MathsQuestforNewSouthWales

Knowledge and skills grid — Stages 3 and 4

NUMBER

Whole numbers

NS3.1 Orders, reads and writes numbers of any size

Knowledge and skills
Students learn about / Maths Quest 7
Exercise/Investigation / Maths Quest 8
Exercise/Investigation
  • applying an understanding of place value and the role of zero to read, write and order numbers of any size
/ 1BPlace value
  • stating the place value of any digit in large numbers
/ 1BPlace value
  • ordering numbers of any size in ascending or descending order
/ 1BPlace value
  • recording large numbers using expanded notation
/ 1BPlace value
  • rounding numbers when estimating
/ 1GEstimation / 1EEstimation
  • recognising different abbreviations of numbers used in everyday contexts; e.g. $350K represents $350000
/ 1BPlace value
  • recognising the location of negative numbers in relation to zero and locating them on a number line
/ 10AIntegers on the number line
  • recognising, reading and converting Roman numerals used in everyday contexts
/ 1ANumber systems from the past
  • identifying differences between the Roman and Hindu-Arabic systems of recording numbers
/ 1ANumber systems from the past

Addition and subtraction

NS3.2 Selects and applies appropriate strategies for addition and subtraction with counting numbers of any size

Knowledge and skills
Students learn about / Maths Quest 7
Exercise/Investigation / Maths Quest 8
Exercise/Investigation
  • selecting and applying appropriate mental, written or calculator strategies to solve addition and subtraction problems
/ 1CAdding and subtracting whole numbers / 1ABasic operations: non-calculator arithmetic
Dartscompetition
  • using a formal written algorithm and applying place value concepts to solve addition and subtraction problems, involving counting numbers of any size
/ 1CAdding and subtracting whole numbers / 1ABasic operations: non-calculator arithmetic
  • using estimation to check solutions to addition and subtraction problems
/ 1GEstimation / 1EEstimation
  • adding numbers with different numbers of digits
/ 1CAdding and subtracting whole numbers / 1ABasic operations: non-calculator arithmetic
Dartscompetition

Multiplication and division

NS3.3 Selects and applies appropriate strategies for multiplication and division

Knowledge and skills
Students learn about / Maths Quest 7
Exercise/Investigation / Maths Quest 8
Exercise/Investigation
  • applying appropriate mental, written or calculator strategies to solve multiplication and division problems
/ 1DMultiplying whole numbers
1EDividing whole numbers / 1ABasic operations: non-calculator arithmetic
  • recognising and using different notations to indicate division
/ 1EDividing whole numbers
  • recording remainders as fractions or decimals, where appropriate
/ 1EDividing whole numbers
  • multiplying three- and four-digit numbers by one-digit numbers using mental or written strategies
/ 1DMultiplying whole numbers
  • multiplying three-digit numbers by two-digit numbers using the extended form (long multiplication)
/ 1DMultiplying whole numbers / 1ABasic operations: non-calculator arithmetic
  • dividing a number with three or more digits by a single-digit divisor using mental or written strategies
/ 1EDividing whole numbers / 1ABasic operations: non-calculator arithmetic
  • using mental strategies to multiply or divide a number by 100 or a multiple of 10
/ 1DMultiplying whole numbers
1EDividing whole numbers
  • finding solutions to questions involving mixed operations
/ 1EDividing whole numbers
1FOrder of operations / 1BOrder of operations
  • determining whether a number is prime or composite by finding the number of factors; e.g. ‘13 has two factors (1 and 13) and therefore is prime; 21 has more than two factors (1, 3, 7, 21) and therefore is composite’
/ 4EPrime numbers
ThesieveofEratosthenes
/ 1CFactors, multiples, prime and composite numbers

Operations with whole numbers

NS4.1 Recognises the properties of special groups of whole numbers and applies a range of strategies to aid computation

Knowledge and skills
Students learn about / Maths Quest 7
Exercise/Investigation / Maths Quest 8
Exercise/Investigation
  • expressing a number as a product of its prime factors
/ 4FPrime factors and factor trees
4GDivisibility
Findingsquarerootsandcuberootswithoutacalculator
  • using index notation to express powers of numbers (positive indices only)
/ 4BIndex notation
4GDivisibility
Findingsquarerootsandcuberootswithoutacalculator / 1DSquares and square roots
  • using the notation for square root and cube root
/ 4ASquares, square roots, cubes and cube roots / 1DSquares and square roots
  • recognising the link between squares and square roots and cubes and cube roots
/ 4ASquares, square roots, cubes and cube roots / 1DSquares and square roots
  • exploring through numerical examples that:
/ 4ASquares, square roots, cubes and cube roots
  • finding square and cube roots of a number expressed as a product of its prime factors
/ Findingsquarerootsandcuberootswithoutacalculator
  • finding square and cube roots of numbers using a calculator, after first estimating
/ 4ASquares, square roots, cubes and cube roots
  • identifying special groups of numbers including figurate numbers, palindromic numbers, Fibonacci numbers in Pascal’s triangle
/ 1HSpecial groups of numbers
Morenumberpatternsinshapes
  • comparing the Hindu-Arabic number system with number systems from different societies past and present
/ 1ANumber systems from the past
Morenumbersystemsofthepast
  • determining and applying tests of divisibility
/ 4GDivisibility
  • using an appropriate non-calculator method to divide two- and three-digit numbers by a two-digit number
/ 1EDividing whole numbers
  • applying a range of mental strategies to aid computation
/ 1DDividing whole numbers

Integers

NS4.2 Compares, orders and calculates with integers

Knowledge and skills
Students learn about / Maths Quest 7
Exercise/Investigation / Maths Quest 8
Exercise/Investigation
  • recognising the direction and magnitude of an integer
/ 10AIntegers on the number line
ComparingtemperaturesWetordryreunion?
  • placing directed numbers on a number line
/ 10AIntegers on the number line
10IDirected numbers on the number line
Mountainclimbing
  • ordering directed numbers
/ 10AIntegers on the number line
Comparingtemperatures
10IDirected numbers on the number line
  • interpreting different meanings (direction or operation) for the + and – signs depending on the context
/ 10AIntegers on the number line
Movies
  • adding and subtracting directed numbers
/ 10DAddition of integers
10ESubtraction of integers
Wetordryreunion?
10KDirected number operations: fractions
10LDirected number operations: decimals
Mountainclimbing
/ 4DSubstituting positive and negative numbers
  • multiplying and dividing directed numbers
/ 10FMultiplication of integers
Numberpatterntable
10GDivision of integers
10KDirected number operations: fractions
10LDirected number operations: decimals / 4DSubstituting positive and negative numbers
  • using grouping symbols as an operator
/ 10HCombined operations / 4DSubstituting positive and negative numbers
  • applying order of operations to simplify expressions
/ 10HCombined operations
10KDirected number operations: fractions / 4DSubstituting positive and negative numbers
  • keying integers into a calculator using the +/– key
/ Graphics calculator tip: Entering a negative number
10ESubtraction of integers
Graphics calculator tip: Division with negative numbers
10GDivision of integers
  • using a calculator to perform operations with integers
/ Graphics calculator tip: Entering a negative number
10ESubtraction of integers
10FMultiplication of integers
Graphics calculator tip: Division with negative numbers
10GDivision of integers
Graphics calculator tip: Finding the square of a number
10HCombined operations
Fractions and decimals

NS3.4 — Unit 1 Compares, orders and calculates with decimals, simple fractions and simple percentages

Knowledge and skills
Students learn about / Maths Quest 7
Exercise/Investigation / Maths Quest 8
Exercise/Investigation
  • modelling thirds, sixths and twelfths of a whole object or collection of objects
/ 5AUnderstanding fractions
  • placing thirds, sixths or twelfths on a number line between 0 and 1 to develop equivalence
/ 5AUnderstanding fractions
  • expressing mixed numerals as improper fractions and vice versa, through the use of diagrams or number lines, leading to a mental strategy
/ 5CImproper fractions and mixed numerals
  • recognising that
/ 5CImproper fractions and mixed numerals
  • using written, diagram and mental strategies to subtract a unit fraction from 1; e.g.
/ Ch 5Are you ready?
Q9
SkillSHEET 5.9
  • using written, diagram and mental strategies to subtract a unit fraction from any whole number; e.g.
/ Ch 5Are you ready?
Q9
SkillSHEET 5.9
  • adding and subtracting fractions with the same denominator; e.g.
/ 5DAdding and subtracting fractions / 1FAddition and subtraction of fractions
  • expressing thousandths as decimals
/ 6APlace value
  • interpreting decimal notation for thousandths
/ 6APlace value
  • comparing and ordering decimal numbers with three decimal places
/ 6BComparing decimals
  • placing decimal numbers on a number line between 0 and 1
/ 1HFractions to decimals, decimals to fractions
  • adding and subtracting decimal numbers with a different number of decimal places
/ 6EAdding decimals
6FSubtracting decimals / 1IAddition and subtraction of decimals
  • multiplying and dividing decimal numbers by single digit numbers and by 10, 100 and 1000
/ 6GMultiplying decimals by a whole number
6IDividing decimals by whole numbers / 1JMultiplication and division of decimals

Fractions and decimals

NS3.4 — Unit 2 Compares, orders and calculates with decimals, simple fractions and simple percentages

Knowledge and skills
Students learn about / Maths Quest 7
Exercise/Investigation / Maths Quest 8
Exercise/Investigation
  • finding equivalent fractions using diagrams and number lines by re-dividing the unit
/ 5AUnderstanding fractions
  • developing a mental strategy for finding equivalent fractions; e.g. multiply or divide the numerator and the denominator by the same number
/ 5AUnderstanding fractions
  • reducing a fraction to its lowest equivalent form by dividing the numerator and the denominator by a common factor
/ 5BSimplifying fractions
  • comparing and ordering fractions greater than one using strategies such as diagrams, the number line or equivalent fractions
/ 5CImproper fractions and mixed numerals
  • adding and subtracting simple fractions where one denominator is a multiple of the other
/ 5DAdding and subtracting fractions / 1F Addition and subtraction of fractions
  • multiplying simple fractions by whole numbers using repeated addition, leading to a rule
/ 5EMultiplying fractions
  • calculating unit fractions of a collection
/ 5EMultiplying fractions
  • representing simple fractions as a decimal and as a percentage
/ 6JConverting fractions to decimals and recurring decimals / 2CFractions to percentages
  • calculating simple percentages (10%, 20%, 25%, 50%) of quantities
/ 2EFinding percentages of an amount using fractions
2HCommon percentages and short cuts

Fractions, decimals and percentages

NS4.3 Operates with fractions, decimals, percentages, ratios and rates

Knowledge and skills
Students learn about / Maths Quest 7
Exercise/Investigation / Maths Quest 8
Exercise/Investigation
Fractions, decimalsandpercentages
  • finding highest common factors and lowest common multiples
/ 5BSimplifying fractions
5DAdding and subtracting fractions
4CMultiples
4DFactors
Howmanytiles? / 1CFactors, multiples, prime and composite numbers
4JFactorising
  • finding equivalent fractions
/ 5AUnderstanding fractions
  • reducing a fraction to its lowest equivalent form
/ 5BSimplifying fractions
  • adding and subtracting fractions using written methods
/ 5DAdding and subtracting fractions
5GMixed operations with fractions
Youwantmorepizza?
10KDirected number operations: fractions / 1FAddition and subtraction of fractions
  • expressing improper fractions as mixed numerals and vice versa
/ 5CImproper fractions and mixed numerals
5GMixed operations with fractions / 1FAddition and subtraction of fractions
  • adding mixed numerals
/ 5DAdding and subtracting fractions
5GMixed operations with fractions
10KDirected number operations: fractions / 1FAddition and subtraction of fractions
  • subtracting a fraction from a whole number
/ 5DAdding and subtracting fractions
5GMixed operations with fractions
  • multiplying and dividing fractions and mixed numerals
/ 5EMultiplying fractions
5FDividing fractions
5GMixed operations with fractions
10KDirected number operations: fractions / 1GMultiplication and division of fractions
  • adding, subtracting, multiplying and dividing decimals (for multiplication and division, limit operators to two digits)
/ 6EAdding decimals
6FSubtracting decimals
6GMultiplying decimals by a whole number
6HMultiplying decimals
6IDividing decimals by whole numbers
6KDividing decimals by decimals
10LDirected number operations: decimals / 1IAddition and subtraction of decimals
1JMultiplication and division of decimals
  • determining the effect of multiplying or dividing by a number less than one
/ 6HMultiplying by decimals
  • rounding decimals to a given number of places
/ 6DRounding / 1HFractions to decimals, decimals to fractions
2HCommon percentages and short cuts
2IFinding percentages using a calculator
2JApplications of percentages
  • using the notation for recurring (repeating) decimals
/ 6JConverting fractions to decimals and recurring decimals
  • converting fractions to decimals (terminating and recurring) and percentages
/ 6JConverting fractions to decimals and recurring decimals / 1HFractions to decimals, decimals to fractions
2CFractions to percentages
2IFinding percentages using a calculator
  • converting terminating decimals to fractions and percentages
/ 6CConverting decimals to fractions / 1HFractions to decimals, decimals to fractions
2DDecimals to percentages
  • converting percentages to fractions and decimals
/ 2APercentages as fractions
2BPercentages as decimals
  • calculating fractions, decimals and percentages of quantities
/ 5EMultiplying fractions
6HMultiplying decimals / 2EFinding percentages of an amount using fractions
Savingsonscooters
2FFinding percentages of an amount using decimals
2HCommon percentages and short cuts
2IFinding percentages using a calculator
2JApplications of percentages
  • increasing and decreasing a quantity by a given percentage
/
Savingsonscooters
2JApplications of percentages
  • interpreting and calculating percentages greater than 100%
/ 2EFinding percentages of an amount using fractions
2FFinding percentages of an amount using decimals
  • expressing profit and/or loss as a percentage of cost price or selling price
/ 2JApplications of percentages
  • ordering fractions, decimals and percentages
/ 5AUnderstanding fractions
6BComparing decimals / 2CFractions to percentages
  • expressing one quantity as a fraction or a percentage of another
/ 5EMultiplying fractions / 2GOne amount as a percentage of another
2IFinding percentages using a calculator
Ratioandrates
  • using ratio to compare quantities of the same type
/ 12AIntroduction to ratios
12BSimplifying ratios
12DComparing ratios
13HScale drawing
  • writing ratios in various forms; e.g. , 4:6, 4 to 6
/ 12AIntroduction to ratios
12BSimplifying ratios
12CProportion
12DComparing ratios
13HScale drawing
  • simplifying ratios; e.g. 4:6 = 2:3, 2 = 1:4,
    0.3:1 = 3:10
/ 12BSimplifying ratios
  • applying the unitary method to ratio problems
/ 12CProportion
  • dividing a quantity in a given ratio
/ 12EIncreasing and decreasing in a given ratio
12FDividing in a given ratio
  • interpreting and calculating ratios that involve more than two numbers
/ 12FDividing in a given ratio
  • calculating speed given distance and time
/ 12ISpeed
  • calculating rates from given information;
    e.g. 150 kilometres travelled in 2 hours
/ 12GRates
Shoppingforthebestbuy
Chance

NS3.5 Orders the likelihood of simple events on a number line from zero to one

Knowledge and skills
Students learn about / Maths Quest 7
Exercise/Investigation / Maths Quest 8
Exercise/Investigation
  • using data to order chance events from least likely to most likely; e.g. roll two dice twenty times and order the results according to how many times each total is obtained
/ 14BExperimental probability
  • ordering commonly used ‘chance words’ on a number line between zero (impossible) and one (certain); e.g. ‘equal chance’ would be placed at 0.5
/ 14AThe language of chance / 14AProbability scale
  • using knowledge of equivalent fractions and percentages to assign a numerical value to the likelihood of a simple event occurring; e.g. there is a five in ten, , 50% or one in two chance of this happening
/ 14CSimple probability
  • describing the likelihood of events as being more or less than a half (50% or 0.5) and ordering the events on a number line
/ SkillSHEET 14.3 / SkillSHEET 14.1
SkillSHEET 14.4
  • using samples to make predictions about a larger ‘population’ from which the sample comes; e.g. predicting the proportion of cubes of each colour in a bag after taking out a sample of the cubes
/ 14BExperimental probability
Inthelongrun—tossingacoin
14CSample spaces and theoretical probability

Probability

NS4.4 Solves probability problems involving simple events

Knowledge and skills
Students learn about / Maths Quest 7
Exercise/Investigation / Maths Quest 8
Exercise/Investigation
  • listing all possible outcomes of a simple event
/ 14BThe sample space
14CSimple probability
14DUsing tables to show sample spaces
Iwin! / 14BExperimental probability
  • using the term ‘sample space’ to denote all possible outcomes; e.g. for tossing a fair die, the sample space is 1, 2, 3, 4, 5, 6
/ 14BThe sample space
14DUsing tables to show sample spaces / 14CSample spaces and theoretical probability
  • assigning probabilities to simple events by reasoning about equally likely outcomes; e.g. the probability of a 5 resulting from the throw of a fair die is
/ 14CSimple probability
14DUsing tables to show sample spaces
Experimentingwithchance
Iwin! / 14AProbability scale
14BExperimental probability
Scissors, paper, rock
Inthelongrun—tossingacoin
14CSample spaces and theoretical probability
  • expressing the probability of a particular outcome as a fraction between 0 and 1
/ 14CSimple probability
14DUsing tables to show sample spaces
Iwin! / 14AProbability scale
14BExperimental probability
14CSample spaces and theoretical probability
  • assigning a probability of zero to events that are impossible and a probability of one to events that are certain
/ 14ALanguage of chance
14CSimple probability / 14AProbability scale
  • recognising that the sum of the probabilities of all possible outcomes of a simple event is 1
/ 14CSample spaces and theoretical probability
14EComplementary events
  • identifying the complement of an event
/ 14EComplementary events
  • finding the probability of a complementary event
/ 14EComplementary events

PATTERNS AND ALGEBRA

Patterns and algebra

PAS3.1a Records, analyses and describes geometric and number patterns that involve one operation using tables and words

Knowledge and skills
Students learn about / Maths Quest 7
Exercise/Investigation / Maths Quest 8
Exercise/Investigation
  • working through a process of building a simple geometric pattern involving multiples, completing a table of values, and describing the pattern in words. This process includes the following steps:
–building a simple geometric pattern using materials
–completing a table of values for the geometric pattern
–describing the number pattern in a variety of ways and recording descriptions using words
–determining a rule to describe the pattern from the table
–using the rule to calculate the corresponding value for a larger number / Geometricpatterns
7AUsing rules
  • working through a process of identifying a simple number pattern involving only one operation, completing a table of values, and describing the pattern in words. This process includes the following steps:
–describing the pattern in a variety of ways and recording descriptions using words
–determining a rule to describe the pattern from the table
–using the rule to calculate the corresponding value for a larger number / Numberpatterns

Patterns and algebra

PAS3.1b Constructs, verifies and completes number sentences involving the four operations with a variety of numbers

Knowledge and skills
Students learn about / Maths Quest 7
Exercise/Investigation / Maths Quest 8
Exercise/Investigation
  • completing number sentences that involve more than one operation by calculating missing values
/ 11AUsing inverse operations
  • completing number sentences involving fractions or decimals
/ 11AUsing inverse operations
  • constructing a number sentence to match a problem that is presented in words and requires finding an unknown
/ 11AUsing inverse operations
  • checking solutions to number sentences by substituting the solution into the original question
/ Ch11Are you ready?
Q6
SkillSHEET 11.8
11D Checking solutions
  • identifying and using inverse operations to assist with the solution of number sentences
/ 11AUsing inverse operations
11BBuilding up expressions

Algebraic techniques

PAS4.1 Uses letters to represent numbers and translates between words and algebraic symbols

Knowledge and skills
Students learn about / Maths Quest 7
Exercise/Investigation / Maths Quest 8
Exercise/Investigation
  • using letters to represent numbers and developing the notion that a letter is used to represent a variable
/ 7BWriting and finding formulas
7CSubstitution
Nutritionpanels
7DProblem solving using algebra
7EAlgebraic expressions
7FExpressions and equations / 4AUsing pronumerals
Patternsandrules
‘Rulesofthumb’
Sonarmeasurements
6ABacktracking — inverse operations
  • using concrete materials such as cups and counters to model expressions
/ 7EAlgebraic expressions
  • recognising and using equivalent algebraic expressions
/ 7FExpressions and equations
  • translating between words and algebraic symbols and between algebraic symbols and words
/ 7CSubstitution
Nutritionpanels
7DProblem solving using algebra
Howhighwillitgrow?
7EAlgebraic expressions
7FExpressions and equations / 4AUsing pronumerals
‘Rulesofthumb’
Sonarmeasurements

Number patterns