Crucial Knowledge – Stage 1 - Number

BIDMAS The order you do calculations in: B rackets I ndices D ivision M ultiplication A ddition 4 Operators Negative Numbers S ubtraction • Addition (or Sum) + • Adding or subtracting – USE A • Subtraction (or Difference) – NUMBER LINE • Multiplication (or Product) x • Multiplying or dividing use the rules • ÷ + x + = + + ÷ + = + + x - = - + ÷ - = - Decimals and - x + = - - ÷ + = - • Different ways of saying part of a whole number - x - = + - ÷ - = + • You can change from one to the other

Prime Numbers Rounding • Have exactly two factors • Decimal places (column after decimal point) • No other whole numbers, except 1 and itself divide into them • Significant Figures (highest value column) Crucial Knowledge – Stage 1 - Number

Highest Common Factor (HCF) Lowest Common Multiple (LCM) Standard Form • Write down all the factors from • A way of writing very BIG or very SMALL numbers the numbers and find the biggest • Think BIG numbers – distance between planets and SMALL numbers value on both lists – This is the – sizes of atoms. Highest Common Factor • BIG numbers have POSITIVE powers and SMALL number have NEGATIVE powers. • Write down all the multiples of Always a multiply Always the number 10 Always a number Positive for very large numbers the two numbers and find the between 1 and 10 smallest on both lists – This is the 4 x 106 Negative for very small numbers Lowest Common Multiple

Percentages Fractions • An amount out of 100 • Multiplying – Multiply top by top and • To Calculate a of an amount (What is 40% of £50) bottom by bottom. Percentage ÷ 100 x amount (40 ÷ 100 x 50 = £20) • Dividing - ‘Keep Change Flip’. • To change to a percentage (you score 4 out of 5 in a test, what • Addition or Subtraction – You need same percentage is this?) bottom number (denominator). Amount you got ÷ what it is out of x 100 (4 ÷ 5 x 100 = 80%) Crucial Knowledge – Stage 2 - Number

Percentage Change Inequalities • If a value goes up, it’s a percentage increase. • Understand inequality symbols < > ≤ ≥ • If a value goes down, it’s a percentage decrease. • List values that satisfy a inequality. • We work out percentage of amount and either add it on • Show by drawing on a number line values that or subtract it from our starting value satisfy inequality. • Or we work out the percentage change by working out the difference in values and dividing by our original value Estimation and then multiplying by 100. • An answer close to the exact answer. • All values are rounded to 1 significant figure. Powers • Follow BIDMAS to get your estimation. • If we multiply powers we add. y3 x y4 = y(3 + 4) = y7 • If we divide powers we subtract. y10 ÷ y6 = y(10 - 6) = y4 Use of Calculator • Anything to the power zero is always 1 • Must be able to use brackets ( ) on calculator to get an answer to multi stage calculations. Product of Primes • Must be able to use powers on calculator. • Any value split into prime numbers MULTIPLIED together. • Must be able to use for Standard Form calculations . • First 5 prime numbers are 2, 3, 5, 7 and 11. • Must be able to use button for all multi tier • Sometimes we put into a VENN diagram to calculate LCM calculations. and HCF. • Must be able to use calculator for percentage calculations. Crucial Knowledge – Stage 3 - Number Recurring Decimals Advanced Powers Upper and Lower Bounds • A decimal with repeating values • A negative power means • Upper is slightly above your values • We indicate the repeating reciprocal (”1 over”) • Lower is slightly below your values numbers with a dot above • 4 = = • Using bounds affects calculations – you ̇ 0. 6 = 0.666666 … • A fractional powers means must find bounds before any calculations ̇ ̇ • Example: 0. 656 = 0.656656656 … find a root ̇ ̇ 0.716 = 0.7161616 … • Must be able to convert recurring • � = � � = � Q: A field measures 34m x 28m both decimals to fractions • More complicated fractions measured to the nearest metre. What is the require using powers and minimum and maximum area the field Fractions roots could have? • Mixed number to improper • 16 = 16 = 4 = 64 Upper Bound Lower Bound 1 13 4 = • Evaluate a negative fractional A: Bounds Length (34m) 34.5m 33.5m 3 3 power in this order Width (28m) 28.5m 27.5m 4 � 3 + 1 = 13 3rd 2nd 2 • Improper to mixed number Maximum Area = 34.5 x 28.5 = 983.25m 13 1 16 2 = 4 Minimum Area = 33.5 x 27.5 = 921.25m 3 3 1st • = ( 16)= 2 = 13 ÷ 3 = 4 ��������� 1 Crucial Knowledge – Stage 1 – and Proportion

Ratio as a measure Equivalent Dividing a given ratio • A ratio is a comparison of parts • Same values but different • The question matches the order of items to • Use a colon (:) to separate parts numbers the order of parts in the ratio. The first of a ratio • Values used can get larger, as thing mentioned gets the first part of the • A colon is read as ‘to’ well as smaller ratio • 2 or 3 parts • Do same to all parts • Find the total number of parts in the ratio • Understand the parts add up 3:6 (+) and stay in proportion x4 x4 • Divide the amount to be shared by the total 12:24 parts (÷) Cancelling ratios • Basic unit conversions Multiply by each part of the ratio (x • Like simplifying fractions • Convert units of length (mm, Example • Look for common factors cm, m, km) Q: Adam and Ben share £45 in the ratio 1:2. • Do the same to both parts of • Be able to convert to common Who gets how much? the ratio unit before calculating A: 1 + 2 = 3 parts in total 3:6 • Convert units of time £45 ÷ 3 = £15 per part ÷3 ÷3 1:2 1:2 • Convert units of measure (ml, l) • Convert units of mass (g, kg, t) x15 x15 15:30 Adam gets £15 and Ben gets £30 Crucial Knowledge – Stage 2 – Ratio and Proportion

Unit conversions Ratio calculations Recipe Scaling • Area conversions • Use a ratio to • Work out we have enough to complete Use the same conversions as measurements up and • How much of something do we need for length, but squared down • Volume conversions • Examples include Example: Use the same conversions as using maps and scale Q: A recipe uses 300g of flour and 150g of butter to for length, but cubed drawings make a cake for 4 people. How much of each • ����� = • Size calculations ingredient is needed to bake a cake for 6 people. • Units for speed include relative to scale and metres per second (m/s) real life A: 6 ÷ 4 = 1.5 (scale factor). and kilometres per hour 300g x 1.5 = 450g flour (kmph) 150 x 1.5 = 225g butter Crucial Knowledge – Stage 3 – Ratio and Proportion

Interest Calculations Proportionality Compound Measures • Compound Interest is an • Values that have a relationship with each other, • ����� = accumulating interest, as one changes, so does the other one changing over time, as a • Y = kx • ������� = growth • y is directly proportional to x. • Depreciation is a reduction Q: If y=24, then x=8 • Compound Interest Work out the value of y when x=2. • �������� = Always multiply Always 1 Number of years A: y = kx 24 = kx8 k=3 and so y = 3x Starting amount a x (1 ± �) So when x = 2 y = 3 x 2 = 6

+ for growth Interest rate as - for depreciation a decimal Inverse Proportionality • A reverse percentage is • Values that have a relationship with each other, as one changes, so does finding the original value the other, but inverse ����� ����� �������� = �100 • � = 100 − % �ℎ���� • y is inversely proportional to x. Q: When y=2, x=3. Work out the value of y when x=18 A: � = 2 = k = 6 and so � = When x = 18 y = 6 ÷ 18 = 1/3 Crucial Knowledge – Stage 1 – Geometry and Measures

Coordinates Terminology Shape • Remember “along the corridor • Edge – Where 2 faces meet then up the stairs” Use of Protractor • Vertices – Where 3 faces meet • X and y values written on the • Measure angles accurately • Face – side of a 3d shape axes • Draw bearings • Quadrilateral – a 4 sided polygon • 4 quadrants • Polygon – a 2d shape with straight Angle Reasoning sides • Angles on straight line = 180o Area and perimeter • Acute – an angle less than 90o • Angles in a triangle = 180o • Perimeter is distance around shape • Obtuse – an angle between 90o • Vertically opposite angles are • Area is space inside a shape (2D), and 180o always equal measure in square units • Reflex – an angle more than 180o • Rectangle ���� = �����ℎ × ����ℎ • Angles in quadrilateral = 360o o • ���� = ���� × ℎ���ℎ� • Angles at a point = 360 Triangle Only use diagonals for perimeter Types of Triangles • ���� = � + � × ℎ���ℎ� • Trapezium Scalene – all sides and angles are different Only use diagonals for perimeter • Isosceles – 2 sides and angles are the same • Circle ���� = � × ������ • Equilateral – 3 sides and angles are the same ������������� = 2� × ������ • Right – contains a right angle Circumference is the perimeter of a circle Crucial Knowledge – Stage 2 – Geometry and Measures

Bearings Pythagoras • 3 digit format • Measure clockwise from Polygons c North, 000o a • A shape with 3 or more straight sides • Be able to draw and add onto • Total Interior Angles = (n-2) x 180 b a diagram • Interior + Exterior = 180o • � + � = � • Measure reflex angles using a • Sum of Exterior = 360o Square root c2 to find Hypotenuse compass • � + � = � Basic transformations • Calculations using North for Square root b2 to find shorter side • Reflections parallel lines Over straight lines (y=, x= ) including Plans and Elevations Angles with parallel lines diagonals (y=x) • Images from 3 different directions • F – Corresponding • Rotations • Front, side and plan Always equal Direction, Distance and Centre • Work out size or volume • Z – Alternate • Translation • Draw 3 images from a 3D drawing ���ℎ� + ���� − Always equal • Draw a 3D image from 3 plans and • C – Co-Interior elevations �� + ���� − Always add to 180o • Enlargement Scale factor and Centre Crucial Knowledge – Stage 3 – Geometry and Measures Loci and Constructions Advanced Volumes • Perpendicular line bisector • ������ = �� Sphere • Angle bisector • ������ = �� Hemisphere • Basic shading of area that • Cone ������ = (���� ���� × ℎ���ℎ�) satisfy a LOCI • ������ = (���� ���� × ℎ���ℎ�) Pyramid Circle Theories • Frustrum – a cone with a cone cut of the top. • Angle facts relating to things in or around a circle Find the volume of the full cone and subtract the • 8 circle theorems volume of the missing cone • Often include Pythagoras’ Theorem and Right angled Trigonometry Advanced Transformations • Negative and Fractional enlargements Similar Shapes • Descriptions of single transformations • Divide 2 similar sides to find a linear scale factor • Area scale factor is the linear scale factor squared Right angled trig • Volume scale factor is the linear scale factor cubed • Identify Hypotenuse, Adjacent and Opposite • Be prepared to redraw diagrams to help. • Identify Sin, Cos or Tan function • ���� = ���� = ���� = • Normal function for sides • Inverse function (Sin-1) etc for angles Crucial Knowledge – Stage 1 – Algebra

Algebra terminology Simplifying – Collecting like terms • 2y means 2 multiplied by the value of ‘y’. • We can only bring ‘like terms’ together to simplify So if y = 5 then 2y = 2 x 5 = 10 the expression • Rewrite to get your ‘like terms together’ • y2 the value of ‘y’ multiplied by itself. Adding and Subtracting So if y = 5 then y2 = 5 x 5 = 25 4a + 3b + 6a – b = 4a + 6a + 3b – b = 10a + 2b 3f2 + 5g2 + 3f2 – 7g2 = 3f2 + 3f2 + 5g2 – 7g2 = 6f2 – 2g2 Substitution • We get rid of our letters by putting number in to create Multiplying and Dividing an answer. 4a x 6a = 24a2 (Multiply numbers and add powers) • We are normally given formula and values to put in, but 30b5 ÷ 5b2 = 6b3 (Divide numbers and subtract powers) sometimes we have to create the expression and then put values in. • We need to know about terminology to do this. Multiplying out single brackets 2 You are told E = ½ mv • Bracket create an order (BIDMAS) Calculate E when m = 10 and v = 2.5 • Brackets are also an invisible multiply E = ½ x 10 x 2.5 x 2.5 6 (a + 3) = 6 x a + 6 x 3 = 6a + 18 E = 31.25 5 (2b – a) = 5 x 2b + 5 x -a = 10b – 5a 2m (3m – 5) = 2m x 3m + 2m x -5 = 6m2 – 10m Crucial Knowledge – Stage 1 – Algebra Solving equations Factorising • To get a numerical answer for a letter • The process of putting things into brackets • We have to do the same to both sides of the • We can have numerical or algebraic factors equals sign • The ‘best’ factor goes on the outside of the brackets • If we move things across the equals sign the • You can check your answer by expanding bracket operator changes to be opposite Factorise 10a + 5b Solve 4y + 1 = 17 ‘best’ factor is 5 so this goes on outside of brackets 5(??????) Move +1 over to become -1 2a + b in brackets because when these are multiplied by 5 4y = 17 -1 so 4y = 16 you get your 10a and 5b Move x4 over to become ÷4 so y = 16÷4 So 5(2a + b) is answer y = 4 Factorise 20a2 + 4a Solve 2(3y + 1) = 20 ‘best’ factor is 4 number wise and a algebra wise it is a so Expand bracket this goes on outside of brackets 4a(??????) 2 x 3y = 6y and 2 x 1 = 2 so 5a + 1 in brackets because when these are multiplied by 4a 6y + 2 = 20 Move +2 over to become -2 you get your 20a2 and 4a 6y = 20 – 2 so 6y = 18 So 4a(5a + 1) is answer Move x6 over to become ÷6 so y = 18÷6 y = 3 Crucial Knowledge – Stage 2 – Algebra Expanding Double Brackets – FOIL Linear sequences • Two brackets with nothing between them • A list of numbers that goes up or down by the same • (x+2)(x+5) – This is a double bracket amount each time • 4(x+2) + 5(x+5) – This is 2 single brackets • Work out Term to Term rule • When expanding them think First Outer Inner Last • Work out your Zero Term • To start with, you get 4 terms out of double brackets • Form your equation for the nth term • You must simplify to 3 or sometimes 2 values • A value appears if a sequence, the nth term equation is solved with an integer answer. Straight line graphs • Remember y = ? (this is horizontal line) Solving linear equations – more advanced • Remember x = ? (this is vertical line) • Fractional or non integer – Follow your normal rules, be • You have to substitute values into equations to plot prepared to give your answer as a fraction, improper the graph fraction or mixed number. It might be positive or • y = mx + c where y = y coordinate, m = gradient (how negative. steep graph is), x = x coordinate and c = intercept • x on both side – Before you start identify the smallest (where we cut y axis) algebra term and do the opposite of this to both sides of • Parallel lines have same gradients the equation. Then, follow your rules to solve as • Gradient is RISE ÷ RUN a positive number we climb normal. and a negative value we ski down Crucial Knowledge – Stage 3 – Algebra

Simultaneous Equations Curved Graphs • When 2 things happen at the same time, • Plot x2 and x3 graphs using substitution with a table of values sometimes you have to form the equations • A + x2 equation gives a smiley face and – x2 equation a frown • You can sometimes take one equation away form • Use quadratic graphs to obtain equation answers by drawing another to solve on your graph • Sometimes you have to cross multiply equations • Remember (SSS) Signs Same Subtract Factorising Quadratics • Follow your solving linear equation rules • Putting into a set of double brackets • Look for number to be product of factor pairs Quadratic Equation • Look for number before ‘x’ to be the sum of factor pairs • Be able to apply equation to solve a quadratic • To solve make either bracket equal to zero ± � = Quadratic Sequences • A list of numbers that goes up or down by a different amount Algebraic Fractions each time • Apply normal rules of fractions • Look for second tier term to term rule each multiple of 2 is • Apply normal rules of solving algebra one x2 • Sometimes you simplify by factorising • Work out first 5 values to this amount of x2 then solve linear • Your answer may still be an algebraic fraction sequence that is the difference between this and original sequence Crucial Knowledge – Stage 1 – Data and Probability

Mean, median, mode and range Displaying data Interpreting data • You must be able to get measures from a list of • Get values from bar values or values in a frequency table charts • Get values from pie • MEAN = Total of values ÷ Number of values Charts • MEDIAN – The middle value when written in size • Use key to get values order from Stem and Leaf • MODE – The value that occurs the most often diagram • RANGE – Maximum value – Minimum value • Use key to get values from Pictogram Sample space diagrams • A list of all possible outcomes from an Probability definition and scale event. We use this to help calculate • Outcome – A possible result of an experiment probabilities • Event – A set of outcomes • Impossible – An outcome that cannot happen Probability and relative frequency • Certain – An event that must happen • A list of all probabilities adds up to 1 • Relative frequency = Crucial Knowledge – Stage 2 – Data and Probability Drawing pie charts Probability trees Stem and leaf diagrams • Angles in a pie chart • Used to show outcomes of multiple events • Pick correct stems = � 360 • All branches add up to 1 • Leaves are always single digits • Multiply along branches to find probabilities • Ascending order • Use a protractor and ruler to • Add multiple routes through tree • Use of key draw accurately • Obtain mean, median, mode Grouped data and range from diagram • Find Mean from a frequency table • Find Estimated Mean from grouped frequency table • Calculated Modal class interval Mean, median, mode and range with missing values • Calculate Median class interval • Be able to calculate missing values from a data set • A class interval means a group of data when given some of the values. Example: The mean of the following 5 numbers is 9: Two way tables [ 6 ] [ 7 ] [ ? ] [ 11 ] [ 13 ] • Values add up vertically and What is the missing number? horizontally • Totals can be given but may Total value = 5 x 9 = 45 need to be calculated Known total = 6 + 7 + 11 + 13 = 37 • Used to simplify information Missing value = 45 – 37 = 8 Crucial Knowledge – Stage 3 – Data and Probability Probability trees with non replacement Listing Number of Outcomes • Draw a probability without being asked to • Be able to list number of outcomes from a written • Change probability on 2nd and potentially 3rd information: event. Use the information given to determine Example: new probabilities. Make sure all branches add up Q: A menu contains 3 starters, 5 mains and 2 desserts. to 1 How many different 3 course meals can be ordered? • Use tree to calculate complicated event outcomes A: 3 x 5 x 2 = 30 different 3 course meals. by multiplying along branches Histograms Box and Whisker Plots Cumulative Frequency curves • Looks like a bar chart with • Displays 5 key pieces of • Plot cumulative frequencies against different width bars ��������� ������� = information. interval’s upper value 1. Minimum • • Complete a table of values 2. Lower Quartile (Q1) Hand drawn curve that passes through 3. Median (Q2) all points • Draw or complete a histogram 4. Upper Quartile (Q3) 5. Maximum • Draw on to obtain values using • Find an estimated mean or • IQR = Q3 – Q1 horizontal and vertical lines median by reading values • Draw a box plot • Understand value you are after is from a histogram • Compare box plots by stating sometimes above or below your drawn which median is larger and on value which IQR is wider.