The history of numbers

New Century Maths 9, Stages 5.2/5.3 2nd edition Exploring numbers Chapter 1, p. 2 Related chapters Year 8: 1 Working with numbers, 4 Percentages, Year 9: 5 Consumer arithmeticearning money, Year 10: 1 Working with number, 3 Surds, 10 Probability. Syllabus reference Number NS5.2.1 Rational numbers: rounds decimals to a specified number of significant figures, expresses recurring decimals in fraction form (p. 67); NS4.3 Fractions, Decimals and Percentages (p. 63). Recommended time: 3 weeks e increase or decrease a quantity by a given percentage 5 identify significant figures, and round numbers to a specified number of decimal places and significant figures 6 estimate answers and use the language of estimation 8 write recurring decimals in fraction form 10 apply the unitary method to fraction and percentage problems Teaching notes and ideas 1 Encourage students to develop a number sense rather than rely upon the calculator too often. Check that answers make sense. Estimate first. 2 Students could investigate irrational numbers. 3 Investigate the decimal and percentage forms of „fraction families‟ such as the eighths and the sixths. What are 16 Introduction This short refresher topic reinforces mostly Stage 4 Number skills. Keep it simple and make the revision suitable to the ability of your class. Throughout this course, Stage 5 students will study similar topic areas but at different depths and emphases. Almost all of the content here has been met before, so teachers can be selective in the skills they will practise with their classes. Reinforce the use of mental computation skills, such as in increasing $140 by 20%. 2 % and 37.5% as fractions? 3 Content The magic of numbers Rational numbers (Stage 5.3) Definition of rational number 3 Fractions 4 Decimals 5 Approximation Rounding using decimal places (Stage 5.2) Rounding using significant figures 6 (Stage 5.2) Estimation 7 Recurring decimals 8 (Stage 5.2) Converting recurring decimals to fractions 9 Percentages 10 The unitary method 11 Revision and mixed problems 1 2 4 Encourage students to learn the fraction equivalents of commonly used percentages. See Skillbank 3B in Chapter 3. 5 Don‟t round if an exact answer (e.g. fraction) is required. 6 For what type of problems is it appropriate to round? What is the difference between 7 cm and 7.0 cm? 7 Syllabus, p.67: „Assess the effect of truncating or rounding during calculations on the accuracy of the results. Appreciate the importance of the number of significant figures in a given measurement.‟ 8 The unitary method is not mentioned explicitly in the syllabus but it is a very important skill. It is quite a powerful method that can be applied to percentages, fractions, decimals, ratios and rates. Technology Use spreadsheets to investigate the decimal form of fractions. Investigate unfamiliar calculator functions: [d/c], [ 1 ], [xy], [ 3 x ], [x!], FIX mode. Skills Students will be able to do the following: 1 identify special groups of numbers 2 define a rational number 3, 4, 7, 9 a perform calculations with fractions, decimals and percentages, using mental and estimation skills, pen-and-paper methods and the calculator b convert between fractions, decimals and percentages and order them c find a fraction or percentage of a quantity d express a quantity as a fraction or percentage of another Students could use spreadsheets to order fractions, decimals and percentages. They should be exposed to a variety of calculator shortcuts to perform percentage operations, such as multiplying by 1.15 to increase a number by 15%. Investigate the [%] key if appropriate. Language (p. 37 of text)  Reinforce the language of approximation: approximate, write correct to, round to n decimal places, nearest tenth.  Note that the syllabus now uses the term rounding rather than rounding off.  When expressing quantities as percentages (or fractions), reinforce to students the importance of the word that follows of in the question.  Why does the unitary method have that name? Teaching notes and ideas Algebra Chapter 2, p. 40 Related chapters Year 8: 2 Algebra, Year 9: 3 Products and factors, 7 Indices, 9 Coordinate geometry, 11 Equations and inequalities. Syllabus reference Patterns and Algebra PAS4.3 Algebraic techniques: uses the algebraic symbol system to simplify, expand and factorise simple algebraic expressions (p. 85); PAS5.2.5 Algebraic techniques (p. 88); PAS5.3.1 Algebraic techniques (p. 92). Recommended time: 2 weeks 1 2 3 4 Introduction This is a Stage 4 Algebra topic in which basic skills such as the arithmetic of algebraic terms, expanding and factorising are reviewed. The content that is new here are the operations with algebraic fractions (Stages 5.2 and 5.3). Algebra continues to be developed through the use of pattern, where students describe and extend a numerical or geometric pattern. The depth of coverage of this revision topic will depend upon the ability level of the class. 5 6 7 Content 1 2 3 4 5 6 7 8 Adding and subtracting algebraic expressions Multiplying and dividing algebraic expressions Substitution in algebraic expressions Expanding algebraic expressions Factorising algebraic expressions (Stage 5.2) Adding and subtracting algebraic fractions (Stage 5.3) Harder problems (Stage 5.2) Multiplying and dividing algebraic fractions Revision and mixed problems 8 9 You could begin this topic by revising the „patterns and algebra‟ approach from Stage 4: matchstick patterns, number sequences, tables of values, translating the pattern rule into algebraic symbols, the use of variables, generalised arithmetic. See Chapter 2 of the Stages 5.1/5.2 text. Investigate pattern in problems such as handshakes, page numbering, angle sum of a polygon, checkerboard squares, creases in paper-folding, Pascal‟s triangle. Demonstrate adding algebraic terms in constructing perimeter formulas and multiplying terms in constructing area formulas. 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. Syllabus, p. 85: „Check expansions and factorisations by performing the reverse process.‟ Harder expansions and factorisations such as binomial products will be covered in the Stage 5.3 Chapter 3 Products and factors. Demonstrate operations with numerical fractions before moving onto algebraic fractions. Stage 5.3 students also add and subtract fractions with binomial (two-term) numerators. Students have already learned to add and subtract simple algebraic expressions in Stage 4. Terms with indices will be met in Chapter 7 Indices. Technology Investigate substitution using spreadsheets and CAS software. Language (p. 59 of text)  Reinforce the meanings of variable, expression, simplify, evaluate and substitute.  The word expand comes from writing out an expression „the long way‟ without grouping symbols. Draw a diagram using rectangles and an array of dots to show equivalences such as 3(n + 2) = 3n + 6.  Expanding an algebraic expression is an application of the distributive property.  Emphasise the difference between expand and factorise, as students will often do the opposite of what is requested.  Syllabus, p.85: „Distinguish between algebraic expressions where letters are used as variables, and equations, where letters are used as unknowns.‟ Skills Students will be able to do the following: 1 a understand and use variables, algebraic terms and expressions b add and subtract like terms to simplify algebraic expressions 2 simplify algebraic expressions that involve multiplication and division 3 substitute into algebraic expressions 4 expand algebraic expressions by removing grouping symbols 5 a factorise a single term, e.g. 6ab  3  2  a  b b factorise algebraic expressions by finding a common factor 6–7 simplify algebraic expressions involving fractions Products and factors Chapter 3, p. 62 Related chapters Year 9: 2 Algebra, Year 10: 4 Equations and inequalities. Syllabus reference Patterns and Algebra PAS5.3.1 Algebraic Techniques: uses algebraic techniques to simplify expressions, expand binomial products and factorise quadratic expressions (p. 92). Recommended time: 4 weeks Introduction This is the first exclusively Stage 5.3 topic, introducing students to more complex algebraic operations involving binomial products. The content is completely new to Year 9 students, so spend considerable time in class examining the patterns found in expansions and practising the abstract algebraic manipulations. The aim is to develop a systematic approach to expansion and factorisation. This is a fairly long topic, so it may be more practical to divide it into separate „expanding‟ and „factorising‟ topics. Content 1 2 3 4 5 6 7 8 9 10 11 12 Expanding binomial products Perfect squares Difference of two squares Simplifying expressions involving binomial products Factorising algebraic expressions Factorising by grouping in pairs Factorising the difference of two squares Factorising a quadratic trinomial Factorising trinomials of the form ax2 + bx + c Factorisation strategies Simplifying algebraic fractions Revision and mixed problems factorising perfect squares and differences of two squares. 3 Demonstrate the equivalence of expansions and factorisations, e.g. (x + 2)(x – 2) = x2 – 4 by substituting a value for x in both sides of the identity. Use a spreadsheet or graphics calculator. 4 Syllabus, p.92: „Check expansions and factorisations by performing the reverse process.‟ 5 Investigate the number of diagonals in a polygon with n sides (handshake problem). 6 Include open-ended questions such as (x …)(x …) = x2 … 5x … or what two terms could be multiplied to give 4a2 + 8a? 7 Evaluate 982 by expanding (100 – 92)2. Evaluate 19 × 21 by expanding (20 – 1)(20 + 1). Investigate the trick for squaring a 2-digit number ending in 5 shown at „Think!‟ at the start of the chapter. 8 Students will use factorising strategies again in the Year 10 topic 4 Equations and inequalities when they solve quadratic equations. The quadratic formula will also be introduced then. 9 Factorising ax2 + bx + c by grouping in pairs is a powerful method because it can be applied to problems where a is negative. 10 With the many types of factorisation, students need to use a systematic approach to decide which method to use. Have them design a poster on this. 11 Encourage students to check that an expression is fully factorised. Include quadratic trinomials where a simple numerical factor can be taken out first, e.g. 2x2 – 10x + 12 = 2(x2 – 5x + 6). Technology Investigate expansions and factorisations using CAS software. Syllabus, p.92: „Interpret statements involving algebraic symbols in other contexts, e.g. spreadsheets.‟ Skills Students will be able to do the following: 1 expand a variety of binomial products (a + b)(c + d) 2–3 recognise and apply special products (a + b)2, (a + b)(a – b) 4 simplify expressions involving binomial products 5–9 factorise expressions using common factors, grouping in pairs, the difference of two squares, perfect squares and quadratic trinomials 10 use a variety of methods to factorise expressions 11 factorise and simplify a variety of more complex algebraic expressions such as fractions Language (p. 95 of text)  Make sure students understand the difference between expand and factorise because sometimes they will do the opposite of what is asked.  binomial = algebraic expression with two terms, e.g. 2ab – b2 or x + 5, from the Latin bi nomen, „two names‟.  trinomial = algebraic expression with three terms, e.g. x2 – x + 4.  monomial = algebraic expression with one term, e.g. 5b3.  quadratic = algebraic expression in which the highest power of x is 2, e.g. 5x2 – 3x + 4. Teaching notes and ideas 1 Describe the process involved when expanding a binomial product. There are many approaches: distributive law, long multiplication, areas of rectangles. 2 Encourage students to look for patterns in their expanded results. This will prove handy when  Investigating geometry Chapter 4, p. 102 Related chapters Year 8: 3 Geometrical figures, Year 9: 12 Congruence and similarity, Year 10: 5 Deductive geometry. Syllabus reference Space and Geometry SGS5.2.1 Properties of geometrical figures: develops and applies results related to the angle sum of interior and exterior angles for any convex polygon (p. 157); SGS4.3 Properties of geometrical figures (p. 154). Recommended time: 3 weeks 3 4 5 6 Introduction This topic revises Stage 4 Space and Geometry, covering properties of triangles and quadrilaterals, including their classification, properties and angle sums. Promote the language of geometry when teaching this topic. The Stage 5.2 work extends students‟ knowledge to polygons and their interior and exterior angle sums (there is no Space and Geometry at Stage 5.1). Although this topic marks the start of more formal geometry, the emphasis is still on discovering properties informally through construction and measurement rather than by deductive proofs using congruent triangles. Nonetheless, reasoning skills should be promoted throughout, with attention given to drawing clear diagrams and setting out solutions logically. 7 8 9 folding, protractor and ruler measurement) rather than by congruent triangle proofs. In how many different ways can you demonstrate the angle sum of a triangle (or quadrilateral)? Include open-ended questions, e.g. an isosceles triangle has one side 10 cm and one angle 25. What might the triangle look like? Examine problems where not enough information is given or too much information is given. Syllabus, p.155: „The properties of special quadrilaterals are important in Measurement. For example, the perpendicularity of the diagonals of a rhombus and a kite allow a rectangle of twice the size to be constructed around them, leading to formulae for finding area.‟ Students should have experience in classifying triangles and quadrilaterals using their properties and minimal conditions, e.g. which quadrilateral‟s diagonals bisect each other? The exterior angle sum of a convex polygon is 360: if you walk around the perimeter of a closed figure, the total of your turns is a revolution. Syllabus, p.155: „This work may be extended to … circles and any closed curve.‟ See Background information, p.157 of syllabus, on how Archimedes used inscribed and circumscribed polygons to calculate the value of . Technology Use dynamic geometry software to investigate the properties of triangles and quadrilaterals. Draw and paint packages are also useful for drawing and manipulating geometrical figures. Content 1 Properties of triangles 2 3 4 5 6 Properties of quadrilaterals (Stage 5.2) Angle sum of a polygon (Stage 5.2) Regular polygons (Stage 5.2) Exterior angle sum of a convex polygon Revision and mixed problems Language (p. 128 of text)  Equilateral comes from the Latin aequus latus, meaning „equal side‟, isosceles from the Greek isos skelos, meaning „equal legs‟, and scalene from the Greek skalenos skelos, meaning „uneven leg‟. Avoid the term base angles in an isosceles triangle as it may be misleading. Instead, use „the angles opposite the equal sides or the two angles next to the uneven side‟. See syllabus, p.160, for the formal definitions of the special triangles and quadrilaterals. Skills Students will be able to do the following: 1 a recognise and classify triangles on the basis of their properties (sides, angles, diagonals, symmetry) b apply the angle sum and exterior angle property of a triangle 2 a recognise and classify the special quadrilaterals (trapeziums, kites, parallelograms, rectangles, squares and rhombuses) on the basis of their properties (sides, angles, diagonals, symmetry) b apply the angle sum of a convex quadrilateral 3–5 a apply the interior and exterior angle sum of a convex polygon b apply angle sum results to find unknown angles   Teaching notes and ideas 1 2 Resources: geometrical instruments, paper, scissors, charts and posters. Properties of triangles and quadrilaterals may be demonstrated informally (by symmetry, paper- Consumer arithmetic – earning money Chapter 5, p. 130 Related chapters Year 8: 4 Percentages, 10 Ratios and rates, Year 9: 1 Exploring number, Year 10: 6 Saving and borrowing. Syllabus reference Number NS5.1.2 Consumer arithmetic: solves consumer arithmetic problems involving earning and spending money (p. 70). Recommended time: 3 weeks 3 Introduction This is a practical topic in which students apply their Number skills to situations involving earning and spending money. This is also the first topic where the content is entirely new and at Stage 5 level. Attention should be given towards making the examples as realistic as possible. Use newspapers to find current prices, wage rates, tax rates, allowances and discounts. The mathematics of saving and borrowing money will be covered in New Century Maths 10 Chapter 6 Saving and borrowing. Discuss types of jobs where overtime, commission and piecework occur. Investigate the advantages and disadvantages of each type of income. 4 Use employment sections of newspapers to compare current wages and salaries of occupations. Use newspaper ads and catalogues to find current prices, discounts and special offers. 5 Explore GST and income tax. 6 Budget the costs of running a school disco, formal, party, excursion or canteen. Investigate the costs involved in owning a mobile phone, running a car, organising a holiday, feeding a family. 7 Why is it cheaper to purchase things in bulk? 8 Investigate vague and misleading forms of advertising, e.g. „from as low as $2.50 per day‟, especially in mobile phone, used car and telemarketing advertisements. 9 Compare special offers from fast food outlets, e.g. pizza, chicken. 10 Use mental arithmetic with percentages to calculate pays and discounts. 11 Does taking off 10% followed by adding 10% give the original price? Content 1 2 3 4 5 6 7 8 9 Wages and salaries Overtime pay Commission and piecework Bonuses, allowances and holiday loadings Gross income and net income Budgeting Best buys Discounts Revision and mixed problems Technology Use spreadsheets to calculate pays, net incomes, budgets, discounts and best buys. Many Internet websites have on-line shopping baskets and financial calculators. Language (p. 162 of text)   Syllabus, p.70: „The abbreviation K comes from the Greek word khilioi meaning thousand. It is used in many job advertisements, e.g. a salary of $80K.‟ p.a. = per annum (Latin) = per year Skills Students will be able to do the following: 1–4 a calculate earnings from wages, salaries, casual work, overtime, commission, piecework and holiday loadings b calculate weekly, fortnightly, monthly and yearly incomes 5 calculate net income from gross income after considering deductions such as taxation and superannuation 6 prepare a budget for a given income 7 calculate the „best buy‟ from a range of products or brands 8 calculate discounts, including successive discounts Teaching notes and ideas 1 2 Resources: job advertisements in newspapers and Internet websites, tax tables, payslips, store catalogues. Liaise with the HSIE/Social Science faculty for Commerce and consumer awareness resources. Trigonometry Chapter 6, p. 166 Related chapters Year 8: 8 Pythagoras‟ theorem, 10 Ratios and rates, 12 Equations and inequalities, 14 Congruent and similar figures, Year 10: 8 Trigonometry (Stage 5.3). Syllabus reference Measurement MS5.1.2 Trigonometry: applies trigonometry to solve problems including those involving angles of elevation and depression (p. 139); MS5.2.3 Trigonometry (bearings) (p. 140). Recommended time: 3 weeks 1 c draw diagrams and use them to solve word problems which involve bearings or angles of elevation and depression Teaching notes and ideas Resources: clinometer, graph paper, magnetic compass, old book of trigonometric tables. 2 Make a clinometer. Calculate the heights of trees, flagpoles and buildings using trigonometry. 3 See Background information, p.140 of syllabus, for links to other topics and KLAs. 4 Investigate the history of the Babylonian base 60 system used in measuring angles (and time). Students have already used the [ ′ ″] button on the calculator in Stage 4 for time calculations. 5 The trigonometric ratios are constant no matter how large the (similar) right-angled triangle. Compare measured values with calculator values. See Worksheet 6-02 Investigating the tangent ratio. 6 Stress that the hypotenuse is a fixed side, while the opposite and adjacent sides depend on the angle. 7 Stage 5.1 students only work in degrees and are given the diagram in trigonometry problems, while Stage 5.2 students work in degrees and minutes, learn about bearings and can be given worded problems without diagrams. 8 Students should set out their solutions properly and use correct trigonometric terminology. Encourage them to „check the reasonableness of answers to trigonometry problems‟ (syllabus, p.140) by making a rough scale drawing. 9 Students need practice in drawing diagrams for a given problem. Have students devise a problem for a given diagram and swap problems. 10 Include open-ended problems: find out everything you can about this triangle. 11 Stage 5.3 students learn about exact ratios, the sine and cosine rules and A = Introduction This Stage 5 Measurement topic is entirely new to Year 9 students, but they have met related areas such as geometry, similar triangles, scale drawings, Pythagoras‟ theorem, ratios and equations at Stage 4. Do not rush throughout this topic – spend some time investigating right-angled triangles and the sine, cosine and tangent ratios before applying them to solve problems. Content 1 2 Naming the sides of a right-angled triangle Comparing the ratios of sides in similar right-angled triangles 3 The trigonometric ratios 4 The calculator in trigonometry 5 Using the calculator to find angles 6 Finding the length of a side 7 More on finding the length of a side 8 Finding an angle 9 Angles of elevation and depression 10 (Stage 5.2) Bearings 11 (Stage 5.2) Problems involving bearings 12 Revision and mixed problems Skills Students will be able to do the following: 1 a identify the hypotenuse, adjacent and opposite sides for a given angle in a right-angled triangle b label the sides of a right-angled triangle in relation to a given angle, e.g. side c is opposite angle C 2 recognise that the ratio of matching sides in similar right-angled triangles is constant for equal angles 3 define the sine, cosine and tangent ratios 4–5 use a calculator to find the trigonometric ratios for angles measured in degrees and minutes 6–8 a use a calculator to find an angle in degrees and minutes, given the trigonometric ratio of the angle b select and use appropriate ratios in right-angled triangles to find unknown sides and angles, including the hypotenuse 9–11 a identify angles of elevation and depression b use three-figure bearings (e.g. 035, 225) and compass bearings (e.g. SSW) 1 ab sin C in Year 10. 2 Technology Use a spreadsheet to compare the ratios of sides of similar right-angled triangles. The trigonometric ratios can be calculated on a spreadsheet, but with angle sizes in radians. Language (p. 212 of text)  Syllabus, p.139: „Emphasis should be placed on correct pronunciation of sin as “sine”. The origin of the word “cosine” is from “complements sine”, so that cos 40 = sin 50.‟ Encourage students to devise mnemonics for the trigonometric ratios and the four compass points. With compass bearings, stress the terminology: „the bearing of P from O‟. Elevated = feeling happy = looking up, Depressed = feeling sad = looking down.    Indices Chapter 7, p. 218 Related chapters Year 8: 1 Working with numbers, 2 Algebra, Year 9: 1 Exploring number, 2 Algebra, Year 10: 1 Working with number, 3 Surds. Syllabus reference Number NS5.1.1 Rational numbers: applies index laws to simplify and evaluate arithmetic expressions and uses scientific notation to write large and small numbers (p. 65); NS5.3.1 Real numbers: performs operations with surds and indices (p. 68); Patterns and algebra PAS5.1.1 Algebraic Techniques: applies the index laws to simplify algebraic expressions (p. 87); PAS5.2.1 Algebraic Techniques: simplifies, expands and factorises algebraic expressions involving negative and fractional indices (p. 88). Recommended time: 3 weeks 1 describe and evaluate numbers written in index form, using terms such as base, power, index and exponent 2–5 develop and use the index laws for multiplying and dividing terms with the same base, and for the power of a base raised to a power 6 establish that a0 = 1 using the index laws 7 develop and use negative indices 8 evaluate a fraction raised to a negative power 9 use index laws to define the fractional indices for square root ( a  a ) and cube root ( a  3 a ) 10 a develop and use fractional indices 1 2 1 3 x  x, x  n xm n 1 n m n b evaluate numerical expressions involving fractional indices and use the x calculator key 11 simplify numeric and algebraic expressions that include index notation 12-13 a express and order numbers in scientific notation b convert numbers expressed in scientific notation to decimal form 14 a enter and read scientific notation on a calculator b calculate with numbers expressed in scientific notation 1 y Introduction In this topic, students revise the concept of index notation before being introduced to the index laws, more complex powers and scientific notation. It examines indices both numerically and algebraically, applying them to numbers as well as variables so that students won‟t make mistakes such as 52 × 56 = 108. More time should be spent on examining the number patterns generated by repeated multiplication so that the different types of powers are more readily understood, especially the negative and fractional indices. There are also opportunities for investigating real applications of big and small numbers. Teaching notes and ideas 1 Use open-ended questions, e.g. find two terms that can be divided to give 27. 2 Syllabus, p.87: „Explain why a3  a2 = a6 is incorrect‟. 3 Common mistakes (syllabus, p.88): 5x0 = 1, 9x5  3x5 = 3x, 2c-4 = Content 1 2 3 4 5 6 7 8 9 Index notation Multiplying terms with the same base Dividing terms with the same base Raising a power to a power (Stage 5.3) Powers of products and quotients The zero index The negative index (Stage 5.2) Algebraic terms with negative indices (Stage 5.3) Negative powers of quotients The fractional index (for square root and cube root) (Stage 5.2) Algebraic terms with fractional indices 1 . Also 2a2 = 4a2, (3b)2 = 3b2. 4 2c 4 Assignment: Research the names of the big numbers or metric prefixes. 5 Investigate the measurement of very small objects and very large objects. How thin is a sheet of paper? 6 Common mistakes: 2 × 104 = 24 or 2 × 10 [EXP] 4 on the calculator. Inform students that the [EXP] key incorporates „[  ] 10 [xy]‟. 7 Syllabus, p.65: „Explain the difference between numerical expressions such as 2 × 104 and 24, particularly with reference to calculator displays.‟ 10 (Stage 5.3) The fractional indices 11 12 13 14 15 1 m and n n Technology Examine powers and scientific notation on a graphics calculator or spreadsheet. Summary of index laws Scientific notation Comparing numbers in scientific notation Calculations in scientific notation Revision and mixed problems Language (p. 262 of text)   For 25, 2 is called the base and 5 is called the power, index or exponent. Syllabus, p.66: „Pay particular attention to how these numbers are said, e.g. 34 is „three to the power of four‟ or „three to the fourth‟, 43 is „four cubed‟ or „four to the power of three‟. Skills Students will be able to do the following: Perimeter, area and volume Chapter 8, p. 266 Related chapters Year 8: 7 The circle, 8 Pythagoras‟ theorem, 11 Area and volume, Year 10: 2 Surface area and volume. Syllabus reference Measurement MS5.1.1 Perimeter and Area: uses formulae to calculate the area of quadrilaterals and find areas and perimeters of simple composite figures (p. 126); MS5.2.1 Perimeter and Area: find areas and perimeters of composite figures (p. 127); MS5.2.2 Surface Area and Volume (p. 132); MS4.1 Perimeter and Area (p. 124); MS4.2 Surface Area and Volume (p. 131). Recommended time: 3 weeks 10 develop and use a formula to find the surface area of a right cylinder 11 a calculate the volumes of right prisms and cylinders b solve problems involving the surface area, volume and capacity of right prisms and cylinders Teaching notes and ideas 1 2 Resources: paper and scissors, 1 cm transparent grid for area problems, nets of solid shapes, summary of area and volume formulas, centicubes. Assignments: investigate paper and envelope sizes, the legal size of an envelope, history of , areas of countries or Australian states, history of measurement, the old imperial system. Discuss the accuracy of measurements. A good starting point is the electronic timing of athletic and swimming events. Investigate Pythagorean triads and the formulas for generating them. Multiplying or dividing a triad by a constant gives another triad. Include perimeter and area problems where extra information is given or Pythagoras‟ theorem must be used. Investigate maximum area problems. Do rectangles or parallelograms with the same perimeter have the same area? Use squares/cubes to explain the relationships between the units for area/volume, e.g. 1 m2 = 1 m × 1 m = 100 cm × 100 cm = 10 000 cm2. 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 diagonals that are perpendicular. The area of a trapezium can be cut up and rearranged into two triangles or one rectangle. Investigate surface area through nets of solids. Find the surface area of a toilet roll by cutting it into a rectangle or parallelogram. Investigate the foil wrap of a Kit Kat chocolate bar. Find the surface area of the „best‟ can. Include open-ended or back-to-front questions, e.g. the volume of a prism is 5400 cm3. What might its dimensions be? Surface area and volume of pyramids, cones and spheres will be covered in the Year 10 Chapter 2 Surface area and volume. According to the syllabus, p.132, the formula for the volume of a right prism also works for oblique prisms. 3 4 5 6 7 8 Introduction This Measurement topic is mostly Stage 4 with some Stage 5. Perimeter, Pythagoras‟ theorem, surface area and volume concepts and calculations are revised while the measurement of area is extended to rhombuses, kites, trapeziums and sectors. Rather than learn a set of facts and formulas, the emphasis is upon understanding each type of measurement and the methods for calculating them. This is achieved by applying these skills to a variety of real problems. Practice in estimating and the use of Pythagoras‟ theorem feature prominently in this topic. Content Metric units Pythagoras’ theorem Perimeter The circumference of a circle (Stage 5.2) Perimeter of a sector 5 Areas of rectangles, triangles and parallelograms 6 (Stage 5.1) Area of a rhombus and kite 7 (Stage 5.1) Area of a trapezium 8 (Stage 5.2) Areas of circles and sectors 9 Surface area 10 (Stage 5.2) Surface area of a cylinder 11 Volume Volume and capacity 12 Revision and mixed problems 1 2 3 4 9 10 11 12 13 14 Skills Students will be able to do the following: 1 convert between metric units 2 use Pythagoras‟ theorem to find lengths and perimeters 3–5 calculate the perimeters and areas of plane shapes 6–7 develop and use formulas to find the areas of a kite, rhombus and trapezium 8 calculate the perimeters and areas of sectors and other circular shapes 9 calculate the surface area of right prisms Technology Use a spreadsheet to find the largest possible rectangle (in area) for a given perimeter. Language (p. 313 of text)  „m2‟ is read „square metres‟, not „metres squared‟. „m3‟ is read „cubic metres‟, not „metres cubed‟. Coordinate geometry Chapter 9, p. 318 Related chapters Year 8: 2 Algebra, 6 Graphing linear equations, 8 Pythagoras‟ theorem, Year 9: 2 Algebra, 8 Perimeter, area and volume, 11 Equations and inequalities, Year 10: 7 Coordinate geometry, 11 Graphs. Syllabus reference Patterns and Algebra PAS5.1.2 Coordinate Geometry: determines the midpoint, length and gradient of an interval joining two points on the number plane and graphs linear relationships (p. 97); Patterns and Algebra PAS5.2.3 Coordinate Geometry: uses formulae to find midpoint, distance and gradient (p. 99); Patterns and Algebra PAS5.3.3 Coordinate Geometry (p. 102). Recommended time: 3 weeks 3 graph vertical and horizontal lines 4 determine the midpoint of an interval from a diagram 5 use the right-angled triangle drawn between two points on the number plane and Pythagoras‟ theorem to determine the length of the interval joining the two points 6 determine whether a line has a positive or negative slope by following the line from left to right 7 a use the relationship gradient = rise to find the run gradient of an interval joining two points b find the gradient of a straight line from a graph by drawing a right-angled triangle 8 use the following formulas to calculate the midpoint, length and gradient of an interval: M(x, y) =  d= m= Introduction This topic marks the start of formal coordinate geometry. Students have already graphed lines and examined gradient and y-intercept informally in New Century Maths 8 Chapter 6 Graphing linear equations, but this Stage 5.2/5.3 topic extends their knowledge to the midpoint, length and gradient of an interval, parallel and perpendicular lines, and the gradient–intercept form of a linear equation y = mx + b.  x1  x 2 y1  y 2  ,  2   2 x2  x1 2   y 2  y1 2 y 2  y1 x 2  x1 Content 1 Graphing linear relationships (Stage 5.3) Using x- and y-intercepts to graph lines 2 Points that satisfy linear relationships 3 Horizontal and vertical lines 4 The midpoint of an interval 5 Finding the length of an interval 6 The gradient of a line 7 Calculating the gradient 8 (Stage 5.2) Formulas for the midpoint, distance and gradient 9 (Stage 5.2) Parallel lines (Stage 5.3) Perpendicular lines 10 (Stage 5.2) The gradient–intercept form of a linear equation 11 (Stage 5.3) Graphing equations of the form y = mx + b 12 Revision and mixed problems 9 a determine that two lines are parallel if their gradients are equal b demonstrate that two lines are perpendicular if the product of their gradients is –1 10 a recognise y = mx + b as representing a straight line with m as the gradient and b as the y-intercept b find the gradient and y-intercept of a straight line from the graph and use them to determine the equation of the line 11 graph equations of the form y = mx + b using the y-intercept (b) and the gradient (m) Teaching notes and ideas 1 Resources: number plane grid paper, graphics calculators. 2 Investigate large, small, positive, negative, zero and fractional gradients. Gradient is also used to describe land, roads and hills in construction and hiking. 3 With the formulas, does swapping x1 with x2 make any difference? What about swapping x1 with y1? 4 Open-ended problems: (a) Find two points that are 2 units apart, (b) If the midpoint of an interval is (1, 4), what could the endpoints of the interval be? Technology Use a graphics calculator, graphing software or spreadsheets to complete tables of values and graph lines. Skills Students will be able to do the following: 1 a graph a variety of linear relationships and identify the x- and y-intercepts b sketch the graph of a line by finding the x- and yintercepts from its equation c graph two intersecting lines on the same set of axes and read off the point of intersection 2 determine whether a point lies on a line by substituting into the equation of the line Language (p. 363 of text)   Remind students that the midpoint is a point and so the answer should be a pair of coordinates. Demonstrate how a negative gradient has a „negative run‟. Exploring data Chapter 10, p. 372 Related chapters Year 8: 9 Collecting and presenting data, 13 Analysing data, Year 10: 9 Analysing data, 10 Probability. Syllabus reference Data DS5.1.1 Data Representation and Analysis: groups data to aid analysis and constructs frequency and cumulative frequency tables and graphs (p. 116); DS5.2.1 Data Analysis and Evaluation (p. 117); DS4.1 Data Representation (p. 114); DS4.2 Data Analysis and Evaluation (p. 115). Recommended time: 4 weeks Introduction This topic revises basic Stage 4 Data concepts introduced in the two New Century Maths 8 chapters, 9 Collecting and presenting data and 13 Analysing data, before extending the knowledge to cumulative frequency, grouped data, interquartile range and box-and-whisker plots. All areas of statistics are met, including sampling and conducting surveys (data collection). Students have already had considerable experience with statistical graphs and calculations in primary and early high school, so a significant portion of this topic is review work. b use the terms cluster and outlier when describing data 5–7 use measures of location (the mean, mode and median) and the range to analyse data that is displayed in a frequency distribution table, stemand-leaf plot, or dot plot 8 a group data into class intervals and construct a frequency table b find the mean, median class and modal class for grouped data 9 construct a frequency histogram and polygon for grouped data 10 construct a cumulative frequency table for ungrouped data 11 a construct a cumulative frequency histogram and polygon (ogive) b use a cumulative frequency polygon to find the median 12 determine the upper and lower quartiles for a set of scores and calculate the interquartile range 13 construct a box-and-whisker plot using the median, the upper and lower quartiles and the extreme values (the „five-point summary‟) Teaching notes and ideas 1 This topic lends itself to investigation projects. The class may be surveyed on a number of characteristics and the data analysed: height, arm span, shoe size, heartbeat rate, reaction time, PDHPE data, number of children in family, hours slept last night, number of letters in first name, number of cars or mobile phones owned at home. Note: there is no Stage 5.3 in Data. See syllabus notes, p.115, for examples of the use of statistics in other school subjects. Newspapers, magazines and the Internet are useful sources of statistical information. Replicate a newspaper survey. An Australian census takes place every five years, in a year ending in 1 or 6. Which is higher, the mean or median price of Australian homes? Open-ended questions: the range of a set of eight scores is 10 and the mode is 3. What might the scores be? Content Stage 4 1 Types of data 2 Collecting data 3 Frequency histograms and polygons 4 Dot plots and stem-and-leaf plots 5 Range 6 Mean, median and mode: measures of location 7 The mean from a frequency table Stage 5.1 8 Grouping data 9 Histograms and polygons using grouped data 10 Cumulative frequency and the median 11 Cumulative frequency polygons and the median Stage 5.2 12 Quartiles and the interquartile range 13 Box-and-whisker plots 14 Revision and mixed problems 2 3 4 5 6 7 Skills Students will be able to do the following: 1 recognise data as quantitative (either discrete or continuous) or categorical 2 a recognise the differences between a census and a sample b formulate, trial and refine key survey questions to generate data that is unbiased, relevant and useful 3 organise data into a frequency distribution table and draw frequency histograms and polygons 4 a draw and use dot plots and stem-and-leaf plots Technology Explore the statistical functions of a calculator, graphics calculator and spreadsheet. Investigate databases and statistical graphing software. A graphics calculator graphs boxplots. Use a spreadsheet to examine the affects of altering data. Language (p. 422 of text)   Reinforce the terminology measures of location and measures of spread. median = middle, e.g. median strip on a road, mode (French) = fashionable, popular. Equations and inequalities Chapter 11, p. 426 Related chapters Year 8: 6 Graphing linear equations, 12 Equations and inequalities, Year 9: 2 Algebra, 8 Coordinate geometry, Year 10: 4 Equations and inequalities, 7 Coordinate geometry. Syllabus reference Patterns and Algebra PAS5.2.2 Algebraic Techniques: solves linear equations and linear inequalities using graphical and analytical methods (p. 90); PAS5.3.2 Algebraic Techniques (p. 95); PAS4.4 Algebraic Techniques (p. 86). Recommended time: 4 weeks solve word problems that result in equations a substitute values into formulas b solve equations arising from substitution into formulas 9 change the subject of a formula 10 find a range of values that satisfy an inequality using strategies such as „guess and check‟ 11–12 solve simple inequalities and represent solutions on the number line 13 change the direction of the inequality when multiplying or dividing by a negative number 7 8 Teaching notes and ideas 1 See p.86 of the syllabus for five models for solving equations: balance method, unknown quantity in „paper cups‟, one-to-one matching (see Skillsheet 11-03 Solving equations with diagrams), „guess, check and improve‟, backtracking. Emphasise the correct setting-out of solutions. The aim is to have the variable on the LHS of the equal sign and the number (answer) on the RHS. Examples of Stage 5.2 equations from syllabus, p.90: Introduction This topic revises and extends the formal and informal methods of solving Stage 4 equations and inequalities, first met in Year 8. This is a fairly technical topic in which students also work with formulas and the application of equations. Stage 5.2 equations and Stage 5.3 skills are also introduced. 2 3 Content Stage 4 1 Linear equations 2 Guess, check and improve 3 Backtracking 4 The balance method Stage 5.2 5 Solving harder equations Equations with variables on both sides Equations with grouping symbols 6 Equations involving algebraic fractions 7 Using equations to solve word problems 8 Formulas 9 (Stage 5.3) Changing the subject of a formula Stage 4 10 Inequalities 11 Representing inequalities on a number line 12 Solving inequalities 13 (Stage 5.3) Multiplying and dividing inequalities by a negative number 14 Revision and mixed problems 2y  3 x x z 3   5,  2,  6  1, 2 3 3 2 3(a + 2) + 2(a – 5) = 10, 3(2t – 5) = 2t + 5, 3r  1 2r  4 .  4 5 4 5 6 With equations involving fractions, denominators should be numerical. When solving a word problem, identify the unknown quantity and call it x, say. After forming and solving the equation, check that its solution sounds reasonable. Examples of Stage 5.2 inequalities from syllabus, p.90: 3x – 1 < 9, 2(a + 4)  24, t4  3 . 5 7 Syllabus, p.95: „Use a numerical example to justify the need to reverse the direction of the inequality when multiplying or dividing by a negative number,‟ for example, multiply both sides of 7 > 3 by -1. Technology Investigate the use of CAS (computer algebra system), spreadsheets and graphics calculators to solve linear equations. Skills Students will be able to do the following: 1 check solutions to equations by substituting 2–3 solve equations using strategies such as „guess, check and improve‟ and backtracking (reverse flow charts) 4 solve equations using algebraic methods that involve up to and including three steps in the solution 5 solve linear equations involving variables on both sides and grouping symbols 6 solve equations involving algebraic fractions Language (p. 461 of text)    expression = „phrase‟ involving terms and arithmetic operations, e.g. 2a + 5. equation = „sentence‟ involving an expression and an equal sign, e.g. 2a + 5 = 13. Some students believe that x < 5 and x  4 mean the same thing. Explain the difference. Congruence and similarity Chapter 12, p. 464 Related chapters Year 8: 3 Geometric figures, 14 Congruent and similar figures, Year 9: 4 Investigating geometry, Year 10: 2 Surface area and volume, 5 Deductive geometry. Syllabus reference Space and Geometry SGS5.2.2 Properties of Geometrical Figures: develops and applies results for proving that triangles are congruent or similar (p. 158); SGS5.3.1 Deductive Geometry (p. 159); SGS5.3.3 Deductive Geometry (p. 162); SGS4.4 Properties of Geometrical Figures (p. 156). Recommended time: 3 weeks b identify that shape, angle size and ratio of matching sides are preserved in similar figures 6 a determine the scale factor for a pair of similar figures b choose an appropriate scale in order to enlarge or reduce a diagram 7 determine whether a pair of triangles are similar 8 calculate unknown sides in a pair of similar figures 9–10 a determine what information is needed to prove that two triangles are similar b write formal proofs for similarity of triangles, preserving the matching order of vertices Teaching notes and ideas 1 Investigate congruence in cultural and religious design patterns. Syllabus, p.156: „Recognise congruent figures in tessellations, art and design work.‟ Prove the properties of the special triangles and quadrilaterals by congruent triangles. Investigate why the geometrical constructions work by analysing the properties of the bisected isosceles triangle and the rhombus. Enlarge diagrams, logos, floor plans and sports fields such as basketball courts. Investigate the scales used in cameras, photocopiers, overhead projectors, telescopes, microscopes, mirrors. Are all equilateral triangles similar? Are all rectangles similar? Are all isosceles triangles similar? When forming a proportion equation involving similar triangles, make x appear in the numerator. Syllabus, p.159: „Students may be interested in reading about the history of deductive geometry, including the work of Euclid and Gauss.‟ Matching areas and volumes of similar figures (Stage 5.3 work) will be met in the Year 10 topic 2 Surface area and volume. Introduction This second Space and Geometry topic revises and extends Stage 4 concepts in congruence and similarity. The tests for congruent triangles are met for the first time, as Stage 5.2 work, with formal proofs being introduced as a Stage 5.3 skill. Geometrical properties should still be discovered informally through construction and measurement, although this topic marks the start of formal deductive reasoning for the Stage 5.3 student. 2 3 Content 1 2 3 Recognising congruent figures (Stage 5.2) Tests for congruent triangles (Stage 5.2) Applying the congruence tests (Stage 5.3) Formal proofs for congruent triangles 4 (Stage 5.3) Proving geometrical results 5 Similarity 6 Enlargements and reductions 7 Recognising similar figures 8 Properties of similar figures 9 (Stage 5.3) Tests for similar triangles 10 (Stage 5.3) Applying similarity 11 Revision and mixed problems 4 5 6 7 Technology Use dynamic geometry to investigate the properties of congruent and similar figures. Skills Students will be able to do the following: 1 match the sides and angles of congruent figures 2 determine what information is needed to show that two triangles are congruent (SSS, SAS, AAS, RHS) 3 a apply the congruency test to justify that two triangles are congruent and find unknown sides and angles b write formal proofs of congruence of triangles, preserving matching order of vertices c apply triangle results to establish properties of special triangles and quadrilaterals 4 construct and write geometrical arguments to prove a general geometrical result 5 a match the sides and angles of similar figures Language (p. 512 of text)  Use matching angles rather than corresponding angles to avoid confusion with the corresponding angles found when a transversal crosses two lines. Syllabus, p.156: „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‟ or „is congruent to‟. In geometry, the word similar has a different meaning to its everyday one.   