distance covered Time taken = average speed D T = S LJR March 2004

D S T D = S X T Problem A car travelled for 2 hours at an average speed of 90 km/h. How far did it travel? D = 90 km X 2 hours = 180 km The car travelled 180 km LJR March 2004

D Problem A car on a 240 km journey can travel at 60 km/h. S T How long will the journey take? D T = T S = 240 km 60 = 4 hours The journey will take 4 hours LJR March 2004

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LJR March 2004

R us i ad Diameter LJR March 2004

Circumference • The formula for the circumference of a circle is: C = d where C is circumference and d is diameter LJR March 2004

Area • The formula for the area of a circle is: A = r 2 where A is area and r is radius LJR March 2004

Calculate the circumference of this circle. 8 cm C = d = 3· 14 8 = 25· 12 cm An approximation for is 3· 14 LJR March 2004

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

LJR March 2004

• Calculate the diameter and radius of a circle with a circumference of 157 m. C = d 157 = 3· 14 d d = 157 ÷ 3· 14 d = 50 m r = 50 ÷ 2 = 25 m LJR March 2004

• Calculate the radius and diameter of a circular slab with an area of 6280 cm 2. A = r 2 6280 = 3· 14 r 2 2 r = 6280 ÷ 3· 14 = 2000 r = 2000 = 44· 72135955 44· 7 cm d = 89· 4 cm LJR March 2004

Composite Shapes • Calculate the Perimeter of this shape C = d = 3· 14 9 9 m = 28· 26 12 m 28· 26 2 = 14· 13 m Perimeter = 14· 13 + 12 + 9 = 47· 13 m LJR March 2004

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

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LJR March 2004

height base Example: Calculate the area of this triangle. 9 cm 4 cm 7 cm LJR March 2004

LJR March 2004

Example: Calculate the area of these shapes. 3 m 4 m 8 m 6 m 9 m LJR March 2004

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

height base Example: Calculate the area of this parallelogram. 5 cm 8 cm 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. 11 cm 10 cm A B 6 cm Area A = 10 × 6 = 60 Area B = 6 × 5 = 30 90 cm 2 6 cm LJR March 2004

12 m Area A = 12 × 11 = 132 11 m Area B = × 6 × 11 = 33 A B 165 m 2 18 m A shape can be split into as many parts as necessary. LJR March 2004

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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 7 6 4 8 2 9 6 2 2 4 6 6 7 8 9 3 3 2 1 7 3 4 5 0 3 0 1 2 3 3 4 5 7 4 1 5 4 8 2 5 4 1 2 4 5 5 8 5 1 2 0 1 5 0 1 1 2 Leaf n = 25 Stem 4|2 means 42 LJR March 2004

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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: LJR March 2004

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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 10 n where 1 a 10 and n is an integer. Positive or negative whole number. LJR March 2004

Normal numbers to scientific notation Examples 700000 decimal point moves 9 places left 530000000 decimal point moves 8 places left 4710000 decimal point moves 6 places left LJR March 2004

You do not need to remember this but it is the reason why we can write small numbers as follows. Examples 0 000008 decimal point moves 10 places right 0 000000692 decimal point moves 7 places right LJR March 2004

Scientific notation to normal numbers. Examples 500000 decimal point moves 9 places right 4700 decimal point moves 3 places right 389000 decimal point moves 5 places right LJR March 2004

Scientific notation to normal numbers. Examples 0 0000004 decimal point moves 7 places left 0 00016 decimal point moves 4 places left 0 0000254 decimal point moves 5 places left LJR March 2004

You must be able to enter and understand scientific notation on a calculator. 4 To enter 3 To enter 7 EXP • 1 EXP +/- 4 On a calculator display 4 1007 3· 1 10 -04 LJR March 2004

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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

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

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Letters are used to represent missing numbers. Expressions An expression contains letters and numbers. x + 3, 2 t – 5, 7 + 4 y 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 =4+3 =7 (ii) 5 x – 7 = 20 – 7 = 13 LJR March 2004

Find an expression for the number of matches in design x. 5 9 13 4 An expression for the no. of matches in design x is 4 x + 1 If a = 3, b = 0, c = 5 and d = 7 find the value of the following expressions (i) 4 b + 2 d (ii) 3 c – 2 a + 5 d = 0 + 14 = 15 – 6 + 35 = 14 = 44 LJR March 2004

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Example: Solve LJR March 2004

Example: Solve LJR March 2004

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Here is some information (or data) – imagine it is a set of test marks belonging to a group of children 19 21 20 17 24 20 16 20 18 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 24 20 16 20 18 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

19 21 20 17 24 20 16 20 18 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 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

Temperatures in ºC 13 13 11 14 17 19 18 11 13 13 14 17 18 19 The m ea sum of n of a set of data is al t the nu l the values mber o divided he f value by s. Unlike th mean u e median and se mode, t gives u s every piec e of da he s an id ea ta. It happen if ther of what woul d e were equal s hares. The sum of the values is 105. The number of values is 7. The mean is 105 7 = 15 LJR March 2004

This set of data shows shoe sizes. Find the mean, median, mode and range. 3 4 6 3 7 5 4 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 sum of the values is 41. The number of values is 9. The mean is LJR March 2004

Here is another set of data. Find the mean, median, mode and range. 19 21 20 17 22 18 28 27 17 18 19 20 21 22 27 28 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 20 • 5 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 0 13 1 21 2 11 11 X 2 goals = 22 3 8 8 X 3 goals = 24 4 6 6 X 4 goals = 24 The sum of the values is: 13 X 0 goals = 0 21 X 1 goal = 21 So the sum of the values is: 0 +21 + 22 + 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 2 dp LJR March 2004

Marks out of 10 Frequency of marks 5 2 6 6 7 9 8 10 9 7 In ge ne there ral, when a data, re n piec es t the v he median of al i +1) te ue of the s ½(n rm. The mode is 8 because this test mark has the highest frequency. The range is 9 – 5 = 4. The value of the range is 4. The total frequency is 34. The median is ½(n +1) value so ½(34 +1) = ½(35) = 17 • 5 The 17 th value is 7 and the 18 th value is 8. The median is 7 • 5 LJR March 2004

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Scatter graphs are used to identify any correlation between two measures. Graph the following data taken from a class of S 3 students. Height (cm) Shoe size 125 130 135 140 145 150 155 160 165 175 2 4 3 5 6 7 8 10 9 11 Does this show a connection between height and shoe size? LJR March 2004

Height and shoe size in S 3 13 12 11 10 This graph shows a strong positive correlation. Shoe size 9 8 7 6 5 4 3 2 1 120 130 140 150 160 170 180 Height (cm) LJR March 2004

28 Exam Marks & Attendance 26 24 22 Exam mark 20 This graph shows a strong negative correlation. 18 16 14 12 10 8 6 4 2 0 2 4 6 8 Absence (in days) 10 12 14 LJR March 2004

Exam marks & Height 28 26 24 22 Exam mark 20 This graph shows no correlation. 18 16 14 12 10 8 6 4 2 0 120 130 140 150 160 Height (cm) 170 180 LJR March 2004

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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 2 x 2 = 23 x 3 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 =2 x 3 x 3 x 2 x 2 = 23 x 3 2 2 x 2 = 23 3 x 3 = 32 LJR March 2004

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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 You can take as many steps as you need 32: 56 16: 28 8: 14 4: 7 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 5 x 3 = 15 15 : 12 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: 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 sweets 25 + 20 = 45 The friends receive 25 and 20 sweets each. LJR March 2004

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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 A reflex angle is greater than 180° but less than 360° A Reflex ABC = 280° B 80° C LJR March 2004

Two lines are perpendicular if they intersect at right angles. A line parallel to the earth’s horizon is horizontal. A line perpendicular to a horizontal is called vertical. LJR March 2004

A D If two angles make a right angle they are said to be complementary. B ABD and DBC are complementary C D If two angles make a straight angle they are said to be supplementary. ABD and DBC are supplementary A B C LJR March 2004

A B D 60° ABD = 90° – 60° = 30° D C DBC = 180° – 40° = 140° A B 40° C Vertically opposite angles are equal. B A • D * E * • C AED = BEC AEB = DEC LJR March 2004

Angles and parallel Lines When two parallel lines are involved F and Z shapes can be used to calculate angles. K A B C E 80° L D G F H Parallel lines (F shape) so HFG = 80° M P N 65° R Q S Parallel lines (Z shape) so PQM = 65° LJR March 2004

Fill in all the missing angles. 117° 63° 63° 117° 180° - 63° = 117° LJR March 2004

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LJR March 2004

Identify the 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

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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

Lord of the Rings 160 Calendar Girls 144 Pirates of the Caribbean 192 The Last Samurai 176 Touching the Void 128 Touching the Void 16% Lord of the Rings 20% 22% The Last Samurai 18% Calendar 24% Girls Pirates of the Caribbean 100% LJR March 2004

Lord of the Rings 160 Calendar Girls 144 Pirates of the Caribbean 192 The Last Samurai 176 Touching the Void 128 Touching the Void 58° Lord of the Rings 72° 79° The Last Samurai 65° Calendar 86° Girls Pirates of the Caribbean 360° LJR March 2004

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Probability is a measure of chance between 0 and 1. Probability of an impossible event is 0. Probability of a certain event is 1. ½ 0 Impossible Unlikely Even chance 1 Most likely Certain 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 6 5 5 The probability of choosing a 5 of diamonds from a pack of cards is 1 out of 52 LJR March 2004

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