King s Year 12 Medium Term Plan for LC3- A-Level Mathematics

Transcription

1 King s Year 12 Medium Term Plan for LC3- A-Level Mathematics Modules Materials needed Progress objectives Algebra, Geometry and Calculus. Text book: Mathematics for A-Level Hodder Education. Calculator. Logarithms and exponentials objectives (AQA F1, F2, F3, F4, F5, F6, F7) Ø Know and use the function a " and its graph, where a is positive. Ø Know and use the function e " and its graph. Ø Know that the gradient of e %" is equal to ke %" and hence understand why the exponential model is suitable in many applications. Ø Know and use the definition of loga x as the inverse of a ", where a is positive and x 0. Ø Know and use the function ln x and its graph. Ø Know and use ln x as the inverse function of e ". Ø Understand and use the laws of logarithms: Ø loga x + loga y = loga xy ; Ø loga x loga, y = loga " 3 ; Ø k loga x = loga x % including, for example k = 1 and k = 5 6 Ø Solve equations of the form a " = b. Ø Use logarithmic graphs to estimate parameters in relationships of the form y = ax 9 and y = kb ", given data for x and y. Ø Understand and use exponential growth and decay; use in modelling (examples may include the use of e in continuous compound interest, radioactive decay, drug concentration decay, exponential growth as a model for population growth); consideration of limitations and refinements of exponential models. Binomial expansion (AQA D1) Ø Understand and use the binomial expansion of (a + bx) 9 for positive integer n ; the notations n! and ncr ; link to binomial probabilities. Ø Understand how to do combinations and selections to solve real-life problems.

2 Proof (AQA A1) Ø Understand and use the structure of mathematical proof, proceeding from given assumptions through a series of logical steps to a conclusion; use methods of proof, including proof by deduction, proof by exhaustion. Ø Disproof by counter example. Ø Proof by contradiction (including proof of the irrationality of 2 and the infinity of primes, and application to unfamiliar proofs). Trigonometry (AQAE1, E3, E5, E7) Ø Understand and use the definitions of sine, cosine and tangent for all arguments; the sine and cosine rules; the area of a triangle in the form 5 ab sin C. 6 Ø Understand and use the sine, cosine and tangent functions; their graphs, symmetries and periodicity] Ø Understand and use tan θ = EFG H. IJE H Ø Understand and use sin 6 θ + cos 6 θ = 1. Ø Solve simple trigonometric equations in a given interval, including quadratic equations in sin, cos and tan and equations involving multiples of the unknown angle. There will be an assessment at the end of LC3, 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 Exponentials and logarithms Chapter 13 Hypothesis 1: x 3 is an exponential function that goes through the point (1, 0) (Point 13.1) ü Understand what an exponential graph is. ü Analyse how to represent exponential growth and decay. ü Evaluate how to model a worded question using exponential functions. Hypothesis 2 and 3: We can use exponentials to calculate compound interest. (Point 13.3) ü Recall what compound interest is and how to calculate it. ü Understand what continuous compound interest is. ü Analyse how to derivate an exponential function. ü Evaluate how to apply exponential functions to solve worded problems. Hypothesis 4 and 5: The four operations cannot be applied to logarithms. (Point 13.2) ü Understand what a logarithm is. ü Understand the laws of logarithms. ü Evaluate how to plot graphs of logarithms. Week 2 Exponentials and logarithms Chapter 13 Hypothesis 1 and 2: Natural logarithms are the inverse of e 10. (Point 13.4) ü Understand what a natural logarithm is. ü Analise how to do calculations with natural logarithms. ü Apply your knowledge to solve worded questions. Hypothesis 3, 4, and 5: Relationships between variables cannot be expressed using an exponential curve. (Point 13.5) ü Recall what inverse and direct proportion can be modelled. ü Analyse how to show relationships using proportion and exponential graphs. ü Apply your knowledge to solve worded real-life problems.

4 Week 3 The binomial expansion Chapter 9 Hypothesis 1: To calculate (x+1) 7 we need to expand brackets. ü Understand the relationship between Pascal s triangle and the binomial expansion. ü Analyse how to find the binomial coefficients for any positive integer value of n. ü Apply your knowledge to calculate factorials. Hypothesis 2 and 3: n! = n(n+1)(n+2) ü Understand how to expand expressions in the form (1+x) n. ü Analyse the relationship between binomial coefficients. ü Apply the binomial theorem to expressions like (a+b) n. Hypothesis 4 and 5: To find the number of possible ways of selecting objects, we need to use factorials. ü Understand what selections and combinations mean. ü Analyse how to calculate selections and combinations in real-life problems. ü Apply your knowledge to solve worded questions.

5 Week 4 Proof Chapter 1 (Year 2 book) Hypothesis 1: It is correct to say that 2n is even n is even. ü Understand what a conjecture is. ü Analyse how to write sufficient and sufficient and necessary conditions. ü Evaluate how to prove mathematical problems using specific notation. Hypothesis 2: To prove a conjecture, we always use the same method: trial and error. ü Understand what proof by direct argument is. Hypothesis 3: The exhaustion method can always be used, regardless of the type of problem. ü Understand what proof by exhaustion is. Hypothesis 4: Contradiction is based on assuming that a false fact is actually true. ü Understand what proof by contradiction is. Hypothesis 5: To complete a proof by counter-example, we need to find an example that supports our conjecture. ü Understand what disproof by counter-example is. Week 5 Trigonometry Chapter 6

6 Hypothesis 1: We only use SOHCAHTOA in right angled triangles. ü Recall how to use the SOH CAH TOA rule. ü Analyse how to apply the rule in different shapes. ü Evaluate how to solve worded-problems. Hypothesis 2, 3 and 4: To prove cosθ 1 1Wsinθ cosθ tanθ we need to use the counter-example method. ü Understand new trigonometric identities. ü Analyse how to use these identities to prove more complex trigonometric identities. ü Apply your knowledge to solve worded questions. Hypothesis 5: The sin and cos graphs are inverse of each other. ü Understand where the cosine, tangent and sine graphs come from. ü Analyse the properties of each of the graphs (asymptotes, maximum points, minimum points ). ü Apply your knowledge of graphs to solve trigonometric problems. Week 6 Trigonometry Chapter 6 Hypothesis 1 and 2: Trigonometric equations have only one solution. ü Recall what the inverse of the sine, cosine and tangent are. ü Analyse how to use trigonometric graphs to find the solution of an equation. ü Evaluate how to solve any trigonometric equation. Hypothesis 3: An angle in a non right angled triangle can be only calculate using the law of sines. ü Recall the law of sines and cosines. ü Understand when to use one rule or another. ü Apply your knowledge to model and solve worded questions.

7 Hypothesis 4: To calculate the area of a triangle we always do bxh. 2 ü Recall how to calculate the area of a non right angled triangle. ü Understand how to prove that bxh = 1 absinθ. 2 2 ü Apply your knowledge of graphs to model and solve worded questions. Week 7 Websites and other resources. The Assessment for LC3 will be done during the last lesson of this week. REACH Week (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