# Essential Mathematics (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 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 1

### Unit 1 Description

This unit provides students with the mathematical skills and understanding to solve problems relating to calculations, applications of measurement, the use of formulas to find an unknown quantity, and the interpretation of graphs. Teachers are encouraged to apply the content of the four topics in this unit – ‘Calculations, percentages and rates’, ‘Measurement’, ‘Algebra’ and ‘Graphs’ – in contexts which are meaningful and of interest to their students. A variety of approaches can be used to achieve this purpose. Two possible contexts which may be used are Mathematics and foods and Earning and managing money. 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 1 Learning Outcomes

#### Learning Outcomes

By the end of this unit students:

• understand the concepts and techniques in calculations, measurement, algebra and graphs
• apply reasoning skills and solve practical problems in calculations, measurement, algebra and graphs
• communicate their arguments and strategies when solving problems using appropriate mathematical language
• interpret mathematical information and ascertain the reasonableness of their solutions to problems.

### Unit 1 Content Descriptions

#### Topic 1: Calculations, percentages and rates

Examples in context

Calculations – for example:

• creating a budget for living at home and for living independently
• using timesheets, which include overtime, to calculate weekly wages
• converting between weekly, fortnightly and yearly incomes.

Percentages – for example:

• expressing ingredients of packaged food as percentages of the total quantity, or per serving size, or per 100 grams
• comparing the quantities, both numerically and in percentage terms, of additives within a product or between similar products, such as flavours
• calculating commissions, including retainers from sales information.

Rates – for example:

• using rates to compare and evaluate nutritional information, such as quantity per serve and quantity per 100g
• calculating heart rates as beats per minute, given the number of beats and different time periods
• applying rates to calculate the energy used in various activities over different time periods
• completing calculations with rates, including solving problems involving direct proportion in terms of rate; for example, if a person works for 3 weeks at a rate of \$300 per week, how much do they earn?
• analysing and interpreting tables and graphs that compare body ratios such as hip height versus stride length, foot length versus height.

#### Calculations:

solve practical problems requiring basic number operations (ACMEM001)

apply arithmetic operations according to their correct order (ACMEM002)

ascertain the reasonableness of answers to arithmetic calculations (ACMEM003)

use leading-digit approximation to obtain estimates of calculations (ACMEM004)

use a calculator for multi-step calculations (ACMEM005)

check results of calculations for accuracy (ACMEM006)

recognise the significance of place value after the decimal point (ACMEM007)

evaluate decimal fractions to the required number of decimal places (ACMEM008)

round up or round down numbers to the required number of decimal places (ACMEM009)

apply approximation strategies for calculations. (ACMEM010)

#### Percentages:

calculate a percentage of a given amount (ACMEM011)

determine one amount expressed as a percentage of another (ACMEM012)

apply percentage increases and decreases in situations; for example, mark-ups, discounts and GST. (ACMEM013)

#### Rates:

identify common usage of rates; for example, km/h as a rate to describe speed, beats/minute as a rate to describe pulse (ACMEM014)

convert units of rates occurring in practical situations to solve problems (ACMEM015)

use rates to make comparisons; for example, using unit prices to compare best buys, comparing heart rates after exercise. (ACMEM016)

#### Topic 2: Measurement

Examples in context

Length – for example:

• determining the dimensions/measurements of food packaging
• determining the length of the lines on a sporting field to find the cost of marking it.

Mass – for example:

• comparing and discussing the components of different food types for the components of packaged food expressed as grams.

Area and volume – for example:

• determining the area of the walls of a room for the purpose of painting
• finding the volume of water collected from a roof under different conditions
• finding the volume of various cereal boxes.

#### Linear measure:

use metric units of length, their abbreviations, conversions between them, and appropriate levels of accuracy and choice of units (ACMEM017)

estimate lengths (ACMEM018)

convert between metric units of length and other length units (ACMEM019)

calculate perimeters of familiar shapes, including triangles, squares, rectangles, and composites of these. (ACMEM020)

#### Area measure:

use metric units of area, their abbreviations, conversions between them, and appropriate choices of units (ACMEM021)

estimate the areas of different shapes (ACMEM022)

convert between metric units of area and other area units (ACMEM023)

calculate areas of rectangles and triangles. (ACMEM024)

#### Mass:

use metric units of mass, their abbreviations, conversions between them, and appropriate choices of units (ACMEM025)

estimate the mass of different objects. (ACMEM026)

#### Volume and capacity:

use metric units of volume, their abbreviations, conversions between them, and appropriate choices of units (ACMEM027)

understand the relationship between volume and capacity (ACMEM028)

estimate volume and capacity of various objects (ACMEM029)

calculate the volume of objects, such as cubes and rectangular and triangular prisms. (ACMEM030)

#### Units of energy:

use units of energy to describe consumption of electricity, such as kilowatt hours (ACMEM031)

use units of energy used for foods, including calories (ACMEM032)

use units of energy to describe the amount of energy in activity, such as kilojoules (ACMEM033)

convert from one unit of energy to another. (ACMEM034)

#### Topic 3: Algebra

Examples in context

Formula substitution – for example:

• using formulas to calculate the volumes of various packaging
• using formulas to find the height of a male (H) given the bone radius (r)
• find weekly wage (W) given base wage (b) and overtime hours(h) at 1.5 times rate (r) W = b + 1.5 × h × r.

#### Single substitution:

substitute numerical values into algebraic expressions; for example, substitute different values of $$x$$ to evaluate the expressions $$\frac{3x}5,\;5(2x-4)$$ (ACMEM035)

#### General substitution:

substitute given values for the other pronumerals in a mathematical formula to find the value of the subject of the formula. (ACMEM036)

#### Topic 4: Graphs

Examples in context

Reading and interpreting graphs – for example:

• analysing and interpreting a range of graphical information about global weather patterns that affect food growth
• interpreting a range of graphical information provided on gas and electricity bills.

Drawing graphs – for example:

• expressing ingredients of particular food types as percentages of the total quantity, or per serving size, or per 100 grams, and presenting the information in different formats; for example, column graphs, and pie graphs
• creating graphs to show the deductions from gross wages such as tax, the Medicare levy and superannuation.

#### Reading and interpreting graphs:

interpret information presented in graphs, such as conversion graphs, line graphs, step graphs, column graphs and picture graphs (ACMEM037)

interpret information presented in two-way tables (ACMEM038)

discuss and interpret graphs found in the media and in factual texts. (ACMEM039)

#### Drawing graphs:

determine which type of graph is best used to display a dataset (ACMEM040)

use spreadsheets to tabulate and graph data (ACMEM041)

draw a line graph to represent any data that demonstrate a continuous change, such as hourly temperature. (ACMEM042)