Mathematics SL

Syllabus content 140 hrs

Requirements

All topics are compulsory. Students must study all the sub-topics in each of the topics in the syllabus as listed in this guide. Students are also required to be familiar with the topics listed as presumed knowledge (PK).

Portfolio 10 hrs

Two pieces of work, based on different areas of the syllabus, representing the following two types of

tasks:

• mathematical investigation

• mathematical modeling.

SYLLABUS DETAILS

Format of the syllabus

The syllabus to be taught is presented as three columns.
• Content: the first column lists, under each topic, the sub-topics to be covered.
• Amplifications/inclusions: the second column contains more explicit information on specific sub-topics listed in the first column. This helps to define what is required in terms of preparing for the examination.
• Exclusions: the third column contains information about what is not required in terms of preparing for the examination.

Although the mathematics SL course is similar in content to parts of the mathematics HL course, there are differences. In particular, students and teachers are expected to take a more sophisticated approach for mathematics HL, during the course and in the examinations. Where appropriate, guidelines are provided in the second and third columns of the syllabus details.

Course of study

Teachers are required to teach all the sub-topics listed for the seven topics in the syllabus. The topics in the syllabus do not need to be taught in the order in which they appear in this guide. Teachers should therefore construct a course of study that is tailored to the needs of their students and that integrates the areas covered by the syllabus, and, where necessary, the presumed knowledge.

Integration of portfolio assignments

The two pieces of work for the portfolio, based on the two types of tasks (mathematical investigation and mathematical modeling), should be incorporated into the course of study, and should relate directly to topics in the syllabus. Full details of how to do this are given in the section on internal assessment.

Time allocation

The recommended teaching time for standard level courses is 150 hours. For mathematics SL, it is expected that 10 hours will be spent on work for the portfolio. The time allocations given in this guide are approximate, and are intended to suggest how the remaining 140 hours allowed for the teaching of the syllabus might be allocated. However, the exact time spent on each topic depends on a number of factors, including the background knowledge and level of preparedness of each student. Teachers should therefore adjust these timings to correspond to the needs of their students.

Use of calculators

Students are expected to have access to a graphic display calculator (GDC) at all times during the course. The minimum requirements are reviewed as technology advances, and updated information will be provided to schools. It is expected that teachers and schools monitor calculator use with reference to the calculator policy. Regulations covering the types of calculators allowed are provided in the Vade Mecum. Further information and advice is provided in the teacher support material.

Mathematics SL information booklet

Because each student is required to have access to a clean copy of this booklet during the examination, it is recommended that teachers ensure students are familiar with the contents of this document from the beginning of the course. The booklet is provided by the IBO and is published separately.

Teacher support materials

A variety of teacher support materials will accompany this guide. These materials will include suggestions to help teachers integrate the use of graphic display calculators into their teaching, guidance for teachers on the marking of portfolios, and specimen examination papers and markschemes. These will be distributed to all schools.

External assessment guidelines

It is recommended that teachers familiarize themselves with the section on external assessment guidelines, as this contains important information about the examination papers. In particular, students need to be familiar with the notation the IBO uses and the command terms, as these will be used without explanation in the examination papers.

Presumed knowledge

General

Students are not required to be familiar with all the topics listed as presumed knowledge (PK) before they start this course. However, they should be familiar with these topics before they take the examinations, because questions assume knowledge of them. Teachers must therefore ensure that any topics designated as presumed knowledge that are unknown to their students at the start of the course are included at an early stage. They should also take into account the existing mathematical knowledge of their students to design an appropriate course of study for mathematics SL. This list of topics is not designed to represent the outline of a course that might lead to the mathematics SL course. Instead, it lists the knowledge, together with the syllabus content, that is essential to successful completion of the mathematics SL course. Students must be familiar with SI (Système International) units of length, mass and time, and their derived units.

Topics

Number and algebra

Geometry

Statistics

Syllabus content

Topic 1—Algebra 8 hrs

Aims

The aim of this section is to introduce students to some basic algebraic concepts and applications. Number systems are now in the presumed knowledge

section.

Details

1.1 Arithmetic sequences and series; sum of finite arithmetic series; geometric sequences and series; sum of finite and infinite geometric series. Sigma notation.

Examples of applications, compound interest and population growth.

1.2 Exponents and logarithms. Laws of exponents; laws of logarithms. Elementary treatment only is required.

Examples:

1.3 The binomial theorem: expansion of

On examination papers: students may determine the binomial coefficients,   by using Pascal’s triangle, or by using a GDC.

Exclude: The formula and consideration of combinations.

Topic 2—Functions and equations 24 hrs

Aims

The aims of this section are to explore the notion of function as a unifying theme in mathematics, and to apply functional methods to a variety of mathematical situations. It is expected that extensive use will be made of a GDC in both the development and the application of this topic.

Details

2.1 Concept of function : domain, range; image (value). Composite functions ; identity function. Inverse function

On examination papers: if the domain is the set of real numbers then the statement “ ” will be omitted. The composite function is defined as f(g(x)). On examination papers: if an inverse function is to be found, the given function will be defined with a domain that ensures it is one-to-one.

Exclude: Formal definition of a function; the terms “one-to-one”, “many-to-one” and “codomain”. Domain restriction.

2.2 The graph of a function; its equation y= f(x) . Function graphing skills: use of a GDC to graph a variety of functions; investigation of key features of graphs. Solution of equations graphically.

On examination papers: questions may be set requiring the graphing of functions that do not explicitly appear on the syllabus. The linear function ax + b is now in the presumed knowledge section. Identification of horizontal and vertical asymptotes. May be referred to as roots of equations, or zeros of functions.

2.3 Transformations of graphs: translations; stretches; reflections in the axes. The graph of y= f−1(x) as the reflection in the line y=x of the graph of y= f(x) .

2.4 The reciprocal function , x ≠ 0: its graph; its self-inverse nature.

2.5 The quadratic function : its graph, y-intercept (0,c) . Axis of symmetry, .  The form : vertex (h,k) . The form :x-intercepts ( p,0) and (q,0) .

Rational coefficients only.

2.6 The solution of , a ≠0. The quadratic formula. Use of the discriminant .

Exclude: On examination papers: questions demanding elaborate factorization techniques will not be set.

Topic 3—Circular functions and trigonometry 16 hrs

Aims

The aims of this section are to explore the circular functions and to solve triangles using trigonometry.

Details

3.1 The circle: radian measure of angles; length of an arc; area of a sector.

Radian measure may be expressed as multiples of PI, or decimals.

3.2 Definition of cosθ and sinθ in terms of the unit circle. Definition of tanθ as sinθ /cosθ. The identity

Given sinθ , finding possible values of cosθ without finding θ. Lines through the origin can be expressed as y = x tanθ , with gradient tanθ .

Exclude: The reciprocal trigonometric functions secθ , cscθ and cotθ .

3.3 Double angle formulae: .

Double angle formulae can be established by simple geometrical diagrams and/or by use of a GDC.

Exclude: Compound angle formulae.

3.4 The circular functions sin x , cos x and tan x : their domains and ranges; their periodic nature; and their graphs. Composite functions of the form f (x)=asin(b(x+c))+d.

On examination papers: radian measure should be assumed unless otherwise indicated by, for example, . Example: f(x)=2cos(3(x−4)) +1. Examples of applications: height of tide, Ferris wheel.

Exclude: The inverse trigonometric functions: arcsin x , arccos x and arctan x .

3.5 Solution of trigonometric equations in a finite interval. Equations of the type asin(b(x+c))=k. Equations leading to quadratic equations in, for example, sin x . Graphical interpretation of the above.

Exclude: The general solution of trigonometric equations.

Topic 4—Matrices 10 hrs

Aims

The aim of this section is to provide an elementary introduction to matrices, a fundamental concept of linear algebra.

Details

4.1 Definition of a matrix: the terms “element”, “row”, “column” and “order”.

Use of matrices to store data.

Exclude: Use of matrices to represent transformations.

4.2 Algebra of matrices: equality; addition; subtraction; multiplication by a scalar. Multiplication of matrices. Identity and zero matrices.

Matrix operations to handle or process information.

4.3 Determinant of a square matrix. Calculation of 2×2 and 3×3determinants. Inverse of a 2× 2 matrix. Conditions for the existence of the inverse of a matrix.

Elementary treatment only. Obtaining the inverse of a 3×3 matrix using a GDC.

Exclude: Cofactors and minors. Other methods for finding the inverse of a 3×3 matrix.

4.4 Solution of systems of linear equations using inverse matrices (a maximum of three equations in three unknowns).

Only systems with a unique solution need be considered.

Topic 5—Vectors 16 hrs

Aims

The aim of this section is to provide an elementary introduction to vectors. This includes both algebraic and geometric approaches.

Details

5.1 Vectors as displacements in the plane and in three dimensions. Components of a vector; column representation. , Algebraic and geometric approaches to the following topics: the sum and difference of two vectors; the zero vector, the vector −v ; multiplication by a scalar, kv ; magnitude of a vector, v ; unit vectors; base vectors i, j, and k; position vectors

Distance between points in three dimensions. Components are with respect to the unit vectors i, j, and k (standard basis). The difference of v and w is v−w=v+(−w) .

5.2 The scalar product of two vectors. . Perpendicular vectors; parallel vectors. The angle between two vectors.

The scalar product is also known as the “dot product” or “inner product”. For non-zero perpendicular vectors v⋅w=0 ; for non-zero parallel vectors

Exclude: Projections.

Exclude: Cartesian form of the equation of a line:

5.4 Distinguishing between coincident and parallel lines. Finding points where lines intersect.

 Awareness that non-parallel lines may not intersect.

Topic 6—Statistics and probability 30 hrs

Aims

The aim of this section is to introduce basic concepts. It may be considered as three parts: descriptive statistics (6.1–6.4), basic probability (6.5–6.8), and modelling data (6.9–6.11). It is expected that most of the calculations required will be done on a GDC. The emphasis is on understanding and interpreting the results obtained.

Details

6.1 Concepts of population, sample, random sample and frequency distribution of discrete and continuous data.

Elementary treatment only.

6.2 Presentation of data: frequency tables and diagrams, box and whisker plots. Grouped data: mid-interval values, interval width, upper and lower interval boundaries, frequency histograms.

Treatment of both continuous and discrete data. A frequency histogram uses equal class intervals.

Exclude: Histograms based on unequal class intervals.

6.3 Mean, median, mode; quartiles, percentiles. Range; interquartile range; variance; standard deviation.

Awareness that the population mean, μ, is generally unknown, and that the sample mean, x , serves as an estimate of this quantity. Awareness of the concept of dispersion and an understanding of the significance of the numerical value of the standard deviation. Obtaining the standard deviation (and indirectly the variance) from a GDC is expected. Awareness that the population standard deviation, σ, is generally unknown, and that the standard deviation of the sample, sn , serves as an estimate of this quantity.

6.4 Cumulative frequency; cumulative frequency graphs; use to find median, quartiles, percentiles.

6.5 Concepts of trial, outcome, equally likely outcomes, sample space (U) and event. The probability of an event A as The complementary events A and A′ (not A); P(A)+P(A′)=1.

Appreciation of the non-exclusivity of “or”. Use of P(A∪B)=P(A)+P(B) for mutually exclusive events.

The term “independent” is equivalent to “statistically independent”. Use of P(A∩B)=P(A)P(B) for independent events.

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

6.9 Concept of discrete random variables and their probability distributions. Expected value (mean), E(X ) for discrete data.

Exclude: Formal treatment of random variables and probability density functions; formal treatment of cumulative frequency distributions and their formulae.

6.10 Binomial distribution. Mean of the binomial distribution.

Exclude: The formula and considerationof combinations. Formal proof of mean.

6.11 Normal distribution. Properties of the normal distribution. Standardization of normal variables.

Appreciation that the standardized value ( z ) gives the number of standard deviations from the mean. Use of calculator (or tables) to find normal probabilities; the reverse process.

Exclude: Normal approximation to the binomial distribution.

Aims

The aim of this section is to introduce students to the basic concepts and techniques of differential and integral calculus and their application.

Details

7.1 Informal ideas of limit and convergence. Definition of derivative as ;  Derivative of , sin x , cos x , tan x ,  and ln x , Derivative interpreted as gradient function and as rate of change.

Only an informal treatment of limit and convergencefor example, 0.3, 0.33, 0.333, ... converges to 1/3.  Use of this definition for derivatives of polynomial functions only. Other derivatives can be justified by graphical considerations using a GDC. Familiarity with both forms of notation, and , for the first derivative. Finding equations of tangents and normals. Identifying increasing and decreasing functions.

7.2 Differentiation of a sum and a real multiple of the functions in 7.1. The chain rule for composite functions. The product and quotient rules. The second derivative.

Familiarity with both forms of notation, and , for the second derivative.

7.3 Local maximum and minimum points. Use of the first and second derivative in optimization problems.

Testing for maximum or minimum using change of sign of the first derivative and using sign of the second derivative.Examples of applications: profit, area, volume.

, Example:

Example: if and y =10 when x = 0 , then .   Only the form ,   Revolution about the x-axis only,

7.6 Kinematic problems involving displacement, s, velocity, v, and acceleration, a.

. Area under velocity–time graph represents distance.

7.7 Graphical behavior of functions: tangents and normals, behavior for large |x| , horizontal and vertical asymptotes. The significance of the second derivative; distinction between maximum and minimum points. Points of inflexion with zero and non-zero gradients.