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Year 8 2015 Mathematics Program

Term 1

1 2 3 4 5 6 7 8 9 10

1. Working With Number(Number and Algebra)

2. Pythagoras(Measurement and Geometry)

3. Algebra(Number and Algebra)

Task 1 – Week 7 (20%)

Term 2

1 2 3 4 5 6 7 8 9 10

4. Geometry(Measurement and Geometry)

Naplan5. Area and Volume

(Measurement and Geometry)6. Fractions and Percentages

(Number and Algebra)

Task 2 – Half Yearly (30%)

Term 3

1 2 3 4 5 6 7 8 9 10

7. Investigating Data(Statistics and Probability)

8. Congruent Figures (Measurement and Geometry)Issue Assignment

9. Probability(Statistics and Probability)

Task 3 – Assignment (10%)

Term 4

1 2 3 4 5 6 7 8 9 10 11

10. Equations (Number and Algebra)

11. Ratio, Rates and Time(Number and Algebra)

(Measurement and Geometry)

12. Graphing linear Equations(Number and Algebra)

Task 4 – Half Yearly Exam (40%).

Pythagoras’ theoremUnit OverviewThis is the first time students meet Pythagoras’ theorem. This is a Year 9 topic in the Australian curriculum but a Stage 4 (Years 7–8) topic in the NSW syllabus: ‘Students should gain an understanding of Pythagoras’ theorem, rather than just being able to recite the formula. Emphasis should be placed upon understanding the theorem and using it to solve problems involving the sides of right-angled triangles.Outcomes

MA4-16 MG applies Pythagoras’ theorem to calculate side lengths in right-angled triangles and solves related problems

MA4-1 WM communicates and connects mathematical ideas using appropriate terminology, diagrams and symbols

MA4-2 WM applies appropriate mathematical techniques 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 3) located at www.nelsonnet.com.auOther important literacy notes for this unit:

Hypotenuse is an ancient Greek word: hypo means ‘under’ while teinousa means ‘stretching’ because the hypotenuse ‘stretches’ under a right angle.

Explain and reinforce the logic behind the ‘converse’ of Pythagoras’ theorem.

‘The meaning of “exact” answer will need to be taught explicitly. Students may find some of the terminology/vocabulary encountered in word problems involving Pythagoras’ theorem difficult to interpret, for example, ‘foot of a ladder’, ‘inclined’, ‘guy wire’.

BLM page 21

Content Quality Teaching Ideas ResourcesSquare roots and surds Students who find this topic difficult may concentrate on solving one

step equations and adding and subtracting numbers that have been squared

NCMACEx 1.1-page 5

Discovering Pythagoras’ theorem identify the hypotenuse as the longest side in any right-angled triangle and also as the side opposite the right angle

establish the relationship between the lengths of the sides of a right-angled triangle in practical ways, including using digital technologies

NCMACEx 1.02-page 7

BLM page 4 BLM page 8

Finding the hypotenuse solve practical problems involving Pythagoras’ theorem, approximating the answer as a decimal and giving an exact answer as a surd

NCMACEx 1.03-page 8

BLM page6 BLM page 10 MOL#3315

Finding a shorter side NCMACEx 1.04-page 12

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MOL#3316

Mixed problems

Harder problems: two-stage or in three-dimensions, for example, longest diagonal in a rectangular prism

Word problems

Length of an interval on the number plane

Irrational numbers, graphing surds on a number line, simplifying surds

X=The real number system, proof that is irrational

NCMACEx 1.05-page 16

MOL#3318 MOL#3319 MOL#3320

Testing for right-angled triangles use the converse of Pythagoras’ theorem to establish whether a triangle has a right angle

R Reasoning (generalising and proving with maths): Proving that a triangle is right-angled given the lengths of its sides


Pythagorean triadsidentify a Pythagorean triad as a set of three numbers such that the sum

of the squares of the first two equals the square of the third

History of Pythagorean triads, properties of Pythagorean triads


BLM page 12 MOL#3321

Pythagoras’ theorem problems PS Problem solving (modelling and investigating with maths): Using Pythagoras’ theorem to solve measurement problems


BLM page 15 BLM page 18

Assessment Research assignment on Pythagoras and Pythagoras’ theorem Matching activities: Pythagoras’ theorem to diagrams Writing activity explaining Pythagoras’ theorem

Registration

Signature: Date:

Evaluation

Working with numbersUnit Overview


This topic revises and extends basic operations with whole numbers, integers, decimals, powers, roots and prime factors, then explores properties of squares and square roots (ab)2 , and the index laws. This is a short refresher topic that reinforces mental, pen-and-paper and calculator skills so don’t dwell too long on particulars. Keep it simple and make the revision suitable to the ability and experience of your Year 8 class. You may even like to set part of this topic as a revision assignment rather than re-teach it all. Ensure that estimating and checking of answers are reinforced during lessons.

Outcomes MA4-4 NA compares, orders and calculates with integers, applying a range of strategies to aid computation MA4-5 NA operates with fractions, decimals and percentages MA4-9 NA operates with positive-integer and zero indices of numerical indices



MA4-3 WM recognises and explains mathematical relationships using reasoning

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 3) located at www.nelsonnet.com.auOther important literacy notes for this unit:

‘3’ is read ‘negative 3’, not ‘minus 3’. Students should not confuse the negative sign with the minus operation.

Reinforce the language of approximation: ‘approximate’, ‘write correct to’, ‘round to’, ‘n decimal places’, ‘nearest tenth’. Note that the NSW syllabus now prefers the term ‘rounding’ to ‘rounding off’.

Terminating means ‘stopping’; recurring means ‘repeating’.

Content Quality Teaching Ideas ResourcesMental calculation

apply the associative, commutative and distributive laws to aid mental computation


Adding and subtracting integers Use the number line to make more visual (E) NCMACEx 2.02-page 39

MOL#8027Multiplying integers

Dividing integers

Do question by ignoring the sign first. Then apply the rules. NCMACEx 2.03-page 42Ex 2.04 – page 44




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Order of operations

apply the order of operations to evaluate expressions involving directed

numbers mentally, including where an operator is contained within the

numerator or denominator of a fraction, for example,


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Decimals round decimals to a specified number of decimal places

When teaching rounding decimals, include more difficult examples, such as rounding 4.8971 to two decimal places



Multiplying and dividing decimals The NSW syllabus says that written multiplication and division of decimals may be limited to operators with two digits.

NCMACEx 2.07-page 54MOL#8050MOL#8051

Terminating and recurring decimals use the notation for recurring (repeating) decimals, for example, 0.33333 …

, 0.345345345 … 0.266666 … C Communicating (describing and representing maths): Using correct

notation for recurring decimals, powers and roots

Investigate the value of . Is it really equal to 1?(X) Convert recurring decimals to fractions (Year 9, Stage 5.3)(X) Investigate patterns in the recurring decimals of the fraction families of the

sixths, sevenths and ninths (X)


Powers and roots find square roots and cube roots of any non-square whole number using a

calculator, after first estimating apply the order of operations to evaluate expressions involving indices,

square and cube roots, with and without a calculator

determine through numerical examples the properties of square roots of


BLM page 29

products: (ab)2 and Irrational numbers, surds, graphing surds on a number line, simplifying

surds(X) How did mathematicians find square roots before calculators and

computers? Investigate Newton’s method.(X)

Some decimals are neither terminating nor recurring. Their digits run endlessly, but without repeating, for example, 1.4142135 . . . and π 3.1415926 . . .

Investigate finding higher powers on the calculator Common mistake: 3. The square root of a number is a single positive

value, so 3 only. However, 3 and the equation x2 9 has two solutions, x 3 or 3.

Prime factors express a number as a product of its prime factors to determine whether its square root or cube root is an integer

As an alternative to factor trees, prime factors can also be extracted by repeated division. See the Skillsheet ‘Prime factors by repeated division’.


Index laws for multiplying and dividing use index notation with numbers to establish the index laws with positive

integral indices and the zero index use index laws to simplify expressions with numerical bases, for example, 52 54 5 57

Investigate the square root of quotients(X)


More index laws Investigate scientific notation(X) NCMACEx 2.12-page 71

Assessment Non-calculator test

Revision assignment

Registration Evaluation

Signature: Date:

AlgebraUnit Overview

The Australian curriculum introduces algebra by generalising number laws and patterns, and in the Year 7 topic Algebra and equations students met elementary concepts such as variables, translating worded statements to algebraic expressions, algebraic abbreviations and substitution. In this Year 8 topic, students meet more formal operations with algebraic terms such as simplifying algebraic expressions, including the processes of expanding and factorising. This topic is fairly technical and abstract so each skill should be taught with care and precision as students may find the concepts difficult. Students should practise and master each skill before moving onto the next one.

Outcomes MA4-8 NA generalises number properties to operate with algebraic expressions

MA4-1 WM communicates and connects mathematical ideas using appropriate terminology, diagrams and symbols MA4-2 WM applies appropriate mathematical techniques to solve problems MA4-3 WM recognises and explains mathematical relationships using reasoning

Literacy Students explore Word Banks using a variety of teaching strategies. (Close passage, definition matching, memory games and spelling tests. Word bank can be accessed via the Nelson Net Text book (Page….) accessed at www.nelsonnet.com.auOther important literacy notes for this unit: Reinforce the meanings of variable, term, expression, simplify, evaluate, substitute, expand and factorise An algebraic term consists of a number and/or a variable, for example, 4p2. An algebraic expression is a ‘phrase’ containing terms and one or more arithmetic operation, for example, 5x 6. An equation is a ‘sentence’ containing an expression, an ‘’ sign and an ‘answer,’ for example, 5x 6 26. The word expand comes from writing out an expression ‘the long way’ without brackets. Draw a diagram using rectangles and an array of dots to show equivalences such as 3(n 2) 3n 6. Students are not required to learn the phrase ‘distributive law’. NSW syllabus: ‘Recognise the role of grouping symbols and the different meanings of expressions, such as 2a 1 and 2(a 1)’. Emphasise the difference between expand and factorise, as students will often do the opposite of what is requested.

Content Quality Teaching Ideas Resources

Prior Knowledge (E) Question/Pretest that students can

a) generalise properties of odd and even numbers, and completes simple number sentences by calculating missing values.

b) analyses and creates geometric number patterns, constructs and completes number sentences, and locates points on the Cartesian plane.

Variables introduce the concept of variables as a

Resources: counters, cubes, cups, blocks, envelopes and other concrete materials for modelling variables (E)

NCMAC

Ex 3.01 page 81



way of representing numbers using letters

extend and apply the laws and properties of arithmetic to algebraic terms and expressions

Stress that a variable does not stand for an object but for the number of objects. Even though we do not know the value of a variable or term, we can still collect them. For example, ‘3 lots of x plus 4 lots of x equals 7 lots of x’.

Some students believe 4a 2b a [4a ] [2b ] a 5a 2b. Encourage them to group each term with the sign before it: 4a [ 2b] [ a] 3a 2b.

From words to algebraic expressions move fluently between algebraic and

word representations as descriptions of the same situation

NCMAC

Ex 3.02 page 84BLM page 43

Substitution create algebraic expressions and evaluate

them by substituting a given value for each variable

Collecting variables

Determine and justify whether a simplified or equivalent expression is correct by substituting a number. (E)

More challenging problems involving substitution and translating worded statements into algebraic expressions(X)

NCMAC

Ex 3.03 page 87Ex 3.04 page 89

Adding and subtracting terms Application of collecting like terms: the formulas for the perimeter of the square and rectangle. Show that variables provide powerful shorthand in this regard. (X)

NCMAC

Ex 3.05 page 93 BLM page 48 MOL#8084

Multiplying terms Common mistakes: 2a a 2, 3b2 3b 3b. Explain that the index 2 belongs to the b only.

NCMAC

Ex 3.06 page 96MOL#8087

Dividing terms NCMAC

Ex 3.07 page 98MOL#8088

Extension: The index laws extend and apply the index laws to

variables, using positive integer indices and the zero index

Negative or fractional indices(X) NCMAC

Ex 3.08 page 99 BLM page 50

MOL#3226#3227#3228

Expanding expressions Binomial expansions (Year 9/Stage 5.2), for example (x 3)(x 2), (x 5)

(x 5), (x 2)2 (X) NCMAC


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#3231#3232Factorising algebraic terms

factorise a single algebraic term, for example, 6ab 3 2 a b

NCMAC

Ex 3.10 page 106MOL#3238

Factorising expressions Factorising by grouping in pairs (X) NCMAC


Factorising with negative terms NSW syllabus: ‘Check expansions and factorisations by performing the

reverse process’. Include examples involving negative terms.

NCMAC

Ex 3.12 page 109

Assessment Writing activity on the use of variables and simplifying algebraic expressions Research assignment or poster on the algebraic rules or the history/meaning of algebra Vocabulary test

Registration

Signature: Date:

Evaluation

GeometryUnit Overview

This topic revises geometrical concepts introduced in Year 7, namely relating to angles, triangles and quadrilaterals, in a more formal way. However, practical activities and correct geometrical terminology should be promoted throughout this topic.

From the NSW syllabus: ‘At this stage in geometry, students should write reasons without the use of abbreviations to assist them in learning new terminology, and in understanding and retaining geometrical concepts’.Outcomes

MA4-17 MG classifies, describes and uses the properties of triangles and quadrilaterals, and determines congruent triangles to find unknown lengths and angles

MA4-18 MG identifies and uses angle relationships, including those related to transversals on sets of parallel lines




Literacy Students explore Word Banks using a variety of teaching strategies. (Close passage, definition matching, memory games and spelling tests. Word bank can be accessed via the Nelson Net Text book (Page….) accessed at www.nelsonnet.com.auOther important literacy notes for this unit:

Equilateral comes from the Latin aequus latus, meaning ‘equal sides’, isosceles comes from the Greek isos skelos, meaning ‘equal legs,’ and scalene comes from the Greek skalenos skelos, meaning ‘uneven leg’.

Avoid using the term ‘base angles’ for isosceles triangles because it may be misleading, depending upon the orientation of the triangle. Instead, use ‘the angles opposite the equal sides’ or ‘the two angles next to the uneven side’.

From the NSW syllabus: ‘The diagonals of a convex quadrilateral lie inside the figure’.

Content Quality Teaching Ideas ResourcesPrior Knowledge(E) Test /quiz students to see if they can manipulate, draw and classify 2

dimensional shapes, including equilateral, isosceles and scalene triangles, special quadrilaterals, pentagons, hexagons and octagons, and describe their properties.

Angle geometry Students calculate missing variables in right angles, straight angles, angles at a point, vertically opposite angles, complementary and supplementary angles. NCMAC


BLM page 61 MOL#8104#3302

Angles on parallel lines Students calculate missing variables in alternate(Z), cointerior(C) and corresponding angles(F) in parallel lines

Students must know what the traversal is .

NCMACEx 4.02-page 123MOL#8106



Line and rotational symmetry identify line and rotational symmetries

Fold pieces of paper in half to demonstrate line symmetry(E) Spin paper or object to demonstrate rotational symmetry(E)

NCMACEx 4.03-page 126BLM page 64MOL#3292

Classifying triangles classify triangles according to their side

and angle properties

Students must become familiar with the terminology obtuse, acute and right to describe triangles by angle and isosceles, scalene and equilateral to describe triangles by their side length.


Classifying quadrilaterals distinguish between convex and non-

convex quadrilaterals (the diagonals of a convex quadrilateral lie inside the figure)

describe squares, rectangles, rhombuses, parallelograms, kites and trapeziums


Properties of quadrilaterals investigate the properties of special

quadrilaterals

classify special quadrilaterals on the basis of their properties

Properties of triangles and quadrilaterals should be demonstrated informally (by symmetry, paper-folding, protractor and ruler measurement), rather than by congruent triangle proofs. (E)

From NSW syllabus: ‘A range of examples of the various triangles and quadrilaterals should be given, including quadrilaterals containing a reflex angle and figures presented in different orientations’.

The properties of special quadrilaterals allow us to develop formulas for finding their areas in the topic Area and volume, for example, the diagonal properties of the kite and rhombus.

NCMACEx 4.06-page 135BLM page 72/75/78

Angle sums of triangles and quadrilaterals justify informally that the interior angle

sum of a triangle is 180°, and that any exterior angle equals the sum of the two interior opposite angles

use the angle sum of a triangle to establish that the angle sum of a quadrilateral is 360°

From syllabus: ‘Students should give reasons when finding unknown angles. For some students, formal setting-out could be introduced. For example,

PQR 70° (corresponding angles, PQ || SR)’. (X)

In how many different ways can you demonstrate the angle sum of a triangle (or quadrilateral)?

Formal proofs in deductive geometry (X)


Extension: Angle sum of a polygon* apply the result for the interior angle sum if a

triangle to find, by dissection, the interior angle sum of polygons with more than three sides

Find the size of one angle in a regular polygon, or the exterior angle sum of a convex polygon (X)


Assessment Writing activity or poster summary on the properties of angles, triangles or quadrilaterals Vocabulary test ‘What quadrilateral am I’ puzzles Research/investigation assignment on properties of triangles or quadrilaterals Assignment on setting out a geometry proof

Language

Registration

Signature: Date:

Evaluation

Area and volumeUnit Overview

This topic revises and extends perimeter, area and volume concepts, with new content including the areas of special quadrilaterals and circles, and conversions between metric units for area and volume. Circle measurement is formally introduced, and after examining the parts and geometrical properties of a circle, students discover the special number π and its role in calculating perimeters and areas of circles and circular shapes.

Outcomes MA4-12 MG calculates the perimeter of plane shapes and the circumference of circles MA4-13 MG uses formulas to calculate the area of quadrilaterals and circles, and converts between units of area

MA4-14 MG uses formulas to calculate the volume of prisms and cylinders, and converts between units of volume




From NSW syllabus: ‘Volume refers to the space occupied by an object or substance. The abbreviation m3 is read “cubic metre(s)” and not “metres cubed”.’ Ensure that students use the correct units for area and volume.

Express area formulas in words as well as algebraically From NSW syllabus: ‘The names for some parts of the circle (centre, radius, diameter, circumference, sector, semicircle and quadrant) are first

introduced in Stage 3 … Pi (π) is the Greek letter equivalent to “p”, and is the first letter of the Greek word perimetron, meaning perimeter. In 1737, Euler used the symbol for pi for the ratio of the circumference to the diameter of a circle.’

Concentric means ‘same centre,’ an annulus is a ring shape bounded by two concentric circles.

Content Quality Teaching Ideas ResourcesPerimeter NCMAC

Ex 5.01-page 155 MOL#8115

Metric units for area F Fluency (applying maths): Selecting correct strategies to

convert between metric units and calculate areas and volumes


Areas of rectangles, triangles and parallelograms

establish the formulas for areas of rectangles, triangles and parallelograms and use these in problem solving

R Reasoning (generalising and proving with maths): Introducing formulas to generalise the rule for calculating perimeters, areas and volumes; analyse relationships for converting between metric units for length, area and volume

Areas may be found by paper-cutting activities and grid overlays: print out the Worksheet ‘1 cm grid paper’ and photocopy it onto an

NCMACEx 5.03-page 161MOL#8117#8118#8121



overhead transparency. (E) Estimate areas of windows, noticeboards, blackboards, desktop,

postage stamps. Mark a square metre or hectare on school grounds. Surface area of a cube, prism and cylinder(X)

Areas of composite shapes Examples of composite shapes: L-shape, T-shape, U-shape, trapezium, semi-circles, annuli and pipes.

Areas of irregular figures: traverse surveys, Simpson’s rule(X)

NCMACEx 5.04-page 163MOL#8119#8120

Area of a trapezium The area of a trapezium can be cut up and rearranged into two triangles or one rectangle.(E)

When proving the formulas for areas of special quadrilaterals, demonstrate the usefulness and power of variables in algebra.(X)


Areas of kites and rhombuses The area of a rhombus or a kite can be cut up and rearranged into two congruent triangles or one rectangle. The area formula actually works for any quadrilateral with perpendicular diagonals.


Parts of a circle investigate the line symmetries and

the rotational symmetry of circles and of diagrams involving circles, such as a sector and a circle with a marked chord or tangent

Circumference of a circle investigate the concept of irrational

numbers, including find the perimeter of quadrants, semi-

circles, sectors and composite figures

C Communicating (describing and representing maths): Describing metric units of area and volume and labelling the parts of a circle

Draw each part of the circle on the board and ask students to describe it in their own words, for example, a sector is like a ‘slice of pizza or cake.’

From the NSW syllabus: ‘The number π is known to be irrational ... At this stage, students only need to know that the digits in its decimal expansion do not repeat (all this means is that it is not a fraction), and in fact have no known pattern.’ 3.141 592 653 589 793 ...

Calculate the perimeter of a regular hexagon inscribed in a circle with the circle’s circumference to demonstrate that π > 3. (X)

Circumference of the Earth, latitude and longitude (small and great circles) on the Earth’s surface (X)

NCMACEx 5.07-page 172Ex5.08-page 177BLM page 92/94/99

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Area of a circle calculate the area of quadrants, semi-

circles, sectors and composite figures

With composite area problems, encourage students to look for opportunities for combining two semicircles. (X)

NCMACEx 5.09-page 185BLM page 97MOL#3284#3285

Metric units for volume NCMAC

Ex 5.10-page 191

Volume of a prism determine if a solid has a uniform

cross-section

Emphasise how area involves multiplying two dimensions or powers of 2 while volume involves three dimensions or powers of 3. Compare the area formula for a circle to that of a square: both involve powers of 2.


Volume of a cylinder Volume of a pyramid or cone (Year 10 Stage 5.3)(X) NCMACEx 5.12-page 197MOL#3286

Volume and capacity connect volume and capacity and their

units of measurement recognise that 1 mL is equivalent to 1

cm3

solve problems involving volume and capacity of right prisms and cylinders


Assessment Practical activity/assignment/test on perimeter/circumference, area and volume Research assignment on the history/progress of π and finding the circumference/area of a circle Open-ended and back-to-front questions: ‘A triangular prism has a volume of

36 cm3. What could its dimensions be?’

Registration

Signature: Date:

Evaluation

Fractions and percentagesUnit OverviewThis topic revises Year 7 concepts in fractions and percentages before introducing operations with percentages and problems involving percentages. Students have been calculating percentages of quantities since primary school but here they will learn the skills necessary for applying percentages to financial situations,

including percentage change, the unitary method, and calculating profit, loss and GST. Although the advancement of computers and the metric system has made decimals more practical than fractions, fraction skills are still applied in areas such as algebraic fractions, solving equations, ratios and similar figures.

Outcomes MA4-5 NA operates with fractions, decimals and percentages MA4-6 NA solves financial problems involving purchasing goods



The word ‘cent’ comes from the Latin centum meaning one hundred, so ‘per cent’ means ‘out of one hundred’. The symbol is a modified

form of ‘ ’. When expressing quantities as percentages, reinforce the importance of identifying what follows ‘of’ in the question, for example, ‘Calculate the discount as a percentage of the marked price’. Students should also be able to differentiate between cost price and selling price. Why does the unitary method have that name?

Content Quality Teaching Ideas Resources Prior knowledge(E) Test/question to ensure students can compare, order and calculate

with fractions, decimals and percentages.Chapter skill check

Fractions compare fractions using equivalence


Adding and subtracting fractions solve problems involving addition and

subtraction of fractions, including those with unrelated denominators

NCMACEx 6.02-page 215MOL#8033#8034#8035#8036#8037#8038

Multiplying and dividing fractions multiply and divide fractions using

efficient written strategies and digital technologies




Percentages, fractions and decimals connect fractions, decimals and

percentages and carry out simple conversions

Fraction and percentage of a quantity find fractions and percentages of

quantities and express one quantity as a fraction or percentage of another, with and without digital technologies

Have students make a collage of newspaper clippings on the applications of percentages. Examine an advertising claim that uses percentages.(significance)

Investigate the percentage forms of ‘fraction families’ such as the eighths and the sixths. What are 16% and 37.5% as fractions?

Applications of percentages: interest rates, cricket statistics (for example, run rate), exam marks, discount, GST, inflation, unemployment, commission, ingredients in food and drink. (significance)

From the NSW syllabus: ‘The GST is levied at a flat rate of 10% on most goods and services, apart from GST-exempt items (usually basic necessities such as milk and bread).’


NCMACEx 6.05-page 2224BLM page 114MOL#8053#8055#8057

#8058#3183

Expressing amounts as fractions and percentages

express one quantity as a fraction of another, with and without digital technologies


Percentage increase and decrease Encourage students to know the percentage equivalents of commonly-used fractions and be able to use their mental computation skills on these. Students should recognise equivalences when calculating, for example, multiplication by 1.05 will increase a number by 5%, multiplication by 0.87 will decrease it by 13%.

Does taking off 10% followed by adding 10% give the original number?

Repeated percentage changes, for example, successive discounts. What percentage change is equivalent to an increase of 10% followed by a decrease of 10%? (X)


Percentages without calculators NCMACEx 6.08-page 235BLM page 124

The unitary method The unitary method is a powerful skill that can be applied to percentages, fractions, decimals, ratios and rates.


Profit, loss and GST NCMACEx 6.10-page 240

BLM page 127MOL#8060

Percentage problems NCMACEx 6.11-page 244BLM page 130

Assessment Collage/poster on the applications of percentages Revision assignment on applications of percentages

Language

Registration

Signature: Date:

Evaluation

Investigating dataUnit OverviewThis topic revises and extends statistical concepts introduced in Year 7, introducing the techniques involved in collecting data. This is a practical topic, and it is expected that some data will be generated from surveys undertaken in class, which can then be used for calculation and analysis. The mass media, including the Internet, is also a rich source of data for statistical investigation.

Outcomes MA4-19 SP collects, represents and interprets single sets of data, using appropriate statistical displays MA4-20 SP analyses single sets of data using measures of location and range



This topic contains much statistical jargon, so a student-created glossary may be useful. Median middle, for example, median strip on a highway, or sounds like ‘medium’, mode (French) fashionable, popular. Population may refer to a collection of items as well as people. Spend considerable time explaining the difference between discrete and continuous data.

Content Quality Teaching Ideas ResourcesOrganising and displaying data

interpret and construct divided bar graphs, sector graphs and line graphs with and without ICT

use a tally to organise data into a frequency distribution table

U Understanding (knowing and relating maths): Knowing the various types of data displays and statistical measures

F Fluency (applying maths): Reading and interpreting graphs, calculating and analysing statistics, comparing data sets

C Communicating (describing and representing maths): Classify and represent data in different forms and make conclusions about data sets after analysing them

Read and comprehend a variety of data displays used in the media and in other school subject areas. Compare the strengths and weaknesses of different forms of data display.’ Each graph should have a title and key or scale.

NCMACEx 7.10 page 257MOL#3326

Types of data recognise data as numerical (either

discrete or continuous) or categorical

NCMACEx 7.02 page 264BLM page 137



The mean and mode Applications of mean: sports averages, rainfall or temperatures, number of matches in a matchbox, market research.

Applications of mode: number of people in Australian family, most popular Australian car, ordering stock for a shop.

NCMACEx 7.03 page 267BLM page 139MOL#8133#3323

The median and range Applications of median: wages, home prices. NCMACEx 7.04 page 271

MOL#8132

Analysing frequency tables NCMACEx 7.05 page 278

Dot plots and stem-and-leaf plots Extension - (Year 10 Stage 5.2) Interquartile range, box-and-whisker plots

NCMACEx 7.06 page 285BLM page 141MOL#4327

Frequency histograms and polygons draw and interpret frequency histograms

and polygons

Extension - Grouped data, class intervals, median class

A histogram is a special type of column graph. Leave a half-column gap at the vertical axis, as the columns are centred on the scores on the horizontal axis.


Sampling NCMACEx 7.08 page 296MOL#4326

Comparing samples and populations Question when it is more appropriate to use the mode or median, rather than the mean, when analysing data. Which is higher, the mean or median price of Australian homes?

Designing survey questions construct appropriate survey questions

and a related recording sheet to collect both numerical and categorical data about an issue of interest

The class may be surveyed on a number of characteristics: height, arm span, shoe size, heartbeat rate, reaction time, number of children in family, number of people living at home, hours slept last night, number of letters in first name, number of cars or mobile phones owned at home, make/colour of car, mode of travel to school, favourite TV/radio station, reaction time, eye/hair colour, birth month or star sign.

NCMACEx 7.09 page 301BLM page 144

Analysing data PS Problem solving (modelling and investigating with maths): Analysing data to solve problems, drawing conclusions

NCMACEx 7.11 page 306

Assessment Include open-ended questions: The range of a set of eight scores is 10 and the mode is 3. What might the scores be? Plan, implement and report on a statistical investigation. Vocabulary test Investigate the use and abuse of statistics and statistical graphs in the media. Research the role of the Australian bureau of Statistics.

Registration

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Evaluation

Congruent FiguresUnit OverviewThis topic introduces the concepts and language associated with congruent figures (especially triangles), building on knowledge learned in past geometry topics. The properties of congruent triangles are to be discovered through construction and measurement, with more formal work such as congruent triangle proofs to be taught in Year 9 as a Stage 5.3 topic. The geometrical constructions are included here because they are based on the properties of special triangles and quadrilaterals, especially the diagonal properties of a rhombus.Outcomes

MA4-18 MG identifies and uses angle relationships, including those related to transversals on sets of parallel lines


Literacy Students explore Word Banks using a variety of teaching strategies. (Close passage, definition matching, memory games and spelling tests. Word bank can be accessed via the Nelson Net Text book (Page….) accessed at www.nelsonnet.com.auOther important literacy notes for this unit: Use matching angles for congruent figures 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.

The mathematical symbol ‘’ means ‘is identical to’ in algebra and ‘is congruent to’ in geometry.

Content Quality Teaching Ideas ResourcesTransformations

describe translations, reflections in an axis, and rotations of multiples of 90° on the Cartesian plane using coordinates

U Understanding (knowing and relating maths): Understanding the concepts of transformation and congruence

NCMACEx 8.01 page 316MOL#8125#8126#8127

Congruent figures name the vertices in matching order when

using the symbol in a congruence statement


Constructing triangles construct triangles using the conditions

for congruence

NCMACEx 8.03 page 328BLM page 154/155MOL#8111

Tests for congruent triangles investigate the minimum conditions

needed, and establish the four tests, for two triangles to be congruent (the SSS, SAS, AAS and RHS rules)

F Fluency (applying maths): Identifying congruent figures and their properties, applying correct transformations, geometrical constructions and congruent triangle tests

C Communicating (describing and representing maths): Using correct notation and terminology for congruent triangles

NCMACEx 8.04 page 334BLM page 153/158/161MOL#4270#4267#4268#4269#3305

Proving properties of triangles and PS Problem solving (modelling and investigating with maths): NCMAC



quadrilaterals use transformations of congruent

triangles to verify some of the properties of special quadrilaterals, including properties of the diagonals

Using geometry to test congruent triangles and prove properties of triangles and quadrilaterals

R Reasoning (generalising and proving with maths): Generalising properties of congruent triangles and using them to prove properties of triangles and quadrilaterals

Students should be encouraged to prove results orally before writing them up. Introduce scaffolds of proofs where students fill in the blanks.

Extension Ideas Similar figures (Year 9)

Formal congruent triangle proofs (Year 9 Stage 5.3)

Ex 8.05 page 339

Constructing parallel and perpendicular lines construct parallel and perpendicular lines

using their properties, a pair of compasses and a ruler, and dynamic geometry software

Extension: Bisecting intervals and angles NCMACEx 8.07 page 344MOL#3312#8106

Assessment Research assignment on congruent and similar figures and their history

Test/assignment on formal setting-out of geometry proof

Vocabulary test

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ProbabilityUnit OverviewThis short topic revises and extends probability concepts learned in Year 7, introducing Venn diagrams and two-way tables as methods of representing sample spaces of more complicated chance situations. There are many opportunities here for class discussion, practical lessons and language activities.

Outcomes MA4-21 SP represents probabilities of simple and compound interest




Literacy Students explore Word Banks using a variety of teaching strategies. (Close passage, definition matching, memory games and spelling tests. Word bank can be accessed via the Nelson Net Text book (Page….) accessed at www.nelsonnet.com.auOther important literacy notes for this unit: How is the word complementary used in this topic similar to its use with ‘complementary angles’ or its everyday English meaning? Carry out language activities on identifying the complement of an event, such as ‘there are fewer than 3 children in a family’. This could be done as a ‘matching pairs’ memory card game. What is the difference between ‘more than 3’ and ‘3 or more’? The NSW syllabus lists the following terms that can be used to describe compound events: at least, at most, not, and, both, not both, or and neither. Also from the NSW syllabus: ‘An event is one or a collection of outcomes. For instance, an event might be that we roll an odd number [on a die], which would include the outcomes 1, 3 and 5. A simple event has outcomes that are equally likely … A compound event is an event which can be expressed as a combination of simple events, for example, drawing a card that is black or a King; throwing at least 5 on a fair six-sided die’.Content Quality Teaching Ideas ResourcesProbability

assign probabilities to the outcomes of events and determine probabilities for events

U Understanding (knowing and relating maths): Knowing the terminology, concepts and notations of probability


Complementary events R Reasoning (generalising and proving with maths): Making generalisations and inferences about probability situations and experiments, including complementary events


Venn diagrams recognise the difference between mutually

exclusive and non-mutually exclusive events

Explore Venn diagrams using attributes of students in the class, for example, brown hair, walks to school. See the NSW syllabus for examples of Venn diagrams and two-way tables.

From the NSW syllabus: ‘Students are expected to be able to interpret Venn diagrams involving three attributes; however students are not expected to construct Venn diagrams involving three attributes’.

NCMACEx 9.03 page 364BLM page 170/172/175MOL#8175#8295#5620



Two-way tables convert representations of the relationship

between two attributes in Venn diagrams to two-way tables


Probability problems solve probability problems involving single-

step experiments such as card, dice and other games

F Fluency (applying maths): Applying probability theory and techniques to solve problems

PS Problem solving (modelling and investigating with maths): Using probability theory to investigate problems, determining sample spaces, analysing the results of a chance experiment


Experimental probability compare observed frequencies across

experiments with expected frequencies

Two-stage or three-stage experiments: making lists, tables, tree diagrams (Year 9)

Investigate common misconceptions about chance, such as if a coin is tossed repeatedly and heads comes up five times in a row then, for the next toss, tails has a better chance than heads.

Investigate the frequency of each letter of the alphabet in print or the Scrabble game.

Investigate games involving dice (Craps, Yahtzee), coins (Two-Up), cards, raffles, spinners, Roulette. Play calculator cricket or noughts-and-crosses on the computer/Internet. Use real or simulated experiments to find probabilities of winning and compare with theoretical probabilities. Investigate the data from past Lotto draws using the NSW Lotteries website (www.nswlotteries.com.au).

. NCMACEx 9.06 page 375BLM page 168/182/185MOL#8135

8 Extension Ideas Counting techniques More complex Venn diagrams, set notation (union vs intersection) Investigate probability expressed as odds (ratio) The addition rule of probability Do not fall into the trap of thinking of (or teaching) probability as being all

about games of chance and gambling. Investigate the applications of probability in insurance, for example, car accidents, home burglaries, life expectancy, quality control or sampling. Use the Internet to find quotes on premiums. What factors affect the chances of a particular car being stolen?

Assessment Vocabulary test or writing activities involving probability

Research/investigation on listing and counting the outcomes of a sample space using Venn diagrams and/or two-way tables

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EquationsUnit OverviewThis short topic revises and builds upon the concept of equations and the algebraic methods for solving them. Students were introduced to equations in the Year 7 topic Algebra and equations, while the algebraic operations of collecting like terms and expanding expressions were learned earlier this year in the Algebra topic. Like many algebra skills, the process of equation-solving is detailed and technical, requiring careful and precise understanding and practice. Aim to teach this topic at a level appropriate to the ability of your class. Solving linear equations graphically will be covered in the topic Graphing linear equations later this year.

Outcomes MA4-10 NA uses algebraic techniques to solve simple linear equations





Algebra comes from the Arabic word ‘al-jabr’, meaning ‘restoration’ or the process of adding the same amount to both sides of an equation. In 825 CE, the Arabic mathematician al Khwarizmi wrote a book called Hisab al-jabr w’al-muqabala (The science of equations).

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.

Content Quality Teaching Ideas ResourcesOne-step equations

solve linear equations using algebraic methods that involve one or two steps in the solution process and which may have non-integer solutions

U Understanding (knowing and relating maths): Learning the techniques for solving equations

R Reasoning (generalising and proving with maths): Using algebraic operations to solve equations

Stress that the goal of solving an equation is to have the variable on its own on the left side of the equation and the value on the right side.

The balancing and backtracking methods of solving equations are quite similar when written algebraically; the difference is in their models (and explanation).

The process of ‘undoing’ (backtracking) or balancing needs to be explained and reinforced early. Use a ‘putting on socks and shoes’ analogy to explain why ‘undoing’ an equation must take place in reverse order. We ‘undo’ the last thing first.

NCMACEx 10.01 page 390BLM page 194/197MOL#8091#8092#8093

Two-step equations Include two-step equations where the variable appears in the second term, for example, 15 2x 7.


Equations with variables on both sides solve linear equations using algebraic

Spreadsheets, graphics calculators and GeoGebra can be used to ‘guess, check and improve’ solutions to equations. CAS calculators can be used to solve equations.




methods that involve at least two steps in the solution process and which may have non-integer solutions

Equations with brackets NCMACEx 10.04 page 399MOL#8095#8096

Equation problems solve real-life problems by using

pronumerals to represent unknowns

PS Problem solving (modelling and investigating with maths): Solving real-life problems by modelling with equations

C Communicating (describing and representing maths): Describing the solution to real-life problems in words after solving an equation

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


Assessment Writing activity comparing and evaluating the different methods of solving an equation

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Ratios, rates and timeUnit OverviewThis topic revises and extends concepts of ratios, rates and time calculations. Ratios compare parts or shares of something, while rates compare quantities expressed in different units, for example, speed compares distance travelled with the time taken. Travel graphs and time calculations are included here because travel graphs also compare distance with time, while many rates include units of time. The new content of this topic is scale maps and plans, dividing a quantity in a given ratio, sketching informal graphs and international time zones. Note that this topic links together concepts in Number, Measurement and Statistics (graphs, timetables).

Outcomes

MA4-7 NA operates with ratios and rates, and explores graphical representation MA4-15 MG performs calculations of time that involve mixed units, and interprets time zones





The symbol for minute is ’. The symbol for second is ”. Their abbreviations are min and s respectively.

The word minute comes from the Latin pars minuta prima, meaning the first (‘prima’) division (‘minuta’) of an hour. In this way, it is related to the alternative meaning and pronunciation of the word minute as ‘tiny’. The word second comes from pars minuta secunda, meaning the second (‘secunda’) division of an hour.

The parts of a ratio are called its terms.

Why does the unitary method have that name?

Content Quality Teaching Ideas ResourcesRatios Ratios can be entered into a calculator using the [a ] fraction key. But,

when simplifying ‘improper ratios,’ it is best to use the [d/c] key to convert the mixed numeral answer to a ‘proper ratio.’ Students should be introduced to the calculator’s degrees-minutes-seconds key for time calculations. Use the Internet to find airline, train and cinema timetables. Put itineraries onto a spreadsheet and calculate different times. Visit Google Maps and analyse its scale.

Extension - Investigate the golden ratio and the golden rectangle: see Just for the Record on page 456 and the NSW syllabus under Proportion

Encourage the class to list instances of ratios, when the parts or shares of a mixture are important: cordial, punch, cake mix, lawn mower fuel, concrete, paste (flour and water), lemonade, milkshake, fertiliser, gear ratios, slopes of hills, probability and betting odds.

Investigate the aspect ratios of TV, computer and cinema screens


Simplifying ratios NCMACEx 11.02 page 416BLM page 216MOL#8072#8073#807



6Ratio problems

recognise and solve problems involving simple ratios


Scale maps and plans For scale drawings, liaise with the TAS and HSIE faculties for plans and maps. Investigate on a map distances between suburbs, towns, world cities.


Dividing a quantity in a given ratio divide a quantity in a given ratio


Rates convert given information rates

For rates, stress that the slash ( / ) indicates the division process and means ‘per’ or ‘out of’.

Encourage students to list examples of rates and the two units being compared: birth rate, population growth, heartbeat, typing speed, fuel consumption, postage rates, metric and currency conversions, download speed, filling a tank, mobile phone costs, classified ads, cost of petrol, meat or fruit, population density, cricket run rate (runs/over), batter’s strike rate (runs/100 balls), bowler’s strike rate (balls/wicket) and other sports statistics.

Investigate population density, population growth, birth rate, death rate, speed, fuel consumption.


Best buys investigate and calculate ‘best buys’

Investigate unit pricing on supermarket shelves, and how sometimes the unit is 100 mL rather than 1 mL (why?). Discuss why the ‘best buy’ is usually the largest item. Since 2009, unit pricing has been compulsory in all Australian supermarkets.


Rate problems Extension Solve harder rate problems, for example, fuel consumption, converting

rates to different units, for example, from km/h to m/s

NCMACEx 11.08 page 436BLM page 224MOL#3197#3198

Speed Extension Investigate speed records and other units of speed such as Mac


Travel graphs use travel graphs to investigate and

compare the distance travelled to and from school

interpret features of travel graphs such as the slopes of lines and the meaning of horizontal lines

R Reasoning (generalising and proving with maths): Making generalisations and inferences about best buys and travel graphs


Sketching informal graphs sketch informal graphs to model familiar

events, for example, noise level during the lesson


Time differences Applications of time calculations: bus/plane trip using timetables, length of movie, payroll (hours worked), sunrise to sunset, length of school or work day.

Extension Research the history of the calendar and/or time measurement: Julian,

Gregorian, Islamic, Chinese, Jewish calendars, daylight saving, International Date Line


International time zones NCMACEx 11.13 page 454MOL#5350

Assessment Design a map or scale drawing.

Poster assignment on applications of ratios or rates

Travel graph ‘tell me a story’ writing activities

Problems involving travel times and time zones

Plan a holiday and create a travel schedule with the times written in 12- or 24-hour time

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Graphing linear equationsUnit OverviewThis algebra topic provides an introduction to coordinate geometry. Students were introduced to the number plane in Years 6–7, but this is the first time they link tables of values and algebraic rules to graphing on a number plane. This topic demonstrates how patterns in number can be represented visually and graphically. More formal coordinate geometry will be examined in Year 9.

Outcomes

MA4-11 NA creates and displays number patterns; graphs and analyses linear relationships; and performs transformations on the Cartesian plane




From the NSW syllabus under Stage 3, Patterns and Algebra 2: ‘The Cartesian plane (commonly referred to as “the number plane”) is named after [René] Descartes who was one of the first to develop analytical [coordinate] geometry on the number plane’.

From the NSW syllabus under Linear Relationships: ‘In Stage 3, students use “position in pattern” and ‘value of term” when describing a pattern from a table of values”, for example, the value of the term is three times the position in the pattern’.

Content Quality Teaching Ideas ResourcesTables of values NCMAC

Ex 12.01 page 464MOL#8081

Finding the rule NCMACEx 12.02 page 466BLM page 238MOL#3214#3215

Finding rules for number patterns use objects to build a geometric pattern, record the results in a table of values, describe the pattern in words and algebraic symbols and represent the relationship on a number grid

NCMACEx 12.03 page 469MOL#3216#3217#3218

The number plane given coordinates, plot points on the

Cartesian plane and find coordinates for a given point

U Understanding (knowing and relating maths): Understanding and relating linear equations, tables of values and the number plane

NCMACEx 12.04 page 473BLM page 241/243/259

Graphing number patterns recognise a given number pattern (including

decreasing patterns), complete a table of values, describe the pattern in words or

C Communicating (describing and representing maths): Representing number patterns algebraically and graphically




algebraic symbols, and represent that relationship on a number grid

Graphing linear equations form a table of values for a linear

relationship by substituting a set of appropriate values for either of the pronumerals and graph the number pairs on the Cartesian plane

extend the line joining a set of points to show that there is an infinite number of ordered pairs that satisfy a linear relationship

Students should be reminded to label the axes and the graphs.

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.

Graphing a linear equation demonstrates how a numerical pattern can be converted to a graphical pattern. Convert the classroom into a coordinate grid system, then ask ‘stand up/hand up all those people whose two coordinates add up to 5’ for a good visual demonstration.

NCMACEx 12.06 page 479BLM page 245MOL#3209#3210#4190

Finding the equation of a line derive a rule for a set of points that has been

graphed on a number plane

F Fluency (applying maths): Using correct strategies to find the equation of a line or number pattern, plotting points on a number plane

R Reasoning (generalising and proving with maths): Finding a general rule for a number pattern or line, solving linear equations graphically

NCMACEx 12.07 page 482BLM page 249/252

Comparing linear graph more than one line on the same set of

axes using ICT and compare the graphs to describe similarities and differences, for example, parallel, pass through the same point

use ICT to graph linear and simple non-linear relationships such as y x3


Solving linear equations graphically Technology - Use a graphics calculator, GeoGebra or spreadsheet software to graph and compare a range of linear equations.


Intersecting lines graph two intersecting lines on the same set

of axes and read off the point of intersection

NCMACEx 12.10 page 495BLM page 255/257

Assessment Report on investigating the graphs of linear equations

Given the line, find the equation

Practical test using a graphics calculator or computer

Graphing test

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