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Maths SL
Workbook for Category 2 IBDP teacher workshop
10-12 October 2013 Amsterdam, the Netherlands
Richard Wade Oliver Bowles
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 brad@philpot.nl
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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
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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
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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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*
* *
*
*
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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.