King s Year 12 Medium Term Plan for LC2- A-Level Mathematics

Transcription

1 King s Year 12 Medium Term Plan for LC2- A-Level Mathematics Modules Algebra, Geometry and Calculus. Materials Text book: Mathematics for A-Level Hodder Education. needed Calculator. Progress objectives Calculus objectives (AQA G1, G2, G3, G4, H1, H2, H3) Ø Understand and use the derivative of f x as the gradient of the tangent to the graph of y = f x at a general point x, y ; the gradient of the tangent as a limit; interpretation as a rate of change; sketching the gradient function for a given curve; second derivatives; differentiation from first principles for small positive integer powers of x. Ø Understand and use the second derivative as the rate of change of gradient. Ø Differentiate x &, for rational values of n, and related constant multiples, sums and differences. Ø Apply differentiation to find gradients, tangents and normal lines, maxima and minima and stationary points. Ø Identify where functions are increasing or decreasing. Ø Understand and use the derivative of sin x and cos x. Ø The second derivative and its connection to convex and concave sections of curves and points of inflection. Ø Differentiate ekx and akx, sin kx, cos kx, tan kx and related sums, differences and constant multiples Ø Understand and use the derivative of ln x. Ø Apply differentiation to find points of inflection. Ø Differentiate using the product rule, the quotient rule and the chain rule, including problems involving connected rates of change and inverse functions. Ø Know and use the Fundamental Theorem of Calculus. Ø Integrate x & (excluding n = 1), and related sums, differences and constant multiples. Ø Evaluate definite integrals; use a definite integral to find the area under a curve. Ø Integrate ekx, 4, sin kx, cos kx and related sums, differences and constant multiples. 5 Ø Use a definite integral to find the area between two curves. Ø Understand and use integration as the limit of a sum. Ø Carry out simple cases of integration by substitution and integration by parts; understand these methods as the inverse processes of the chain and product rules respectively. (Integration by substitution includes finding a suitable substitution

2 and is limited to cases where one substitution will lead to a function which can be integrated; integration by parts includes more than one application of the method but excludes reduction formulae). Geometry objectives (AQA C1, C2) Ø Understand how to use vectors in two dimensions. Ø Calculate the magnitude and direction of a vector and convert between component form and magnitude/direction form. Ø Add vectors diagrammatically and perform the algebraic operations of vector addition and multiplication by scalars, and understand their geometrical interpretations. Ø Understand and use position vectors; calculate the distance between two points represented by position vectors. Ø Use vectors to solve problems in pure mathematics and in context, including forces. There will be an assessment at the end of LC2, followed by a week of GAP work to reteach any content missed or that needs to be learned better. Nevertheless, pupils will complete several knowledge checks at the end of every two units. Overview of units and lesson allocation.

3 Week 1 Differentiation Chapter 10 Hypothesis 1: The gradient of the tangent at a point in a curve is the same as the gradient of the chord that joins two points of that curve. (Point 10.1) ü Understand the gradient of the tangent as a limit (first principles). ü Analyse how to calculate the gradient of a curve using approximation or tendency to a limit. ü Apply your knowledge to calculate the gradient of different monomial curves. Hypothesis 2: To calculate the limit, we can make h converge to any value. (Point 10.9) ü Understand what convergence means. ü Analyse how to calculate the derivative using first principles. ü Apply your knowledge to calculate the derivative of simple polynomial curves using first principles. Hypothesis 3: To calculate the gradient of a curve we must calculate the gradient of the tangent. (Point 10.2) ü Identify from previous work what the derivative of a monomial such as x 6 is. ü Analyse how to calculate the derivative of polynomial functions such as y = 2x 6 x ü Apply your knowledge to differentiate polynomials involving brackets and fractions. Hypothesis 4: We can only calculate the equation of the tangent working out m and c in the equation y=mx+c. (Point 10.3) ü Recall the different ways of finding the equation of a line. ü Analyse how to use derivatives in order to calculate the equation of the tangent to a curve at a point. ü Evaluate how to calculate the equation of the normal line of a curve at a point from the equation of the tangent. ü Hypothesis 5: A function can only either increase or decrease, but never do both. (Point 10.4 and 10.5) ü Understand what an increasing/decreasing function is. ü Analyse how the gradient of a curve can determine the growth of a function or graph. Apply your knowledge to find minimum and maximum points of a curve in order to sketch the curve.

4 Week 2 Differentiation Chapter 10 Hypothesis 1: We cannot use the rule used to calculate the derivative of polynomials to calculate the derivative of x = 4t 5?3 t t 2. (Point 10.6) ü Understand how to transform functions such as square roots and fractions into polynomials. ü Analyse how to differentiate more complicated polynomials including negative and fractional powers. ü Apply your knowledge to solve worded questions. v Synoptic knowledge check from previous weeks. Hypothesis 2: We have to use the first principles to calculate the derivative of functions such as sine and cosine. (Y13 content) ü Understand the derivatives of the cos x, sin x and tan x. ü Analyse what the graphs of e 5 and ln x look like. ü Evaluate how to calculate the derivatives of more complex functions such as e 5, a F5, sin kx, cos kx, tan kx, ln x. Hypothesis 3: The quotient rule can be replaced by the product rule. (Y13 content) ü Understand the product, quotient and chain rules for simple functions such as sin 3x. ü Analyse what the rules would look like for more complicated functions such as 7x 6 cos 4x 8 x. ü Apply your knowledge to calculate the derivatives of different functions. Hypothesis 4: The quotient rule can be replaced by the product rule. (Point 10.7 and Y13 content) ü Understand what the second derivative in x means for polynomial curves and how to calculate it. ü Analyse how to determine if a point is a maximum or a minimum using the second derivatives. ü Evaluate what type of curvature a function has depending on the values of the second derivative. Hypothesis 5: Only a maximum can be a point of inflection. (Y13 content) ü Understand how many points of inflection a polynomial curve can have. ü Analyse the points of inflection in a curve using derivatives. ü Apply your knowledge to sketch graphs of different polynomial curves.

5 Week 3 Integration Chapter 11 Hypothesis 1 and 2: To integrate a polynomial we calculate the derivative where it came from. ü Understand what integration means. ü Analyse how to integrate simple polynomials in the form of x n. ü Apply your knowledge to calculate integrals of polynomials of any grade including fractions. v Synoptic knowledge check from previous weeks. Hypothesis 3 and 4: To find the area beneath a curve we approximate the area to a 2 D shape and then work out its area. ü Understand what an indefinite and a definite integral are as well as their differences. ü Analyse why we can use integration to calculate the area beneath a graph. ü Evaluate how to calculate the area beneath a graph using integration. Hypothesis 5: The area below the X axis is just the area above the axis but negative. ü Understand what the area below the graph is, linked to what we learned in the previous lessons. ü Analyse what the differences are between calculating the area above and below the X axis. ü Apply your knowledge to calculate the area between the X axis and a negative function.

6 Week 4 Integration Chapter 11 Hypothesis 1: To integrate a polynomial with a negative power we need to use the rule to convert negative indices into fractions first. ü Recall the rules of negative and fractional indices. ü Analyse how we could convert a polynomial with fractions into a polynomial with negative indices. ü Apply your knowledge of powers to integrate negative powers. v Synoptic knowledge check from previous weeks. Hypothesis 2 and 3: To integrate the sine we need to calculate the cosine. (Y13 content) ü Recall what an indefinite and a definite integral are as well as their differences. ü Analyse how to integrate ekx, 4, sin (kx), cos kx and related sums, differences and constant multiples. 5 ü Evaluate how to calculate the area beneath the graphs above using integration. Hypothesis 4: To solve an integral by substitution we can use any other function. (Y13 content) ü Understand what integrating by substitution means. ü Analyse how to find a change in variable to complete the right substitution. ü Apply your knowledge to calculate integrals by substitution. Hypothesis 5: To solve an integral by parts we just split the integral into smaller ones. (Y13 content) ü Understand what integrating by parts means. ü Analyse how to identify when to use this method to solve integration. ü Apply your knowledge to calculate integrals by parts. Week 5 Vectors Chapter 12 Hypothesis 1: Having a vector in 2D, we cannot use the sin and cos rules, but we can use SOHCAHTOA. ü Understand new terminology for vectors. ü Understand trigonometric rules to calculate cos and sin. ü Evaluate how to convert between component form and magnitude-direction form.

7 v Synoptic knowledge check from previous weeks. Hypothesis 2 and 3: We always use BIDMAS with vectors. ü Recall how to multiply a vector by a scalar and what the negative vector is. ü Analyse how to add and subtract vectors using their component form. ü Evaluate how to calculate unit vectors. Hypothesis 4 and 5: A hexagon has nothing to do with vectors. ü Recall everything learned about vectors this week. ü Analyse how to express one vector as a result of a calculation between another two. ü Apply your knowledge to solve questions involving 2D shapes. Week 6 Week 7 During week 6, pupils will revise all topics seen from September in order to cover any gaps they may have. Ø Surds and Indices. Ø Quadratic functions. Ø Equations and inequalities. Ø Coordinate Geometry. Ø Polynomials. Ø Graphs and transformations. Ø Differentiation Ø Integration Ø Vectors The Assessment for LC1 and LC2 will be done this week too. REACH (1H) (Review & Recap, Evaluate & Endeavour, Attainment & Achievement, Challenge yourself, Hone your skills) Improvement time to be allocated in one lesson every week from week 2 so as to learn from mistakes done the week before. Individual feedback will be provided as well as personalised improvement questions depending on what each pupil has done wrong in the assessment.

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