Stage 5.3 Quadratic Equations by pyb17727

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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

								
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