Year 8 Foundation – Unit 1 -Number

Adding and Subtracting Negatives Factors and MultiplesHCF and LCM using Prime Factors Powers Roots, cube roots

Next Steps

Try some GCSE or wordedquestions that require you to identify whether the LCM or the HCF is required.Find HCF and LCM with VennDiagrams

Powers/Indices and Roots

Powers: Squaring: 32 = 3 x 3 = 9Cubing: 43 = 4 x 4 x 4 = 64To the power 4: 5⁴ = 5 x 5 x 5 x 5 = 625Remember: a° = 1Roots:Square root - √121 = 11 or -11 Cube root – 3√64 = 4 (because 4x4x4 = 64)

Key words

Multiple- appears in times tablesFactor- A whole number that ‘goes’ into another (by multiplying it by another whole number)HCF -The biggest number that divides exactly into both numbersLCM - two numbers is the smallest number that appears in the multiplication tables of both numbers. Indices/ Powers- numbers that tell us how many times a number has to be multiplied by itself Product – MultiplyPrime Number - can be divided evenly only by 1 or itself.Roots -the original value that has been squared or cubed to get the result

Multiple or Factor?

Multiples of 10: 10, 20, 30, 40, 50, 60 etc.Factors of 10:1, 10 2, 5

1 and 10 are a factor pair of 10 because 1 x 10 =

10

Calculating with Negative Numbers

Adding and subtracting negative numbers:e.g. 10 + - 3 = 710 - - 9 = 19-9 + - 3 = -12 - 7 - - 3 = - 4

Multiplying and dividing negative numbers: e.g.

7 x – 4 = - 28- 6 x – 4 = 24- 35 ÷ 5 = - 7-40 ÷ -8 = 5

HCF and LCM using prime factors

We do this by using a prime factor tree. The number s repeatedly split into factor pairs until only prime numbers are left at the end of the branches. →

The HCF of 24 and 60 is found by comparing the prime factors and multiplying the numbers common to both lists.

So the HCF of 24 and 60 is 2 x 2 x 3

= 12.

The LCM of 24 and 60 is found by multiplying together all the prime factors of both numbers. However if a number occurs in both lists we only count it once.

So the LCM of 24 and 60 is2 x 2 x 2 x 3 x 5 = 120.

Hint: This sum is easy if you first do 2 x 5 = 10 then 2 x 2 x 3 = 12. Then 12 x 10 =120.

Year 8 Foundation – Unit 2 –Circles

Parts circle, Calculate radius and diameter, Circumference, Area of a circle and semi-circle,

Next Steps

Calculate perimeter and area of compound shapes with circlesFind the arc length Find the sector area

Key Words

Circumference – the perimeter of the circle Diameter – the line from one end of the circumference to the other that crosses the centre (radius x 2) Radius – a line from the centre to the circumference . It is half of the diameter (diameter ÷ 2)

Segment – a line running across the4 circle creates two of themChord – a line that cuts across the circle and doesn’t touch the centreTangent – a line that touches the outside of the circle at one point Arc – a section of the circumferenceSector – a slice of a circle made of two radii.

Semi Circles

Area of a = πr² / 2semi-circle = (π x 3²)/2

= 14.1cm² (1d.p)

3cm

Radius and Diameter

Radius = 5cmDiameter = 5 x 2

= 10cm

Diameter = 12cmRadius = 12 ÷ 2

= 6cm

Year 8 Foundation – Unit 3 -Sequences

Fibonacci sequence Triangle NumbersArithmetic sequences (nth term)Substitution into nth term

Next Steps

Find the nth term of the quadratic sequences

Key Words

Fibonacci - series of numbers:0, 1, 1, 2, 3, 5, 8, 13, 21, 34, ...The next number is found by adding up the two numbers before it.Triangle Numbers - This sequence comes from a pattern of dots that form a equilateral triangle: 1, 3, 6, 10, 15, 21Linear Sequence -A number pattern which increases (or decreases) by the same amount each timeTerm-to-term rule- how you get from one term of a sequence to the nextSubstitute- to put in place of another

Fibonacci Sequence

The Fibonacci sequence is a series of numbers that begins with 1, 1 and from then on each term is formed by adding the two terms before it.1, 1, 2, 3, 5, 8, 13, 21, 34. 55

1 + 1 = 21 + 2 = 32 + 3 = 53 + 5 = 8 etc.

Identify sequences

2,4,6,8,10 – evens1, 4, 9, 16, 25 – square numbers1, 8, 27, 64, 125 – cube numbers2, 3, 5, 7, 11 – prime numbers

Finding the nth term of a linear sequence

A linear sequence increases or decreases by thesame amount each time.

The nth term rule is a way of finding any term ina linear sequence, where n is the position in the sequence.How to find the nth term of a sequence:1) Find the common difference or term-to-term rule

2) Write out the times tables of the term-to-term rule above you sequence. Find the difference between this and the original sequence.

e.g.

The term-to-term rule is + 5.The first term of the sequence is 2, the

second term is 7 etc.

This is what we put in front of the n

5n

5 10 15 20 25 30 35 -3 -3 -3 -3 -3 -3 -3

We the add or subtract this:

5n - 3

If the term-to-term rule is negative, you would have a negative number in front of n, and your times tables would be negative.

Substitution into the nth term rule

If you want to find any term of a linear sequence from the nth term rule, you substitute in the term you would like to find e.g. 50th term, n = 501000th term, n = 1000You can then substitute your value of n into the nth term rulee.g. what is the 25th term of a sequence with an nth term rule of 5n-4n= 255 x (25) – 4 = 125 – 4 = 121

Triangle Numbers

Triangle numbers come from a sequence of dots that form an equilateral triangle. The first five triangle numbers are:1, 3, 6, 10, 15, 21

Arithmetic Sequence

In an Arithmetic Sequence the difference between one term and the next is a constant.In other words, we just add the same value each time ... infinitely.1, 4, 7, 10, 13, 16, 19, 22, 25, ...

This sequence has a difference of 3 between each number.

Year 8 Foundation – Unit 4 –Geometry 1

Properties of quadrilaterals,Interior/ Exterior angles of regular polygons,Translations with vectors

Next Steps

Try questions including algebra

Key Words

Polygon - is any flat shape with straight sidesIrregular Polygon - does not have all sides equal and all angles equalQuadrilaterals – just means "four sides", it is 2D shape, closed (the lines join up), and has straight sides.Interior angles – are the angles inside the shapeExterior – the angle formed outside the shape, when a side is extended Translation - simply means moving without rotating, resizing or anything elseVector (for translations) – the movement along the x and y axis, like a coordinate

Properties of quadrilaterals

Interior angles in a quadrilateral add up to 360°

Square Rhombus

Parallelogram Kite

Trapezium

a + b = 180° and c + d = 180°

Angles in Polygons

For an n-sided regular polygon:Sum of Interior angles: 180(n-2)°Each interior angle: sum/nSum of Exterior angles: 360°Each Exterior angles: 360°/n

Example: Hexagon (6 sided shape, n=6)Sum of Interior: 180 x (6-2) ° = 720°Each Interior: 720°/6 = 120°Sum of Exterior: 360°Each Exterior: 360°/6 = 60°

Translations

Every point of the shape must move the same distance and in the same directionVector(-12) A has moved 12 spaces left (-12) A has moved 12 spaces down

(x) : movement right (+) and left (-) (y) : movement up (+) and down (-)

Rotation

There are 3 key parts of information:1. Centre of rotation2. Direction: clockwise or

anticlockwise3. Amount of turn: 90, 180,

270, 360

Rotation of 90° clockwise

> and >> show parallel sides

Exterior Interior

A

A’

Quadrilateral Example

Step 1 – identify The shape Trapezium Step 2 – Apply rulesOf the shape A + 70 = 180B + 125 = 180 Step 3 = find the angles A = 180 – 70

= 110°B = 180 – 125

= 55°

Check you have the right answer by adding all the angles together to get 360°70+125+110+55 = 360°

Step 1 – KiteStep 2e = fe + f + 112 + 60 = 360Step 3 Because e = f

e + e + 112 + 60 = 3602e + 112 + 60 = 3602e = 188e = 94 therefore f = 94°

Angle bisectors

Next Steps

Accurately construct a 30o angle without the use of a protractor

Key words

Construct = Accurately drawBisector – Cut throughQuadrilateral – four sided shape

Year 8 Foundation – Unit 4 – Geometry 2

Construct accurately circles, perpendicular bisectors, angle bisectors.Construct the locus of the pathAngles in parallel lines and angle facts

Perpendicular bisector -A line which bisects (cuts in half) another line, and is perpendicular (at right

angles) to that line.

The path of a point that moves according to a certain rule is a locus. The plural of locus is loci.

Angles in Parallel Lines

Angle Facts

Angles vertically opposite are the same so a = b

Volume of a cuboid

Surface Area of a CubeSurface Area of a CuboidYear 8 Foundation – Unit 5 –2D & 3D Shapes

Area and perimeterArea of a trapeziumCompound shapesSurface area of cubes and cuboidsVolume of cuboids,

Volume of prisms

Next Steps

Calculating capacityCalculating volume and surface area of compound prism

Key Words

Area -The size of a surface.Perimeter - The distance around a two-dimensional shape.Volume - The amount of 3-dimensional space an object occupies. Units of volume include: Metric: cubic centimetres (cm3), cubic metres (m3), litresPrism - 3d shape with two identical ends and flat sides where the cross section is the same all along its length like a loaf of bread

Triangular Prisms

Volume: Find the cross sectional area and multiply by the depth

Surface Area: Find the area of all faces and add together

1) Find cross sectional area½ x 8 x 3 = 12cm2

2) Multiply by the depth12 x 12 = 144cm3

1)Find the area of all facesFront face = 12cm2 (x 2 as back face the same)Bottom = 12 x 8 = 96cm2

Rectangular faces = 5 x 12 = 60cm2(x 2 as two faces)2)Add together 24 + 96 + 120 = 240cm2

5cm

A triangular prism has 5 faces – make sure you have found them all

Area

Area and Perimeter

Year 8 Foundation – Unit 6 – Graphs

Graphs of linear equations Work out a gradient, y = mx + c,Draw graphs of simple quadratic equations, Draw real life graphs. Key Words

Linear = A straight line graphQuadratic = Means that the graph has an x2

Quadratic graph = curved graphParabola = A U shape Intercept = Where a graph crosses the y axisGradient = How steep a graph is

So the graph of y = 2x + 5 will cross the y axis at + 5 and will have a gradient of 2.Key Point - If two lines are parallel they will have the same gradient.

Gradient

The gradient can be drawn or calculated by going one square across and seeing how many squares you go up or down. For example on the blue graph on the left, from 5 on the y axis, go across 1 and you will see it goes up 2 squares; so the gradient is 2 (m = 2). Gradient = difference in y/difference

in xThe orange graph is x = 2 because the coordinates of any point on the line will be (2, ?) for example, (2, 1)It has a gradient of 0 (m = 0)

X = 2

Simple Quadratic Equations

A function which involves x² is called a quadratic function which create a graph of a ‘U’ shape (parabola). Equation: y = x²

X -3 -2 -1 0 1 2 3

Y = x² 9 4 1 0 1 4 9

Note:(-3)² = (-3) x (-3)

= 9-ive x –ive = +ive

Next Steps

Solving equations graphically

Drawing GraphsStep 1 – Draw a

table upStep 2 – Plot co-

ordinates on axesStep 3 - Join the

points

Real Life Graphs -Distance/time

The most common real life graph showsdistance travelled (on the y-axis) against time (on the x-axis).

The gradient (or slope) of the graph represents the speed.

Percentage (Non Calc)

Year 8 Foundation – Unit 7 –Percentage

Percentage of an amount (cal and non cal)Calculate percentage change, Percentage increase / decrease using multipliersSimple interest

Next Steps

Look at compound interest Appreciate the multiplicative nature of percentage increase and decrease.

Key Words

Percentage - is the "result obtained by multiplying a quantity by a percent". Multiplier – a number that is used to find the result of increasing or decreasing an amount by a percentageInterest – a percentage of money added onto savings or debt Simple interest - is when the same percentage of the original amount is paid each year.

Percentage increase and decrease

Increase £30 by 6% Step 1 Find the multiplier. 100% plus the extra 6%. This is a total of 106%.Convert into a decimal to get the multiplier106/100 = 1.06.Step 2 Multiply the amount by the multiplier30 x 1.06 = £31.8

Decrease 50kg by 7%Step 1 7% less than the original 100%,so it represents 100% – 7% = 93% of the original.

This is a multiplier of 0.9 (93/100)Step 2 50 x 0.93 = 46.5kg

Percentage Change

A percentage change may be a percentage increase or a percentage decrease.

There were 160 smarties in the box, but now there are 116, what is the percentage change? 44 x 100 = 27.5 27.5% decrease

160

Simple Interest

Simple interest pays the same interest each year.Elizabeth has £400 to invest. Account A pays 6.5% simple interest if she leaves her money in the account for 3 years. 6.5% of £400 = £263 x £26 = £78After 3 years, she will have £400 + £78 = £478

Always refer to the question to check if the amount has increase or decreased

Percentage of Amount No Calc/Calc

Convert the percentage to a decimal, eg 10% = 0.1 <= This is your Mulitplier

Do AMOUNT x Multiplier

0.1 x 54 = 5.4 kg. This is easier for random percentages that are not 1,5,10,25,50 etc.

Amount as % of Another &Percentage Change

If they are not in the same units, convert so they are

If the amount being expressed is smaller <100%If the amount being expressed is larger >100%

Percentages to Fractions/Decimals

Percent to Decimal – Divide by 100Decimal to Percent – Multiply by 100Percentage to Fraction:Put percentage over 100, simplify if poss.Fraction to Percentage: Either-- Try to get the denominator to 100- Bus stop to divide the fraction (to get a

decimal) and then x100

Year 8 Foundation –Unit 8– Interpreting

Data

Interpret pie chartsConstruct pie charts by scaling methodUse scatter graphs to estimateConstruct scatter graphs. Describe strength of correlation

Key Words

Pie Chart – a representation of information in circle form divided into sectors, in which the relative sizes of the areas of the sectors correspond to the relative sizes of the quantitiesSector – A piece of a circle that looks like a slice of pie. Its mathematical definition is ‘a plane figure bounded by two radii and the included arc of a circle.’Scatter graphs are used to show whether there is a relationship between two sets of data. Points are plottedone by one on the graphCorrelation between two sets of data means they are connected in some way Line of Best Fit – a straight line that best represents the data

Pie Charts

Scatter Graphs

The sets of data are on separate axis’ and you plot the points one by one on the graph.

Positive correlation as one quantity increases so does the other

Negative correlation as one quantity increases the other decreases

No correlation where both quantities vary and there is no clear relationship.

We can estimate values with a line of best fit, for example:As temperature increases so does the sales of ice cream generally. At 20°C there would be approximately $450 worth of sales.

Next Steps

Comparing sets of data, to know which chart to use for different data sets

Year 8 Foundation – Unit 9 – Expressions

Simplify expressions (with powers)Use expressions with area of compound shapesExpand single brackets and simplifyFactorise (numbers)

Next Steps

Expand and factorise double brackets

Key Words

Expand – Removing the brackets by multiplying out.Factorise – Adding brackets back in by dividing out common factorsSimplification – An easier, more efficient or more attractive way to write an expression a + 2b + 4c - 3a + 6c = -2a + 2b + 10cRoots - where it crosses the x-axisIntercept - where it crosses the y-axis

Simplify

Only collect terms with same powers together

2x² + 3x + 7y² - x² - 6y = x² + 3x + 7y ² - 6y

Expanding brackets

Factorising

This is the opposite of expanding out brackets,to put them back in.By extracting a highest common factor we can factorise an expression, say 9y + 81.The biggest number to fit into 9 and 81 exactly (highest common factor) is 9. This goes on the outside of the bracket.9 (y + __) To get to 81, it is 9 x 9 so:9y + 81 = 9 (y + 9)

Area Expression

A

B

Area of A = 3 x y = 3ycm²

Area of B = 6 x 4= 24cm²

Total Area = 3y + 24 cm²

6

Year 8 Foundation – Unit 10 –Congruence

Recognise congruent shapesCongruent Triangles using SSSDecide/Prove congruence by working out missing angles.Express simple scale diagrams with a ratioSimplify scales.Scale on Drawings and Maps.

Next Steps

Discover similar shapes

Key Words

Congruent – shapes that arethe same size and angles Scale Drawing - A drawing that shows a real object with accurate sizes reduced or enlarged by a certain amount (called the scale which is represented by a ratio)

Congruent Shapes

Congruent shape have the same size. Reflection, rotations and translations produce congruent shapes. For example:

Scale drawings

Scale drawing allows us to draw large objects on a smaller scale while keeping them accurate.All scale drawings must have a scale written on them. Scales are usually expressed as ratios.Example: 1cm : 100cmThe ratio 1cm:100cm means that for every 1cm on the scale drawing the length will be 100cm in real life

Map scales

Some maps have scales on. This means that they are drawn in proportion. You can calculate the real distances between things by using the scale. E.g. on the picture below, for the distance between Raleigh and Charlotte, draw a line between them and measure how many centimetres apart they are. Then multiply this by 75, because the scale shows 1cm : 75km. This then tells you how many km apart they are.

The ratio for this scale drawing would be 8:32 for the length Simplify 8:32

1:4

Congruent Triangles

Congruent Triangles - Three sides equal (SSS) Triangles are congruent if all three sides in one triangle are congruent to the corresponding sides in the other.

7cm

7cm

Year 8 Foundation – Unit 11 – Fraction and Decimals

Adding and subtracting fractions,Multiply a fraction by an integer Divide with unit fractionsConvert fractions to decimalsMultiply and divide decimals.

Next Steps

Convert recurring decimals into fractionsMultiplying fractions and fractions

Dividing fractions by fractions

Add and Subtract Fractions

Golden rules1. The denominators must be the

same2. Only add/subtract the numerators3. Whatever you do the denominator

you must do to the numerator

Example – same denominators 3 - 2 = 17 7 7

Example – different denominators 1 + 12 3Step 1 - Make the denominators the sameFind the LCM (lowest common multiple) of 2 and 3, which is 6.Make 6 the new denominator 1 x3 + 1 x2 = 3 + 22 x3 3 x2 6 6Step 2 – Add/Subtract the fractions3 + 2 = 56 6 6

Convert fractions into decimals

You convert a fraction by dividing the numerator by the denominator. Use bus stop method to complete division if necessary. 3

8= 3 ÷ 8

3

8= 0.375

Multiply and Dividing Decimals

Multiplying two decimals

0.45 x 0.6

Step 1 - Multiply both decimal numbers to

make them whole numbers

0.45 x 100 = 450.6 x 10 = 6Step 2 – Complete the calculation with the

whole numbers

45x 6270

3

Step 3 – Divide answer by the amount you multiplied the decimals to make it a whole number 270 ÷ 100 = 2.72.7 ÷ 10 = 0.27

Dividing two decimals number 3.65 ÷ 0.2Step 1 – write the division as a fraction 3.650.2Step 2 – make the decimal numbers integers (make sure you multiply the numerator by the same amount you have multiplied the denominator)3.65 x100 = 3650.2 x100 20Step 3 – do the division (use bus stop if necessary)365 ÷ 20 = 18.25

Multiplying and diving fractions

Multiplying fractions by integers

Dividing by unit fractions

There are 6 halves in 3, so 3 ÷1

2= 6

Key words

Fraction – a way of writing numbers which are not integers.Reciprocal – the reciprocal of a fraction is the original fraction turned upside downInteger – a whole number.Numerator – The top number of a fractionDenominator – the bottom number of a fractionEquivalent fractions – the same fraction written in different ways.

Unit fractions – is a rational number

written as a fraction where the numerator is one and the denominator is a positive integer e.g. ½, ¼, and 1/8 etc.

Year 8 Foundation – Unit 12 – Ratio and Proportion

Recognise a ratio and Unitary ratio.

Calculate values in direct proportion (straight

multiplication)

Draw graphs to represent direct proportion

Understand inverse proportion through the use

of speed distance time.

.

Next Steps

Look at real life examples of inverse proportion and form equations and graphs

Key Words

Ratio – A way of comparing unitsAlgebraic notation – to write in algebraDirectly proportional: as one amount increases, another amount increases at the same rate. The symbol for "directly proportional" is ∝Inversely Proportional: when one value decreases at the same rate that the other increases.

Ratio

A ratio compares values. It says how much of one thing there is compared to another thing.

For every 1 red bead there are 3 blue beads.1 to 3 -> 1 : 3 -> Red:BlueThis is called a unitary ratio (when 1:k)

Simplify ratios by finding The highest common factor

In both numbers and Dividing by that number

Direct Proportion

Example: you are paid £20 an hourHow much you earn is directly proportional to how many hours you work (Work more hours, get more pay; in direct proportion)

Earnings ∝ Hours worked

E ∝ HForm an equation (what do you multiply hours by to get earnings)

E = KH (K is constant - you multiply hours by 20 to know the earnings therefore K = 20)

E = 20HIf you wanted to know how much money 7 hours will get you, h = 7, substitute into the equation: E = 20 x 7 -> E = 140 -> £140If you earned £230, how many hours did you work? 260 = 20 x H -> H = 13 -> 13 hours

Hours 1 2 3 4 5

Earnings 20 40 60 80 100

Direct Proportion Graph

A graph of values of two values in direct proportion always has these two properties:1. The points are in a straight line.2. The line passes through the origin (0,0)(Look to left for the data)M = 1.6K

Inverse Proportion

Example: speed and travel timeSpeed and travel time are Inversely Proportional because the faster we go the shorter the time.•As speed goes up, travel time goes down•And as speed goes down, travel time goes up

The distancewould be

120km (20x6).

If you doubled your speed (20 -> 40 km/h) then the time taken halves (6 -> 3 hours)

The graph shows the numbers from the table.

The points are not in a straight line.A smooth curve has been drawn

through them.

Year 8 Foundation – Unit 13 –Simplifying Numbers

Standard form (conversions only)Multiplying and dividing by 10, 100 and 1000Rounding to powers of 10Hard estimation problemsRound to any number of significant figures

Next Steps

Work with negative and fractional powersmultiply Standard Form numbersUse Standard Form in contextual questions

Key Words

Standard form/ Standard form index- A way of writing a very small or a very large number. A number represented in the form A x 10n where A is between 1 (included) and 10 (not included) and n is an integer (whole number).Significant Figure- The first number or digit with meaningEstimate - To find a value that is close enough to the right answerInteger – a whole number

Converting numbers to and from standard form

Writing ordinary numbers as standard form.

These numbers are now written in standard form.a x 10ⁿWhere 1 ≤ a < 10n is a positive or negative integer

Standard form is often used to write very large Or very small numbers.

Significant Figures

e.g. round 3268 to 1 sfthe first significant figure is a 3, which represents 300, so we need to round to the nearest thousand

e.g. round 0.0058 to 1 sfthe first significant figure is a 5, which represents 0.005, so we need to round to the nearest thousandth 0.0058 look at the next digit, it is greater than 4 so we round up to 0.006

Rounding to powers of 10

Powers of 1010¹ = 1010² = 10 x 10 = 10010³ = 10 x 10 x 10 = 100010⁴ = 10 x 10 x 10 x 10 = 10000

Round 435 to 10¹The tens value is 30435The digit next to 3 is 5 so round the number up to 440

Round 3875 to 10³The thousands value is 30003875 The digit next to 3 is 8 so round the number up to 4000

Round 985431 to 10⁴The ten thousands value is 80000, 984431The digit next to 8 is 4 so stay at 980000

Round 9,925,631 to 10⁶The millions value is 90000009925631The digit next to 9 is 9 so round up to 10,000,000

Estimation Problems

First round to 1 sf and then calculate, remembering BIDMAS

Question708 – 43.84.32 + 3.9

Answer660 ÷ 20 = 33

Rounded700 – 4042 + 4

Calculations660 = 660

16 + 4 20

Multiply and Divide by 10, 100 and 1000

Year 8 Foundation – Unit 14–Equations and Formulae

Solve equations (2 step)Solve formulae (variable on both sides), Rearrange formulae. Forming and solving equationsSolving equations with brackets and fractionsRearrange with factorising.

Next Steps

Practice substituting into more complex formulae including powers, roots and fractions and doing the inverse.

Key Words

Formula – A formula is a rule involving two or more variables.Rearrange – To move around Subject – What the formula is about e.g. Speed - distance / time; speed is the subject.Substitute – To replace the letters with numbers.Rule – Something

2 Step Equations

To solve two step equations, you want to isolate your variable by itself. To do so, you must use your knowledge of inverse operations.

Including Fractions

b

You must do the inverse

𝑥

3+ 3 = 10

-3 -3𝑥

3= 7

x 3 x 3x = 21

𝑥+2

3= 12

x3 x3x +2 = 7

-2 -2x = 5

Variable On Both Sides

Put the variables on one side and then solve

Step 3. Solve the equation as seen previously

Forming and Solving Equations

I am think of a number. I subtract 5 from it and then divide the result by 4. The answer is 7. What number did I think of to start with? Form an equation thenx – 5 = 7

4

The triangle to the right is Isosceles.Calculate the lengths of the sides if theperimeter is 68cm.

Step 1 – form an equation by adding all the sides together2x – 1 + 2x -1 + x =68Step 2 – Simplify the equation by collect like terms 2x – 1 + 2x -1 + x =685x -2 = 68Step 3 – solve the equation as seenpreviously

Solve the equationx – 5 = 7

4x4 x4

x - 5 = 28+5 +5

x = 33

x

2x - 1

2x – 1 This side is the same

REMEMBER: perimeter is found by adding all

the sides up

Rearrange

Sometimes we will need to rearrange a formula to find the value of a subject.For example: area of a rectangle = length x width

A = L X WWe can rearrange the equation to find the length, therefore we want L as the subject (by itself) use the same steps of solving an equation A = L X W÷W ÷WA = L Therefore length is found by W dividing the area by the width

Substitution into formulae

A formula expresses a relationship between two variables, and are often used to find something e.g. the formula for the area of a rectangle is length x width (A = lw). When substituting into a formula, just replace the letters with the given values.e.g.Helen is an electrician. She uses the following formula to work out what to charge a client:C= 25 + 26h, where c is the cost in pounds and h is the number of hours she has worked. If she works 10 hours, how much does she charge?C = 25 + 26hC = 25 + 26(10) C = 25 + 260C = 285So she charged £285

Ensure you use BIDMAS to do the operations in the

correct order.

Year 8 Foundation – Unit 15 –Probability

Probability (single event)Experimental Probability Joint Events (Venn Diagrams, 2 way tables)

Next Steps

Harder probability questions with Venn diagrams – understanding and/or rules

Key words

Probability = The chance of somethinghappeningMutually exclusive = Can’t happen at the same timeOutcome = the end resultIntersection = the area that overlaps on a venn diagram

Venn diagrams

Venn diagrams represent a number of things that may happen together, e.g. having an adidas bag and blonde hair. The centre is called the intersection as it shows the amount that satisfies both conditions, e.g has blonde hair and an adidas bag. The outside shows how many do not satisfy any of the conditions e.g brown hair and a puma bag. You can use venn diagrams to calculate probability of an outcome

The intersection is where we have items both from Set A and Set B, these can be found in the section that overlaps.Probability of A and B = {6, 7, 9, 12}.Using Venn diagrams for probabilities

In the above example:P(A∩B) is 4/13.P(A∪B) is 13/13 = 1Work out P(A′), i.e. NOT A, is 5/13

2-way probability tables

A two-way table links two variables:

Sometimes you will have to fill in the gaps for the table. In total there are 38 people that prefer Basketball. Therefore to find the amount of a females, find the difference between 38 and 22 (38 -22 = 16). 16 females prefer basketball.The probability of football being someone’s favourite sport to watch on TV is 52/150 (which will cancel to 26/75)

Sample space diagrams

When you are calculating the probability of 2 things that are happening at the same time you may use a sample space diagram. For example imagine I rolled 2 dice at the same time and added together the scores. I could show it in a sample space to the right. I can then use that to calculate the probability of the outcomes.

Probability

Probability will always add up to one. So if the probability of it raining tomorrow is 0.4 then the probability of it not raining tomorrow is 0.6. This is because 0.4 + 0.6 = 1.

Probability of getting a 6 on a die =1

6

Experimental probability is when you role the dice more than once and note how many times you get each number. For example I could only get 6 twice if I rolled it 24 times, therefore the

probability would be 2

6.