# 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.

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## 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.

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)

## Unit 2

### Unit 2 Description

This unit provides students with the mathematical skills and understanding to solve problems related to representing and comparing data, percentages, rates and ratios, the mathematics of finance, and time and motion. Teachers are encouraged to apply the content of the four topics in this unit – ‘Representing and comparing data’, ‘Percentages’, ‘Rates and ratios’ and ‘Time and motion’ – in a context which is 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 cars and Mathematics and independent living. 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 2 Learning Outcomes

By the end of this unit, students:

• understand the concepts and techniques used in representing and comparing data, percentages, rates and ratios, and time and motion
• apply reasoning skills and solve practical problems in representing and comparing data, percentages, rates and ratios, and time and motion
• communicate their arguments and strategies when solving mathematical and statistical problems using appropriate mathematical or statistical language
• interpret mathematical and statistical information and ascertain the reasonableness of their solutions to problems.

### Unit 2 Content Descriptions

#### Topic 1: Representing and comparing data

Examples in context
• analysing and interpreting a range of statistical information related to car theft, car accidents and driver behaviour
• using statistics and graphs to find the number of people in each blood type, given the population percentages of blood types in different countries
• using blood usage statistics to predict the amount of blood needed at different times of the year
• using blood donation statistics to predict how much blood will be needed and when.

#### Classifying data:

identify examples of categorical data (ACMEM043)

identify examples of numerical data. (ACMEM044)

#### Data presentation and interpretation:

display categorical data in tables and column graphs (ACMEM045)

display numerical data as frequency distributions, dot plots, stem and leaf plots, and histograms (ACMEM046)

recognise and identify outliers (ACMEM047)

compare the suitability of different methods of data presentation in real-world contexts. (ACMEM048)

#### Summarising and interpreting data:

identify the mode (ACMEM049)

calculate measures of central tendency, the arithmetic mean and the median (ACMEM050)

investigate the suitability of measures of central tendency in various real-world contexts (ACMEM051)

investigate the effect of outliers on the mean and the median (ACMEM052)

calculate and interpret quartiles, deciles and percentiles (ACMEM053)

use informal ways of describing spread, such as spread out/dispersed, tightly packed, clusters, gaps, more/less dense regions, outliers (ACMEM054)

calculate and interpret statistical measures of spread, such as the range, interquartile range and standard deviation (ACMEM055)

investigate real-world examples from the media illustrating inappropriate uses, or misuses, of measures of central tendency and spread. (ACMEM056)

#### Comparing data sets:

compare back-to-back stem plots for different data-sets (ACMEM057)

complete a five number summary for different datasets (ACMEM058)

construct box plots using a five number summary (ACMEM059)

compare the characteristics of the shape of histograms using symmetry, skewness and bimodality. (ACMEM060)

#### Topic 2: Percentages

Examples in context
• calculating stamp duty costs involved in buying a car, using percentages and tables
• calculating depreciation of a vehicle over time
• using statistics and graphs to find the number of people in each blood type, given the population percentages of blood types in different countries.

#### Percentage calculations:

review calculating a percentage of a given amount (ACMEM061)

review one amount expressed as a percentage of another. (ACMEM062)

#### Applications of percentages:

determine the overall change in a quantity following repeated percentage changes; for example, an increase of 10% followed by a decrease of 10% (ACMEM063)

calculate simple interest for different rates and periods. (ACMEM064)

#### Topic 3: Rates and ratios

Examples in context

Rates – for example:

• using rates to find fuel consumption for different vehicles under different driving conditions
• calculating food, clothing, transport costs per day, week or month using tables, spreadsheets, and estimation
• calculating clothing costs per week or month using tables, spreadsheets, and estimation.

Ratios – for example:

• discussing various ratios used in bicycle gears
• comparing ratios such as people per household.

#### Ratios:

demonstrate an understanding of the elementary ideas and notation of ratio (ACMEM065)

understand the relationship between fractions and ratio (ACMEM066)

express a ratio in simplest form (ACMEM067)

find the ratio of two quantities (ACMEM068)

divide a quantity in a given ratio (ACMEM069)

use ratio to describe simple scales. (ACMEM070)

#### Rates:

review identifying common usage of rates such as km/h (ACMEM071)

convert between units for rates; for example, km/h to m/s, mL/min to L/h (ACMEM072)

complete calculations with rates, including solving problems involving direct proportion in terms of rate (ACMEM073)

use rates to make comparisons (ACMEM074)

use rates to determine costs; for example, calculating the cost of a tradesman using rates per hour, call-out fees. (ACMEM075)

#### Topic 4: Time and motion

Examples in context

Time – for example:

• calculating reaction times through experiments.

Distance – for example:

• calculating distances travelled to school and the time taken, considering different average speeds.

Speed – for example:

• calculating stopping distances for different speeds by using formulas for different conditions such as road type, tyre conditions and vehicle type.

#### Time:

use units of time, conversions between units, fractional, digital and decimal representations (ACMEM076)

represent time using 12-hour and 24-hour clocks (ACMEM077)

calculate time intervals, such as time between, time ahead, time behind (ACMEM078)

interpret timetables, such as bus, train and ferry timetables (ACMEM079)

use several timetables and electronic technologies to plan the most time-efficient routes (ACMEM080)

interpret complex timetables, such as tide charts, sunrise charts and moon phases (ACMEM081)

compare the time taken to travel a specific distance with various modes of transport (ACMEM082)

#### Distance:

use scales to find distances, such as on maps; for example, road maps, street maps, bushwalking maps, online maps and cadastral maps (ACMEM083)

optimise distances through trial-and-error and systematic methods; for example, shortest path, routes to visit all towns, and routes to use all roads. (ACMEM084)

#### Speed:

identify the appropriate units for different activities, such as walking, running, swimming and flying (ACMEM085)

calculate speed, distance or time using the formula speed = distance/time (ACMEM086)

calculate the time or costs for a journey from distances estimated from maps (ACMEM087)

interpret distance-versus-time graphs (ACMEM088)

calculate and interpret average speed; for example, a 4-hour trip covering 250 km. (ACMEM089)

## Unit 3

### Unit 3 Description

This unit provides students with the mathematical skills and understanding to solve problems related to measurement, scales, plans and models, drawing and interpreting graphs, and data collection. Teachers are encouraged to apply the content of the four topics in this unit – ‘Measurement’, ‘Scales, plans and models’, ‘Graphs’ and ‘Data collection’ – 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 and design and Mathematics and medicine. 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 3 Learning Outcomes

By the end of this unit, students:

• understand the concepts and techniques used in measurement, scales, plans and models, graphs, and data collection
• apply reasoning skills and solve practical problems in measurement, scales, plans and models, graphs, and data collection
• communicate their arguments and strategies when solving mathematical and statistical problems using appropriate mathematical or statistical language
• interpret mathematical and statistical information and ascertain the reasonableness of their solutions to problems.

### Unit 3 Content Descriptions

#### Topic 1: Measurement

Examples in context
• calculating and interpreting dosages for children and adults from dosage panels on medicines, given age or weight
• calculating and interpreting dosages for children from adults’ medication using various formulas (Fried, Young, Clark) in milligrams or millilitres
• calculating surface areas of various buildings to compare costs of external painting.

#### Linear measure:

review metric units of length, their abbreviations, conversions between them, estimation of lengths, and appropriate choices of units

(ACMEM090)

calculate perimeters of familiar shapes, including triangles, squares, rectangles, polygons, circles, arc lengths, and composites of these.

(ACMEM091)

find the area of irregular figures by decomposition into regular shapes (ACMEM094)

#### Area measure:

review metric units of area, their abbreviations, and conversions between them (ACMEM092)

use formulas to calculate areas of regular shapes, including triangles, squares, rectangles, parallelograms, trapeziums, circles and sectors (ACMEM093)

find the surface area of familiar solids, including cubes, rectangular and triangular prisms, spheres and cylinders (ACMEM095)

find the surface area of pyramids, such as rectangular- and triangular-based pyramids (ACMEM096)

use addition of the area of the faces of solids to find the surface area of irregular solids. (ACMEM097)

#### Mass:

review metric units of mass (and weight), their abbreviations, conversions between them, and appropriate choices of units (ACMEM098)

recognise the need for milligrams (ACMEM099)

convert between grams and milligrams. (ACMEM100)

#### Volume and capacity:

review metric units of volume, their abbreviations, conversions between them, and appropriate choices of units (ACMEM101)

recognise relations between volume and capacity, recognising that $$1\mathrm c\mathrm m^3=1\mathrm m\mathrm L$$ and $$1\mathrm m^3=1\mathrm k\mathrm L$$ (ACMEM102)

use formulas to find the volume and capacity of regular objects such as cubes, rectangular and triangular prisms and cylinders (ACMEM103)

use formulas to find the volume of pyramids and spheres. (ACMEM104)

#### Topic 2: Scales, plans and models

Examples in context
• drawing scale diagrams of everyday two-dimensional shapes
• interpreting common symbols and abbreviations used on house plans
• using the scale on a plan to calculate actual external or internal dimensions, the lengths of the house and the dimensions of? particular rooms
• using technology to translate two-dimensional house plans into three-dimensional buildings
• creating landscape designs using technology.

#### Geometry:

recognise the properties of common two-dimensional geometric shapes and three-dimensional solids (ACMEM105)

interpret different forms of two-dimensional representations of three-dimensional objects, including nets and perspective diagrams (ACMEM106)

use symbols and conventions for the representation of geometric information; for example, point, line, ray, angle, diagonal, edge, curve, face and vertex. (ACMEM107)

#### Interpret scale drawings:

interpret commonly used symbols and abbreviations in scale drawings (ACMEM108)

find actual measurements from scale drawings, such as lengths, perimeters and areas (ACMEM109)

estimate and compare quantities, materials and costs using actual measurements from scale drawings; for example, using measurements for packaging, clothes, painting, bricklaying and landscaping. (ACMEM110)

#### Creating scale drawings:

understand and apply drawing conventions of scale drawings, such as scales in ratio, clear indications of dimensions, and clear labelling (ACMEM111)

construct scale drawings by hand and by using software packages. (ACMEM112)

#### Three dimensional objects:

interpret plans and elevation views of models (ACMEM113)

sketch elevation views of different models (ACMEM114)

interpret diagrams of three-dimensional objects. (ACMEM115)

#### Right-angled triangles:

apply Pythagoras’ theorem to solve problems (ACMEM116)

apply the tangent ratio to find unknown angles and sides in right-angled triangles (ACMEM117)

work with the concepts of angle of elevation and angle of depression (ACMEM118)

apply the cosine and sine ratios to find unknown angles and sides in right-angled triangles (ACMEM119)

solve problems involving bearings. (ACMEM120)

#### Topic 3: Graphs

Examples in context
• interpreting graphs showing growth ranges for children (height or weight or head circumference versus age)
• interpreting hourly hospital charts showing temperature and pulse
• interpreting graphs showing life expectancy with different variables.

#### Cartesian plane:

demonstrate familiarity with Cartesian coordinates in two dimensions by plotting points on the Cartesian plane (ACMEM121)

generate tables of values for linear functions, including for negative values of $$x$$ (ACMEM122)

graph linear functions for all values of $$x$$ with pencil and paper and with graphing software. (ACMEM123)

#### Using graphs:

interpret and use graphs in practical situations, including travel graphs and conversion graphs (ACMEM124)

draw graphs from given data to represent practical situations (ACMEM125)

interpret the point of intersection and other important features of given graphs of two linear functions drawn from practical contexts; for example, the ‘break-even’ point. (ACMEM126)

#### Topic 4: Data collection

Examples in context
• analysing data obtained from medical sources, including bivariate data.

#### Census:

investigate the procedure for conducting a census (ACMEM127)

#### Surveys:

understand the purpose of sampling to provide an estimate of population values when a census is not used (ACMEM129)

investigate the different kinds of samples; for example, systematic samples, self-selected samples, simple random samples (ACMEM130)

investigate the advantages and disadvantages of these kinds of samples; for example, comparing simple random samples with self-selected samples. (ACMEM131)

#### Simple survey procedure:

identify the target population to be surveyed (ACMEM132)

investigate questionnaire design principles; for example, simple language, unambiguous questions, consideration of number of choices, issues of privacy and ethics, and freedom from bias. (ACMEM133)

#### Sources of bias:

describe the faults in the collection of data process (ACMEM134)

describe sources of error in surveys; for example, sampling error and measurement error (ACMEM135)

investigate the possible misrepresentation of the results of a survey due to misunderstanding the procedure, or misunderstanding the reliability of generalising the survey findings to the entire population (ACMEM136)

investigate errors and misrepresentation in surveys, including examples of media misrepresentations of surveys. (ACMEM137)

#### Bivariate scatterplots:

describe the patterns and features of bivariate data (ACMEM138)

describe the association between two numerical variables in terms of direction (positive/negative), form (linear/non-linear) and strength (strong/moderate/weak). (ACMEM139)

#### Line of best fit:

identify the dependent and independent variable (ACMEM140)

find the line of best fit by eye (ACMEM141)

use technology to find the line of best fit (ACMEM142)

interpret relationships in terms of the variables (ACMEM143)

use technology to find the correlation coefficient (an indicator of the strength of linear association) (ACMEM144)

use the line of best fit to make predictions, both by interpolation and extrapolation (ACMEM145)

recognise the dangers of extrapolation (ACMEM146)

distinguish between causality and correlation through examples. (ACMEM147)