# The Following Are the 7 Areas of Study with Approximate Recommended Hours of Instruction

**IB MATHEMATICS SL**

*Course Description: IB Mathematics SL is an advanced study of mathematics, designed to prepare the student for the IB Math SL Exam and additional Calculus, either AP Calculus AB or BC. It is a rigorous course of study specifically designed for that student who expects to go on to study subjects which have significant mathematical content. The class will include a portfolio, consisting of two assignments, one on mathematical investigation and the other on mathematical modelling.*

**The following are the 7 Areas of study with approximate recommended hours of instruction:**

Algebra8 hrs

**Functions & Equations24 hrs**

**Circular functions & trigonometry16 hrs**

Matrices10 hrs

Vectors16 hrs

**Statistics & probability30 hrs**

Calculus36 hrs

** Total = 140 hrs**

**AIMS & SUBTOPICS**

**1. ALGEBRAIC CONCEPTS**

**1.1.aArithmetic sequences & series**

**1.1.bGeometric sequences & series**

**1.1.cSigma Notation**

**1.2.aExponents & Logarithms**

**1.2.bLaws of Exponents & Logarithms**

**1.2.cChange of Base Formula and application**

**1.3.aBinomial Theorem: expansion of **,

n N

- FUNCTIONS & EQUATIONS

**2.1.aConcept of function ** with

**Domain & Range**

2.1.bComposite functions

2.1.cInverse function

2.2.aThe graph of a function = f(x)

2.2.bWindow settings on GDC to view both global

and local behavior

2.2.cSolutions of equations graphically with “root”

concept with f(x) = 0 or “intersect”

concept with f(x) = g(x)

2.3.aTransformations of graphs

2.3.a.1Vertical translations y = f(x) + b

2.3.a.2Horizontal translations y = f(x –a)

2.3.a.3Vertical dilations (stretches) y = pf(x)

2.3.a.4Horizontal dilations y = f(x/q)

2.3.a.5X-axis Reflections y = - f(x)

2.3.a.6Y-axis Reflections y = f( -x )

2.3.bGraphs of as a reflection of y = f(x)

in the line y = x

2.4.aThe reciprocal function y = 1/x and its self-

inverse nature

2.5.aThe quadratic function f(x) =

with its graph, the y-intercept (0, c) and the axis of symmetry x =

2.5.b The quadratic function in the form

f(x) = with vertex at (h,k)

2.5.cThe quadratic function in the form

f(x) = a(x – p)(x – q) with

x-intercepts at ( p, 0) and ( q, 0)

2.6.a Solutions to with use of the discriminant and the quadratic formula.

2.7.aThe exponential function

and its graph

2.7.bThe logarithmic function

and its graph

2.7.cApplication of

2.7.dSolution of using logarithms

2.8.aThe exponential function

2.8.bThe natural log function

2.8.c Manipulation/Application of with compound interest and growth/decay models

*CIRCULAR FUNCTIONS & TRIGONOMETRY*

3.1.aRadian measure versus degree measure of an angle

3.1.bLength of an arc of a circle

3.1.cArea of a sector of a circle

3.2.aDefinitions of sin and cos in terms of the

unit circle

3.2.bDefinition of tan and use of y = x ( tan )

3.2.cThe Pythagorean Identity

and itstransformations

3.3.aDouble angle formulas for sin 2 and cos 2

3.4.aDomain, Range , and graphs of the basic three circular

functions sin x, cos x, and tan x

3.4.bComposite functions of the form

f(x) = a sin (b(x + c)) + d

3.5.aSolutions of trig equations in a restricted interval using both graphical and analytic means

3.5.bEquations of the type a sin(b(x + c)) = k with

graphical interpretation

3.5.cQuadratic equations involving trig functions

3.6.aSolving Right triangle: finding all parts

3.6.bLaw of Cosines

3.6.cLaw of Sines and the Ambiguous Case

3.6.dThe area of a triangle using

with application

- MATRICES

4.1.aDefinition of a matrix with understanding of the terms “element”, “row”, “column”, and “order”

4.2.aAlgebra of Matrices: Addition, Subtraction,

Multiplication by a scalar

4.2.bMultiplication of matrices

4.2.cIdentity and Zero matrices

4.3.aDeterminant of a square matrix

4.3.bCalculation of 2 X 2 and 3 X 3 determinants

4.3.cInverse of a 2 X 2 matrix

4.3.dConditions for the existence of the inverse of a

matrix

4.4.aSolution of systems of linear equations using

inverse matrices with a maximum of three

equations in three unknowns

- Vectors

5.1.aVectors in two dimensions and 3-D

5.1.bComponents of a vector with column representation

with i,j, & k unit vectors

5.1.cAlgebraic and Geometric approaches to:

5.1.c.1Sum and Difference of two vectors

5.1.c.2Multiplication by a scalar, kv

5.1.c.3Magnitude of a vector

5.1.c.4Unit vectors, base vectors i, j , and k

5.1.c.5Position vectors

5.2.aScalar product or dot product of two vevtors:

5.2.bParallel and perpendicular vectors

5.2.cAngle between two vectors

5.3.aRepresentation of a line r = a + tb

5.3.bThe angle between two lines

5.4.aDistinguishing between intersecting and parallel

lines

5.4.bFinding points where lines intersect

- STATISTICS & PROBABILITY

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

6.2.aPresentation of data including:

6.2.a.1Frequency tables & diagrams

6.2.a.2Box & whisker plots

6.2.a.3Grouped data

6.2.a.4mid-interval values

6.2.a.5interval width

6.2.a.6upper and lower interval

boundaries

6.2.a.7frequency histograms

6.3.aMean, Median, Mode, Quartiles, & Percentiles

6.3.bRange, inter-quartile range, variance, and standard

deviation

6.4.aCumulative frequency and cumulative frequency

graphs used to find median, quartiles, and

percentiles

6.5.aConcepts of trial, outcomes, equally likely

outcomes, sample space, and event

6.5.bProbability of an event

6.5.cComplementary events A and A’ (not A);

P(A) + P(A’) = 1

6.6.aCombined events, the formula:

where

for mutually exclusive events

6.7.aConditional Probability, the formula:

6.7.bIndependent Events, the definition:

6.8.aUse of Venn diagrams to solve problems

6.9.aConcept of discrete random variables and their

probability distributions

6.9.b Expected value (mean), E(X) for discrete data

6.10.a Binomial distribution

6.10.b Mean of the binomial distribution

6.11.a Normal distribution

6.11.b Properties of the normal distribution

6.11.c Standardization of normal variables

7. CALCULUS- an informal introductory approach is used in developing the history of and utility for modern day

calculus

A. The derivative definitions:

f(x) - f(a)

1. f’(a) = lim ------

x --->a x - a

f(x + h) - f(x)

2. f’(x ) = lim ------

h ---> 0 h

B. derivatives of elementary functions

C. derivatives of sums, products, and quotients

D. the chain rule

E. derivatives of implicitly defined functions

F. related rates applications

G. use of graphing calculator to explore the tangent

question

- instantaneous change vs. average rate of change => the tangent vs. the secant
- Indefinite integration with application to acceleration and velocityfunctions

J. Integration by substitution

- Numerical approximation techniques including the trapezoidal rule

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