# Higher Scheme of Work Year

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1 Higher Scheme of Work Year

2 Assumed knowledge Understand and use positive numbers and negative integers, both as positions and translations on a number line answers to a division sum using multiplication eg use inverse operations Multiply and divide whole numbers by a given multiple of 10 Put digits in the correct place in a decimal number Compare the sizes of fractions Recall and use properties of angles angles at a point angles at a point on a straight line perpendicular lines vertically opposite angles Distinguish between scalene, isosceles, equilateral, and right-angled triangles Understand and use the angle properties of triangles Draw parallel lines Draw circles and arcs to a given radius Measure sides of a rectangle to work out perimeter or area Find the perimeter of rectangles and triangles Draw, label and scale axes Recall the definition of a circle and identify (name) and draw the parts of a circle Understand related terms of a circle Draw a circle given the radius or diameter 2

3 General Topic Higher Tier Objectives Grade Clip: on mathswatchvle By the end of the module students should be able to 1. Number Identify factors, multiples and prime number 3 Find the prime factor decomposition of positive integers 3 Clip 78 Find the common factors and common multiples of two numbers 3 Find the Highest Common Factor (HCF) and the Least Common Multiple (LCM) of 3 Clip 79/80 two numbers Recall integer squares from 2 2 to and the corresponding square roots 3 Clip 81 Recall the cubes of 2, 3, 4, 5 and 10 3 Clip 81 Use index notation for squares and cubes 3 Use brackets and the hierarchy of operations (BIDMAS) 3 clip 75 Use index laws to simplify and calculate the value of numerical expressions involving multiplication and division of integer powers, and powers of a power 4 Clip 131 Use a calculator efficiently including index buttons, root buttons and fraction button. 3 Clip 77 Understand how to find the reciprocal of a number 3 Clip 76 Use index laws to simplify and calculate numerical expressions involving powers, 5 Clip 154 eg ( ) 2 4, 4 0, 8 2/3 3

4 2. Expressions and sequences Use notation and symbols correctly Higher Tier Write an expression Manipulate algebraic expressions by collecting like terms 2 Clip 33 Use index laws to simplify algebraic expressions 4 Clip 131 Substitute numbers into a formula 3 Clip 95 Recognise sequences of odd and even numbers Generate simple sequences of numbers, squared integers and sequences derived from diagrams Describe the term-to-term definition of a sequence in words Identify which terms cannot be in a sequence Generate specific terms in a sequence using the position-to-term and term-toterm rules Apply index laws for negative indices Apply index laws for fractional indices Find the n th term of an arithmetic sequence 3 Clip 103 Use the n th term of an arithmetic sequence to generate terms 3 Clip 102 Recognise and use sequences of triangular, square and cube numbers, Fibonacci type sequences, quadratic sequences and geometric sequences 4 Clip 141 4

5 Find the nth term of a quadratic sequence and generate terms 8/9 Clip 214 Recognise and use geometric sequences where the common ratio may be a surd 3. Fractions 4. Decimals and estimation Find equivalent fractions Write a fraction in its simplest form Find fractions of an amount Convert between mixed numbers and improper fractions Add and subtract fractions 3 Clip 71 Multiply and divide fractions including mixed numbers 3 Clip 73/74 Find fractions of amounts 3 Clip 72 Multiply and divide by any number between 0 and 1 3 Clip 66/67 Multiply and divide decimal numbers by whole numbers and decimal numbers (up to 2 d.p.) eg Convert between fractions and decimals and recognise when a fraction will 3 Clip 84 convert to a terminating or recurring decimal Understand and order integers and decimals 3 Clip 70 Revise rounding whole numbers to the nearest, 10, 100, 1000, Round decimals to appropriate numbers of decimal places or significant figures 2 Clip 32 3 Clip 90 Check their calculations by rounding, eg Clip 91 Using the answer to a calculation to work out others Convert recurring decimals to fractions and understand the proof of this 6 7 Clip 177 Clip 189 5

6 Calculate the upper and lower bounds of calculations, particularly when working with measurements Find the upper and lower bounds of calculations involving perimeter, areas and volumes of 2-D and 3-D shapes Find the upper and lower bounds in real life situations using measurements given to appropriate degrees of accuracy Give the final answer to an appropriate degree of accuracy following an analysis of the upper and lower bounds of a calculation 4 Clip 132 8/9 Clip 206 8/9 Clip 206 8/9 Clip 206 Revision, Chapter 2 and 4 Assessments, DIRT lesson 5. Angles and polygons Mark parallel lines on a diagram Use the standard convention for labelling sides and angles of polygons Understand and use the angle properties of parallel lines 3 Clip 120 Use the properties of corresponding and alternate angles Understand and use the angle properties of intersecting lines Give reasons for angle calculations Know and use the properties of special triangles 3 Clip 122 Use the size/angle properties of isosceles and equilateral triangles. Recall and use these properties of angles in more complex problems Derive and use the angle sum of a triangle is 180º 3 Clip 121 Understand the proof that the angle sum of a triangle is 180º Understand a proof that the exterior angle of a triangle is equal to the sum of the interior angles at the other two vertices 6

7 Understand and use the angle properties of quadrilaterals Explain why the angle sum of a quadrilateral is 360º 3 Clip 123 Understand what angles of elevation and depression are Understand, draw and measure bearings 3 Clip 124 Calculate bearings and solve bearings problems Calculate and use the sums of the interior angles of polygons 3 Clip 123 Use geometric language appropriately and recognise and name pentagons, hexagons, heptagons, octagons and decagons 6. Collecting and recording data Use the sum of the exterior angles of any polygon is 360º Use the sum of the interior angle and the exterior angle is 180º Calculate and use the angles of regular polygons Use the sum of the interior angles of an n sided polygon Use the angle sums of irregular polygons Specify the problem and plan Decide what data to collect and what statistical analysis is needed Collect data from a variety of suitable primary and secondary sources Understand that there are different types of data: qualitative, quantitative, discrete, continuous 2 Clip 63 Use suitable data collection techniques Process and represent the data Interpret and discuss the data Discuss how data relates to a problem Understand how different sample sizes may affect the reliability of conclusions drawn Identify which primary data they need to collect and in what format, including grouped data Consider fairness Understand sample and population 4 Clip 142, 152 7

8 Select and justify a sampling scheme and a method to investigate a population, including random and stratified sampling Use stratified sampling Design and use data-collection sheets for grouped, discrete and continuous data 5 Clip Measure Collect data using various methods Sort, classify and tabulate data and discrete or continuous quantitative data Group discrete and continuous data into class intervals of equal width Extract data from lists and tables Design and use two-way tables for discrete and grouped data Use information provided to complete a two way table 2 Clip 61 Convert between units of measure in the same system. 3 Clip 112 Use the relationship between distance, speed and time to solve problems Convert between metric units of speed eg km/h to m/s Know that density is found by mass volume Use the relationship between density, mass and volume to solve problems, eg find the mass of an object with a given volume and density Convert between metric units of density eg kg/m to g/cm Calculate speed when, eg fractions of an hour must be entered as fractions or as decimals Calculate compound measures including pressure in numerical and algebraic contexts Revision, Assessments Chapter 6 and 7, DIRT lesson 4 Clip Congruence, symmetry and similarity 8

9 Understand and use SSS, SAS, ASA and RHS conditions to prove the congruence of triangles using formal arguments, and to verify standard ruler and a pair of compasses constructions Recognise line and rotational symmetry of 2D shapes. Draw lines of symmetry. Find order of rotational symmetry. Draw shapes with given line/rotational symmetry. Understand similarity of triangles and of other plane figures, and use this to make geometric inferences Formal geometric proof of similarity of two given triangles Recognise that all corresponding angles in similar figures are equal in size when the lengths are not Solve similar triangles problems. 5 Clip Clip Expanding brackets and factorising Multiply a single term over a bracket 3 Clip 93 Multiply out multiple single brackets and simplify expressions Factorise algebraic expressions by taking out common factors 3 Clip 94 Expand double brackets 4 Clip 134 Factorise quadratic expressions 5 Clip 157 Factorise quadratic expressions using the difference of two squares 5 Clip 158 Expand three binomials 6 Clip 178 9

10 10. Area and volume 1 Higher Tier Calculate perimeters of shapes made from triangles and rectangles 2 Clip 52 Calculate areas of shapes made from triangles and rectangles 2 Clip 53 Recall and use the formulae for the area of a triangle, rectangle and a 2 Clip 54 parallelogram Find the area of a parallelogram and a trapezium 2 Clip 55, 56 Calculate perimeter and area of compound shapes made from triangles, rectangles and other shapes Know the parts of a circle 3 Clip 116 Find circumferences of circles and areas enclosed by circles 3 Clip 118, 117 Recall and use the formulae for the circumference of a circle and the area enclosed by a circle Use π or use the π button on a calculator Find the perimeters and areas of semicircles and quarter circles Use 2-D representations of 3-D shapes Use isometric grids Draw nets and show how they fold to make a 3-D solid 2 Clip 44 Understand and draw front and side elevations and plans of shapes made from simple solids Given the front and side elevations and the plan of a solid, draw a sketch of the 3-D solid 2 Clip 51 10

11 Know and use formulae to calculate the surface areas and volumes of cuboids and right-prisms Solve a range of problems involving surface area and volume, eg given the volume and length of a cylinder find the radius Find the volume of a cylinder and surface area of a cylinder 3 Clip 114, 119 Revision, Assessment of chapters 9 and Averages and range Calculate mean, mode, median and range for small data sets 2 Clip 62 Consider outliers when calculating the range of a distribution Recognise the advantages and disadvantages between measures of average Calculate averages and range from frequency tables (Use Σx and Σfx) 3 Clip Constructions and loci Estimate the mean for large data sets with grouped data (and understand that it is an estimate) Compare distributions and make inferences, using the shapes of distributions and measures of average and spread, including median and quartiles 6 Clip 186 Use straight edge and a pair of compasses to do standard constructions 4 Clip 145, 146 Construct triangles including an equilateral triangle 4 Clip 147 Understand, from the experience of constructing them, that triangles satisfying SSS, SAS, ASA and RHS are unique, but SSA triangles are not Construct the perpendicular bisector of a given line 4 Clip

12 Construct the perpendicular from a point to a line 4 Clip 146 Construct the perpendicular from a point on a line 4 Clip 146 Construct the bisector of a given angle 4 Clip 145 Construct angles of 60º, 90º, 30º, 45º Construct a regular hexagon inside a circle Construct diagrams of everyday 2-D situations involving rectangles, triangles, perpendicular and parallel lines Draw and construct diagrams from given information Construct: -a region bounded by a circle and an intersecting line a given distance from a point and a given distance from a line equal distances from 2 points or 2 line segments regions which may be defined by nearer to or greater than Find and describe regions satisfying a combination of loci 5 Clip 165 Use and interpret maps and scale drawings Read and construct scale drawings drawing lines and shapes to scale Estimate lengths using a scale diagram 13. Linear equations Solve simple linear equations 4 Clip 135 Solve linear equations, with integer coefficients, in which the unknown appears on either side or on both sides of the equation 12

13 Solve linear equations that include brackets, those that have negative signs occurring anywhere in the equation, and those with a negative solution Solve linear equations in one unknown, with integer or fractional coefficients Understand the symbol Set up linear equations from word problems 4 Clip 137 Select an expression/identity/equation/formula from a list 14. Percentages Understand that a percentage is a fraction in hundredths Convert between fractions, decimals and percentages 3 Clip 85 Write one number as a percentage of another number 3 Clip 88, 89 Calculate the percentage of a given amount 3 Clip 86, 87 Find a percentage increase/decrease of an amount 3 Clip 108 Use a multiplier to increase by a given percent over a given time, eg increases 64 by 10% over 8 years Calculate simple and compound interest 3 Clip 111 Find a reverse percentage, eg find the original cost of an item given the cost after a 10% deduction Revision, Assessment of chapters 13 and 14, DIRT lesson 3 Clip

14 15. Graphs Higher Tier Use axes and coordinates to specify points in all four quadrants in 2-D and 3-D Identify points with given coordinates Identify coordinates of given points (NB: Points may be in the first quadrant or all four quadrants) Find the coordinates of points identified by geometrical information in 2-D and 3- D Find the coordinates of the midpoint of a line segment, AB, given the coordinates 4 Clip 133 of A and B Plot and draw graphs of straight lines with equations of the form 3 Clip 96 y = mx + c Recognise that equations of the form y = mx + c correspond to straight-line graphs in the coordinate plane Find the gradient of a straight line from a graph 3 Clip 97 Find the gradient of a straight line from its equation Explore the gradients of parallel lines and lines perpendicular to each other Write down the equation of a line parallel or perpendicular to a given line Use the fact that when y = mx + c is the equation of a straight line then the gradient of a line parallel to it will have a gradient of m and a line perpendicular to this line will have a gradient of 1 m 8/9 Clip 208 Interpret and analyse a straight line graph and generate equations of lines parallel and perpendicular to the given line Find the equation of a line through two points or through one point given gradient Construct and interpret distance time graphs 4 Clip

15 Interpret areas under graphs and gradients of graphs in real-life contexts (e.g. recognise that the area under a velocity time graph represents displacement) Draw straight line graphs for real-life situations ready reckoner graphs conversion graphs fuel bills, eg gas and electric - fixed charge (standing charge) and cost per unit Analyse problems and use gradients to interpret how one variable changes in relation to another Interpret and analyse a straight-line graph 16. Ratio and proportion Use ratios 2 Clip 38 Write ratios in their simplest form Divide a quantity in a given ratio 3 Clip 106 Solve a ratio problem in a context Solve word problems about ratio and proportion Calculate an unknown quantity from quantities that vary in direct or inverse proportion Express a multiplicative relationship between two quantities as a ratio or fraction 2 Clip 42 Write a ratio as a linear function 3 Clip 107 Revision, Assessments on chapters 15, 16 DIRT lesson 15

16 17. Transformations Higher Tier Understand translation as a combination of a horizontal and vertical shift including signs for directions Translate a given shape by the vector Reflect shapes in a given mirror line; parallel to the coordinate axes and then y = x or y = x 2 Clip 50 2 Clip 48 Clip 148 Clip 181 Understand that shapes produced by translation, rotation and reflection are congruent to the image Understand rotation as a (anti clockwise) turn about a given origin 2 Clip 49 Describe and transform 2-D shapes using single rotations Understand that rotations are specified by a centre and an (anticlockwise) angle Find the centre of rotation Rotate a shape about the origin, or any other point Describe and transform 2-D shapes using combined rotations, reflections, translations, or enlargements Use congruence to show that translations, rotations and reflections preserve length and angle, so that any figure is congruent to its image under any of these transformations Distinguish properties that are preserved under particular transformations 6 Clip 182 Describe a transformation 16

17 Enlarge shapes by a given scale factor from a given point; using positive and negative scale factors greater and less than one Find the centre of enlargement 4 6 Recognise that enlargements preserve angle but not length, linking to similarity 18. Processing, representing and interpreting data Draw and from pie charts: find the total frequency and find the frequency represented by each sector Produce: composite bar charts, comparative and dual bar charts, pie charts, frequency polygons, histograms with equal class intervals and frequency diagrams for grouped discrete data, scatter graphs, line graphs, frequency polygons for grouped data, grouped frequency tables for continuous data Interpret: composite bar charts, comparative and dual bar charts, pie charts, scatter graphs, frequency polygons and histograms Recognise simple patterns, characteristics and relationships in line graphs and frequency polygons Present findings from databases, tables and charts 3 Clip Clip 64 From line graphs, frequency polygons and frequency diagrams: read off frequency values, calculate total population, find greatest and least values Find the median from a histogram or any other information from a histogram, such as the number of people in a given interval Draw histograms 7 Clip 205 Find the median from a histogram From histograms: complete a grouped frequency table and understand and define frequency density Draw cumulative frequency tables and graphs 6 Clip

18 Use cumulative frequency graphs to find median, quartiles and interquartile range Draw box plots from a cumulative frequency graph 6 Clip 187 Compare the measures of spread between a pair of box plots/cumulative frequency graphs Interpret box plots to find median, quartiles, range and interquartile range Revision, Assessments on chapters 17, 18 DIRT lesson 18

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