Download Math - Algebra

Learning Ladders
Algebra/Pattern and Function
Step
30
Year
DP
Algebra
Math Studies
Math Standard Level
Solving the nth term of a GP given any two terms
Applying rate of change techniques in kinematics.
Finding the nth term of both AP and GP given three
consecutive terms
Finding the area between two curves
Finding the number of terms given the sum of the first
n terms of a GP using GDC
Using the derivative in order to define when the
function decreases or increases.
Using the rate of change concept to maximize or
minimize the volume or Surface area.
Finding the volume of a function when the area is
rotated around the x-axis.
Finding the intersection of two lines using vector form
Finding the angle between two lines using the dot
product
Finding the coefficient of any term when multiplying
two binomials using the Binomial theorem.
Math Higher Level
Proof by mathematical induction the nth derivative as
well as the De Moivre’s Theorem
Understanding the relationship between complex
roots.
Writing a Polynomial function using the factor theorem
Finding the cross product of two vectors and use it to
find the area of a triangle.
Finding the shortest distance between a point and a
plane
Finding the shortest distance between two planes.
Finding the angle between two planes
Writing the equation of a plane passing through three
points
Finding the intersection of three planes with different
solutions (Unique, no or infinite solutions)
Implicit differentiation including product rule and
quotient rule
Proving the maximum, minimum and inflexion points
by table of signs and second derivatives
Graph a function given its derivative
Solving kinematics questions with complex
displacement functions and use graphs to interpret the
data
Solve complex rate of change worded problem using
chain rule.
Finding the volume when the area between two curves
is rotated around the y-axis
Integrating functions by parts
Integrating functions using partial fraction
Proving convergence and divergence sequences and
series using all provided tests
Prove the Taylor and Mclaurin series expansion by
proving that the error is 0 (Lagrange Theorem)
Solving differential equation using Integrating factor
29
DP
Understanding how to make a set of factors, square
numbers, prime numbers
Solving the nth term of an arithmetic sequence given
any two terms.
Solving the nth term of a geometric sequence given
the first term and any other term.
Finding the number of terms given the Sum of the first
n terms of an AP
Finding the derivative of ( )
Finding the equation of the normal at any given point
Solving problems involving both AP and GP
Finding the number of terms given the sum of the first
n terms of a GP both algebraically and graphically
Finding the derivative using the Chain Rule and apply it
to different functions
Identifying when a function increases, decreases or
stationary.
Finding the inflexion point (Proof not required)
Finding the coordinates of intersection of a function
with its tangent given only the slope of the tangent
and the function.
Finding the coefficient of any term when multiplying
two binomials using the Binomial theorem.
Proof by mathematical induction that a series is true
for k.
Writing a complex number in all three forms
Finding complex roots
Finding a Polynomial function given real or/and
complex roots
Using the De Moivre’s Theorem to solve complex roots
Finding the intersection of two lines using vector form
and prove if they skew or not
Finding the area under a curve bounded by two
vertical lines.
Finding the cross product of two vectors
Finding the total areas
Finding the shortest distance between a point and a
line
Finding the angle between two vectors using the dot
product
Finding the angle between two lines
Writing the equation of a line in vector form given the
position vector and the direction vector.
Finding the coefficient of any term using Binomial
Theorem.
Writing the equation of a plane given a position and
containing two curves
Finding the intersection of two planes
Using the first principle method to differentiate
rational functions
Implicit differentiation not including product rule
Finding the maximum, minimum and inflexion points
Graph a derivative of a function
Solving kinematics questions with complex
displacement functions
Solve complex rate of change worded problem with all
data given
Finding area between two curves
Finding the volume when the area between two curves
is rotated around the x-axis
Integrating functions using substitution
Applying the l’Hopital Rule
Proving convergence and divergence sequences and
series using most tests
Using the Taylor and McLaurin expansion to deduce
the expansion of different functions
Solving differential equation using substitution method
Draw the slope filed of any differential equation
28
DP
Expressing numbers in the form of
and is an integer.
, where
Understanding the number sets.
Substituting values in the Arithmetic Sequence and
Series formulas when the sequence is defined
Substituting values in the Geometric Sequence and
Series formulas when the sequence is defined
Finding the derivative of a Polynomial function
( ( )
)
Finding the gradient at a given point
Finding the equation of the tangent when the function
Solving the nth term of an arithmetic sequence given
any two terms.
Solving an AP and GP problems both algebraically and
graphically.
Solving the nth term of a geometric sequence given
the first term and any other term.
Finding the coefficient of any term using Binomial
Theorem.
Finding the number of terms given the Sum of the first
n terms of an AP
Finding the sum from the Sigma notation
Finding the derivative of
( )
Proof by mathematical induction if an expression is
divisible by an integer.
Finding the modulus of complex number
Graphing a complex number.
Applying both product and quotient rules
Finding the equation of the tangent and normal at any
rd
Solving up to 3 degree polynomial equation using the
factor theorem
(Polynomial) and point are both given.
given point
Finding the component and modulus of a 3-D vector.
Finding the coordinates of the stationary point(s) of a
polynomial function.
Finding the components of a vector in vector and
Cartesian form.
Finding the equation of a line in all three forms
Finding the magnitude of a vector
Adding and subtracting two 2D-vectors both
algebraically and graphically
Finding the intersection of two lines using vector form
Writing the equation of a plane given a position and
normal vectors
Applying the scalar product formula
Using the first principle method to differentiate simple
functions
Finding the integral of a polynomial function
Finding the equation of tangent and normal
Finding the area under a curve bounded by two
vertical lines (above the x-axis)
Differentiating simple functions
) using the Binomial Theorem or
Expand (
Pascal’s Triangle
Product, quotient and chain rule
Finding the maximum and minimum points
Solving kinematics questions with simple displacement
functions
Solve very simple rate of change worded problem
Finding the area under a curve
Finding the volume when the area is rotated around
the x-axis.
Integrating simple functions with no sophisticated
methods
Finding the limits of any sequence
Proving convergence and divergence sequences and
series using some tests
Expand Taylor and McLaurin series to a certain degree
Solving separable differential equation
Use Euler’s Method to approximate the value of the
derivative at any point
27
11
Regular Track
Quadratics
Advanced Track
Binomial Expansions
(Benchmarked against IGCSE Additional Mathematics Topic #3 Quadratics)

Rational Equations(all four operations)
2

Graphing more challenging quadratic functions where coefficient of x can
be negative, positive or fractions by hand, using GDC and using autograph

Solving challenging problems using quadratic optimization
Sequences (Back tracked from IBDP)



th
Quadratic Sequences – finding the n term
th
Arithmetic sequences – finding the n term and sum for challenging
sequences
th
Geometric sequences - finding the n term and sum for challenging
sequences
Graphing
(Benchmarked against IGCSE Additional Mathematics Topic #18 Graphs of
Functions)

Solving systems of equations by graphing manually and finding the point
of intersection and by elimination and substitution. Can solve challenging
word problems by constructing equations.
(Benchmarked against IGCSE Additional Mathematics Topic #12 Binomial
Expansions)

Expand difficult binomial expansions using the general term nCra
a and b are whole numbers and fractions)
(n-r) r
b (where
Differentiation and Integration
(Benchmarked against IGCSE Additional Mathematics Topic #15 Differentiation and
Integration)



Differentiate polynomial functions with positive, negative and fractional
powers
Product Rule and Quotient Rule for differentiating polynomials apply
differentiation to gradients, tangents and normals, stationary points,
connected rates of change, small increments and approximations and
practical maxima and minima problems
apply differentiation and integration to kinematics problems that involve
displacement, velocity and acceleration of a particle moving in a straight line
with variable or constant acceleration, and the use of x -t and v -t graphs
Inequalities Linear Programming
(Benchmarked against IGCSE Mathematics Topic #25 Linear Programming)

Solving linear inequalities for

Graphing systems of linear inequalities on a graph and shading in/out the
region with vertical, horizontal, sloping lines where one has to draw lines
from the given equations.
Functions
(Benchmarked against IGCSE Additional Mathematics Topic #2 Functions)


26
11
Transforming y=f(x)
Challenging Inverse functions
Quadratics
Binomial Expansions
(Benchmarked against IGCSE Additional Mathematics Topic #3 Quadratics)

Rational Equations(multiplication and division)
2

Graphing of quadratic functions where the coefficient of x is an integer by
hand, using GDC and using autograph

Solving simple problems using quadratic optimisation
(Benchmarked against IGCSE Additional Mathematics Topic #12 Binomial
Expansions)

Expand simple binomial expansions using the general term nCra
when a and b are whole numbers)
(n-r) r
b ( only
Sequences (Back tracked from IBDP)



Number Sequences
th
Arithmetic sequences – finding the n term and the sum of simple
sequences
th
Geometric sequences - finding the n term and sum of simple sequences
Graphing
(Benchmarked against IGCSE Additional Mathematics Topic #18 Graphs of
Functions)

Solving systems of equations by graphing manually and finding the point
of intersection and by elimination and substitution.
Differentiation and Integration
(Benchmarked against IGCSE Additional Mathematics Topic #15 Differentiation and
Integration)





Inequalities Linear Programming
Differentiate polynomial functions with positive and negative powers
Product Rule and Quotient Rule for differentiating polynomials apply
differentiation to gradients, tangents and normals, stationary points and
practical maxima and minima problems
discriminate between maxima and minima by any method
understand integration as the reverse process of differentiation
apply differentiation to kinematics problems that involve displacement,
velocity and acceleration of a particle moving in a straight line with constant
acceleration.
(Benchmarked against IGCSE Mathematics Topic #25 Linear Programming)

Graphing inequalities on the number line

Solving linear inequalities

Graphing systems of linear inequalities on a graph and shading in/out the
region with vertical, horizontal and sloping lines.
Functions
(Benchmarked against IGCSE Additional Mathematics Topic #18)

Composite Functions with linear and quadratic functions.
Logarithms and Exponential Functions
(Benchmarked against IGCSE Additional Mathematics Topic #7)

Use the laws of logarithms
25
11
Quadratics
Binomial Expansions
(Benchmarked against IGCSE Additional Mathematics Topic #3 Quadratics)

Rational Equations(multiplication)

Solve quadratic equations using completing the square method

Using the discriminant to determine the nature of roots

Graphing of quadratic functions using GDC and using autograph
Sequences (Back tracked from IBDP)
(Benchmarked against IGCSE Additional Mathematics Topic #12 Binomial
Expansions)

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

Number Sequences - linear
th
Quadratic Sequences – finding the n term
th
Arithmetic sequences – finding the n term
th
Geometric sequences - finding the n term


n
Use the binomial Theorem for the expansion of (a+b) for positive values of n
Is familiar with the general term
(n-r) r
b
nCra
Differentiation and Integration
(Benchmarked against IGCSE Additional Mathematics Topic #15 Differentiation and
Integration)


Understand the idea of a derived function
Use the notation f’(x) and f’’(x)
Graphing
(Benchmarked against IGCSE Additional Mathematics Topic #18 Graphs of
Functions)

Solving systems of equations elimination and substitution

Solving by using the GDC and finding the point of intersection of both
linear and polynomial equations



Differentiate polynomial functions with positive powers
Product Rule differentiating polynomials apply differentiation to gradients,
tangents and stationary points, and practical maxima and minima problems
Knows that integration is the backward process of differentiation.
Inequalities Linear Programming
(Benchmarked against IGCSE Mathematics Topic #25 Linear Programming)

Graphing systems of linear inequalities on a graph and shading in/out the
region with vertical and horizontal lines
Functions
(Benchmarked against IGCSE Additional Mathematics Topic #18)

Relations and functions, Function notation

Composite Functions.

Simple Inverse functions
Logarithms and Exponential Functions
(Benchmarked against IGCSE Additional Mathematics Topic #7)

Logarithms-can move between exponential equations to logarithmic
equations.
Introduction to Calculus
(Back Tracked from DP)

Differentiating Polynomials
24
10
Quadratic Equations
Quadratics
(Benchmarked against IGCSE Mathematics Extended Topic #24 )

Challenging questions on Factorizing by Splitting

Complex Rational Equations

Solve linear Inequalities and also graph it

Problem Solving using challenging unfamiliar situations using linear
equations

Problem Solving using challenging unfamiliar situations quadratics

Sketching the quadratic Graph

Axis of symmetry and vertex of the quadratic graph.
(Benchmarked against IGCSE Additional Mathematics Topic #3 Quadratics)
Indices



Solve quadratic inequalities by graphing
Solve quadratic inequalities
2
find the maximum or minimum value of the quadratic function f: x =ax + bx +
c by completing the square for coefficients other than 1.
Functions
(Benchmarked against IGCSE Additional Mathematics Topic #2 Functions)

–1
Inverse functions f (x) to describe their inverses
Polynomials
(Benchmarked against IGCSE Mathematics Extended Topic #23 Indices )

Evaluate negative and fractional indices

Exponential Functions

Solve problems involving growth & decay

Solve more difficult exponential equations

Operation with surds using all 4 operations.

Rationalizing the denominator with more than 2 terms
(Benchmarked against IGCSE Additional Mathematics Topic #5 Factors of
Polynomials)

Solve cubic equations
Partial Fractions (Additional for DP)

Separate algebraic fractions where one is a repeated factor and one is a linear
factor in the denominator-also involves cases where factorizing is involved
Logarithmic and Exponential Functions
(Benchmarked against IGCSE Additional Mathematics Topic #7)

Know and use the laws of logarithms including change of base
x

Solve exponential equations of the form a =b

know simple properties and graphs of the logarithmic and exponential
x
functions including ln x and e
23
10
Algebraic Representation
Quadratics
(Benchmarked against IGCSE Mathematics Topic #20 Algebraic Representation and
Formulae)

Formula rearrangement

Substitution after rearrangement
(Benchmarked against IGCSE Additional Mathematics Topic #3 Quadratics)
Quadratic Equations
(Benchmarked against IGCSE Mathematics Extended Topic #24 )
2

Factorizing Quadratic Trinomials with coefficient of x more than 1

Simple Rational Equations

Solving linear inequalities

Problem Solving using difficult quadratics in familiar situations
Indices
(Benchmarked against IGCSE Mathematics Extended Topic #23 Indices )

Evaluate negative indices

Solve problems involving depreciation and appreciation

Solve exponential equations

Properties of Radicals

Addition and Subtraction of Radicals

Multiplication and Division of Radicals

Rationalizing the denominator where the denominator has only 1 term



Factorizing more difficult quadratic functions
Solve inequalities by drawing graphs
Use the maximum or minimum value of f(x) to sketch the graph or determine
the range for a given domain

Discriminant and conditions for f(x) = 0 to have: (i) two real roots, (ii) two
equal roots, (iii) no real roots and the related conditions for a given line to (i)
intersect a given curve, (ii) be a tangent to a given curve, (iii) not intersect a
given curve
2

find the maximum or minimum value of the quadratic function f(x) =ax + bx +
2
c by completing the square with coefficients of x being 1.
Functions
(Benchmarked against IGCSE Additional Mathematics Topic #2 Functions)


–1
Inverse functions notation f (x)
Composite Functions as defined by gf(x) = g(f(x))
Partial Fractions (Additional for DP)


Separate algebraic fractions with two linear factors in the denominator
Separate algebraic fractions where one is a repeated factor and one is a linear
factor in the denominator
Logarithmic and Exponential Functions
(Benchmarked against IGCSE Additional Mathematics Topic #7)

Exponential graphs
22
10
Algebraic Representation
Quadratics
(Benchmarked against IGCSE Mathematics Topic #20 Algebraic Representation and
Formulae)

Formula Substitution

Formula rearrangement simple problems

Substitution after rearrangement simple direct problems
(Benchmarked against IGCSE Additional Mathematics Topic #3 Quadratics)
Quadratic Equations
(Benchmarked against IGCSE Mathematics Extended Topic #24 )



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




Factorizing Expressions with Four Terms
2
Factorizing Quadratic Trinomials with the coefficient of x is 1
Factorizing by Splitting
Solving linear equations
Linear Inequalities
Problem Solving using liner equations
Solving Simple Quadratic Equations
Solving by Factorizing
Problem Solving using quadratics
Using The Quadratic Formula
Indices
(Benchmarked against IGCSE Mathematics Extended Topic #23 Indices )

Use index laws to simplify exponential expressions

Solve problems involving compound interest

Solve simple exponential equations

Basic operations with radicals




Draw linear and quadratic functions, find roots, vertex and line of symmetry
Factorising difficult quadratic functions
Solve quadratic inequalities using quadratic graphics
Represent inequalities graphically use the maximum or minimum value of f(x)
to sketch the graph or determine the range for a given domain

Discriminant and conditions for f(x) = 0 to have: (i)two distinct real roots, (ii)
one root, (iii) no real roots
2

Find the maximum or minimum value of the quadratic function f : x =ax + bx
+ c by any method
Functions
(Benchmarked against IGCSE Additional Mathematics Topic #2 Functions)


Use function notation, e.g. f(x) = 3x – 5, f(x)= 3x – 5 to describe simple
functions
–1
and the notation f (x) to describe their inverses; also composite functions
Simultaneous Equations
(Benchmarked against IGCSE Additional Mathematics Topic #6 Simultaneous Eq)

Solve simultaneous equations in two unknowns with one linear
equation and one quadratic
Polynomials
(Benchmarked against IGCSE Additional Mathematics Topic #5 Factors of
Polynomials)

Know and use the remainder and factor theorem

Find factors of polynomials
Partial Fractions (Additional for DP)

Separate algebraic fractions with two linear factors in the denominator
Logarithmic and Exponential Functions
(Benchmarked against IGCSE Additional Mathematics Topic #7)

Know and use the laws of logarithms
x

Solve equations of the form a =b

Simple Exponential graphs
21
9
9As3
Find the inverse of a linear function (Covered with Functions in Step 20 and 221)
9As6
Use systematic trial and improvement methods to find approximate solutions of equations such as x² + 2x = 20 (1, 2 and 7)
9As7
Construct functions arising from real-life problems; draw and interpret their graphs
IGCSE (Topic #23) More difficult Indices including Negative and Fractional powers
IGCSE (Topic #10) Direct and Indirect Variation in more complex word problems-
20
9
IGCSE (Topic )
Simplifying Surds and Rationalizing-More difficult expressions
IGCSE (Topic 24)
Inequalities and Number Lines (Finding the range of values to fit two inequalities)
IGCSE (Topic 3)
Solving Quadratic Equations by factorizing and quadratic formula including negative numbers and fractions)
IGCSE (Topic 19)
Equations of Straight Lines-parallel and perpendicular lines
9Ae1
Know the origins of the word algebra and its links to the work of the Arab mathematician Al’Khwarizmi
9Ae8
Construct and solve linear equations with integer coefficients (with and without brackets, negative signs anywhere in the equation, positive or negative solution);
solve a number problem by constructing and solving a linear equation
9Ae9
Solve a simple pair of simultaneous linear equations by eliminating one variable
9Ae10
Expand the product of two linear expressions of the form x ± n and simplify the corresponding quadratic expression
9Ae11
Understand and use inequality signs; construct and solve linear inequalities in one variable; represent the solution set on a number line
9As1
Generate terms of a sequence using term-to-term and position-to-term rules
9As2
Derive an expression to describe the nth term of an arithmetic sequence
9As4
Construct tables of values and plot the graphs of linear functions, where y is given implicitly in terms of x, rearranging the equation into the form y = mx + c; know
the significance of m and find the gradient of a straight line graph
9As5
Find the approximate solutions of a simple pair of simultaneous linear equations by finding the point of intersection of their graphs
9As8
Use algebraic methods to solve problems involving direct proportion, relating solutions to graphs of the equations
IGCSE (Topic #23) Indices including Negative and Fractional powers
IGCSE (Topic #10 ) Ratios-dividing in given ratios, increasing and decreasing by given ratios
IGCSE (Topic #10) Direct and Indirect Variation in straight forward word problems
IGCSE (Topic )
Simplifying Surds and Rationalizing
IGCSE (Topic #21) Simplifying Algebraic Fractions (Addition, Subtraction, Multiplication and Divisions)
19
9
IGCSE (Topic 24)
Inequalities and Number Lines-
IGCSE (Topic 3)
Solving Quadratic Equations by factorizing and quadratic formula with small integers
IGCSE (Topic 19)
Equations of Straight Lines-parallel and perpendicular lines in simple situations
9Ae2
Use index notation for positive integer powers; apply the index laws for multiplication and division to simple algebraic expressions
9Ae4
Simplify or transform algebraic expressions by taking out single-term common factors
9Ae5
Add and subtract simple algebraic fractions
9Ae6
Derive formulae and, in simple cases, change the subject; use formulae from mathematics and other subjects
18
8
8As4
Construct tables of values and use all four quadrants to plot the graphs of linear functions, where y is given explicitly in terms of x; recognize that equations of the
form y = mx + c correspond to straight-line graphs
17
8
8Ae6
Substitute positive and negative integers into formulae, linear expressions and expressions involving small powers, e.g. 3x² + 4 or 2x³, including examples that
lead to an equation to solve
8As1
Generate terms of a linear sequence using term-to-term and position-to-term rules; find term-to-term and position-to-term rules of sequences, including spatial
patterns
8As2
Use a linear expression to describe the nth term of a simple arithmetic sequence, justifying its form by referring to the activity or practical context from which it
was generated
16
8
8As3
Express simple functions algebraically and represent them in mappings
9Ae3
Construct algebraic expressions
9Ae7
Substitute positive and negative numbers into expressions and formulae
8Ae1
Know that letters play different roles in equations, formulae and functions; know the meanings of formula and function
8Ae2
Know that algebraic operations, including brackets, follow the same order as arithmetic operations; use index notation for small positive integer powers
8Ae3
Construct linear expressions
8Ae4
Simplify or transform linear expressions with integer coefficients; collect like terms; multiply a single term over a bracket
8Ae5
Derive and use simple formulae, e.g. to convert degrees Celsius (°C) to degrees Fahrenheit (°F)
8Ae7
Construct and solve linear equations with integer coefficients (unknown on either or both sides, without or with brackets)
15
7
7Ae6
Substitute positive integers into simple linear expressions/formulae
7Ae7
Construct and solve simple linear equations with integer coefficients (unknown on one side only), e.g. 2x = 8, 3x + 5 = 14, 9 - 2x = 7
7As3
Represent simple functions using words, symbols and mappings
7As4
Generate coordinate pairs that satisfy a linear equation, where y is given explicitly in terms of x; plot the corresponding graphs; recognize straight-line graphs
parallel to the x- or y-axis
14
13
12
11
10
9
8
7
6
5
4
3
2
1
7
7
6
6
5
5
4
4
3
3
2
2
1
1
7Ae5
Derive and use simple formulae, e.g. to change hours to minutes
7As2
Generate sequences from spatial patterns and describe the general term in simple cases
7Ae1
Use letters to represent unknown numbers or variables; know the meanings of the words term, expression and equation
7Ae2
Know that algebraic operations follow the same order as arithmetic operations
7Ae3
Construct simple algebraic expressions by using letters to represent numbers
7Ae4
Simplify linear expressions, e.g. collect like terms; multiply a constant over a bracket
7As1
Generate terms of an integer sequence and find a term given its position in the sequence; find simple term-to-term rules
NA part of Number
NA part of Number
NA part of Number
NA part of Number
NA part of Number
NA part of Number
NA part of Number
NA part of Number
NA part of Number
NA part of Number
NA part of Number
NA part of Number