Year 10 (5.1) 2015 Mathematics Program

Term 1

1 2 3 4 5 6 7 8 9 10

1. Interest and Depreciation

2. Coordinate Geometry 3. Surface Area and Volume

Task 1 – 20%

Term 2

1 2 3 4 5 6 7 8 9 10

4. Indicies5. Investigating Data

6. Equations and Inequalities

Task 2 – Half Yearly (30%)

Term 3

1 2 3 4 5 6 7 8 9 10

7. Graphs 8. Trigonometry 10. Probability

Task 3 – Assignment (10%)

Term 4

1 2 3 4 5 6 7 8 9 10 11

11. GeometryTBC

Task 4 – Half Yearly Exam (40%)

INTEREST AND DEPRECIATIONUnit OverviewIn this short Financial Mathematics topic, students revise the mathematics of earning an income and paying income tax from Year 9 before being introduced to the concept of compound interest, with Stage 5.2 students learning the compound interest formula and depreciation. Half of this topic is actually unique to the NSW syllabus and does not appear in the national Australian curriculum, but it has been retained so that Stage 5.1 students can be more financially literate with the mathematics of earning, saving and borrowing. Classroom examples should be as realistic as possible, with current rates being found on the Internet.

Outcomes MA5.1-4 NA solves financial problems involving earning, spending and investing money

MA5.1-1 WM uses appropriate terminology, diagrams and symbols in mathematical contexts MA5.1-2 WM selects and uses appropriate strategies to solve problems MA5.1-3 WM provides reasoning to support conclusions that are appropriate to the context

Literacy Students learn word bank using a variety of different teaching strategies for example close passage, definition matching exercise, memory games and spelling

tests. Word bank can be found at NCM 8 (pg ) located at www.nelsonnet.com.au

Some students have difficulty differentiating between interest and interest rate when answering questions. Note that ‘flat interest’ = ‘simple interest’.

Content Quality Teaching Ideas Resourceso Earning an income (5.1)

solve problems involving earning money

calculate weekly, fortnightly, monthly and yearly earnings

calculate earnings from wages, overtime, commission and pieceworkcalculate annual leave loading

Make problems as realistic as possible. Some students may be starting part-time jobs now and earning incomes.

Back-to-front problems, for example, given the final pay after annual leave loading or overtime pay was added, find the original pay

Use spreadsheets or graphics calculators to calculate incomes, tax, interest and depreciation

o Income tax (5.1)determine annual taxable income using current tax

Collect job advertisements and interest rates from newspapers and websites, tax tables, payslips, savings and loans brochures from banks and

ratesuse published tables or online calculators to determine the weekly, fortnightly or monthly tax to be deducted from a worker’s pay under the Australian ‘pay-as-you-go’ (PAYG) taxation system

credit unions, depreciation tables from tax guides, spreadsheets. Calculating tax refunds or debts

o Simple interest (5.1)apply the simple interest formula I = PRN to solve problems related to investing money at simple interest ratessolve problems involving simple interest

Students should learn the skill of expressing the interest rate, R, as a decimal. They should not round the value of R when calculating interest compounded monthly.

o Term payments (5.1)calculate the cost of buying expensive items by paying an initial deposit and making regular repayments that include simple interest

Collect examples of term payments and interest rates from store catalogues such as Harvey Norman and The Good Guys. Compare total paid with cash price.

Assessment Practical or problem-solving test/assignment Collage/poster/case study on the different ways of earning money. Compound interest assignment comparing different interest rates, principals or compounding periods. Spreadsheet or graphics calculator test.Registration

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Evaluation

COORDINATE GEOMETRY

Unit OverviewThis algebra topic revises and extends coordinate geometry concepts and skills introduced in the Year 9 topic, Coordinate geometry and graphs. It examines intervals and lines on the number plane as well as various forms of the equation of a straight line. The general form of a linear equation is met for the first time, as well as the equations of parallel and perpendicular lines. There is much scope for using graphing software such as GeoGebra in this topic. Note that the formulas for the length, midpoint and gradient of an interval are no longer part of the Stage 5.2 course, amd students will meet non-linear graphs in the Graphs topic.

Outcomes MA5.1-6 NA determines the midpoint, gradient and length of an interval, and graphs linear relationships

MA5.1-1 WM uses appropriate terminology, diagrams and symbols in mathematical contexts MA5.1-3 WM provides reasoning to support conclusions that are appropriate to the context

Literacy Students learn word bank using a variety of different teaching strategies for example close passage, definition matching exercise, memory games and spelling

tests. Word bank can be found at NCM 8 (pg ) located at www.nelsonnet.com.au

Why does the gradient-intercept equation have that name?

Content Quality Teaching Ideas Resourceso Length, midpoint and gradient of an interval

find the distance between two points located on the Cartesian plane using a range of strategies, including graphing softwarefind the midpoint and gradient of a line segment (interval) on the Cartesian plane using a range of strategies, including graphing software

Develop the idea of the midpoint as an average. Remind students that the midpoint is a point and so the answer should be a pair of coordinates

o Parallel and perpendicular lines (5.1)determine that parallel lines have equal gradients(STAGE 5.2) determine that straight lines are perpendicular if the product of their gradients is -1

Identify the x- and y-intercepts of a line.

o Graphing linear equations (5.1)sketch linear graphs using the coordinates of two pointsdetermine whether a point lies on a line by

All points that lie on the line have coordinates that satisfy the linear equation. Points that don’t lie on the line do not satisfy the equation.

substitutionAssessment Practical graphing test using pen-and-paper or technology. Graphing test or graphics calculator test.Registration

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Evaluation

SURFACE AREA AND VOLUME

Unit OverviewThis topic revises and extends surface area and volume concepts met in Year 9. Rather than learn a set of facts and formulas, the emphasis is upon understanding each idea met in this topic. This is achieved by applying the skills to a variety of real problems. Stage 5.1 students intending to study the Mathematics General course next year should cover the Stage 5.2 content, either now or at the end of the year as an option topic. As this is a Measurement topic, there are opportunities for investigation, practical work and open-ended problem-solving. Practice in estimating, the correct setting-out of solutions and the rounding of answers should feature prominently in the teaching of this topic.

Outcomes MA5.1-8 MG calculate the areas of composite shapes, and the surface areas of rectangular and triangular prisms MA4.14MG uses formulas to calculate the volumes of prisms and cylinders and converts between units of volume

MA5.1-1 WM uses appropriate terminology, diagrams and symbols in mathematical contexts MA5.1-2 WM selects and uses appropriate strategies to solve problems

Literacy Students learn word bank using a variety of different teaching strategies for example close passage, definition matching exercise, memory games and spelling

tests. Word bank can be found at NCM 8 (pg ) located at www.nelsonnet.com.au

Content Quality Teaching Ideas Resourceso Areas of composite shapescalculate the areas of composite figures by dissection into triangles, special quadrilaterals, quadrants, semi-circles and sectors

chart of area, surface area and volume formulas, nets or models of solid shapes, paper, scissors, measuring containers for capacity.

o Surface area of a rectangular and triangular prisms

solve problems involving the surface areas of right prisms

Emphasise how area involves multiplying two dimensions or powers of 2 while volume involves three dimensions or powers of 3.

Include surface area problems where Pythagoras’ theorem must be applied to find a slant height.

o Volumes of prisms and cylinderssolve problems involving volume and capacity of right prisms and cylindersfind the volumes of solids that have uniform cross-sections that are sectors, including semi-circles and

The formula for the volume of a right prism and cylinder also works for oblique prisms and cylinders, provided that the perpendicular height is used.

Find applications of surface area and volume in building and construction, e.g. backyard pool, packing material.

quadrants

o Revision and mixed problemssolve problems involving volume and capacity of right prisms and cylindersfind the volumes of solids that have uniform cross-sections that are sectors, including semi-circles and quadrants

Include problems where extra information is given, or composite solids are involved.

Include back-to-front problems where the surface area or volume is given.

Link to algebra: show that the formula for the surface area of a cylinder SA = 2πr2 + 2πrh may be factorised to SA = 2πr(r + h) for easier calculation.

Assessment Practical activity/assignment/test on area, surface area and volume. Open-ended and back-to-front questions: ‘The volume of a triangular prism is 540 cm3. What might its dimensions be?’ Research project.

Language From NSW syllabus: ‘Students are expected to be able to determine whether the prisms and cylinders referred to in practical problems are closed or open (one end

only or both ends), depending on the context’. From NSW syllabus: ‘The abbreviation m2 is read as 'square metre(s)' and not 'metre(s) squared' or 'metre(s) square'. A right prism has side faces that are rectangular and perpendicular to its cross-section. An oblique prism has side faces that are parallelograms and that are not

perpendicular to its cross-section. Similarly, a right cylinder has its axis (of rotation) perpendicular to its cross-section. An oblique cylinder’s axis is not perpendicular to its cross-section.

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INDICES

Unit OverviewThis topic mostly revises algebra skills from the Year 9 topics Algebra and Indices, before introducing factorising (monic) quadratic expressions of the form x2 + bx +c. Note that fractional indices are not part of the Stage 5.2 course, only negative indices are. This topic is fairly technical and abstract so each skill should be revised with care and precision appropriate to the level of the class. Students should practise and master each skill before moving onto the next one.

Outcomes MA5.1-5 NA operates with algebraic expressions involving positive-integer and zero indices, and establishes the meaning of negative indices for numerical bases

MA5.1-1 WM uses appropriate terminology, diagrams and symbols in mathematical contexts MA5.1-3 WM provides reasoning to support conclusions that are appropriate to the context

Literacy Students learn word bank using a variety of different teaching strategies for example close passage, definition matching exercise, memory games and spelling

tests. Word bank can be found at NCM 8 (pg ) located at www.nelsonnet.com.au For 24, 2 is called the base and 4 is called the power, index or exponent. From the NSW syllabus: ‘Teachers should use fuller expressions before shortening them, for example, 24 should be expressed as “2 raised to the power of 4”, before

“2 to the power of 4” and finally “2 to the 4”.Content Quality Teaching Ideas Resourceso The index lawssimplify algebraic products and quotients using index lawsThe zero indexNegative indices from numerical bases

Open-ended question: find two terms that can be divided to give 27. Verify the index laws by using a calculator. Explain why a particular algebraic

sentence, for example, a3 × a2 = a6, is incorrect. For 24, 2 is called the base and 4 is called the power, index or exponent. From the NSW syllabus: ‘Teachers should use fuller expressions before

shortening them, for example, 24 should be expressed as “2 raised to the power of 4”, before “2 to the power of 4” and finally “2 to the 4”.

AssessmentWriting activity on the use of variables and simplifying algebraic expressionsResearch assignment or poster on the algebraic rules or the history/meaning of algebraVocabulary test

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Evaluation

INVESTIGATING DATAUnit Overview

In this mostly Stage 5.2 topic, students consolidate their statistical skills by meeting interquartile range, box plots, bivariate data and scatter plots. The shape of a frequency distribution is revised first, but the rest of the topic will be new to students. The objective of this topic is to compare statistical measures for different sets of data. Aim to include analysis of data from class surveys and students’ own experiences. Because this is an interpretation and investigation topic, there is much scope for writing and literacy activities.

Outcomes MA5.1-12 SP uses statistical displays to compare sets of data, and evaluates statistical claims made in the media

MA5.1-1 WM uses appropriate terminology, diagrams and symbols in mathematical contexts MA5.1-3 WM provides reasoning to support conclusions that are appropriate to the context

Literacy Students learn word bank using a variety of different teaching strategies for example close passage, definition matching exercise, memory games and spelling

tests. Word bank can be found at NCM 8 (pg ) located at www.nelsonnet.com.au

This topic contains much statistical jargon, so a student-created glossary may be useful. Strictly speaking, the term bi-modal does not mean ‘two modes’. A bi-modal distribution actually has two ‘peaks’, with the higher one being the mode. However, in

this context, ‘mode’ has the same meaning as ‘peak’. Reinforce the terminology measures of location and measures of spread. Name the five measures found in a five-number summary.

Content Quality Teaching Ideas Resourceso The mean, median, mode and rangecalculate mean, median, mode and range for sets of data, and interpret these statistics in the context of data. investigate the effect of individual data values, including outliers, on the mean and median

Resources: Graphics calculator, statistical and graphing software, spreadsheets, databases, newspapers and magazines, Australian Bureau of Statistics (www.abs.gov.au), Bureau of Meteorology (www.bom.gov.au).

o Comparing data sets compare data displays using

mean, median and range to describe and interpret numerical data sets in terms of location (centre) and spread

o The shape of a frequency distributiondescribe data using terms, including ‘skewed’, ‘symmetric’ and ‘bi-modal’

o Statistics in the mediaEvaluate statistical reports in the media and other places by linking claims to displays, statistics and representative data

This topic lends itself to investigation projects. The class may be surveyed on a number of characteristics and the data analysed: height, arm span, shoe size, heartbeat rate, reaction time, health and PE data, number of children in family, number of people living at home, hours slept last night, number of letters in first name, number of vehicles/TV sets/mobile phones owned at home.

Examine the statistics from the sports page of a newspaper or website Find newspaper articles in which statistics have been misinterpreted. Students

could analyse the statistics used in media claims or use statistics to justify an argument themselves.

EXTENSION IDEAS Standard deviation (Stage 5.3) Grouped data, class intervals, median class, cumulative frequency graphs (no

longer part of syllabus) Replicate or implement a major statistical investigation.

Assessment Plan, implement and report on a statistical investigation. Vocabulary test, Statistical graphs and displays test. Investigate the use and abuse of statistics and statistical graphs in the media. Research the role of the Australian Bureau of Statistics or the Australian Census.Registration

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Evaluation

EQUATIONS AND INEQUALITIES

Unit OverviewThis topic revises equation-solving skills from the Year 9 topic, Equations before introducing students to quadratic equations of the form x2 + bx + c = 0 and linear inequalities. Stage 5.1 students should simply revise Stage 4 work but those who intend to study the Mathematics General course next year should also learn to solve equations with algebraic fractions and apply equations and formulas, either here or as an option topic at the end of the year. Simultaneous equations will be covered in a separate topic, while harder quadratic equations will be met in the Stage 5.3 option topic Quadratic equations and the parabola. Like many algebra skills, the process

of equation-solving is detailed and technical, requiring careful and precise understanding and practice, so don’t rush through this topic.

Outcomes MA5.2-8 NA solves linear and simple quadratic equations, linear inequalities and linear simultaneous equations, using analytical and graphical techniques

MA5.2-3 WM constructs arguments to prove and justify results

Literacy Students learn word bank using a variety of different teaching strategies for example close passage, definition matching exercise, memory games and spelling

tests. Word bank can be found at NCM 8 (pg ) located at www.nelsonnet.com.au

An algebraic expression refers to a ‘phrase’ containing terms and arithmetic operations, such as 2a + 5, while an algebraic equation refers to a ‘sentence’ involving an expression and an equals sign, such as 2a + 5 = 13.

Encourage students to set out their solutions to equations neatly with equals signs aligned in the same column. quadratic = algebraic expression in which the highest power of x is 2, eg 5x2 – 3x + 4. Some students believe that x < 5 and x 4 mean the same thing. Explain the difference.

Content Quality Teaching Ideas Resourceso Equationssolve linear equations using algebraic techniques

Students have been solving equations since Year 8. Emphasise the correct setting-out of solutions. The aim is to have the variable on the LHS of the equal sign, and the numerical answer on the RHS.

Encourage students to check solutions to equations and inequalities by substituting back.

o Equations with algebraic fractionssolve linear equations involving simple algebraic fractions

Examples of Stage 5.2 equations with algebraic fractions from NSW syllabus:

(denominators should be numerical).

o Equation problemssolve real-life problems by using pronumerals to represent unknowns

When solving a word problem, identify the unknown quantity and call it x, say. After solving, check that its solution sounds reasonable.

o Equations and formulassubstitute values into formulas to determine an

Examples of formulas: perimeter and area, circle formulas, speed, metric conversions (for example, Celsius to Fahrenheit), Pythagoras’ theorem, angle sum

unknownsolve problems involving linear equations, including those derived from formulas

of a polygon, E = mc2.

o Graphing inequalities on a number linerepresent simple inequalities on the number line

o Solving inequalitiessolve linear inequalities and graph their solutions on a number line

Assessment Writing activity comparing and evaluating the different methods of solving an equation. Writing activity describing the process of solving an inequality.

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GRAPHSUnit OverviewThis algebra topic revises and extends concepts in proportion and non-linear graphs from the Year 9 topic Coordinate geometry and graphs. Last year, students met the idea of direct proportion and the graphs of simple parabolas and circles, but here they are introduced to inverse proportion, conversion graphs and the exponential curve. There is much scope for using graphing software such as GeoGebra in this topic.

Outcomes

MA5.1-7 NA graphs simple non-linear relationships

MA5.1-1 WM uses appropriate terminology, diagrams and symbols in mathematical contexts MA5.1-3 WM provides reasoning to support conclusions that are appropriate to the context

Literacy Students learn word bank using a variety of different teaching strategies for example close passage, definition matching exercise, memory games and spelling

tests. Word bank can be found at NCM 8 (pg ) located at www.nelsonnet.com.au

Why is it called direct proportion? Why is it called inverse proportion? y = ax is called an exponential equation because a is the base and x is the power, index or exponent.

Content Quality Teaching Ideas Resourceso Direct proportionsolve problems involving direct proportion and explore the relationship between graphs and equations corresponding to simple rate problems

All points that lie on the graph have coordinates that satisfy its equation. Points that don’t lie on the graph do not satisfy the equation.

Use a graphics calculator, graphing software or spreadsheets to complete tables of values and graph linear and non-linear equations.

o Inverse proportionidentify and describe everyday examples of inverse (indirect) proportiono Conversion graphsinterpret and use conversion graphs to convert from one unit to anothero The parabolagraph simple non-linear relations, with and without the use of digital technologiesgraph parabolic relationships of the form y = ax2 and y = ax2 + c(STAGE 5.2) determine the x-coordinate of a point on a

When graphing, encourage students to label axes, use a suitable scale and label the graph.

number plane grid paper, graphics calculator, graphing software, spreadsheets. Graphing parabolas of the form y = ax2 + bx + c (Stage 5.3 option topic) Graphing hyperbolas and cubic curves (Stage 5.3)

parabola, given the y-coordinate of the point

o The exponential curvegraph exponential relationships of the form y = ax

(STAGE 5.2) sketch, compare and describe simple exponential curves of the form y = ±a±x + c◊

The parabola is a conic section formed by the intersection of a cone by a plane that cuts it at a steeper angle to its base than its axis. The path of a projectile (object thrown) is a parabola, as is the shape of a satellite dish, concave lens or car headlight. The path of some comets is a parabola

o The circlesketch circles of the form x2 + y2 = r2

o Identifying graphsmatch graphs of straight lines, parabolas, circles and exponentials to the appropriate equations

Compound interest and population growth can be modelled by exponential equations and graphs.

EXTENSION IDEASConsider a back deck or padio?Assessment Practical graphing test using pen-and-paper or technology. Matching equations to their graphs.Registration

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Evaluation

Topic 8TRIGONOMETRYUnit OverviewThis topic revises right-angled trigonometry from Year 9 before introducing angles of elevation and depression, and bearings. Do not rush through this topic—spend some time reviewing the sine, cosine and tangent ratios before applying them to solve problems. Stage 5.1 students work with angles in degrees only, while Stage 5.2 students work in degrees and minutes. Ensure that students receive plenty of practice in setting out their work correctly.

Outcomes MA5.1-1 WM uses appropriate terminology, diagrams and symbols in mathematical contexts MA5.1-2 WM selects and uses appropriate strategies to solve problems MA5.1-3 WM provides reasoning to support conclusions that are appropriate to the context MA5.1-10 MG applies trigonometry, given diagrams, to solve problems, including problems involving angles of elevation and depression MA5.2-1 WM selects appropriate notations and conventions to communicate mathematical ideas and solutions MA5.2-2 WM interprets mathematical or real-life situations, systematically applying appropriate strategies to solve problems MA5.2-13 MG applies trigonometry to solve problems, including problems involving bearings

Content Quality Teaching Ideas ResourcesPythagoras’ theorem investigate Pythagoras’ theorem and its

application to solving simple problems involving right-angled triangles

Investigate the history of the Babylonian base 60 system used in measuring angle size (and time). Students have already used the degrees-minutes-seconds button on the calculator for time calculations in Stage 4.

The trigonometric ratios Students could verify their answers to trigonometric problems using scale drawings.

Students should set out their solutions properly and use correct trigonometric terminology. Encourage them to check the reasonableness of answers to trigonometric problems by making a rough scale drawing. Students need practice in drawing diagrams for a given problem. Have students devise a problem for a given diagram and swap problems.

Finding an unknown side select and use appropriate trigonometric

ratios in right-angled triangles to find unknown sides, where the given angle is measured in degrees

(STAGE 5.2) find the lengths of unknown sides in right-angled triangles where the given angle is measured in degrees and minutes◊

apply trigonometry to solve right-angled triangle problems

Make a clinometer. Calculate the heights of trees, flagpoles and buildings using trigonometry.

Finding an unknown angle select and use appropriate trigonometric

ratios in right-angled triangles to find

unknown angles correct to the nearest degree

(STAGE 5.2) find the size in degrees and minutes of unknown angles in right-angled trianglesAngles of elevation and depression

solve a variety of practical problems involving angles of elevation and depression, including problems for which a diagram is not provided

Problems involving angles of elevation and depression usually require the tangent ratio. Also discuss the effect of the observer’s height.

Bearings (5.2) interpret three-figure bearings (for

example, 035°, 225°) and compass bearings (for example, SSW)

Problems involving bearings (5.2) solve a variety of practical problems

involving bearings, including problems for which a diagram is not provided

Harder problems involving angles of elevation and depression and bearings. (5.3)

Revision and mixed problems The exact ratios, complementary relations

such as cos 25° = sin 65°, trigonometry of obtuse angles (Year 10 Stage 5.3)

The sine, cosine and tangent graphs (Year 10 Stage 5.3)

Assessment Practical test involving clinometers Research project on the history or applications of trigonometryLanguage From the NSW syllabus: ‘The word “trigonometry” is derived from two Greek words meaning “triangle” and “measurement”’. Stress that the hypotenuse is a fixed side in a right-angled triangle, while the opposite and adjacent sides depend upon the angle quoted. From the NSW syllabus: ‘Emphasis should be placed on correct pronunciation of sin as “sine”.’ Encourage students to devise mnemonics for the trigonometric ratios. The word minute comes from the Latin pars minuta prima, meaning the first (prima) division of a degree or hour. The word second comes

from pars minuta secunda, meaning the second (secunda) division of a degree or hour. With compass bearings, stress the terminology: ‘the bearing of P from O.’ See syllabus Language notes (Stage 5.2) for more details.

Elevated = feeling happy = looking up, Depressed = feeling sad = looking down. From the NSW syllabus: ‘Students may find some of the terminology encountered in word problems involving trigonometry difficult to

interpret, eg “base/foot of the mountain”, “directly overhead”, “pitch of a roof”, “inclination of a ladder”. Teachers should provide students with a variety of word problems and they should explain such terms explicitly’

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Topic 9:SIMULTANEOUS EQUATIONSUnit Overview

In this short Stage 5.2 algebra topic, students are introduced to linear simultaneous equations and three different methods for solving them: graphical method, elimination method and substitution method, building upon previous work on algebra and graphing linear equations. As mentioned previously, the process of equation-solving is detailed and technical, and even more so for simultaneous equations, requiring careful and precise understanding and practice, so don’t rush through this topic.

Outcomes

MA5.2-1 WM selects appropriate notations and conventions to communicate mathematical ideas and solutions MA5.2-2 WM interprets mathematical and real-life situations, systematically applying appropriate strategies to solve problems MA5.2-8 NA solves linear and simple quadratic equations, linear inequalities and linear simultaneous equations, using analytical and

graphical techniquesContent Quality Teaching Ideas Resources

Solving simultaneous equations graphically (5.2)

solve linear simultaneous equations by finding the point of intersection of their graphs

Investigate the use of CAS (computer algebra system) calculators/software, websites such as Wolfram Alpha, spreadsheets and graphics calculators to solve simultaneous equations.

Encourage students to check solutions by substituting back. For word problems, check that the solution sounds reasonable

Using technology and websites to solve simultaneous equations

The elimination method (5.2) solve linear simultaneous equations

using appropriate algebraic techniques

When solving a word problem, students need practice in identifying the unknown quantity and calling it x, say.

When solving simultaneous equations, students often forget to give the solution for both variables x and y.

The substitution method (5.2) Open-ended question: if x + 2y = 9, what are some possible values of x and y?

Simultaneous equation problems (5.2) generate and solve linear simultaneous

equations from word problems and interpret the results

Simultaneous equations involving linear and non-linear equations (Stage 5.3 option topic Quadratic equations and the parabola)

Revision and mixed problems Graphing software and graphics calculators allow students to graph two linear equations and to display the coordinates of the point of intersection of their graphs’.

Assessment

Writing activity comparing and evaluating the different methods of simultaneous equations. Simultaneous equations test/assignment comparing different methods of solution.

Language Why are they called simultaneous equations? Why do the three methods of solving simultaneous equations have those names?

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Topic 10:PROBABILITYUnit Overview This topic revises Year 9 probability theory before Stage 5.2 students tackle the more advanced concepts of multi-step experiments, dependent events and conditional probability. The focus is upon interpreting descriptions of events using the words ‘and’, ‘or’, ‘without replacement’ and ‘given that’, so there are many opportunities for class discussion and language activities. Tree diagrams to represent the sample space of multi-step experiments are introduced, so spend considerable time teaching and practising drawing these as students (even in Years 11-12) often have difficulty understanding them.

Outcomes• MA5.1-1 WM uses appropriate terminology, diagrams and symbols in mathematical contexts• MA5.1-2 WM selects and uses appropriate strategies to solve problems• MA5.1-3 WM provides reasoning to support conclusions that are appropriate to the context• MA5.1-13 SP calculates relative frequencies to estimate probabilities of simple and compound events• MA5.2-1 WM selects appropriate notations and conventions to communicate mathematical ideas and solutions• MA5.2-2 WM interprets mathematical or real-life situations, systematically applying appropriate strategies to solve problems• MA5.2-3 WM constructs arguments to prove and justify results• MA5.2-17 SP describes and calculates probabilities in multi-step chance experiments

Content Quality Teaching Ideas Resources Relative frequency

o calculate relative frequencies from given or collected data to estimate probabilities of events involving ‘and’ or ‘or’

o Do not assume that all students have had experience with the properties of playing cards: suits, colours, deck of 52. Be sensitive to religious and cultural differences in attitudes towards gambling.

Resources: Dice, coins, counters, spinners, playing cards, probability simulation software

Venn diagramso represent events in Venn diagrams and

solve related problemso describe events using language of ‘at

least,’ exclusive ‘or’ (A or B but not both), inclusive ‘or’ (A or B or both) and ‘and’

o calculate probabilities of events from data contained in Venn diagrams

o Counting techniques, the birthday problem (X)o Investigating the probability of winning games of chance and

gambling (X)o Investigate the use of probability in insurance, for example, life

expectancy (X)

Two-way tableso represent events in two-way tables and

solve related problemso calculate probabilities of events from

data contained in two-way tables

o Probability tree diagrams that have probability values listed on branches, addition and product rules(X)

o calculate probabilities of simple and

compound events in two- and three-step chance experiments, with and without replacement

(5.2)

o describe the results of two- and three-step chance experiments, with and without replacement, assign probabilities to outcomes, and determine probabilities of events

o distinguish informally between

dependent and independent events, and recognise that for independent events A and B, P(A and B) = P(A) × P(B)

o distinguish informally between

dependent and independent events, and recognise that for independent events A and B, P(A and B) = P(A) × P(B)

o calculate probabilities of events where

a condition is given that restricts the sample space, eg given that a number less than 5 has been rolled on a fair six-sided die, calculate the probability that this number was a 3

o critically evaluate conditional statements used in descriptions of chance situations

Graph the results of a probability experiment on a dot plot or histogram.

If a coin is tossed seven times and comes up heads each time, what is the probability that the next toss is also a head?

Revision and mixed problemsAssessment Writing and comprehension activities on describing events involving mutually exclusive and overlapping activities Experimental probability investigation or simulation Research project on the applications or history of probability, for example, insurance premiums, planning for roads and new communities

Language Students should know the difference between an outcome and an event: an event contains one or more outcomes of an experiment. Inclusive ‘or’ = A or B or both, exclusive ‘or’ = A or B but not both, mutually exclusive means A and B are not overlapping and cannot both

happen What is the difference between ‘at least 4’ and ‘4 or more’? Students (even in Year 12) often think that the two phrases mean the same thing. Note that in the new syllabus the term ‘multi-step experiment’ replaces ‘multi-stage experiment’. Clearly explain the difference between ‘with

replacement’ and ‘without replacement’. From the NSW syllabus (Stage 4): A compound event is an event that can be expressed as a combination of simple events, eg drawing a

card that is black or a King from a standard set of playing cards, throwing at least 5 on a standard six-sided die.

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Evaluation

Topic 11:GEOMETRYUnit Overview This short topic revises Year 9 geometry theory with angle sums of polygons, similar figures and the similar triangles tests and introduces formal congruent triangle proofs to Stage 5.2 students, including proving properties of triangles and quadrilaterals. Stage 5 marks the start of more formal deductive geometry. Promote the correct use of language in reasoning, with attention given to drawing clear diagrams and setting out proofs and solutions carefully.

Outcomes MA5.1-1 WM uses appropriate terminology, diagrams and symbols in mathematical contexts MA5.1-2 WM selects and uses appropriate strategies to solve problems MA5.1-3 WM provides reasoning to support conclusions that are appropriate to the context MA5.1-11 MG describes and applied the properties of similar figures and scale drawings MA5.2-1 WM selects appropriate notations and conventions to communicate mathematical ideas and solutions MA5.2-2 WM interprets mathematical or real-life situations, systematically applying appropriate strategies to solve problems MA5.2-3 WM constructs arguments to prove and justify results

MA5.2-14 MG calculates the angle sum of any polygon and uses minimum conditions to prove triangles are congruent or similar

Content Quality Teaching Ideas Resources

o apply the result for the interior angle sum of a triangle to find, by dissection, the interior angle sum of polygons with more than three sides

Properties of triangles and quadrilaterals may be demonstrated informally (by symmetry, paper-folding, protractor and ruler measurement) but now also by congruent triangle proofs.

Students should have experience in classifying triangles and quadrilaterals using their properties and minimal conditions, for example, which quadrilateral’s diagonals bisect each other?

The exterior angle sum of a convex polygon is 360°: if you walk around the perimeter of a closed figure, the total of your turns should be a revolution.

There is much scope in this topic to use dynamic geometry software such as GeoGebra. The Internet is full of dynamic geometry animations and applets that demonstrate the properties shown in this topic. The Math Open Reference website www.mathopenref.com contains animations demonstrating the tests for congruent and similar triangles.

o establish that the sum of the exterior

angles of any convex polygon is 360°

Congruent triangle proofso write formal proofs of the congruence

of triangles, preserving matching order of vertices

Proving properties of triangles and

quadrilateralso use the congruence of triangles to

prove properties of special triangles and quadrilaterals

o use the enlargement transformation to

explain similarity

figures

o solve problems using ratio and scale factors in similar figures

Tests for similar triangles (5.2)o investigate the minimum conditions

needed, and establish the four tests, for two triangles to be similar

Assessment Writing activity or poster summary on the properties of triangles, quadrilaterals and polygons Vocabulary test Research/investigation assignment on properties of polygons or similar figures. Writing activities, especially in identifying congruent and similar triangles or in writing a proof Test or assignment on setting out a geometrical proof correctly. Proving properties of geometrical figures by congruent triangles. Practical activities/projects using similar triangles.

Language Use matching angles rather than corresponding to avoid confusion with corresponding angles found when a transversal crosses two lines.

From the NSW syllabus: ‘This syllabus has used “matching” to describe angles and sides in the same position: however, the use of the word “corresponding” is not incorrect.’

Encourage students to set out their geometrical answers logically, step-by-step and giving reasons. From the NSW syllabus: ‘If students abbreviate geometrical reasons that they use in deductive geometry, they must take care not to

abbreviate the reasons to such an extent that the meaning is lost’. The mathematical symbol ‘≡’ means ‘is identical to’ in algebra and ‘is congruent to’ in geometry. In geometry, the word similar has a different meaning to its everyday one. Remember to name the vertices of congruent and similar figures in matching order.Be wary that in NSW, there is a continual debate on whether the tests for similar triangles can be abbreviated by initials in the same way as the

tests for congruent triangles. The Australian curriculum lists these abbreviations in its glossary (using AAA for ‘equiangular’), but the NSW syllabus does not formally acknowledge them.

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