# Essential Mathematics

### 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 used to describe much of the physical world.

### Links to Foundation to Year 10

For all content areas of Essential Mathematics, the proficiency strands of Understanding, Fluency, Problem solving and Reasoning from the F–10 curriculum are still very much applicable and should be inherent in students’ learning of the subject.

### 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 Essential Mathematics

Essential Mathematics has four units each of which contains a number of topics. It is intended that the topics be taught in a context relevant to students’ needs and interests. In Essential Mathematics, students use their knowledge and skills to investigate realistic problems of interest which involve the application of mathematical relationships and concepts.

## Unit 4

### Unit 4 Description

This unit provides students with the mathematical skills and understanding to solve problems related to probability, Earth geometry and time zones, and loans and compound interest. Teachers are encouraged to apply the content of the three topics in this unit – ‘Probability and relative frequencies’, ‘Earth geometry and time zones’ and ‘Loans and compound interest’ – in a context which is meaningful and of interest to the students. A variety of approaches can be used to achieve this purpose. Two possible contexts which may be used in this unit are Mathematics of finance and Mathematics of travelling. However, as these contexts may not be relevant to all students, teachers are encouraged to find suitable contexts relevant to their particular student cohort.

It is assumed that an extensive range of technological applications and techniques will be used in teaching this unit. The ability to choose when and when not to use some form of technology, and the ability to work flexibly with technology, are important skills.

### Unit 4 Learning Outcomes

By the end of this unit, students:

• understand the concepts and techniques used in probability and relative frequencies, earth geometry and time zones, loans and compound interest
• apply reasoning skills and solve practical problems in probability and relative frequencies, earth geometry and time zones, loans and compound interest
• communicate their arguments and strategies when solving mathematical problems using appropriate mathematical or statistical language
• interpret mathematical information and ascertain the reasonableness of their solutions to problems.

### Unit 4 Content Descriptions

#### Topic 1: Probability and relative frequencies

Examples in context
• using data to calculate the relative frequencies of the different countries of origin of visitors to a particular tourist venue or country
• using data to calculate the relative frequencies of the amounts of household expenditure is this sentence incomplete?

#### Probability expressions:

interpret commonly used probability statements, including ‘possible’, ‘probable’, ‘likely’, ‘certain’ (ACMEM148)

describe ways of expressing probabilities formally using fractions, decimals, ratios, and percentages. (ACMEM149)

#### Simulations:

perform simulations of experiments using technology (ACMEM150)

recognise that the repetition of chance events is likely to produce different results (ACMEM151)

identify relative frequency as probability (ACMEM152)

identify factors that could complicate the simulation of real-world events. (ACMEM153)

#### Simple probabilities:

construct a sample space for an experiment (ACMEM154)

use a sample space to determine the probability of outcomes for an experiment (ACMEM155)

use arrays or tree diagrams to determine the outcomes and the probabilities for experiments. (ACMEM156)

#### Probability applications:

determine the probabilities associated with simple games (ACMEM157)

determine the probabilities of occurrence of simple traffic-light problems. (ACMEM158)

#### Location:

locate positions on Earth’s surface given latitude and longitude using GPS, a globe, an atlas, and digital technologies (ACMEM159)

find distances between two places on Earth on the same longitude (ACMEM160)

find distances between two places on Earth using appropriate technology. (ACMEM161)

#### Time:

understand the link between longitude and time (ACMEM162)

solve problems involving time zones in Australia and in neighbouring nations, making any necessary allowances for daylight saving (ACMEM163)

solve problems involving Greenwich Mean Time and the International Date Line (ACMEM164)

find time differences between two places on Earth (ACMEM165)

solve problems associated with time zones; for example, internet and phone usage (ACMEM166)

solve problems relating to travelling east and west, incorporating time zone changes. (ACMEM167)

#### Topic 3: Loans and compound interest

Examples in context
• using formula, graphs and spreadsheets to calculate the outcomes of investment accounts with compound interest
• using percentages, rates and spreadsheets to investigate personal loan calculations
• calculating and analysing the costs, hidden traps, advantages and disadvantages of payment plans with interest free periods, using rates and percentages.

#### Compound interest:

review the principles of simple interest (ACMEM168)

understand the concept of compound interest as a recurrence relation (ACMEM169)

consider similar problems involving compounding; for example, population growth (ACMEM170)

use technology to calculate the future value of a compound interest loan or investment and the total interest paid or earned (ACMEM171)

use technology to compare, numerically and graphically, the growth of simple interest and compound interest loans and investments (ACMEM172)

use technology to investigate the effect of the interest rate and the number of compounding periods on the future value of a loan or investment. (ACMEM173)

#### Reducing balance loans (compound interest loans with periodic repayments):

use technology and a recurrence relation to model a reducing balance loan (ACMEM174)

investigate the effect of the interest rate and repayment amount on the time taken to repay a loan. (ACMEM175)