1
Maths Quest 9 for New South Wales 5.3 pathway
Knowledge and skills grid
NUMBER
Rational numbers
NS5.1.1 Applies index laws to simplify and evaluate arithmetic expressions and uses scientific notation to write large and small numbers
Knowledge and skills
Students learn about
/ MQ 9 NSW 5.3 pathwayExercise / Investigation
- describing numbers written in index form using terms such as base, power, index, exponent
2B Powers and bases
- evaluating numbers expressed as powers of positive whole numbers
2A What are indices?
Thechessboardproblem
2B Powers and bases- establishing the meaning of the zero index and negative indices e.g. by patterns
2G Negative indices
- writing reciprocals of powers using negative indices
- translating numbers to index form (integral indices) and vice versa
Thechessboardproblem
2B Powers and bases
- developing index laws arithmetically by expressing each term in expanded form
2D Division using indices
2G Negative indices
- using index laws to simplify expressions
2D Division using indices
2E Zero index
2F Raising a power to a power
2G Negative indices
- using index laws to define fractional indices for square and cube roots
- writing square roots and cube roots in index form
- recognising the need for a notation to express very large or very small numbers
- expressing numbers in scientific notation
AlphaCentauri
- entering and reading scientific notation on a calculator
AlphaCentauri
- using index laws to make order of magnitude checks for numbers in scientific notation
- converting numbers expressed in scientific notation to decimal form
2I Scientific notation
- ordering numbers expressed in scientific notation
Rational numbers
NS5.2.1 Rounds decimals to a specified number of significant figures, expresses recurring decimals in fraction form and converts rates from one set of units to another
Knowledge and skills
Students learn about
/ MQ 9 NSW 5.3 pathwayExercise / Investigation
- identifying significant figures
- rounding numbers to a specified number of significant figures
1G Further estimation and calculator use
Worldpopulation
- using the language of estimation appropriately, including: rounding, approximate, level of accuracy
1G Further estimation and calculator use
- using symbols for approximation e.g.
1G Further estimation and calculator use
- determining the effect of truncating or rounding during calculations on the accuracy of the results
- writing recurring decimals in fraction form using calculator and non-calculator methods
- converting rates from one set of units to another
Real numbers
NS5.3.1 Performs operations with surds and indices
Knowledge and skills
Students learn about
/ MQ 9 NSW 5.3 pathwayExercise / Investigation
- defining a rational number
- distinguishing between rational and irrational numbers
Brakingdistances
- using a pair of compasses and a straight edge to construct simple rationals and surds on the number line
- defining real numbers
- demonstrating that is undefined for x < 0, = 0 for x = 0, and is the positive square root of x when x > 0
- using the following results for x, y > 0:
, ,
- using the four operations of addition, subtraction, multiplication and division to simplify expressions involving surds
6E Multiplication and division of surds
- expanding expressions involving surds
- rationalising the denominator of surds of the form
- using the index laws to demonstrate the reasonableness of the definitions for fractional indices
MQ10forNSW5.3pathway
- translating expressions in surd form to expressions in index form and vice versa
- evaluating numerical expressions involving fractional indices
- using the key on a calculator
- evaluating a fraction raised to the power of –1, leading to
Consumer arithmetic
NS5.1.2 Solves consumer arithmetic problems involving earning and spending money
Knowledge and skills
Students learn about
/ MQ 9 NSW 5.3 pathwayExercise / Investigation
- calculating earnings for various time periods from different sources, including:
11C Commission and royalties
11E Loadings and bonuses
- calculating income earned in casual and part-time jobs, considering agreed rates and special rates for Sundays and public holidays
- calculating weekly, fortnightly, monthly and yearly incomes
- calculating net earnings considering deductions such as taxation and superannuation
- calculating simple interest using the formula
I=PRT where
where I is the interest, P the principal, R the annual interest rate and T the number of years
- applying the simple interest formula to problems related to investing money at simple interest rates
- calculating compound interest for two or three years by repeated multiplication using a calculator e.g. a rate of 5% per annum leads to repeated multiplication by 1.05
- calculating compound interest on investments using a table
- calculating and comparing the cost of purchasing goods using:
- calculating a ‘best buy’
Consumer arithmetic
NS5.2.2 Solves consumer arithmetic problems involving compound interest, depreciation, and successive discounts
Knowledge and skills
Students learn about
/ MQ 9 NSW 5.3 pathwayExercise / Investigation
- calculating the result of successive discounts
- calculating compound interest on investments and loans using repetition of the formula for simple interest
- determining and using the formula for compound interest, A = P (1 + R)n, where A is the total amount, P is the principal, R is the interest rate per period as a decimal and n is the number of periods
- using the compound interest formula to calculate depreciation
- comparing the cost of loans using flat and reducible interest for a small number of repayment periods
Probability
NS5.1.3 Determines relative frequencies and theoretical probabilities
Knowledge and skills
Students learn about
/ MQ 9 NSW 5.3 pathwayExercise / Investigation
- repeating an experiment a number of times to determine the relative frequency of an event
Probabilityexperiments
Investigatingrelativefrequency
Simulatingdaysoftheweek
- estimating the probability of an event from experimental data using relative frequencies
Investigatingrelativefrequency
14C Experimental probability14E Estimating probability
- expressing the probability of an event from experimental data using relative frequencies
- expressing the probability of an event A given a finite number of equally likely outcomes as
where n is the total number of outcomes in the sample space
- using the formula to calculate probabilities for simple events
14C Experimental probability
14D Theoretical probability of an event
- simulating probability experiments using random number generators
Probability
NS5.3.2 Solves probability problems involving compound events
Knowledge and skills
Students learn about
/ MQ 9 NSW 5.3 pathwayExercise / Investigation
- distinguishing informally between dependent and independent events
- sampling with and without replacement in two-stage experiments
- analysing two-stage events through constructing organised lists, tables and/or tree diagrams
- solving two-stage probability problems including instances of sampling with and without replacement
- finding probability of compound events using organised lists, tables or diagrams
PATTERNS AND ALGEBRA
Algebraic techniques
PAS5.1.1 Applies the index laws to simplify algebraic expressions
Knowledge and skills
Students learn about
/ MQ 9 NSW 5.3 pathwayExercise / Investigation
- using the index laws previously established for numbers to develop the index laws in algebraic form
2D Division using indices
2F Raising a power to another power
- establishing that a0 = 1 using the index laws
- simplifying algebraic expressions that include index notation
2D Division using indices
2E Zero index
2F Raising a power to another power
2G Negative indices
Algebraic techniques
PAS5.2.1 Simplifies, expands and factorises algebraic expressions involving fractions and negative and fractional indices
Knowledge and skills
Students learn about
/ MQ 9 NSW 5.3 pathwayExercise / Investigation
- simplifying algebraic expressions involving fractions
4E Algebraic fractions
- applying the index laws to simplify expressions involving pronumerals
2D Division using indices
2E Zero index
2F Raising a power to another power
2G Negative indices
- establishing that
- using index laws to assist with the definition of the fractional index for square root given and then
- using index laws to assist with the definition of the fractional index for cube root
- using index notation and the index laws to establish that
- applying the index laws to simplify algebraic expressions such as
2D Division using indices
2E Zero index
2F Raising a power to another power
2G Negative indices
- expanding, by removing grouping symbols, and collecting like terms where possible, algebraic expressions such as
15C More complicated expansions
- factorising, by determining common factors, algebraic expressions such as
Algebraic techniques
PAS5.2.2 Solves linear and simple quadratic equations, solves linear inequalities and solves simultaneous equations using graphical and analytical methods
Knowledge and skills
Students learn about
/ MQ 9 NSW 5.3 pathwayExercise / Investigation
Linearandquadraticequations
- solving linear equations
5C Solving equations with pronumerals on both sides
5D Solving linear equations with grouping symbols
- solving word problems that result in equations
Thecostofconcrete
Maximumviewingarea- exploring the number of solutions that satisfy simple quadratic equations of the form x2 = c
- solving simple quadratic equations of the form ax2 = c
- solving equations arising from substitution into formulae
Maximumviewingarea
Linearinequalities- solving inequalities
Simultaneousequations
- solving simultaneous equations using non-algebraic methods, such as ‘guess and check’, setting up tables of values or looking for patterns
Solvingsimultaneousequationsusingguess, checkandimprove
- solving linear simultaneous equations by finding the point of intersection of their graphs
Howmanycockatoosandkangaroos?
- solving simple linear simultaneous equations using an analytical method
17C Algebraic solutions of simultaneous equations — elimination method 1
17D Algebraic solutions of simultaneous equations — elimination method 2
Cramer’sruleforsimultaneousequations
Simultaneousequationsin3unknowns
- generating simultaneous equations from simple word problems
Concerthallseating
Algebraic techniques
PAS5.3.1 Uses algebraic techniques to simplify expressions, expand binomial products and factorise quadratic expressions
Knowledge and skills
Students learn about
/ MQ 9 NSW 5.3 pathwayExercise / Investigation
- simplifying algebraic expressions, including those involving fractions
4B Simplifying algebraic expressions
4C Using grouping symbols
4D Simplifying expressions with grouping symbols
4E Algebraic fractions
15A Binomial products
15D Applications
- expanding binomial products by finding the area of rectangles
- using algebraic methods to expand a variety of binomial products
15C More complicated expansions
15D Applications
- recognising and applying the special products
(a ± b)2 = a ± 2ab + b2 / 15B Special products
Usingexpandingformulastosquarelargenumbers
15C More complicated expansions
HigherorderexpansionsandPascal’striangle
- factorising expressions:
difference of two squares
perfect squares
trinomials
grouping in pairs for four-term expressions / 15E The highest common factor
15F More factorising using the highest common factor
15G Factorising using the difference of two squares rule
15H Quadratic trinomials
Mousepaddimensions
15I More quadratic trinomials15J Mixed factorising practice
15K Simplifying algebraic fractions multiplication and division
Equalornotequal?
15L Simplifying algebraic fractions addition and subtraction
- using a variety of methods, including combinations of the above, to factorise expressions
15J Mixed factorising practice
- factorising and simplifying a variety of more complex algebraic expressions
Equalornotequal?
15L Simplifying algebraic fractions addition and subtractionAlgebraic techniques
PAS5.3.2 Solves linear, quadratic and simultaneous equations, solves and graphs inequalities, and rearranges literal equations
Knowledge and skills
Students learn about
/ MQ 9 NSW 5.3 pathwayExercise / Investigation
Linear, quadraticandsimultaneousequations
- using analytical and graphical methods to solve a range of linear equations, including equations that involve brackets and fractions
5C Solving equations with pronumerals on both sides
5D Solving linear equations with grouping symbols
5F Solving more complex equations
- solving problems involving linear equations
Thecostofconcrete
Maximumviewingarea
- developing the quadratic formula
- solving equations of the form ax2 + bx+c = 0 using:
completing the square
the quadratic formula / 16B Solving quadratic equations by using factors
16C Solving quadratic equations by completing the square
16D Solving quadratic equations by using the quadratic formula
16E Problems and applications using quadratic equations
- solving a variety of quadratic equations
- identifying whether a given quadratic equation has no solution, one solution or two solutions
- checking the solutions of quadratic equations
- generating quadratic equations from problems
Isthepriceright?
16E Problems and applications using quadratic equations- solving problems involving quadratic equations
16E Problems and applications using quadratic equations
Flyingdolphin
- solving quadratic equations resulting from substitution into formulae
Flyingdolphin
- using analytical methods to solve a variety of simultaneous equations, including those that involve a first degree equation and a second degree equation
Inequalities
- using <, >, , , to generate linear inequalities from problems
- solving linear inequalities analytically, including changing the direction of the inequality when multiplying or dividing by a negative number in inequalities
- solving problems involving inequalities
17G Solving simultaneous inequalities
Literalequations
- changing the subject of a formula, using examples from other strands and other subjects
- determining restrictions on the values of variables implicit in the original formula and after rearrangement of the formula
Understandingvariables
- replacing variables with other expressions
- using variable substitution to simplify expressions and equations so that specific cases can be seen to belong to general categories
- interpreting expressions and equations given additional information
Coordinate geometry
PAS5.1.2 Determines the midpoint, length and gradient of an interval joining two points on the number plane and graphs linear and simple non-linear relationships from equations
Knowledge and skills
Students learn about
/ MQ 9 NSW 5.3 pathwayExercise / Investigation
Midpoint, lengthandgradient
- determining the midpoint of an interval from a diagram
- graphing two points to form an interval on the number plane and forming a right-angled triangle by drawing a vertical side from the higher point and a horizontal side from the lower point
- using the right-angled triangle drawn between two points on the number plane and Pythagoras’ theorem to determine the length of the interval joining the two points
- using the right-angled triangle drawn between two points on the number plane and the relationship
gradient =
to find the gradient of the interval joining two points
Gradientandy-intercept
- determining whether a line has a positive or negative slope by following the line from left to right – if the line goes up it has a positive slope and if it goes down it has a negative slope
Gradientandy-intercept
- finding the gradient of a straight line from the graph by drawing a right-angled triangle after joining two points on the line
Gradientandy-intercept
Graphsofrelationships
- constructing tables of values and using coordinates to graph vertical and horizontal lines
- identifying the x- and y-intercepts of graphs
Gradientandy-intercept
10D Sketching straight line graphs
Predictingaperson’sheight
- identifying the x-axis as the line y = 0
10D Sketching straight line graphs
- identifying the x-axis as the line x = 0
10D Sketching straight line graphs
- graphing a variety of linear relationships on the number plane by constructing a table of values and plotting coordinates using an appropriate scale
Gradientandy-intercept
Predictingaperson’sheight
ARomanaqueduct
- graphing simple non-linear relationships
- determining whether a point lies on a line by substituting into the equation of the line
Coordinate geometry
PAS5.2.3 Uses formulae to find midpoint, distance and gradient and applies the gradient/intercept form to interpret and graph straight lines
Knowledge and skills
Students learn about
/ MQ 9 NSW 5.3 pathwayExercise / Investigation
Midpoint, distanceandgradientformulae
- using the average concept to establish the formula for the midpoint, M, of the interval joining two points (x1, y1) and (x2, y2) on the number plane
- using the formula to find the midpoint of the interval joining two points on the number plane
- using Pythagoras’ theorem to establish the formula for the distance, d, between two points (x1, y1) and (x2, y2) on the number plane
- using the formula to find the distance between two points on the number plane
10I Midpoint of a line segment
- using the relationship
gradient =
to establish the formula for the gradient, m, of an interval joining two points (x1, y1) and (x2,y2) on the number plane
- using the formula to find the gradient of an interval joining two points on the number plane
ARomanaqueduct