Download pre-IB Mathematics syllabus outline

pre-IB Mathematics syllabus outline
future Maths SL/HL
future Maths HL only
1. Numbers
10. Numbers
2. Logic
11. Sets
3. Sets
12. Quadratics and polynomials
4. Statistics
5. Linear function
6. Functions
7. Quadratic function
8. Trigonometry
9. Geometry
Krzysztof Sikora
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1. Numbers
knowledge
expected basic abilities (students should / should be able to...)
exclusions
notation
1.1. subsets of R
distinguish whether a number is real, rational, irrational, integer, natural
1.2. prime and coprime
numbers
1.3. composite numbers
state whether a natural number is prime;
express a composite number as a product of its prime factors
divisibility tests (by 2,3,4,5,6,9,10)
1.4. factors, remainders,
multiples
write an integer as a product of its prime factors;
find lowest (least) common multiple and highest common factor (greatest
common divisor) of two integers;
use Euclid’s algorithm
LCM ,
GCD(HCF )
1.5. fractions & decimals
express the decimal as a fraction and fraction as a decimal (incl. recurring
decimals)
1.2345345 · · · =
= 1.23̇45̇ =
= 1.2345
1.6. absolute value
find the absolute value of a real number;
use geometric representation of an absolute value;
solve an equation which could be simplified to the one with single absolute
value of a linear expression of a variable (e.g. |2x + 3| = 4);
identify equations involving absolute value that have no solutions (e.g.
|4x − 5| + 1 = 0)
1.7. percentages
1.8. exponents and roots
1.9. formulae for
simplified
multiplication and
the Pascal’s triangle
R, Q, Z, N
equations with
more than one
absolute value,
inequalities - HL
only
solve various word problems
operate on integer indices and negative bases;
operate on roots of different degree;
express roots as powers with rational indices;
operate on numbers and algebraic expressions involving powers and roots
use efficiently formulae involving squares of sums, squares of differences,
differences of squares;
evaluate a square of a sum of 3 terms;
evaluate a cube of a sum / difference of 2 terms;
use Pascal’s triangle to evaluate higher
√ of simple sums;
√ powers
rationalise denominators in the form a ± b;
2
simplifying
fractions with cube
roots in
denominators;
|...|
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2. Logic
knowledge
expected basic abilities (students should / should be able to...)
2.1. the types of single
and compound logic
sentences
2.2. the truth tables for
each type of a logic
sentence;
2.3. tautology,
contradiction;
2.4. basic laws of logic;
2.5. negations of
compound sentenes
create compound sentences
2.6. the types of
quantifiers
use quantifiers in logic sentences, make them negative and state if the sentences
with quantifiers are true or false
2.7. de Morgan’s laws
use de Morgan’s laws to determine if the logic sentences are tautologies or
contradictions
use the 0 − 1 method (truth table with 2 or 3 single sentences) to determine if
the logic sentences are tautologies or contradictions;
create a negation of the provided logic sentences (incl. implication and
equivalence);
provide the basic laws of logic and prove them using 0 − 1 method (truth table)
3
exclusions
notation
∴
∧, ∨
¬(∼)
⇒, ⇐, ⇔
∀, ∃
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3. Sets
knowledge
expected basic abilities (students should / should be able to...)
3.1. basic operations on
sets (complement,
union, intersection,
difference)
3.2. subset
operate on given sets incl. standard number sets,
recognize duality with logic,
use set builder notation
3.3. Venn diagrams
use Venn diagrams to justify equalities on sets
3.4. de Morgan’s laws
use de Morgan’s laws to justify equalities on sets;
proof of de Morgan’s laws
3.5. number of elements
of a set (power of a
set)
solve problems on numbers of elements of various intersecting sets up to 3 sets
3.6. intervals
perform the same operations and use the same properties as for sets in general;
use the IB notation;
mark the intervals on the real number line
exclusions
list all subsets of a given finite set;
state whether a set is a subset of the other one
formal proofs
notation
∈, 6∈
⊂, ⊆, 6⊂, 6⊆
∩, ∪, \
∅, R, Q, Z, N
A0 , U
n(A), (Ā¯)
x ∈]a, b[
A = [a, +∞[
4. Statistics
knowledge
4.6. types of data
4.7. graphical
representation of
data
4.8. comparing data
expected basic abilities (students should / should be able to...)
understand the difference between discrete and continuous data
use the diagrams (stem and leaf diagrams, box-plots, histograms, cumulative
frequency curves) to find median, quartiles etc.
use GDC for creating graphs and evalutions
comment on the data and on the diagrams
4.9. measures of tendency find the mean, median and mode,
draw conclusions from mean, median, mode,
4.10. measures of spread
find range, quartiles, interquartile range and percentiles,
use GDC for evalutions
4
exclusions
creating graphs /
charts manually
outliers
notation
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5. Linear function
knowledge
5.1. definition and
properties
5.2. general equation of
the line + vectors
5.3. parallel and
perpendicular lines
expected basic abilities (students should / should be able to...)
exclusions
notation
recognize the equation y = ax + b;
understand the meaning of gradient (slope), y-intercept and x-intercept both in
theory and in real-life situations;
sketch the line with given gradient and a point;
find the gradient of a line segment with given end points;
find equation of the line when gradient and point or two points are given
recognize the equation Ax + By + C = 0;
recognize whether two equations describe the same line;
sketch the line with given equation;
find midpoint of a line segment, distance between 2 points or point and line
find a vector from a point to a point
find a unit vector or a vector with given magnitude parallel or perpendicular to
a given vector or line
find equation of a line parallel or perpendicular to the line given passing
through the point given
5.4. applications of linear equations
form and solve linear equations and inequalities with one variable;
form and solve systems of two linear equations with two variables;
sketch regions described with inequalities;
use GDC to find solutions of equations / inequalities / systems of equations
6. Functions
knowledge
expected basic abilities (students should / should be able to...)
exclusions
6.1. elementary functions know the basic functions (incl. abs. value, linear, quadratic, x 7→ √x, x3 , 1 );
x
sketch their graphs and state their properties
6.2. transformations of
graphs of functions
transform a graph of a function and rearrange its equation;
HL only:
compose transformations incl. translations, reflections in both coordinate axes, composition of 2
absolute value (y = |f (x)| and y = f (|x|)), dilations;
horizontal
state a sequence of transformations needed to transform one function into the
transformations:
other
y = f (x) → y = f (ax + b)
5
notation
y = f (x)
x 7→ f (x)
f : x 7→ y
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knowledge
6.3. basic properties
6.4. solving equations
expected basic abilities (students should / should be able to...)
find zeroes of a function;
read from the graph where a function is increasing / decreasing / constant;
find from the graph and from the equation whether a function is one-to-one /
even or odd;
know which of the basic functions are even / odd
exclusions
notation
increase / decrease
from an equation
1−1
solve equations of the type f (x) = g(x) both with and without GDC
7. Quadratic function
knowledge
7.1. definition and
properties
7.2. quadratic equations
expected basic abilities (students should / should be able to...)
recognize different forms of quadratic function:
general form y = ax2 + bx + c;
product form y = a(x − x1 )(x − x2 );
vertex form y = a(x − p)2 + q;
rearrange expressions to change from one form into any other;
factorise quadratic expression to find its zeroes;
complete the square in a quadratic expression to find zeroes;
use quadratic formula to find zeroes;
7.3. parabola
find the x-and y-intercepts, vertex (translation vector) and line of symmetry to
sketch the graph of a quadratic function;
use completing the square to find a vertex
7.4. quadratic
inequalities
solve quadratic inequalities with help of a sketch of graph
7.5. applications of
quadratic functions
use quadratics in solving real-life problems incl. optimisation;
ax2 +bx+c
investigate functions of the form y = dx
2 +ex+f
use GDC in all above
6
exclusions
notation
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8. Trigonometry
knowledge
expected basic abilities (students should / should be able to...)
8.1. Deegres and radians
understand the meaning of radian measure,
change degrees to radians and radians to degrees
8.2. Trigonometric ratios
of an acute angle
know the definition of sine, cosine and tangent as ratio of sides of a triangle
(SOHCAHTOA);
sin α
use basic identities, sin2 α + cos2 α = 1, tan α = cos
;
α
know range of values of trigonometric ratios;
know values of sin x, cos x and tan x for x ∈ {0, π6 , π4 , π3 , π2 },
use GDC to find a trig ratio or its inverse;
use trig ratios in solving simple geometric problems.
8.3. Trigonometric
functions of a real
number
use an idea of an orientated angle;
use the unit circle to evaluate trigonometric ratios of any number (or to express
it as a ±ratio of sides of a triangle);
recognize the quadrant of an angle from signs of trig ratios.
8.4. Trigonometric
equations
solve trigonometric equations over limited domain;
find exact answers when possible and use GDC to find approximations;
solve equations that invlove linear or quadratic expressions of a single
trigonometric ratio, e.g. 3 sin2 x + sin x − 2 = 0.
8.5. Trigonometry in
geometry
use the formula A = 12 ab sin C for area of a triangle;
be familiar and use the concepts of arc, sector, chord, tangent and segment,
circle, its centre and radius, area and circumference;
evaluate lengths of arcs and areas and perimeters of sectors and segments;
find angles and lengths of sides of triangles using sine rule and cosine rule;
consider all possible cases (sketch and solve analitically);
solve questions invloving bearings and compass directions.
7
exclusions
cot x,
sec x,
csc x
notation
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9. Geometry
knowledge
expected basic abilities (students should / should be able to...)
9.1. Geometry of a plane understand and apply simple geometric transformations: translation, reflection,
rotation, enlargement;
evaluate areas and perimeters of simple plane figures: polygons, circles;
use congruence and similarity in solving geometric problems, including the
concept of scale factor of an enlargement.
know properties of triangles and quadrilaterals, including parallelograms,
rhombuses, rectangles, squares, kites and trapeziums (trapezoids); compound
shapes.
9.2. 3-dimensional shapes evaluate volumes and surface areas of cuboids, pyramids, spheres, cylinders and
cones;
be familiar with classification of prisms and pyramids, including tetrahedra.
9.3. Coordinate geometry be familiar with the concepts of dimension for point, line, plane and space,
ordered pairs (x, y), origin, axes;
know and use equations of lines;
use formulea for a mid-point of a line segment and distance between two points
in the Cartesian plane and in three dimensions.
8
exclusions
notation
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10. Numbers
knowledge
expected basic abilities (students should / should be able to...)
10.1. binomial theorem
− 3yx2 )4 ,
expand integer powers of sums, e.g. ( 2x
y2
find a power for which expansion fullfils given conditions
10.2. introduction to
logarithms
10.3. absolute value
equations and
inequalities
solve equations involving more than one absoulte value
solve inequalities of the form e.g. |ax + b| > |cx + d|,
solve inequalities graphically.
10.4. complex numbers
add, subtract, multiply and divide complex numbers in cartesian form,
represent complex numbers in the complex plane (Argand diagram).
10.5. mathematical
induction
prove statements by mathematical induction,
use sigma notation for the sums.
know and use the definition of a logarithm,
change logarithmic equation into an exponential one and reversly, √
evaluate exact values of logarithms without a calculator, e.g. log4 8 2,
evaluate approximate values of logarithms with use of GDC,
simplify expressions involving logarithms (incl. adding, subtracting,
multiplying by a constant and changing bases),
solve simple logarithmic and exponential equations.
exclusions
notation
graphs of
logarithmic and
exponential
functions,
inequalities
11. Sets
knowledge
11.1. operations on sets
expected basic abilities (students should / should be able to...)
use union, intersection, difference and symmetric difference of sets,
use commutative, associative and distributive properties (knowing which of
them can be applied and which cannot).
9
exclusions
notation
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12. Quadratics and polynomials
knowledge
expected basic abilities (students should / should be able to...)
12.1. Vieta’s formulea
state the signs of solutions of a quadratic equation without solving it,
state the conditions for which an equation has e.g. 2 distinct negative solutions.
12.2. algebraic fractions
operate on fractions with linear, quadratic or cubic numerators and
denominators
simplify fractions by factorising numerators and denominators,
state the domain of an expression involving algebraic fractions,
add, subtract, multiply and divide algebraing fractions,
solve equations involving algebraic fractions
12.3. equation of a circle
know and use an equation of a circle,
find the centre and the radius of a circle defined by an equation in expanded
form,
find intersections of a circle and a line.
12.4. introduction to
polynomials
add, subtract, divide and multiply polynomials,
P (x)
R(x)
change an algebraic fraction D(x)
into the form Q(x) + D(x)
, where
deg R < deg D,
show that a number is a solution of a polynomial equation without solving it,
find a remainder when a polynomial is divided by ax + b without dividing,
factorize cubic polynomials by grouping terms,
e.g. 2x3 − 5x2 − 4x + 10 =
= x2 (2x − 5) − 2(2x − 5) = (2x − 5)(x2 − 2),
factorize cubic polynomials when one of the zeroes is given.
12.5. introduction to
complex numbers
find compex solutions of polynomial equations with real coefficients.
10
exclusions
notation