# Mathematical Methods (Version 8.4)

### Rationale

Mathematics is the study of order, relation and pattern. From its origins in counting and measuring it has evolved in highly sophisticated and elegant ways to become the language now used to describe much of the modern world.

### Links to Foundation to Year 10

In Mathematical Methods, there is a strong emphasis on mutually reinforcing proficiencies in Understanding, Fluency, Problem solving and Reasoning.

### Representation of General capabilities

The seven general capabilities of Literacy, Numeracy, Information and Communication technology (ICT) capability, Critical and creative thinking, Personal and social capability, Ethical understanding, and Intercultural understanding are identified where they offer opportunities to add depth and richness to student learning.

### Structure of Mathematical Methods

Mathematical Methods is organised into four units. The topics broaden students’ mathematical experience and provide different scenarios for incorporating mathematical arguments and problem solving. The units provide a blending of algebraic and geometric thinking.

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## Unit 4

### Unit 4 Description

The calculus in this unit deals with derivatives of logarithmic functions. In probability and statistics, continuous random variables and their applications are introduced and the normal distribution is used in a variety of contexts. The study of statistical inference in this unit is the culmination of earlier work on probability and random variables. Statistical inference is one of the most important parts of statistics, in which the goal is to estimate an unknown parameter associated with a population using a sample of data drawn from that population. In Mathematical Methods statistical inference is restricted to estimating proportions in two-outcome populations.

Access to technology to support the computational aspects of these topics is assumed.

### Unit 4 Learning Outcomes

By the end of this unit, students:

• understand the concepts and techniques in calculus, probabilty and statistics
• solve problems in calculus, probability and statistics
• apply reasoning skills in calculus, probability and statistics
• interpret and evaluate mathematical and statistical information and ascertain the reasonableness of solutions to problems.
• communicate their arguments and strategies when solving problems.

### Unit 4 Content Descriptions

#### Logarithmic functions:

define logarithms as indices: $$a^x=b$$ is equivalent to $$x=\log_ab$$ i.e. $$a^{\log_ab}=b$$ (ACMMM151)

establish and use the algebraic properties of logarithms (ACMMM152)

recognise the inverse relationship between logarithms and exponentials: $$y=a^x$$ is equivalent to $$x=\log_ay$$ (ACMMM153)

interpret and use logarithmic scales such as decibels in acoustics, the Richter Scale for earthquake magnitude, octaves in music, pH in chemistry (ACMMM154)

solve equations involving indices using logarithms (ACMMM155)

recognise the qualitative features of the graph of $$y=\log_ax$$ $$(a>1)$$ including asymptotes, and of its translations $$y=\log_ax+b$$ and $$y=\log_a{(x+c)}$$ (ACMMM156)

solve simple equations involving logarithmic functions algebraically and graphically (ACMMM157)

identify contexts suitable for modelling by logarithmic functions and use them to solve practical problems. (ACMMM158)

#### Calculus of logarithmic functions:

define the natural logarithm $$\ln x=\log_ex$$ (ACMMM159)

recognise and use the inverse relationship of the functions $$y=e^x$$ and $$y=\ln x$$ (ACMMM160)

establish and use the formula $$\frac d{dx}\left(\ln x\right)=\frac1x$$ (ACMMM161)

establish and use the formula $$\int\frac1xdx=\ln\;x\;+c$$ for $$x>0$$ (ACMMM162)

use logarithmic functions and their derivatives to solve practical problems. (ACMMM163)

#### General discrete random variables:

use relative frequencies and histograms obtained from data to estimate probabilities associated with a continuous random variable (ACMMM164)

understand 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 (ACMMM165)

recognise the expected value, variance and standard deviation of a continuous random variable and evaluate them in simple cases (ACMMM166)

#### General continuous random variables:

understand the effects of linear changes of scale and origin on the mean and the standard deviation. (ACMMM167)

#### Normal distributions:

identify contexts such as naturally occurring variation that are suitable for modelling by normal random variables (ACMMM168)

recognise features of the graph of the probability density function of the normal distribution with mean $$\mu$$ and standard deviation $$\sigma$$ and the use of the standard normal distribution (ACMMM169)

calculate probabilities and quantiles associated with a given normal distribution using technology, and use these to solve practical problems. (ACMMM170)

#### Random sampling:

understand the concept of a random sample (ACMMM171)

discuss sources of bias in samples, and procedures to ensure randomness (ACMMM172)

use graphical displays of simulated data to investigate the variability of random samples from various types of distributions, including uniform, normal and Bernoulli. (ACMMM173)

#### Sample proportions:

understand the concept of the sample proportion $$\widehat p$$ as a random variable whose value varies between samples, and the formulas for the mean $$p$$ and standard deviation $$\sqrt[{}]{(p(1-p)/n}$$ of the sample proportion $$\widehat p$$ (ACMMM174)

examine the approximate normality of the distribution of $$\widehat p$$ for large samples  (ACMMM175)

simulate repeated random sampling, for a variety of values of $$p$$ and a range of sample sizes, to illustrate the distribution of $$\widehat p$$ and the approximate standard normality of $$\frac{\widehat p\;-p}{\sqrt[{}]{(\widehat p(1-\widehat p)/n}}$$ where the closeness of the approximation depends on both $$n$$ and $$p$$ (ACMMM176)

#### Confidence intervals for proportions:

the concept of an interval estimate for a parameter associated with a random variable (ACMMM177)

use the approximate confidence interval $$\left(\widehat p-z\sqrt[{}]{(\widehat p(1-\widehat p)/n},\;\;\widehat p+z\sqrt[{}]{(\widehat p(1-\widehat p)/n}\right),$$ as an interval estimate for $$p$$, where $$z$$ is the appropriate quantile for the standard normal distribution (ACMMM178)

define the approximate margin of error $$E=z\sqrt[{}]{(\widehat p(1-\widehat p)/n}$$ and understand the trade-off between margin of error and level of confidence (ACMMM179)

use simulation to illustrate variations in confidence intervals between samples and to show that most but not all confidence intervals contain $$p$$ (ACMMM180)