Maths GCSE Checklist

Edexcel GCSE Maths Linear Exam Topic List - HIGHER NUMBER Add, subtract, multiply, divide c Write numbers in words c Write numbers from words c Add, subtract, multiply, divide whole numbers, integers, negatives, fractions, and decimals and numbers in index form c Multiply and divide any number between 0 and 1. c Divide decimals up to 2 decimal places c Solve a problem involving division by a decimal (up to two decimal places) c Know the fraction-to-decimal conversion of familiar fractions Order numbers c Put in order of size, integers, decimals and fractions c Understand and use positive and negative numbers on a number line Factors, multiples and primes Understand the terms; c Odd and even c Factor c Multiple c Common factor c Highest common factor c Least (lowest) common multiple c Prime number c Be able to identify factors, multiples and primes from a list of numbers c Express a number as a product of prime factors (factor tree) c Find common multiples or common factors of two numbers c Find the highest common factor (HCF) or the lowest common multiple (LCM) of two numbers. Squares, square roots, cubes and cube roots c Know all the square numbers from 2² = 4 up to 15² = 225 c Know all the cube numbers from 2³ = 8 up to 5³ = 125 and also 10³ = 1000 Index notation c Use index notation for squares and cubes, eg. 5³ c Use index notation for powers of 10, eg. 106 c Understand indices in calculations Index laws c Multiply and divide by adding or subtracting indices c Calculate using index laws when indices are fractions or negative c Understand that for any number n, nº = 1 c Understand that n-1 = 1 / n c Understand that n½ = √n c Understand that n⅓ = 3√n Standard form c Understand numbers written in standard form c Write large or small numbers in standard form c Convert between standard form and normal form c Understand and use standard form on a calculator Equivalent fractions and adding and subtracting fractions c Find equivalent fractions c Simplify a fraction to its simplest form c Convert between improper fractions and mixed numbers c Add and subtract fractions Decimals, including recurring decimals c Know fraction to decimal conversions for simple fractions c Convert between fractions and decimals c Understand that all recurring decimals are exact fractions, and that some exact fractions are recurring decimals c Convert between recurring decimals and fractions c Know how to convert from recurring decimal to fraction using a proof Percentages c Understand percentages c Convert between fractions, decimals and percentages Using fractions, decimals and percentages c Find a fraction of a quantity c Find a percentage of a quantity c Use decimals to find quantities c Use a multiplier to increase of decrease a quantity (eg. use x 1.05 to increase by 5%, or 0.88 to decrease by 12%)) Percentages and proportional change c Use percentages to calculate and use o VAT o Simple interest o Income tax o Compound interest o Depreciation o Prices after an increase or decrease o Percentage profit and loss c Find the original amount, given the new amount and the percentage change c Calculate repeated proportional change c Use a multiplier raised to a power to calculate repeated proportional change c Use a multiplier to increase or decrease by a percentage Direct and indirect proportion c Calculate an unknown quantity where quantities are in direct proportion c Calculate an unknown quantity where quantities are in inverse proportion Fractions, decimals and percentages c Find one number as a fraction of another number c Find one number as a percentage of another number c Multiply using percentages as operators Number operations and the relationships between them, including order of operations and inverse operations c Understand multiplying and dividing, and that one is the inverse of the other c Use inverse operations c Understand the use of brackets in calculations c Understand the hierarchy of operations (BIDMAS) c Solve word problems c Understand and find reciprocals c Understand that the inverse of raising to the power of n is the same as raising to the power of 1 over n c Understand and use 1 over a number is the inverse of multiplying by that number c Use reverse percentage calculations Ratio c Write a ratio in its simplest form c Divide a quantity in a given ratio c Solve problems using ratios Use surds and π in exact calculations c Use surds (roots) and π in calculations without a calculator, leaving the surd or π in the answer, eg. give an answer of 25 π c Give an answer to a Pythagoras question as √17 c Manipulate surds in calculations, eg. (3 - √3)² c Rationalise a denominator, ie. manipulate so that there is no longer a surd on the bottom of the fraction Rounding and approximation c Round to the nearest integer (whole number) c Round numbers to any given power of 10 c Round to a number of decimal places c Round to a number of significant figures c Estimate the answer to a calculation by using rounding Upper and lower bounds Find the upper and lower bound of a calculation, especially in the calculation of: c measurements c perimeter c area c volume c Give a final answer to a calculation to an appropriate degree of accuracy using upper and lower bounds Use a calculator effectively c Simple and complex calculations, including involving time or money c Use the following functions c +, -, x, ÷ c x² and √x c memory functions c brackets c x to the power of y c x to the power of 1 over y c brackets c trigonometrical functions c Understand that rounding too early can causes inaccuracy c Understand numbers shown in standard form, and be able to enter numbers in standard form c Calculate in standard form c Use for dividing to do reverse percentage calculations c Use a multiplier and the power key to calculate exponential growth or decay ALGEBRA Algebraic notation c Understand notation and symbols used in algebra c Understand the difference between “expression”, “formula”, “equation” and “identity” c Be able to select an expression, formula, equation or identity from a list c Be able to write an expression to solve a problem Manipulate algebraic expressions c Simplify by collecting like terms c Multiply out a single bracket c Factorise a single bracket by taking out a common factor c Expand two brackets c Factorise quadratics into two brackets c Factorise quadratics using the difference of two squares, eg. 4y² - 25 = (2y + 5)(2y - 5) c Simplify algebraic expressions by cancelling, adding, subtracting and multiplying c Use index laws, including fractional, zero and negative powers, and powers raised to another power Solve linear equations c Set up simple equations for a problem c Rearrange simple equations c Solve simple equations c Solve equations with the unknown on either side c Solve equations with the unknown on both sides c Solve equations that include brackets c Solve equations with negatives, including negative answers c Solve equations involving fractions Solve simultaneous equations with two unknowns c Use elimination to solve simultaneous equations c Use substitution to solve simultaneous equations c Draw straight line graphs and find the solution from the intersection of the two graphs c Write simultaneous equations for a problem Solve quadratic equations c Solve quadratic equations by factorisation c Solve quadratic equations by completing the square c Solve quadratic equations using the quadratic formula Using formulae c Derive formulae c Substitute numbers (positive or negative) into a formula, including formulae with x² or x³ terms c Change the subject of a simple formula c Change the subject of a formula where the subject appears on both sides of the formula c Change the subject of a formula that includes a power of the subject Solve linear inequalities c Solve a simple linear inequality with one variable c Show the solution to a linear inequality with one variable on a number line c Show the solution to several inequalities with two variables on a graph Trial and improvement c Use trial and improvement to find an approximate solution to an equation Sequences c Understand odd and even numbers c Generate number sequences from diagrams c Describe the rule for a number sequence c Find a particular term in a sequence, or explain why a particular number is not in a sequence Nth term of a sequence c Find the nth term expression for a sequence c Use the nth term expression to find a number in the sequence Coordinates c Use axes and coordinates, both positive and negative in 2D c Understand and plot points in four quadrants c Use axes and coordinates in 3D c Find the coordinates of a point c Plot a point given the coordinates, in 2D or 3D c Find the mid-point of a line c Calculate the length of a line using coordinates Graphs c Draw, label and add a scale to axes c Understand that an equation of the form y = mx + c corresponds to a straight line graph c Plot straight line graphs from their equations c Plot and draw a graph of an equation in the form y = mx + c c Find the gradient of a straight line graph c Find the gradient of a straight line graph from its equation c Understand that a graph of an equation in the form y = mx + c has gradient of m and a y intercept of c (ie. crosses the y axis at c) c Understand how the gradient of a real life graph relates to the relationship between the two variables Gradients of parallel and perpendicular lines c Understand how the gradients of parallel lines are related c Understand how the gradients of perpendicular lines are related c Understand that if the gradient of a graph in the form y = mx + c is m, then the gradient of a line perpendicular to it will be c Generate equations of a line parallel or perpendicular to a straight line graph Simultaneous equations (one linear and one quadratic) c Find the intersection of a linear and a quadratic graph to find (approximate) solutions to simultaneous equations c Solve simultaneous equations (one linear, one quadratic in one variable) by elimination c Solve simultaneous equations where one equation is of the form x² + y² = r² Other graphs c Plot, sketch or recognise graphs of cubic functions c Plot, sketch or recognise graphs of y = 1/x c Plot, sketch or recognise graphs of y = kx for integer values of x c Plot, sketch or recognise graphs of y = sin x and y = cos x from -360° to +360° c Draw or plot other mathematical functions c Recognise or analyse other mathematical functions Graphs of loci c Construct the graphs of simple loci including the circle, x² + y² = r² c Find the points of intersection of a circle and a straight line c Apply understanding of loci to construct graphs based on circles and perpendicular lines Graphs from quadratic and other functions c Generate points for quadratic functions c Plot graphs of quadratic functions c Find (approximate) solutions to a quadratic equation from the graph of its function c Find (approximate) solutions to simultaneous equations, one quadratic and one linear from the intersections of their graphs Real life graphs c Plot a linear graph c Interpret information on linear and non-linear graphs Direct and inverse proportion c Set up equations to solve word problems involving direct proportion c Set up equations to solve word problems involving indirect proportion c Understand and use graphs of equations involving direct and indirect proportion Transformation of functions Apply to the graph of y = f(x) the following transformations: c y = f(x) + a c y = f(ax) c y = f(x + a) c y = a f(x) for linear, quadratic and sine and cosine functions, f(x) c Apply the following transformations to functions: c reflection c rotation c enlargement c translation c Analyse transformations of functions and write them algebraically GEOMETRY Angles on intersecting lines, in triangles and quadrilaterals, and on parallel lines c Angles round a point add up to 360° c Angles on a straight line add up to 180° c Perpendicular lines c Know the properties of scalene, isosceles, equilateral and right-angled triangles c Angles in a triangle add up to 180° c Angle properties of intersecting lines, and vertically opposite angles are equal c Be able to mark parallel lines on a diagram c Corresponding angles in parallel lines c Alternate angles in parallel lines c Calculate angles and give reasons c Explain why the angle sum of a quadrilateral is 360° c Understand a proof that the angle sum of a triangle is 180° c Understand the proof that the exterior angle of a triangle of a triangle is equal to the sum of the interior angles at the other two vertices c Calculate angles in more complex problems Interior and exterior angles of polygons c Calculate the sum of interior angles in a polygon c Understand the polygon names; hexagon, heptagon, octagon and decagon c Use the angle sum of an irregular polygon in a problem c Calculate and use the sum of the interior angles of a regular polygon c Understand and use fact that the exterior angles of a polygon add up to 360° c Understand and use the fact that the interior and exterior angles at one vertex of a polygon add up to 180° c Be able to calculate the exterior angle of a regular polygon c Be able to calculate the interior angle of a regular polygon c Be able to deduce the number of sides of a regular polygon, given one of its angles c Understand tessellations of regular and irregular polygons c Tessellate combinations of polygons c Explain why some shapes tessellate and some do not Properties of quadrilaterals Remember the definitions and properties (including symmetry) of special quadrilaterals, ie. c Square c Rectangle c Parallelogram c Trapezium c Rhombus c Kite c List or classify quadrilaterals by their properties Reflection and rotation symmetry in 2D shapes c Recognise reflection symmetry and be able to draw lines of symmetry on a shape c Recognise rotation symmetry of 2D shapes c Identify the order of rotational symmetry of a shape c Complete a diagram given the line or lines of symmetry c State a line of symmetry on a grid as a simple algebraic equation, eg. x = 2 or y = x c Complete diagrams with a given order of rotational symmetry Congruence and similarity c Understand that angles in similar shapes are the same c Prove the congruence of triangles using SSS, SAS, ASA and RHS and formal argument c Understand SSS, SAS, ASA and RHS in ruler and compass constructions c Understand similarity of triangles and other 2D shapes, c Use understanding of similar figures in problems c Prove formally that two triangles are similar Pythagoras’ theorem c Understand and use Pythagoras’ theorem in triangles c Understand and use Pythagoras’ theorem in 3D problems c Understand the language associated with 3D shapes, including diagonals of a cuboid c Use Pythagoras’ theorem to calculate the length of a diagonal of a cuboid Trigonometry c Understand and remember trigonometric relationships in right angled triangles c Use trigonometry in 2D problems c Use trigonometry in 3D problems c Use trigonometry to find the angle between a line and a plane c Find angle of elevation and angle of depression c Use the sine rule to solve 2D and 3D problems c Use the cosine rule to solve 2D and 3D problems Parts of a circle c Draw a circle with compasses, given either the diameter or radius Understand and remember parts of a circle: c Centre c Radius c Diameter c Chord c Circumference c Tangent c Arc c Sector c Segment Circle theorems and their proofs c Prove and use each of the circle theorems: c Tangent is perpendicular to the radius at the point the tangent meets the circle c Two tangents from a point are equal in length c Angle subtended from an arc at the centre is twice the angle at the circumference c Angle in a semicircle is a right angle c Angles in the same segment are equal c Opposite angles of a cyclic quadrilateral add up to 180° c Alternate segment theorem c Perpendicular from the centre to a chord bisects the chord Using 2D diagrams to represent 3D shapes c Use isometric grids c Draw nets and show how they fold to make a 3D solid shape c Understand and draw front and side elevations and plans of simple solids c Draw a sketch of a 3D solid shape given the front and side elevations and plan of the solid Transformations Rotations c Rotate a 2D shape around the origin or other point c Understand that a rotation is defined by an angle, direction and a centre of rotation c Find the centre of rotation c Understand that a rotation produces a shape congruent to the original Reflections c Understand and describe reflections c Identify the mirror line for a reflection, and find its equation c Understand that a reflection produces a shape congruent to the original Translations c Understand and use translations c Understand that translations are defined by a distance and a direction using vector notation c Translate a shape by a given vector c Understand that a translation produces a shape congruent to the original Enlargements c Understand that an enlargement is defined by a centre of enlargement and a scale factor c Understand that angles remain the same in an enlargement c Enlarge a shape using (0, 0) or any other point as the centre c Enlarge a shape by a positive scale factor c Enlarge a shape by a fractional scale factor c Enlarge a shape by a negative scale factor c Find the centre of a given enlargement c Identify the scale factor of a given enlargement Combined transformations c Describe a transformation using a combination of rotation, reflection, translation or enlargements. Straight edge and compass constructions c Construct a given triangle c Construct an equilateral triangle c Understand that SSS, SAS, ASA and RHS triangles are unique but ASS ones are not c Construct a perpendicular bisector of a line c Construct a perpendicular from a point to a line c Construct a perpendicular from a point on a line c Bisect an angle c Construct angles of 60°, 90°, 30° and 45° c Construct parallel lines c Draw circles and arcs of a given radius c Construct a regular hexagon inside a circle c Construct diagrams involving any of the above c Construct diagrams from given information Loci c Construct a region bounded by a circle and an intersecting line c Construct a loci of a given distance from a point and a given distance from a line c Construct a loci of equal distances from two points c Construct a loci of equal distances from two lines c Identify regions defined by “nearer to” or “greater than” c Find or describe regions satisfying a combination of loci Perimeter and area c Measure shapes to find perimeter or area c Find the perimeter of a rectangle or triangle c Use a formula to find the area of a rectangle c Use a formula to find the area of a triangle c Use a formula to find the area of a parallelogram c Use a formula to find the area of a trapezium c Calculate the perimeter and area of compound shapes made from triangles, rectangles and other shapes c Find the surface area of shapes such as prisms or pyramids by using the formulae for triangles, rectangles and other shapes Area of a triangle c Calculate the area of a triangle using the formulae A = ½ ab sinC Circumference and area of a circle c Find circumference of a circle using C = πd or C = 2πr c Find the area of a circle using A = πr² c Use π = 3.142 or the π button on a calculator c Find the perimeter and area of semcircles and quarter circles c Calculate the length of an arc c Calculate the area of a sector c Give answers in terms of π if required c Find the surface area of a cylinder Volumes of prisms c Use the formula to calculate the volume of a cuboid c Calculate volume of a prism, such as a triangular prism c Calculate the volume of a prism made from cuboids c Find the volume of a cylinder Complex shapes and solids c Find the surface area of cubes, cuboids, cones, pyramids, spheres and hemispheres c Find the volumes of cones, pyramids, spheres and hemispheres, frustrums c Find the surface area or volume of a compound solid made up of other solid shapes, eg. a cuboid with pyramid on top, or cyclinder with cone on top. c Use volumes in complex problems c Find the area of a segment of a circle given the radius and length of the chord Vectors c Understand and use vector notation c Add or subtract two vectors c Multiply a vector by a number c Calculate the result of two vectors c Solve problems using vectors c Use vectors in geometrical proofs MEASURES Maps and scale drawings c Use, interpret and construct maps and scale drawings c Draw lines and shapes to scale c Estimate lengths using a scale diagram Enlargement of shapes, including solids c Understand the effect of enlargement on perimeter, area and volume c Understand and use the fact that area and volume are affected differently by an enlargement c Know the relationship between linear, area and volume scale factors when one 2D or solid shape is an enlargement of another Interpretation and accuracy c Read and interpret scales on measuring equipment c Know the relationships between seconds, minutes, hours, days, weeks, months and years c Use 12 and 24 hour clock times c Calculate time intervals c Recognise inaccuracy of measurement, and choose appropriate units of measurement c Understand that choice of unit affects accuracy c Understand that measurements given to a whole unit may be up to half a unit inaccurate in either direction Converting measurements c Know conversion factors between different metric units c Convert between metric units c Convert between imperial units, given the conversion factor c Know rough imperial/metric equivalents as follows c 1 kg = 2.2 pounds c 1 litre = 1¾ pints c 4.5 litres = 1 gallon c 8 km = 5 miles c 30 cm = 1 foot c Convert between imperial and metric measures using the above conversion factors c Convert between metric measurements of area c Convert between metric measurements of volume c Convert between different metric units of speed, eg. metres per second and km per hour c Convert between metric units of volume and metric units of capacity, eg. 1 cm³ = 1 ml Estimation of measures c Make estimates of measurements c Choose appropriate units for estimates of measurements Bearings c Use 3 figure bearings to specify direction c Mark a point on a diagram, given a bearing and distance from another point c Measure or draw a bearing on a map or scale plan c Given a bearing of one point from another, find the bearing of the first point from the second Compound measures c Understand and use compound measures, including speed and density Measure and draw lines and angles c Measure and draw straight lines to the nearest mm c Measure and draw angles to the nearest degree Drawing using a ruler and protractor c Make accurate drawings of triangles and other 2D shapes using ruler and protractor c Make an accurate scale drawing from a diagram c Use accurate drawing to solve bearings problems STATISTICS Data handling c Decide on what data and analysis may be required for a problem c Data collection c Presenting data c Discuss data Bias c Identify why data may be biased, and know how to minimise bias c Understand the implications of different sizes of samples Designing a survey c Identify what data is needed c Consider fairness of a survey c Understand sample and population c Design a question for a survey c Criticise questions for a survey c Understand random sampling c Understand stratified sampling c Calculate numbers needed for stratified sampling Design data collection methods c Design and use a data collection sheet, including one for continuous data c Sort and classify data, and put data into a table c Group data into class intervals with equal width Tables and lists c Extract data from tables and lists Two-way tables c Design two-way tables c Complete a two-way table Charts and diagrams Draw the following charts or diagrams c Bar chart c Dual bar chart c Pie chart c Histogram with equal class intervals c Frequency polygon c Frequency diagram for grouped discrete data c Scatter graph c Line graph c Frequency polygon for grouped data c Grouped frequency table for continuous data c Stem and leaf diagram c Two-sided stem and leaf diagram c Cumulative frequency table c Cumulative frequency graph c Box plots (from raw data, or when given the median and quartiles) c Histograms with unequal class intervals, using frequency density Types of average and range Calculate the following c Mean c Mode c Modal class c Median c Interval containing the median c Range c Estimate the mean of grouped data using mid-points of intervals c Find median, quartiles and interquartile range for grouped data c Estimate the mean for grouped data c Find median, quartiles and interquartile range from a cumulative frequency graph c Find median, quartiles and interquartile range from a box plot Interpreting graphs and diagrams Understand and find information from c pie charts c stem and leaf diagrams c scatter graphs c frequency polygons c box plots c cumulative frequency diagrams c histograms c Find the median or other information from a histogram, for example the number of people in a particular group c Find information from line graphs, frequency polygons and frequency diagrams c Find information from pie charts c Find median, mode, range and interquartile range from stem and leaf diagrams c Estimate values and find median, quartiles and interquartile range from a cumulative frequency graph c Complete a frequency table from a histogram c Understand and define frequency density Patterns in data c Find patterns in data c Find exceptions in data c Explain an isolated point on a scatter graph Lines of best fit c Draw a line of best fit c Understand positve, negative and no correlation c Understand that correlation does not always imply one thing causes the other c Predict values using a line of best fit c Understand that “no correlation” does not necessarily mean no relationship between the values, just no linear relationship Comparing data c Compare two sets of data using shapes of distributions c Compare two sets of data using averages and spread, such as median, range and quartiles c Compare spread using box plots or cumulative frequency graphs c Compare two pie charts c Compare data from dual bar charts c Understand the advantages and disadvantages of different types of average Using calculators c Calculate mean using the correct key on a scientific calculator c Σx and Σfx or calculation of the line of best fit PROBABILITY Probability language and the probability scale c Impossible, unlikely, even chance, likely and certain events c Mark events or probabilities on a 0 to 1 probability scale c Write probabilities as fractions, decimals or percentages Estimates of probability and relative frequency c Find probabilities of events using dice, spinners, coins c Understand and use relative frequency as estimates of probability c Calculate an estimate of how many times an event will occur, given its probability and the number of trials Listing events c List the outcomes for one or two events c Use and draw diagrams to show all possibilities Mutually exclusive outcomes c Understand that the sum of all the mutually exclusive outcomes is 1 c Know that if P is a probability of an outcome occurring, then 1 - P is the probability of the same outcome not occurring c Fill in a missing probability in a table c Know and use the fact that, for mutually exclusive events, P(A OR B) = P(A) + P(B) Independent events c Know that, for independent events, P(A AND B) = P(A) x P(B) c Understand the difference in calculation for selection of an object with or without replacement Tree diagrams c Draw a probability tree diagram c Calculate probability of compound events from a tree diagram Experimental data and theoretical probability c Compare experimental data with theoretical probability c Understand that the same experiment repeated can have different results, and that increasing sample size increases accuracy c Compare results from different sample sizes