# GCSE Maths Linear Higher Tier Grade Descriptors

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1 GSE Maths Linear Higher Tier escriptors Fractions /* Find one quantity as a fraction of another Solve problems involving fractions dd and subtract fractions dd and subtract mixed numbers Multiply and divide fractions Round numbers to significant figures Find the reciprocal of a number Find upper and lower bounds Use upper and lower bounds in calculations Percentages ompare harder percentages fractions and decimals Work out more difficult percentages of given quantities Increase or decrease by a given percentage Express one quantity as a percentage of another Work out a percentage increase or decrease Work out compound interest Understand how to use successive percentages Use a multiplier raised to a power to solve problems involving repeated percentage change Work out reverse percentage problems ecimals Multiply and divide decimals onvert fractions to decimals Round numbers to different degrees of accuracy, decimal places and significant figures Recognise that recurring decimals are exact fractions and that some exact fractions are recurring decimals onvert recurring decimals to fractions Ratio and Proportion Use ratio notation, including reduction to its simplest from and its links to fraction notation ivide a quantity in a given ratio Solve simple ratio and proportion problems, such as finding the ratio of teachers to students in a school ivide a quantity in a given ratio Solve simple ratio & proportion problems such as finding the ratio of teachers to students in a school Solve more complex ratio and proportion problems such as sharing money in the ratio of peoples ages Solve ratio and proportion problems using the unitary method. Solve more complex ratio & proportion problems Solve ratio & proportion problems using the unitary method Prime Factors Find the LM of two simple numbers Find the HF of two simple numbers Write a number as the product prime factors find the LM of two or more numbers Find the HF of two or more numbers

2 * Sequences Surds Write the terms of a sequence or a series of diagrams given a rule or nth term Write the nth term of a linear sequence or a series of diagrams Understand the definition of rational and Irrational numbers Know the rules of surds i.e. ab = a x b and a/b = a b Find factors of a surd in order to simplify. Rationalise the denominator of a surd Simplify surds such as (3-5)2 in the form a + b 5 Indices and Standard Form Use the terms square, positive square root, negative square root, cube and cube root Recall integer squares from 2x2 to 15x15 and the corresponding square roots Recall the cubes of 2,3,4,5, and 10 and the corresponding cube roots Use index notation and index laws for positive powers Use index notation and index laws for negative powers onvert between ordinary and standard index form numbers Use standard index form for calculations involving multiplication and division Use index notation and index laws for fractional powers such as 16 ½ Working with Symbols Expand brackets such as 4(x 3) Factorise an expression such as 6x + 8 Expand and simplify an expression such as 3(3x 7)-2(3x + 1) Expand and simplify two brackets such as (x 3)(x + 5) Expand and simplify two brackets such as (3x + 1)(2x 3) Simplify fractions such as x + x 3 5 Equations and formulae Set up and solve more complicated equations such as 3x-4 = 5+x or 2(5x + 1) = 28 Substitute numbers into more complicated formulae such as =(+1)/9 erive more complex formulae istinguish between an expression, an equation, an identity and a formula Solve an equation such as 3x + 2 = 6 x or 4(2x 1) = 20 Represent and interpret inequalities on a number line Solve an equation involving fractions Rearrange linear formulae such as p = 3q+5 Solve equations such as x 3 + x = 12 using systematic trial and improvement methods Solve equations such as (2x-1)/6 + (x+3)/3 = 5/2 Rearrange formulae involving brackets, indices, fractions and square roots

3 Inequalities Solve an inequality such as 2x 7 < 9 Find the integer solutions of an inequality such as -8 < 2n 5 Solve an inequality such as 3x x Represent linear inequalities in two variables such as x + y < 7, as a region on a graph. Simultaneous equations Solve a pair of simultaneous equations such as x+3y=9 and 3x-2y=5 Solve a pair or linear equations graphically; identifying the point of intersection as the solution Solve a pair of simultaneous equations such as y=4x+5 and y=x² Find the point of intersection of a linear and a quadratic equation; recognising that the solution could be found from the points of intersection of the graphs Quadratic functions Factorise an expression such as x²+5x+14 or x²-9 Solve an equation such as x²-5x+14=0 Factorise an expression such as x 2 5x +14 or x 2 9 Solve an equation such as x 2 5x +14 = 0 Factorise an expression such as 3x 2 + 7x + 2 or 3x 2 27 Solve an equation such as 3x 2 + 7x + 2 = 0 by factorising erive an algebraic proof using reasoning and logic. Graphs raw the graph of a line such as y = 3x 5 without being given a table of values Solve problems such as finding where the line y = 3x 5 crosses the line y = 4 Find the gradients of straight line graphs Find the midpoint of a line segment such as the line from (1,5) to (3,7) raw graphs of quadratics such as y=x 2 +2x + 1 Use a quadratic graph to estimate x- and y-values, giving answers to an appropriate degree of accuracy raw graphs of harder quadratics such as y=2x 2-7x + 5 Find the gradient and equation of a line through two points such as (0,3) and (5,13) Find the equation of a line parallel to another line, such as y = 3x 5 passing through a given point omplete tables for, and draw graphs of, cubic functions and use the graphs to solve equations Real-life Graphs Interpret real life graphs Find simple average speed from distance time graphs Recognise from a distance time graph when the fastest average speed takes place Find the average speed in km/h from a distance time graph with time in minutes iscuss and interpret graphs modelling real situations ngles and classifying shapes Recognise corresponding, alternate and interior angles on parallel lines Understand and use three figure bearings lassify a quadrilateral using geometric properties alculate exterior and interior angles of a regular polygon

4 rea and volume Find the area of a triangle, trapezium and parallelogram onvert between square units such as changing 2.6m2 to cm2 Find the area and perimeter of shapes made from triangles and rectangles alculate the circumference and area of a circle /* /* Work out the perimeter and area of a semicircle onvert between cube units such as changing 3.7m 3 to cm 3 Find the volume of prisms including cylinders Find the surface area of simple prisms alculate the lengths of arcs of circles alculate the area of sectors of circles Solve problems involving surface areas and volumes of pyramids, cones and spheres 3- shapes, coordinates raw the front, plan and side elevations of a solid on squared paper Use 3- coordinates Pythagoras theorem Use Pythagoras theorem to find the third side of a right-angled triangle Use Pythagoras theorem to prove that a triangle is right-angled Find the distance between two points from their coordinates Use Pythagoras theorem in 3- problems Trigonometry Use sine, cosine and tangent to calculate a side in a right-angled triangle Use sine, cosine and tangent to calculate an angle in a right-angled triangle Use trigonometry to solve problems, including those involving bearings Use trigonometry to find sides and angles in three dimensions Use the sine and cosine rules to solve 2- problems alculate the area of a triangle using ½absin Measures alculate average speeds Use compound measures, such as speed Recognise that measurements to the nearest unit may be inaccurate by up to one half unit in either direction Use compound measures such as density, for example find density given cross-sectional area length and mass Properties of circles Know the angle and tangent properties of a circles angle at centre is twice than point on circumference angle at circumference in a semicircle is 90 degrees angles subtended by same arc are equal opposite angles in a cyclic quadrilateral add up to 180 the radius or diameter of a circle bisects a chord at right angle tangents meeting at a point outside a circle are equal in length. Understand the angle and tangent properties of a circle Understand the alternate segment theorem

5 onstruction and Loci raw a quadrilateral such as a kite, parallelogram or rhombus with given measurements Understand that giving the lengths of two sides and a non-included angle may not produce a unique triangle Understand the idea of a locus onstruct perpendicular bisectors and angle bisectors Match one angle and one side of congruent triangles given some dimensions onstruct the locus of points equidistant from two fixed points onstruct the locus of points equidistant from two fixed lines Solve loci problems, for example the locus of points less than 3cm from a point P onstruct perpendicular lines from a point to a line, perpendicular at a point on a line and an angle of 60 o Match sides and angles of similar triangles Vectors dd, subtract and multiply vectors Use addition, subtraction and multiplication of vectors to solve simple geometric problems Understand the relationship between parallel vectors Reflections, rotations and translations Reflect shapes in lines parallel to the axes, such as x=2 and y=-1 Rotate shapes about the origin escribe fully reflections in a line and rotations about the origin Translate a shape using a description such as 4 units right and 3 units down Reflect shapes in lines such as y=x and y=-x Rotate shapes about any point escribe fully reflections in any line parallel to the axes, y=x or y=-x and rotations about any point Find the centre of a rotation and describe it fully Transform shapes by a combination of translation, rotation and reflection Translate a shape by a vector Use congruence to show that translations, reflections and rotations preserve length and angle Enlargements Enlarge a shape by a positive scale factor from a given centre ompare the area of an enlarged shape with the original area Find the ratio of corresponding lengths in similar shapes and identity this as the scale factor of enlargement Use ratios in similar shapes to find missing lengths Enlarge a shape by a fractional scale factor Enlarge a shape by a negative scale factor ompare lengths, areas and volumes of enlarged shapes Use the effect of enlargement on perimeter area and volume in calculations ollecting ata Understand and name different types of data esign and use data collection sheets, surveys and questionnaires esign and use two-way tables for discrete and grouped data Understand and name other types of data collection methods Identify possible sources of bias in the design and use of data collection sheets and questionnaires Understand the data handling cycle Understand that increasing sample size generally leads to better estimates

6 * Statistical Measures alculate the mean for a frequency distribution Find the modal class for grouped data Find the mean for grouped data Find the median class for grouped data. Find the upper and lower quartiles and calculate the inter-quartile range for a frequency distribution Representing data onstruct a stem and leaf diagram (ordered) onstruct a histogram for data with equal class intervals onstruct and interpret a line graph onstruct a frequency polygon onstruct and interpret a cumulative frequency diagram for continuous or grouped data Use a cumulative frequency diagram to estimate median and interquartile range onstruct and interpret a box plot ompare two sets of data using a box plot referencing average and spread Scatter Graphs raw a scatter graph by plotting points on a graph Interpret the scatter graph raw a line of best fit on the scatter graph Interpret the line of best fit Identify the strength and type of correlation Probability Use a two-way table to find a probability Understand mutually exclusive events Identify different mutually exclusive events and know, if they cover all possibilities, then the sum of their probabilities is 1. Use probability to estimate outcomes for a population Understand and use relative frequency raw tree diagrams Understand independent and non-independent events Find probabilities of successive independent events Find probabilities of successive dependent events

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