Maths Quest 7 for New South Wales

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

• describing numbers written in index form using terms such as base, power, index, exponent

• evaluating numbers expressed as powers of positive whole numbers

Thechessboardproblem

• establishing the meaning of the zero index and negative indices e.g. by patterns

• writing reciprocals of powers using negative indices

• translating numbers to index form (integral indices) and vice versa

• developing index laws arithmetically by expressing each term in expanded form

• using index laws to simplify expressions

• 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

• 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

• identifying significant figures

• rounding numbers to a specified number of significant figures

Worldpopulation

• using the language of estimation appropriately, including: rounding, approximate, level of accuracy

• using symbols for approximation e.g. 

• 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

• 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

• 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

• calculating earnings for various time periods from different sources, including:

• 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

• 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

• 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

• 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

• simulating probability experiments using random number generators

Probability

NS5.3.2 Solves probability problems involving compound events

Knowledge and skills

Students learn about

• 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

• using the index laws previously established for numbers to develop the index laws in algebraic form

• establishing that a0 = 1 using the index laws

• simplifying algebraic expressions that include index notation

Algebraic techniques

PAS5.2.1 Simplifies, expands and factorises algebraic expressions involving fractions and negative and fractional indices

Knowledge and skills

Students learn about

• simplifying algebraic expressions involving fractions

• applying the index laws to simplify expressions involving pronumerals

• 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

• expanding, by removing grouping symbols, and collecting like terms where possible, algebraic expressions such as

• 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

• solving linear equations

• solving word problems that result in equations

Thecostofconcrete

• 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

• solving inequalities

• solving simultaneous equations using non-algebraic methods, such as ‘guess and check’, setting up tables of values or looking for patterns

• solving linear simultaneous equations by finding the point of intersection of their graphs

• solving simple linear simultaneous equations using an analytical method

• 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

• simplifying algebraic expressions, including those involving fractions

• expanding binomial products by finding the area of rectangles

• using algebraic methods to expand a variety of binomial products

• recognising and applying the special products

• factorising expressions:

Mousepaddimensions

• using a variety of methods, including combinations of the above, to factorise expressions

• factorising and simplifying a variety of more complex algebraic expressions

Equalornotequal?

Algebraic techniques

PAS5.3.2 Solves linear, quadratic and simultaneous equations, solves and graphs inequalities, and rearranges literal equations

Knowledge and skills

Students learn about

• using analytical and graphical methods to solve a range of linear equations, including equations that involve brackets and fractions

• solving problems involving linear equations

Thecostofconcrete

Maximumviewingarea

• developing the quadratic formula

• solving equations of the form ax2 + bx+c = 0 using:

• 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?

• solving problems involving quadratic equations

• solving quadratic equations resulting from substitution into formulae

• using analytical methods to solve a variety of simultaneous equations, including those that involve a first degree equation and a second degree equation

• 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

• 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

• 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

• 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

• finding the gradient of a straight line from the graph by drawing a right-angled triangle after joining two points on the line

• constructing tables of values and using coordinates to graph vertical and horizontal lines

• identifying the x- and y-intercepts of graphs

• identifying the x-axis as the line y = 0

• identifying the x-axis as the line x = 0

• graphing a variety of linear relationships on the number plane by constructing a table of values and plotting coordinates using an appropriate scale

• 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

• 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

• 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