King s Year 12 Medium Term Plan for LC1- A-Level Mathematics Modules Algebra, Geometry and Calculus. Materials Text book: Mathematics for A-Level Hodder Education. needed Calculator. Progress objectives Algebra objectives (AQA B1, B2, B3, B4, B5, B6, B7, B9) Ø Understand and use the laws of indices for all rational exponents. Ø Use and manipulate surds, including rationalising the denominator. Ø Work with quadratic functions and their graphs; the discriminant of a quadratic function, including the conditions for real and repeated roots; completing the square; solution of quadratic equations including solving quadratic equations in a function of the unknown. Ø Solve simultaneous equations in two variables by elimination and by substitution, including one linear and one quadratic equation. Ø Manipulate polynomials algebraically, including expanding brackets and collecting like terms, factorisation and simple algebraic division; use of the factor theorem. Ø Solve linear and quadratic inequalities in a single variable and interpret such inequalities graphically, including inequalities with brackets and fractions. Ø Express solutions through correct use of and and or, or through set notation. Ø Represent linear and quadratic inequalities such as y > x + 1 and y > ax ' + bx + c graphically. Ø Understand and use graphs of functions; sketch curves defined by simple equations including polynomials. Ø y = + and y = + (including their vertical and horizontal asymptotes); interpret algebraic solution of equations,,- graphically; use intersection points of graphs to solve equations. Ø Understand and use proportional relationships and their graphs. Ø Understand the effect of simple transformations on the graph of y = f x including sketching associated graphs: y = af x, y = f x + a, y = f x + a, y = f ax.
Geometry objectives (AQA C1, C2) Ø Understand and use the equation of a straight line, including the forms y y 1 = m x x 1 and ax + by + c = 0; gradient conditions for two straight lines to be parallel or perpendicular. Ø Be able to use straight line models in a variety of contexts. Ø Understand and use the coordinate geometry of the circle including using the equation of a circle in the form (x a) ' + (y b) ' = r ' ; completing the square to find the centre and radius of a circle; use of the following properties: Ø The angle in a semicircle is a right angle. Ø The perpendicular from the centre to a chord bisects the chord. Ø The radius of a circle at a given point on its circumference is perpendicular to the tangent to the circle at that point. Calculus objectives (AQA G1, G2, G3, G4) 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 8, 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. There will be no formal assessment in LC1. Nevertheless, pupils will complete several knowledge checks at the end of every two units.
Overview of units and lesson allocation. Week 1 Surds and Indices Chapter 2 Hypothesis 1: To rationalise an expression we need to multiply by its denominator. ü Understand how to multiply and simplify surds. ü Analyse how to rationalise expressions with surds in the denominator. ü Apply your knowledge to real-life questions and proof. 2 Hypothesis 2: 4 B2/3 = 4 3 = 8 ü Recall the laws of indices for multiplication, division, negative indices and fractional indices. ü Analyse how to apply the previous rules to mixed base powers and more difficult expressions such as 3x(x + 7) I - 2(x + 7) K/'. ü Evaluate what happens with a power such as 0.5 h when h.
Quadratic functions - Chapter 3 Hypothesis 3: To plot a quadratic graph using the calculator, we always use the same settings. ü Understand how to use a graphical calculator to produce a quadratic graph and solve problems quicker. ü Analyse how to factorise quadratic equations. ü Evaluate how to plot quadratic graphs using the factorised expression of their quadratic equation. Hypothesis 4: It is useful to factorise a quadratic expression using the completing the square method in order to plot it. ü Recall how to write a quadratic equation using the completing the square method. ü Analyse how to find the turning point or vertex of a parabola from the completing the square method. ü Apply your knowledge of equations to solve a quadratic from the completing the square form. Hypothesis 5: We need to solve a quadratic equation to know the number of solutions it has. ü Recall how to solve a quadratic equation using the formula. ü Analyse how the discriminant can help is plot a quadratic graph. ü Apply your knowledge to real-life questions and worded problems. Week 2 Simultaneous equations Chapter 4 Hypothesis 1: We can easily find the solutions for a pair of simultaneous equations by using trial and error. v Knowledge check from week 1 ü Recall how to solve a pair of linear equations by substitution or elimination. ü Understand how to use a graphical calculator to solve linear simultaneous equations. ü Analyse how the solution of a pair of linear simultaneous equations can be found graphically. Hypothesis 2: A pair of simultaneous equations will always give us only two solutions, one for x and one for y. ü Understand how to solve simultaneous equations involving a quadratic equation. ü Analyse how to solve simultaneous equations involving a quadratic equation using a calculator. ü Evaluate how many possible solutions this type of simultaneous equations could have.
Polynomials Chapter 7 Hypotheses 3 and 4: The dominant term does not determine what will happen when x. ü Recall how to add and multiply polynomials. ü Understand how the degree of the polynomial affects the number of turning points. ü Evaluate how to sketch polynomials. Hypothesis 5: If we divide a polynomial of order 5 by a quadratic factor, the result will be a polynomial of order 7. ü Understand how to divide polynomials. ü Evaluate how to use division of polynomials to factorise one of grade 3 and above. ü Apply your knowledge of division to sketch polynomials. Week 3 Hypothesis 1: To use the factor theorem, we need to complete the table of values for our polynomial equation. ü Understand the difficulty of solving a polynomial equation of order 3 and above. ü Analyse how to solve polynomial equations using the factor theorem. ü Apply your knowledge to solve real-life worded questions Inequalities Chapter 4 Hypothesis 2: To solve a quadratic inequality we always need to plot it. ü Recall how to solve linear inequalities. ü Understand how to solve quadratic inequalities graphically. ü Analyse how to solve quadratic inequalities algebraically. Hypothesis 3: The solution of inequalities can be written in a simple sentence. ü Understand how to express solutions of inequalities using and, or or set notation. ü Analyse how to represent linear and quadratic inequalities graphically. ü Apply your knowledge to solve real-life questions.
Graphs and Transformations Chapter 8 Hypothesis 4: Proportion cannot be shown on a graph. ü Identify each type of graph with their own graph. ü Analyse reciprocal graphs and their properties. ü Evaluate how proportion can be shown using graphs. Hypothesis 5: Given the function y = x 2, the function y = (x-2) 2 means I must plot the same graph but 2 units lower. ü Understand what a translation does to a graph and how to express it algebraically. ü Analyse what a reflected graph would look like. ü Evaluate how to express a reflection algebraically. Week 4 Hypothesis 1: Graphs can be translated in more than one way. ü Recall how a translated graph can be written algebraically. ü Analyse what the algebraic expression for a two-way translated graph is. ü Apply your knowledge to solve worded questions. Hypothesis 2: Trigonometric graphs follow the same transformation rules as any other graphs. ü Recall the graphs of the trigonometric functions (sin, cos and tan). ü Analyse how trigonometric graphs are affected by translations and reflections. ü Evaluate how to plot a variety of transformations in functions. Coordinate Geometry Chapter 5 Hypothesis 3: To calculate the gradient of a line we divide the change in x by the change in y. ü Recall how to calculate the midpoint of a segment. ü Understand how to calculate the gradient of any line or segment. ü Apply your knowledge of gradients to parallel and perpendicular lines.
Hypothesis 4: To find the equation of a line we always need to plot it. ü Recall how to plot a straight line given its equation. ü Analyse how to determine the equation of a line given: The gradient and the coordinates of a point. The gradient and the y intercept. Two points. ü Apply your knowledge to solve real-life questions. Hypothesis 5: Pythagoras Theorem cannot be applied to circles. ü Understand how to rearrange the equation of a circle to have it in different forms. ü Analyse how to find the radius of a circle given a point. ü Apply your knowledge of circle theorems to solve circle problems algebraically. Week 5 Hypotheses 1 and 2: We need to use simultaneous equations to find the intersection points of two curves. ü Recall how to solve simultaneous equations. ü Analyse how to calculate the intersection points (if any) between a line and a curve and between two lines. ü Apply your knowledge of simultaneous equations to narrow down the number of intersection points. Differentiation Chapter 10 Hypothesis 3: 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 4: 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 5: 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 K is. ü Analyse how to calculate the derivative of polynomial functions such as y = 2x K x ' + 5. ü Apply your knowledge to differentiate polynomials involving brackets and fractions. Week 6 Hypothesis 1: 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 2: 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. ü Apply your knowledge to find minimum and maximum points of a curve in order to sketch the curve. Hypothesis 3: We cannot use the rule used to calculate the derivative of polynomials to calculate the derivative of x = 4t 5 B3 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. Hypothesis 4 and 5: 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, and ln x look like. ü Evaluate how to calculate the derivatives of more complex functions such as e,, a W,, sin kx, cos kx, tan kx, ln x.
Week 7 Hypothesis 1: 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 K cos 4x ' x. ü Apply your knowledge to calculate the derivatives of different functions. Hypothesis 2 and 3: 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 4: 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. Hypothesis 1: Derivatives can only be used in theoretical mathematics. (Point 10.8) ü Understand the different real-life situations where using derivatives is needed. ü Analyse how to identify when to use derivatives to solve real-life problems. ü Apply your knowledge to solve different real-life questions using derivatives. Websites and other resources. https://revisionmaths.com/advanced-level-maths-revision http://www.s-cool.co.uk/a-level/maths https://www.thestudentroom.co.uk/content.php?r=3534-a-level-maths https://mrbartonmaths.com/students/a-level/a-level-notes.html