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Maths SL Workbook for Category 2 IBDP teacher workshop 10-12 October 2013 Amsterdam, the Netherlands Richard Wade Oliver Bowles

IBDP Maths SL Cat.2 Workbook

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This is the workbook used during the IBDP teacher-training workshop for Philpot Education from 10-12 October 2013 by Richard Wade and Oliver Bowles.

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Page 1: IBDP Maths SL Cat.2 Workbook

Maths SL

Workbook for Category 2 IBDP teacher workshop

10-12 October 2013 Amsterdam, the Netherlands

Richard Wade Oliver Bowles

   

Page 2: IBDP Maths SL Cat.2 Workbook

Author’s note

 Date  of  publication:  1  October  2013    While  the  contents  of  this  workbook  are  not  in  copyright,  the  author  kindly  asks  that  you  respect  his  work.  Rather  than  photocopying  or  redistributing  this  work  digitally,  please  contact  the  author  directly  for  an  updated,  original  version  of  this  workbook.  Furthermore  you  are  kindly  asked  to  give  the  author  credit  when  citing  his  ideas  or  text.      Brad  Philpot    Director  of  Philpot  Education  [email protected]  

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Workshop Leader Agenda Amsterdam International Community School

Amsterdam, the Netherlands 10-12 October 2013

Workshop:  Maths  SL  Workshop  leader:  Richard  Wade  and  Oliver  Bowles    Category:    2  

Thursday,  10  October   Time      

Session  1   9:00  –  10:30   Introductions,  Syllabus  changes  &  challenging  Topics.  A  summary  and  discussion  of  the  changes  to  the  syllabus.  What  are  the  challenging  topics  from  the  course?  (An  area  to  return  to  later  in  the  workshop  with  some  answers!)  

Coffee  break   10:30  –  11:00    

Session  2   11:00  –  12:30   The  Exploration  1.  A  chance  to  get  to  grips  with  the  new  assessment  criteria,  grade  some  IAs  and  understand  how  can  we  get  our  students  on  the  right  side  of  the  boundaries.  

Lunch   12:30  –  13:30    

Session  3   13:30  –  15:00   The  Exploration  2.  How  to  provide  inspiration  for  explorations,  develop  a  timeline,  prepare  students  for  extended  mathematical  writing,  avoid  plagiarism,  …  We  will  share  ideas  and  strategies  for  managing  the  new  IA.  

Coffee  break   15:00  –  15:15    

Session  4   15:15  –  16:30   Engaging  Students.  Providing  starters,  inspiration  and  hooks  to  engage  learners.  

 Friday,  11  October   Time      

Session  5   9:00  –  10:30   External  Assessment.  Avoiding  the  pitfalls.  

Coffee  break   10:30  –  11:00    

Session  6   11:00  –  12:30   The  Great  Share.  Delegates  will  be  encouraged  to  share  their  best  resources  and  practice.  

Lunch   12:30  –  13:30    

Session  7   13:30  –  15:00   TOK  &  Challenging  Topics.  Some  models  for  delivering  TOK  and  activities  for  the  SL  Mathematics  class.  Challenging  topics  will  be  a  chance  to  return  to  some  of  the  topics  mentioned  in  session  1.  

Coffee  break   15:00  –  15:15    

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Session  8   15:15  –  16:30   30  Great  Ideas  for  Using  Technology  in  the  Standard  Level  Mathematics  Classroom.  30  quick  fire  ideas  followed  by  some  time  to  get  some  practice.  

 Saturday,  12  October   Time      

Session  9   9:00  –  10:30   Technology  Workshop  A  chance  to  explore  and  gain  some  expertise  in  Geogebra  and  Autograph.  

Coffee  break   10:30  –  11:00    

Session  10   11:00  –  12:30   GDC  Activities  for  competent  Users  and  Wrap  Up  These  activities  aim  to  show  the  possibilities  of  using  the  GDC  as  a  tool  for  exploration  rather  than  just  a  calculating  device.  The  wrap  up  might  be  give  an  opportunity  to  finish  off  some  topics  from  previous  sessions.  

 

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IBDP  Mathematics  Standard  Level  Cat  2  Workshop  (Experienced  Teachers)  

10-­‐12  October  2013  Amsterdam  International  Community  School  

 

 

       

 

 

Workshop  Leader:  Richard  Wade  

 

   

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Contents  Summary  of  changes  to  SL  Syllabus  .......................................................................................................  4  

Course  Plan  ............................................................................................................................................  6  

External  Assessment  ..............................................................................................................................  8  

Summary  of  Examiner’s  Reports  .......................................................................................................  8  

Maths  SL  specimen  papers  ..............................................................................................................  11  

Assessed  Work  .................................................................................................................................  23  

External  Assessment  ........................................................................................................................  27  

Internal  Assessment  ............................................................................................................................  35  

Your  responsibilities  as  a  teacher  ...................................................................................................  35  

Exemplars  ........................................................................................................................................  35  

Sample  1  –  The  Magic  Circle  in  Basketball  .......................................................................................  36  

Assessment  criteria  ..........................................................................................................................  52  

Applying  the  assessment  criteria  .....................................................................................................  52  

Achievement  levels  ..........................................................................................................................  53  

Exploration  Marking  Grid  ................................................................................................................  57  

Notes  ...........................................................................................................................................  57  

Sample  2  –  SA  of  Water  in  tilted  cylinder  ........................................................................................  58  

Exploration  Marking  Grid  ................................................................................................................  69  

Notes  ...........................................................................................................................................  69  

The  Exploration  –  Long  Term  Planning  ............................................................................................  70  

The  Exploration  –  Short  Term  Planning  ...........................................................................................  71  

Exploration  –  Deciding  on  a  Topic  ...................................................................................................  72  

Self-­‐Assessment  ...............................................................................................................................  73  

Ideas  for  the  Exploration  .................................................................................................................  76  

Resources  ............................................................................................................................................  78  

International  Mindedness  -­‐  Calculus  ...............................................................................................  78  

TOK  –  Randomness  and  Pseudo  randomness  .................................................................................  79  

Images  .............................................................................................................................................  81  

Trigonometry  –  3-­‐2-­‐1  Blastoff!  ........................................................................................................  83  

Trigonometry  –  Trigonometry  Function  Family  ...............................................................................  84  

Statistics  -­‐  Hunt  the  Box  Plot!  ..........................................................................................................  85  

Probability  –  Dependent  Events  ......................................................................................................  86  

Probability  –  Crazy  Dice  ...................................................................................................................  87  

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Calculus  –  Parabola  Proof  ................................................................................................................  90  

Calculus  –  Investigating  Areas  under  Graphs  ..................................................................................  91  

Research  –  Aligning  Teaching  for  Constructive  Learning  .....................................................................  93  

 

 

   

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Summary  of  changes  to  SL  Syllabus  (first  exams  2014)  

The  process  of  change  started  in  2006.  Throughout  the  review  process,  teachers  were  encouraged  to  provide   feedback,   starting   with   questionnaires   on   the   current   courses,   and   then   comments   on  proposed  changes.  Suggestions  for  syllabus  change  varied  tremendously,  and  were  all  considered  by  the  review  team.  During  the   initial  stages,   it  was  proposed  that  the   IA  be  a  modelling  task,  and  as  such,   it   was   felt   appropriate   to   include   the   correlation   and   regression   subtopic   in   Statistics   and  Probability.   Discussions   on   matrices   suggested   that   they   needed   to   be   extended   to   include  transformations,   but   given   the   time   constraints,   this   was   not   feasible.   Thus,   to   make   room   for  correlation  and  regression,   it  was  agreed  to  remove  matrices  (the  HL  team  independently  came  to  the  same  conclusion).  

Teaching  and  Learning  

The  new  syllabus  includes  more  of  an  emphasis  on  modelling  and  classroom  inquiry  techniques.  

External  Assessment  

External   assessment   is   largely   unchanged.   The   number   of  marks   available   on   each   section  will   be  worth  exactly  45  marks  (previously  this  was  approximately  45  marks).  Statistical  tables  are  no  longer  allowed   in   examinations.   Nothing   stops   teachers   using   them   to   introduce   the   topic   of   normal  distribution  for  example.  

Internal  Assessment  

Internal  assessment  has  completely  changed  from  the  portfolio  to  a  student  centred  exploration.  The  major  change  is  the  ownership  of  the  tasks  from  the  IB  developed  tasks  to  ones  from  student  and/or  teacher  interest.  Ideas  for  work  for  internal  assessment  should  arise  out  of  classroom  experience  as  topics  are  introduced.  

The  following  are  some  of  the  points  made  in  the  curriculum  review  reports.    

• Ideas  for  work  for  internal  assessment  should  arise  out  of  classroom  experience  as  topics  are  introduced,  and  not  be  generated  by  the  IB.  

• Teachers  reluctant  to  produce  their  own  tasks  due  to  perceived  problems  with  moderation  

• Teachers  reluctant  to  share  their  own  developed  tasks  with  the  IB  as  then  they  would  have  a  limited  shelf  life  

• Plagiarism  and  the  perception  of  plagiarism      

   

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Syllabus  Content  The  syllabus  is  now  a  good  deal  clearer  and  it  includes  links  to  Aims,  Applications,  International  Mindedness  and  TOK.    Here  are  the  main  changes  to  the  syllabus:    Algebra  (+1hour):  

𝑛𝑟  should  be  found  using  both  the  formula  and  technology.  

Functions  and  Equations:  More  emphasis  on  applications  to  everyday  use.  A  direct  reference  to  

graphing  the  rational  function,  f(x)  =  !"!!!"!!

 

Circular  Functions  &  Trigonometry:  Exact  values  of  trigonometric  ratios  of    0, !!, !!, !!, !!  and  their  

multiples.  

Matrices  (-­‐10hours):  excluded  

Vectors:  no  change  

Statistics  &  Probability  (+5hours):  No  statistical  tables,  Requirement  to  use  GDC  for  statistical  Values,  Outlier  is  defined  as  more  than  1.5  IQR  ×  from  the  nearest  quartile,  Effect  of  constant  changes  to  the  original  data,  Regression  and  correlation  included  (not  in  HL  Core),  Variance  of  the  binomial  distribution.  

Calculus  (+4hours):  Limit  notation,  Higher  derivatives,𝑓!(𝑥)  ,  Integration  by  substitution  

 

   

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Course  Plan  Any   school   will   have   to   devise   a   programme   of   study   that   suits   the   students,   groupings   and   any  national  requirements.  Where  I  work  students  are  taught  in  separate  groups  for  HL,  SL  and  studies.  However,  since  students  often  have  unreasonably  high  expectations  of  themselves  there  are  always  a   number   of   students   who   change   groups   throughout   the   first   year   of   the   course.   As   such   a  programme  has  been  devised  for  the  three  courses  that  allows.  Here  is  a  possible  outline.  

Year  1  Module  1  –  Number  &  Algebra    Arithmetic  &  Geometric  Sequences  and  Series,  Sigma  Notation  and  Infinite  Series  Indices  and  Surds,Graphs  of  Exponential  Functions,  Investigating  e  Logarithms  &  the  Logarithmic  Function  The  Binomial  Expansion  

Extended  problem  –  e.g.  Lacsap’s  Fractions  

Module  2  –  Functions  Introduction  to  Function  Notation,  domain  &  range  Composite  Functions  The  Inverse  Function  Transforming  Functions  Quadratic  Functions  &  The  Discriminant  

Module  3  –  Circular  Functions  &  Trigonometry  The  circle:  radian  measure  of  angles;  length  of  an  arc;  area  of  a  sector.  The  Solution  of  Triangles  –  recap  Sine,  Cosine  Rule,  area  of  triangle  Unit  Circle  Pythagorean  Identities  Double  Angle  Formulae  Transforming  Trig  Functions  Modelling  using  Trig  Functions  Solving  Trig  Equations  

Extended  problem  –  e.g.  modelling  sunrise  times  Introduce  the  Exploration  Module  4  –  Differentiation  Introducing  Rates  of  Change  Limits  Differentiation  from  1st  Principles  Differentiating  Polynomials  Equations  of  Tangents  and  Normals  Higher  derivatives  Stationary  Points  Non-­‐stationary  points  of  inflexion  Small  Angle  Approximations  Differentiating  Trig  Function  Chain  Rule  Differentiating  Exponential  &  Log  Functions  Product  &  Quotient  Rule  Optimisation      

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Module  5  –  Statistics  &  Probability  Manipulation  and  Presentation  of  statistical  Data  Standard  Deviation  Experimental  &  theoretical  Probability  Independent  &  dependent  Events  &  Probability  trees  Venn  Diagrams  

Year  2  Module  5  –  Statistics  &  Probability  (continued)  Recap  Y1  probability  Laws  of  Probability  Conditional  Probability  Regression  and  correlation  Discrete  Random  Variables  Binomial  Distribution  Normal  Distribution  

Module  6  –  Integration    Recap  Differentiation  Area  under  a  Graph  investigation  Antidifferentiation  Finding  C  Definite  Integrals  Areas  under  Graphs  Volumes  of  Revolution  Integration  by  Recognition  Integration  by  Substitution  Kinematics  Exploration    Module  7  –  Vectors  Notation,  Scalar  Multiple  ,  adding  &  subtracting  Length  of  a  Vector,  midpoints,  distances,  position  vectors  &  3D  Vectors  Angle  between  2  vectors,  Scalar  (Dot)  Product  The  Vector  Equation  of  a  Straight  Line  2D  &  3D  Applications  -­‐  The  Velocity  Vector  of  a  Moving  Object  Lines  –  intersecting,  coincident  &  parallel    

 

 

 

   

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External  Assessment  

Summary  of  Examiner’s  Reports  Reading   subject   reports   from   a   particular   examination   can   be   quite   useful,   but   looking   at   subject  reports  over  a  number  of  sessions  can  be  quite  revealing.  That  is,  many  problems  are  recurring.  Here  is  a  summary  of  the  examiner’s  reports  from  May  2010  until  November  2012  

Recommendations  in  order  of  frequency  reported:  

Command   Terms   –   there   appears   to   be   a   lack   of   understanding   of   the   command   terms   such   as  "show   that",   "find",   or   "write   down",   “sketch”,   “hence”.   For   example,   for   “Show   that”   –  working  backwards  from  a  result  is  not  acceptable.  “Sketch”  –  no  need  to  use  graph  paper  but  include  zeros,  maximum  and  minimum  points,  and  domain  and  end  point,  etc.  

Numerical   Approaches   –   On   paper   2   candidates   often   use   analytical   approaches   when   graphical  approach   is   required.   E.g.   Students   are   not   using   graphical   approaches   to   derivatives.   Analytical  relationships  are  primarily  examined  on  Paper  1.  

Past  Paper  Practice  –  There  is  evidence  to  suggest  that  candidates  are  not  getting  sufficient  practice  in  past  paper  questions,  e.g.  time  management  issues,  looking  at  marks  awarded  for  clues,  etc.  

Showing  working  -­‐  Communication  when  using  GDC  still  needs  more  emphasis;  “found  using  GDC”  is  insufficient   working.   Including   sketches   and   equations   entered   into   the   GDC   will   ensure   follow  through  marks  can  be  awarded  if  errors  are  made  in  previous  parts  

Mode  of  calculator  –  degree  or  radian  often  misused.  

Calculator  notation  –  candidates  should  avoid  using  calculator  notation  like  binompdf,  etc.  

Rounding  -­‐  candidates  need  to  avoid  premature  rounding.  Some  confusion  about  3s.f.  with  students  rounding  to  3  d.p.  

Numeracy  skills  -­‐  Numeracy  skills  sometimes  weak  e.g.  addition  and  multiplication  of  fractions  

 

Difficult  Topics  

Here  is  a  list  of  topics  that  students  find  difficult.  Again  the  repetition  of  certain  topics  is  revealing!  

 

• Recognizing  the  sign  of  a  trigonometric  ratio  for  an  angle  not  in  the  first  quadrant  • Finding  an  axial  intercept  for  a  vector  equation  in  three  dimensions  • Using  the  discriminant    • Vector  geometry  • Using  the  chain  rule  to  find  a  derivative  • Reasoning  skills  • Correct  use  of  parenthesis  when  expanding  a  binomial  • Understanding  and  use  of  the  command  terms  “sketch”  and  “show  that”  

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• Sketching  graphs  carefully  to  show  important  points  and  using  the  correct  domain  • Normal  distribution  especially  finding  the  value  of  a  standardized  variable  • Conditional  probability  and  the  meaning  of  independent  events  • Finding  maximum  velocity  from  displacement  function  • Area  between  two  curves  • Use  of  a  graphic  display  calculator  (GDC)  to  evaluate  definite  integrals  • using  the  binomial  theorem  with  a  general  exponent    • analyzing  circular  functions  • conditional  probability  and  probability  of  compound  events  • transformations  of  functions  • Binomial  and  conditional  probability.    • “Show  that”  questions.  • Matrix  algebra.  • Chain  rule.  • Probabilities  involving  more  than  one  event  • Discrete  probability  distributions  • Working  with  double-­‐angle  formulas  • Using  the  discriminant  to  determine  the  nature  of  the  roots  of  a  quadratic  • Finding  the  parameters  of  a  trigonometric  function  • Chain  rule  differentiation  • Finding  the  equation  of  the  tangent  to  a  curve  at  a  point  • Rules  of  logarithms  • Combining  transformations  of  functions  (especially  stretch  parallel  to  the  y-­‐axis)  • Binomial  probability  • Ambiguous  case  of  sine  rule  • Solving  equations  that  involve  logarithms  • Recognition  of  integration  of  velocity  to  find  distance  • “Show  that”  questions  • Interpreting  the  second  derivative  as  a  rate  of  change  • understanding  Venn  diagrams  • working  with  rules  of  logarithms  • transformation  of  functions  • kinematics  • area  between  two  functions  with  different  boundaries  • finding  the  total  range  of  two  sets  of  values  • normal  distribution  • direction  vectors  • recognizing  binomial  distribution  • using  the  graphic  display  calculator  (GDC)  to  solve  algebraically  complicated    • equations  • show  that  questions  • trigonometric  values  for  angles  such  as  π  ,  0,  3π/2  • conditional  probability  and  finding  probabilities  using  a  tree  diagram  • integration  of  functions  of  the  form  f  ax  b  (  )  • working  with  logarithms  • quadratic-­‐type  trigonometric  equations  

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• transformations  of  functions  • “show  that”  and  “hence”  in  the  command  terms  • basic  computation  and  algebraic  manipulation  • Obtaining  relevant  statistical  values  from  a  GDC  • Graphical  solutions  of  equations  • Solution  of  a  system  of  linear  equations  on  a  GDC  • Relationships  between  f,  f  '  and  f  "  • Use  of  a  trigonometric  model  • Giving  precise  explanations  for  mathematical  situations  • Finding  a  unit  vector  in  the  direction  of  another  vector  • Working  with  trigonometric  functions  of  certain  angles  (0,  π/2,  π,  and  3π/2)  • Relating  the  derivative  to  the  gradient  of  a  curve  • Applying  logarithm  properties  • Interpreting  second  derivative  from  the  concavity  of  a  graph  • Concept  of  the  constant  of  integration    • Conditional  and  combined  probability  • Algebraic  manipulation  and  arithmetic  with  fractions  • A  considerable  number  of  candidates  still  find  challenging  the  use  of  the  GDC  as  a  tool  to  

find  information  (i.e.  standard  deviation,  local  maximum  and  minimum  points.    

   

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Maths  SL  specimen  papers    A  full  specimen  paper  1  and  paper  2  and  associated  mark  scheme  are  provided  at  http://occ.ibo.org  under   assessment.   Given   that   external   assessment   is   virtually   unchanged   the   focus   here   will   be  specimen   questions   on   the   new   syllabus   items   to   give   us   a   chance   to   get   to   grips  with  what   the  examiners   have   install   for   our   students!   The   markscheme   is   included   with   a   sample   student  response  for  you  to  assess.  

Source:  IBO  

Paper  1  

 

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Paper  2  

 

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Assessed  Work  Qu  2  

 

Qu  4  

 

Qu  5  

 

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Qu  7  

 

Paper  2  Qu  5  

 

   

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Qu  8  

 

   

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Qu  10  

 

 

 

 

 

 

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External  Assessment    Below  is  a  list  of  past  paper  exam  questions  from  the  Functions  module.  The  focus  of  this  session  is  on  differentiation,  difficult  topics,  the  mark  scheme  and  where  and  how  students   loose  marks  and  what  we  can  do  to  address  this.  Each  group  will  have  one  module  to  look  over.  The  aim  is  to  classify  each   question,   or   part   of   a   question   (as   is   most   appropriate)   with   a   corresponding   IB   level   (see  below)  with   a   view   to   analysing  why   students  may  produce   an   incorrect   response:   (a)   conceptual  understanding  (b)  problem  solving  skills  (c)  rigour  (d)  memory  (e)  lack  of  fluency/precision  with  basic  algebra  skills  etc.    

1   very  poor   3   mediocre   5   good   7   excellent  2   poor   4   satisfactory   6   very  good      

8. f (x) = 4 sin .

For what values of k will the equation f (x) = k have no solutions?

(Total 4 marks)

9. The diagram represents the graph of the function

f : x (x – p)(x – q).

(a) Write down the values of p and q.

(b) The function has a minimum value at the point C. Find the x-coordinate of C.

(Total 4 marks)

10. (a) Express f (x) = x2 – 6x + 14 in the form f (x) = (x – h)2 + k, where h and k are to be

determined.

(b) Hence, or otherwise, write down the coordinates of the vertex of the parabola with equation y = x2 – 6x + 14.

(Total 4 marks)

⎟⎠⎞⎜

⎝⎛ +

23 πx

x

y

C

212–

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1. A ball is thrown vertically upwards into the air. The height, h metres, of the ball above the ground after t seconds is given by

h = 2 + 20t – 5t2, t ≥ 0

(a) Find the initial height above the ground of the ball (that is, its height at the instant when it is released).

(2)

(b) Show that the height of the ball after one second is 17 metres. (2)

(c) At a later time the ball is again at a height of 17 metres.

(i) Write down an equation that t must satisfy when the ball is at a height of 17 metres.

(ii) Solve the equation algebraically. (4)

(d) (i) Find .

(ii) Find the initial velocity of the ball (that is, its velocity at the instant when it is released).

(iii) Find when the ball reaches its maximum height.

(iv) Find the maximum height of the ball. (7)

(Total 15 marks)

3. The diagram shows the graph of y = f (x), with the x-axis as an asymptote. 4 marks

(a) On the same axes, draw the graph of y =f (x + 2) – 3, indicating the coordinates of

the images of the points A and B. (b) Write down the equation of the asymptote to the graph of y = f (x + 2) – 3.

thdd

A(–5, –4)

B(5, 4)

y

x

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2. (a) Factorize x2 – 3x – 10.

(b) Solve the equation x2 – 3x – 10 = 0. Total 4 marks

4. Let f (x) = , and g (x) = 2x. Solve the equation

(f –1 g)(x) = 0.25. (Total 4 marks)

6. Consider the function

(a) Determine the inverse function f –1. (b) What is the domain of f –1?

(Total 4 marks) Problem-Solving 7. In the diagram below, the points O(0, 0) and A(8, 6) are fixed. The angle varies as the point P(x, 10) moves along the horizontal line y = 10.

Diagram to scale

(a) (i) Show that (ii) Write down a similar expression for OP in terms of x.

(2) (b) Hence, show that

(3) (c) Find, in degrees, the angle when x = 8.

(2) (d) Find the positive value of x such that .

(4) Let the function f be defined by

(e) Consider the equation f (x) = 1.

(i) Explain, in terms of the position of the points O, A, and P, why this equation has a solution.

(ii) Find the exact solution to the equation. (5)

(Total 16 marks)

x

1–,1: ≥+ xxxf

AP̂O

y

x

A(8, 6)

O(0, 0)

y=10P( , 10)x

.8016–AP 2 += xx

,)}100)(8016–{(

408–AP̂Ocos 22

2

++√

+=

xxxxx

AP̂O

°=60AP̂O

.150,)}100)(8016–{(

408–AP̂Ocos)( 22

2

≤≤++√

+== x

xxxxxxf

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5. The diagram below shows part of the graph of the function

The graph intercepts the x-axis at A(–3, 0), B(5, 0) and the origin, O. There is a minimum point at P and a maximum point at Q.

(a) The function may also be written in the form where a < b. Write down the value of

(i) a;

(ii) b. (2)

(b) Find

(i) f ʹ′(x);

(ii) the exact values of x at which f '(x) = 0;

(iii) the value of the function at Q. (7)

(c) (i) Find the equation of the tangent to the graph of f at O.

(ii) This tangent cuts the graph of f at another point. Give the x-coordinate of this point.

(4)

(d) Determine the area of the shaded region. (2)

(Total 15 marks)

.152–: 23 xxxxf ++

40

–20–15–10–5

5101520253035

–3 –2 –1 1 2 3 4 5

y

xA

P

B

Q

),–()–(–: bxaxxxf

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11. The diagrams show how the graph of y = x2 is transformed to the graph of y = f (x) in three steps.

For each diagram give the equation of the curve.

(Total 4 marks)

y

y

y

y

0

0

0

0

x

xx

xy=x2

4

1

1 1

1

3

7

(a)

(b) (c)

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13. The diagram shows the graph of the function y = ax2 + bx + c.

Complete the table below to show whether each expression is positive, negative or zero.

Expression positive negative zero a c b2 – 4ac b

(Total 4 marks)

14. Two functions f and g are defined as follows: f (x) = cos x, 0 ≤ x ≤ 2π; g (x) = 2x + 1, x ∈ .

Solve the equation (g f)(x) = 0. (Total 4 marks)

15. The diagram shows three graphs.

A is part of the graph of y = x.

B is part of the graph of y = 2x.

C is the reflection of graph B in line A.

Write down

(a) the equation of C in the form y =f (x); (b) the coordinates of the point where C cuts the x-axis.

(Total 4 marks)

y

x

y

x

B

A

C

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16. The quadratic equation 4x2 + 4kx + 9 = 0, k > 0 has exactly one solution for x. Find the value of k.

(Total 4 marks)

17. Three of the following diagrams I, II, III, IV represent the graphs of

(a) y = 3 + cos 2x

(b) y = 3 cos (x + 2)

(c) y = 2 cos x + 3.

Identify which diagram represents which graph.

(Total 4 marks)

x

–π π12

– –π 12

–π 32

–π

y

2

1

–1

–2

x

–π π12

– –π 12

–π 32

–π

y

3

2

1

–3

y

x

4

2

–π π12

– –π 12

–π 32

–π

x

–π π12

– –π 12

–π 32

–π

5

4

3

2

1

y

I

III

II

IV

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19. The function f is given by f (x) = Find the domain of the function. (Total 4 marks)

18. The function f is given by

f (x) = , x ∈ , x ≠ 3.

(a) (i) Show that y = 2 is an asymptote of the graph of y = f (x). (2)

(ii) Find the vertical asymptote of the graph. (1)

(iii) Write down the coordinates of the point P at which the asymptotes intersect. (1)

(b) Find the points of intersection of the graph and the axes. (4)

(c) Hence sketch the graph of y = f (x), showing the asymptotes by dotted lines. (4)

(d) Show that fʹ′ (x) = and hence find the equation of the tangent at

the point S where x = 4. (6)

(e) The tangent at the point T on the graph is parallel to the tangent at S.

Find the coordinates of T. (5)

(f) Show that P is the midpoint of [ST]. (l)

(Total 24 marks)

 

 

 

 

 

.)2(n 1 −x

312

−+

xx

2)3(7

−−x

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Internal  Assessment  As   a   requirement  of   the   course,   students   have   to   complete   a  mathematical   exploration.   Students  choose   from  a  wide   variety  of   activities,   for   example  modelling,   investigations  and  applications  of  mathematics   and   produce   a   short   written   report.     This   report   of   6   to   10   pages   will   be   assessed  formally  by  the  teacher,  and  then  it  may  be  sent  off  to  an  external  examiner  for  moderation.    This  piece  of  coursework  will  provide  20%  of  the  overall  grade  in  this  subject.      Your  responsibilities  as  a  teacher  The  teacher  has  10  main  responsibilities.  

During  the  process:  

• to  advise  students  in  choosing  an  appropriate  topic  for  an  exploration  • to  provide  opportunities  for  students  to  learn  the  skills  related  to  exploration  work  • to  ensure  that  students  understand  the  assessment  criteria  and  how  they  will  be  applied  • to  encourage  and  support  students  throughout  the  research  and  writing  of  explorations  • to  provide  students  with  feedback.  • At  the  end  of  the  process:  • to  verify  the  accuracy  of  all  calculations  • to  assess  the  work  accurately,  annotating  it  appropriately  to  indicate  where  achievement  

levels  have  been  awarded  • to  ensure  that  the  relevant  form  from  the  Handbook  of  procedures  for  the  Diploma  

Programme  has  been  completed,  justifying,  with  comments,  the  marks  awarded  • to  ensure  that  the  relevant  form  from  the  Handbook  of  procedures  for  the  Diploma  

Programme  has  been  signed  by  both  the  student  and  the  teacher,  declaring  that  the  exploration  is  the  student’s  own  work  

• to  ensure  that  students  fully  understand  the  strengths  and  weaknesses  of  the  exploration.      

Exemplars  See  teacher  support  material  (TSM)  at  http://occ.ibo.org  under  General  Documents  for  support  on  the  Exploration.  There  you  can  find  9  assessed  samples  with  the  following  titles  and  scores  out  of  20.  

• Example  1:  Breaking  the  code  (15/20)  • Example  2:  Euler’s  Totient  Theorem  (16/20)  • Example  3:  Minesweeper  (5/20)  • Example  4:  Modelling  musical  chords  (9/20)  • Example  5:  Newton–Raphson  (11/20)  • Example  6:  Florence  Nightingale  (20/20)  • Example  7:  Modelling  rainfall  (16/20)  • Example  8:  Spirals  in  Nature  (16/20)  • Example  9:  Tower  of  Hanoi  (14/20)  

 

 

   

   

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Here  are  two  further  samples  from  IBO  

Sample  1  –  The  Magic  Circle  in  Basketball    

 

 

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Source:  www.occ.ibo.org  

   

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Assessment  criteria  Each  exploration  should  be  assessed  against  the  following  five  criteria.  

Criterion  A   Communication  

Criterion  B   Mathematical  presentation  

Criterion  C   Personal  engagement  

Criterion  D   Reflection  

Criterion  E   Use  of  mathematics  

The  descriptions  of  the  achievement  levels  for  each  of  these  five  assessment  criteria  follow  and  it  is  important  to  note  that  each   achievement   level   represents   the  minimum  requirement   for   that   level   to   be   awarded.   The   final   mark   for   each  exploration  is  obtained  by  adding  together  the  achievement  levels  awarded  for  each  criterion  A–E.  It  should  be  noted  that  the  descriptors  for  criterion  E  are  different  for  mathematics  SL  and  mathematics  HL.  

The  maximum  possible  mark  is  20.  

Applying  the  assessment  criteria  The   method   of   assessment   used   is   criterion   referenced,   not   norm   referenced.   That   is,   the   method   of   assessing   each  exploration   judges   students  by   their  performance   in   relation   to   identified   assessment   criteria   and  not   in   relation   to   the  work  of  other  students.  

Each   exploration   submitted   for  mathematics   SL   or  mathematics  HL   is   assessed   against   the   five   criteria   A  to  E.   For   each  assessment   criterion,   different   levels   of   achievement   are   described   that   concentrate   on   positive   achievement.   The  description  of  each  achievement  level  represents  the  minimum  requirement  for  that  level  to  be  achieved.  

The  aim  is  to  find,  for  each  criterion,  the  level  descriptor  that  conveys  most  adequately  the  achievement  level  attained  by  the  student.  

Teachers  should  read  the  description  of  each  achievement  level,  starting  with  level  0,  until  one  is  reached  that  describes  a  level  of  achievement  that  has  not  been  reached.  The  level  of  achievement  gained  by  the  student  is  therefore  the  preceding  one,  and  it  is  this  that  should  be  recorded.  

For  example,  when  considering  successive  achievement  levels  for  a  particular  criterion,   if  the  description  for   level  3  does  not  apply,  then  level  2  should  be  recorded.  

For  each  criterion,  whole  numbers  only  may  be  recorded;  fractions  and  decimals  are  not  acceptable.  

The  highest  achievement  levels  do  not  imply  faultless  performance,  and  teachers  should  not  hesitate  to  use  the  extremes,  including  0,  if  they  are  appropriate  descriptions  of  the  work  being  assessed.  

A   student  who   attains   a   high   level   of   achievement   in   relation   to   one   criterion  will   not   necessarily   attain   high   levels   of  achievement  in  relation  to  the  other  criteria.  Similarly,  a  student  who  attains  a  low  level  of  achievement  for  one  criterion  will   not   necessarily   attain   low   achievement   levels   for   the   other   criteria.   Teachers   should   not   assume   that   the   overall  assessment  of  the  students  will  produce  any  particular  distribution  of  marks.  

It   is  expected  that  the  assessment  criteria  be  available  to  students  at  all  times.  Descriptors  of  the  achievement  levels  for  each  assessment  criterion  are  given   in  the  tables   in  the  following  section.  Within  the  tables,   for  each  achievement   level,  there  is  a  link  to  an  exploration  within  this  TSM  that  achieved  that  level  for  that  particular  criterion.  

Students  should  be  made  aware  that  they  will  not  receive  a  grade  for  mathematics  SL  or  mathematics  HL  if  they  have  not  submitted  an  exploration.  

   

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Achievement  levels  

Criterion  A:  Communication  

This   criterion   assesses   the   organization   and   coherence   of   the   exploration.   A   well-­‐organized   exploration   contains   an  introduction,  has  a  rationale  (which   includes  explaining  why  this  topic  was  chosen),  describes  the  aim  of  the  exploration  and  has  a  conclusion.  A  coherent  exploration  is  logically  developed  and  easy  to  follow.  

Graphs,  tables  and  diagrams  should  accompany  the  work  in  the  appropriate  place  and  not  be  attached  as  appendices  to  the  document.  

Achievement  level  

Descriptor  

0   The  exploration  does  not  reach  the  standard  described  by  the  descriptors  below.  

1   The  exploration  has  some  coherence.  

2   The  exploration  has  some  coherence  and  shows  some  organization.  

Example  1  

3   The  exploration  is  coherent  and  well  organized.  

Example  8  

4   The  exploration  is  coherent,  well  organized,  concise  and  complete.  

Example  9  

Criterion  B:  Mathematical  presentation  

This  criterion  assesses  to  what  extent  the  student  is  able  to:  

• use  appropriate  mathematical  language  (notation,  symbols,  terminology)  

• define  key  terms,  where  required  

• use  multiple  forms  of  mathematical  representation  such  as  formulae,  diagrams,  tables,  charts,  graphs  and  models,  where  appropriate.  

Students  are  expected  to  use  mathematical  language  when  communicating  mathematical  ideas,  reasoning  and  findings.  Students  are  encouraged  to  choose  and  use  appropriate  ICT  tools  such  as  graphic  display  calculators,  screenshots,  graphing,  spreadsheets,  databases,  drawing  and  word  processing  software,  as  appropriate,  to  enhance  mathematical  communication.  

Achievement  level  

Descriptor  

0   The  exploration  does  not  reach  the  standard  described  by  the  descriptors  below.  

1   There  is  some  appropriate  mathematical  presentation.  

Example  4  

2   The  mathematical  presentation  is  mostly  appropriate.  

Example  9  

3   The  mathematical  presentation  is  appropriate  throughout.  

Example  1  

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Criterion  C:  Personal  engagement  

This  criterion  assesses  the  extent  to  which  the  student  engages  with  the  exploration  and  makes  it  their  own.  Personal  engagement  may  be  recognized  in  different  attributes  and  skills.  These  include  thinking  independently  and/or  creatively,  addressing  personal  interest  and  presenting  mathematical  ideas  in  their  own  way.  

Achievement  level  

Descriptor  

0   The  exploration  does  not  reach  the  standard  described  by  the  descriptors  below.  

1   There  is  evidence  of  limited  or  superficial  personal  engagement.  

Example  3  

2   There  is  evidence  of  some  personal  engagement.  

Example  5  

3   There  is  evidence  of  significant  personal  engagement.  

Example  7  

4   There  is  abundant  evidence  of  outstanding  personal  engagement.  

Example  6  

Criterion  D:  Reflection  

This  criterion  assesses  how  the  student  reviews,  analyses  and  evaluates  the  exploration.  Although  reflection  may  be  seen  in  the  conclusion  to  the  exploration,  it  may  also  be  found  throughout  the  exploration.  

Achievement  level  

Descriptor  

0   The  exploration  does  not  reach  the  standard  described  by  the  descriptors  below.  

1   There  is  evidence  of  limited  or  superficial  reflection.  

Example  5  

2   There  is  evidence  of  meaningful  reflection.  

Example  8  

3   There  is  substantial  evidence  of  critical  reflection.  

Example  6  

 

   

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Criterion  E:  Use  of  mathematics  

The  achievement  levels  and  descriptors  for  criterion  E  are  different  for  mathematics  SL  and  mathematics  HL.  

SL  only  

This  criterion  assesses  to  what  extent  students  use  mathematics  in  the  exploration.  

Students  are  expected  to  produce  work  that  is  commensurate  with  the  level  of  the  course.  The  mathematics  explored  should  either  be  part  of  the  syllabus,  or  at  a  similar  level  or  beyond.  It  should  not  be  completely  based  on  mathematics  listed  in  the  prior  learning.  If  the  level  of  mathematics  is  not  commensurate  with  the  level  of  the  course,  a  maximum  of  two  marks  can  be  awarded  for  this  criterion.  A  piece  of  mathematics  can  be  regarded  as  correct  even  if  there  are  occasional  minor  errors  as  long  as  they  do  not  detract  from  the  flow  of  the  mathematics  or  lead  to  an  unreasonable  outcome.    

Achievement  level  

Descriptor  

0   The  exploration  does  not  reach  the  standard  described  by  the  descriptors  below.  

Example  3  

1   Some  relevant  mathematics  is  used.  

2   Some  relevant  mathematics  is  used.  Limited  understanding  is  demonstrated.  

3   Relevant  mathematics  commensurate  with  the  level  of  the  course  is  used.  Limited  understanding  is  demonstrated.  

Example  4  

4   Relevant  mathematics  commensurate  with  the  level  of  the  course  is  used.  The  mathematics  explored  is  partially  correct.  Some  knowledge  and  understanding  are  demonstrated.  

Example  9  

5   Relevant  mathematics  commensurate  with  the  level  of  the  course  is  used.  The  mathematics  explored  is  mostly  correct.  Good  knowledge  and  understanding  are  demonstrated.  

Example  8  

6   Relevant  mathematics  commensurate  with  the  level  of  the  course  is  used.  The  mathematics  explored  is  correct.  Thorough  knowledge  and  understanding  are  demonstrated.  

Example  2  

 

   

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HL  only  

This  criterion  assesses  to  what  extent  and  how  well  students  use  mathematics  in  the  exploration.  

Students  are  expected  to  produce  work  that  is  commensurate  with  the  level  of  the  course.  The  mathematics  explored  should  either  be  part  of  the  syllabus,  or  at  a  similar  level  or  beyond.  It  should  not  be  completely  based  on  mathematics  listed  in  the  prior  learning.  If  the  level  of  mathematics  is  not  commensurate  with  the  level  of  the  course,  a  maximum  of  two  marks  can  be  awarded  for  this  criterion.  The  mathematics  can  be  regarded  as  correct  even  if  there  are  occasional  minor  errors  as  long  as  they  do  not  detract  from  the   flow   of   the   mathematics   or   lead   to   an   unreasonable   outcome.   Sophistication   in   mathematics   may   include  understanding  and  use  of  challenging  mathematical  concepts,  looking  at  a  problem  from  different  perspectives  and  seeing  underlying   structures   to   link   different   areas   of  mathematics.   Rigour   involves   clarity   of   logic   and   language  when  making  mathematical  arguments  and  calculations.  Precise  mathematics  is  error-­‐free  and  uses  an  appropriate  level  of  accuracy  at  all  times.    

Achievement  level  

Descriptor  

0   The  exploration  does  not  reach  the  standard  described  by  the  descriptors  below.  

Example  3  

1   Some  relevant  mathematics  is  used.  Limited  understanding  is  demonstrated.  

Example  4  

2   Some  relevant  mathematics  is  used.  The  mathematics  explored  is  partially  correct.  Some  knowledge  and  understanding  are  demonstrated.  

Example  5  

3   Relevant  mathematics  commensurate  with  the  level  of  the  course  is  used.  The  mathematics  explored  is  correct.  Good  knowledge  and  understanding  are  demonstrated.  

Example  6  

4   Relevant  mathematics  commensurate  with  the  level  of  the  course  is  used.  The  mathematics  explored  is  correct  and  reflects  the  sophistication  expected.  Good  knowledge  and  understanding  are  demonstrated.  

Example  1  

5   Relevant  mathematics  commensurate  with  the  level  of  the  course  is  used.  The  mathematics  explored  is  correct  and  reflects  the  sophistication  and  rigour  expected.  Thorough  knowledge  and  understanding  are  demonstrated.  

Example  7  

6   Relevant  mathematics  commensurate  with  the  level  of  the  course  is  used.  The  mathematics  explored  is  precise  and  reflects  the  sophistication  and  rigour  expected.  Thorough  knowledge  and  understanding  are  demonstrated.  

Example  2  

 

Source:  occ.ibo.org    

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Exploration  Marking  Grid    

Criterion   Maximum  Mark  

Comment   Mark  

A   4      

B   3      

C   4      

D   3      

E   6      

Total   20    

Notes  ……………………………………………………………………………………………………………………………………………………………

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Sample  2  –  SA  of  Water  in  tilted  cylinder    

 

 

 

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Exploration  Marking  Grid    

Criterion   Maximum  Mark  

Comment   Mark  

A   4      

B   3      

C   4      

D   3      

E   6      

Total   20    

Notes  ……………………………………………………………………………………………………………………………………………………………

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The  Exploration  –  Long  Term  Planning    

Year  1  Term  1   Teacher  explains  to  students  the  requirements  of  the  exploration  and  discusses  plagiarism.  

Year  1  Term  1   Students  get  practice  in  extended  problems  

Year  1  Term  2   Students  look  at  exemplars  of  explorations  from  the  TSM.  

Students  brainstorm  ideas  for  possible  areas  of  interest.  

Year  1  Term  2   The  class  start  a  short  practice  research  question  on  modelling,  investigation  or  application  decided  by  the  teacher  (the  same  piece  will  be  undertaken  by  all  the  students).      This  piece  could  have  been  adapted  from  one  of  the  old  portfolio  assignments.  Some  lesson  time  is  given  to  starting  this  and  students  will  be  required  to  complete  it  for  homework.  

Year  1  Term  2   Submit  short  practice  research  question.  Peer  assessment  is  used  to  mark  the  practice  exploration  giving  the  students  an  opportunity  to  gain  familiarity  with  the  assessment  criteria.    

If  an  investigation  is  attempted,  emphasis  will  need  to  be  given  that  investigations  are  not  the  only  type  of  exploration  that  can  be  carried  out.    Students  should  be  given  another  opportunity  to  look  at  possible  modelling  and  application  exemplars.  

Year  1  Term  3   Students  brainstorm  and  use  mind  mapping  techniques  to  develop  a  focus  area  for  their  research  question.  

Year  1  Term  3   Individual  teacher  interview  with  student  to  discuss  and  develop  the  area  of  interest.  

Year  2  Term  1   Students  submit  the  exploration  topic  with  a  brief  outline  description.    

They  will  describe  the  aims  in  class  inviting  discussion  and  comment  from  their  peers.  

Year  2  Term  1   Students  start  work  on  their  exploration.  

  Guidance  given  to  students  to  ensure  the  research  is  appropriate  and  well-­‐focussed  (students  may  consult  with  the  teacher  throughout  the  whole  process).  

Year  2  Term  1   Submit  first  draft.    

Year  2  Term  1   Teacher  gives  advice  in  terms  of  the  way  the  work  could  be  improved.  

Students  amend  and  complete  their  exploration.  

Year  2  Term  1   Students  submit  final  draft.  

  Student  signs  the  coversheet  to  confirm  that  the  work  is  his  or  her  authentic  work  

Year  2  Term  2   Teacher  assesses  work  using  the  prescribed  IB  markscheme.    

Year  2  Term  2   A  short  interview  with  the  student  to  check  the  content  and  the  authenticity  of  the  work.  

  Teacher  signs  the  cover  sheet.  

Year  2  Term  2   The  work  is  internally  moderated  to  ensure  consistency  of  assessment.  

 

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The  Exploration  –  Short  Term  Planning  

It  is  envisaged  that  10  hours  of  class  time  and  approximately  10  hours  outside  class  be  spent  on  the  exploration.  

Choosing  a  focus/topic:  2  weeks  

Class  time:  2–3  hours  

This  will  involve  introductory  lesson(s)  leading  to  each  student  having  a  focused  aim  to  their  exploration.  The  purpose  and  scope  of  the  exploration  should  be  explained.  In  doing  this,  teachers  could  demonstrate  in  various  ways  how  a  stimulus  will  be  used.  The  list  below  shows  the  wide  range  of  stimuli  that  are  suitable  as  starting  points  to  generate  an  idea  as  a  focus  for  the  exploration.  

It  could  also  be  useful  to  look  at  an  example  of  one  or  more  stimuli  and  discuss  with  students  how  this  could  lead  to  a  focus  for  a  mathematical  exploration.  An  example  of  a  “mind  map”  starting  from  the  stimulus  “water”  is  included  below  to  exemplify  how  this  process  could  develop.  

Examples  of  explorations  from  the  TSM  and  other  sources  could  be  looked  at  to  demonstrate  to  students  what  is  expected  of  them.  

At  the  end  of  this  period,  each  student  should  have  decided  on  a  focus  for  the  exploration  and  have  a  preliminary  plan  of  how  to  approach  it.  This  could  involve  describing  the  aims  in  class  and  inviting  discussion  and  comment  from  fellow  students.  

Draft  exploration:  3  weeks  

Class  time:  4–5  hours  

Class  time  could  be  used  for  writing  the  exploration  (though  it  is  envisaged  that  students  will  also  spend  time  outside  lessons  researching  and  writing  their  exploration  and  preparing  a  draft  to  submit).  Discussion  among  their  peers  and  with  the  teacher  is  encouraged,  but  it  is  essential  that  the  written  draft  exploration  submitted  is  the  student’s  own  work  and  they  should  be  prepared  to  explain  any  aspects  of  their  work.  Teachers  may  also  utilize  this  time  to  review  and  comment  on  drafts.  

Teacher  to  review  and  comment  on  drafts:  4–8  weeks  

Class  time:  1–2  hours  

This  draft  should  be  reviewed  by  the  teacher  and  comments  made  on  the  strengths  and  weaknesses  of  the  work.  This  first  draft  must  not  be  heavily  annotated  or  edited  by  the  teacher,  but  is  an  opportunity  for  students  to  receive  further  guidance  on  the  exploration.  This  time  could  be  used  by  students  to  consider  and  possibly  discuss  the  implications  of  this  draft.  

Final  writing:  2  weeks  

Class  time:  1–2  hours  

The  student  will  now  have  a  short  period  in  which  to  finalize  the  exploration  based  on  the  draft  and  the  advice  given.  During  this  time,  the  student  can  discuss  their  work  with  the  teacher,  but  the  final  document  must  be  exclusively  their  own  work.  It  is  after  this  stage  that  the  work  will  be  marked  by  the  teacher.  

Source:  occ.ibo.org  

   

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Exploration  –  Deciding  on  a  Topic    

Name:          

1. Chosen  stimulus    

2. My  mind  map    

 

 

 

 

 

 

 

 

 

 

 

 

 

3. Chosen  Topic        

4. Why  I’m  interested  in  this  topic.        

5. List  mathematical  topics  likely  to  be  included      

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Self-­‐Assessment    

A:  COMMUNICATION  (4)    

This  criterion  assesses  the  organization  and  coherence  of  the  exploration.  A  well-­‐organized  exploration  contains  an  introduction,  has  a  rationale  (which  includes  explaining  why  this  topic  was  chosen),  describes  the  aim  of  the  exploration  and  has  a  conclusion.  A  coherent  exploration  is  logically  developed  and  easy  to  follow.  

Graphs,  tables  and  diagrams  should  accompany  the  work  in  the  appropriate  place  and  not  be  attached  as  appendices  to  the  document.  

0   The  exploration  does  not  reach  the  standard  described  by  the  descriptors  

1   The  exploration  has  some  coherence.  

2   The  exploration  has  some  coherence  and  shows  some  organization.  

3   The  exploration  is  coherent  and  well  organized.  

4   The  exploration  is  coherent,  well  organized,  concise  and  complete.  

B:  MATHEMATICAL  PRESENTATION  (3)  

This  criterion  assesses  to  what  extent  the  student  is  able  to:  

• use  appropriate  mathematical  language  (notation,  symbols,  terminology)  • define  key  terms,  where  required  • use  multiple  forms  of  mathematical  representation  such  as  formulae,  diagrams,  tables,  charts,  

graphs  and  models,  where  appropriate.  Students  are  expected  to  use  mathematical  language  when  communicating  mathematical  ideas,  reasoning  and  findings.  

Students  are  encouraged  to  choose  and  use  appropriate  ICT  tools  such  as  graphic  display  calculators,  screenshots,  graphing,  spreadsheets,  databases,  drawing  and  word  processing  software,  as  appropriate,  to  enhance  mathematical  communication.  

0   The  exploration  does  not  reach  the  standard  described  by  the  descriptors.  

1   There  is  some  appropriate  mathematical  presentation.  

2   The  mathematical  presentation  is  mostly  appropriate.  

3   The  mathematical  presentation  is  appropriate  throughout.  

   

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C:  PERSONAL  ENGAGEMENT  (4)  

This  criterion  assesses  the  extent  to  which  the  student  engages  with  the  exploration  and  makes  it  their  own.  Personal  engagement  may  be  recognized  in  different  attributes  and  skills.  These  include  thinking  independently  and/or  creatively,  addressing  personal  interest  and  presenting  mathematical  ideas  in  their  own  way.  

0   The  exploration  does  not  reach  the  standard  described  by  the  descriptors.  

1   There  is  evidence  of  limited  or  superficial  personal  engagement.  

2   There  is  evidence  of  some  personal  engagement.  

3   There  is  evidence  of  significant  personal  engagement.  

4   There  is  abundant  evidence  of  outstanding  personal  engagement.  

D:  REFLECTION  (3)  

This  criterion  assesses  how  the  student  reviews,  analyses  and  evaluates  the  exploration.  Although  reflection  may  be  seen  in  the  conclusion  to  the  exploration,  it  may  also  be  found  throughout  the  exploration.  

0   The  exploration  does  not  reach  the  standard  described  by  the  descriptors.  

1   There  is  evidence  of  limited  or  superficial  reflection.  

2   There  is  evidence  of  meaningful  reflection.  

3   There  is  substantial  evidence  of  critical  reflection.  

E:  USE  OF  MATHEMATICS  (6)        

This  criterion  assesses  to  what  extent  and  how  well  students  use  mathematics  in  the  exploration.  SL  Only.  Students  are  expected  to  produce  work  that  is  commensurate  with  the  level  of  the  course.  The  mathematics  explored  should  either  be  part  of  the  syllabus,  or  at  a  similar  level  or  beyond.  It  should  not  be  completely  based  on  mathematics  listed  in  the  prior  learning.  If  the  level  of  mathematics  is  not  commensurate  with  the  level  of  the  course,  a  maximum  of  two  marks  can  be  awarded  for  this  criterion.  A  piece  of  mathematics  can  be  regarded  as  correct  even  if  there  are  occasional  minor  errors  as  long  as  they  do  not  detract  from  the  flow  of  the  mathematics  or  lead  to  an  unreasonable  outcome  

0   The  exploration  does  not  reach  the  standard  described  by  the  descriptors.  

1   Some  relevant  mathematics  is  used.  

2   Some  relevant  mathematics  is  used.  Limited  understanding  is  demonstrated.  

3   Relevant  mathematics  commensurate  with  the  level  of  the  course  is  used.    Limited  understanding  is  demonstrated.  

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4   Relevant  mathematics  commensurate  with  the  level  of  the  course  is  used.  The  mathematics  explored  is  partially  correct.  Some  knowledge  and  understanding  are  demonstrated.  

5   Relevant  mathematics  commensurate  with  the  level  of  the  course  is  used.  The  mathematics  explored  is  mostly  correct.  Good  knowledge  and  understanding  are  demonstrated.  

6   Relevant  mathematics  commensurate  with  the  level  of  the  course  is  used.  The  mathematics  explored  is  correct.  Thorough  knowledge  and  understanding  are  demonstrated.  

 

   

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Ideas  for  the  Exploration  Two  key  areas  to  consider  in  the  choice  of  a  topic  

1) Will  it  involve  mathematics  at  a  commensurate  level  to  the  course?  2) Will  it  offer  opportunities  for  student  reflection?  

Applications   –   these   might   appeal   to   students   since   they   may   well   have   an   interest   and   some  background  knowledge.  However,  a  good  deal  of  guidance  will  be  required  to  ensure  that  relevant  mathematics  commensurate  with  the  level  of  the  course  is  used!  

• Bar  codes  • Deforestation  • Engineering  • Football  • Global  warming  • Internet  security  • Investments  &  mortgages  • Pendulum  • Roller  coasters  • Skiing  • The  stock  market  • Triangulation  • Weather  Predictions  • Wind  turbines  

Mathematical   Topics   -­‐   It   is   possible   for   students   to   explore   some   mathematical   topics   that   lie  outside   the   SL   syllabus.  More   guidance   in   the   first   instance   will   be   required   from   the   teacher   in  explaining  what  these  topics  might  entail  but  there  should  be  plenty  of  resources  available.  The  level  of  mathematics  by  definition  should  be  high  enough  but  students  might  need  to  be  encouraged  to  show  reflection  in  their  work.  

• Central  Limit  Theorem  • Complex  Numbers  • Critical  Path  Analysis  • Differential  Equations  • Imaginary  numbers  • Matrices  • Numerical  Solution  of  Equations  • Polar  Coordinates  • Proof  by  Induction  • Taylor  Series  • The  Travelling  Salesman  • Transportation  Problems  • Hyperbolic  functions  

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Modelling   –   these   could   well   provide   a   simple   task   for   students   lacking   inspiration!   Often   the  challenge   is   to   find   the   data   to  model   in   the   first   place.  Will   students   be   able   to   get   across   their  ‘personal  engagement’?  

• BMI  • Population  growth  • Projectiles  • Spread  of  Viruses  • Sunrise  and  Sunset  times  • Tides  

 

The  occ  provides  the  following  very  broad  stimuli  for  the  exploration  

sport   archaeology   computers   algorithms  

cell  phones   music   sine   musical  harmony  

motion   e   electricity   water  

space   orbits   food   volcanoes  

diet   Euler   games   symmetry  

architecture   codes   the  internet   communication  

tiling   population   agriculture   viruses  

health   dance   play   π  

geography   biology   business   economics  

physics   chemistry   ITGS   psychology  

 

 

 

   

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Resources  

International  Mindedness  -­‐  Calculus  Given   the   significant  breakthroughs   that   they  made   in   this   area  of  mathematics   in   the   late17th  Century   it   is  often  considered  that  the  forefathers  of  Calculus  were  Newton  and  Leibnitz.    However,  during  the  17th  Century  many  other  European  mathematicians  contributed  to  the  ideas  of  the  derivative  and  long  before  that  ancient  Greek   mathematicians   contributed   to   the   idea   of   infinitesimally   small   amounts.   Between   this   many   other  mathematicians   from   around   the   world   contributed   to   the   ideas   that   formed   the   roots   of   calculus.     This  activity   aims   for   students   to   discover   the   international   nature   of   mathematics   and   different   cultures   have  contributed  over  time  to  its  development.    Lesson  Idea  1. The  following  names  should  be  put  on  the  board:    Gottfried  Leibniz,  Isaac  Newton  

 Students  are  brainstormed  to  find  out  what  they  know  about  these  people.They  might  need  some  guidance  to  understand  that  they  made  some  contribution  to  the  discovery  or  invention  of  calculus.  

 2. Students  are  put  into  pairs  and  given  one  of  the  following  names.  If  class  size  prevents  all  of  the  names  

being  used  it  should  be  ensured  that  a  mixture  of  nationalities  is  chosen.    

Al-­‐Biruni   Gottfried  Leibniz   Liu  Hui  Archimedes     Isaac  Barrow     Madhava  of  Sangamagrama    Bhaskaracharya   Isaac  Newton     Pierre  de  Fermat  Bonaventura  Cavalieri   James  Gregory   Zeno    Gilles  de  Roberval   Johannes  Kepler   Zu  Chongzhi    

 3. Using  the  internet  (or  other)  as  a  resource  tool  they  should  find  out  the  nationality  of  their  mathematician  

and  what  contribution  they  made  to  the  development  of  calculus.  4. The  teacher  provides  a  large  poster  of  a  world  map  and  students  are  asked  to  place  the  name  of  their  

mathematician  (&  picture)  on  the  poster  with  brief  description  of  discovery  made  by  that  mathematician.  5. Finally  the  teacher  makes  the  connections  between  them.  

The  poster  may  be  left  on  the  wall  of  the  classroom  for  display.  

   

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TOK  –  Randomness  and  Pseudo  randomness  Theory  of  Knowledge  is  concerned  with  how  we  know  what  we  claim  to  know.    Probability  and  randomness  is  an  interesting  and  often  counterintuitive  area  of  the  subject  and  people  are  often  poor  judges  of  randomness.    Hence  his  topic  provides  a  rich  source  of  discussion  about  our  knowledge  and  understanding  of  the  world.  Is  the   fact   that   randomness   is   often   surprising   or   counterintuitive   evidence   that   mathematics   exists  independently   of   us?   This   could   provide   an   interesting   debate   on   whether   mathematics   is   discovered   or  invented  (not  explored  here).  

This  lesson  discusses  whether  it  is  possible  to  have  truly  random  events  (non-­‐deterministic)  and  compares  random  and  pseudo-­‐random  events.  

Lesson  Outline  

I  roll  a  dice  6  times  and  get  the  following  results:  

1  4  2  6  5  3  

1  5  2  2  2  6  

6  6  6  6  6  6  

Discuss  these  results.    Look  at  the  following  2  images.  Which  one  do  you  think  looks  more  randomly  distributed?  

 

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Images  courtesy  of  Peter  Coles  

1. The  teacher  shows  the  class  100  numbers.    These  numbers  have  been  generated  ‘randomly’  by  a  spreadsheet  or  other.  Discuss:  Are   these  numbers   random?  A  number  or   sequence   is   said   to  be  random  if  it  is  non-­‐deterministic.  Discuss:  Are  these  numbers  random?  

2. Students   may   now   consider   pseudo-­‐random   numbers   and   how   computers   use   algorithms   to  generate   them   (since   a  machine   is   deterministic   it   cannot   use   an   algorithm   to   generate   truly  random  numbers.  

3. Students  in  pairs  roll  a  dice  10  times.    They  share  their  results  with  the  class.  Are  these  sequence  of   numbers   truly   random  or   could   they   ever   be   considered  deterministic?  Discuss  Newtonian  physics.    Are  they  random  or  do  we  just  not  know  enough  information  to  predict  the  outcome?  

4. Is  a  roulette  wheel  random?  Teacher  explains  the  case  of  a  roulette  wheel  scam  in  2004:  Three  Eastern  Europeans  gamblers  were  using  lasers  in  mobile  phones  to  predict  the  ball’s  behaviour.    It   is  thought  that  they  were  able  to  reduce  the  odds  of  winning  from  37:1  to  6:1  by  predicting  the  region  of  the  roulette  wheel  in  which  the  ball  would  land.    

5. The  class  could  investigate  some  of  the  tests  for  sequences  of  random  numbers  

a. The  frequency  test  

b. The  chi  squared  test  

c. The  serial  test  

d. The  poker  test  

e. The  gap  test  

If  time  allows  students  could  test  a  large  number  of  digits  of  π  (online  tests  exist  to  do  this).  The  digits  of  π  appear  random  and  it  passes  the  random  sequence  tests  yet  π  is  determined!  Tests  can  give  evidence  for  randomness  but  never  prove  it.    

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Images  

 

 

 

 

Source  :  XKCD.COM    

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Lesson  Activities  

Trigonometry  –  3-­‐2-­‐1  Blastoff!  

 

 

Borrow  a  water  powered  rocket  from  your  science  department.  

Find  a  large  space  and  stand  well  back  

Measure  distance  from  launch  

Pump  rocket  

Measure  angle  of  elevation  to  zenith  of  rocket  using  clinometer  

Calculate  the  height  attained  by  the  rocket  

 

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Trigonometry  –  Trigonometry  Function  Family  Reproduce  the  following  graphs!  

 

 

A  help  video  to  show  you  how  you  might  use  Geogebra  to  reproduce  these  graphs  can  be  found  here:  http://www.youtube.com/watch?v=o4dXkLzBDBY  

For  more  questions  like  this  and  the  complete  activity  visit  www.teachmathematics.net  

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Statistics  -­‐  Hunt  the  Box  Plot!    

 The  following  box  plots  have  been  made  using  data  sets  with  10  pieces  of  data.    The  view  window  is  always  the  same.    Can  you  create  a  data  set  that  will  reproduce  each  of  the  graphs?    

*  

*  

*     *  

*    

*  

   

   

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Probability  –  Dependent  Events    

 

 

A  hat  contains  5  red  and  3  blue  tickets.  To  win  the  game  you  need  to  draw  a  red  ticket.    

Player  one  draws  a  ticket.  The  ticket  is  not  replaced  in  the  hat.  Player  two  then  draws  a  ticket.    

Is  it  better  to  play  first  or  second?  

 

 

Here’s  some  help  to  figure  it  out.  

1. Find  the  probability  that  the  first  ticket  is  red.    

2. Find  the  probability  that    a. The  second  ticket  is  red  if  the  first  ticket  was  blue.  b. The  second  ticket  is  red  if  the  first  ticket  was  red.  

 3. What  is  the  probability  that  the  second  ticket  is  red?  4. What  do  you  notice  about  this  result?  5. Is  this  a  coincidence  or  would  it  happen  with  different  numbers  of  tickets  in  the  hat?  6. Imagine  a  hat  with  m  red  and  n  blue  tickets.  

 

 

Teacher  note:  It  is  always  good  to  play  this  game  in  class  first!  

   

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Probability  –  Crazy  Dice    

You   should   have   played   with   these   dice   (or   used   the  simulation)   and   have   found   out   that   red   is   stronger   than  blue,  blue  is  stronger  than  green  BUT  green  is  stronger  than  red!    This  doesn’t  mean  that  red  will  ALWAYS  beat  blue  in  a  single  game,  but  over  a  number  of  rolls  it  will  generally  do  better.  

Now,   it   is   time   to   analyse   these   dice   and   discover   why  certain  dice  will  win  over  others.  

 

One  Roll    

RED  Vs  BLUE  

RED  and  BLUE  go  head  to  head.    RED  should  win  but  with  what  probability?  

 

a. Here  is  a  possibility  space  for  all  the  36  possible  outcomes.    In  it  you  can  note  the  winners  for  each  

outcome.  Copy  and  complete  the  table.  

 

 

 

b. Use  it  to  find  the  probability  of  RED  winning  against  BLUE.    

   

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c. You  can  also  use  a  probability  tree  to  represent  the  rolls  of  the  two  dice.    Complete  it  and  use  it  to  find  the  probability  that  RED  wins  

 

 

 

 

 

 

 

The  answers  to  b  and  c  should  be  the  same!  

BLUE  Vs  GREEN  

Use  one  of  the  above  methods  to  show  that  the  probability  that  BLUE  

beats  GREEN  is  !"!".  

 GREEN  Vs  RED  

Find  the  probability  that  GREEN  beats  RED.  

 

 

2. Two  Rolls  Imagine  playing   the   same  game,  but   this   time  you   roll   your   chosen  dice   twice  and  add   the   scores  together  to  get  your  total.    Your  opponent  does  the  same  and  the  highest  total  score  wins.   If  RED  beats  BLUE  with  one  roll  then  surely  it  is  even  more  likely  to  beat  it  with  two  rolls?      

a. Use   the   following  possibility   space   to   find   the  possible  outcomes   from   two  rolls  of  the  RED  dice.  

Probability  (score  8)  =    

Probability  (score  13)  =    

Probability  (score  18)  =    

 

   

4  

9  

2  

2  7  

7  

Red  wins  56  

36  

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4  

 9  

14  

b. Find  the  possible  outcomes  from  two  rolls  of  the  BLUE  dice.    

Probability  (score  4)  =    

Probability  (score  9)  =    

Probability  (score  14)  =    

 

 

 

c. Use   a   probability   tree   like   this   to   find   the   probability   of   TWO   RED   rolls  beating  TWO  BLUE  rolls.  

 

                     

 d. How  does  this  result  compare  to  ONE  roll?  e. What  is  the  probability  of  TWO  BLUE  rolls  beating  TWO  GREEN  rolls?  f. What  about  GREEN  against  RED?  g. How  do  the  results  for  TWO  rolls  compare  to  ONE  roll?  

 3. Extension  

a. In  a  one  roll  contest  can  you  create  another  dice  that  will  beat  red  but  lose  to  green?  Make  this  dice  yellow.  

b. In  a  one  roll  contest  can  you  create  another  dice  that  will  beat  red  but  lose  to  yellow?  Make  this  dice  magenta.  

 

 

Full  activity  can  be  found  at  www.teachmathematics.net  

18  

13  

8  

4  

 9  

14  

4    9  

14  

4  

4  

4  

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Calculus  –  Parabola  Proof    

 

   

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Calculus  –  Investigating  Areas  under  Graphs      

A. Linear  Functions    

Find  the  shaded  areas  below  

 

     

 

 

 

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B. Quadratic  Functions  Calculating  exact  areas  under  curves  is  difficult,  but  you  can  make  estimates.  

Here  is  the  graph  of  y=x².      

 

We  can  divide  the  area  under  the  graph  into  5  rectangles.    Calculate  the  total  area  of  these  rectangles.  

 

 

 

 

This  estimate  of  the  area  can  be  improved  by  increasing  the  number  of  rectangles,  to  10  for  example  

 

 

 

 

 

Or  even  more…  

The  more  rectangles  we  divide  the  area   into  the  better  the  estimate  of  the  area  becomes,  but  the  more  arduous  the  calculation  gets!    Fortunately  you  can  use  Technology  to  make  these  estimates  for  you.      

You  can  use  Geogebra  to  make  estimates  of  the  area  bounded  by  the  curve  by  using  the  commands  LowerSum  and  UpperSum  

Use  Geogebra  to  find  the  area  bounded  by  the  curve  y=x²  and    

a) x=0  and  x=1    b) x=0  and  x=b    c) x=a  and  x=b            

You  may  wish  to  use  the  regression  tool  analysis  on  your  GDC  to  help  you  find  this  formula  

C. Cubic  Functions  Use  Geogebra  to  an  estimate  of  the  area  bounded  by  the  graph  of  y=x3  and    x=a  and  x=b  

D. Generalising  What  would  be  the  area  of  bounded  by  the  graph  of  y=xn  and    x=a  and  x=b?  

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Research  –  Aligning  Teaching  for  Constructive  Learning  By  John  Biggs  

Summary  

'Constructive  alignment'   starts  with   the  notion   that   the   learner  constructs  his  or  her  own   learning  through   relevant   learning   activities.   The   teacher's   job   is   to   create   a   learning   environment   that  supports   the   learning  activities  appropriate   to  achieving   the  desired   learning  outcomes.  The  key   is  that  all  components  in  the  teaching  system  -­‐  the  curriculum  and  its  intended  outcomes,  the  teaching  methods  used,  the  assessment  tasks  -­‐  are  aligned  to  each  other.  All  are  tuned  to  learning  activities  addressed  in  the  desired  learning  outcomes.  The  learner  finds  it  difficult  to  escape  without  learning  appropriately.  

Biography  

John   Biggs   obtained   his   Ph.D.   from   the   University   of   London   in   1963,   and   has   held   Chairs   in  Education   in  Canada,  Australia,  and  Hong  Kong.  He  retired   in  1995  to  act  as  a  consultant   in  Higher  Education,  and  has  been  employed  in  this  capacity  in  many  institutions  in  Australia,  Hong  Kong,  and  the  United  Kingdom.  

Keywords  

intended   learning  outcomes,  constructive  alignment,  criterion-­‐referenced  assessment,   teaching   for  active  learning,  systems  approach  to  teaching.  

Introduction  

Teaching  and  learning  take  place  in  a  whole  system,  which  embraces  classroom,  departmental  and  institutional   levels.  A  poor  system  is  one   in  which  the  components  are  not   integrated,  and  are  not  tuned  to  support  high-­‐level  learning.  In  such  a  system,  only  the  'academic'  students  use  higher-­‐order  learning  processes.   In  a  good  system,  all  aspects  of   teaching  and  assessment  are  tuned  to  support  high   level   learning,   so   that   all   students   are   encouraged   to   use   higher-­‐order   learning   processes.  'Constructive  alignment'  (CA)  is  such  a  system.  It  is  an  approach  to  curriculum  design  that  optimises  the  conditions  for  quality  learning.  

For  an  example  of  a  poor  system,  here  is  what  a  psychology  undergraduate  said  about  his  teaching:  

'I  hate  to  say  it,  but  what  you  have  got  to  do  is  to  have  a  list  of  'facts';  you  write  down  ten  important  points   and  memorize   those,   then   you'll   do   all   right   in   the   test   ...   If   you   can   give   a   bit   of   factual  information  -­‐  so  and  so  did  that,  and  concluded  that  -­‐  for  two  sides  of  writing,  then  you'll  get  a  good  mark.'  Quoted  in  Ramsden  (1984:  144)  

The  problem  here  was  not  the  student.  In  fact,  this  student  liked  writing  extended  essays,  and  finally  graduated   with   First   Class   Honours,   but   he   was   contemptuous   of   these   quick   and   snappy  assessments.   So   in   psychology,   he   made   a   strategic   decision   to   memorise,   knowing   that   it   was  enough   to   get   him   through,   saving   his   big   guns   for   his  major   subject.   The   problem   here  was   the  assessment:  it  was  not  aligned  with  the  aims  of  teaching.  

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So  often   the   rhetoric   in   courses   and  programmes   is   all   that   it   should  be,   stating   for   example   that  students  will  graduate  with  a  deep  understanding  of  the  discipline  and  the  ability  to  solve  problems  creatively.  Then  they  are  told  about  creative  problem  solving  in  packed  lecture  halls  and  tested  with  multiple-­‐choice   tests.   It's   all   out   of   kilter,   but   such   a   situation   is   not,   I   strongly   suspect,   all   that  uncommon.  

What  is  constructive  alignment?  

'Constructive  alignment'  has  two  aspects.  The  'constructive'  aspect  refers  to  the  idea  that  students  construct  meaning  through  relevant  learning  activities.  That  is,  meaning  is  not  something  imparted  or   transmitted   from   teacher   to   learner,   but   is   something   learners   have   to   create   for   themselves.  Teaching  is  simply  a  catalyst  for  learning:  

'If   students   are   to   learn   desired   outcomes   in   a   reasonably   effective   manner,   then   the   teacher's  fundamental   task   is   to  get  students   to  engage   in   learning  activities   that  are   likely   to   result   in   their  achieving   those  outcomes...   It   is   helpful   to   remember   that  what   the   student  does   is   actually  more  important  in  determining  what  is  learned  than  what  the  teacher  does.'  (Shuell,  1986:  429)  

The   'alignment'  aspect   refers   to  what   the   teacher  does,  which   is   to  set  up  a   learning  environment  that  supports  the  learning  activities  appropriate  to  achieving  the  desired  learning  outcomes.  The  key  is   that   the   components   in   the   teaching   system,   especially   the   teaching   methods   used   and   the  assessment   tasks,   are   aligned  with   the   learning   activities   assumed   in   the   intended  outcomes.   The  learner   is   in   a   sense   'trapped',   and   finds   it   difficult   to   escape  without   learning  what   he   or   she   is  intended  to  learn.  

In  setting  up  an  aligned  system,  we  specify  the  desired  outcomes  of  our  teaching  in  terms  not  only  of  topic   content,   but   in   the   level   of   understanding  we  want   students   to   achieve.  We   then   set   up   an  environment   that  maximises   the   likelihood   that   students  will   engage   in   the   activities   designed   to  achieve   the   intended   outcomes.   Finally,   we   choose   assessment   tasks   that   will   tell   us   how   well  individual  students  have  attained  these  outcomes,   in  terms  of  graded  levels  of  acceptability.  These  levels  are  the  grades  we  award.  

There  are  thus  four  major  steps:    

1.  Defining  the  intended  learning  outcomes  (ILOs);    

2.  Choosing  teaching/learning  activities  likely  to  lead  to  the  ILOs;    

3.  Assessing  students'  actual  learning  outcomes  to  see  how  well  they  match  what  was  intended;    

4.  Arriving  at  a  final  grade.  

Defining  the  ILOs  

When   we   teach   we   should   have   a   clear   idea   of   what   we   want   our   students   to   learn.   More  specifically,  on  a  topic  by  topic  basis,  we  should  be  able  to  stipulate  how  well  each  topic  needs  to  be  understood.   First,   we   need   to   distinguish   between   declarative   knowledge   and   functioning  knowledge.  

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Declarative   knowledge   is   knowledge   that   can   be   'declared':   we   tell   people   about   it,   orally   or   in  writing.   Declarative   knowledge   is   usually   second-­‐hand   knowledge;   it   is   about   what   has   been  discovered.  Knowledge  of  academic  disciplines  is  declarative,  and  our  students  need  to  understand  it  selectively.  Declarative  knowledge  is,  however,  only  the  first  part  of  the  story.  

We  don't  acquire  knowledge  only  so  that  we  can  tell  other  people  about  it;  more  specifically,  so  that  our  students  can  tell  us  -­‐   in  their  own  words  of  course  -­‐  what  we  have  recently  been  telling  them.  Our   students  need   to  put   that  knowledge   to  work,   to  make   it   function.  Understanding  makes  you  see  the  world  differently,  and  behave  differently  towards  that  part  of  the  world.  We  want  lawyers  to  make  good  legal  decisions,  doctors  to  make  accurate  diagnoses,  physicists  to  think  and  behave  like  physicists.   After   graduation,   all   our   students,   whatever   their   degree   programmes,   should   see   a  section   of   their  world   differently,   and   to   behave   differently   towards   it,   expertly   and  wisely.   Thus,  simply   telling  our   students  about   that  part  of   the  world,  and  getting   them  to   read  about   it,   is  not  likely   to   achieve   our   ILOs   with   the  majority   of   students.   Good   students   will   turn   declarative   into  functioning  knowledge  in  time,  but  most  will  not  if  they  are  not  required  to.  

Accordingly,  we   have   to   state   our   objectives   in   terms   that   require   students   to   demonstrate   their  understanding,   not   just   simply   tell   us   about   it   in   invigilated  exams.   The   first   step   in   designing   the  curriculum   objectives,   then,   is   to   make   clear   what   levels   of   understanding   we   want   from   our  students  in  what  topics,  and  what  performances  of  understanding  would  give  us  this  knowledge.  

It   is   helpful   to   think   in   terms   of   appropriate   verbs.   Generic   high   level   verbs   include:   Reflect,  hypothesise,  solve  unseen  complex  problems,  generate  new  alternatives.  

Low   level   verbs   include:  Describe,   identify,  memorise,   and   so   on.   Each   discipline   and   topic  will   of  course   have   its   own   appropriate   verbs   that   reflect   different   levels   of   understanding,   the   topic  content  being  the  objects  the  verbs  take.    

Incorporating  verbs  in  our  intended  learning  outcomes  gives  us  markers  throughout  the  system.  The  same  verbs  need  to  be  embedded   in   the   teaching/learning  activities,  and   in   the  assessment   tasks.  They  keep  us  on  track.  

Choosing  teaching/learning  activities  (TLAs)  

Teaching   and   learning   activities   in  many   courses   are   restricted   to   lecture   and   tutorial:   lecture   to  expound  and  package,  and  tutorial  to  clarify  and  extend.  However,  these  contexts  do  not  necessarily  elicit   high   level   verbs.   Students   can   get   away   with   passive   listening   and   selectively   memorising.  There  are  many  other  ways  of  encouraging  appropriate   learning  activities   (Chapter  5,  Biggs  2003),  even   in   large  classes   (Chapter  6,  op.   cit.),  while  a   range  of  activities   can  be   scheduled  outside   the  classroom,   especially   but   not   only   using   educational   technology   (Chapter   10,   op   cit.).   In   fact,  problems   of   resourcing   conventional   on-­‐campus   teaching,   and   the   changing   nature   of   HE,   are  coming  to  be  blessings   in  disguise,  forcing   learning  to  take  place  outside  the  class,  with   interactive  group  work,   peer   teaching,   independent   learning   and  work-­‐based   learning,   all   of  which   are   a   rich  source  of  relevant  learning  activities.  

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Assessing  students'  learning  outcomes  

Faulty  assumptions  about  and  practices  of  assessment  do  more  damage  by  misaligning  teaching  than  any  other  single   factor.  As  Ramsden   (1992)  puts   it,   the  assessment   is   the  curriculum,  as   far  as   the  students  are  concerned.  They  will  learn  what  they  think  they  will  be  assessed  on,  not  what  is  in  the  curriculum,   or   even   on   what   has   been   'covered'   in   class.   The   trick   is,   then,   to   make   sure   the  assessment  tasks  mirror  the  ILOs.  

 

To   the   teacher,   assessment   is   at   the   end   of   the   teaching-­‐learning   sequence   of   events,   but   to   the  student   it   is   at   the  beginning.   If   the  curriculum   is   reflected   in   the  assessment,  as   indicated  by   the  downward  arrow,  the  teaching  activities  of  the  teacher  and  the  learning  activities  of  the  learner  are  both  directed  towards  the  same  goal.  In  preparing  for  the  assessments,  students  will  be  learning  the  curriculum.  The  cynical  game-­‐playing  we  saw  in  our  psychology  undergraduate  above,  with  his  'two  pages  of  writing',  is  pre-­‐empted.  

Matching   individual   performances   against   the   criteria   is   not   a   matter   of   counting   marks   but   of  making   holistic   judgments.   This   is   a   controversial   issue,   and   is   dealt   with   in   more   detail   in   Biggs  (2003,  Chapters  8  and  9).   Just   let  me  say  here   that   the   ILOs  cannot   sensibly  be  stated   in   terms  of  marks  obtained.  Intended  outcomes  refer  to  sought-­‐for  qualities  of  performance,  and  it  is  these  that  need  to  be  stated  clearly,  so  that  the  students'  actual  learning  outcomes  can  be  judged  against  those  qualities.  If  this  is  not  done,  we  are  not  aligning  our  objectives  and  our  assessments.  

Conclusion  

Constructive  alignment  is  more  than  criterion-­‐reference  assessment,  which  aligns  assessment  to  the  objectives.  CA  includes  that,  but  it  differs  (a)  in  talking  not  so  much  about  the  assessment  matching  the  objectives,  but  of  first  expressing  the  objectives  in  terms  of  intended  learning  outcomes  (ILOs),  which  then  in  effect  define  the  assessment  task;  and  (b)  in  aligning  the  teaching  methods,  with  the  intended  outcomes  as  well  as  aligning  just  the  assessment  tasks.  

References    

Biggs,  J.B.  (2003).  Teaching  for  quality  learning  at  university.  Buckingham:  Open  University  Press/Society  for  Research  into    

Higher  Education.  (Second  edition)  Ramsden,  P.  (1984).  The  context  of  learning.  In  F.  Marton,  D.  Hounsell,  and  N.  Entwistle,  N.  (eds),  The  Experience  of  Learning.  Edinburgh:  Scottish  Academic  Press.    

Ramsden,  P.  (1992).  Learning  to  teach  in  higher  education.  London:  Routledge.    

Shuell,  T.J.  (1986).  Cognitive  conceptions  of  learning.  Review  of  Educational  Research,  56,  411-­‐436.