UAIS IB Mathematics HL course outline

Name of the teacher who prepared the outline:

Tiffany McNair

Course description:

This course caters for students with a good background in mathematics who are competent in a range of analytical and technical skills. The majority of these students will be expecting to include mathematics as a major component of their university studies, either as a subject in its own right or within courses such as physics, engineering and technology. Others may take this subject because they have a strong interest in mathematics and enjoy meeting its challenges and engaging with its problems. Students should be expected to understand mathematics conceptually as well as in application. A component of Math HL is the need for the student to make connections between several topics of mathematics in order to problem solve.

The course focuses on developing important mathematical concepts in a comprehensible, coherent and rigorous way. This is achieved by means of a carefully balanced approach. Students are encouraged to apply their mathematical knowledge to solve problems set in a variety of meaningful contexts. Development of each topic should feature justification and proof of results. Students embarking on this course should expect to develop insight into mathematical form and structure, and should be intellectually equipped to appreciate the links between concepts in different topic areas. They should also be encouraged to develop the skills needed to continue their mathematical growth in other learning environments. Students are expected to be fluent in mathematics with and without a graphics display calculator.

Mathematics is in a sense an international language, and, apart from slightly differing notation, mathematicians from around the world can communicate within their field. Many of the foundations of modern mathematics were laid many centuries ago by Arabic, Greek, Indian and Chinese civilizations, among others and the language of mathematics continues to develop with technology. To demonstrate the universality of mathematics in a historical context, Math HL students will learn about mathematicians and the historical context in which they worked.

The internally assessed component, the exploration, offers students the opportunity for developing independence in their mathematical learning. Students are encouraged to take a considered approach to various mathematical activities and to explore different mathematical ideas. The exploration also allows students to work without the time constraints of a written examination and to develop the skills they need for communicating mathematical ideas. It is essential for students to communicate their appreciation for the beauty of mathematics and its place in the universe as well as a connection to a persona interest of theirs. Students must also sit for three externally graded examinations; with and without a GDC.

Year 1Topic (from syllabus) Sections Timeframe

Algebra (1)(covered partially)

1.1 – Sequences and series; sum of finite arithmetic series; geometric sequences and series; sum of finite and infinite geometric series; Sigma notation; Applications.

1.2 – Exponents and Logarithms; Laws of Exponents; Change of Base1.3 – Counting Principles, including permutations and combinations; The binomial theorem:

expansion of (a+b )n , n∈N 1.4 – Proof by Mathematical Induction

1.5 – Complex numbers: the numberi=√−1 ; the terms real part, imaginary part, conjugate. Cartesian formz=a+bi . Sums, products and quotients of complex numbers.

1.8 – Conjugate roots of polynomial equations with real coefficients.1.9 – Solutions of systems of linear equations, including cases where there is a unique

solution, an infinity of solutions or no solution.

20 hours

Optional Links to ToK Optional Links to Internationalismo What is Zeno’s dichotomy paradox? How far can

mathematical facts be from intuition?o How did Gauss add up integers from 1 to 100?

Discuss the idea of mathematical intuition as the basis for formal proof.

o The nature of mathematics. The unforeseen links between Pascal’s triangle, counting methods and the coefficients of polynomials. Is there an underlying truth that can be found linking these?

o Debate over the validity of the notion of “infinity”: finitists such as L. Kronecker consider that “a mathematical object does not exist unless it can be constructed from natural numbers in a finite number of steps”.

o Prove √2 is irrationalo How many different tickets are possible in a lottery? What

does this tell us about the ethics of selling lottery tickets to those who do not understand the implications of these large numbers?

o Mathematics, sense, perception and reason. If we can find

o The chess legend (Sissa ibn Dahir).o Aryabhatta is sometimes considered the “father

of algebra”. Compare with al-Khawarizmi.o Pascal’s triangle. Attributing the origin of a

mathematical discovery to the wrong mathematician.

o The so-called “Pascal’s triangle” was known in China much earlier than Pascal.

o Koch’s Snowflake

solutions in higher dimensions, can we reason that these spaces exist beyond our sense perception?

o Nature of mathematics and science. What are the different meanings of induction in mathematics and science?

o Knowledge claims in mathematics. Do proofs provide us with completely certain knowledge? Who judges the validity of a proof?

Functions (2) 2.1 – Concept of function f : x↦ f (x ): domain, range, image (value); Odd and even

functions; Composite functions f ∘g ; Identity function; One-to-one and many-to-one

functions; Inverse function f−1

, including domain restriction. Self-inverse functions.

2.2 – The graph of a function; its equation y=f ( x ) ; Investigation of key features of graphs, such as maximum and minimum values, intercepts, horizontal and vertical asymptotes and symmetry, and consideration of domain and range; The graphs of the functions,

y=|f ( x )| and y=f (|x|) ; The graph of y= 1

f ( x ) from y=f ( x ) .2.3 – Transformations of graphs; translations; stretches; reflections in the axes; The graph of

the inverse as the reflection in the line y=x .

2.4 – The rational function x↦ ax+b

cx+d and its graph; The function x↦ ax , a>0 and its

graph; The function y=loga x , x>0 and its graph.2.5 – Polynomial functions and their graphs; The factor and remainder theorems; The

fundamental theorem of algebra2.6 – Solving quadratic equations using the quadratic formula; Use of the discriminant

Δ=b2−4 ac to determine the nature of the roots; Solving polynomial equations both graphically and algebraically; Sum and product of the roots of polynomial equations;

Solution of ax=b using logarithms; Use of technology to solve a variety of equations,

including those where there is no appropriate analytic approach.

2.7 – Solutions ofg ( x )≥f ( x ) ; Graphical or algebraic methods, for simple polynomials up

22 hours

to degree 3; Use of technology for these and other functions.

Optional Links to ToK Optional Links to Internationalismo Is zero the same as “nothing”?o How accurate is a visual representation of a

mathematical concept? (Limits of graphs in delivering information about functions and phenomena in general, relevance of modes of representation.)

o Mathematics and the knower. To what extent should mathematical knowledge be consistent with our intuition?

o Mathematics and the world. Some mathematical constants (pi, e, Fibonacci numbers) appear consistently in nature. What does this tell us about mathematical knowledge?

o The nature of mathematics. Is mathematics simply the manipulation of symbols under a set of formal rules?

o The nature of mathematics and science. Were logarithms an invention or discovery?

o Short-term loans at high interest rates. How can knowledge of mathematics result in individuals being exploited or protected from extortion?

o Mathematics and knowledge claims. Does studying the graph of a function contain the same level of mathematical rigor as studying the function algebraically (analytically)?

o The phrase “exponential growth” is used popularly to describe a number of phenomena. Is this a misleading use of a mathematical term?

o Economics SL/HL – shifts in demand and supply curves.

o The development of functions, Rene Descartes (France), Gottfried Wilhelm Leibniz (Germany) and Leonhard Euler (Switzerland)

o The Babylonian method of multiplication:

ab=(a+b )2−a2−b2

2o Sulba Sutras in ancient India and the Bakhshali

Manuscript contained an algebraic formula for solving quadratic equations.

o The Legend of the Ambalappuzha Paal Payasamo Euler, John Napiero The notation for functions was developed by a

number of different mathematicians in the 17th and 18th centuries. How did notation we use today become internationally accepted?

Trigonometry (3) 3.1 – The circle: radian measure of angles; Length of an arc; Area of a sector.3.2 – Definition ofsin θ , cosθ and tanθ in terms of the unit circle. Exact values of sin, cos

22 hours

and tan of 0 , π

6, π

4, π

3, π

2 and their multiples; Definition of the reciprocal trigonometric

ratioscscθ ,secθ andcot θ ; Pythagorean identities: cos2 θ+sin2 θ=1 , 1+ tan2 θ=sec2 θ , 1+cot2θ=csc2 θ .

3.3 – Compound angle identities; Double angle identities.

3.4 – Composite functions of the formf ( x )=a sin (b ( x+c ) )+d ; Applications.

3.5 – The inverse functionsx↦ arcsin x , x↦arccos x , x↦ arctan x ; their domains and ranges; their graphs.

3.6 – Algebraic and graphical methods of solving trigonometric equations in a finite interval, including the use of trigonometric identities and factorization.

3.7 – The cosine rule; The sine rule including the ambiguous case; Area of a triangle as12

ab sin C; Applications.

Optional Links to ToK Optional Links to Internationalismo Which is a better measure of angle: radian or

degree? What are the “best” criteria by which to decide?

o Trigonometry was developed by successive civilizations and cultures. How is mathematical knowledge considered from a sociocultural perspective?

o Non-Euclidean geometry: angle sum on a globe greater than 180°.

o What other measures of angle are there? Which is the most natural unit of angle measure?

o Nature of mathematics. If the angles of a triangle can add up to less than 180°, 180° or more than 180°, what does this tell us about the “fact” of the angle sum of a triangle and about the nature of mathematical knowledge?

o Mathematics and the knower. Why do we use radians? (The arbitrary nature of degree measure versus radians as real numbers and the implications of using these two measures on

o Seki Takakazu calculating π to ten decimal places.o Hipparchus, Menelaus and Ptolemy.o Why are there 360 degrees in a complete turn?

Links to Babylonian mathematics.o The first work to refer explicitly to the sine as a

function of an angle is the Aryabhatiya of Aryabhata (ca. 510).

o Cosine rule: Al-Kashi and Pythagoras.o The origin of degrees in the mathematics of Mesopotamia and

why we use minutes and seconds for time.o Why did Pythagoras link the study of music and mathematics?o The use of triangulation to find the curvature of the Earth in

order to settle a dispute between England and France over Newton’s gravity.

o Michael Faraday and electric currento The origin of the word “sine”.o Euclid’s axioms as the building blocks of

the shape of sinusoidal graphs.)o Mathematics and knowledge claims. If trigonometry is based

on right triangles, how can we sensibly consider trigonometric ratios of angles greater than a right angle?

o Mathematics and knowledge claims. How can there be an infinite number of discrete solutions to an equation?

o Mathematics and the world. Music can be expressed using mathematics. Does this mean that music is mathematical, that mathematics is musical or that both are reflections of a common “truth”?

Euclidean geometry. Link to non-Euclidean geometry.

Calculus (6)(covered partially)

6.1 – Informal ideas of limit and convergence; Definition of derivative from first principles

f ' ( x )= limh→0 ( f ( x+h )−f (x )

h ); The derivative interpreted as a gradient function and as

rate of change; Finding equations of tangents and normal; Identifying increasing and decreasing functions; The second derivative; Higher derivatives.

6.2 – Derivatives of xn ,sin x ,cos x ,tan x , e x ,ln x ; Differentiation of sums and multiple

functions; The product and quotient rules; The chain rule for composite functions; Related rates of change; Implicit differentiation; Derivatives of sec x ,csc x ,cot x , ax ,loga x ,arcsin x ,arccos x ,arctan x

6.3 – Local maximum and minimum values; Optimization problems; Points of inflexion with zero and non-zero gradients; Graphical behavior of functions, including the relationship

between the graphs of f , f ', f ''

6.4 – Indefinite integration as anti-differentiation; Indefinite integral ofxn ,sin x ,cos x , ex;

Other indefinite integrals using the results of 6.2; The composite of any of these with a linear function.

6.5 – Anti-differentiation with a boundary condition to determine the constant of integration; Definite integrals; Area of a region enclosed by a curve and the x-axis or y-axis in a given interval; areas of regions enclosed by curves; Volumes of revolution about the x-axis or y-axis.

38 hours

Optional Links to ToK Optional Links to Internationalismo What value does the knowledge of limits have? Is

infinitesimal behavior applicable to real life?o The nature of mathematics. Does the fact that Leibniz and

Newton came across the calculus at similar times support the argument that mathematics exists prior to its discovery?

o Mathematics and the knower. What does the dispute between Newton and Leibniz tell us about human emotion and mathematical discovery?

o Mathematics and knowledge claims. Euler was able to make important advances in mathematical analysis before calculus had been put on a solid theoretical foundation by Cauchy and others. However, some work was not possible until after Cauchy’s work. What does this tell us about the importance of proof and the nature of mathematics?

o Mathematics and the real world. The seemingly abstract concept of calculus allows us to create mathematical models that permit human feats, such as getting a man on the Moon. What does this tell us about the links between mathematical models and physical reality?

o Is optimization unique to mathematics? How does mathematics fit into the scientific method? Does mathematics have a prescribed method of its own? Is mathematics a science?

o Opportunities for discussing hypothesis formation and testing, and then the formal proof can be tackled by comparing certain cases, through an investigative approach

o Mathematics, sense, perception and reason. If we can find solutions in higher dimensions, can we reason that these spaces exist beyond our sense perception?

o Successful calculation of the volume of the pyramidal frustum by ancient Egyptians (Egyptian Moscow papyrus).

o Use of infinitesimals by Greek geometers.o Accurate calculation of the volume of a cylinder

by Chinese mathematician Liu Hui.o Ibn Al Haytham: first mathematician to calculate

the integral of a function, in order to find the volume of a paraboloid.

o How the Greeks’ distrust of zero meant that Archimedes’ work did not lead to calculus.

Statistics and Probability (5)

(partially covered)

5.1 – Concepts of population, sample, random sample and frequency distribution of discrete and continuous data; Grouped data; mid-interval values, interval width, upper and lower interval boundaries; Mean, variance, standard deviation.

20 hours

5.2 – Concepts of trial, outcome, equally likely outcomes, sample space (U) and event; The

probability of event A asP( A )=

n( A )n(U ) ; The complementary events A and A’ (not A);

Use of Venn diagrams, tree diagrams, counting principles and tables of outcomes to solve problems.

5.3 – Combined events, the formula forP( A∪B ) ; Mutually exclusive events.

5.4 – Conditional probability; the definition P( A|B )=

P( A∩B )P( B) ; Independent events; the

definition P( A|B )=P ( A )=P ( A|B ' ); Use of Bayes’ theorem for a maximum of 3 events.

Optional Links to ToK Optional Links to Internationalismo Do different measures of central tendency

express different properties of the data?o How easy is it to lie with statistics?o Is mathematics useful to measure risks?o Does the use of statistics lead to an overemphasis on attributes

that can easily be measured over those that cannot?o The nature of mathematics. Why have mathematics and

statistics sometimes been treated as separate subjects?o Misleading statistics in media reports.o The nature of knowing. Is there a difference between

information and data?o In what ways can mathematics model the world

without using functions? How does a knowledge of probability theory affect decisions we make? Do ethics play a role in the use of mathematics?

o Can gambling be considered as an application of mathematics? (This is a good opportunity to generate a debate on the nature, role and ethics of mathematics regarding its applications.)

o Mathematics and knowledge claims. Is independence as defined in probabilistic terms the same as that found in normal experience?

o The St Petersburg paradox, Chebychev, Pavlovsky.

o Discussion of the different formulae for variance.o Blaise Pascal and Pierre de Fermat

o Analyzing charts and graph for bias and deceptiono Why has it been argued that theories based on the calculable

probabilities found in casinos are pernicious when applied to everyday life (eg economics)?

Year 2Topic Sections Timeframe

Vectors (4)(partially covered)

4.1 – Concept of a vector; Representation of vectors using directed line segments; Unit

vectors; base vectors i, j, k; Components of a vector: v = (v1

v2

v3)=v1 i+v2 j+v3 k

; Algebraic and geometric approaches to the following:

the sum and difference of two vectors; the zero vector 0, the vector –v; multiplication by a scalar kv; magnitude of a vector, |v|;

position vectors O A⃗=a ;

A B⃗=b – a 4.2 – The definition of the scalar product of two vectors; Properties of the scalar product:

v . w = w . v

10 hours

u . (v + w) = u . v + u . w (kv) . w = k(w . v) v . v = |v|2

The angle between two vectors; Perpendicular vectors; Parallel vectors. 4.5 – The definition of the vector product of two vectors; Properties of the vector product:

v w = - w v; u (v + w) = u . v + u . w; (kv) w = k(w v); v v = 0;

Geometric interpretation of | v w |Optional Links to ToK Optional Links to Internationalism

o Vectors are used to solve many problems in position location. This can be used to save a lost sailor or destroy a building with a laser-guided bomb.

o Are algebra and geometry two separate domains of knowledge? (Vector algebra is a good opportunity to discuss how geometrical properties are described and generalized by algebraic methods.)

o The nature of mathematics. Why this definition of scalar product?

o Mathematics and knowledge claims. You can perform some proofs using different mathematical concepts. What does this tell us about mathematical knowledge?

Algebra (1)(completed)

1.5 – The terms modulus and argument. 1.6 – Modulus–argument (polar) form z = r(cos + i sin). The complex plane.1.7 – Powers of complex numbers: de Moivre’s theorem; nth roots of a complex number

10 hours

Optional Links to ToK Optional Links to Internationalismo Mathematics and the knower. Do the words

imaginary and complex make the concepts more difficult than if they had different names?

o Mathematics and the world. Why does i appear in so many fundamental laws of physics?

o The nature of mathematics. Has i been invented

or was it discovered?o Mathematics and the knower. Why might it be

said that e iπ+1=0 is beautiful?o The nature of mathematics. Was the complex plane already

there or was it used to represent complex numbers geometrically?

Vectors (4)(completed)

4.3 – Vector equation of a line in two and three dimensions: r = a + b; Simple applications to kinematics; The angle between two lines

4.4 – Coincident, parallel, intersecting and skew lines, distinguishing between these cases; Points of intersections.

4.6 – Vector equation of a plane r = a + b + c; Use of normal vector to obtain the formr . n = a . n; Cartesian equation of a plane ax + by + cz = d.

4.7 – Intersections of: a line with a plane, two planes, three planes; Angle between: a line and a plane, two planes.

14 hours

Optional Links to ToK Optional Links to Internationalismo Modelling linear motion in three dimensions.o The nature of mathematics. Why might it be

argued that vector representation of lines is superior to Cartesian?

o Physics SL/HL – magnetic force and field.o Mathematics and the knower. Why are symbolic

representations of three-dimensional objects easier to deal with than visual representations? What does this tell us about our knowledge of mathematics in other dimensions?

o How do we relate a theory to the author? Who developed vector analysis: JW Gibbs or O Heaviside?

Calculus (6)(completed)

6.6 – Kinematic problems involving displacement s, velocity v, and acceleration a; Total distance travelled.

6.7 – Integration by substitution; Integration by parts.10 hours

Optional Links to ToK Optional Links to Internationalismo Does the inclusion of kinematics as core

mathematics reflect particular cultural heritage?o Investigate attempts by Indian mathematicians (500–1000 CE)

to explain division by zero.

Statistics and Probability (5)

(completed)

5.5 – Concepts of discrete and continuous random variables and their probability distributions; Definition and use of probability density functions; Expected value (mean), mode, median, variance and standard deviation; Applications.

5.6 – Binomial distribution, its mean and variance; Poisson distribution, its mean and variance.5.7 – Normal distribution; Properties of the normal distribution; Standardization of normal

variables.

16 hours

Optional Links to ToK Optional Links to Internationalismo Mathematics and knowledge claims. To what extent can we

trust mathematical models such as the normal distribution?o Expected gains to insurance companies.o Use of probability methods in medical studies to assess risk

factors for certain diseases.o Why might the misuse of the normal distribution lead to

dangerous inferences and conclusions?o Mathematics and the real world. Is the binomial distribution

ever a useful model for an actual real-world situation?

o De Moivre’s derivation of the normal distribution and Quetelet’s use of it to describe l’homme moyen.

Option: Statistics and Probability (7)

7.1 – Cumulative distribution functions for both discrete and continuous distributions; Geometric distribution; Negative binomial distribution; Probability generating functions for discrete random variables; Using probability generating functions to find the mean, variance and the distribution of the sum of n independent random variables.

7.2 – Linear transformation of a single random variable. Mean of linear combinations of n random variables; Variance of linear combinations of n independent random variables; and variance; Expectation of the product of independent random variables.

7.3 – Unbiased estimators and estimates; Comparison of unbiased estimators based on

variances; X as an unbiased estimator forμ ; S2

as an unbiased estimator forσ2

.7.4 – A linear combination of independent normal random variables is normally distributed. In

particularX ~ N ( μ , σ2 )⇒ X ~ N (μ , σ2

n ); The Central Limit Theorem.

7.5 – Confidence intervals for the mean of a normal population.

48 hours

7.6 – Null and alternative hypothesis H0 and H1; Significance level; Critical regions, p-values, one-tailed and two-tailed tests; Type I and Type II errors, including calculation of their probabilities; Testing hypotheses for the mean of a normal population.

7.7 – Introduction to bivariate distributions; Covariance and (population) product moment

correlation coefficient ρ ; Proof that ρ=0 in the case of independence and ±1 in the case of a linear relations between X and Y; Definition of the (sample) product moment correlation coefficient R in terms of n paired observations on X and Y. Its application to the estimation of ρ

Optional Links to ToK Optional Links to Internationalismo Statistical compression of data files.o Mathematics and the world. In the absence of

knowing the value of a parameter, will an unbiased estimator always be better than a biased one?

o Mathematics and the world. “Without the central limit theorem, there could be no statistics of any value within the human sciences.”

o Nature of mathematics. The central limit theorem can be proved mathematically, but its truth can be confirmed by its applications.

o Mathematics and the world. Claiming Brand A is better on average than Brand B can mean very little if there is a large overlap between the confidence intervals of the two means.

o Mathematics and the world. In practical terms, is saying that a result is significant the same as saying it is true?

o Mathematics and the world. Does the ability to test only certain parameters in a population affect the way knowledge claims in the human sciences are valued?

o When is it more important not to make a Type I error and when is it more important not to make a Type II error?

o Mathematics and the world. Given that a set of

o The negative binomial distribution is also known as Pascal’s distribution. Why?

o The physicist Frank Oppenheimer wrote: “Prediction is dependent only on the assumption that observed patterns will be repeated.” This is the danger of extrapolation. There are many examples of its failure in the past world-wide, eg climate change and the spread of disease.

data may be approximately fitted by a range of curves, where would we seek for knowledge of which equation is the “true” model?

o The correlation between smoking and lung cancer was “discovered” using mathematics. Science had to justify the cause.

Links to the Learner ProfileThe Mathematics Learner Profile taken from: henricowarriors.org/.../2011/12/The-Mathematics-Learner-Profile.pdf

Assessment:

IB External Assessment PAPER 1

o 120 minute exam consisting of two sections and no calculator is allowed Section A consisting of short response questions and Section B consisting of extended response

questions Both sections are based on the whole syllabus, Topics 1 – 6

PAPER 2o 120 minute exam consisting of two sections and calculator is allowed

Section A consisting of short response questions and Section B consisting of extended response questions

Both sections are based on the whole syllabus, Topics 1 – 6 PAPER 3

o 60 minute exam and calculator is allowed Consisting of extended response questions Questions are based on the Optional Topic 7

The external assessment is 80% of the overall grade towards the IB diploma. These papers are given in May according to the IB schedule. The marks are given externally and are awarded for method, accuracy, answers and reasoning, including

interpretation. These papers will be completed at the end of the students’ second year.

IB Internal Assessment Internal assessment is an integral part of the course and is compulsory for all students. It enables students to demonstrate the application of their skills and knowledge, and to pursue their personal interests, without the time limitations and other constraints that are associated with written examinations.

Internal assessment in Mathematics HL is an individual exploration. This is a piece of written work that involves investigating an area of mathematics. It is marked according to five assessment criteria (Communication, Mathematical Presentation, Personal Engagement, Reflection, Use of Mathematics).

The process in the Mathematics HL class occurs entirely during the students’ second year, after the completion of the core syllabus topics

Introduction (2 Hours)o Students are given the rules and guidelines o Students are provided the rubric o Students are given time to read through exemplars and compare their predicted

mark for each with the IB assigned mark Stimuli (.5 Hour)

o Students participate in multiple activities (mind mapping, graphic organizers, etc.) to help generate topic ideas

Topic, Research and Data (1 Hour)o Students choose a possible topic based on previous stimuli activity o Students are given time to either find data to model or research a topic to investigateo Individual meetings with the instructor to decide on the feasibility of the chosen topic

Rough Draft (6 Hours)o Students are to type out a rough draft and include the following subtopics

Introduction – why this topic Rationale – individual and global value Aim – mathematic s to be used Data/Research – tables/graphs/research Mathematics – all work shown and screenshots of the GDC Analysis/Reflection – results and meaning of mathematics

Conclusion including limitations and further research

o Rough draft due is before the end of the junior year and must be submitted to turnitin.com

Final Draft (.5 Hour)o Students will have an individual meeting with the instructor discussing strengths and

weaknesses of the rough drafto Students will not be provided with an annotated versiono Students will have 9 – 10 days to turn in a final draft to the instructor and turnitin.com

All work submitted to the IB for moderation or assessment must be authenticated by a teacher, and must not include any known instances of suspected or confirmed malpractice. Each student must sign the coversheet for internal assessment to confirm that the work is his or her authentic work and constitutes the final version of that work. Once a student has officially submitted the final version of the work to a teacher (or the coordinator) for internal assessment, together with the signed coversheet, it cannot be retracted.

In-Class Formative AssessmentFormative Assessment is used to recognize achievements and difficulties at the beginning or during a course, so that teachers and students can take appropriate action. This type of assessment forms an integral part of all learning.

Observations Questioning Discussion Graphic Organizers Self-Assessment

Think Aloud Talk to the Text Mind Map White-Boarding Practice Problems

In-Class Summative AssessmentSummative assessment is used to summarize and record overall achievement at the end of a course, for promotion and certification. Most ‘high stakes’ tests and external examinations are designed for this purpose. Summative assessment is also used to evaluate the relative effectiveness of a particular course, teaching method, or even an institution.

End of Topic tests built from questions from the IB test bank Topic Homework Sets (take-home quizzes) built from the IB test bank Midterm exams Final Exams ACT/SAT testing M-STEP Testing

Resources:

- Haese & Harris Math HL 2nd edition for students- Cambridge Math HL Topic7 – Option: Statistics and Probability for students - Haese & Harris Math HL 2rd edition Worked Solutions as a resource material- Haese & Harris Math HL 2rd edition Exam Preparation and Practice Guide as a resource material- IBID Press 3rd Edition Math HL as teacher resource material- Agnesi to Zeno – 100 vignettes from History of Math, Sanderson and Smith as teacher resource material

(Internationalism)- The Mathematical Experience, Davis and Hersh as teacher resource material (Internationalism)- The Little Book of Mathematical Principles, Theories and Things, Solomon as teacher resource material

(Internationalism)- Brooks/Cole, Cengage Learning Calculus 10th edition as a resource material- IB QuestionBank (CD) and past IB Examinations available through IBO

- Online Curriculum Center- TI-84Plus Graphics Display Calculator