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Essential Mathematics

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Unit 1

Unit 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.

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.

Content Descriptions

Examples in Context Topic 1: Calculations, percentages and rates

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)

Examples in Context Topic 2: Measurement

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)

Examples in Context Topic 3: Algebra

Single substitution:

substitute numerical values into algebraic expressions; for example, substitute different values of x to evaluate the expressions 3x5, 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)

Examples in Context Topic 4: Graphs

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)

 

Unit 2

Unit 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.

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.

Content Descriptions

Examples in Context Topic 1: Representing and comparing data

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)

Examples in Context Topic 2: Percentages

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)

Examples in Context Topic 3: Rates and ratios

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)

Examples in Context Topic 4: Time and motion

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 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.

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.

Content Descriptions

Examples in Context Topic 1: Measurement

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)

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

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
1cm3=1mL and 1m3=1kL (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)

Examples in Context Topic 2: Scales, plans and models

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)

Examples in Context Topic 3: Graphs

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)

Examples in Context Topic 4: Data collection

Census:

investigate the procedure for conducting a census (ACMEM127)

investigate the advantages and disadvantages of conducting a census. (ACMEM128)

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)

 

Unit 4

Unit 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.

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.

Content Descriptions

Examples in Context Topic 1: Probability and relative frequencies

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)

Topic 2: Earth geometry and time zones

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)

Examples in Context Topic 3: Loans and compound interest

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)

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F-10 Curriculum

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