Resource materials - Irene McCormack Catholic College
Irene McCormack Catholic CollegeMathematicsYear 12 Methods (ATAR)Unit 3 & 4Program 2020Resource materialsMethods 12: Nelson Senior Maths for the Australian CurriculumGraphics Calculator – Casio Classpad 400Other: WACE Revision Series Mathematics Methods (Academic Task Force, OT Lee), Methods 3 & 4 (Sadler), Past WA Exams and WACE Exam papers, Mathspace.Assessment StructureSemester OneAssessment TypeWeighting2 Investigations10% total3 Response 20% total1 Examination (Unit 3 only)15%Semester TwoAssessment TypeWeighting2 Investigations10% total3 Response 20% total1 Examination (Units 3 and 4)25%Additional items permitted in assessmentsIn all response assessments students are permitted to have the SCSA Formula Sheet for the entire assessment. One A4 page of notes (both sides permitted) may be used in any resource enabled sections (calculator sections)In all examinations students are permitted to have the SCSA Formula Sheet for the entire assessment. Two A4 page of notes (both sides permitted) may be used in any resource enabled sections (calculator sections)Personal notes in assessments must not have folds, white out, liquid paper or anything glued on them. They will be confiscated and no additional time allowed. Student’s results may be altered or cancelled to reflect this anisation of ContentUnit 3This unit contains the three topics:3.1 Further differentiation and applications3.2 Integrals3.3 Discrete random variablesThe study of calculus continues by introducing the derivatives of exponential and trigonometric functions and their applications, as well as some basic differentiation techniques and the concept of a second derivative, its meaning and applications. The aim is to demonstrate to students the beauty and power of calculus and the breadth of its applications. The unit includes integration, both as a process that reverses differentiation and as a way of calculating areas. The fundamental theorem of calculus as a link between differentiation and integration is emphasised. Discrete random variables are introduced, together with their uses in modelling random processes involving chance and variation. The purpose here is to develop a framework for statistical inference..Unit 4This unit contains the three topics:4.1The logarithmic function4.2 Continuous random variables and the normal distribution4.3 Interval estimates for proportionsThe logarithmic function and its derivative are studied. Continuous random variables are introduced, and their applications examined. Probabilities associated with continuous distributions are calculated using definite integrals. In this unit, students are introduced to one of the most important parts of statistics, namely, statistical inference, where the goal is to estimate an unknown parameter associated with a population using a sample of that population. In this unit, inference is restricted to estimating proportions in two-outcome populations. Students will already be familiar with many examples of these types of populations.Semester One – 2019Unit 3 Learning OutcomesBy the end of this unit, students:understand the concepts and techniques in calculus, probability and statisticssolve problems in calculus, probability and statisticsapply reasoning skills in calculus, probability and statisticsinterpret and evaluate mathematical and statistical information and ascertain the reasonableness of solutions to municate their arguments and strategies when solving problems.WeekContent DescriptionResources & AssessmentsTerm 1Week 1 – 2(7 Hours)Exponential functionsestimate the limit of ah-1h as h→0, using technology, for various values of a >0 identify that e is the unique number a for which the above limit is 1 establish and use the formula ddxex=ex use exponential functions of the form Aekx and their derivatives to solve practical problemsTrigonometric functionsestablish the formulas ddxsinx=cosx and ddxcosx=-sinx by graphical treatment, numerical estimations of the limits, and informal proofs based on geometric constructions use trigonometric functions and their derivatives to solve practical problemsChapter 1 (Nelson Maths)INVESTIGATION 1Term 1Week 3 – 4(6 Hours)Differentiation rulesexamine and use the product and quotient rules examine the notion of composition of functions and use the chain rule for determining the derivatives of composite functions apply the product, quotient and chain rule to differentiate functions such as xex, tanx, 1xn , xsinx, e-xsinx and fax-bChapter 1TEST 1 Term 1Week 5 - 6(6 Hours)General discrete random variables3.3.1develop the concepts of a discrete random variable and its associated probability function, and their use in modelling data 3.3.2use relative frequencies obtained from data to obtain point estimates of probabilities associated with a discrete random variable 3.3.3identify uniform discrete random variables and use them to model random phenomena with equally likely outcomes 3.3.4examine simple examples of non-uniform discrete random variables3.3.5identify the mean or expected value of a discrete random variable as a measurement of centre, and evaluate it in simple cases 3.3.6identify the variance and standard deviation of a discrete random variable as measures of spread, and evaluate them using technology 3.3.7examine the effects of linear changes of scale and origin on the mean and the standard deviation3.3.8use discrete random variables and associated probabilities to solve practical problems Chapter 2Term 1Week 6 – 7(7 Hours)The second derivative and applications of differentiationuse the increments formula: δy≈dydx×δx to estimate the change in the dependent variable y resulting from changes in the independent variable x apply the concept of the second derivative as the rate of change of the first derivative functionidentify acceleration as the second derivative of position with respect to timeexamine the concepts of concavity and points of inflection and their relationship with the second derivativeapply the second derivative test for determining local maxima and minimasketch the graph of a function using first and second derivatives to locate stationary points and points of inflectionsolve optimisation problems from a wide variety of fields using first and second derivativesChapter 3Term 1Week 8(5 Hours)Anti-differentiation3.2.1identify anti-differentiation as the reverse of differentiation 3.2.2use the notation fxdx for anti-derivatives or indefinite integrals 3.2.3establish and use the formula xndx=1n+1xn+1+c for n≠-1 3.2.4establish and use the formula exdx=ex+c 3.2.5establish and use the formulas sinxdx=-cosx+c and cosxdx=sinx+c3.2.6identify and use linearity of anti-differentiation 3.2.7determine indefinite integrals of the form fax-bdx 3.2.8identify families of curves with the same derivative function 3.2.9determine fx, given f'x and an initial condition fa=bTEST 2Chapter 6.01, 6.02Extra resources neededTerm 1Week 9 – 10 (7 hours)Term 2Week 1 - 2 (7 hours)Definite integrals3.2.10examine the area problem and use sums of the form ifxi δxi to estimate the area under the curve y=f(x) 3.2.11identify the definite integral abfxdx as a limit of sums of the form ifxi δxi 3.2.12interpret the definite integral abfxdx as area under the curve y=fx if fx>0 3.2.13interpret abfxdx as a sum of signed areas 3.2.14apply the additivity and linearity of definite integralsFundamental theorem3.2.15examine the concept of the signed area function Fx=axftdt 3.2.16apply the theorem: F'x=ddxaxftdt=fx, and illustrate its proof geometrically 3.2.17develop the formula abf'xdx=fb-f(a) and use it to calculate definite integralsApplications of integrationcalculate total change by integrating instantaneous or marginal rate of changecalculate the area under a curve 3.2.20calculate the area between curves 3.2.21determine displacement given velocity in linear motion problems3.2.22determine positions given linear acceleration and initial values of position and velocity.Chapter 4 INVESTGATION 2Chapter 6.0.3-6.0.7Term 22 – 4(10 hours)Bernoulli distributions3.3.9use a Bernoulli random variable as a model for two-outcome situations 3.3.10identify contexts suitable for modelling by Bernoulli random variables 3.3.11determine the mean p and variance p1-pof the Bernoulli distribution with parameter p 3.3.12use Bernoulli random variables and associated probabilities to model data and solve practical problemsBinomial distributions3.3.13examine the concept of Bernoulli trials and the concept of a binomial random variable as the number of ‘successes’ in n independent Bernoulli trials, with the same probability of success p in each trial 3.3.14identify contexts suitable for modelling by binomial random variables 3.3.15determine and use the probabilities PX=x=nxpx1-pn-x associated with the binomial distribution with parameters n and p; note the mean np and variance np1-p of a binomial distribution3.3.16use binomial distributions and associated probabilities to solve practical problems Chapter 5TEST 3Term 2Week 4 EXAM REVISIONWeek 5 - 6EXAMSSemester Two – 2019 Unit 4 Learning OutcomesBy the end of this unit, students:understand the concepts and techniques in calculus, probability and statisticssolve problems in calculus, probability and statisticsapply reasoning skills in calculus, probability and statisticsinterpret and evaluate mathematical and statistical information and ascertain the reasonableness of solutions to municate their arguments and strategies when solving problems.WeekContent DescriptionResource &AssessmentTerm 2Week 7 - 8 Logarithmic functions4.1.1define logarithms as indices: ax=b is equivalent to x=logab i.e. alogab=b 4.1.2establish and use the algebraic properties of logarithms 4.1.3examine the inverse relationship between logarithms and exponentials: y=ax is equivalent to x=logay 4.1.4interpret and use logarithmic scales 4.1.5solve equations involving indices using logarithms 4.1.6identify the qualitative features of the graph of y=logax (a>1), including asymptotes, and of its translations y=logax+b and y=loga(x-c) 4.1.7solve simple equations involving logarithmic functions algebraically and graphically 4.1.8identify contexts suitable for modelling by logarithmic functions and use them to solve practical problems. Chapter 7Investigation 3 WeekContent DescriptionResources & AssessmentTerm 2Week 9 - 10Calculus of the natural logarithmic functiondefine the natural logarithm lnx=logex examine and use the inverse relationship of the functions y=ex and y=lnx 4.1.11establish and use the formula ddxlnx=1x 4.1.12establish and use the formula 1xdx=ln x +c, for x>0 4.1.13determine derivatives of the form ddxlnf(x) and integrals of the form f'xfxdx, for f (x)>04.1.14use logarithmic functions and their derivatives to solve practical problemsTerm 3Week 1 - 2General continuous random variables4.2.1use relative frequencies and histograms obtained from data to estimate probabilities associated with a continuous random variable 4.2.2examine the concepts of a probability density function, cumulative distribution function, and probabilities associated with a continuous random variable given by integrals; examine simple types of continuous random variables and use them in appropriate contexts 4.2.3identify the expected value, variance and standard deviation of a continuous random variable and evaluate them using technology 4.2.4examine the effects of linear changes of scale and origin on the mean and the standard deviation Chapter 8Test 4 WeekContent DescriptionResources & AssessmentTerm 3Week 2 – 3Normal distributions4.2.5identify contexts, such as naturally occurring variation, that are suitable for modelling by normal random variables4.2.6identify features of the graph of the probability density function of the normal distribution with mean μ and standard deviation σ and the use of the standard normal distribution4.2.7calculate probabilities and quantiles associated with a given normal distribution using technology, and use these to solve practical problems Chapter 8Test 5Term 3Week 4 - 5 Random sampling4.3.1examine the concept of a random sample 4.3.2discuss sources of bias in samples, and procedures to ensure randomness 4.3.3 use graphical displays of simulated data to investigate the variability of random samples from various types of distributions, including uniform, normal and Bernoulli.Chapter 9Investigation 4Term 3Week 6 – 7 Sample proportions4.3.4examine the concept of the sample proportion p as a random variable whose value varies between samples, and the formulas for the mean p and standard deviation p1-pn of the sample proportion p examine the approximate normality of the distribution of p for large samples 4.3.6 simulate repeated random sampling, for a variety of values of p and a range of sample sizes, to illustrate the distribution of p and the approximate standard normality of p –pp1-pn where the closeness of the approximation depends on both n and pChapter 9WeekContent DescriptionResources & AssessmentTerm 3Week 8Confidence intervals for proportions4.3.7examine the concept of an interval estimate for a parameter associated with a random variable 4.3.8use the approximate confidence interval p-zp1-pn, p+zp1-pn as an interval estimate for p, where z is the appropriate quantile for the standard normal distribution 4.3.9define the approximate margin of error E=zp1-pn and understand the trade-off between margin of error and level of confidence 4.3.10use simulation to illustrate variations in confidence intervals between samples and to show that most, but not all, confidence intervals contain pChapter 10Test 6Term 3Week 9 - 10Revision/ExaminationsExamination Week 2 of Holiday or week 1 of Term 4 – Units 3 and 4 ................
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