Mathscape 10 Teaching Program Page 1. Stage 5 MATHSCAPE 10. Term Chapter

Transcription

1 Mathscape 10 Teaching Program Page 1 Stage 5 MATHSCAPE 10 Term Chapter Time 1 1. Consumer arithmetic weeks / 8 hrs. Trigonometry weeks / 8 hrs 3. Simultaneous equations weeks / 8 hrs 4. Graphs in the number plane weeks / 8 hrs 5. Volume and surface area weeks / 8 hrs 6. Congruence and similarity weeks / 8 hrs 3 7. Data representation and evaluation weeks / 8 hrs 8. Products and factors 3 weeks / 1 hrs 9. Surds weeks / 8 hrs Quadratic equations weeks / 8 hrs 11. Further trigonometry 3 weeks / 1 hrs 1. Further coordinate geometry 3 weeks / 1 hrs 14. Co-ordinate geometry weeks / 8 hrs Published by Macmillan Education Australia. Macmillan Education Australia 006.

2 Mathscape 10 Teaching Program Page Chapter 1. Consumer arithmetic Substrand Consumer Arithmetic Duration weeks / 8 hours Tet References Mathscape 10: Ch 1: Consumer arithmetic (pages 1-33) CD Reference Key Ideas Solves consumer arithmetic problems involving earning and spending money. Calculate simple interest and find compound interest using a calculator and tables of values Use compound interest formula. Solve consumer arithmetic problems involving compound interest, depreciation and successive discounts. Outcomes NS5.1. (p70): Solve simple consumer problems including those involving earning and spending money NS5.. (p71): Solves consumer arithmetic problems involving compound interest, depreciation, and successive discounts Working Mathematically Students learn to compare simple interest with compound interest in practical situations eg loans (Applying Strategies) interpret spreadsheets or tables when comparing simple interest and compound interest on an investment over various time periods (Applying Strategies, Communicating) realise the total cost and/or hidden costs involved in some types of purchase arrangements (Applying Strategies) make informed decisions related to purchases eg determining the best mobile phone plan for a given situation (Applying Strategies) solve problems involving discounts and compound interest (Applying Strategies) Knowledge and Skills Students learn about calculating simple interest using the formula I = PRT where r R = where I is the 100 interest, P the principal, R the annual interest rate and T the number of years applying the simple interest formula to problems related to investing money at simple interest rates Teaching, Learning and Assessment TRY THIS: Costs of raising a child (p4): Problem solving Consumer Price Inde (p10): Research Assignment Depreciating Value (p14): Problem Solving Home Loan (p6): Spreadsheet eercise Published by Macmillan Education Australia. Macmillan Education Australia 006.

3 Mathscape 10 Teaching Program Page 3 calculating compound interest for two or three years by repeated multiplication using a calculator eg a rate of 5% per annum leads to repeated multiplication by 1.05 calculating compound interest on investments using a table calculating and comparing the cost of purchasing goods using: cash credit card lay-by deferred payment buying on terms loans determining and using the formula for compound interest, A = P ( 1 + R) n, where A is the total amount, P is the principal, R is the interest rate per period as a decimal and n is the number of periods using the compound interest formula to calculate depreciation comparing the cost of loans using flat and reducible interest for a small number of repayment periods FOCUS ON WORKING MATHEMATICALLY: (p7) The cost of living in Australia EXTENSION ACTIVITIES, LET S COMMUNICATE, REFLECTING (p9) CHAPTER REVIEW (p30): A collection of problems to revise the chapter. Technology Published by Macmillan Education Australia. Macmillan Education Australia 006.

4 Mathscape 10 Teaching Program Page 4 Chapter. Trigonometry Substrand Trigonometry Duration weeks / 8 hours Tet References Mathscape 10: Ch : Trigonometry (pages 34 68) CD Reference Key Ideas Solve further trigonometry problems including those involving three-figure bearings Outcomes MS5..3 (p 140): Applies trigonometry to solve problems including those involving bearings Working Mathematically Students learn to solve simple problems involving three-figure bearings (Applying Strategies, Communicating) recognise directions given as SSW, NE etc (Communicating) solve practical problems involving angles of elevation and depression (Applying Strategies) check the reasonableness of answers to trigonometry problems (Reasoning) interpret directions given as bearings (Communicating) find the angle between a line with a positive gradient and the -ais in the coordinate plane by using a right-angled triangle formed by the rise and the run, from the point where the line cuts the -ais to another point on the line (Reasoning, Reflecting) Knowledge and Skills Students learn about Further Trigonometric Ratios of Acute Angles using a calculator to find trigonometric ratios of a given approimation for angles measured in degrees and minutes using a calculator to find an approimation for an angle in degrees and minutes, given the trigonometric ratio of the angle Further Trigonometry of Right-Angled Triangles finding unknown sides in right-angled triangles where the given angle is measured in degrees and minutes Teaching, Learning and Assessment TRY THIS: Playground Swing (p48): Problem solving Hot Air Balloon (p53): Problem solving The Mariner s Compass (p60): Practical FOCUS ON WORKING MATHEMATICALLY: (p61) Orienteering EXTENSION ACTIVITIES, LET S COMMUNICATE, REFLECTING (p63) CHAPTER REVIEW (p65): A collection of problems to revise the chapter. Published by Macmillan Education Australia. Macmillan Education Australia 006.

5 Mathscape 10 Teaching Program Page 5 using trigonometric ratios to find unknown angles in degrees and minutes in rightangled triangles using three-figure bearings (eg 035º, 5º) and compass bearings eg SSW drawing diagrams and using them to solve word problems which involve bearings or angles of elevation and depression Technology Published by Macmillan Education Australia. Macmillan Education Australia 006.

6 Mathscape 10 Teaching Program Page 6 Chapter 3. Simultaneous equations Substrand Algebraic Techniques Duration weeks / 8 hours Tet References Mathscape 10: Ch 3: Simultaneous Equations (pages 69 91) CD Reference Key Ideas Solve simultaneous equations using graphical and analytical methods for simple eamples Outcomes PAS5.. (p 90): Solves linear and simple quadratic equations, solves linear inequalities and solves simultaneous equations using graphical and analytical methods Working Mathematically Students learn to use graphics calculators and spreadsheet software to plot pairs of lines and read off the point of intersection (Applying Strategies) solve linear simultaneous equations resulting from problems and interpret the results (Applying Strategies, Communicating) Knowledge and Skills Students learn about Simultaneous Equations solving simultaneous equations using non-algebraic methods, such as guess and check, setting up tables of values or looking for patterns solving linear simultaneous equations by finding the point of intersection of their graphs solving simple linear simultaneous equations using an analytical method eg solve the following 3 a + b =17 Teaching, Learning and Assessment TRY THIS: Find the values (p79): Problem Solving A Pythagorean Problem (p83): Problem Solving FOCUS ON WORKING MATHEMATICALLY: (p87) Eploring for water, oil and gas- The density of air-filled porous rock EXTENSION ACTIVITIES, LET S COMMUNICATE, REFLECTING (p89) CHAPTER REVIEW (p91): A collection of problems to revise the chapter. a b = 8 generating simultaneous equations from simple word problems Published by Macmillan Education Australia. Macmillan Education Australia 006.

7 Mathscape 10 Teaching Program Page 7 Technology Published by Macmillan Education Australia. Macmillan Education Australia 006.

8 Mathscape 10 Teaching Program Page 8 Chapter 4. Graphs in the number plane Substrand Coordinate Geometry Duration weeks / 8 hours Tet References Mathscape 10: Ch 4:Graphs in the number plane (pages 9 137) CD Reference Key Ideas Apply the gradient/intercept form to interpret and graph straight lines Draw and interpret graphs including simple parabolas and hyperbolas Outcomes PAS5..3 (p 99): Uses formulae to find midpoint, distance and gradient and applies the gradient/intercept form to interpret and graph straight lines PAS5..4 (p 101): Draws and interprets graphs including simple parabolas and hyperbolas Working Mathematically Students learn to determine the difference in equations of lines that have a negative gradient and those that have a positive gradient (Reasoning) match equations of straight lines to graphs of straight lines and justify choices (Communicating, Reasoning) compare similarities and differences between sets of linear relationships eg y = 3, y = 3 +, y = 3 1 y =, y =, y = 3 =, y = (Reasoning) sort and classify a mied set of equations of linear relationships into groups to demonstrate similarities and differences (Reasoning, Communicating) conjecture whether a particular equation will have a similar graph to another equation and graph both lines to test the conjecture (Questioning, Applying Strategies, Reasoning) eplain the effect on the graph of a line of changing the gradient or y -intercept (Reasoning, Communicating) use a graphics calculator and spreadsheet software to graph a variety of equations of straight lines, and compare and describe the similarities and differences between the lines (Applying Strategies, Communicating, Reasoning) apply knowledge and skills of linear relationships to practical problems (Applying Strategies) apply ethical considerations when using hardware and software (Reflecting) identify parabolic shapes in the environment (Reflecting) Published by Macmillan Education Australia. Macmillan Education Australia 006.

9 Mathscape 10 Teaching Program Page 9 describe the effect on the graph of y = of multiplying by different constants or of adding different constants (Reasoning, Communicating) discuss and predict the equation of a parabola from its graph, with the main features clearly marked, using computer graphing software (Communicating) 1 describe the effect on the graph of y = of multiplying by different constants (Reasoning, Communicating) k eplain what happens to the y -values of the points on the hyperbola y = as the -values get very large (Reasoning, Communicating) k eplain what happens to the y -values of the points on the hyperbola y = as the -values get closer to zero (Reasoning, Communicating) sort and classify a set of graphs, match each graph to an equation, and justify each choice (Reasoning, Communicating) eplain why it may be useful to include small and large numbers when constructing a table of values 1 eg For y =, why do we need to use more than the integers 1,, 3, and 4 for? (Reasoning, Communicating) use a graphics calculator and spreadsheet software to graph, compare and describe a range of linear and non-linear relationships (Applying Strategies, Communicating) Knowledge and Skills Students learn about constructing tables of values and using coordinates to graph straight lines of the form y = m + b ie gradient/intercept form recognising equations of the form y = m + b as representing straight lines and interpreting the -coefficient (m) as the gradient and the constant (b) as the y -intercept rearranging an equation in general form (a + by + c = 0) to the gradient/intercept form graphing equations of the form y = m + b using the y -intercept (b) and the gradient (m) determining that two lines are parallel if their gradients are equal finding the gradient and the y -intercept of a straight line from the graph and using them to determine the equation of the line generating simple quadratic relationships, compiling tables of values and graphing equations of the form y = a and y = a + c Teaching, Learning and Assessment TRY THIS: The Parkes telescope (p116): Problem Solving Flagpoles (p10): Problem Solving Gas under pressure (p14): Problem Solving Maimum Area (p130): Problem Solving FOCUS ON WORKING MATHEMATICALLY: (p130) Sydney to Alice by Boeing 737 EXTENSION ACTIVITIES, LET S COMMUNICATE, REFLECTING (p13) CHAPTER REVIEW (p134): A collection of problems to revise the chapter. Published by Macmillan Education Australia. Macmillan Education Australia 006.

10 Mathscape 10 Teaching Program Page 10 generating simple hyperbolic relationships, compiling tables of values and graphing equations of the form k y = for integral values of k identifying graphs of straight lines, parabolas and hyperbolas matching graphs of straight lines, parabolas and hyperbolas to the appropriate equations Technology Published by Macmillan Education Australia. Macmillan Education Australia 006.

11 Mathscape 10 Teaching Program Page 11 Chapter 5. Volume and surface area Substrand Surface Area and Volume Duration weeks / 8 hours Tet References Mathscape 10: Ch 5: Volume and Surface Area (pages ) CD Reference Key Ideas Find surface area of right cylinders and composite solids Find the volume of right pyramids, cones, spheres and composite solids Outcomes MS5.. (p 13): Applies formulae to find the surface area of right cylinders and volume of right pyramids, cones and spheres and calculates the surface area and volume of composite solids Working Mathematically Students learn to solve problems relating to volumes of right pyramids, cones and spheres (Applying Strategies) dissect composite solids into several simpler solids to find volume (Applying Strategies) solve practical problems related to volume and capacity eg find the volume of a swimming pool with a given rectangular surface and a given trapezoidal side (Applying Strategies) solve practical problems related to surface area eg compare the amount of packaging material needed for different shapes (Applying Strategies) Knowledge and Skills Students learn about Surface Area of Right Cylinders developing a formula to find the surface area of right cylinders SA = πr + πrh for closed cylinders where r is the length of the radius and h is the perpendicular height finding the surface area of right cylinders calculating the surface area of composite solids involving right cylinders and prisms Volume of Right Pyramids, Cones and Spheres using the fact that a pyramid has one-third the volume of a prism with the same base and the same perpendicular height Teaching, Learning and Assessment TRY THIS: Can we use an iceburg? (p153): Research Problem Magic Nails (p158): Problem Solving Paper cup (p163): Problem Solving Cones, Spheres and Cylinders (p171): Problem Solving FOCUS ON WORKING MATHEMATICALLY: (p171) Archimedes finds the volume and surface area of a sphere EXTENSION ACTIVITIES, LET S COMMUNICATE, REFLECTING (p173) CHAPTER REVIEW (p174): A collection of problems to revise the chapter. Published by Macmillan Education Australia. Macmillan Education Australia 006.

12 Mathscape 10 Teaching Program Page 1 using the fact that a cone has one-third the volume of a cylinder with the same base and the same perpendicular height 1 using the formula V = Ah to find the volume of pyramids and cones where A is 3 the base area and h is the perpendicular height 3 using the formula 4 V = π r to find the volume of spheres where r is the length of 3 the radius finding the dimensions of solids given their volume and/or surface area by substitution into a formula to generate an equation finding the volume of prisms whose bases can be dissected into triangles, special quadrilaterals and sectors finding the volume of composite solids Technology Published by Macmillan Education Australia. Macmillan Education Australia 006.

13 Mathscape 10 Teaching Program Page 13 Chapter 6. Congruence and similarity Substrand Properties of Geometrical Figures Duration weeks / 8 hours Tet References Mathscape 10: Ch 6: Congruence and Similarity (pages 181 3) CD Reference Key Ideas Identify similar triangles and describe their properties Apply tests for congruent triangles Use simple deductive reasoning in numerical and non-numerical problems Verify the properties of special quadrilaterals using congruent triangles Outcomes SGS5.. (p 158): Develops and applies results for proving that triangles are congruent or similar Working Mathematically Students learn to apply the properties of congruent and similar triangles to solve problems, justifying the results (Applying Strategies, Reasoning) apply simple deductive reasoning in solving numerical and non-numerical problems (Applying Strategies, Reasoning) eplain why any two equilateral triangles, or any two squares, are similar, and eplain when they are congruent (Communicating, Reasoning) investigate whether any two rectangles, or any two isosceles triangles, are similar (Applying Strategies, Reasoning) use dynamic geometry software to investigate the properties of geometrical figures (Applying Strategies, Reasoning) Knowledge and Skills Students learn about Congruent Triangles determining what information is needed to show that two triangles are congruent If three sides of one triangle are respectively equal to three sides of another triangle, then the two triangles are congruent (SSS). If two sides and the included angle of one triangle are respectively equal to two sides and the included angle of another triangle, then the two triangles are congruent (SAS). If two angles and one side of one triangle are respectively equal to two angles and the matching side of another triangle, then the two triangles are congruent (AAS). Teaching, Learning and Assessment TRY THIS: An investigation of triangles (p179): Introductory investigation Four Congruent Farms (p186): Problem solving Triangle angles (p191): Problem solving Intersecting parallelograms (p196): Proof Similar triangles (p10): Problem solving Ratio in triangles (p 16): Problem solving FOCUS ON WORKING MATHEMATICALLY: (p0) The rail link from Alice to Darwin Published by Macmillan Education Australia. Macmillan Education Australia 006.

14 Mathscape 10 Teaching Program Page 14 If the hypotenuse and a second side of one right-angled triangle are respectively equal to the hypotenuse and a second side of another right-angled triangle, then the two triangles are congruent (RHS). applying the congruency tests to justify that two triangles are congruent applying congruent triangle results to establish properties of isosceles and equilateral triangles eg If two sides of a triangle are equal, then the angles opposite the equal sides are equal. Conversely, if two angles of a triangle are equal, then the sides opposite those angles are equal. If three sides of a triangle are equal then each interior angle is 60º. applying congruent triangle results to establish some of the properties of special quadrilaterals, including diagonal properties eg the diagonals of a parallelogram bisect each other applying the four triangle congruency tests in numerical eercises to find unknown sides and angles Similar Triangles identifying the elements preserved in similar triangles, namely angle size and the ratio of corresponding sides determining whether triangles are similar applying the enlargement or reduction factor to find unknown sides in similar triangles calculating unknown sides in a pair of similar triangles Technology EXTENSION ACTIVITIES, LET S COMMUNICATE, REFLECTING (p3) CHAPTER REVIEW (p5): A collection of problems to revise the chapter. Published by Macmillan Education Australia. Macmillan Education Australia 006.

15 Mathscape 10 Teaching Program Page 15 Chapter 7. Data analysis and evaluation Substrand Data Analysis and Evaluation Duration weeks / 8 hours Tet Reference Mathscape 10: Ch 7: Data Analysis and Evaluation (pages 33 85) CD Reference Key Ideas Determine the upper and lower quartiles of a set of scores Construct and interpret bo-and-whisker plots Find the standard deviation of a set of scores using a calculator Use the terms skew and symmetrical to describe the shape of a distribution Outcomes DS5..1 (p 117): Uses the interquartile range and standard deviation to analyse data Working Mathematically Students learn to compare two or more sets of data using bo-and-whisker plots drawn on the same scale (Applying Strategies) compare data with the same mean and different standard deviations (Applying Strategies) compare two sets of data and choose an appropriate way to display these, using back-to-back stem-and-leaf plots, histograms, double column graphs, or bo-andwhisker plots (Communicating, Applying Strategies) analyse collected data to identify any obvious errors and justify the inclusion of any scores that differ remarkably from the rest of the data collected (Applying Strategies, Reasoning) use spreadsheets, databases, statistics packages, or other technology, to analyse collected data, present graphical displays, and discuss ethical issues that may arise from the data (Applying Strategies, Communicating, Reflecting) use histograms and stem-and-leaf plots to describe the shape of a distribution (Communicating) recognise when a distribution is symmetrical or skewed, and discuss possible reasons for its shape (Communicating, Reasoning) Knowledge and Skills Students learn about determining the upper and lower quartiles for a set of scores Teaching, Learning and Assessment TRY THIS: Rowing outliers (p46): Graph interpretation Published by Macmillan Education Australia. Macmillan Education Australia 006.

16 Mathscape 10 Teaching Program Page 16 constructing a bo-and-whisker plot using the median, the upper and lower quartiles and the etreme values (the five-point summary ) finding the standard deviation of a set of scores using a calculator using the mean and standard deviation to compare two sets of data comparing the relative merits of measures of spread: range interquartile range standard deviation using the terms skewed or symmetrical when describing the shape of a distribution Cliometrics (p50): Research Two standard deviations (p61): Research Correlation (p66): Research FOCUS ON WORKING MATHEMATICALLY: (p74) The imprisonment of indigenous people EXTENSION ACTIVITIES, LET S COMMUNICATE, REFLECTING (p75) CHAPTER REVIEW (p79): A collection of problems to revise the chapter. Technology Published by Macmillan Education Australia. Macmillan Education Australia 006.

17 Mathscape 10 Teaching Program Page 17 Chapter 8. Products and factors Substrands Algebraic Techniques Duration 3 weeks / 1 hours Tet References Mathscape 10: Ch 8: Products and Factors (pages 86 34) CD Reference Key Ideas Use algebraic techniques to simplify epressions, epand binomial products and factorise quadratic epressions Outcomes PAS5.3.1 (p 91): Uses algebraic techniques to simplify epressions, epand binomial products and factorise quadratic epressions Working Mathematically Students learn to determine and justify whether a simplified epression is correct by substituting numbers for pronumerals (Applying Strategies, Reasoning) check epansions and factorisations by performing the reverse process (Reasoning) interpret statements involving algebraic symbols in other contets eg spreadsheets (Communicating) Knowledge and Skills Students learn about epanding binomial products by finding the area of rectangles eg 8 hence ( + 8)( + 3) = = using algebraic methods to epand a variety of binomial products, such as ( + )( 3) (y + 1) (3a 1)(3a + 1) Teaching, Learning and Assessment TRY THIS: Proof (p91): Investigation Consecutive Numbers (p94): Investigation Market Garden (p98): Problem Solving Squaring Fives (p30): Investigation Difference of Two Squares (p309): Challenge Problem FOCUS ON WORKING MATHEMATICALLY: (p318) Products and Factors by paper folding and cutting EXTENSION ACTIVITIES, LET S COMMUNICATE, REFLECTING (p319) CHAPTER REVIEW (p31): A collection of problems to revise the chapter. Published by Macmillan Education Australia. Macmillan Education Australia 006.

18 Mathscape 10 Teaching Program Page 18 recognising and applying the special products (a + b)(a b) = a b (a ± b) = a ± ab + b factorising epressions: - common factors - difference of two squares - perfect squares - trinomials - grouping in pairs for four-term epressions Technology Published by Macmillan Education Australia. Macmillan Education Australia 006.

19 Mathscape 10 Teaching Program Page 19 Chapter 9. Surds Substrand Real Numbers Duration weeks / 8 hours Tet References Mathscape 10: Ch 9: Surds (pages ) CD Reference Key Ideas Define the system of real numbers distinguishing between rational and irrational numbers Perform operations with surds Use integers and fractions for inde notation Convert between surd and inde form Outcomes NS5.3.1 (p 68): Performs operations with surds and indices Working Mathematically Students learn to eplain why all integers and recurring decimals are rational numbers (Communicating, Reasoning) eplain why rational numbers can be epressed in decimal form (Communicating, Reasoning) demonstrate that not all real numbers are rational (Communicating, Applying Strategies, Reasoning) solve numerical problems involving surds and/or fractional indices (Applying Strategies) eplain why a particular sentence is incorrect eg (Communicating, Reasoning) Knowledge and Skills Students learn about defining a rational number: A rational number is the ratio b a of two integers where b 0. distinguishing between rational and irrational numbers using a pair of compasses and a straight edge to construct simple rationals and surds on the number line Teaching, Learning and Assessment TRY THIS: Greater Number (p331): Problem Solving Imaginary Numbers (p337): Etension Eact Values (p343): Problem Solving FOCUS ON WORKING MATHEMATICALLY: (p344) Fibonacci Numbers and the Golden Mean Published by Macmillan Education Australia. Macmillan Education Australia 006.

20 Mathscape 10 Teaching Program Page 0 defining real numbers: Real numbers are represented by points on the number line. Irrational numbers are real numbers that are not rational. demonstrating that is undefined for < 0, = 0 for = 0, and is the positive square root of when > 0 using the following results for, y > 0: ( ) y = = y = = using the four operations of addition, subtraction, multiplication and division to simplify epressions involving surds epanding epressions involving surds such as ( 5 ). y 3 + or ( 3)( + 3) rationalising the denominators of surds of the form a b c d using the inde laws to demonstrate the reasonableness of the definitions for fractional indices 1 n m n = = n n m y EXTENSION ACTIVITIES, LET S COMMUNICATE, REFLECTING (p346) CHAPTER REVIEW (p348): A collection of problems to revise the chapter. Technology Published by Macmillan Education Australia. Macmillan Education Australia 006.

21 Mathscape 10 Teaching Program Page 1 Chapter 10. Quadratic equations Substrand Algebraic Techniques Duration weeks / 8 hours Tet References Mathscape 10: Ch 10: Quadratic Equations (pages ) CD Reference Key Ideas Solve quadratic equations by factorising, completing the square, or using the quadratic formula Solve a range of inequalities and rearrange literal equations Solve simultaneous equations including quadratic equations Outcomes PAS5.3. (p 93): Solves linear, quadratic and simultaneous equations, solves and graphs inequalities, and rearranges literal equations Working Mathematically Students learn to eplain why a particular value could not be a solution to an equation (Applying Strategies, Communicating, Reasoning) determine quadratic epressions to describe particular number patterns such as y = + 1 for the table (Applying Strategies, Communicating) choose the most appropriate method to solve a particular quadratic equation (Applying Strategies) solve quadratic equations and discuss the possible number of roots for any quadratic equation (Applying Strategies, Communicating) Knowledge and Skills Students learn about developing the quadratic formula b ± = b 4ac a solving equations of the form a + b + c = 0 using: - factors Teaching, Learning and Assessment TRY THIS: Forming a quadratic equation (p356): Problem Solving Chocolate Time! (p363): Problem Solving Touching Circles (p370): Investigation FOCUS ON WORKING MATHEMATICALLY: (p370)the long distance flight of arrows under gravity Published by Macmillan Education Australia. Macmillan Education Australia 006.

22 Mathscape 10 Teaching Program Page - completing the square - the quadratic formula solving a variety of quadratic equations such as 3 ( y ) = = 0 ( 4) = 4 identifying whether a given quadratic equation has no solution, one solution or two solutions checking the solutions of quadratic equations generating quadratic equations from problems solving problems involving quadratic equations solving quadratic equations resulting from substitution into formulae = 9 EXTENSION ACTIVITIES, LET S COMMUNICATE, REFLECTING (p37) CHAPTER REVIEW (p374): A collection of problems to revise the chapter. Technology Published by Macmillan Education Australia. Macmillan Education Australia 006.

23 Mathscape 10 Teaching Program Page 3 Chapter 11. Further trigonometry Substrand Trigonometry Duration 3 weeks / 1 hours Tet References Mathscape 10: Ch 11: Further Trigonometry (pages ) CD Reference Key Ideas Determine the eact trigonometric ratios for 30, 45, 60 Apply relationships in trigonometry for complementary angles and tan in terms of sin and cos Determine trigonometric ratios for obtuse angles Sketch sine and cosine curves Eplore trigonometry with non-right-angled triangles: sine rule, cosine rule and area rule Solve problems involving more than one triangle using trigonometry Outcomes MS5.3. (p 141): Applies trigonometric relationships, sine rule, cosine rule and area rule in problem solving Working Mathematically Students learn to solve problems using eact trigonometric ratios for 30º, 45º and 60º (Applying Strategies) solve problems, including practical problems, involving the sine and cosine rules and the area rule eg problems related to surveying or orienteering (Applying Strategies) use appropriate trigonometric ratios and formulae to solve two-dimensional trigonometric problems that require the use of more than one triangle, where the diagram is provided, and where a verbal description is given (Applying Strategies) recognise that if given two sides and an angle (not included) then two triangles may result, leading to two solutions when the sine rule is applied (Reasoning, Reflecting, Applying Strategies, Reasoning) eplain what happens if the sine, cosine and area rules are applied in right-angled triangles (Reasoning) ask questions about how trigonometric ratios change as the angle increases from 0 to 180 (Questioning) recognise that if sin A 0 then there are two possible values for A, given 0º A 180º (Applying Strategies, Reasoning) find the angle of inclination, θ, of a line in the coordinate plane by establishing and using the relationship gradient = tan θ (Reasoning, Reflecting) Published by Macmillan Education Australia. Macmillan Education Australia 006.

24 Mathscape 10 Teaching Program Page 4 Knowledge and Skills Students learn about Further Trigonometry with Right-Angled Triangles proving and using the relationship between the sine and cosine ratios of complementary angles in right-angled triangles ( 90 ) ( 90 ) cos A = sin A sin A = cos A proving that the tangent ratio can be epressed as a ratio of the sine and cosine ratios sinθ tan θ = cosθ determining and using eact sine, cosine and tangent ratios for angles of 30, 45, and 60 The Trigonometric Ratios of Obtuse Angles establishing and using the following relationships for obtuse angles, where A : 0 90 ( 180 A) sin A ( 180 A) = cos A ( 180 A) = tan A sin = cos tan drawing the sine and cosine curves for at least 0 A 180 finding the possible acute and/or obtuse angles, given a trigonometric ratio The Sine and Cosine Rules and the Area Rule proving the sine rule: In a given triangle ABC, the ratio of a side to the sine of the opposite angle is a constant. a b c = = sin A sin B sin C using the sine rule to find unknown sides and angles of a triangle, including in problems in which there are two possible solutions for an angle proving the cosine rule: In a given triangle ABC a = b + c bccos A Teaching, Learning and Assessment TRY THIS: Trigonometric Curves (p393): Investigation Double Trouble (p400): Problem Solving Eact Length (p409): Problem Solving Eact area of a segment (p418): Etension Chord and Radius (p45): Etension FOCUS ON WORKING MATHEMATICALLY: (p46) An investigation of the cosine rule EXTENSION ACTIVITIES, LET S COMMUNICATE, REFLECTING (p49) CHAPTER REVIEW (p430): A collection of problems to revise the chapter. Published by Macmillan Education Australia. Macmillan Education Australia 006.

25 Mathscape 10 Teaching Program Page 5 b + c a cos A = bc using the cosine rule to find unknown sides and angles of a triangle proving and using the area rule to find the area of a triangle: In a given triangle ABC 1 Area = absin C drawing diagrams and using them to solve word problems that involve non-rightangled triangles Technology Published by Macmillan Education Australia. Macmillan Education Australia 006.

26 Mathscape 10 Teaching Program Page 6 Chapter 1. Further coordinate geometry Substrand Co-ordinate Geometry Duration 3 weeks / 1 hours Tet References Mathscape 10: Ch 1: Coordinate Geometry (pages ) CD Reference Key Ideas Use a diagram to determine midpoint, length and gradient of an interval joining two points on the number plane. Graph linear and simple non-linear relationships from equations Use midpoint, distance and gradient formulae. Apply the gradient/intercept form to interpret and graph straight lines. Outcomes PAS5.1. (p 97): Determines the midpoint, length and gradient of an interval joining two points on the number plane and graphs linear and simple non-linear relationships from equations. PAS5..3 (p 99): Uses formulae to find midpoint, distance and gradient and applies the gradient/intercept form to interpret and graph straight lines. Working Mathematically Students learn to describe the meaning of the midpoint of an interval and how it can be found (Communicating) describe how the length of an interval joining two points can be calculated using Pythagoras theorem (Communicating, Reasoning) eplain the meaning of gradient and how it can be found for a line joining two points (Communicating, Applying Strategies) distinguish between positive and negative gradients from a graph (Communicating) relate the concept of gradient to the tangent ratio in trigonometry for lines with positive gradients (Reflecting) eplain the meaning of each of the pronumerals in the formulae for midpoint, distance and gradient (Communicating) use the appropriate formulae to solve problems on the number plane (Applying Strategies) use gradient and distance formulae to determine the type of triangle three points will form or the type of quadrilateral four points will form and justify the answer (Applying Strategies, Reasoning) eplain why the following formulae give the same solutions as those in the left-hand column (Reasoning, Communicating) 1 ) + ( y1 ) d = ( y and y1 y m = 1 Published by Macmillan Education Australia. Macmillan Education Australia 006.

27 Mathscape 10 Teaching Program Page 7 determine the difference in equations of lines that have a negative gradient and those that have a positive gradient (Reasoning) match equations of straight lines to graphs of straight lines and justify choices (Communicating, Reasoning) compare similarities and differences between sets of linear relationships eg y = 3, y = 3 +, y = 3 1 y =, y =, y = 3 =, y = (Reasoning) sort and classify a mied set of equations of linear relationships into groups to demonstrate similarities and differences (Reasoning, Communicating) conjecture whether a particular equation will have a similar graph to another equation and graph both lines to test the conjecture (Questioning, Applying Strategies, Reasoning) eplain the effect on the graph of a line of changing the gradient or y -intercept (Reasoning, Communicating) use a graphics calculator and spreadsheet software to graph a variety of equations of straight lines, and compare and describe the similarities and differences between the lines (Applying Strategies, Communicating, Reasoning) apply knowledge and skills of linear relationships to practical problems (Applying Strategies) apply ethical considerations when using hardware and software (Reflecting) Knowledge and Skills Students learn about Midpoint, Length and Gradient determining the midpoint of an interval from a diagram graphing two points to form an interval on the number plane and forming a rightangled triangle by drawing a vertical side from the higher point and a horizontal side from the lower point using 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 using the right-angled triangle drawn between two points on the number plane and the relationship rise gradient = run to find the gradient of the interval joining two points determining whether a line has a positive or negative slope by following the line Teaching, Learning and Assessment TRY THIS: 3 Minute mile? (p445): Problem Solving Car hire (p451): Problem Solving Temperature Rising (p454): Problem Solving Money Inequality (p458): Problem Solving FOCUS ON WORKING MATHEMATICALLY: (p468) The tower of terror EXTENSION ACTIVITIES, LET S COMMUNICATE, REFLECTING (p469) CHAPTER REVIEW (p471): A collection of problems to revise the chapter. Published by Macmillan Education Australia. Macmillan Education Australia 006.

28 Mathscape 10 Teaching Program Page 8 from left to right if the line goes up it has a positive slope and if it goes down it has a negative slope finding the gradient of a straight line from the graph by drawing a right-angled triangle after joining two points on the line Graphs of Relationships constructing tables of values and using coordinates to graph vertical and horizontal lines such as = 3, = 1 y =, y = 3 identifying the - and y -intercepts of graphs identifying the -ais as the line y = 0 identifying the y -ais as the line = 0 graphing a variety of linear relationships on the number plane by constructing a table of values and plotting coordinates using an appropriate scale eg graph the following: y = y = + y = 5 y = y = 3 graphing simple non-linear relationships eg y = y = + y = determining whether a point lies on a line by substituting into the equation of the line Midpoint, Distance and Gradient Formulae using the average concept to establish the formula for the midpoint, M, of the interval joining two points ( 1, y 1 ) and (, y ) on the number plane Published by Macmillan Education Australia. Macmillan Education Australia 006.

29 Mathscape 10 Teaching Program Page y1 + y M (, y) =, using the formula to find the midpoint of the interval joining two points on the number plane using Pythagoras theorem to establish the formula for the distance, d, between two points ( 1, y 1 ) and (, y ) on the number plane d = 1) + ( y 1) ( y using the formula to find the distance between two points on the number plane using the relationship rise gradient = run to establish the formula for the gradient, m, of an interval joining two points ( 1, y 1 ) and (, y ) on the number plane y m = using the formula to find the gradient of an interval joining two points on the number plane Gradient/Intercept Form constructing tables of values and using coordinates to graph straight lines of the form y = m + b ie gradient/intercept form recognising equations of the form y = m + b as representing straight lines and interpreting the -coefficient (m) as the gradient and the constant (b) as the y - intercept rearranging an equation in general form (a + by + c = 0) to the gradient/intercept form graphing equations of the form y = m + b using the y -intercept (b) and the gradient (m) determining that two lines are parallel if their gradients are equal finding the gradient and the y -intercept of a straight line from the graph and using them to determine the equation of the line y 1 1 Published by Macmillan Education Australia. Macmillan Education Australia 006.

30 Mathscape 10 Teaching Program Page 30 Technology Published by Macmillan Education Australia. Macmillan Education Australia 006.