# New Higher-Proposed Order-Combined Approach. Block 1. Lines 1.1 App. Vectors 1.4 EF. Quadratics 1.1 RC. Polynomials 1.1 RC

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1 New Higher-Proposed Order-Combined Approach Block 1 Lines 1.1 App Vectors 1.4 EF Quadratics 1.1 RC Polynomials 1.1 RC Differentiation-but not optimisation 1.3 RC Block 2 Functions and graphs 1.3 EF Logs and Log graphs and exp 1.1 EF Trig graphs 1.3 EF Integration 1.4 RC Recurrence 1.3 App Block 3 Circle 1.2 Ap Trig eqt 1.2 RC TRIG -plus wave function 1.2 EF Application of Differentiation 1.4 App Application of Integration 1.4 App

2 UNIT 1 Higher Maths The Straight Line 10 Periods (a) Finding gradient, using m=tanx Heinemann Ex 1A Maths In Action P4,5 Maths In Action P4,5 (b) Establishing collinearity, finding perpendicular and parallel lines (c) Using Y=mx+c, Re-arrange to General Equation Ax+By+C=0 He Ex 1B, 1C, 1D P6, 7 P6, 7 He Ex 1E, 1F P10 P10 (d) Using y-b=m(x-a) He Ex 1G P11,12,13 P11,12,13 (e) Equation of Perpendicular Bisector, finding midpoint (f) Equation of altitude and median He Ex 1I P12, 13 P12, 13 He Ex1K, 1M P12, 13 P12, 13 (g) Intersecting Lines/Problem Solving He Ex 1N, 1O P14,15,16 P14,15,16, Pegasus Assessment App 1.1 Teaching Notes The vocabulary within this topic e.g. median, altitude and perpendicular bisector should be emphasized and re-emphasized. Re-inforce prior knowledge e.g. parallel lines, equations of horizontal,vertical lines, proving if points lie on a line etc

3 Higher Maths Vectors 11 Periods (d) Find unit vectors and position vectors. Ex 13F, 13G P203, 204, P203, 204 (e) Verify collinearity. Ex 13I P (f) Use Section Formula Ex 13K P205 P205 (g) Find 3D vectors and use properties Ex 13L, 13M, 13N P197, 198 P197, 198 (h) Use scalar product Ex 13O, 13P P209, 210 P209, 210 (i) Calculate angle between vectors (j) Perform calculations for perpendicular vectors (k) Perform calculations w.r.t. properties of scalar product Ex 13Q P212, 213 P212, 213 Ex 13R P214, 215 P214, 215 Ex 13T, 13U P214, 215 P214, 215 Assessment EF 1.4 Teaching Notes Pupils have worked with vectors through N5-a brief review of previous learning should support learning (f) Example: A(3,2), B(7,14). Find the coordinates of P which divides AB in the ratio 1:3.A diagram should be added her!!, is this the best method? m:n = 1:3A(3,2) B(7,14) x P = = 4 y P = 4 4 = 5 So P is point (4,5)

4 Higher Maths Quadratic Functions 7 Periods (a) Sketch a Quadratic Function identifying y intercept, zeros, axis of symmetry and turning point (b) Use the Completing The Square method (c) Solve Quadratic Inequations by sketching the curve) (d) State nature of roots using the Discriminant (e) Determine tangency using the Disciminant or by Factorising Heinemann Ex 8B, 8C Maths In Action P33 Maths In Action P33 Scho lar Ex 8D MIA P30 MIA P30 Ex 8F MIA P118,119 MIA P118,119 Ex 8H, 8I MIA P118,119 MIA P118,119 Ex 8J MIA P120 MIA P120 (f) Problem Solving Ex 8K MIA P123 MIA P123, Pegasys Assessment RC 1.1 Note should be assessed with polynomials Assessment Standard Teaching Notes When stating solutions to problems pupils should use correct language Emphasise tangency and link to circle

5 Higher Maths Polynomials 8 Periods (a) Find degree, roots, coefficients. (b) Use nested form to evaluate. (c) Use synthetic division to find quotient and remainder. (d) Use synthetic division to factorise polynomials. (e) Find values of p and q in a polynomial. (f) Solve polynomial equations. (g) Identify graphs. (h) Sketch graphs of polynomial eqns of order 3 and 4. (i) Find approximate roots. Ex 7A (orally) Ex 7B Exs 7C, 7D Ex 7E Ex 7F Ex 7F Ex 7G Ex 7H Ex 7G Assessment RC 1.1 Should be assessed with Quadratics

6 Higher Maths Differentiation Periods (a) Find/Evaluating the Derivative Heinemann Ex 6D, 6E, 6F Maths In Action P62, 64 Maths In Action P62, 64 of y=x n, y=ax n (b) Find/Evaluating Derivatives of Products and Quotients Ex 6G P63, 65 MMS Bk8 P63, 65 (c) Find equations of tangents Ex 6J P67,68 P67,68 (h) Sketch graph of Derived Function Ex 6P P69,70 P69,70 Link with previous learning of graphs and inverse functions Find derivatives of sinx and cosx Heinemann Ex 14B Maths In Action P221, 222 Maths In Action P221, 222 Find Derivatives of (x+a) n, Ex 14D, 14E, 14F, 14G P225 P225 (ax+b) n Use the Chain Rule Ex 14H P226, 227, 228 P226, 227, 228 Teacher Notes A revision of indices will be needed to begin with. The main focus should be on the gradient of a tangent. Using first principles, work through a couple of examples, then give the general rule: if f(x) = x n, then f (x) =nx n 1 1. Common mistakes made with f(x) = 3 2x type. f(x) = = 3 x not 2 x 3 2x 2 More notes required here e.g. a simple way of understanding the chain rule. Assessment Teachers can assess each assessment standard individually or test as a complete unit

7 UNIT 2 Higher Maths Graphs And Functions 5 Periods (a) Understand and use Set notation Heinemann Ex 2A. Exs on board Maths In Action P20 Maths In Action P20 Identify Set Range and Domain (b) Find formula and evaluate Composite Functions and inverse function (c) Identify and draw graphs of Ex 2C P1, 2 P1, 2 Ex 3P, Also MIA P34, 35, 36 P34, 35, 36 y=f(x)+a, y=f(x+a), y=-f(x), y=f(-x), y=kf(x), y=f(kx), 2f(x), f(x+3), f(-x), 2-f(x). Inverse function Assessment EF 1.3 The domain is the set of values x can take. Ask what values x cannot be. E.g. f(x) = For range, ask what f(x) cannot be. And WHY! Pupils should be able to find algebraically and draw Inverse functions 1 x 2. Domain { x : x 2, x R} Pupils should spend time drawing and identifying graphs using small whiteboards or ICT etc rather than jotter work! Inclusion of Trig graphs and unusual graphs (e.g. logs and exp) will support learning and future topics

8 Higher Maths Trigonometric Graphs And Equations 6 Periods (a) Identify/Sketch graphs of Trigonometric Functions. Introduce period and amplitude. Heinemann Ex 4A(orally), 4B Maths In Action P50, 51, 52 Maths In Action P50, 51, 52 (b) Calculate angles in Radians. Ex 4C P45, 46 P45, 46 Convert between Radians and Degrees (c) Use triangles to find exact values (d) Solve Trigonometric equations graphically (e) Solve Trigonometric equations algebraically (f) Solve compound angle equations Ex 4E, P48,49 P48,49 Ex 4G P53 P53 Ex 4H P53,54 P53,54 Ex 4I P55 P55 Assessment RC1.2 Should be assessed after all Trig completed Teaching Notes Pupils should learn exact values from the graphs and from the 30,45,60 triangles ie they should start questions by drawing the triangles Graph sketching should re-inforce the learning from previous topic again use whiteboards to allow pupils to sketch graphs Pupils should be able to solve Trig equations with radians and degrees-some examples here

9 Higher Maths The Wave Function 6 Periods (a) Interpret and draw trigonometric graphs Ex 16A Maths In Action P50, 51 Maths In Action P50, 51 (b) Express acosx+bsinx in the form kcos(x-a) (c) Express asinx+bsinx in other forms including multiple angles Ex 16C, 16D P252, 253 P252, 253 Ex 16E, 16F P253 P253 (d) Find max and min values Ex 16G P254, 255, 256, 257 P254, 255, 256, 257 (e) Solving wave equations Ex 16H P258 P258 Assessment Standards EF1.2- RC1.2 Should be assessed with trig formula etc Teaching notes Should build on from the previous sketching of graphs and solving equations Methodology example of solving an equation Please provide an example-please refer to the SQA marking scheme when providing an example to ensure that pupils gain full credit for their working

10 Higher Maths Trigonometry 8 Periods (a) Expand sin(x+y) and sin(x-y) in order to find exact values (b) Expand cos(x+y), and cos(x-y) in order to find exact values Ex 11B, 11C P P Ex 11D P P (c) Prove Trig. identities Ex 11E P153 P153 (d) Expand sin2x and cos2x in order to find exact values (e) Solve trigonometric equations using double angle formulae Ex 11G P158, 159, 160 P158, 159, 160 Ex 11H P161, 162 P161, 162 Assessment standards RC 1.2 EF 1.2 Should be assessed with wave function Trig identities need now to developed within this topic Previous learning e.g. sin 2 x + cos 2 x = 1 and Tan a = Sin a/cos a should be re-visited Time should be spent at the end of this topic providing pupils with the opportunity to practice all types of trig equations (including previous topics) so that they can identify the appropriate strategy for different equations Assessment Unit 2 can be broken down into Assessment standards or taken as a whole group of topics

11 Higher Maths Logarithms And Exponentials 9 periods (a) Interpret and draw exponential graphs (b) Carry out exponential growth and decay calculations (c) Evaluate/Solve exponential functions to the base e (d) Use properties of logarithms to simplify/evaluate/solve (e) Solve equations using natural logarithms (f) Find a logarithmic formula from experimental data Heinemann Ex 15B Maths In Action P39 Maths In Action P39 Ex 15C P236, 237, 238 P236, 237, 238 Ex 15D P239, 240, 245 P239, 240, 245 Ex 15E, 15F, 15G P242, 243 P242, 243 Ex 15H P244, 245 P244, 245 Ex 15I, 15J P247 P247

12 Higher Maths Basic Integration 8 Periods (a) Find Indefinite Integrals for Ex 9F, 9G, 9H MIA P126 P126 x n and ax n (b) Integrate with Products and Quotients (c) Find area using definite integrals, Fundamental Theorem of Calculus Ex 9I P127, 128 P127, 128 Ex 9K, 9L P128, 129 P128, 129 (d) Find area above and below x axis Ex 9M, 9N P133, 134, 135, 136 P133, 134, 135, 136 (e) Find area between two graphs Ex 9O, 9P P137, 138 P137, 138 (f) Find the solution to Differential Equations Ex 9Q P129, 130 P129, 130 Integrate sinx and cosx Ex 14C P230 P230 Integrate (ax+b) n Ex 14J P229, 230 P229, 230 Integrate sin(ax+b) and cos(ax+b) Ex 14K P231 P231 Teaching notes The integral should be introduced as the anti-derivative, e.g. 3x² dx = x³, so x² dx = 1 x³. Do a few examples, then introduce + C. 3

13 Higher Maths Recurrence Relations 6 Periods (a) Construct/Evaluate formula for nth term in a sequence Heinemann Ex 5A Maths In Action P88, 89 Maths In Action P88, 89 (b) Find/Use a Recurrence Relation Ex 5B, 5C P91 P91 (c) Construct/Evaluate a Linear Recurrence Relation (d) Establishing if Limit exists Calculating limit Ex 5D P92 MIA Prep. For Ass. P15 P92 MIA Prep. For Ass. P15 Ex 5H (2 periods) P91 MIA Prep. For Ass. P16 P91 MIA Prep. For Ass. P16 (e) Solving Recurrence Relations to find a and b Ex 5I P92 P92 Assessment Standard App 1.3 Teaching Note Time should be spent solving problems and becoming familiar with the language used within the questions Example of finding the limit methodology should be given here e.g. L = 0.5 L + 0.4

14 UNIT 3 Higher Maths The Circle 8 Periods (a) Use the Distance Formula Heinemann Ex 12A,12B Maths In Action P2, 3 Maths In Action P2, 3 (b) Find the equation of a circle with centre at the origin and radius r (c) Find the equation of a circle with centre (a,b) and radius r (d) Use the general equation of circle to find g, f, c and then r (e) Find point(s) of intersection for a line and circle (f) Find the point of intersection and equation of tangent to a circle Ex 12D P168, 169 P168, 169 Ex 12F P170, 171 P170, 171 Ex 12G, 12H P172, 173 P172, 173 Ex 12I, Ex 12J P174, 175 P174, 175 Ex 12K, 12L P176, 177, 178 P176, 177, 178 Assessment App 1.2 Teaching notes Review previous knowledge of circles including angles /diameter and tangents/properties of a Kite/Rhombus etc Pupils should be encouraged to solve problems by drawing diagrams, looking for isosceles triangles, right angled triangles and if required looking for additional information to improve diagrams to allow problems to be attempted Previous learning within quadratics and polynomials support solving problems and should be emphasized

15 Higher Maths Applications of Differentiation 13 Periods (d) Determine whether functions are increasing or decreasing Ex 6L P73 P73 Rate of change problem (e) Find Stationary Points Ex 6L P72 P72 Determine Nature (f) Sketch curve by identifying x- intercept, y intercept, stationary points, large values of + and - x (g) Determine max and min values of function for closed interval (h) Sketch graph of Derived Function Ex 6N P74, 75 MMS Bk8 P74, 75 Ex 6O P76, 77 P76, 77 Ex 6P P69,70 P69,70 Link to inverse functions and previous graph work Perform optimisation calculations Ex 6Q, 6R P76, 77, 78 P76, 77, 78 Teaching notes Exam technique in difficult optimisation problems-could include- attempting the proof at the end of the question after completing finding the x value etc!!

16 Higher Maths Application of Integration 8 Periods Find area using definite integrals, Fundamental Theorem of Calculus Ex 9K, 9L P128, 129 P128, 129 Find area above and below x axis Ex 9M, 9N P133, 134, 135, 136 P133, 134, 135, 136 Find area between two graphs Ex 9O, 9P P137, 138 P137, 138 Find the solution to Differential Equations Ex 9Q P129, 130 P129, 130 Assessment App 1.2 Assessed with differentiation application Further Notes HW-could we produce a standard set of HWs? revising current and previous topics as we go through the course-including 1 or 2 reasoning questions/ past paper type questions METHODOLOGY We should provide further guidance on effective methodology-bringing together the experience of all our teachers-providing a a more consistent approach and support to less experienced teachers Activities-link to good teaching activities-hyperlinks?-video clips?-smart board resources Assessment Support Assessments and re-assessments to be provided plus a recording tracking grid Could we also produce student assessment review sheets-ie what I have learned what I have still to learn and a link to resources to assist learning?

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