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NZC - Mathematics and Statistics Phase 4 (Years 9–10)

Knowledge overview and teaching sequence for Phase 4 (Years 9-10) of the Mathematics and Statistics Learning Area. From 1 January 2026 this content is part of the statement of official policy relating to teaching, learning, and assessment of Mathematics and Statistics in all English medium state and state-integrated schools in New Zealand.

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About this resource

Knowledge overview and teaching sequence for Phase 4 (Years 9-10) of the Mathematics and Statistics Learning Area. From 1 January 2026 this content is part of the statement of official policy relating to teaching, learning, and assessment of Mathematics and Statistics in all English medium state and state-integrated schools in New Zealand.

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Number

Knowledge

The facts, concepts, principles, and theories to teach

Practices

The skills, strategies, and applications to teach

During Year 9

During Year 10

During Year 9

During Year 10

Number structures and operations

  • A number written in scientific notation has the form a × 10k, where 1 ≤ a < 10 and k is an integer.
  • Repeated division can be summarised using exponent notation with a negative exponent.
  • There are an infinite number of rational numbers between any two numbers; these can be represented by terminating decimals, recurring decimals, and fractions.
  • Multiplying a fraction by an equivalent form of 1, such as 3—3, results in an equivalent fraction.
  • When giving a fraction as an answer, there should be a positive or negative integer in the numerator and a positive integer in the denominator.
  • Numbers, including fractions, decimals, and percentages, can be represented using number lines.
  • Non-repeating, infinite decimals are irrational numbers; some of them are represented by special symbols, such as √2 and π.
  • The terms index, power, and exponent are used interchangeably.
  • For the number an, a represents the base, and n represents the exponent.
  • Exponent rules govern how operations involving exponents work and include:
    • am × an = am+n (the product-of-exponents rule)
    • am—an = am− n (the quotient-of-exponents rule)
    • (am)n = am×n (the exponent-of-exponents rule)
    • a−m = 1—am, (a ≠ 0) (the negative exponent rule)
    • a0 = 1 (a ≠ 0) (the zero exponent rule)
  • Only like roots can be added and subtracted; multiples of a root are represented with coefficients (e.g. √3 + √3 = 2√3).
  • There are rules for working with roots, including not leaving roots in a denominator:
    • √a × √a = a
    • √a × √b = √ab
    • √a ÷ √a = 1
    • √a ÷ √b = √a—b = √a—√b × √b—√b = √ab—b
  • Identifying, reading, writing, representing, comparing, ordering, and converting between fractions, decimals, and percentages
  • Recording, comparing, and ordering whole and decimal numbers using scientific notation (e.g. 3.14 × 103)
  • Finding equivalent fractions, simplifying fractions, and converting between improper fractions and mixed numbers
  • Expressing remainders from division as fractions or decimals, depending on the context
  • Identifying powers of 2 through to 210
  • Converting between negative powers and unit fractions (e.g. 3−2 = 1—9 )
  • Approximately locating roots on the number line with reference to the closest perfect square (e.g. √48 is between √36 = 6 and √49 = 7, but closer to 7)
  • Recording, comparing, ordering, and calculating with numbers in scientific notation
  • Identifying irrational numbers (e.g. 3√10, π)
  • Generalising about whether square and cube roots of whole numbers are rational or irrational
  • Calculating using integer exponents
  • Calculating exactly using fractions, roots, and multiples of π
  • Rounding and estimation support efficiently predicting results and checking the reasonableness of calculations.
  • The rules for identifying significant figures are:
    • all non-zero digits are significant
    • zeros appearing anywhere between two non-zero digits are significant
    • leading zeros are not significant
    • trailing zeros are significant if there is a decimal point present, and are not significant otherwise
    • exact numbers have an unlimited number of significant figures.
  • For numbers written in scientific notation as a × 10k, the number of significant figures is determined by applying the rules to the value of a.
  • Using rounding and estimation to predict results and to check the reasonableness of calculations
  • Using rounding, including to specified significant figures, and estimation to predict results and to check the reasonableness of calculations
  • Rounding to the degree of precision required for the context
  • The order of operations is important when evaluating or forming expressions. Operations are done as follows:
    1. grouped operations (e.g. expressions under a square root, involving the numerator of a fraction, or inside brackets)
    2. exponents or powers
    3. multiplication and division, from left to right
    4. addition and subtraction, from left to right.
  • A mnemonic, such as GEMA—Grouped (e.g. √32 + 42), Exponents (e.g. (−2)3), Multiplicative (× and ÷), Additive (+ and −)—can be used to remember the order of operations.
  • Every non-zero number has a multiplicative inverse (reciprocal), and their product is 1 (e.g. 5 and 1—5 are reciprocals, so 5 × 1—5 = 1—5 × 5 = 1).
  • Generalising about exponents of 0 and 1
  • Adding, subtracting, multiplying, and dividing integers
  • Generalising the rule for dividing by a fraction by starting with dividing a whole number by a fraction
  • Adding, subtracting, multiplying, and dividing fractions and decimals
  • Connecting multiplying or dividing decimals with multiplying or dividing fractions (e.g. 0.3 × 0.15 = 3—10 × 15—100).
  • Checking for equivalence in expressions involving negative numbers (e.g. (−3)2 ≠ −32, −2 + 3 = 3 + (−2), 2 × (−3) = (−3) × 2 = (−2) × 3, 2—−3 = −2—3 = −2—3
  • Adding, subtracting, multiplying, and dividing positive and negative numbers, including fractions and decimals
  • Evaluating positive integer exponents for positive and negative numbers (e.g. 35, (−1)4)
  • Percentages are a way of expressing a fraction of 100.
  • Percentages can be used to proportionally increase or decrease a quantity by multiplication and can be presented as decimal multipliers.
    • A percentage increase can be described by the additional percentage or the percentage of the final amount compared to the original amount (e.g. a 20% increase represents 120% of the original amount).
    • A percentage decrease can be described by the percentage lost or the percentage of the final amount compared to the original amount (e.g. a 20% decrease represents 80% of the original amount).
  • Ratios show part-to-part or part-to-whole comparisons of two or more quantities.
  • Ratios can be scaled up or down or simplified.
  • A rate proportionally compares two quantities that have different units of measure; when working with rates, ‘per’ means ‘for every’ in day-to-day contexts.
  • Finding a fraction or percentage of a number
  • Finding the whole amount, given a fraction or percentage (e.g. 20% of an amount is 30. What is the original amount?)
  • Expressing a number as a fraction or percentage of another number
  • Increasing or decreasing a number by a given proportion
  • Representing proportional relationships using whole-number ratios, including reducing the ratios to their simplest form
  • Dividing a quantity into two parts, given the part:part or part:whole ratio
  • Finding equivalent ratios and rates by scaling up or down
  • Applying a proportional increase or decrease to a number
  • Calculating the percentage increase or decrease between two numbers (e.g. What is the percentage increase between 50 and 75?)
  • Comparing and using ratios and rate (e.g. finding speed, given distance and time)

Financial mathematics

  • Percentages, ratios, rates, and proportions are often applied in financial situations.
  • Applying percentage mark-ups and discounts
  • Calculating simple interest and GST on dollar amounts (e.g. finding 15% GST on $432)
  • Converting New Zealand dollars into other currencies, and vice versa
  • Finding proportions of costs (e.g. the price of 400 g of an item, given the cost per kilogram)
  • Calculating compound interest on dollar amounts, by calculating simple interest month by month for short time periods (e.g. How much do you have after 3 months if you invest $100 at a 2.5%-per-month interest rate?)

Algebra

Knowledge

The facts, concepts, principles, and theories to teach

Practices

The skills, strategies, and applications to teach

During Year 9

During Year 10

During Year 9

During Year 10

Equations and relationships

  • The properties of operations (commutative, distributive, associative, inverse, and identity) and the order of operations apply to numbers and variables.
  • When operating on or writing equations with fractions, fractions of magnitude greater than 1 are usually written as improper fractions.
  • Simplifying and manipulating algebraic expressions involving sums, products, differences, and positive integer powers, by:
    • collecting like terms
    • factorising using common factors
    • expanding products, including multiplying a single term by a bracketed term.
  • Generalising the properties of operations with variables (e.g. multiplication is distributive over subtraction)
  • Multiplying or dividing by −1 in inequalities (e.g.−3 < 5)
  • Forming and solving linear equations with rational number coefficients and linear inequalities with positive coefficients
  • Using substitution to find the value of an expression or a formula, given the values of its variables
  • Rearranging formulae (e.g. solving P = 2l + 2w for w)
  • Simplifying and manipulating algebraic expressions involving sums, products, differences, and positive integer powers, by:
    • collecting like terms
    • factorising using common factors
    • factorising quadratic expressions with a leading coefficient of 1
    • expanding products, including multiplying a single term by a bracketed term, and multiplying two expressions each of the form ax + b, where a and b are integers
    • factorising by grouping (i.e. using the distributive law) (e.g. x2x − 8 = x2 + 4x − 2x − 8 = x(x + 4) − 2(x + 4) = (x − 2)(x + 4))
  • Forming and solving linear equations and linear inequalities with rational number coefficients
    (e.g. −2—5x + 5 ≤ −10), giving exact or rounded solutions, and representing the solution on a number line
  • Solving quadratic equations that are factorised or of the form x2 + c = 0 (where c is an integer), and connecting the solutions to the x-intercepts of the related graph
  • Substituting into, rearranging, and simplifying expressions or formulae that involve squares or square roots
    (e.g. A = πr2, c2 = a2 + b2)

  • Multiplying or dividing by a negative number reverses an inequality.
  • The constant rate of change of a linear graph is the vertical change (how far it goes up or down) divided by the horizontal change (how far it moves sideways).
  • Finding square roots of numbers and solving quadratic equations are related but have some differences.
    • Any positive real number has two square roots: one positive (the principal square root) and one negative (e.g. for 16, 4 and −4).
    • The square root operation √ refers specifically to the principal square root (e.g. √16 = 4).
    • There are 0, 1, or 2 real-number solutions to x2 = a, where a is a number.
  • The zero product property states that if two expressions multiply to be zero, then at least one expression must be zero (e.g. if ab = 0 then either a or b is 0, or if (x − a)(x − b) = 0 then either (x − a) = 0 or (x − b) = 0). 
  • There are specific factorising relationships that are useful to recognise:
    • x(x + a) = x2 + ax
    • difference of two squares: (x + a)(x − a) = x2 − a2
    • square of a sum: (x + a)2 = x2 + 2ax + a2
    • square of a difference: (x − a)2 = x2 − 2ax + a2.
  • The solution to a linear inequality is a set of values which may be represented with a number line.
  • For a specific straight line, the gradient, m, and y-intercept, c, are fixed, and x varies with y according to the rule y = mx + c.
  • The y-intercept touches the y-axis and has coordinates (0,c).
  • Interpreting rules of the form y = mx + c and using a combination of substitution and tables to plot points from the linear graph, connecting the points to form a line
  • Identifying the sign of m from tables of values, and linear graphs
  • Identifying the value of c for a straight line, from tables of values and from linear graphs
  • Using tables and graphs in the coordinate plane (showing all four quadrants), and diagrams to recognise the relationship between the ordinal position and its corresponding element in a linear pattern developing a rule for the pattern in words and making conjectures about further elements in the pattern
  • Identifying the constant increase or decrease in a linear pattern, using variables and algebraic notation to represent the rule in an equation, and drawing on the rule to make conjectures
  • Interpreting and graphing linear equations in the form y = mx + c, using the gradient and y-intercept
  •  Calculating the gradient and y-intercept of a line, using a graph
  • Comparing the relative magnitude of m in two or more linear graphs, using the concept of steepness and relating it to the magnitude of m
  • Finding the equation of a line, given two points or the gradient and a single point 
  • Determining the effect on graphs in the coordinate plane of changing the coefficient of x2 and the fixed value c, for a range of quadratic equations of the form y = ax2 or y = x2 + c, where a is a positive integer and c is an integer
  • In the equation of a line y = mx + c, m and c represent constants (they are unchanging), y and x can vary, and all the values of x and y that satisfy the equation create an infinite number of points that form the line.
  • The gradient is a ratio that can be interpreted as the ‘steepness’ of a linear graph.
    • A positive gradient slopes upwards when the graph is read from left to right.
    • A negative gradient slopes downwards when the graph is read from left to right.
    • A horizontal line has a gradient equal to zero. Its equation will be y = c.
  • The gradient m of a straight line can be determined with the formula m = rise—run = Δy—Δx .
  • A vertical line has an infinite gradient. This cannot be expressed in the formula m = rise—run as it becomes m = rise—0, which cannot be evaluated.
  • The equation of a vertical line is x = b, where the x-intercept is (b,0).

Measurement

Knowledge

The facts, concepts, principles, and theories to teach

Practices

The skills, strategies, and applications to teach

During Year 9

During Year 10

During Year 9

During Year 10

Measuring

  • A solution to a calculation cannot be more precise than the least precise number used in that calculation.
  • Estimating, calculating, converting, and accurately representing measurements
  • Estimating, calculating, converting, and accurately representing measurements using significant figures

  • The number of significant figures in a measurement is the number of digits that contribute to the degree of accuracy of the measurement, given as the number of digits known with certainty plus one uncertain digit.
  • When calculating, it is best to use at least one more significant figure than is required in the final solution, and round at the end of the whole calculation.
  • Conversions between different-sized metric units may be needed to give the appropriate units for a measurement or calculation.
  • The metric prefixes kilo–, mega–, giga–, and tera– signify a unit that is one thousand, one million, one billion, and one trillion times larger than the base unit.
  • The metric prefixes centi–, milli–, micro–, and nano– signify a unit one hundredth, one thousandth, one millionth, and one billionth the size of the base unit.
  • Derived units (e.g. cm2, km/h) reflect a relationship—a product or quotient—between two different measurements.
  • Selecting and using appropriate measurement units for a given context, converting between metric units if necessary and using appropriate prefixes
  • Converting between metric units, and using the appropriate prefixes in the metric system (e.g. kilo–, mega–, centi–, milli–, micro–)
  • The constant π is found by dividing a circle’s circumference by its diameter.
  • For a circle of radius r, the circumference is 2πr.
  • The area of a circle is given by A = πr2.
  • The surface area of a solid object is a measure of the total area that the surface of the object occupies.
  • The general formula for the volume of a prism is V = Al, where A is the consistent cross-sectional area and l is perpendicular to the plane of the cross-sectional area.
  • Finding:
    • the perimeter of 2D shapes
    • the circumference of circles
    • the area of parallelograms, trapeziums, and kites, relating the formulae used to the formula for a rectangle
  • Deriving the formulae for the perimeter of half and quarter circles from the formula for a full circle
  • Calculating the perimeter of half circles and quarter circles
  • Finding:
    • the area of circles and composite shapes that include circles or semicircles
    • the surface area and volume or capacity of prisms, pyramids, and cylinders
  • Deriving the formulae for the area of half and quarter circles from the formula for a full circle
  • Deriving the formulae for the surface area of cubes, rectangular prisms, and cylinders
  • Calculating the area of half circles and quarter circles
  • Calculating the surface area of cubes, rectangular prisms, triangular prisms, cylinders, and composite figures
  • Calculating the volume of cylinders and irregular prisms with a consistent cross-sectional area

  • Resizing (enlarging or reducing) a shape changes its perimeter, area, or volume proportionally according to the dimensions of the units; linear metric conversions must be squared to convert area and cubed to convert volume.

  • Scaling a shape by a factor, and determining the scale factor for the scaled shape’s area or volume
  • For right-angled triangles, Pythagoras’ theorem states that the square of the hypotenuse (longest side) is equal to the sum of the squares of the other two sides.
  • If (a,b,c) is a Pythagorean triple, then so is (ka,kb,kc), where k is a positive integer.
  • Using Pythagoras’ theorem to:
    • verify that given side lengths in a right-angled triangle satisfy the theorem
    • find the length of the hypotenuse in a right-angled triangle, given the lengths of the other two sides
  • Proving Pythagoras’ theorem (e.g. by rearranging four congruent right-angled triangles into a square)
  • Finding another Pythagorean triple from a given Pythagorean triple
  • Using Pythagoras’ theorem to:
    • find the length of an unknown side in a right-angled triangle
    • check if a triangle has a right angle
    • calculate the distance between two points in the coordinate plane, yielding the distance formula
      d = √(x2 − x1)2 + (y2 − y1)2
  • There is a fixed relationship between speed, distance, and time: speed = distance—time.
  • In position-time graphs, the gradient represents speed.
  • Finding distance, given speed and time
  • Finding time, given distance and speed
  • Finding speed, distance, or time, given any two of the measurements
  • Decimal measures are used for very small durations (e.g. milliseconds); the rest of time measurement uses a different system, based principally on 12 and 60.
  • Reasoning about duration using different units of time, including decimal fractions of milliseconds where appropriate

Geometry

Knowledge

The facts, concepts, principles, and theories to teach

Practices

The skills, strategies, and applications to teach

During Year 9

During Year 10

During Year 9

During Year 10

Shapes

  • A circle is the path traced out by a point moving in a plane and always a fixed distance (the radius) from a central point.
  • Angles between parallel lines and a transversal can be corresponding, co-interior, or alternate; corresponding angles are equal, and alternate angles are equal.
  • In similar shapes, corresponding angles are equal and the lengths of corresponding sides are proportional.
  • Congruent shapes are identical in shape and size.
  • Identifying and describing parts of a circle (e.g. a chord; the diameter, radius, and circumference) and how they relate to each other
  • Reasoning about unknown angles in situations involving intersecting and parallel lines and transversals.
  • Verifying that two lines are parallel, using angles at the intersections of a transversal
  • Using the properties of similarity in 2D shapes, including right-angled triangles, to find unknown lengths and angles

Spatial Reasoning

  • A set of points in a plane can be transformed by translation, reflection about a line, and rotation about a fixed point.
  • Representing and constructing 3D shapes, including rectangular and triangular prisms and pyramids, from nets and plan views drawings
  • Transforming 2D shapes in the coordinate plane by translation, reflection about a given line of symmetry, and rotation about a given point by a multiple of 90 degrees
  • Representing and constructing 3D shapes, including cylinders, from nets
  • Transforming 2D shapes, including composite shapes, by resizing them by any scale factor

Statistics

Knowledge

The facts, concepts, principles, and theories to teach

Practices

The skills, strategies, and applications to teach

During Year 9

During Year 10

During Year 9

During Year 10

Developing knowledge from data

  • Multivariate data is data in a set that has more than two variables.
  • Data can be collected from observational studies in which the observers do not alter or control the behaviour of the subjects.
  • Statistical questions clearly identify the variable, group of interest, and the intent of an investigation.
    • A summary investigation is about a group.
    • A comparison investigation compares a variable across two clearly identified groups.
    • A relationship investigation looks for a connection between paired numerical or paired categorical variables.
    • A time-series investigation looks at a variable over time.
  • Primary data is data that is collected first-hand.
  • Secondary data is data collected by someone else.
  • It is not always possible to get data from the entire population (as in a census). To make inferences about a population without a census, sampling is used.
  • Samples must be taken randomly from the population, otherwise there will be bias in the data, leading to inaccurate and misleading statistics. Samples are ideally chosen using simple random sampling, in which each item of the population has an equal probability of being chosen.
  • When sampling from a population, the distribution for a variable varies from sample to sample. To make a reliable inference about what is happening in the population, sample sizes need to be:
    • about 1,000 for categorical variables (with the sample obtained using technology)
    • at least 30 for numerical variables.
  • Planning and collecting multivariate data to respond to a statistical question and where at least one variable is categorical and at least one is numerical
  • Calculating the five-point-summary for numerical data:
    • the minimum value
    • the value of quartile 1, or Q1
    • the value of the median or quartile 2, or Q2
    • the value of quartile 3, or Q3
    • the maximum value
  • Calculating the interquartile range as IQR = Q3 − Q1
  • Planning and collecting multivariate data to respond to a statistical question using a sample or census
  • Reasoning why a mean or median would be a better measure of central tendency for a given statistical question

Visualisation of data

  • A distribution is formed from all the possible values of a variable and their frequencies. It can be shown using data visualisations that show patterns, trends, and variations and that include dot plots, bar graphs, frequency tables, box plots, histograms, time-series graphs, scatter plots, and two-way tables.
  • A good data visualisation should allow viewers to discern the variable(s) and who the data was collected from, and then, depending on the type of visualisation, additional information such as frequency, proportions, patterns or trends, and units for numerical variables.
  • Creating multiple data visualisations for an investigation
  • Selecting appropriate scales for data
  • In relationship investigations:
    • sometimes one variable is thought of as predictive of the other variable; then the response or dependent variable is on the y-axis, and the ‘predictive’, explanatory, or independent variable is on the x-axis
    • an eyeballed line or curve of best fit can be added for paired numerical data.
  • For relationship investigations, drawing an eyeballed line or curve of best fit to predict possible y-values (the response variable) for given x-values (the explanatory variable)

Interpretation of data

  • Elements of chance affect the certainty of results from observational studies and experiments.
  • Uncertainty should be taken into account when making claims.
  • Critically considering data visualisations, including those from contemporary media, to see if they support or misrepresent the data
  • Data visualisations need to be critically assessed to see if they support or misrepresent the data.
  • To compare data on the same variable from two groups in the same population, the 75%-to-50% comparison rule for informal inferences is used. If the groups are called A and B, and If more than 50% of group B’s data is larger than 75% of group A’s data, then we can make the claim that B tends to be larger than A back in the population.
  • An interpolation involves making predictions within the range of a numerical data variable.
  • An extrapolation involves making predictions outside the range of a numerical data variable.
  • Communicating findings in context to answer an investigative question, using evidence
  • Providing possible explanations for findings
  • Comparing findings to initial conjectures or assertions and existing knowledge
  • Evaluating findings and data-collection methods to check whether claims or statements are supported by the data
  • Communicating findings in context to answer an investigative question, using evidence and with an awareness of variability
  • Making an informal inference in comparative situations about what might be happening in the population, based on visual considerations and using the 75%-to-50% comparison rule
  • Making informal predictions from scatter plots in relationship situations

Probability

Knowledge

The facts, concepts, principles, and theories to teach

Practices

The skills, strategies, and applications to teach

During Year 9

During Year 10

During Year 9

During Year 10

Experimental and theoretical probability

  • Some chance-based situations, such as tossing a non-regular 3D shape, can only be explored through probability experiments.
  • Results from sets of repeated trials for the same experiment may vary.
  • The Law of Large Numbers states that as the number of trials in a chance experiment increases, the experimental probability will approach the experiment’s theoretical probability.
  • Lists, tables, two-way tables, and tree diagrams are useful systematic methods for generating all possible outcomes.
  • In joint events, events can be dependent or independent.
  • Probabilities for joint events cannot simply be added, because doing so would double-count outcomes that are common to both events.
  • Mutually exclusive events cannot occur together.
  • The estimated probability of an event from an experiment is the number of times the event happens divided by the total number of trials in the experiment (i.e. the relative frequency for that event).
  • Carrying out a chance experiment, including running simulations for a large number of trials using digital tools
  • Systematically listing outcomes for the sample space
  • Comparing experimental probability (from at least 30 trials) to theoretical probability for a chance experiment, and explaining why they differ and how increasing the number of trials reduces this difference
  • Carrying out chance experiments of at least 100 trials and comparing the experimental probability of each individual outcome to its theoretical probability, in order to demonstrate the Law of Large Numbers
  • Creating and describing data visualisations for the distribution of observed outcomes from a chance experiment 
  • Calculating probability estimates for different outcomes

The language of Mathematics and Statistics for Years 9–10

Year 9

Students will be taught the following new words:

Year 10

Students will be taught the following new words:

Number

  • GST
  • index
  • irrational number
  • like roots
  • original amount
  • precision.
  • rate
  • reciprocal
  • recurring
  • scientific notation
  • simple interest.
  • compound interest
  • principal square root
  • significant figures.

Algebra

  • expanding
  • gradient, slope
  • intercept.
  • linear relationship
  • rate of change.
  • operator
  • quadratic equation, relationship
  • zero product property.

Measurement

  • accuracy
  • chord
  • congruent
  • derived unit
  • hypotenuse.
  • mega–, giga–, tera–
  • micro–, nano–
  • Pythagorean triple
  • speed.
  • distance formula
  • resizing
  • scale factor
  • surface area.

Geometry

  • alternate, co-interior, or corresponding angles.
  • intersect
  • transversal.
  • similarity.

Statistics

  • comparison investigation, relationship investigation, summary investigation, time-series investigation 
  • distribution 
  • explanatory variable.
  • line or curve of best fit
  • multivariate data
  • population
  • quartile.
  • sampling
  • informal inference
  • interpolation, extrapolation
  • 75%-to-50% comparison rule.

Probability

  • elements of chance
  • joint events
  • mutually exclusive.
  • probability estimate
  • simulation.

Word or phrase

Description

Abstraction

The process of identifying and extracting the fundamental structures, patterns, or properties of a mathematical or statistical concept, detaching it from its original context to create a more general idea.

Additive identity

Zero will not change the value when added to a number. For example, 16 + 0 = 16.

Algebraic expression

A single mathematical expression that can be a number or a variable that may or may not have exponents, or combinations of these that is written as products or quotients. Examples include 8, x2, 8x2, 8—x2, or 32a7m—bc4.

Algorithm

A set of step-by-step instructions to perform a computation.

Arithmetically (growing pattern)

A description of a pattern that grows when each term increases or decreases by adding or subtracting a constant value.         

Associative property

A property of operation where three or more numbers can be added or multiplied in any grouping without changing the result. For example, (4 + 3) + 7 = 4 + (3 + 7) because 7 + 7 = 4 + 10, and (4 × 3) × 5 = 4 × (3 × 5).

Attribute

A geometric characteristic or feature of an object or common feature of a group of objects — such as size, shape, colour, number of sides.

Base ten

Our number value system with ten digit symbols (0-9); the place value of a digit in a number depends on its position; as we move to the left, each column is worth ten times more, with zero used as a placeholder; to the right, the system continues past the ones’ column, to create decimals (tenths, hundredths, thousandths); the decimal point marks the column immediately to the right as the tenths column.    

Benchmarks

A reference point that we can use for comparison or estimation. For example, “My finger is about one centimetre wide.”

Bivariate data

Bivariate data is data in a set that has two variables for each subject.

Categorical variables

A variable that classifies objects or individuals into groups or categories. For example, hair colour, breed of dog.

Chance

The likelihood that an outcome will occur.

Claim

A statement of interpretation drawn from mathematical or statistical data.

Coefficient

In an algebraic term, the coefficient is the number that multiplies the variable. For example, in the term 3y, 3 is the coefficient.

Commutative property

A property of operations where numbers can be added or multiplied in any order without changing the result. For example, 5 + 6 = 6 + 5 or 7 × 8 = 8 × 7.

Compose, decompose, recompose

Compose is to make a shape using other shapes. Decompose is to break a shape into other shapes. Recompose is to form the broken pieces of a shape into its original shape.

Conditional probabilities

The possibility of an event or outcome happening, based on the existence of a previous event or outcome.

Conjecture

A mathematical or statistical statement whose truth or otherwise is yet to be determined through analysis or computation.

Constant

A fixed value or a specific unchanging number in an equation, function, or expression.

Data

A collection of facts, numbers, or information; the individual values of which are often the results of an experiment or observations.

Data collection methods

Questions asked to get the data; carefully posed to ensure that the data will help to answer the intended investigative question.

Data visualisations

A graphical, tabular, or pictorial representation of information or data.

Discrete materials

Separate objects that can be counted and grouped. For example, counters or ice block sticks.

Discrete numerical variables

Variables that can be counted and have a limited range of possibilities. For example, number of students in each team or the result of rolling a die.

Distribution

In mathematics, distribution describes spreading terms out equally across an expression; in statistics, distribution describes how data values are spread across the range of values collected.

Element (in a repeating pattern)

In a repeating pattern, an element is the repeating core.
In a growing pattern, an element is a section of the pattern. For example, in the Fibonacci sequence 1, 1, 2, 3, 5, 8, the next element is the sum of the previous two elements (5 + 8 = 13).

Equation

A number statement that contains an equal sign. The expressions on either side of the equal sign have the same value (are equal).

Equivalent fractions

Fractions that represent the same value or number. For example, 1—2, 2—4, 3—6, and 4—8 are equivalent fractions because they represent the same number.

Estimate

A rough judgement of quantity, value, or number. In statistics, an assessment of the value of an existing, but unknown, quantity. In probability, an estimate is the probability that results from the outcome of an experiment

Event

One or more outcomes from a probability activity, situation, or experiment.

Evidence

Information, findings, data that support (prove) a statement or argument.

Expression

Two or more terms involving numbers and/or variables connected by operations. Expressions do not include an equal or inequality sign.

Function

The expression or equation which describes the relationship between two variables, where every x value has a unique y value. This can be written using mathematical notation or shown as a graph in the XY plane.

Geometrically

A description of a pattern that grows when each term increases or decreases by multiplying or dividing a constant value.  

Group of interest

Who the data is collected from in a statistics investigation.

Growing pattern

A pattern where there is a constant increase or decrease between each term. For example, 5, 10, 15, 20.

Inequality

A mathematical statement in which one number or expression is greater or less than another.

Inference

Making a conclusion based on evidence and reasoning.

Informal unit

A non-standard unit used to measure. For example, blocks, pens, fingers. The informal units used should all be the same size.

Inverse operations

The opposite operation, so addition is inverse to subtraction, and multiplication is inverse to division. They are useful to check calculations. For example, to check 4 × 5 = 20, we can see if 20 ÷ 5 = 4.

Investigative question

A question that guides inquiry in an investigative study.

Irrational number

A real number that cannot be expressed as a ratio of two integers. Examples are π and √4.

Justify

Use previously accepted statements and mathematical reasoning, evidence, or proof to explain statements about a conjecture.

Number sentence

An equation or inequality expressed using numbers and mathematical symbols. For example, 10 + 10 = 3 + 7 + 5 + 5.

Ordinal

The numerical position of the element in the sequence. For example, first, second, third, and so on.

Orientation

The angle at which an object is positioned.

Outcome

A possible result of a trial of a probability activity or a situation involving an element of chance.

Population

Group of individuals, items, or data used for an investigative study.

Primary data

Data collected first-hand for a specific purpose. For example, a survey, experiment, or interview. (See also Secondary data).

Probability experiment

A test that can be carried out multiple times in the same way (trials). The outcome of each trial is recorded.

Quantifying

Expressing a quantity using numbers.

Question
  • Investigative
  • Interrogative
  • Survey
  • Data Collection
  • Analysis (Analytical Question)

Rational number

A real number that can be expressed as a ratio of two integers. This includes integers, decimals, and fractions.

Rationalise

The process of eliminating roots or imaginary numbers from the denominator of a fraction.

Reasoning

Analysing a situation and using mathematical and statistical methods to arrive at a finding or conclusion.

Reciprocal

The inverse of a number or function, also known as the multiplicative inverse, for example, the reciprocal of 3—x is x—3.

Relative frequency

The number of times an event occurs divided by the total number of possible outcomes.

Repeating pattern

A pattern containing a 'unit of repeat'. For example, red, green, blue, red, green, blue.

Secondary data

Data collected by someone else, or a process, and/or obtained from another source. For example, online, books, other researchers. (See also Primary data).

Similarity

This is used to describe figures that have the same features, but different sizes. For example, two triangles both having angles of 40°, 60° and 80° but different side lengths.

Subitise

Instantly recognise the number of items in an arrangement without counting.

Tangible and intangible

Tangible is an object that can be touched. For example, a group of blocks. Intangible is a quality or measurement that cannot be touched. For example, colour or length.

Term (in a pattern)

One of the numbers in a pattern or sequence. For example, for 2, 4, 6, 8, the second term is 4.

Theoretical probability

A calculation of how likely an event is to occur in a situation involving chance.

Uncertainty

In probability, when the chance of an event occurring is unknown.

Unit of repeat

The part of a repeating pattern that repeats. The part is made up of several elements.

Variables (statistics and algebra)

A property or quantity that can take on different values. In statistics, a variable represents characteristics that may vary among individuals or over time. In algebra, a variable typically represents an unknown value or a quantity that can change within a given context.

Variation

The differences seen in the values of a property for different individuals or at different times.

Visualisation

The process of creating a mental or visual representation of data, concepts, or ideas. It can involve mentally imagining or manipulating information or visually representing data.

Links to mathematics and statistics supports and resources:

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