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Stage 5.3 Quadratic Equations PA 5.3.2 Outcome: Solves quadratic equations and simultaneous equations, solves and graphs inequalities, and rearranges literal equations 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 Background Information Language binomial product completing the square difference of two squares expanded form factors highest common factor perfect square quadratic trinomial Quakers Hill High School Year 10 (Stage 5.3) Mathematics Program 1 Knowledge and Skills Strategies Resources Working Mathematically Students learn about Students learn to • developing the quadratic formula a) Review Year 9 Algebra - algebraic • New Signpost Mathematics 10 Stage 5.3 – • choose the most appropriate method to expressions, indices, equations, inequations, page 7, 14, 17, 23 solve a particular quadratic equation ! b ± b 2 ! 4ac simultaneous equations, binomial products x= (Applying Strategies) 2a and identities, factorising by grouping in • New Signpost Mathematics 10 Stage 5.3 - • solve quadratic equations and discuss the • solving equations of the form pairs, using the difference of two squares Chapter 2 and 10 possible number of roots for any ax 2 + bx + c = 0 using: and quadratic trinomials quadratic equation (Applying Strategies, - factors Communicating) - completing the square b) solution of quadratics by factorising • determine quadratic expressions to - the quadratic formula describe particular number patterns such c) solution of quadratics by completing the • solving a variety of quadratic equations square as y = x 2 + 1 for the table such as 3x 2 = 4 d) solution of quadratics by quadratic 0 1 2 3 4 5 2 x ! 8x ! 4 = 0 formula x( x ! 4) = 4 1 2 5 10 17 26 2 e) mixed examples and choosing the best (Applying Strategies, Communicating) ( y ! 2) = 9 method to solve quadratic equations • identifying whether a given quadratic • graph simultaneous equations to find equation has no solution, one solution or f) application of solving quadratics to solutions and compare this method with two solutions problem solving analytic methods (Applying Strategies, Reflecting, • checking the solutions of quadratic g) simultaneous equations involving a Communicating equations • generating quadratic equations from quadratic problems h) Literal equations • solving problems involving quadratic equations i) Understanding variables – variable • solving quadratic equations resulting substitution, factorising using a change of from substitution into formulae. variable, expanding using a change of variable Quakers Hill High School Year 10 (Stage 5.3) Mathematics Program 2 Knowledge and Skills Strategies Resources Working Mathematically • using analytical methods to solve a variety of simultaneous equations, including those that involve a first degree equation and a second degree equation, such as 3 x ! 4 y = 2 and 2 x + y = 3 y = x2 and y = x y = x2 – x – 2 and y = x + 6 Literal Equations • changing the subject of a formula, using examples from other strands and other subjects eg make r the subject of 1 = 1 + 1 , x r s make b the subject of x = b 2 ! 4ac • determining restrictions on the values of variables implicit in the original formula and after rearrangement of the formula eg consider what restrictions there would be on the variables in the equation Z = ax 2 and what additional restrictions are assumed if the equation is rearranged to x= Z a Quakers Hill High School Year 10 (Stage 5.3) Mathematics Program 3 Knowledge and Skills Strategies Resources Working Mathematically Understanding Variables • replacing variables with other expressions eg find an expression for x2 + 4 if x = 2at • using variable substitution to simplify expressions and equations so that specific cases can be seen to belong to general categories eg substitute u for x 2 to solve x 4 ! 13x 2 + 36 = 0 • interpreting expressions and equations given additional information ASSESSMENT SUGGESTIONS Test Class test Group Work Assignment Written - which includes as many concepts as possible. Quakers Hill High School Year 10 (Stage 5.3) Mathematics Program 4 Stage 5.3 Probability NS 5.1.3 and 5.3.2 Outcome: Determines relative frequencies and theoretical probabilities Solves probability problems involving compound events Key Ideas Determine relative frequencies to estimate probabilities Determine theoretical probabilities Solve probability problems including two-stage and compound events Background Information This topic links with relative frequency in the Data strand. Software programs could be used for simulation experiments to demonstrate that the relative frequency gets closer and closer to the theoretical probability as the number of trials increases. Venn diagrams may be used as a tool in solving some probability problems. The mathematical analysis of probability was prompted by the French gentleman gambler, the Chevalier de Meré. Over the years, the Chevalier had consistently won money betting on at least one six in four rolls of a die. He felt that he should also win betting on at least one double six in 24 rolls of two dice, but in fact regularly lost. In 1654 he asked his mathematician friend Pascal to explain why. This question led to a famous correspondence between Pascal and the renowned mathematician Fermat. The Chevalier’s change of fortune is explained by the fact that the chance of at least one six in four rolls of a die is 51.8%, while the chance of at least one double six in 24 rolls of two dice is 49.1%. Language Complementary events experimental probability mutually exclusive outcome Probability random Sample simulation survey theoretical complement Independent tree diagram union universal set Venn diagram Quakers Hill High School Year 10 (Stage 5.3) Mathematics Program 5 Knowledge and Skills Strategies Resources Working Mathematically Students learn about Students learn to a) Review year 8 probability – theoretical New Signpost Mathematics 10 Stage 5.3 – • repeating an experiment a number of • recognise and explain differences probability, complementary events page 11 times to determine the relative between relative frequency and n(E) • New Signpost Mathematics 9 Stage 5.3 - frequency of an event P(E) = theoretical probability in a simple n(S) Chapter 4 • estimating the probability of an event experiment (Communicating, from experimental data using relative Reasoning) b) experimental probabilities, relative • New Signpost Mathematics 10 Stage 5.3 - frequencies Chapter 3 • apply relative frequency to predict future frequency • expressing the probability of an event A ! experimental outcomes (Applying given a finite number of equally likely •spreadsheet simulation of dice throwing or Strategies) c) organising outcomes of compound outcomes as events – tables and tree diagrams. other computer game / simulation. BASIC • design a device to produce a specified number of favourable outcomes program relative frequency eg a four-coloured P( A) = circular spinner n d) dependent and independent events • random number function on calculator. (Applying Strategies) where n is the total number of outcomes e) probability using tree and dot diagrams • recognise that probability estimates in the sample space become more stable as the number of • using the formula to calculate f) probability using tables and Venn trials increases (Reasoning) probabilities for simple events diagrams • recognise randomness in chance • simulating probability experiments situations (Communicating) using random number generators. g) simulation experiments • apply the formula for calculating • distinguishing informally between probabilities to problems related to dependent and independent events card, dice and other games (Applying • sampling with and without replacement in Strategies) two-stage experiments eg drawing two • critically evaluate statements on chance counters from a bag containing 3 blue, and probability appearing in the media 4 red and 1 white counter and/or in other subjects (Reasoning) • analysing two-stage events through • evaluate the likelihood of winning a prize constructing organised lists, tables in lotteries and other competitions and/or tree diagrams (Applying Strategies, Reasoning) • solving two-stage probability problems • evaluate games for fairness including instances of sampling with (Applying Strategies, Reasoning) and without replacement Quakers Hill High School Year 10 (Stage 5.3) Mathematics Program 6 Knowledge and Skills Strategies Resources Working Mathematically • finding probabilities of compound events • identify common misconceptions related using organised lists, tables or diagrams to chance events eg if you get four tails eg the table below represents data in a row when tossing a coin, there is a collected on 300 athletes and compares greater chance that the next outcome is height with weight – what is the a head (Applying Strategies) probability of choosing a light, short • recognise the use of probability by athlete from the population represented governments and companies eg in in the table? demography, insurance, planning for Heavy Light roads (Reflecting) Tall 70 20 90 Short 50 160 210 120 180 300 ASSESSMENT SUGGESTIONS Test Class test Group Work Assignment Written - which includes as many concepts as possible. Quakers Hill High School Year 10 (Stage 5.3) Mathematics Program 7 Stage 5.3 Consumer Arithmetic NS 5.1.2 and 5.2.2 Outcome: Solves consumer arithmetic problems involving compound interest, depreciation, and successive discounts Key Ideas Use compound interest formula Solve consumer arithmetic problems involving compound interest, depreciation and successive discounts Background Information Internet sites may be used to find commercial rates for home loans and ‘home loan calculators’ The work in this topic links with Commerce Language Income salary wage gross net allowance bonus Commission consumer deduction discounts earnings employer overtime percentage piecework retainer taxation superannuation depreciation reducible Quakers Hill High School Year 10 (Stage 5.3) Mathematics Program 8 Knowledge and Skills Strategies Resources Working Mathematically Students learn about Students learn to a) Year 9 review - saving money New Signpost Mathematics 10 Stage 5.3 - • calculating simple interest using the • compare simple interest with compound Chapter 4 formula interest in practical situations eg loans b) Simple interest formula and problems r (Applying Strategies) I = PRT where R = involving formula • use a prepared spreadsheet or table to 100 chart a superannuation policy over years of where I is the interest, P the principal, R c) Compound interest – reading a table contributions given different rates of • interpret spreadsheets or tables when the annual interest rate and T the number and using repeatd multiplication on interest, reducible home loan spreadsheet. comparing simple interest and of years calculator or repitition of simple interest compound interest on an investment • applying the simple interest formula to formula. Us of compound interest formula. • use a spreadsheet or tables of compound over various time periods problems related to investing money at interest to compare interest, eg compare the (Applying Strategies, Communicating) simple interest rates d) Depreciation including formula interest earned on $10 000 invested at 8% simple interest per annum for three years to • realise the total cost and/or hidden costs • calculating compound interest for two or e) Reducible interest $10 000 invested at 8% per annum involved in some types of purchase three years by repeated multiplication compounded annually over the same time arrangements (Applying Strategies) using a calculator eg a rate of 5% per f) Borrowing money – reducible interest period. annum leads to repeated multiplication and flat rate interest. by 1.05 • calculating compound interest on g) Home loans • given an imaginary sum of money investments using a table. ($5000), purchase shares in two or three • calculating compound interest on h) Buying on terms – calculating the cost companies, estimate the gains to be had investments and loans using repetition of buying a car over two months, chart the actual profit/loss of the formula for simple interest obtained, compare to the profit/loss which • determining and using the formula for could have been made if other shares had compound interest, A = P ( + R ) n , 1 been bought. Set the information up on a spreadsheet and use chart to graph and where A is the total amount, P is the predict gains or losses over a 1 year period. 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 • calculating the result of successive discounts Quakers Hill High School Year 10 (Stage 5.3) Mathematics Program 9 Knowledge and Skills Strategies Resources Working Mathematically • comparing the cost of loans using flat and reducible interest for a small number of repayment periods ASSESSMENT SUGGESTIONS Test Class test Group Work Looking through newspapers to find and compare jobs and salaries Assignment Written – look in resources section and set as a written assignment Assignment Multi media presentation of concepts learnt Quakers Hill High School Year 10 (Stage 5.3) Mathematics Program 10 Stage 5.3 Number Plane Graphs and Coordinate Geometry PA 5.3.4 Outcome: Draws and interprets graphs including simple parabolas and hyperbolas Draws and interprets a variety of graphs including parabolas, cubics, exponentials and circles and applies coordinate geometry techniques to solve problems Key Ideas Draw and interpret graphs including simple parabolas and hyperbolas Draw and interpret a variety of graphs including parabolas, cubics, exponentials and circles Solve coordinate geometry problems Background Information Graphics calculators and various computer programs facilitate the investigation of the shapes of curves and the effect on the equation of multiplying by, or adding, a constant. This topic could provide opportunities for modelling. 1 For example, the hyperbola y = for x > 0 , models sharing a prize between x people, or length of a rectangle given area k and breadth x. x Links to other key learning areas and real life examples of graphs eg exponential graphs used for population growth in demographics, radioactive decay, town planning, etc. This topic could provide opportunities for modelling. For example, y = 1.2 x for x ! 0 , models the growth of a quantity beginning at 1 and increasing 20% for each unit increase in x. Links to other key learning areas and real life examples of graphs eg exponential graphs used for population growth in demographics, radioactive decay, town planning, etc. This topic could provide opportunities for modelling. x For example, y = 1.2 for x ! 0 , models the growth of a quantity beginning at 1 and increasing 20% for each unit increase in x. Language Parabola circle cubic exponential equation graph hyperbola symmetry ertex concavity concave up concave down maximum minimum asymptote intercepts axis of symmetry Quakers Hill High School Year 10 (Stage 5.3) Mathematics Program 11 Knowledge and Skills Strategies Resources Working Mathematically Students learn about Students learn to • generating simple quadratic relationships, a) Review Year 9 coordinate geometry -– • New Signpost Mathematics 10 Stage 5.3 • identify parabolic shapes in the compiling tables of values and graphing review distance formula, midpoint formula, – page 20 environment (Reflecting) equations of the form gradient of a straight line, lines parallel to • New Signpost Mathematics 10 Stage 5.3 the axes. – Chapter 5 • describe the effect on the graph of y = x 2 y = ax2 and y = ax2 + c of multiplying by different constants or of • generating simple hyperbolic b) parabola – plotting points with table of adding different constants (Reasoning, relationships, compiling tables of values values add/ subtract a constant to see effect, • Use of Graphing software or graphics Communicating) and graphing equations of the form positive and negative coefficient of x, calculator to draw curves. • discuss and predict the equation of a k graphs of (x + a) 2 parabola from its graph, with the main y= x features clearly marked, using computer for integral values of k c) Parabolas of the for y = ax2 + bx + c graphing software (Communicating) • identifying graphs of straight lines, - finding x and y intercept 1 "b • describe the effect on the graph of y = parabolas and hyperbolas - axis of symmetry x = x • matching graphs of straight lines, 2a of multiplying by different constants parabolas and hyperbolas to the - finding vertex (Reasoning, Communicating) appropriate equations. - finding max or min value • explain what happens to the y -values of ! k k the points on the hyperbola y = as the x Graphs d) hyperbola general form y = and y= x x • identifying a variety of graphs from their k -values get very large (Reasoning, equations - plotting points, asymptotes. Communicating) x • finding x - and y -intercepts for the graph • explain what happens to the y -values of of ! e) exponential graphs y = ax k the points on the hyperbola y = as the x y = ax2 + bx + c given a, b and c plotting points and the effect of the x ! constant -values get closer to zero (Reasoning, Communicating) • graphing a range of parabolas, including where the equation is given in the form f) circle x2 + y 2 = r 2 centre (0, 0 ) radius r • sort and classify a set of graphs, match each graph to an equation, and justify y = ax 2 + bx + c g) cubic equations of the form: each choice for various values of a, b and c y = ax3 + c (Reasoning, Communicating) plotting points to see effect of the constants. • explain why it may be useful to include small and large numbers when Quakers Hill High School Year 10 (Stage 5.3) Mathematics Program 12 Knowledge and Skills Strategies Resources Working Mathematically constructing a table of values • finding the equation of the axis of 1 symmetry of a parabola by: h) miscellaneous graphs – review equation eg ‘For y = , why do we need to use x of straight line – match equations to graphs more than the integers 1, 2, 3, and 4 for - finding the midpoint of the interval x?’ joining the points at which the parabola cuts (Reasoning, Communicating) the x -axis • use a graphics calculator and spreadsheet - using the formula software to graph, compare and describe a b range of linear and non-linear x=! 2a relationships • finding the coordinates of the vertex of a (Applying Strategies, Communicating) parabola by: - finding the midpoint of the interval joining the points at which the • describe the graph of a parabola from its parabola cuts the x -axis and substituting equation (Communicating) - completing the square • investigate and describe similarities and - using the formula for the axis of differences between the graphs of a symmetry to obtain variety of parabolas such as the x -coordinate and substituting to y = x2 obtain the y = x2 ± 2 y -coordinate y = ( x ! 2) 2 • identifying and using features of y = ( x + 2)( x ! 2) parabolas and their equations to assist in sketching quadratic relationships eg x - (Questioning, Applying Strategies, and y -intercepts, vertex, axis of Reasoning, Communicating) symmetry and concavity • investigate the graphs of parabolas of the • graphing equations of the form following forms to determine features y = ax 3 + d and describing the effect on y = ax 2 the graph of different values of a and d y = ax 2 + k y = ( x ± a) 2 y = ( x ± a) 2 + k (Applying Strategies) • sort and classify a set of equations, match Quakers Hill High School Year 10 (Stage 5.3) Mathematics Program 13 Knowledge and Skills Strategies Resources Working Mathematically each equation to a graph, and justify each • sketching, comparing and describing choice the key features of simple exponential (Applying Strategies, Reasoning) curves such as • discuss and predict a possible equation y = 2x from a given graph and check using y = !2 x technology (Applying Strategies, Reasoning) y = 2!x • draw and compare graphs using a y = !2 ! x graphics calculator and/or a computer graphing package • recognising and describing the (Applying Strategies) algebraic equations that represent circles • compare and contrast a mixed set of with centre the origin and radius r graphs and determine possible equations • using Pythagoras’ theorem to establish from key features the equation of a circle, centre the origin, radius r and graph equations of the form eg x2 + y2 = r 2 y=2 y = 2! x • solving a variety of problems by applying coordinate geometry formulae and y = 2x2 reasoning y = ( x ! 2) 2 y = x3 ! 2 y = 2x 2 x + y2 = 4 (Applying Strategies, Reasoning, Communicating) • determine whether a particular point is inside, on, or outside a circle (Applying Strategies, Reasoning) • derive the formula for the distance between two points (Applying Strategies, Quakers Hill High School Year 10 (Stage 5.3) Mathematics Program 14 Knowledge and Skills Strategies Resources Working Mathematically Reasoning) • show that two intervals with equal gradients and a common point form a straight line and use this to show that three points are collinear (Applying Strategies, Reasoning) • use coordinate geometry to investigate and describe the properties of triangles and quadrilaterals (Applying Strategies, Reasoning, Communicating) • use coordinate geometry to investigate the intersection of the perpendicular bisectors of the sides of acute-angled triangles (Applying Strategies, Reasoning, Communicating) • show that four specified points form the vertices of particular quadrilaterals (Applying Strategies, Reasoning, Communicating) Quakers Hill High School Year 10 (Stage 5.3) Mathematics Program 15 Stage 5.3 Surface Area and Volume MS 5.3.1 Outcome: Applies formulae to find the surface area of pyramids, right cones and spheres Key Ideas Apply formula for surface area of pyramids, right cones and spheres. Background Information Pythagoras’ theorem is applied here to right-angled triangles in three-dimensional space. The work here requires a sound knowledge of polyhedra covered in the Space and Geometry strand. The results concerning ratios of matching areas and volumes in similar three-dimensional figures are linked with work on similar two- dimensional figures in the Space and Geometry strand (see page 156). There is also a link with PDHPE issues such as why babies dehydrate so quickly and why mice eat so much. The focus in this section is on right cones and right pyramids. Dealing with the oblique version of these objects is difficult and is mentioned only as a possible extension. The area of the curved surface of a hemisphere is 2πr2 which is twice the area of its base. This may be a way of making the formula for the surface area of a sphere look reasonable to students. Deriving the relationship between the surface area and the volume of a sphere by dissection into infinitesimal pyramids may be an extension activity for some students. Similarly, some students may investigate as an extension, the surface area of a sphere by projection of infinitesimal squares onto a circumscribed cylinder. Language Composite solid prism pyramid slant height surface area volume Quakers Hill High School Year 10 (Stage 5.3) Mathematics Program 16 Knowledge and Skills Strategies Resources Working Mathematically Students learn about Students learn to Volume of Right Pyramids, Cones and a) Review year 9 measurement - formula • New Signpost Mathematics 10 Stage 5.3 • apply Pythagoras’ theorem to problems Spheres for area of shapes. – page 16 involving surface area (Applying • using the fact that a pyramid has one-third • New Signpost Mathematics 10 Stage 5.3 Strategies) the volume of a prism with the same b) review surface area of prisms including – chapter 6 base and the same perpendicular height cylinder • solve problems involving the surface area • using the fact that a cone has one-third and volume of solids (Applying the volume of a cylinder with the same c) Surface area of a pyramid Strategies) base and the same perpendicular height 1 d) Surface area of a cone • find surface area of composite solids eg a • using the formula V = Ah to find the cylinder with a hemisphere on top 3 e) Surface area of a sphere (Applying Strategies) volume of pyramids and cones where A is the base area and h is the perpendicular height f) Volumes of a pyramid 4 • using the formula V = !r 3 to find the g) Volume of a cone 3 volume of spheres where r is the length h) Volume of a sphere of the radius • finding the dimensions of solids given i) Practical applications of volume and their volume and/or surface area by surface area substitution into a formula to generate an equation i) Practical problems involving volume • finding the volume of prisms whose bases j) Take the ratio of intervals, surface area can be dissected into triangles, special and volumes to determine the ratios of quadrilaterals and sectors similar solids. (see knowledge and skills) • finding the volume of composite solids. Quakers Hill High School Year 10 (Stage 5.3) Mathematics Program 17 Knowledge and Skills Strategies Resources Working Mathematically Surface Area of Pyramids, Right Cones and Spheres • identifying the perpendicular and slant height of pyramids and right cones • using Pythagoras’ theorem to find slant height, base length or perpendicular height of pyramids and right cones • devising and using methods to calculate the surface area of pyramids • developing and using the formula to calculate the surface area of cones Curved surface area of a cone = !rl where r is the length of the radius and l is the slant height • using the formula to calculate the surface area of spheres Surface area of a sphere = 4!r 2 where r is the length of the radius • finding the dimensions of solids given their surface area by substitution into a formula to generate an equation. ASSESSMENT SUGGESTIONS Test Class test Group Work Looking through newspapers to find and compare jobs and salaries Assignment Multi media presentation of concepts learnt Quakers Hill High School Year 10 (Stage 5.3) Mathematics Program 18 Stage 5.3 Statistics DS 5.2.1 Outcome: Uses the interquartile range and standard deviation to analyse data Key Ideas Determine the upper and lower quartiles of a set of scores Construct and interpret box-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 Background Information Many school subjects make use of graphs and data eg in PDHPE students might review published statistics on road accidents, drownings etc. In Stage 4 Design and Technology, students are required, in relation to marketing, to ‘collect information about the needs of consumers in relation to each Design Project’. The group investigation could relate to aspects of the PDHPE syllabus eg ‘appraise the values and attitudes of society in relation to lifestyle and health’. In Geography, range is used when discussing aspects such as temperature and is given by stating the maximum and minimum values. This is different to the use of ‘range’ in mathematics where the difference is calculated for the range. In Geography, use is made of a computer database of local census data. Also, students collect information about global climatic change, greenhouse gas emission, ozone depletion, acid rain, waste management and carbon emissions. In Science, students carry out investigations to test or research a problem or hypothesis; they collect, record and analyse data and identify trends, patterns and relationships. Many opportunities occur in this topic to implement aspects of the Key Competencies (see Cross-curriculum Content): - collecting, analysing and organising information - communicating ideas and information - planning and organising activities - working with others and in teams - using mathematical ideas and techniques - solving problems, and - using technology. Language box-and-whisker-plot cumulative frequency histogram dot plot five-point summary frequency histogram interquartile range lower quartile maximum value measures of spread median minimum value range outliers upper quartile standard deviation skew Quakers Hill High School Year 10 (Stage 5.3) Mathematics Program 19 Knowledge and Skills Strategies Resources Working Mathematically Students learn about Students learn to • compare two or more sets of data using • determining the upper and lower a) review range median, mode, mean from • New Signpost Mathematics 10 Stage 5.3 box-and-whisker plots drawn on the quartiles for a set of scores a list of numbers, table, stem and leaf plot – page 22 same scale – include fx column and cf column. (Applying Strategies) • constructing a box-and-whisker plot using the median, the upper and lower • New Signpost Mathematics 10 Stage 5.3 • compare data with the same mean and quartiles and the extreme values (the b) measures of spread – chapter 7 different standard deviations (Applying ‘five-point summary’) - interquartile range and use of Strategies) cumulative frequency diagram. • Use of spreadsheet functions to find • finding the standard deviation of a set of - box and whisker plots to find measures of spread. • compare two sets of data and choose an scores using a calculator quartiles appropriate way to display these, using • using the mean and standard deviation to back-to-back stem-and-leaf plots, - standard deviation compare two sets of data histograms, double column graphs, or • comparing the relative merits of c) comparing sets of data looking at shapes box-and-whisker plots (Communicating, measures of spread: of distribution Applying Strategies) range • analyse collected data to identify any interquartile range obvious errors and justify the inclusion standard deviation. of any scores that differ remarkably from • using the terms ‘skewed’ or the rest of the data collected ‘symmetrical’ when describing the shape (Applying Strategies, Reasoning) of a distribution • 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 Quakers Hill High School Year 10 (Stage 5.3) Mathematics Program 20 ASSESSMENT SUGGESTIONS Test Class test Group Work Collection and analysis of data Assignment Written - which includes as many concepts as possible. Quakers Hill High School Year 10 (Stage 5.3) Mathematics Program 21 Stage 5.3 Similarity SGS 5.2.2 Outcome: Develops and applies results for proving that triangles are congruent or similar. Constructs geometrical arguments using similarity tests for triangles Key Ideas Identify similar triangles and describe their properties Use simple deductive reasoning in numerical and non-numerical problems Background Information The definitions of the trigonometric ratios depend upon the similarity of triangles eg any two right-angled triangles in which another angle is 30º must be similar Students are expected to give reasons to justify their results. For some students formal setting out could be introduced. For students proceeding to Stage 5.3 outcomes, this material could be combined with the more formal Euclidean approach in SGS5.3.1 and SGS5.3.2 Language similarity scale drawing matching angles matching sides scale factor similar figures similar triangles superimpose reduction factor ratio Quakers Hill High School Year 10 (Stage 5.3) Mathematics Program 22 Knowledge and Skills Strategies Resources Working Mathematically Students learn about Students learn to • identifying the elements preserved in a) definition of similarity • New Signpost Mathematics 10 Stage 5.3 – • apply the properties of similar triangles to similar triangles, namely angle size and chapter 8 solve problems, justifying the results the ratio of corresponding sides b) similar triangles – matching angles, ratio (Applying Strategies, Reasoning) • determining whether triangles are similar of matching sides • apply simple deductive reasoning in • applying the enlargement or reduction solving numerical and non-numerical factor to find unknown sides in similar c) using the scale factor to find the problems (Applying Strategies, triangles unknown sides. Reasoning) • calculating unknown sides in a pair of • explain why any two equilateral triangles, similar triangle or any two squares, are similar, • determining what information is needed d) similar triangle proofs (Communicating, Reasoning) to establish that two triangles are similar • investigate whether any two rectangles, or If the three sides of one triangle are e) ratio of sides and areas of similar figures any two isosceles triangles, are similar proportional to the three sides of (Applying Strategies, Reasoning) another triangle, then the two f) similar solids and ratio of areas on faces • use dynamic geometry software to triangles are similar. and volumes investigate the properties of geometrical figures (Applying Strategies, Reasoning) If two sides of one triangle are • prove statements about geometrical proportional to two sides of another figures (Reasoning, Communicating, triangle, and the included angles are Applying Strategies) equal, then the two triangles are • solve problems using deductive reasoning similar. (Reasoning, Applying Strategies) If two angles of one triangle are • make, refine and test conjectures respectively equal to two angles of (Questioning, Communicating, Applying another triangle, then the two Strategies, Reasoning) triangles are similar. • state possible converses of known results, If the hypotenuse and a second side of a and examine whether or not they are also right-angled triangle are proportional true (Communicating, Applying to the hypotenuse and a second side of Strategies, Reasoning) another right-angled triangle, then the two triangles are similar. • use dynamic geometry software to Quakers Hill High School Year 10 (Stage 5.3) Mathematics Program 23 Knowledge and Skills Strategies Resources Working Mathematically • writing formal proofs of similarity of investigate and test conjectures about triangles in the standard four- or five-line geometrical figures format, preserving the matching order of (Applying Strategies, Reasoning) vertices, identifying the similarity factor when appropriate, and drawing relevant conclusions from this similarity • proving and applying further theorems using similarity, in particular The interval joining the midpoints of two sides of a triangle is parallel to the third side and half its length. Conversely, the line through the midpoint of a side of a triangle parallel to another side bisects the third side. ASSESSMENT SUGGESTIONS Test Class test Group Work Assignment Written - which includes as many concepts as possible. Quakers Hill High School Year 10 (Stage 5.3) Mathematics Program 24 Stage 5.3 Further Trigonometry MS 5.3.2 Outcome: Applies trigonometric relationships, sine rule, cosine rule and area rule in problem solving Key Ideas Determine the exact 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 Explore trigonometry with non-right-angled triangles: sine rule, cosine rule and area rule Solve problems involving more than one triangle using trigonometry Background Information The origin of the word ‘cosine’ is from ‘complements sine’, so that cos 40° = sin 50°. The tangent ratio can be interpreted as the gradient of a line in the coordinate plane. Students studying circle geometry in the Space and Geometry strand will be able to apply their trigonometry to many problems, making use of the right-angles between a chord and a radius bisecting it, between a tangent and a radius at the point of contact, and in a semicircle. The work with bearings links to orienteering in PDHPE and map work in Stage 5 Geography. Students could have practical experience in using clinometers for finding angles of elevation and depression and in using magnetic compasses for bearings. Students may need encouragement to set out their solutions carefully and to use the correct mathematical language and suitable diagrams. Students need to recognise the 16 points of a mariner’s compass (eg SSW) for comprehension of compass bearings in everyday life eg weather reports. When setting out their solutions related to finding unknown lengths and angles, students should be advised to give a simplified exact answer eg 4 25 sin 42º metres or sin A = 7 , then give an approximation correct to a specified or sensible level of accuracy. The origin of the word ‘cosine’ is from ‘complements sine’, so that cos 40° = sin 50º. The sine and cosine rules and the area rule are closely linked with the standard congruence tests for triangles. These are the most straightforward ways to proceed: Given an SAS situation, use the cosine rule to find the third side. Given an SSS situation, use the alternative form of the cosine rule to find an angle. Given an AAS situation, use the sine rule to find each unknown side. Given an ambiguous ASS situation (the angle non-included), use the sine rule to find the sine of the unknown angle opposite the known side - there may then be two solutions for this angle. Alternatively, use the cosine rule to form a quadratic equation for the unknown side. The cosine rule is a generalisation of Pythagoras’ theorem. The sine rule is linked to the circumcircle and to circle geometry. The definitions of the trigonometric functions in terms of a circle provide the link between Cartesian and polar coordinates. Note that the angle concerned is turned anti-clockwise from the positive x -axis (East). This is not the same as the angle used in navigation (clockwise from North). Quakers Hill High School Year 10 (Stage 5.3) Mathematics Program 25 The formula gradient = tan θ is a formula for gradient in the coordinate plane. Circle geometry and the trigonometric functions are closely linked. First, Pythagoras’ theorem becomes the equation of a circle in the coordinate plane, and such a circle is used to define the trigonometric functions for general angles. Secondly, the sine and cosine rules are closely linked with the circle geometry theorems concerning angles at the centre and circumference and cyclic quadrilaterals. Many formulae relating the sides, diagonals, angles and area of cyclic quadrilaterals are now accessible. The trigonometric functions here could be redefined for the general angle using a circle in the coordinate plane - this allows the sine and cosine functions to be plotted for a full revolution and beyond so that their wave nature becomes clear. The intention, however, of this section is for students to become confident using the sine and cosine rules and area rule in practical situations. For many students it is therefore more appropriate to justify the extension of the trigonometric functions to obtuse angles only, either by plotting the graphs and continuing them in the obvious way, or by taking the identities for 180º – θ as definitions. Whatever is done, experimentation with the calculator should be used to confirm this extension. Language angle of depression angle of elevation bearing cosine rule sine rule trigonometric ratios obtuse Quakers Hill High School Year 10 (Stage 5.3) Mathematics Program 26 Knowledge and Skills Strategies Resources Working Mathematically Students learn about Students learn to Further Trigonometry with Right- • New Signpost Mathematics 10 Stage 5.3 – • solve problems using exact trigonometric a) Show sine and cosine as complements chapter 9 Angled Triangles ratios for 30º, 45º and 60º (Applying and tan is the ratio of sin x and cos x. Strategies) • proving and using the relationship between the sine and cosine ratios of b) Exact values of trig ratios • solve problems, including practical complementary angles in right-angled problems, involving the sine and cosine c) Trig ratios of obtuse angles rules and the area rule eg problems triangles d) Sketching of trig ratios related to surveying or orienteering ( cos A = sin 90 o ! A ) (Applying Strategies) sin A = cos( 90 o ! A) e) Finding the angles given the trig ratio • use appropriate trigonometric ratios and • proving that the tangent ratio can be f) Sine rule formulae to solve two-dimensional expressed as a ratio of the sine and trigonometric problems that require the cosine ratios g) Cosine rule use of more than one triangle, where the sin ! diagram is provided, and where a verbal tan ! = h) Area of any triangle description is given (Applying cos! Strategies) • determining and using exact sine, cosine i) Problems involving more than one • recognise that if given two sides and an and tangent ratios for angles of 30°, triangle angle (not included) then two triangles 45°, and 60° may result, leading to two solutions when the sine rule is applied The Trigonometric Ratios of Obtuse (Reasoning, Reflecting, Applying Angles Strategies, Reasoning) • establishing and using the following • explain what happens if the sine, cosine relationships for obtuse angles, where and area rules are applied in right- 0o ! A ! 90o : angled triangles (Reasoning) ( ) sin 180 o ! A = sin A • ask questions about how trigonometric cos(180 o ! A)= ! cos A ratios change as the angle increases from 0° to 180° (Questioning) tan ( 180 o ! A)= ! tan A • recognise that if sin A ≥ 0 then there are • drawing the sine and cosine curves for at two possible values for A, given 0º ≤ A least 0o ! A ! 180o ≤ 180º • finding the possible acute and/or obtuse (Applying Strategies, Reasoning) angles, given a trigonometric ratio. Quakers Hill High School Year 10 (Stage 5.3) Mathematics Program 27 Knowledge and Skills Strategies Resources Working Mathematically The Sine and Cosine Rules and the Area Rule • find the angle of inclination, θ, of a line • proving the sine rule: In a given triangle in the coordinate plane by establishing ABC, the ratio of a side to the sine of and using the relationship gradient = the opposite angle is a constant. tan θ (Reasoning, Reflecting) • 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 2 = b 2 + c 2 ! 2bc cos A b2 + c2 ! a2 cos A = 2bc • 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 Area = 1 ab sin C 2 • drawing diagrams and using them to solve word problems that involve non- right-angled triangles Quakers Hill High School Year 10 (Stage 5.3) Mathematics Program 28 Stage 5.3 Circle Geometry SG 5.3.4 Outcome: Applies deductive reasoning to prove circle theorems and to solve problems Key Ideas Deduce chord, angle, tangent and secant properties of circles Background Information As well as solving arithmetic and algebraic problems in circle geometry, students should be able to reason deductively within more theoretical arguments. Diagrams would normally be given to students, with the important information labelled on the diagram to aid reasoning. Students would sometimes need to produce a clear diagram from a set of instructions. Attention should be given to the logical sequence of theorems and to the types of arguments used. Memorisation of proofs is not intended. Ideally, every theorem presented should be preceded by a straight-edge-and-compasses construction to confirm it, and then proven, in a manner appropriate to the student’s work level, by way of an exercise or an investigation. The tangent-and-radius-theorem is difficult to justify at this Stage, and is probably better taken as an assumption as indicated above. This topic may be extended to examining the converse of some of the theorems related to cyclic quadrilaterals, leading to a series of conditions for points to be concyclic. However, students may find these results difficult to prove and apply. The angle in a semicircle theorem is also called Thales’ theorem because it was traditionally ascribed to Thales (c 624–548 BC) by the ancient Greeks, who reported that it was the first theorem ever proven in mathematics. The use of dynamic geometry software enables students to investigate chord, angle, tangent and secant properties. Language arc chord circumference diameter radius radii secant sector segment semicircle tangent subtends Quakers Hill High School Year 10 (Stage 5.3) Mathematics Program 29 Knowledge and Skills Strategies Resources Working Mathematically Students learn about Students learn to • identifying and naming parts of a circle a) circle parts and terminology – angle at • New Signpost Mathematics 10 Stage 5.3 – (centre, radius, diameter, the centre, angle at ther circumference, chapter 11 • apply circle theorems to prove that the circumference, sector, arc, chord, angles in the same segment, angles standing angle in a semicircle is a right angle secant, tangent, segment, semicircle) on the same arc, subtends (Applying Strategies, Reasoning) • using terminology associated with angles • apply circle theorems to find unknown in circles such as subtend, standing on b) use of compass to construct circles given angles and sides in diagrams (Applying the same arc, angle at the centre, angle centre and tangent. Strategies, Reasoning) at the circumference, angle in a • find the centre of a circle by construction segment c) Circle properties (see knowledge and (Applying Strategies) • identifying the arc on which an angle at skills) • construct tangents to a circle (Applying the centre or circumference stands - chord properties of circles Strategies) • demonstrating that at any point on a - angle properties of circles • use circle and other theorems to prove circle, there is a unique tangent to the - tangent properties of circles geometrical results and in problem circle, and that this tangent is - further circle properties solving perpendicular to the radius at the point (Applying Strategies, Reasoning) of contact d) deductive exercises involving the circle • using the above result as an assumption when proving theorems involving tangents • proving and applying the following theorems: Chord Properties Chords of equal length in a circle subtend equal angles at the centre and are equidistant from the centre. The perpendicular from the centre of a circle to a chord bisects the chord. Conversely, the line from the centre of a circle to the midpoint of a chord is perpendicular to the chord. The perpendicular bisector of a chord of a circle passes through the centre. Given any three non-collinear points, the point of intersection of the Quakers Hill High School Year 10 (Stage 5.3) Mathematics Program 30 Knowledge and Skills Strategies Resources Working Mathematically perpendicular bisectors of any two sides of the triangle formed by the three points is the centre of the circle through all three points. When two circles intersect, the line joining their centres bisects their common chord at right angles. Angle properties The angle at the centre of a circle is twice the angle at the circumference standing on the same arc. The angle in a semicircle is a right angle. Angles at the circumference, standing on the same arc, are equal. The opposite angles of cyclic quadrilaterals are supplementary. An exterior angle at a vertex of a cyclic quadrilateral is equal to the interior opposite angle. Tangents and secants The two tangents drawn to a circle from an external point are equal in length. The angle between a tangent and a chord drawn to the point of contact is equal to the angle in the alternate segment. When two circles touch, their centres and the point of contact are collinear. The products of the intercepts of two intersecting chords of a circle are equal. Quakers Hill High School Year 10 (Stage 5.3) Mathematics Program 31 Knowledge and Skills Strategies Resources Working Mathematically The products of the intercepts of two intersecting secants to a circle from an external point are equal. The square of a tangent to a circle from an external point equals the product of the intercepts of any secants from the point. ASSESSMENT SUGGESTIONS Test Class test Group Work Assignment Written - which includes as many concepts as possible. Quakers Hill High School Year 10 (Stage 5.3) Mathematics Program 32 Stage 5.3 Polynomials # PAS 5.3.7 Outcome: Recognises, describes and sketches polynomials, and applies the factor and remainder theorems to solve problems Key Ideas Add, subtract, multiply and divide polynomials Apply the factor and remainder theorems Background Information Optional topic Language Constant term continuous degree factor factor theorem leading coefficient Leading term monic polynomial remainder theorem root zero Quakers Hill High School Year 10 (Stage 5.3) Mathematics Program 33 Knowledge and Skills Strategies Resources Working Mathematically Students learn about Students learn to • recognising a polynomial expression a) ) definition of a polynomial • New Signpost Mathematics 10 Stage 5.3 – an x n + an!1 x n!1 + ... + a1 x + a0 chapter 13 • recognise linear, quadratic and cubic and using the terms degree, leading b) sum and difference of polynomials expressions as examples of term, leading coefficient, constant term polynomials and relate sketching of and monic polynomial appropriately c) multiplying and dividing polynomials by these curves to factorising polynomials a linear expression. and finding the zeros (Applying • using the notation P(x) for polynomials Strategies, Reasoning, Reflecting, and P(c) to indicate the value of P(x) Communicating) for x = c d) Remainder and factor theorems • determine the importance of the sign of • adding and subtracting polynomials and e) Solving polynomial equations the leading term of the polynomial on multiplying polynomials by linear the behaviour of the curve as x " ±! expressions f) Sketching polynomials and curves (Applying Strategies, Reasoning, • dividing polynomials by linear related to polynomials Communicating) expressions to find the quotient and • use a graphics calculator or software remainder, expressing the polynomial package to sketch polynomials of odd as the product of the linear expression and even degree and investigate the and another polynomial plus a relationship between the number of remainder zeros and the degree of the polynomial eg P( x) = ( x ! a) Q ( x) + c (Applying Strategies, Communicating) • verifying the remainder theorem and • describe the key features of a polynomial using it to find factors of polynomials so the graph of the polynomial can be • using the factor theorem to factorise drawn from the description (Applying certain polynomials completely Strategies, Communicating) ie if ( x ! a) is a factor of P(x), then P(a) • discuss the similarities and differences =0 between the graphs of two polynomials • using the factor theorem and long such as y = x 3 + x 2 + x, y = x 3 + x 2 + x + 1 division to find all zeros of a simple (Applying Strategies, Reasoning, polynomial and hence solve P( x) = 0 Communicating) (degree ! 4 ) • stating the number of zeros that a polynomial of degree n can have Quakers Hill High School Year 10 (Stage 5.3) Mathematics Program 34 Knowledge and Skills Strategies Resources Working Mathematically • sketching the graphs of quadratic, cubic and quartic polynomials by factorising and finding the zeros • determining the effect of single, double and triple roots of a polynomial equation on the shape of the curve • using the leading term, the roots of the equation and the x - and y -intercepts to sketch polynomials • using the sketch of y = P(x) to sketch y = ! P( x) y = P(! x) y = P( x) + c y = aP( x) Quakers Hill High School Year 10 (Stage 5.3) Mathematics Program 35