Teaching Program Year 8

New Century Maths 8 teaching program (p. 1)

Teaching program New Century Maths 8 for the Australian Curriculum

Year 8 topics

Week SEMESTER 1 Week SEMESTER 2

Term 1 1

1. Pythagoras theorem (Measurement and Geometry)

Term 3 1

7. Investigating data (Statistics and Probability)

2

2

3

3

4

2. Working with numbers (Number and Algebra)

4

8. Congruent figures (Measurement and Geometry)

5

5

6

6

9. Probability (Statistics and Probability)

7

3. Algebra (Number and Algebra)

7

8

8

9

9

Lost time

10

Lost time

10

Term 2 1

4. Geometry (Measurement and Geometry)

Term 4 1

10. Equations (Number and Algebra)

2

2

3

3

4

5. Area and volume (Measurement and Geometry)

4

11. Ratios, rates and time (Number and Algebra,

5

5

Measurement and Geometry)

6

6

7

6. Fractions and percentages (Number and Algebra)

7

12. Graphing linear equations

8

8

(Number and Algebra)

9

9

10

Lost time

10

Lost time

CURRICULUM STRANDS Number and Algebra Measurement and Geometry Statistics and Probability


Year 7 topics

Week SEMESTER 1 Week SEMESTER 2 Term 1

1 1. Integers

(Number and Algebra) Term 3

1 7. Decimals

(Number and Algebra) 2

2

3

3

4

2. Angles (Measurement and Geometry)

4

8. Area and volume (Measurement and Geometry)

5

5

6

6

7

3. Whole numbers (Number and Algebra)

7

9. The number plane (Number and Algebra,

8

8


9

9

Lost time

10

Lost time 10

Term 2 1

4. Fractions and percentages (Number and Algebra)

Term 4 1

10. Analysing data (Statistics and Probability)

2

2

3

3

4

5. Algebra and equations (Number and Algebra)

4

11. Probability (Statistics and Probability)

5

5

6

6

12. Ratios, rates and time (Number and Algebra,

7

6. Geometry (Measurement and Geometry)

7


8

8

9

9

Lost time

10

Lost time 10

CURRICULUM STRANDS Number and Algebra Measurement and Geometry Statistics and Probability


1. PYTHAGORAS THEOREM Time: 3 weeks (Term 1, Week 1) Text: New Century Maths 8, Chapter 1, p.2 NSW and Australian Curriculum references: Measurement and Geometry Right-angled triangles (Pythagoras) / Real numbers

investigate the concept of irrational numbers, including (8NA186) Right-angled triangles (Pythagoras) / Pythagoras and trigonometry

investigate Pythagoras theorem and its application to solving simple problems involving right-angled triangles (NSW Stage 4 / 9MG222)

NSW Stage 4 outcomes A student:

MA4-1 WM communicates and connects mathematical ideas using appropriate terminology, diagrams and symbols MA4-2 WM applies appropriate mathematical techniques to solve problems MA4-16 MG applies Pythagoras theorem to calculate side lengths in right-angled triangles and solves related problems

INTRODUCTION This 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. CONTENT 1 Square roots and surds 8NA186 U 2 Discovering Pythagoras theorem 9MG222 U F R C

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 3 Finding the hypotenuse 9MG222 U F

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

4 Finding a shorter side 9MG222 U F 5 Mixed problems 9MG222 F 6 Testing for right-angled triangles 9MG222 U F R

use the converse of Pythagoras theorem to establish whether a triangle has a right angle 7 Pythagorean triads 9MG222 U F

identify 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

8 Pythagoras theorem problems 9MG222 F PS 9 Revision and mixed problems RELATED TOPICS Year 7: Whole numbers, Algebra and equations, Geometry Year 8: Working with numbers, Geometry, Area and volume, Congruent figures Year 9: Pythagoras theorem, Trigonometry, Coordinate geometry PROFICIENCY STRANDS / WORKING MATHEMATICALLY U = Understanding (knowing and relating maths): Understanding how the sides of a right-angled triangle are related by

Pythagoras theorem F = Fluency (applying maths): Selecting appropriate techniques involving Pythagoras theorem PS = Problem solving (modelling and investigating with maths): Using Pythagoras theorem to solve measurement

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

sides C = Communicating (describing and representing maths): Describing and explaining Pythagoras theorem in words and


as a formula EXTENSION IDEAS

Perigals dissection and other formal proofs of Pythagoras theorem Pythagoras and the Pythagoreans, history of Pythagoras theorem Harder problems: two-stage or in three-dimensions, for example, longest diagonal in a rectangular prism Word problems History of Pythagorean triads, properties of Pythagorean triads Length of an interval on the number plane Irrational numbers, graphing surds on a number line, simplifying surds The real number system, proof that 2 is irrational

TEACHING NOTES AND IDEAS Pythagoras theorem was actually discovered by others, centuries before Pythagoras was born around 580 BCE. Use knotted rope to show how ancient Egyptians builders made a 3-4-5 triangle to create a right angle. State Pythagoras theorem in words and as a formula. Stress that it works for right-angled triangles only. Emphasise correct

setting-out of solutions. Check answers. Obviously its wrong if the hypotenuse is shorter than one of the other sides.

There are different formulas for creating Pythagorean triads, such as (p2 q2, 2pq, p2 + q2), (n,2

12 n , 2

12 +n ) for odd n, (2n

+ 1, 2n2 + 2n, 2n2 + 2n + 1). Multiplying or dividing a triad by a constant gives another triad: we can use this to create decimal triads such as (2.8, 9.6, 10).

Pythagorean triads (useful for triangle problems): (3, 4, 5) (5, 12, 13) (6, 8, 10) (7, 24, 25) (8, 15, 17) (9, 12, 15) (9, 40, 41) (10, 24, 26) (11, 60, 61) (12, 16, 20) (12, 35, 37) (13, 84, 85) (14, 48, 50) (15, 20, 25) (15, 36, 39) (16, 30, 34) (16, 63, 65) (18, 24, 30) (18, 80, 82) (20, 21, 29) (20, 48, 52) (20, 99, 101) (21, 28, 35) (21, 72, 75) (24, 32, 40) (24, 45, 51) (24, 70, 74) (25, 60, 65) (27, 36, 45) (28, 45, 53) (28, 96, 100) (30, 40, 50) (30, 72, 78) (32, 60, 68) (33, 44, 55) (33, 56, 65) (35, 84, 91) (36, 48, 60) (36, 77, 85) (39, 52, 65) (39, 80, 89) (40, 42, 58) (40, 75, 85) (40, 96, 104) (42, 56, 70) (45, 60, 75) (48, 55, 73) (48, 64, 80) (48, 90, 102) (51, 68, 85) (54, 72, 90) (56, 90, 106) (57, 76, 95) (60, 63, 87) (60, 80, 100) (60, 91, 109) (63, 84, 105) (65, 72, 97) (66, 88, 110) (69, 92, 115) (72, 96, 120) (80, 84, 116).

ASSESSMENT IDEAS Research assignment on Pythagoras and Pythagoras theorem. Matching activities: Pythagoras theorem to diagrams. Writing activity explaining Pythagoras theorem. TECHNOLOGY Spreadsheets can be used to find unknown sides or generate Pythagorean triads. Use the Internet to research the history of Pythagoras and irrational numbers. Use dynamic geometry software to explore and prove Pythagoras theorem. LANGUAGE 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. From the NSW syllabus: 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.


2. WORKING WITH NUMBERS Time: 3 weeks (Term 1, Week 4) Text: New Century Maths 8, Chapter 2, p.36 NSW and Australian Curriculum references: Number and Algebra Indices / Number and place value

Use index notation with numbers to establish the index laws with positive integral indices and the zero index (8NA182) Investigate index notation and represent whole numbers as products of powers of prime numbers (7NA149)

Computation with Integers / Number and place value Carry out the four operations with rational numbers and integers, using efficient mental and written strategies and

appropriate digital technologies (8NA183) Fractions, Decimals and Percentages / Real numbers

Investigate terminating and recurring decimals (8NA184) NSW Stage 4 outcomes A student:

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 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

INTRODUCTION 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 ab and the index laws. This is a short refresher topic that reinforces mental, pen-and-paper and calculator skills so dont 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. CONTENT 1 Mental calculation 7NA151 U F R

apply the associative, commutative and distributive laws to aid mental computation 2 Adding and subtracting integers 8NA183 U F PS R 3 Multiplying integers 8NA183 U F R 4 Dividing integers 8NA183 U F R 5 Order of operations 8NA183 U F

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, 15 915 3+

6 Decimals 8NA183 U F round decimals to a specified number of decimal places

7 Multiplying and dividing decimals 8NA183 U F PS R 8 Terminating and recurring decimals 8NA184 U F R C

use the notation for recurring (repeating) decimals, for example, 0.33333 = 0.3 , 0.345345345 = 0.345 , 0.266666 = 0.26

9 Powers and roots 7NA149 U F R C 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 products: (ab)2 and ab 10 Prime factors 7NA149 U F R

express a number as a product of its prime factors to determine whether its square root or cube root is an integer 11 Index laws for multiplying and dividing 8NA182 U F R

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


12 More index laws 8NA182 U F R 13 Revision and mixed problems RELATED TOPICS Year 7: Integers, Whole numbers, Fractions and percentages, Decimals, Ratios, rates and time Year 8: Algebra, Fractions and percentages, Ratios, rates and time Year 9: Working with numbers, Indices PROFICIENCY STRANDS / WORKING MATHEMATICALLY U = Understanding (knowing and relating maths): Understanding operations with numbers, including powers, roots and

the index laws F = Fluency (applying maths): Using appropriate methods for evaluating numerical expressions PS = Problem solving (modelling and investigating with maths): Using operations with integers and decimals to solve

real-life problems R = Reasoning (generalising and proving with maths): Discovering general properties of numbers and operations with

numbers C = Communicating (describing and representing maths): Using correct notation for recurring decimals, powers and

roots EXTENSION IDEAS Investigate the square root of quotients Investigate the history of calculation methods, for example, Italian multiplication Irrational numbers, surds, graphing surds on a number line, simplifying surds. Investigate the value of 0.9 . Is it really equal to 1? Convert recurring decimals to fractions (Year 9, Stage 5.3). Investigate scientific notation. How did mathematicians find square roots before calculators and computers? Investigate Newtons method. TEACHING NOTES AND IDEAS Revise number and calculator skills through problems, puzzles and games. Encourage students to develop number sense. Analysing properties of numbers leads to the study of pattern and early algebra work. Fractions, percentages, ratios and rates will be covered later this year. The NSW syllabus says that written multiplication and division of decimals may be limited to operators with two digits. When teaching rounding decimals, include more difficult examples, such as rounding 4.8971 to two decimal places. Investigate patterns in the recurring decimals of the fraction families of the sixths, sevenths and ninths. Some decimals are neither terminating nor recurring. Their digits run endlessly, but without repeating, for example, 2 =

1.4142135 and = 3.1415926 Investigate finding higher powers on the calculator. As an alternative to factor trees, prime factors can also be extracted by repeated division. See the Skillsheet Prime factors

by repeated division.

Common mistake: 9 = 3. The square root of a number is a single positive value, so 9 = 3 only. However, 9 = -3 and the equation x2 = 9 has two solutions, x = 3 or -3.

In Year 8, the index laws are applied to numerical expressions only. The index laws in algebraic form will be covered in Year 9 or in the Year 8 topic Algebra as extension work.

ASSESSMENT IDEAS Non-calculator test. Revision assignment. TECHNOLOGY Not all calculators are the same: teachers will need to look for subtle differences in the locations and functions of keys. Use calculators to evaluate mixed expressions, including the use of the parentheses and ANS keys, but beware of cheap calculators that do not follow order of operations rules. Students can use the spreadsheet to round or order decimals, or convert fractions to terminating and recurring decimals.


LANGUAGE -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.


3. ALGEBRA Time: 3 weeks (Term 1, Week 7) Text: New Century Maths 8, Chapter 3, p.88 NSW and Australian Curriculum references: Number and Algebra Algebraic Techniques 1 and 2 / Patterns and algebra

Extend and apply the distributive law to the expansion of algebraic expressions (8NA190) Factorise algebraic expressions by identifying numerical factors (8NA191) Factorise algebraic expressions by identifying algebraic factors (NSW Stage 4) Simplify algebraic expressions involving the four operations (8NA192)


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

INTRODUCTION 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 will the concepts difficult. Students should practise and master each skill before moving onto the next one. CONTENT 1 Variables 7NA175 U F R C

introduce the concept of variables as a way of representing numbers using letters extend and apply the laws and properties of arithmetic to algebraic terms and expressions

2 From words to algebraic expressions 7NA177 U F PS R C move fluently between algebraic and word representations as descriptions of the same situation

3 Substitution 7NA176 U F PS create algebraic expressions and evaluate them by substituting a given value for each variable

4 Collecting variables 8NA192 U F R C 5 Adding and subtracting terms 8NA192 U F R C 6 Multiplying terms 8NA192 U F R C 7 Dividing terms 8NA192 U F R C 8 Extension: The index laws 9NA212 U F R C

extend and apply the index laws to variables, using positive integer indices and the zero index 9 Expanding expressions 8NA190 U F R 10 Factorising algebraic terms 8NA191 U F R

factorise a single algebraic term, for example, 6ab = 3 2 a b 11 Factorising expressions 8NA191 U F R 12 Factorising with negative terms 8NA191 U F R 13 Revision and mixed problems RELATED TOPICS Year 7: Algebra and equations Year 8: Working with numbers, Equations, Graphing linear equations Year 9: Algebra, Indices, Equations PROFICIENCY STRANDS / WORKING MATHEMATICALLY U = Understanding (knowing and relating maths): Learning algebraic concepts and operations F = Fluency (applying maths): Applying general rules effectively to simplify algebraic expressions PS = Problem solving (modelling and investigating with maths): Using expressions and formulas to represent and solve

problems


R = Reasoning (generalising and proving with maths): Using algebra to represent, generalise and simplify pattern in numbers

C = Communicating (describing and representing maths): Describing and representing general properties of numbers algebraically

EXTENSION IDEAS More challenging problems involving substitution and translating worded statements into algebraic expressions Binomial expansions (Year 9/Stage 5.2), for example (x + 3)(x 2), (x + 5)(x 5), (x + 2)2 Factorising by grouping in pairs Negative or fractional indices TEACHING NOTES AND IDEAS Resources: counters, cubes, cups, blocks, envelopes and other concrete materials for modelling variables From the NSW syllabus: To gain an understanding of algebra, students must be introduced to the concepts of pronumerals,

expressions, unknowns, equations, patterns, relationships and graphs in a wide variety of contexts. For each successive context, these ideas need to be redeveloped. Students need gradual exposure to abstract ideas as they begin to relate to algebraic terms to real situations.

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.

Determine and justify whether a simplified or equivalent expression is correct by substituting a number. Common mistakes: 2a a = 2, 3b2 = 3b 3b. Explain that the index 2 belongs to the b only. Application of collecting like terms: the formulas for the perimeter of the square and rectangle. Show that variables provide

a powerful shorthand in this regard. For simplifying algebraic terms, include mixed exercises so that students experience all four operations and identify which

rule to use. Include terms that are constants or which have powers. NSW syllabus: Check expansions and factorisations by performing the reverse process. Include examples involving

negative terms. ASSESSMENT IDEAS 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 TECHNOLOGY Note that spreadsheet formulas are written differently to algebraic formulas. CAS (Computer Algebra Systems) can be used to simplify, expand or evaluate algebraic expressions. LANGUAGE 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.


4. GEOMETRY Time: 3 weeks (Term 2, Week 1) Text: New Century Maths 8, Chapter 4, p.130 NSW and Australian Curriculum references: Measurement and Geometry Properties of Geometrical Figures 1 / Geometric reasoning

Identify corresponding, alternate and co-interior angles when two straight lines are crossed by a transversal, and the relationships between them (7MG164)

Investigate conditions for two lines to be parallel and solve simple numerical problems using reasoning (7MG165) Classify triangles according to their side and angle properties and describe quadrilaterals (7MG165) Demonstrate that the angle sum of a triangle is 180 and use this to find the angle sum of a quadrilateral (7MG166)


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 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

INTRODUCTION 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. CONTENT 1 Angle geometry 6MG141 U F PS R C

investigate angles on a straight line, angles at a point and vertically opposite angles, and use results to find unknown angles

2 Angles on parallel lines 7MG163, 7MG164 U F PS R C 3 Line and rotational symmetry 7MG181 U F C

identify line and rotational symmetries 4 Classifying triangles 7MG165 U F R C

classify triangles according to their side and angle properties 5 Classifying quadrilaterals 7MG165 U F R C

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

6 Properties of quadrilaterals 7MG165 U F R C investigate the properties of special quadrilaterals classify special quadrilaterals on the basis of their properties

7 Angle sums of triangles and quadrilaterals 7MG166 U F PS R 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

8 Extension: Angle sum of a polygon NSW STAGE 5.2 U F PS R 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 9 Revision and mixed problems RELATED TOPICS Year 7: Angles, Geometry Year 8: Pythagoras theorem, Area and volume, Congruent figures Year 9: Geometry, Congruent and similar figures


PROFICIENCY STRANDS / WORKING MATHEMATICALLY U = Understanding (knowing and relating maths): Learning geometrical concepts, definitions, terminology and notation F = Fluency (applying maths): Applying correct procedures, language and notation to solve geometrical problems PS = Problem solving (modelling and investigating with maths): Finding unknown angles in geometrical problems R = Reasoning (generalising and proving with maths): Using logic and reasoning to explore and deduce geometrical

ideas and properties C = Communicating (describing and representing maths): Classifying angles, triangles and quadrilaterals and describing

their properties, including symmetries EXTENSION IDEAS Investigate the history of geometry and Euclid. From NSW syllabus: Students who recognise class inclusivity and minimum requirements for definitions may address this

Stage 4 content concurrently with content in Stage 5 Properties of Geometrical Figures where properties of triangles and quadrilaterals are deduced from formal definitions. For example, is a parallelogram a trapezium?

The formal definitions and tests for special quadrilaterals (Stage 5.3). See the NSW syllabus (Stage 5.3 Properties of Geometrical Figures) on introducing more formal definitions of the special triangles and quadrilaterals.

Find the size of one angle in a regular polygon, or the exterior angle sum of a convex polygon. Formal proofs in deductive geometry.

TEACHING NOTES AND IDEAS Resources: rulers, set squares, protractors, paper and scissors, charts and posters, geometry and drawing software. 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). Give examples and counter-examples of the types of triangles and quadrilaterals and ask students to describe them in their

own words. You may like to give the meaning first, then the name. Properties of triangles and quadrilaterals should be demonstrated informally (by symmetry, paper-folding, protractor and

ruler measurement), rather than by congruent triangle proofs. 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. In how many different ways can you demonstrate the angle sum of a triangle (or quadrilateral)? Proving properties of quadrilaterals by similar triangles will be covered in the topic Congruent figures. ASSESSMENT IDEAS 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 TECHNOLOGY 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 of angles, triangles and quadrilaterals shown in this topic. LANGUAGE 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.


5. AREA AND VOLUME Time: 3 weeks (Term 2, Week 4) Text: New Century Maths 8, Chapter 5, p.170 NSW and Australian Curriculum references: Measurement and Geometry Area / Using units of measurement

Choose appropriate units of measurement for area and volume and convert from one unit to another (8MG195) Find perimeters and areas of parallelograms, trapeziums, rhombuses and kites (8MG196) Investigate the relationship between features of circles such as circumference, area, radius and diameter; use formulas to

solve problems involving circumference and area (8MG197) Volume / Using units of measurement

Develop the formulas for volumes of rectangular and triangular prisms and prisms in general; use formulas to solve problems involving volume (8MG198)


MA4-1 WM communicates and connects mathematical ideas using appropriate terminology, diagrams and symbols MA4-2 WM applies appropriate mathematical techniques to solve problems MA4-12 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

INTRODUCTION 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 and its role in calculating perimeters and areas of circles and circular shapes. CONTENT 1 Perimeter 8MG196 U F PS R 2 Metric units for area 8MG195 U F PS R C 3 Areas of rectangles, triangles and parallelograms 7MG159 U PS R

establish the formulas for areas of rectangles, triangles and parallelograms and use these in problem solving 4 Areas of composite shapes 7MG159 U F PS 5 Area of a trapezium 8MG196 U PS R 6 Areas of kites and rhombuses 8MG196 U PS R 7 Parts of a circle 8MG197 U C

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

8 Circumference of a circle 8NA186, 8MG197 U F PS R investigate the concept of irrational numbers, including find the perimeter of quadrants, semi-circles, sectors and composite figures

9 Area of a circle 8MG197 U F PS R calculate the area of quadrants, semi-circles, sectors and composite figures

10 Metric units for volume 8MG195 U F PS R C 11 Volume of a prism 8MG198 U F PS R C

determine if a solid has a uniform cross-section 12 Volume of a cylinder 8MG198 U F PS R 13 Volume and capacity 6MG138 U F PS

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

14 Revision and mixed problems RELATED TOPICS Year 7: Geometry, Area and volume Year 8: Pythagoras theorem, Geometry Year 9: Geometry, Surface area and volume, Congruent and similar figures


PROFICIENCY STRANDS / WORKING MATHEMATICALLY U = Understanding (knowing and relating maths): Learning measurement concepts, terminology and techniques F = Fluency (applying maths): Selecting correct strategies to convert between metric units and calculate areas and

volumes PS = Problem solving (modelling and investigating with maths): Solving problems involving measurement, perimeter,

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

parts of a circle EXTENSION IDEAS Herons formula for the area of a triangle with sides of length a, b and c. Areas of irregular figures: traverse surveys, Simpsons rule. Surface area of a cube, prism and cylinder. History of , formulas for generating the value of . Area formula involving d rather than r. Area of an ellipse. Calculate the perimeter of a regular hexagon inscribed in a circle with the circles circumference to demonstrate that > 3. Circumference of the Earth, latitude and longitude (small and great circles) on the Earths surface. Volume of a pyramid or cone (Year 10 Stage 5.3). TEACHING NOTES AND IDEAS Resources: 1 cm grid paper, cube blocks, cardboard grid diagrams of plane shapes, nets and models of prisms, discs, coins,

cups, CDs and lids for circumference measurement activities, compasses, string, measuring tape, ruler, trundle wheel, paper and scissors for dissection activities, geometry/drawing software.

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

Estimate areas of windows, noticeboards, blackboards, desktop, postage stamps. Mark a square metre or hectare on school grounds.

Examples of composite shapes: L-shape, T-shape, U-shape, trapezium, semi-circles, annuli and pipes. 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. The area of a trapezium can be cut up and rearranged into two triangles or one rectangle. When proving the formulas for areas of special quadrilaterals, demonstrate the usefulness and power of variables in algebra. 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. 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 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

With composite area problems, encourage students to look for opportunities for combining two semi-circles. ASSESSMENT IDEAS Practical activity/assignment/test on perimeter/circumference, area and volume. Research assignment on the history/progress of 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? TECHNOLOGY Drawing and animation software may be used to demonstrate area and volumes of geometrical figures. Also search for animations and applets from the Internet. LANGUAGE 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, semi-circle and quadrant) are first introduced in Stage 3 Pi () 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.


6. FRACTIONS AND PERCENTAGES Time: 3 weeks (Term 2, Week 7) Text: New Century Maths 8, Chapter 6, p.234 NSW and Australian Curriculum references: Number and Algebra Fractions, Decimals and Percentages / Real numbers

Solve problems involving the use of percentages, including percentage increases and decreases, with and without digital technologies (8NA187)

Financial Mathematics / Money and financial mathematics Solve problems involving profit and loss, with and without digital technologies (8NA189) Investigate and calculate Goods and Services Tax (GST), with and without digital technologies (NSW Stage 4)


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

INTRODUCTION This 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. CONTENT 1 Fractions 7NA152 U F

compare fractions using equivalence 2 Adding and subtracting fractions 7NA153 U F R

solve problems involving addition and subtraction of fractions, including those with unrelated denominators 3 Multiplying and dividing fractions 7NA154 U F PS

multiply and divide fractions using efficient written strategies and digital technologies 4 Percentages, fractions and decimals 7NA157 U F C

connect fractions, decimals and percentages and carry out simple conversions 5 Fraction and percentage of a quantity 7NA158 U F C

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

6 Expressing amounts as fractions and percentages 7NA155, 7NA158 U F C express one quantity as a fraction of another, with and without digital technologies

7 Percentage increase and decrease 8NA187 U F PS C 8 Percentages without calculators 8NA187 U F PS R 9 The unitary method 8NA187 U F PS R C 10 Profit, loss and GST 8NA189 U F PS R C 11 Percentage problems 8NA187 U F PS C 12 Revision and mixed problems RELATED TOPICS Year 7: Fractions and percentages, Decimals, Ratios, rates and time Year 8: Working with numbers, Probability, Equations, Ratios, rates and time PROFICIENCY STRANDS / WORKING MATHEMATICALLY U = Understanding (knowing and relating maths): Learning the concepts, notations and operations of fractions and

percentages F = Fluency (applying maths): Applying appropriate fraction and percentage operations to different situations


PS = Problem solving (modelling and investigating with maths): Solve a variety of real-life problems using fractions and percentages, including financial problems

R = Reasoning (generalising and proving with maths): Finding shortcuts for calculating with fractions and percentages by looking for general patterns

C = Communicating (describing and representing maths): Converting between fractions, decimals and percentages; interpreting and writing worded answers to problems

EXTENSION IDEAS Repeated percentage changes, for example, successive discounts. What percentage change is equivalent to an increase of

10% followed by a decrease of 10%? Investigate interest rates and the method and formula for calculating simple and compound interest. TEACHING NOTES AND IDEAS Resources: newspaper cuttings of applications of percentages, for example, interest rates, GST, statistical graphs, opinion

polls. Have students make a collage of newspaper clippings on the applications of percentages. Examine an advertising claim that

uses percentages. 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%.

Investigate the percentage forms of fraction families such as the eighths and the sixths. What are 32 , 16 % and 37.5% as

fractions? Encourage students to develop a number sense rather than rely upon the calculator too often. Check that answers make

sense. Estimate first. Applications of percentages: interest rates, cricket statistics (for example, run rate), exam marks, discount, GST, inflation,

unemployment, commission, ingredients in food and drink. Does taking off 10% followed by adding 10% give the original number? The unitary method is a powerful skill that can be applied to percentages, fractions, decimals, ratios and rates. 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). ASSESSMENT IDEAS Collage/poster on the applications of percentages. Revision assignment on applications of percentages. TECHNOLOGY Use spreadsheets to convert between fractions, decimals and percentages and to order fractions, decimals and percentages. You could investigate the percentage format on a spreadsheet. Some calculators have a [%] key: 16 [] 25 [%] gives 25% of 16; 5 [] 40 [%] gives 5 out of 40 as a percentage; 150 [] 13 [%] [] decreases 150 by 13%. LANGUAGE 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 100

.

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?


7. INVESTIGATING DATA Time: 3 weeks (Term 3, Week 1) Text: New Century Maths 8, Chapter 7, p.282 NSW and Australian Curriculum references: Statistics and Probability Data Collection and Representation / Data representation and interpretation

Identify and investigate issues involving numerical data collected from primary and secondary sources (7SP169) Construct and compare a range of data displays including stem-and-leaf plots and dot plots (7SP170) Explore the practicalities and implications of obtaining data through sampling using a variety of investigative processes

(8SP206) Investigate techniques for collecting data, including census, sampling and observation (8SP284)

Single Variable Data Analysis / Data representation and interpretation Calculate mean, median, mode and range for sets of data, and interpret these statistics in the context of data (7SP171) Describe and interpret data displays using median, mean and range (7SP172) Investigate the effect of individual data values, including outliers, on the mean and median (8SP207) Explore the variation of means and proportions of random samples drawn from the same population (8SP293)


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 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

INTRODUCTION This 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. CONTENT 1 Organising and displaying data 7SP170 U F PS R C

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

2 Types of data 8SP284 U F R C recognise data as numerical (either discrete or continuous) or categorical

3 The mean and mode 7SP171 U F PS R 4 The median and range 7SP171 U F PS R 5 Analysing frequency tables 7SP170, 7SP172 U F PS R 6 Dot plots and stem-and-leaf plots 7SP170, 7SP172 U F PS R C 7 Frequency histograms and polygons 7SP170, 7SP172 U F PS R C

draw and interpret frequency histograms and polygons 8 Sampling 8SP206, 8SP284, 7SP169 U F C 9 Designing survey questions 8SP206, 7SP169 U F PS C

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

10 Comparing samples and populations 8SP293, 7SP169 U F PS R C 11 Analysing data 7SP172, 8SP207 U F PS R C 12 Revision and mixed problems RELATED TOPICS Year 7: Fractions and percentages, Analysing data Year 8: Fractions and percentages Year 9: Investigating data


PROFICIENCY STRANDS / WORKING MATHEMATICALLY 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 PS = Problem solving (modelling and investigating with maths): Analysing data to solve problems, drawing conclusions R = Reasoning (generalising and proving with maths): Making generalizations and drawing conclusions from statistical

displays and measures C = Communicating (describing and representing maths): Classify and represent data in different forms and make

conclusions about data sets after analysing them EXTENSION IDEAS (Year 10 Stage 5.2) Interquartile range, box-and-whisker plots. Grouped data, class intervals, median class. Replicate or implement a major statistical investigation. TEACHING NOTES AND IDEAS Resources: ruler, compasses, graph paper, graphs and tables from newspapers, statistical yearbooks and census data from

the Australian Bureau of Statistics, spreadsheets, statistical and graphing software, accident statistics. From NSW syllabus: Dot plots and line graphs are first introduced in Stage 3. Students construct, describe and interpret

column graphs in Stages 2 and 3; however, Stage 4 is the first Stage in which histograms, divided bar graphs and sector (pie) graphs are encountered.

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. Applications of median: wages, home prices. 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. 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. Newspapers, magazines and the Internet are useful sources of statistical information. Replicate a newspaper survey. Examples of surveys: TV/radio ratings, opinion polls, phone polls, CD sales, quality control. Survey the number of left-

handed or blue-eyed students in the class or Year group and use this to estimate the number with the same feature in the school or whole of Australia.

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.

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?

Do more surnames begin with AM or NZ? Sometimes, a sample is biased because it is too small or does not represent the population accurately, for example, men

only, adults only. ASSESSMENT IDEAS 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. TECHNOLOGY Explore the statistical and graphing features of a spreadsheet, GeoGebra, Fx-Stat, graphics/CAS calculators or software. Visit the Australian Bureau of Statistics CensusAtSchool website www.abs.gov.au/censusatschool or purchase their CD-ROMs. LANGUAGE 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.


8. CONGRUENT TRIANGLES Time: 2 weeks (Term 3, Week 4) Text: New Century Maths 8, Chapter 8, p.342 NSW and Australian Curriculum references: Measurement and Geometry Properties of Geometrical Figures 2 / Geometric reasoning

Define congruence of plane shapes using transformations (8MG200) Develop the conditions for congruence of triangles (8MG201) Establish properties of quadrilaterals using congruent triangles and angle properties, and solve related numerical

problems using reasoning (8MG202) NSW Stage 4 outcomes A student:

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

INTRODUCTION This 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. CONTENT 1 Transformations 7MG181 U F R C

describe translations, reflections in an axis, and rotations of multiples of 90 on the Cartesian plane using coordinates 2 Congruent figures 8MG200 U F R C

name the vertices in matching order when using the symbol in a congruence statement 3 Constructing triangles 8MG201 U F PS R

construct triangles using the conditions for congruence 4 Tests for congruent triangles 8MG201 U F PS R C

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

5 Proving properties of triangles and quadrilaterals 8MG202 U F PS R C use transformations of congruent triangles to verify some of the properties of special quadrilaterals, including properties

of the diagonals 6 Extension: Bisecting intervals and angles U F R 7 Constructing parallel and perpendicular lines 7MG163 U F

construct parallel and perpendicular lines using their properties, a pair of compasses and a ruler, and dynamic geometry software

8 Revision and mixed problems RELATED TOPICS Year 7: Geometry Year 8: Pythagoras theorem, Geometry Year 9: Geometry, Congruent and similar figures PROFICIENCY STRANDS / WORKING MATHEMATICALLY U = Understanding (knowing and relating maths): Understanding the concepts of transformation and congruence F = Fluency (applying maths): Identifying congruent figures and their properties, applying correct transformations,

geometrical constructions and congruent triangle tests PS = Problem solving (modelling and investigating with maths): 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 C = Communicating (describing and representing maths): Using correct notation and terminology for congruent

triangles EXTENSION IDEAS Similar figures (Year 9) Formal congruent triangle proofs (Year 9 Stage 5.3) TEACHING NOTES AND IDEAS Resources: geometrical instruments, dynamic geometry software, reference and summary charts. Investigate congruence in cultural and religious design patterns. From NSW syllabus: Congruent figures are embedded in a

variety of designs, for example, tapa cloth, Aboriginal designs, Indonesian ikat designs, Islamic designs, designs used in ancient Egypt and Persia, window lattice, woven mats and baskets.

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

ASSESSMENT IDEAS Research assignment on congruent and similar figures and their history Test/assignment on formal setting-out of geometry proof Vocabulary test TECHNOLOGY The Math Open Reference website www.mathopenref.com contains animations demonstrating geometrical constructions and the tests for congruent triangles. From NSW syllabus: Dynamic geometry software or prepared applets are useful tools for investigating properties of congruent figures through transformations. LANGUAGE 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.


9. PROBABILITY Time: 3 weeks (Term 3, Week 6) Text: New Century Maths 8, Chapter 9, p.384 NSW and Australian Curriculum references: Statistics and Probability Probability 1 / Chance

Identify complementary events and use the sum of probabilities to solve problems (8SP204) Probability 2 / Chance

Describe events using language of at least, exclusive or (A or B but not both), inclusive or (A or B or both) and and (8SP205)

Represent events in two-way tables and Venn diagrams and solve related problems (8SP292) NSW Stage 4 outcomes A student:

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 MA4-21 SP represents probabilities of simple and compound interest

INTRODUCTION This 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. CONTENT 1 Probability 7SP168 U F PS C

assign probabilities to the outcomes of events and determine probabilities for events 2 Complementary events 8SP204 U F R C 3 Venn diagrams 8SP205, 8SP292 U F PS R C

recognise the difference between mutually exclusive and non-mutually exclusive events 4 Two-way tables 8SP205, 8SP292 U F PS R C

convert representations of the relationship between two attributes in Venn diagrams to two-way tables 5 Probability problems 8SP205, 8SP292 U F PS R

solve probability problems involving single-step experiments such as card, dice and other games 6 Experimental probability 6SP146 U F PS R

compare observed frequencies across experiments with expected frequencies 7 Revision and mixed problems RELATED TOPICS Year 7: Analysing data, Probability Year 8: Fractions and percentages, Interpreting data Year 9: Probability PROFICIENCY STRANDS / WORKING MATHEMATICALLY U = Understanding (knowing and relating maths): Knowing the terminology, concepts and notations of probability 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 R = Reasoning (generalising and proving with maths): Making generalisations and inferences about probability

situations and experiments, including complementary events C = Communicating (describing and representing maths): Expressing probabilities as fractions, decimals and

percentages, describing complementary events, representing sample spaces on Venn diagrams and two-way tables EXTENSION IDEAS Two-stage or three-stage experiments: making lists, tables, tree diagrams (Year 9) Counting techniques More complex Venn diagrams, set notation (union vs intersection)


Investigate probability expressed as odds (ratio) The addition rule of probability

TEACHING NOTES AND IDEAS Resources: Dice, coins, counters, spinners, playing cards, probability simulation software. 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. Reinforce the ideas of randomness and equally likely outcomes. Discuss the claim: Since traffic lights can show red, amber

or green, the probability that a light shows red is 1/3. 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. 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. Collect newspaper or Internet articles involving chance, or test a chance game to see if it is fair. 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).

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 IDEAS 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. TECHNOLOGY Random numbers can be generated on a calculator, graphics or CAS calculator, or spreadsheet. The Internet, spreadsheets and other software may be used ti simulate a chance situation such as a lotto draw, coin tosses or dice throws. LANGUAGE 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.


10. EQUATIONS Time: 3 weeks (Term 4, Week 1) Text: New Century Maths 8, Chapter 10, p.418 NSW and Australian Curriculum references: Number and Algebra Equations / Linear and non-linear relationships

Solve simple linear equations (7NA179) Solve linear equations using algebraic and graphical techniques, and verify solutions by substitution (8NA194) Solve simple quadratic equations (NSW Stage 4)


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

INTRODUCTION This 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. CONTENT 1 One-step equations 7NA179 U F R

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

2 Two-step equations 7NA179 U F R 3 Equations with variables on both sides 8NA194 U F R

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

4 Equations with brackets 8NA194 U F R 5 Simple quadratic equations x2 = c NSW U F R C 6 Equation problems 8NA194 U F PS R C

solve real-life problems by using pronumerals to represent unknowns 7 Revision and mixed problems RELATED TOPICS Year 7: Algebra and equations Year 8: Algebra Year 9: Equations PROFICIENCY STRANDS / WORKING MATHEMATICALLY U = Understanding (knowing and relating maths): Learning the techniques for solving equations F = Fluency (applying maths): Selecting correct techniques for solving equations PS = Problem solving (modelling and investigating with maths): Solving real-life problems by modeling with equations R = Reasoning (generalising and proving with maths): Using algebraic operations to solve equations C = Communicating (describing and representing maths): Describing the solution to real-life problems in words after

solving an equation EXTENSION IDEAS

Equations involving x2 or 1x

Harder formulas and word problems, constructing formulas


Equations with the unknown in the denominator Non-linear equations, for example, squares and square roots Simultaneous equations TEACHING NOTES AND IDEAS Resources: counters, cups, cubes, blocks, envelopes, two-pan balance scales and other concrete materials for modelling

variables in equations. From the NSW syllabus: Distinguish between algebraic expressions where pronumerals are used as variables, and

equations where pronumerals are used as unknowns Include two-step equations where the variable appears in the second term, for example, 15 2x = 7. 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. When solving a word problem, identify the unknown quantity and call it x, say. After solving, check that its solution sounds

reasonable. ASSESSMENT IDEAS Writing activity comparing and evaluating the different methods of solving an equation. TECHNOLOGY Spreadsheets, graphics calculators and GeoGebra can be used to guess, check and improve solutions to equations. CAS calculators can be used to solve equations. LANGUAGE 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 wal-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.


11. RATIOS, RATES AND TIME Time: 3 weeks (Term 4, Week 4) Text: New Century Maths 8, Chapter 11, p.444 NSW and Australian Curriculum references: Number and Algebra, Measurement and Geometry Proportion / Real numbers

Solve a range of problems involving rates and ratios, with and without digital technologies (8NA188) Proportion / Linear and non-linear relationships

Investigate, interpret and analyse graphs from authentic data (7NA180) Time / Using units of measurement

Solve problems involving duration, including using 12- and 24-hour time within a single time zone (8MG199) Solve problems involving international time zones (NSW Stage 4)


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 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

INTRODUCTION This topic revised and extends concepts in 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 are 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). CONTENT 1 Ratios 8NA188 U C 2 Simplifying ratios 8NA188 U F 3 Ratio problems 8NA188 U F PS C

recognise and solve problems involving simple ratios 4 Scale maps and plans 8NA188 U F PS C 5 Dividing a quantity in a given ratio 8NA188 U F

divide a quantity in a given ratio 6 Rates 8NA188 U C

convert given information rates 7 Best buys 7NA174 U F PS R C

investigate and calculate best buys 8 Rate problems 8NA188 U F PS C 9 Speed 8NA188 U F PS C 10 Travel graphs 7NA180 U F PS R C

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

11 Sketching informal graphs 7NA180 U F PS R C sketch informal graphs to model familiar events, for example, noise level during the lesson

12 Time differences 8MG199 U F PS C 13 International time zones NSW U F PS C 14 Revision and mixed problems RELATED TOPICS Year 7: Ratios, rates and time Year 8: Fractions and percentages, Graphing linear equations Year 9: Working with numbers


PROFICIENCY STRANDS / WORKING MATHEMATICALLY U = Understanding (knowing and relating maths): Learning the concepts and operations involving ratios, rates and time F = Fluency (applying maths): Applying appropriate concepts and skills to situations PS = Problem solving (modelling and investigating with maths): Solving real-life problems using ratios, rates, travel

graphs and time calculations R = Reasoning (generalising and proving with maths): Making generalizations and inferences about best buys and travel

graphs C = Communicating (describing and representing maths): Describing and interpret relationships using ratios, scale

diagrams, rates and travel graphs, and represent time and time differences in various ways EXTENSION IDEAS Investigate the golden ratio and the golden rectangle: see Just for the Record on page 456 and the NSW syllabus under

Proportion Solve harder rate problems, for example, fuel consumption, converting rates to different units, for example, from km/h to

m/s Investigate speed records and other units of speed such as Mach Research the history of the calendar and/or time measurement: Julian, Gregorian, Islamic, Chinese, Jewish calendars,

daylight saving, International Date Line TEACHING NOTES AND IDEAS Resources: supermarket catalogues for best buys, tables of data showing rates such as fuel consumption or birth rates,

stopwatch, 24-hour clock, calendars, timetables, map with world time zones 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. For scale drawings, liaise with the TAS and HSIE faculties for plans and maps. Investigate on a map distances between

suburbs, towns, world cities. 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), batters strike rate (runs/100 balls), bowlers strike rate (balls/wicket) and other sports statistics.

Investigate population density, population growth, birth rate, death rate, speed, fuel consumption. 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. Applications of time calculations: bus/plane trip using timetables, length of movie, payroll (hours worked), sunrise to

sunset, length of school or work day ASSESSMENT IDEAS 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

TECHNOLOGY Ratios can be entered into a calculator using the [ab/c] fraction key. However, when simplifying improper ratios, use the [d/c] key to convert the mixed numeral answer to a proper ratio. Students should be introduced to the calculators 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. LANGUAGE 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?


12. GRAPHING LINEAR EQUATIONS Time: 3 weeks (Term 4, Week 7) Text: New Century Maths 8, Chapter 12, p.500 NSW and Australian Curriculum references: Number and Algebra Linear Relationships / Linear and non-linear relationships

Plot linear relationships on the Cartesian plane with and without the use of digital technologies (8NA193) Solve linear equations using algebraic and graphical techniques, and verify solutions by substitution (8NA194)


MA4-1 WM communicates and connects mathematical ideas using appropriate terminology, diagrams and symbols MA4-3 WM recognises and explains mathematical relationships using reasoning MA4-11 NA creates and displays number patterns; graphs and analyses linear relationships; and performs

transformations on the Cartesian plane

INTRODUCTION This 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. CONTENT 1 Tables of values 8NA193 U F 2 Finding the rule 8NA193 U F R C 3 Finding rules for number patterns 8NA193 F PS R C

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

4 The number plane 7NA178 U F C given coordinates, plot points on the Cartesian plane and find coordinates for a given point

5 Graphing number patterns 8NA194 U F R C recognise a given number pattern (including decreasing patterns), complete a table of values, describe the pattern in words

or algebraic symbols, and represent that relationship on a number grid 6 Graphing linear equations 8NA193 U F R C

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

7 Finding the equation of a line 8NA194 U F R C derive a rule for a set of points that has been graphed on a number plane

8 Comparing linear equations NSW U F PS R C 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

9 Solving linear equations graphically 8NA194 U F R C 10 Intersecting lines 8NA194 U F R C

graph two intersecting lines on the same set of axes and read off the point of intersection 11 Revision and mixed problems RELATED TOPICS Year 7: Algebra and equations, The number plane Year 8: Algebra, Equations Year 9: Equations, Coordinate geometry PROFICIENCY STRANDS / WORKING MATHEMATICALLY U = Understanding (knowing and relating maths): Understanding and relating linear equations, tables of values and the

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


PS = Problem solving (modelling and investigating with maths): Identifying similarities and differences between two or more lines

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

C = Communicating (describing and representing maths): Representing number patterns algebraically and graphically EXTENSION IDEAS The gradient of a line Graphing of curves: parabola, cubic, exponential, hyperbola. Use of a graphics calculator or GeoGebra. Simultaneous equations Elementary coordinate geometry: distance and midpoint Applications of linear functions, for example, profit function TEACHING NOTES AND IDEAS Resources: number plane grid paper and activities/puzzles, graphics calculators or software, spreadsheet software. 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 dont 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.

ASSESSMENT IDEAS Report on investigating the graphs of linear equations Given the line, find the equation Practical test using a graphics calculator or computer Graphing test

TECHNOLOGY Use a graphics calculator, GeoGebra or spreadsheet software to graph and compare a range of linear equations. LANGUAGE From the NSW syllabus under Stage 3, Patterns and Algebra 2: The Cartesian plane (commonly referred to as the number

plane) is named after [Ren] 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.

Time: 3 weeks (Term 1, Week 1) Text: New Century Maths 8, Chapter 1, p.2INTRODUCTIONCONTENTRELATED TOPICSPROFICIENCY STRANDS / WORKING MATHEMATICALLYEXTENSION IDEASTEACHING NOTES AND IDEASASSESSMENT IDEASTECHNOLOGYLANGUAGETime: 3 weeks (Term 1, Week 4) Text: New Century Maths 8, Chapter 2, p.36INTRODUCTIONCONTENTRELATED TOPICSPROFICIENCY STRANDS / WORKING MATHEMATICALLYEXTENSION IDEASTEACHING NOTES AND IDEASASSESSMENT IDEASTECHNOLOGYLANGUAGETime: 3 weeks (Term 1, Week 7) Text: New Century Maths 8, Chapter 3, p.88INTRODUCTIONCONTENTRELATED TOPICSPROFICIENCY STRANDS / WORKING MATHEMATICALLYEXTENSION IDEASTEACHING NOTES AND IDEASASSESSMENT IDEASTECHNOLOGYLANGUAGETime: 3 weeks (Term 2, Week 1) Text: New Century Maths 8, Chapter 4, p.130INTRODUCTIONCONTENTRELATED TOPICSPROFICIENCY STRANDS / WORKING MATHEMATICALLYEXTENSION IDEASTEACHING NOTES AND IDEASASSESSMENT IDEASTECHNOLOGYLANGUAGETime: 3 weeks (Term 2, Week 4) Text: New Century Maths 8, Chapter 5, p.170INTRODUCTIONCONTENTRELATED TOPICSPROFICIENCY STRANDS / WORKING MATHEMATICALLYEXTENSION IDEASTEACHING NOTES AND IDEASASSESSMENT IDEASTECHNOLOGYLANGUAGETime: 3 weeks (Term 2, Week 7) Text: New Century Maths 8, Chapter 6, p.234INTRODUCTIONCONTENTRELATED TOPICSPROFICIENCY STRANDS / WORKING MATHEMATICALLYEXTENSION IDEASTEACHING NOTES AND IDEASASSESSMENT IDEASTECHNOLOGYLANGUAGETime: 3 weeks (Term 3, Week 1) Text: New Century Maths 8, Chapter 7, p.282INTRODUCTIONCONTENTRELATED TOPICSPROFICIENCY STRANDS / WORKING MATHEMATICALLYEXTENSION IDEASTEACHING NOTES AND IDEASASSESSMENT IDEASTECHNOLOGYLANGUAGETime: 2 weeks (Term 3, Week 4) Text: New Century Maths 8, Chapter 8, p.342INTRODUCTIONCONTENTRELATED TOPICSPROFICIENCY STRANDS / WORKING MATHEMATICALLYEXTENSION IDEASTEACHING NOTES AND IDEASASSESSMENT IDEASTECHNOLOGYLANGUAGETime: 3 weeks (Term 3, Week 6) Text: New Century Maths 8, Chapter 9, p.384INTRODUCTIONCONTENTRELATED TOPICSPROFICIENCY STRANDS / WORKING MATHEMATICALLYEXTENSION IDEASTEACHING NOTES AND IDEASASSESSMENT IDEASTECHNOLOGYLANGUAGETime: 3 weeks (Term 4, Week 1) Text: New Century Maths 8, Chapter 10, p.418INTRODUCTIONCONTENTRELATED TOPICSPROFICIENCY STRANDS / WORKING MATHEMATICALLYEXTENSION IDEASTEACHING NOTES AND IDEASASSESSMENT IDEASTECHNOLOGYLANGUAGETime: 3 weeks (Term 4, Week 4) Text: New Century Maths 8, Chapter 11, p.444INTRODUCTIONCONTENTRELATED TOPICSPROFICIENCY STRANDS / WORKING MATHEMATICALLYEXTENSION IDEASTEACHING NOTES AND IDEASASSESSMENT IDEASTECHNOLOGYLANGUAGETime: 3 weeks (Term 4, Week 7) Text: New Century Maths 8, Chapter 12, p.500INTRODUCTIONCONTENTRELATED TOPICSPROFICIENCY STRANDS / WORKING MATHEMATICALLYEXTENSION IDEASTEACHING NOTES AND IDEASASSESSMENT IDEASTECHNOLOGYLANGUAGE

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