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) Naplan 5. 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
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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
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
NCMACEx 1.06-page 19
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
NCMACEx 1.07-page 21
BLM page 12 MOL#3321
Pythagoras’ theorem problems PS Problem solving (modelling and investigating with maths): Using Pythagoras’ theorem to solve measurement problems
NCMACEx 1.08-page 22
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
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
MA4-1 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 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’.
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,
NCMACEx 2.05-page 47
MOL#8030
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
NCMACEx 2.06-page 50
BLM page 28 MOL#8052
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)
NCMACEx 2.08-page 57
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
NCMACEx 2.09-page 60
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’.
NCMACEx 2.10-page 64
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)
NCMACEx 2.11-page 69
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)
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
Ex 3.09 page 102 BLM page 52
MOL#8089#3230
#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
Ex 3.11 page 107 BLM page 53
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
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:
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
NCMACEx 4.01-page 119
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
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.
NCMACEx 4.04-page 128BLM page 67MOL#3304
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
NCMACEx 4.05-page 132MOL#8109
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)
NCMACEx 4.07-page 139BLM page 70MOL#8107
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)
NCMACEx 4.08-page 144MOL#8110
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
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 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:
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.
Metric units for area F Fluency (applying maths): Selecting correct strategies to
convert between metric units and calculate areas and volumes
NCMACEx 5.02-page 158
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
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)
NCMACEx 5.05-page 167MOL#3275
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.
NCMACEx 5.06-page 170MOL#3276
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)
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.
NCMACEx 5.11-page 193MOL#8122
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
NCMACEx 5.13-page 1199BLM page 106MOL#8124
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
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:
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
NCMACEx 6.01-page 213MOL#8031
Adding and subtracting fractions solve problems involving addition and
subtraction of fractions, including those with unrelated denominators
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).’
express one quantity as a fraction of another, with and without digital technologies
NCMACEx 6.06-page 227MOL#8031
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)
NCMACEx 6.07-page 232MOL#3184#3185
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.
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
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:
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.
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.
NCMACEx 7.07 page 292MOL#3329
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?
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.
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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
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: 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.
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
NCMACEx 8.02 page 322
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
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.
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
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: 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
NCMACEx 9.01 page 355BLM page 176MOL#8134
Complementary events R Reasoning (generalising and proving with maths): Making generalisations and inferences about probability situations and experiments, including complementary events
NCMACEx 9.02 page 361
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’.
Two-way tables convert representations of the relationship
between two attributes in Venn diagrams to two-way tables
NCMACEx 9.04 page 368
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
NCMACEx 9.05 page 373
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).
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
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:
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.
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.
Two-step equations Include two-step equations where the variable appears in the second term, for example, 15 2x 7.
NCMACEx 10.02 page 393BLM page 199/202MOL#8094
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.
NCMACEx 10.05 page 405BLM page 205/208MOL#8102
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
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:
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
recognise and solve problems involving simple ratios
NCMACEx 11.03 page 419BLM page 218MOL#8077
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.
NCMACEx 11.04 page 422BLM page 221MOL#8078
Dividing a quantity in a given ratio divide a quantity in a given ratio
NCMACEx 11.05 page 430MOL#8075
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.
NCMACEx 11.06 page 432MOL#3196
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.
NCMACEx 11.07 page 434MOL#8061
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
NCMACEx 11.09 page 438BLM page 227MOL#4117
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
NCMACEx 11.10 page 442BLM page 228MOL#3337
Sketching informal graphs sketch informal graphs to model familiar
events, for example, noise level during the lesson
NCMACEx 11.11 page 447
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
NCMACEx 11.12 page 451BLM page 231MOL#3263
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
MA4-1 WM communicates and connects mathematical ideas using appropriate terminology, diagrams and symbols
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:
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.
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
NCMACEx 12.08 page 487
Solving linear equations graphically Technology - Use a graphics calculator, GeoGebra or spreadsheet software to graph and compare a range of linear equations.
NCMACEx 12.09 page 491
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