Standard Grade Prelim Revision The topics we will be revising are: 1. Wednesday 17th November – Pythagoras 2. Monday 22nd November – Statistics 3. Tuesday 23rd November – Volume 4. Wednesday 24th November – Area 5. Thursday 25th November – Time / Distance / Speed 6. Monday 29th November – Linear Equations (Finding a Rule) 7. Tuesday 30th November – Probability 8. Wednesday 1st December – Calculating % (Non-Calculator) 9. Thursday 2nd December - Fractions NOTE: We will try and stick to this schedule as far as we can. Supported Study can and should be used for any other revising MONDAY – THURSDAY 3:30PM – 4:45 PM

Right Angled Triangles Standard Grade Prelim Revision Pythagoras’ Theorem Pythagoras’ Theorem can ONLY be used on Right Angled Triangles

Pythagoras’ Theorem Standard Grade Prelim Revision The Hypotenuse The name given to the longest side on a Right Angled Triangle, and the side opposite the right angle is called? The Hypotenuse

a² + b² = c² Pythagoras’ Theorem Standard Grade Prelim Revision c a b Pythagoras’ Theorem allows us to calculate a missing length of a right angled triangle when we know 2 of its lengths. The formula is c a² + b² = c² a b

Pythagoras’ Theorem Standard Grade Prelim Revision Example : Calculate the length of the hypotenuse on each of the 3 right angled triangles below. 3.7cm 8cm 5cm 2.5cm 4cm 11cm

Pythagoras’ Theorem Standard Grade Prelim Revision Is this triangle right angled? Explain with working 30cm 12cm 24cm

Pythagoras’ Theorem Standard Grade Prelim Revision Pythagoras’ theorem can also be used to calculate one of the smaller sides of a right angled triangle.. If a² + b² = c² Then by using balance method c a a² = c² - b² or b b² = c² - a²

Pythagoras’ Theorem Standard Grade Prelim Revision Calculate the lengths of the missing sides of these right angled triangles 15cm 4cm 9cm 24cm 11cm 8cm

Pythagoras’ Theorem Standard Grade Prelim Revision A rectangular picture measuring 540mm by 200mm is placed diagonally in a cuboid shaped box as shown The box has 530mm by 200mm, Calculate the height of the box.

Pythagoras’ Theorem Standard Grade Prelim Revision What to remember : 1. Pythagoras can only be used on Right Angled Triangles. 2. If you are given a question with 2 given lengths and a right angled triangle is involved, Pythagoras will be required. 3. When calculating a smaller side of a right angled triangle, always take the other smaller side away from the hypotenuse

Starter Questions Standard Grade Prelim Revision BEANS 10cm The cardboard box above contains 4 cans of baked beans (below). Calculate the volume of one can of beans. Calculate the volume of the box 5cm

Statistics Standard Grade Prelim Revision Construction of Scattergraph When two quantities are strongly connected we say there is a strong between them. correlation A is a line that leaves roughly half of the points on one side of the line, and roughly half of the points on the other. best fit line Best fit line x x Best fit line Strong positive correlation Strong negative correlation

Statistics Standard Grade Prelim Revision Construction of Scattergraph Draw in the best fit line Standard Grade Prelim Revision From the best fit line, estimate the value of a car aged 5 years Statistics Construction of Scattergraph Is there a correlation? If yes, what kind? Age Price (£1000) 3 1 2 4 5 9 8 7 6 Strong negative correlation

Mean (Average) Find the mean of the set of data 1, 26, 3, 1, 2, 1, 1 Can you see that this is not the most suitable of averages since five out of the six numbers are all below the mean of 5

Median (Average) = 6.5 Median = 1, 4, 5, 8, 10, 13 Median = 5 + 8 2 Find the median of the set of data 1, 13, 5, 8, 10, 4 Median = 1, 4, 5, 8, 10, 13 The median lies between 5 and 8 so…….. Median = 5 + 8 = 6.5 2

Mode (Average) Find the mode of the set of data 1, 26, 3, 1, 2, 1, 1 Mode = 1

Range Find the median of the set of data 1, 26, 3, 1, 2, 1, 1 Range = 26 – 1 = 25

Different Averages Example : Find the mean, median, mode and range for the set of data. 10, 2, 14, 1, 14, 7

Sum of Tally is the Frequency 12, 14, 12, 17, 9, 19, 21, 12, 22 Frequency tables Raw data can often appear untidy and difficult to understand. Organising data into frequency tables can make it much easier to make sense of the data. Data Tally Frequency llll represents a tally of 5 Sum of Tally is the Frequency

Frequency tables Diameter Tally Frequency 56 57 58 59 60 61 62 lll llll 3 4 9 13 10 5 4 From the data, we can then calculate the Range, Mode and Median Range = Largest - Smallest Mode = Most common number = 62 - 56 = 59 = 6

Frequency tables 59 Diameter Tally Frequency 56 57 58 59 60 61 62 lll llll 3 4 9 13 10 5 4 The median is harder to calculate…….. To calculate the median in a frequency table we add each frequency up……. 3 + 4 + 9 + 13 + 10 + 5 + 4 = 48 Then divide by 2…… 48 ÷ 2 = 24 ….therefore the median is the 24th value 59

We call this column the cumulative frequency column Frequency Tables Working Out the Mean Example : This table shows the number of brothers and sisters of pupils in an S2 class. No of Siblings (S) Freq. (f) S x f Adding a third column to this table will help us find the total number of siblings and the ‘Mean’. 9 0 x 9 =0 1 13 1 x 13 = 13 2 6 2 x 6 = 12 Total Cumulative frequency ÷ total Frequency column 3 1 3 x 1 = 3 5 1 5 x 1 = 5 Totals 30 33

Starter Questions Q1. Factorise a) 48 – 12s b) 3t + 27t c) 9x + 54 Q2. Calculate the height (h) of the tower h 48º 452m 23

A short cut ! Volume = 6 x 3 x 4 = 72 cm³ Volume = length x breadth height Area of rectangle breadth length Volume = 6 x 3 x 4 = 72 cm³ Volume = length x breadth x height

Example 1 27cm 5 cm 18 cm Working Volume = l x b x h V = 18 x 5 x 27 Heilander’s Porridge Oats V = 18 x 5 x 27 V = 2430 cm³ 27cm 5 cm 18 cm

Example 2 Working Volume = l x b x h V = 2 x 2 x 2 V = 8 cm³ 2cm

Liquid Volume Volume = l x b x h = 1 cm³ I’m a very small duck! How much water does this hold? 1 cm 1 cm 1 cm Volume = l x b x h = 1 cm³ A cube with volume 1cm³ holds exact 1 millilitre of liquid. A volume of 1000 ml = 1 litre.

Volume of a Cylinder Volume = Area x height = πr2 = πr2h h x h The volume of a cylinder can be thought as being a pile of circles laid on top of each other. Volume = Area x height h = πr2 x h Cylinder (circular Prism) = πr2h

Volume of a Cylinder V = πr2h = π(5)2x10 = 250π cm Example : Find the volume of the cylinder below. 5cm Cylinder (circular Prism) 10cm V = πr2h = π(5)2x10 = 250π cm

Surface Area of a Cylinder 2πr h Total Surface Area = 2πr2 + 2πrh The surface area of a cylinder is made up of 2 basic shapes can you name them. Cylinder (circular Prism) 2πr Curved Area =2πrh Top Area =πr2 Roll out curve side  h Bottom Area =πr2 2 x Circles Rectangle Total Surface Area = 2πr2 + 2πrh

Surface Area of a Cylinder = 2π x (3 x 3) + 2π x 3 x 10 Example : Find the surface area of the cylinder below: 3cm Surface Area = 2πr2 + 2πrh 10cm = 2π x (3 x 3) + 2π x 3 x 10 = 2π x 9 + 2π x 30 Cylinder (circular Prism) = 245.04cm2

Surface Area of a Cylinder Radius = 1diameter 2 Example : A net of a cylinder is given below. Find the curved surface area only! 6cm Curved Surface Area = 2πrh Rectangle Only!! = 2 x π x 3 x 9 9cm = 169.64 cm2