Rationale
Global aims
General objectives
Introduction
Contexts
To meet the requirements of the notice and justification criterion, applicants must demonstrate that they are working fully for each question. This section highlights the intent of the syllabus in relation to the subject and indicates how candidates should develop their understanding of the subject. This section provides learning experiences that can be effective in using the subject to achieve the general objectives of the subject.
Candidates must develop an understanding of average and instantaneous rates of change and of the derivative as a function. Candidates are encouraged to develop an understanding of the use of differentiation as a tool in a range of situations involving the optimization of continuous functions. interpret a graph as the rate of increase or decrease of a function; relate these observations to the behavior of the derivative.
To meet the requirements of the Communication and Reasoning criterion, candidates must show complete working methods for each question. The level of achievement will be based on the exit standard for each of the three criteria Knowledge and Procedures, Modeling and Problem Solving, and Communication and Reasoning.
Objectives
Organisation
- Introduction
- Time allocation
- Sequencing
- Technology
The subject material is organized into seven topics, which are discussed in detail in section 5. For teaching centers preparing candidates for the external examination, the recommended number of hours of teaching in the subject developed from this syllabus is 150 to 200 hours. Recommended assumed times are given in parentheses to indicate the relative emphasis of each topic.
After examining the subject and the relevant learning experiences required to achieve the overall objectives, a spiraling and integrated sequence of learning experiences should be developed that enables candidates to see connections between the different subjects of mathematics, rather than seeing them as separate to be considered. The order in which the topics are presented in the syllabus is not intended to indicate an order of learning, but some topics contain subject matter that is developed and expanded in the subject matter of other subjects. The topic should be revisited and spiralled, so that candidates internalize their knowledge before developing it further.
The advantage of mathematical technology in mathematics is that it allows the exploration of the concepts and processes of mathematics. For example, graphing calculators allow candidates to research, explore and more easily understand concepts, especially graphs. Specifically, technology that supports mathematics enables candidates to solve more diverse life problems.
It can be used in statistics to examine larger data sets and quickly produce a variety of graphical representations and summary statistics, freeing candidates to look for patterns, spot anomalies in the data, and make informed observations.
Topics
Introduction
The topics
Derive the formula for solving a general quadratic equation and solve squares by completing the square. Find solutions of trigonometric equations within a given domain such as —2π≤θ≤ 2π for equations such as 2 sin θ = –0.7, 2 sin2 θ = cos θ. Note that the fraction of radioactive material remaining after time t is e -kt, where k is a positive constant; investigate the relationship between k and the half-life of the material.
Develop the derivative of the function of ax and loga x to identify the meaning of the exponential constant e. Use logarithms to solve equations, such as the time it takes for an investment to double for a given rate of interest. Investigate the time at which the amount of the intermediate reaches a maximum in a simple two-step radioactive decay; that is, the original substance decays into an intermediate substance, which in turn decays into an inert substance.
Plot the logarithm of Australia's population at censuses (a) from 1891 to 1933 (b) from 1947 to 1971 and (c) from 1971 to 1991; recognize that the linear tendencies of the plot indicate power/exponential relationships in the original. Use life-related situations such as enclosing a rectangular area with fixed-length fencing to demonstrate the need for calculus to determine optimal values. Use zero values of the derivative to find local optima and points of horizontal bending when sketching curves of simple functions (the solutions must not depend on the factor theorem).
Explore applications of shortest distance problems such as: a water hydrant inside a property boundary must be connected to a main running along the property boundary, assuming the boundary is straight; find the shortest length of pipe required to connect the faucet to the main. Candidates should develop an understanding of the concept of integration as a process by which a "whole" can be obtained from the sum of a large number of parts. Use integration to determine appropriate values in contexts such as learning curves, consumer surplus, Lorenz curves.
Candidates should develop a working knowledge of the concepts involved in describing, summarizing, comparing and modeling data and some basic concepts in using data to estimate probabilities and parameters and to answer simple questions. Organize your data set using a variety of approaches, such as summary statistics and graphical displays. Given the data set, produce a concise summary of the main information in the data, referring to graphical representations and summary statistics.
Use binomial probabilities in real-life situations such as investigating the effectiveness of a new drug. Examine the use of summary statistics in, for example, newspapers, articles, TV programs such as weather reports and advertisements, government reports.
Assessment
- Summative assessment
- Exit criteria
- Special consideration
- Awarding levels of achievement
The provisions on special considerations are described in more detail in the annual Handbook for External Senior Examinations, available on QSA's website at The process of reaching a judgment about a candidate's answers to examination questions is essentially a process of matching the candidate's answers against the syllabus standards associated with exit criteria. Information on how manuscripts are assessed can be found in the annual Handbook for External Senior Examinations, available on the QSA website.
Once standards have been established for each of the three criteria, the following table is used to determine the level of performance, with A representing the highest standard and E the lowest. VHA The candidate must achieve a standard A in two of the two exit criteria and no less than a standard B in the remaining criterion. HA The candidate must achieve Standard B in two of the two exit criteria and no less than a Standard C in the remaining criterion.
SA The candidate must achieve Standard C in two of the two exit criteria, one of which must be the knowledge and procedure criterion and no less than Standard D in the remaining criterion. LA The candidate must achieve Standard D in two of the two exit criteria, one of which must be the Knowledge and procedure criterion. The overall quality of a candidate's performance across the full spectrum within each context and across subjects generally shows.
Resources
QSA website
Textbooks and other resources
The bottom quartile, Q1, is the median of the bottom 9 observations, which is 114, and the top quartile, Q3, is the median of the top 9. An extreme value in the observations, for example, an observation that is outside the box in the box-and-whisker plot, or a point far from the line of best fit. An exploratory technique that simultaneously ranks the data and gives an idea of its distribution.
Characteristics that describe the sample of observations, for example, mean, median, or standard deviation. The area under the curve on the x-axis between the limits x = a and x = b can be approximated by dividing the interval [a, b] into n subintervals of equal length. The way in which observations differ (vary) from each other, often measured by the standard deviation or range.
