MATHS METHODS UNIT 1 JUNE REVISION

A Checklist of Maths Methods Skills and

Understandings

The following have been taken from the syllabus details from each topic covered in this unit.
You can use the list as a basis for revising your grasp of these skills and understandings.

Linear Relationships/Coordinate Geometry

By hand: (these will be tested in the non-calculator section of assessments)

-solve linear equations and inequations

-sketch linear graphs
-substitute into and rearrange formulae
-determine the gradient of a line
-determine the equation of a line given 2 points or gradient and a point
-determine the length of a line segment
-determine the co-ordinates of a midpoint of a line segment
-determine the equation of lines parallel and perpendicular to a given line through some other point
-set up and solve systems of simultaneous linear equations involving 2 unknowns

Understand …
-what is meant by independent and dependent variables
-that a linear relationship is one involving a constant addition to the dependent variable as the
independent variable changes by a constant amount
-how to recognise a linear relationship from a table of values
-the general form of a rule describing a linear graph and the development of
-that the gradient of a straight line is the rate of change of the dependent variable (vertical axis) with
respect to the independent variable (horizontal axis) and how to determine gradient given two points or
an angle of inclination (the relationship m = tan )
-that the gradient of straight line is constant
-what is meant by a sketch graph
-that two ‘bits’ of information are required to determine a linear graph/rule
-the relationship between rule and graph
-what is meant by the terms ‘parallel’, ‘perpendicular’, ‘line segment’, ‘midpoint’, ‘collinear – (of points)
lying in the same straight line’.
-the gradient conditions for lines to be parallel or perpendicular

Use the Calculator to:

-solve linear equations, inequations and linear simultaneous equations in up to 4 unknowns inc.
graphical solution
-rearrange formulae to make another variable the subject
-sketch linear graphs with an appropriate WINDOW
-find axis intercepts
-find the y value for a particular x
-find points of intersection

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Polynomials - Quadratics
By hand: (these will be tested in the non-calculator section of assessments)
 substitute integer, simple rational and exact irrational values into rules
 rearrange simple algebraic equations and inequalities
 expand and factorise quadratic expressions with integer coefficients (including difference of perfect
squares and perfect squares)
 express ax2 + bx + c in completed square form where a, b and c are integers
 sketch graphs of quadratic functions
 determine the images of point(s) or rule(s) under transformation in the Cartesian plane (dilation from
the x -axis or from the y-axis, reflection in the x or y or both axes and translations (moving left/right,
up/down)

Understand….
-what a polynomial is and how to recognise one
-what is meant by the terms ‘degree’, ‘term’, ‘co-efficient’, ’variable’, ’constant’, ‘quadratic’, ‘cubic’,
‘quartic’, ‘monomial’, ‘binomial’, ‘trinomial’
-how to name a polynomial using P(x) notation, evaluate P(a) for various values of ‘a’ and perform
algebraic manipulations such as writing down an expression for 3P(x), [P(x)]2 etc
-how to expand up to three brackets (that produce cubic polynomials, for example)
-how to equate co-efficients (and the meaning of the symbol )
-how to recognise a quadratic relationship from a table of values (constant second difference)
-the forms of a quadratic equation (expanded, factorised, completed square) and be able to change
from one to the other
-the symmetrical nature of the quadratic graph and its features (in particular finding the TP by finding

halfway between. x-ints or completing the square or x = and knowing which method is more
efficacious in each case)
-how to use a variety of analytical, graphical and numerical approaches to determine and verify
solutions to equations over a specified interval (inc. the general formula and determining which method
of solution is the more appropriate in the given situation)
-the usefulness of the discriminant in determining the number & type of solutions to a quadratic
equation and the related number of x -intercepts for its graph
-the meaning of transformation as it applies to graphs and the types of transformation: translation,
dilation, reflection.
-how to use matrices to find the images of points in the Cartesian plane that have been translated,
dilated away from the co-ordinate axes or reflected in the axes or the line y = x
-the relationship between various constants and the related transformations of the parabola when its
equationn is in completed square form
-how to apply all of the above to situations that are reducible to quadratic modelling
-how to solve simultaneous equations involving linear/quadratic and quadratic/quadratic (by hand when
reducible to simple equations that can be solved by simple algebra)

Technology Objectives:
- Define polynomials such as P(x) = x3 + 2x then evaluate or determine the expressions such as P(–3),
P(a), P(x+1), [P(x)]2 etc
NOTE: Always best to Clear a-z (Menu, 1, 4) or New problem before starting each polynomial definition
- Use Algebra menu: Solve, Factor, Expand, Zeros, Complete the Square
- Use Solve to solve simultaneous equations
- Use Solve to re-arrange equations
- Use Algebra menu to efficiently and competently recognise equivalent expressions
- Use Graphing screen to effectively and competently sketch graphs, include being able to adjust
WINDOW to see relevant features
- On Graphing screen use Menu 5,1 to determine the y value when an x value is entered or Menu, 6 to
determine the Zero, Minimum, Maximum, Intersection correctly as well as knowing when to use them

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Polynomials - Cubics/Quartics
By hand: (these will be tested in the non-calculator section of assessments)
 express a cubic polynomial, with integer co-efficients, as the product of a linear factor ( x – a), where a
is an integer, and a quadratic factor
 use the factor theorem and null factor law to solve equations
 sketch the power functions y = x3 and y = x4 and simple transformations of these
 draw graphs of polynomials to degree 3 and degree 4 (when in factored form)… (not location of TPs
for degree 3 and 4 unless they are on the x-axis)

Understand….
- that cubic refers to a polynomial of degree 3 and quartic to degree 4
- the various forms of equations for cubic and quartic polynomials (expanded form, factor form, power
function form)
- that there are two graphical forms for polynomials with degree ≥ 3: the one when the equation can be
written in power function form (ie y = a (x – b)n + c ) and in factor form eg. y = ax(x – b)(x – c), y = a(x –
b)(x – c)2, y = a(x – b)(x – c)3
- the graphical forms are for cubics and quartics and their features (shape, TPs, SPOIs, axis intercepts)
- the relationship between factors and x-intercepts of the related graph include the significance of
repeated factors
- how to determine the equations for given graphs using information such as x-ints and y-int or SPOI
and y-int or…
- how to expand three brackets to create a cubic polynomial
- how to factorise using SOC/DOC (Sum of cubes/ Difference of Cubes)
- that not all cubic polynomials can be factorised
- that, for those that can, the constants in the factor brackets must multiply to give the constant term
in the expanded form
- that finding the values of x that makes each factor bracket zero gives the values of x for which the
polynomial has a value of zero
- the factor theorem
- the remainder theorem
- how to find the quadratic factor once a linear factor is found for cubics
- how to solve polynomial equations algebraically, graphically
- how to apply the above knowledge to solve problems set in real-life situations
- that the domain is the set of x values permissible in a situation
- that the range is the set of y values permissible
- how to give domain and range in set notation, interval notation or using the names of sets such as R,
Z, N etc
eg. x  R\ {2}
- the concept of maximal domain as the ‘default’ domain if one isn’t given
- the use algebraic, graphical and numerical approaches, including the factor theorem and the bisection
method algorithm, to determine and verify solutions to equations over a specified interval

Technology Objectives:
- Use Algebra meny: Solve, Factor, Expand, Zeros, Fraction Tools  Common Denominator (to write ,for

example, as a single fraction), propFrac or Expand (to do divisions such as

)
- Use Algebra menu to efficiently and competently recognise equivalent expressions
- Use the Graphing Screen to effectively and competently sketch graphs inc. being able to adjust
WINDOW to see relevant features
- Enter Piecewise function into the Graphing screen including domain restrictions
- On Graphing screen use Menu 5,1 to determine the y value when an x value is entered or Menu, 6 to
determine the Zero, Minimum, Maximum, Intersection correctly as well as knowing when to use
them

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Power Functions
By hand: (these will be tested in the non-calculator section of assessments)

 sketch graphs of the power functions y = xn for n = –2, –1, , 1, 2, 3, 4 and simple transformations
of these

Understand….
- what is meant by the term ‘power function’ and how these are different to polynomials
- that the graphs of power functions for n  N form patterns of shape related to whether the power is
odd or even and can identify key features of these graphs
- that n = –1 produces a reciprocal linear graph known as the hyperbola and can identify its key
features
- that n = – 2 produces a reciprocal quadratic graph known as the truncus (in Victoria only) and can
identify its key features
- what an asymptote is and how it is indicated graphically
- what is meant by the first, second, third and fourth quadrants of the Cartesian plane
- what is meant by the branch of an hyperbola or truncus and knowing that a sketch should show at
least one point on each branch

- that n = produces the square root graph and n = produces the cube root graph as well as be able
to identify its key features

Technology Objectives:

- Use Algebra Menu: Solve, Zeros, propFrac/Expand (to do divisions such as in order to sketch the
associated hyperbola)
- Use Algebra menu to efficiently and competently recognise equivalent expressions
- Use Graphing Screen to effectively and competently sketch graphs inc. being able to adjust WINDOW
to see relevant features
- Use of Table to investigate tables of values for input equations
- On Graphing screen use Menu 5,1 to determine the y value when an x value is entered or Menu, 6 to
determine the Zero, Minimum, Maximum, Intersection correctly as well as knowing when to use
them

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Relations and Functions
Understand….
- that a relation is any set of ordered pairs (ie. between two variables)
- that a relation can be described by a set of ordered pairs, an equation linking the variables
(dependent variable always the subject), a written description (the height of the plant was 3 cm more
than twice its age, in weeks) or a graph.
- that relations can be classified according to type: how the dependent variable is related to the
independent (ie. one-one, one-many, many-one and many-many)
- that ‘many’ in this sense refers to more than one
- that domain is the set of x values and range the set of y values
- that a function is a type of relation that produces a single y value for whatever x value is put in. For
this reason, functions are most applicable to real-life situations in finance, science etc.
- how to tell if a relation is or is not a function given a set of ordered pairs, an equation or a graph
(vertical line test)
- that function notation, f(x), is similar to polynomial notation and how to use it
- the meaning of ‘image’ in relation to functions
- the meaning of implied or maximal domains and restricted domains
- how to determine the maximal domain for a given function rule
(using knowledge of not square rooting a negative and not being able to divide by zero)
- the meaning of mapping notation (not the co-domain)
- the equation for a circle with centre at the origin and transformations of this in centre-radius form
- how to sketch arcs of circles using domain restrictions
- the piecewise rule and how to sketch its graph and determine its domain and range
- what constitutes inverse operations for number and how, when these operations are performed
consecutively, this essentially is a return to the original state
- the meaning of an inverse relation
- that the graph of the inverse relation is formed by reflecting the original graph in the line y = x
- how to graph an inverse relation
- the notation f –1(x) for an inverse function
- that the domain of the original relation becomes the range of the inverse and vice-versa

Technology Objectives:
- Be able to define functions
- Use Algebra Menu: Solve (including solving equations such as f(x) = x and re-arranging equations for y
in order to graph them and finding inverse rules), Zeros
- Use Algebra menu to efficiently and competently recognise equivalent expressions
- Use Graphing Screen to effectively and competently sketch graphs and their inverses include being
able to adjust WINDOW to see relevant features

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Transformations
Understand….

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Probability
The addition principle we want this or this or this

The multiplication principle we want this and this to occur

Arrangements in a row, on a shelf, words, numbers

if there are identical items we need to reduce the
number of arrangements by completing a division

Factorial

Permutations – arrangements, order IS important AB is different to BA

Combinations – select, choose, group, order is NOT important

As the order is not important we need to reduce the number of
possibilities by dividing by the number of identical arrangements

More difficult combinations When given a restriction – put it in first

When certain items need to be together or not together
Pascal’s Triangle

Probability =

Understand….
 set up probability simulations, and describe the notions of randomness and variability, and their
relation to events
 apply counting techniques to solve probability problems and calculate probabilities for compound
events, by hand in simple cases

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Rates of Change
Understand….
- what is meant by a rate of change (how one quantity changes in relation to another)
- what is meant by a constant rate of change and what is meant by a non-constant rate of change
- what situations would lead to each of the above
- how mathematicians use what they already know to assist them make sense of a new situation – ie.
how to use the knowledge of being able to find gradient for a straight line (in the graphical case of
constant rate of change) in investigating rate of change for non-constant situations
- what is meant by average rate of change and instantaneous rate of change and how to determine
these for graphs and for situations where equations are given
- the difference between distance & displacement, speed and velocity
- the difference between average velocity and instanteous velocity and how to find these.
- that velocity is the rate of change of displacement and acceleration is the rate of change of velocity
- how to create a v/t graph from a displacement /time graph by thinking about how the gradient
changes
- how to analyse a v/t graph

Technology Objectives:
- use of graphing screen to draw a Tangent line or to find the instantaneous gradient at a point on a
curve of given equation

Where to from here?

 Have you done your own summary sheet for each topic?
 Have you completed the Chapter Review questions in each topic?
 Chapter 8 – Revision of Chapters 2 – 7 has plenty of revision questions
 Chapter 12 - Revision of Chapters 9 – 11 … only do questions related to Chapter 10

My hard work
has paid off 😊

And so …

Best wishes for your

upcoming Exams!

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