AQA Mathematics - Higher Tier

Transcription

1 Q Mathematics - Higher Tier General Topic and resources No. of lessons Trigonometry 1 Indices and Standard form 4 Probability 4/5 Objectives Grade y the end of the module students should be able to 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 the terms square, positive and negative square root, cube and cube root Recall integer squares from to and the corresponding square roots Recall the cubes of,, 4, 5 and 10 Use index notation and index laws for positive and negative powers 1 4 Use index notation and index laws for fractional powers such as 16 4 Use index notation and index laws for fractional powers such as 16 Use standard index form with and without a calculator onvert between ordinary and standard index form representations Use probability to estimate outcomes for a population Use relative frequency to find probabilities Use a two-way table to find a probability Understand mutually exclusive events Use the fact that the probabilities of mutually exclusive events add up to 1 Understand dependent and independent outcomes omplete a tree diagram Use tree diagrams to find probabilities of successive independent events raw tree diagrams and use them to find probabilities of successive dependent events omments Two-Tier GSE Mathematics - Medium Term Plan

2 Fractions 5 Pythagoras theorem ollecting data 5 rea and volume (1) 4/5 Q Mathematics - Higher Tier o calculations with simple fractions involving subtraction o calculations with mixed numbers o calculations with simple fractions involving division Use Pythagoras theorem to find the hypotenuse of a right-angled triangle Use Pythagoras theorem to find any side of a right-angled triangle Use Pythagoras theorem to find the height of an isosceles triangle Use Pythagoras theorem in practical problems Find the distance between two points from their coordinates Use Pythagoras theorem in - problems lassify and know the difference between various types of data esign and use data collection sheets and questionnaires Identify possible sources of bias in the design and use of data collection sheets and questionnaires Use a variety of different sampling methods Use stratified sampling methods Find the area of a triangle, parallelogram, kite and trapezium Find the area and perimeter of compound shapes alculate the circumference of a circle to an appropriate degree of accuracy Find the perimeter of a semicircle alculate the area of a circle to an appropriate degree of accuracy Find the area of a semicircle alculate volumes of triangular prisms, parallelogram-based prisms and cylinders onvert between measures of volume Solve problems involving surface areas of prisms and cylinders onvert between measures of area Two-Tier GSE Mathematics - Medium Term Plan

3 Use of symbols Exer 9 9 oordinates Properties of circles Representing data 1 Equations Q Mathematics - Higher Tier Multiply out expressions with brackets Expand (and simplify) harder expressions Factorise expressions Expand (and simplify) quadratic expressions Factorise quadratic expressions Simplify rational expressions Factorise harder quadratic expressions Simplify harder rational expressions raw lines such as y = x + Solve problems involving straight lines Find the midpoint of a line segment Use and understand coordinates in three dimensions Use the angle properties of a circle Prove the angle properties of a circle Use the tangent / chord properties of a circle Prove the tangent / chord properties of a circle Use and prove the alternate segment theorem onstruct a stem-and-leaf diagram (ordered) onstruct a frequency diagram Inteprete a time series graph onstruct a time series graph and plot the moving average Use the trend line to estimate other values onstruct and interpret a cumulative frequency diagram Use a cumulative frequency diagram to estimate the median and interquartile range onstruct and interpret a box plot ompare two sets of data using box plots Solve equations such as x 4 = 5 + x Solve equations such as (5x + 1) = 8 Solve equations such as x 1 = (x 5) 7 x x Solve equations such as = or = = 6 x 4 Solve equations such as x 1 x 5 Two-Tier GSE Mathematics - Medium Term Plan

4 Statistical measures Sequences Properties of polygons Surds ecimals Graphs of linear functions Representing data Q Mathematics - Higher Tier 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 Write the terms of a sequence or a series of diagrams given the nth term Write the nth term of a sequence or a series of diagrams lassify a quadrilateral by geometric properties alculate exterior and interior angles of a regular polygon Simplify surds, such as ( 5 ) in the form a + b 5 Rationalise the denominator of a surd such as Multiply two decimals such as ivide a number by a decimal such as 1 0. and.8 divided by 0.07 onvert decimals to fractions and fractions to decimals onvert recurring decimals to fractions and fractions to recurring decimals Identify recurring and terminating decimals Solve problems involving graphs, such as finding where the line = x + y crosses the line y = 1 Recognise the equations of straight-line graphs such as y = 4 x + Find the gradients of straight-line graphs Explore the gradients of parallel straight-line graphs Explore the gradients of perpendicular straight-line graphs onstruct and interpret a histogram with unequal class intervals 5 Inequalities Solve inequalities such as x, 9 and 1 n < 0 Solve linear inequalities such as 4x < 10 and 4x < x + 7 Represent sets of solutions on the number line Two-Tier GSE Mathematics - Medium Term Plan

5 Q Mathematics - Higher Tier Solve linear inequalities such as x + 1 > 5x Solve a set of linear inequalities in two variables and represent the solution as a region of a graph Similarity and congruence Quadratic functions Trigonometry Rounding + dimensions Match sides and angles of similar triangles, given some dimensions Find the area of a - shape, given the area of a similar shape and the ratio Find the volume of a - solid, given the volume of a similar solid and the ratio Match one side and one angle of congruent triangles, given some dimensions Prove that two triangles are congruent Prove the construction theorems Solve quadratic equations such as x 8x + 15 = 0 by factorisation 4 Solve equations such as + = x + x 1 Solve equations such as x x 1 = 0 by using the quadratic formula Write quadratic expressions in forms like ( x + a) + b (that is, complete the square) Use completing the square to solve equations and find maximum and minimum values Sketch and draw trigonometric graphs Understand the graphs of trigonometric functions for angles of any size Estimate answers to calculations involving division Estimate answers to calculations involving division by numbers less than one Round to a given number of significant figures Find minimum and maximum values raw a scatter graph by plotting points on a graph Interpret the scatter graph Two-Tier GSE Mathematics - Medium Term Plan

6 Q Mathematics - Higher Tier Scatter graphs Identify the type and strength of the correlation raw a line of best fit on the scatter graph Integers Interpret the line of best fit Find the least common multiple (LM) of two or more numbers Find the highest common factor (HF) of two or more numbers Recognise prime numbers Write a number as a product of prime factors Find the reciprocal of a number Spring Quadratic Graphs Simultaneous equations Vectors raw graphs of simple quadratic functions such as y = x + 4 y = x and raw graphs of harder quadratic functions such as y = x x + 1 Find the points of intersection of quadratic graphs with lines Use graphs to find the approximate solutions of quadratic equations Use the points of intersection of a quadratic graph such as y = x x 4 with lines such as y = x + 1 to solve equations like x x 4 = x + 1 and simplify this to x 4x 5 = 0 Solve a pair of simultaneous equations in two unknowns such as x + y = 5 and x y = 4 Know that each equation can be represented by a line on a graph and that the point of intersection of the lines is the solution Solve a pair of simultaneous equations where one is linear and one is non-linear such as y = x and y = x Solve a pair of simultaneous equations where one is linear and one is non-linear such as x + 5 y = 1 and x + y = 1 dd, subtract and multiply vectors to solve vector geometry problems Two-Tier GSE Mathematics - Medium Term Plan

7 ubic, exponential and trigonometric functions Ratio and proportion Translation & enlargements Q Mathematics - Higher Tier Understand the relationship between parallel and perpendicular vectors Solve more difficult vector geometry problems omplete tables for, and draw graphs of cubic functions Use cubic graphs to solve equations Solve cubic equations by drawing appropriate lines on graphs omplete tables for, and draw graphs of reciprocal functions Use reciprocal graphs to solve equations Plot and sketch graphs of exponential functions Recognise the shapes of graphs of functions Solve simple ratio and proportion problems, such as finding the ratio of teachers to students in a school Solve more complex ratio and proportion problems, such as sharing out money between two groups in the ratio of their numbers Solve ratio and proportion problems using the unitary method alculate proportional changes using a multiplier Solve direct and inverse proportion problems Interpret the graphs of direct and inverse proportion relationships Translate a shape using a description such as 4 units right and units down 4 Translate a shape by a vector such as Transform shapes by a combination of translation, reflection and rotation ompare the area of an enlarged shape with the original shape Enlarge a shape by a positive scale factor from a given centre Enlarge a shape by a fractional scale factor Enlarge a shape by a negative scale factor ompare areas and volumes of enlarged shapes istinguish between formulae for perimeter, area and volume by considering dimensions S Use trigonometry to find sides and angles in three dimension Two-Tier GSE Mathematics - Medium Term Plan

8 Trigonometry in d Percentages Q Mathematics - Higher Tier Find the angle between a line and a plane Increase or decrease a quantity by a given percentage Understand how to use successive percentages Work out compound interest Express one quantity as a percentage of another Work out a percentage increase or decrease Work out reverse percentage problems Quadratic sequences Find the nth term for a quadratic sequence Rotations and reflections Trial and improvement Real life graphs Reflect shapes in lines such as x = or y = 1 Rotate shapes about the origin Reflect shapes in the lines y = x and y = x Rotate shapes about any point escribe fully reflections and rotations about the origin escribe fully reflections and rotations about any point Find the centre of a rotation and describe it fully ombine reflections and rotations Identify reflection symmetry in - solids Year 11 Form and solve equations such as x + x = 1 using trial and improvement methods alculate simple average speeds from distance time graphs alculate complex average speeds from distance time graphs Interpret velocity time graphs iscuss and interpret graphs modelling real situations Measures Solve more difficult speed problems Understand and use compound measures such as speed and density Two-Tier GSE Mathematics - Medium Term Plan

9 Q Mathematics - Higher Tier Recognise accuracy in measurements given to the nearest whole unit Find the upper and lower bounds of simple calculations involving quantities given to a particular degree of accuracy Find the upper and lower bounds of more difficult calculations with quantities given to a various degrees of accuracy Substitute numbers into more complicated formulae such as Formulae ( + 1) = 9 Find a solution to a problem by forming an equation and solving it Rearrange linear formulae such as p = q + 5 Rearrange formulae that include brackets, fractions and square roots Rearrange formulae where the variable appears twice onstruction raw a quadrilateral such as a kite or a parallelogram with given measurements Understand that giving the lengths of two sides and a non-included angle may not produce a unique triangle onstruct the perpendicular bisector of a line onstruct the bisector of an angle onstruct the perpendicular from a point to a line onstruct the perpendicular from a point on a line onstruct angles of 60 and 90 onstruct and recognise the nets of - solids such as pyramids and triangular prisms raw plans and elevations of - solids Loci Understand the idea of a locus onstruct accurately loci, such as those of points equidistant from two fixed points Solve loci problems, such as identifying points less than cm from a point P onstruct the graphs of loci, including the circle x + y = r Solve simultaneous equations graphically, such as y = x 1 and x + y = 16 Two-Tier GSE Mathematics - Medium Term Plan

10 Transforming functions lgebraic Proofs Q Mathematics - Higher Tier Solve simultaneous equations graphically, such as y = x 1 and x + y = 7 Transform the graphs of y = f ( x), such as linear, quadratic, cubic, sine and cosine functions, using the transformations y = f ( x) + a, y = f ( x + a), y = af ( x) and y = f ( ax) ecide with a reason whether a harder statement is true or false Identify a counter example Understand the difference between a demonstration and a proof Show step-by-step deductions in providing a basic algebraic explanation Show step-by-step deductions in providing a full mathematical explanation erive simple algebraic proofs using reasoning erive harder algebraic proofs using reasoning and logic Two-Tier GSE Mathematics - Medium Term Plan