skip to main content
New NSW Syllabuses

Glossary

The following glossary was developed by the Australian Curriculum, Assessment and Reporting Authority (ACARA) (as accessed from the ACARA website at www.acara.edu.au on 9 July 2012) and is included here to assist teachers in the implementation of the Mathematics K–10 Syllabus.

For any additional information regarding the glossary, please see the ACARA website. Enquiries regarding the content of the glossary should be directed to ACARA.

Go to a b c d e f g h i J k l m n o p q r s t u v w X Y Z
adjacent angles

Two angles at a point are called adjacent if they share a common ray and a common vertex, and lie on opposite sides of the common ray.

Hence, in the diagram \(\angle AOC\) and \(\angle BOC\) are adjacent.

The image shows an acute angle AOB. It is bisected by the line CO.

algebraic expression

An algebraic expression is formed by combining numbers and algebraic symbols using arithmetic operations. The expression must be constructed unambiguously according to the rules of algebra. For example, \({a^2} + 3ab - 2{b^2}\) and \(\left( {x + 1} \right){e^x}\) are algebraic expressions, but \(2x +\div 3y\) is not because it is incomplete.

algebraic fraction

An algebraic fraction is a fraction in which both the numerator and denominator are algebraic expressions.

algebraic term

An algebraic term is an algebraic expression that forms a 'separable' part of some other algebraic expression. For example, \({x^2}\) and \(5{x^{ - 1}}\) are terms in the inequality \({x^2} \le 5{x^{ - 1}}\), and \(2,{\rm{ }}3x,{\rm{ }}5{x^2}\) are terms of the polynomial \(2 + 3x + 5{x^2}\).

alternate angles

In each diagram below, the two marked angles are called alternate angles (since they are on alternate sides of the transversal).

2 sets of non-parallel lines & transversal. Alternate unequal angles are marked in each diagram.

If the lines AB and CD are parallel, then each pair of alternate angles is equal.

A transversal intersects two parallel lines. A pair of alternate equal angles is marked.

angle

An angle is the figure formed by two rays sharing a common endpoint, called the vertex of the angle.

Imagine that the ray OB is rotated about the point O until it lies along OA. The amount of turning is called the size of the angle AOB.

The image shows an acute angle AOB. The angle is marked by an arc at O.

angle classification

Angles are classified according to their size.

We say that

  • an angle with size \(\alpha \) is acute if \(0^\circ < \alpha < 90^\circ\),
  • an angle with size \(\alpha \) is obtuse if \(90^\circ < \alpha < 180^ \circ\),
  • an angle with size \(\alpha \) is reflex if \(180^ \circ < \alpha < 360^ \circ\).
angles of elevation and depression

When an observer looks at an object that is lower than 'the eye of the observer', the angle between the line of sight and the horizontal is called the angle of depression.

2 lines from 1 point - horizontal line and line going down. Resultant angle = angle of depression.

When an observer looks at an object that is higher than 'the eye of the observer', the angle between the line of sight and the horizontal is called the angle of elevation.

2 lines from 1 point - a horizontal line and line going up. Resultant angle = angle of elevation.

array

An array is an ordered collection of objects or numbers. Rectangular arrays are commonly used in primary mathematics.

associative

A method of combining two numbers or algebraic expressions is associative if the result of the combination of three objects does not depend on the way in which the objects are grouped.

For example, addition of numbers is associative and the corresponding associative law is:

\(\left( {a + b} \right) + c = a + \left( {b + c} \right)\) for all numbers \(a\), \(b\) and \(c\).

Multiplication is also associative: \(\left( {ab} \right)c = a\left( {bc} \right)\) for all numbers \(a\), \(b\) and \(c\), but subtraction and division are not, because, for example, \(\left( {7 - 4} \right) - 3 \ne 7 - \left( {4 - 3} \right)\) and \(\left( {12 \div 6} \right) \div 2 \ne 12 \div \left( {6 \div 2} \right)\).

back-to-back stem-and-leaf plot

A back-to-back stem-and-leaf plot is a method for comparing two data distributions by attaching two sets of 'leaves' to the same 'stem' in a stem-and-leaf plot.

For example, the stem-and-leaf plot below displays the distribution of pulse rates of 19 students before and after gentle exercise.

A method for comparing two data distributions by attaching two sets of leaves to the same stem.

bi-modal data

Bi-modal data is data whose distribution has two modes.

bivariate data

Bivariate data is data relating to two variables, for example, the arm spans and heights of 16-year-olds, the sex of primary school students and their attitude to playing sport.

bivariate numerical data

Bivariate numerical data is data relating to two numerical variables, for example height and weight.

box plot

The term box plot is a synonym for a box-and-whisker plot.

A box-and-whisker plot is a graphical display of a five-number summary.

In a box-and-whisker plot, the 'box' covers the interquartile range (IQR), with 'whiskers' reaching out from each end of the box to indicate maximum and minimum values in the data set. A vertical line in the box is used to indicate the location of the median.

The box-and-whisker plot below has been constructed from the five-number summary of the resting pulse rates of 17 students.

Graphical display of a 5-number summary. Box covers the interquartile range; whiskers show min & max

The term 'box-and-whisker plot' is commonly abbreviated to 'box plot'.

capacity

Capacity is a term used to describe how much a container will hold. It is often used in relation to the volume of fluids. Units of capacity (volume of fluids or gases) include litres and millilitres.

Cartesian coordinate system

Two intersecting number lines are intersecting at right angles at their origins to form the axes of the coordinate system.

The plane is divided into four quadrants by these perpendicular axes called the x-axis (horizontal line) and the y-axis (vertical line).

The position of any point in the plane can be represented by an ordered pair of numbers \(\left( {x,\,y} \right)\). These are called the coordinates of the point. This is called the Cartesian coordinate system. The plane is called the Cartesian plane.

The point with coordinates (4, 2) has been plotted on the Cartesian plane shown. The coordinates of the origin are (0, 0).

A point on a Cartesian plane is represented by an ordered pair of numbers (X and Y)

categorical variable

A categorical variable is a variable whose values are categories.

Examples: blood group is a categorical variable; its values are: A, B, AB or O. So too is construction type of a house; its values might be brick, concrete, timber, or steel.

Categories may have numerical labels, for example, for the variable postcode the category labels would be numbers like 3787, 5623, 2016, etc, but these labels have no numerical significance. For example, it makes no sense to use these numerical labels to calculate the average postcode in Australia.

census

A census is an attempt to collect information about the whole population.

chord

A chord is a line segment (interval) joining two points on a circle.

 

A line PQ joins 2 points on the circumference of a circle but does not go through the centre.

circle

The circle with centre O and radius r is the set of all points in the plane whose distance from O is r.

Image shows a circle with centre labelled O. A radius is drawn to point A. The radius is labelled r.

The line segment OA (interval OA) is also called a radius of the circle.

Putting the point of a pair of compasses at the centre and opening the arms to the radius can draw a circle.

co-interior angles

In each diagram the two marked angles are called co-interior angles and lie on the same side of the transversal.

2 sets of non-parallel lines & transversal. 1 pair of co-interior angles is marked in each diagram.

If the lines AB and CD are parallel then \(\alpha  + \beta  = {180^ \circ }\).

A transversal intersects two parallel lines. A pair of co-interior angles is marked alpha and beta.

Co-interior angles formed by parallel lines are supplementary.

Conversely, if a pair of co-interior angles is supplementary then the lines are parallel.

column graph

A column graph is a graph used in statistics for organising and displaying categorical data.

To construct a column graph, equal width rectangular bars are constructed for each category with height equal to the observed frequency of the category as shown in the example below, which displays the hair colours of 27 students.

The image is a column graph showing numbers of students and their hair colour.

Column graphs are frequently called bar graphs or bar charts. In a bar graph or chart, the bars can be either vertical or horizontal.

common factor

A common factor (or common divisor) of a set of numbers or an algebraic expression is a factor of each element of that set.

For example, 6 is a common factor of 24, 54 and 66, and \(x + 1\) is a common factor of \({x^2} - 1\) and \({x^2} + 5x + 4\).

commutative

A method of combining two numbers or algebraic expressions is commutative if the result of the combination does not depend on the order in which the objects are given.

For example, addition of numbers is commutative, and the corresponding commutative law is:

\(a + b = b + a\) for all numbers \(a\) and \(b\).

Multiplication is also commutative: \(ab = ba\) for all numbers \(a\) and \(b\), but subtraction and division are not, because, for example, \(5 - 3 \ne 3 - 5\) and \(12 \div 4 \ne 4 \div 12\).

complementary angles

Two angles that add to 90° are called complementary. For example, 23° and 67° are complementary angles.

complementary events

Events \(A\) and \(B\) are complementary events, if \(A\) and \(B\) are mutually exclusive and \(P\left( A \right) + P\left( B \right) = 1\).

composite number

A natural number that has a factor other than 1 and itself is a composite number.

compound interest

The interest earned by investing a sum of money (the principal) is compound interest if each successive interest payment is added to the principal for the purpose of calculating the next interest payment.

For example, if the principal \(\$ P\) earns compound interest at the rate of r per period, then after n periods the principal plus interest is \(\$ P{\left( {1{\rm{ }} + r} \right)^n}\).

cone

A cone is a solid that is formed by taking a circle called the base and a point not in the plane of the circle, called the vertex, which lies above or below the circle and joining the vertex to each point on the circle.

If the vertex is directly above or below the centre of the circular base, we call the cone a right cone.

The height of the cone is the distance from the vertex to the centre of the circular base.

The slant height of a cone is the distance from any point on the circle to the vertex.

The image is a right cone showing radius (r), height (h), & slant height (marked with an arrow).

congruence

Two plane figures are called congruent if one can be moved by a sequence of translations, rotations and reflections so that it fits exactly on top of the other figure.

Two figures are congruent when we can match every part of one figure with the corresponding part of the other figure. For example, the two figures below are congruent.

Matching intervals have the same length, and matching angles have the same size.

A quadrilateral with two right-angles; all angles are named. Mirror-image is shown.

congruent triangles

The four standard congruence tests for triangles

Two triangles are congruent if:

SSS: the three sides of one triangle are respectively equal to the three sides of the other triangle, or

SAS: two sides and the included angle of one triangle are respectively equal to two sides and the included angle of the other triangle, or

AAS: two angles and one side of one triangle are respectively equal to two angles and the matching side of the other triangle, or

RHS: the hypotenuse and one side of one right-angled triangle are respectively equal to the hypotenuse and one side of the other right-angled triangle.

continuous variable

A continuous variable is a numerical variable that can take any value that lies within an interval. In practice, the values taken are subject to the accuracy of the measurement instrument used to obtain these values.

Examples include height, reaction time to a stimulus and systolic blood pressure.

corresponding angles

In each diagram the two marked angles are called corresponding angles.

4 sets of non-parallel lines transversed. Unequal corresponding angles are marked on each image.

If the lines are parallel, then each pair of corresponding angles is equal.

A transversal intersects two parallel lines. A pair of corresponding angles is marked at QGD & GFB.

Conversely, if a pair of corresponding angles is equal, then the lines are parallel.

cosine

In any right-angled triangle,

\(\cos \theta  = \frac{\rm{adjacent}}{\rm{hypotenuse}}\) where \(0^\circ < \theta < 90^\circ \).

Right-angled triangle with sides labelled hypotenuse, opposite and adjacent; 1 angle is marked theta.

cosine rule

In any triangle \(ABC\),

\({c^2} = {a^2} + {b^2}-2ab\cos C\).

The image shows a scalene triangle ABC. AB is labelled c, BC is labelled a and AC is labelled b.

counting number

The counting numbers are the non-negative integers, that is, one of the numbers 0, 1, 2, 3, …

Sometimes it is taken to mean only a positive integer.

A natural number is a positive integer or counting number. The natural numbers are 1, 2, 3, … The set of natural numbers is usually denoted by \(\mathbb{N}\).

counting on

Counting a collection, or reciting a sequence of number words, from a point beyond the beginning of the sequence.

For example, when a child has counted that there are 6 objects in a collection and is then asked 'How many?' after several more are added, the child might count on from 6 saying '7, 8, 9, ...' to reach the total. This is considered a more sophisticated strategy than counting the whole collection from 1.

cylinder

A cylinder is a solid that has parallel circular discs of equal radius at the ends. Each cross-section parallel to the ends is a circle with the same radius, and the centres of these circular cross-sections lie on a straight line, called the axis of the cylinder.

A cylinder with cross-section marked by a broken line and labelled. Axis is also marked and labelled.

data

Data is a general term for a set of observations and measurements collected during any type of systematic investigation.

Primary data is data collected by the user. Secondary data is data collected by others. Sources of secondary data include web-based data sets, the media, books, scientific papers, etc.

data display

A data display is a visual format for organising and summarising data.

Examples include box plots, column graphs, frequency tables and stemplots.

decimal

A decimal is a numeral in the decimal number system.

For example, the decimal expansion of \(6{\textstyle{3 \over 4}}\) is 6.75. The integer part is 6 and the fractional part is 0.75.

A decimal is terminating if the fractional part has only finitely many decimal digits. It is non-terminating if it has infinitely many digits.

For example, 6.75 is a terminating decimal, whereas 0.3161616…, where the pattern 16 repeats indefinitely, is non-terminating.

Non-terminating decimals may be recurring, that is, contain a pattern of digits that repeats indefinitely after a certain number of places.

For example, 0.3161616… is a recurring decimal, whereas 0.101001000100001…, where the number of 0s between the 1s increases indefinitely, is not recurring.

It is common practice to indicate the repeating part of a recurring decimal by using dots or lines as superscripts.

For example, 0.3161616… could be written as \(0.3\dot 1\dot 6\) or \(0.3\bar 1\bar 6\).

The decimal number system is the base 10, place-value system most commonly used for representing real numbers. In this system positive numbers are expressed as sequences of Arabic numerals 0 to 9, in which each successive digit to the left or right of the decimal point indicates a multiple of successive powers (respectively positive or negative) of 10.

For example, the number represented by the decimal 12.345 is the sum \(1 \times {10^1} + 2 \times {10^0} + 3 \times {10^{ - 1}} + 4 \times {10^{ - 2}} + 5 \times {10^{ - 3}}\).

denominator

In the fraction \(\frac{a}{b}\), b is the denominator. It is the number of equal parts into which the whole is divided in order to obtain fractional parts. For example, if a line segment is divided into 5 equal parts, each of those parts is one-fifth of the whole and corresponds to the unit fraction \(\frac{1}{5}\).

diameter

A diameter is a chord passing through the centre of a circle.

The word diameter is also used for the length of the diameter.

difference

A difference is the result of subtracting one number or algebraic quantity from another.

distributive

Multiplication of numbers is distributive over addition because the product of one number with the sum of two others equals the sum of the products of the first number with each of the others. This means that we can multiply two numbers by expressing one (or both) as a sum and then multiplying each part of the sum by the other number (or each part of its sum).

For example, \(8 \times 17 = 8 \times \left( {10 + 7} \right) = 8 \times 10 + 8 \times 7 = 80 + 56 = 136\).

This distributive law is expressed algebraically as follows:

\(a\left( {b + c} \right) = ab + ac\), for all numbers aand c.

divisible

In general, a number or algebraic expression x is divisible by another y if there exists a number or algebraic expression q of a specified type for which \(x = yq\).

A natural number m is divisible by a natural number n if there is a natural number q such that \(m = nq\).

For example, 12 is divisible by 4 because 12 = 3 x 4.

dot plot

A dot plot is a graph used in statistics for organising and displaying numerical data.

Using a number line, a dot plot displays a dot for each observation. Where there is more than one observation, or observations are close in value, the dots are stacked vertically. If there is a large number of observations, dots can represent more than one observation. Dot plots are ideally suited for organising and displaying discrete numerical data.

The dot plot below displays the number of passengers observed in 32 cars stopped at a traffic light.

A dot plot diagram showing number of occurrences in each group 0-7 (passengers).

Dot plots can also be used to display categorical data, with the numbers on the number line replaced by category labels.

element

An element of a set is a member of that set. For example, the elements of the set \(\left\{ {2, \, 3, \, 4, \, 6, \,8} \right\}\) are the numbers 2, 3, 4, 6 and 8. We write \(x \in S\) to indicate that \(x\) is a member of the set \(S\).

enlargement (dilation)

An enlargement (or dilation) is a scaled up (or down) version of a figure in which the transformed figure is in proportion to the original figure. The relative positions of points are unchanged and the two figures are similar.

In the diagram below, triangle \(A'B'C'\) is the image of triangle ABC under the enlargement with enlargement factor 2 and centre of enlargement O.

A scaled version of a figure, where the new figure is in proportion to the original figure.

equally likely outcomes

Equally likely outcomes occur with the same probability.

For example, in tossing a fair coin, the outcome 'head' and the outcome 'tail' are equally likely.

In this situation, \(P\left( {{\rm{head}}} \right) = P\left( {{\rm{tail}}} \right) = 0.5\).

equation

An equation is a statement that asserts that two numbers or algebraic expressions are equal in value. An equation must include an equals sign. For example, 3 + 14 = 11 + 6.

An identity is an equation involving algebraic expressions that is true for all values of the variables involved.

For example, \({x^2}-4 = \left( {x-2} \right)\left( {x + 2} \right)\).

An identity is an equation that is true for all values of the variables involved.

For example, \({x^2}-{y^2} = \left( {x-y} \right)\left( {x + y} \right)\).

equivalent fractions

Two fractions \(\frac{a}{b}\) and \(\frac{c}{d}\) are equivalent if they are equal, that is, \(ad = bc\).

Equivalent fractions are alternative ways of writing the same fraction.

For example, \(\frac{1}{2} = \frac{2}{4} = \frac{3}{6} =  ...\)

estimate

In statistical terms, an estimate is information about a population extrapolated from a sample of the population.

For example, the mean number of decayed teeth in a randomly selected group of eight-year-old children is an estimate of the mean number of decayed teeth in eight-year-old children in Australia.

even number

A whole number is even if it is divisible by 2.

event

An event is a subset of the sample space for a random experiment.

For example, the set of outcomes from tossing two coins is \(\left\{ {{\rm{HH, HT, TH, TT}}} \right\}\), where \(\rm{H}\) represents a 'head' and \(\rm{T}\) a 'tail'.

For example, if A is the event 'at least one head is obtained', then \(A = \left\{ {{\rm{HH, HT, TH}}} \right\}.\)

expected frequency

An expected frequency is the number of times that a particular event is expected to occur when a chance experiment is repeated a number of times. For example, if the experiment is repeated n times, and on each of those times the probability that the event occurs is p, then the expected frequency of the event is np.

For example, suppose that a fair coin is tossed 5 times and the number of heads showing recorded. Then the expected frequency of 'heads' is \(\frac{5}{2}\).

This example shows that the expected frequency is not necessarily an observed frequency, which in this case is one of the numbers 0, 1, 2, 3, 4 or 5.

expression

Two or more numbers or variables connected by operations. For example, \(17-9,\, 8 \times \left( 2 + 3 \right),\, 2a + 3b\) are all expressions.

Expressions do not include an equals sign.

factor

In general, a number or algebraic expression x is a factor (or divisor) of another y if there exists a number or algebraic expression q of a specified type for which \(y = xq\).

A natural number m is a factor of a natural number n if there is a natural number q such that \(n = mq\).

For example, 4 is a factor of 12 because \(12 = 3 \times 4\).

A polynomial \(a\left( x \right)\) is divisible by a polynomial \(b\left( x \right)\) if there is a polynomial \(q\left( x \right)\) for which \(a\left( x \right) = b\left( x \right)q\left( x \right)\).

For example, \(x - 2\) is a factor of \({x^2}-6x + 8\) because \({x^2}-6x + 8 = \left( {x-4} \right)\left( {x-2} \right)\).

factor and remainder theorem

According to the factor theorem, if \(p\left( x \right)\) is a polynomial and \(p\left( a \right) = 0\) for some number a, then \(p\left( x \right)\) is divisible by \(x-a\).

This follows easily from the remainder theorem, because for \(p\left( x \right) \div \left( {x-a} \right)\) the remainder is \(p\left( a \right)\). So if \(p\left( a \right) = 0\), the remainder is \(0\) and \(p\left( x \right)\)is divisible by \(x-a\).

The factor theorem can be used to obtain factors of a polynomial.

For example, if \(p\left( x \right) = {x^3}-3{x^2} + 5x-6\), then it is easy to check that \(p\left( 2 \right) = {2^3}-3 \times {2^2} + 5 \times 2-6 = 0\). So by the factor theorem \(x - 2\) is a factor of \(p\left( x \right) = {x^3} - 3{x^2} + 5x - 6\).

According to the remainder theorem, if a polynomial \(p\left( x \right)\) is divided by \(x - a\), where \(a\) is any real number, the remainder is \(p\left( a \right)\). That is, \(p\left( x \right) = q\left( x \right)\left( {x - a} \right) + p\left( a \right)\), for some polynomial \(q\left( x \right)\).

factorise

To factorise a number or algebraic expression is to express it as a product.

For example, 15 is factorised when expressed as a product: \(15 = 3 \times 5\), and \({x^2} - 3x + 2\) is factorised when written as a product: \({x^2} - 3x + 2 = \left( {x - 1} \right)\left( {x - 2} \right)\).

five-number summary

A five-number summary is a method for summarising a data set using five statistics: the minimum value, the lower quartile, the median, the upper quartile and the maximum value.

fraction

The fraction \(\frac{a}{b}\) (written alternatively as \(a\) / \(b\)), where \(a\) is a non-negative integer and \(b\) is a positive integer, was historically obtained by dividing a unit length into \(b\) equal parts and taking \(a\) of these parts.

For example, \(\frac{3}{5}\) refers to 3 of 5 equal parts of the whole, taken together.

In the fraction \(\frac{a}{b}\) the number \(a\) is the numerator and the number \(b\) is the denominator.

It is a proper fraction if \(a < b\) and an improper fraction otherwise.

frequencies

Frequency, or observed frequency, is the number of times that a particular value occurs in a data set.

For grouped data, it is the number of observations that lie in that group or class interval.

frequency distribution

A frequency distribution is the division of a set of observations into a number of classes, together with a listing of the number of observations (the frequency) in that class.

Frequency distributions can be displayed in tabular or graphical form.

frequency table

A frequency table lists the frequency (number of occurrences) of observations in different ranges, called class intervals.

The frequency distribution of the heights (in cm) of a sample of 42 people is displayed in the frequency table below.

Height (cm)

Class interval

Frequency

155–<160

3

160–<165

2

165–<170

9

170–<175

7

175–<180

10

180–<185

5

185–<190

5

190–<195

1

function

A function f assigns to each element of one set S precisely one element of a second set T.

The functions most commonly encountered in elementary mathematics are real functions of real variables. For such functions, the domain and codomain are sets of real numbers.

Functions are usually defined by a formula for \(f\left( x \right)\) in terms of x.

For example, the formula \(f\left( x \right) = {x^2}\) defines the 'squaring function' that maps each real number x to its square x2.

gradient

If \(A\left( {{x_1},\,\,{y_1}} \right)\) and \(B\left( {{x_2},\,\,{y_2}} \right)\) are points in the plane, \({x_2} - {x_1} \ne 0\), the gradient of the line segment (interval) \(AB = \frac{{{\rm{rise}}}}{{{\rm{run}}}} = \frac{{{y_2} - {y_1}}}{{{x_2} - {x_1}}}\).

Using an x y axis, the gradient of a line is shown to be the same at any point on this line.

The gradient of a line is the gradient of any line segment (interval) within the line.

highest common factor

The highest common factor (HCF), greatest common factor (GCF) or greatest common divisor (GCD) of a given set of natural numbers is the common divisor of the set that is greater than each of the other common divisors.

For example, 1, 2, 3 and 6 are the common factors of 24, 54 and 66 and 6 is the greatest common divisor.

histogram

A histogram is a statistical graph for displaying the frequency distribution of continuous data.

A histogram is a graphical representation of the information contained in a frequency table. In a histogram, class frequencies are represented by the areas of rectangles centred on each class interval. The class frequency is proportional to the rectangle's height when the class intervals are all of equal width.

The histogram below displays the frequency distribution of the heights (in cm) of a sample of 42 people with class intervals of width 5 cm.

Histogram is a graphical representation of information in a frequency table.

hyperbola (rectangular)

The graph of \(y = \frac{1}{x}\) is called a rectangular hyperbola. The x- and y-axes are asymptotes as the curve gets as close as we like to them.

Image shows the two curves of y equals 1 over x drawn on a Cartesian plane.

independent event

Two events are independent if knowing the outcome of one event tells us nothing about the outcome of the other event.

independent variable

When investigating relationships in bivariate data, the explanatory variable is the variable that may explain or cause a difference in the response variable.

For example, when investigating the relationship between the temperature of a loaf of bread and the time it has spent in a hot oven, temperature is the response variable and time is the explanatory variable.

With numerical bivariate data it is common to attempt to model such relationships with a mathematical equation and to call the response variable the dependent variable and the explanatory variable the independent variable.

When graphing numerical data, the convention is to display the response (dependent) variable on the vertical axis and the explanatory (independent) variable on the horizontal axis.

When there is no clear causal link between the events, the classification of the variables as either the dependent or independent variable is quite arbitrary.

index

Index is synonymous with exponent.

The exponent or index of a number or algebraic expression is the power to which the latter is to be raised. The exponent is written as a superscript. Positive integral exponents indicate the number of times a term is to be multiplied by itself. For example, \({a^3} = a \times a \times a\).

index law

Index laws are rules for manipulating indices (exponents). They include

\({x^a} \times {x^b} = {x^{a + b}}\);   \({x^a} \div {x^b} = {x^{a - b}}\);   \({\left( {{x^a}} \right)^b} = {x^{ab}}\);   \({x^a}{y^a} = {\left( {xy} \right)^a}\);  \({x^0} = 1\);  \({x^{ - a}} = \frac{1}{{{x^a}}}\);  and  \({x^{\frac{1}{a}}} = \sqrt[a]{x}\).

inequality

An inequality is a statement that one number or algebraic expression is less than (or greater than) another. There are four types of inequalities:

  • a is less than b is written \(a < b\),
  • a is greater than b is written \(a > b\),
  • a is less than or equal to b is written \(a \le b\), and
  • a is greater than or equal to b is written \(a \ge b\).
informal unit

Informal units are not part of a standardised system of units for measurement. For example, an informal unit for length could be paperclips of uniform length. An informal unit for area could be uniform paper squares of any size. Informal units are sometimes referred to as nonstandard units.

integer

The integers are the 'whole numbers' \( \ldots  \, ,-3,-2,-1,\, 0, \,1, \,2, \,3, \, ...\)
The set of integers is usually denoted by \(\mathbb{Z}\). Integers are basic building blocks in mathematics.

interquartile range

The interquartile range (IQR) is a measure of the spread within a numerical data set. It is equal to the upper quartile \(\left( {{Q}_{3}} \right)\) minus the lower quartile \(\left( {{Q}_{1}} \right)\); that is, \(IQR = {Q_3} - {Q_1}\).

The IQR is the width of an interval that contains the middle 50% (approximately) of the data values. To be exactly 50%, the sample size must be a multiple of four.

interval

An interval is a certain type of subset of the number line.

A finite interval is the set of all real numbers between two given real numbers called the endpoints of the interval. The endpoints may or may not be included in the interval.

irrational number

An irrational number is a real number that is not rational. Some commonly used irrational numbers are \(\pi \), \(e\) and \(\sqrt 2 \).

The Euler number is an irrational real number whose decimal expansion begins \(e = 2.718281828...\)

irregular shape

An irregular shape can be a polygon. A polygon that is not regular is irregular.

kite

A kite is a quadrilateral with two pairs of adjacent sides of equal length.

Two quadrilaterals (convex and non-convex) each with two pairs of adjacent sides of equal length.

A kite may be convex, as shown in the diagram above to the left, or non-convex, as shown above to the right. The axis of the kite is shown.

line segment (interval)

If A and B are two points on a line, the part of the line between and including A and B is called a line segment or interval.

The distance AB is a measure of the size or length of AB.

A & B are 2 points on a line. The part of line between A & B is a segment or interval.

Any point A on a line divides the line into two pieces called rays. The ray AP is that ray which contains the point P (and the point A). The point A is called the vertex of the ray and it lies on the ray.

line symmetry

A plane figure F has line symmetry in a line m if the image of F under the reflection in m is F itself. The line m is called the axis of symmetry.

The axis of symmetry is drawn through the centre of the face from the top to the bottom.

linear equation

A linear equation is an equation involving just linear terms, that is, polynomials of degree 1. The general form of a linear equation in one variable is \(ax + b = 0\).

location (statistics)

A measure of location is a single number that can be used to indicate a central or 'typical value' within a set of data.

The most commonly used measures of location are the mean and the median, although the mode is also sometimes used for this purpose.

logarithm

The logarithm of a positive number x is the power to which a given number b, called the base, must be raised in order to produce the number x. The logarithm of x, to the base b is denoted by \({\log _b}x\).

Algebraically: \({\log _b}x = y \leftrightarrow {b^y} = x\).

For example, \({\log _{10}}100 = 2\) because \({10^2} = 100\), and \({\log _2}\left( \frac{1}{32} \right) = -5\) because \(2^{ - 5} = \frac{1}{32}\).

many-to-one correspondence

A many-to-one correspondence is a function or mapping that takes the same value for at least two different elements of its domain. For example, the squaring function \(x \mapsto {x^2}\) is many-to-one because \({x^2} = {\left( { - x} \right)^2}\) for all real numbers \(x\).

mean

The arithmetic mean of a list of numbers is the sum of the data values divided by the number of numbers in the list.

In everyday language, the arithmetic mean is commonly called the average.

For example, for the following list of five numbers \(\left\{ {2,\,\,3,\,\,3,\,\,6,\,\,8} \right\}\) the mean equals \(\frac{{2 + 3 + 3 + 6 + 8}}{5} = \frac{{22}}{5} = 4.4\).

median

The median is the value in a set of ordered data that divides the data into two parts. It is frequently called the 'middle value'.

Where the number of observations is odd, the median is the middle value.

For example, for the following ordered data set with an odd number of observations, the median value is 5.

1, 3, 3, 4, 5, 6, 8, 9, 9

Where the number of observations is even, the median is calculated as the mean of the two central values.

For example, in the following ordered data set, the two central values are 5 and 6, and median value is the mean of these two values, 5.5.

1, 3, 3, 4, 5, 6, 8, 9, 9, 10

The median provides a measure of location of a data set that is suitable for both symmetric and skewed distributions and is also relatively insensitive to outliers.

midpoint

The midpoint \(M\) of a line segment (interval) \(AB\) is the point that divides the segment into two equal parts.

Let \(A\left( {x_1},\,{y_1} \right)\) and \(B\left( x_2,\, y_2 \right)\) be points in the Cartesian plane. Then the midpoint \(M\) of line segment \(AB\) has coordinates \(\left( \frac{x_1 + \, x_2}{2},\frac{y_1 + \, y_2}{2} \right)\).

This can be seen from the congruent triangles below.

The Midpoint (M) of a line segment is the point that divides the segment into 2 equal parts.

mode

The mode is the most frequently occurring value in a set of data. There can be more than one mode. When there are two modes, the data set is said to be bi-modal.

The mode is sometimes used as a measure of location.

monic

A monic polynomial is one in which the coefficient of the leading term is 1.

For example, \({x^3} + 2{x^2} - 7\) is monic, but \(4{x^2} - x + 1\) is not.

multiple

A multiple of a number is the product of that number and an integer.

A multiple of a real number x is any number that is a product of x and an integer. For example, 4.5 and –13.5 are multiples of 1.5 because \(4.5 = 3 \times 1.5\) and \( - 13.5 = - 9 \times 1.5\).

multiplication

Multiplicative situations are problems or contexts that involve multiplication (or division). Calculating the number of seats in a theatre that has 30 rows of 24 seats, finding equivalent fractions, and working with ratios and percentages are all multiplicative situations.

mutually exclusive events

Two events \(A\) and \(B\) are mutually exclusive if one is incompatible with the other; that is, if they cannot be simultaneous outcomes in the same chance experiment.

For example, when a fair coin is tossed twice, the events '\(HH\)' and '\(TT\)' cannot occur at the same time and are, therefore, mutually exclusive.

In a Venn diagram, as shown below, mutually exclusive events do not overlap.

The image shows a Venn diagram with 2 separate circles. The circles are labelled A and B.

net

A net is a plane figure that can be folded to form a polyhedron.

One possible net for a cube is shown below.

Net of a cube - 4 squares joined in a row; 2 squares are joined to top and bottom of 3rd square.

number line

A number line gives a pictorial representation of real numbers.

numeral

A figure or symbol used to represent a number. For example, –3, 0, 45, IX.

numerator

In the fraction \(\frac{a}{b}\), is the numerator. If an object is divided into b equal parts, then the fraction \(\frac{a}{b}\) represents a of these parts taken together.

For example, if a line segment is divided into 5 equal parts, each of these parts is one-fifth of the whole and 3 of these parts taken together corresponds to the fraction \(\frac{3}{5}\).

numerical data

Numerical data is data associated with a numerical variable.

numerical variables

Numerical variables are variables whose values are numbers, and for which arithmetic processes such as adding and subtracting, or calculating an average, make sense.

A discrete numerical variable is a numerical variable, each of whose possible values is separated from the next by a definite 'gap'. The most common numerical variables have the counting numbers 0, 1, 2, 3, … as possible values. Others are prices, measured in dollars and cents.

Examples include the number of children in a family or the number of days in a month.

odd number

An odd number is an integer that is not divisible by 2.

one-to-one correspondence

In early counting development one-to-one correspondence refers to the matching of one and only one number word to each element of a collection.

More generally it refers to a relationship between two sets such that every element of the first set corresponds to one and only one element of the second set.

operation

The process of combining numbers or expressions. In the primary years operations include addition, subtraction, multiplication and division. In later years operations include substitution and differentiation.

order of operations

A convention for simplifying expressions that stipulates that multiplication and division are performed before addition and subtraction and in order from left to right. For example, in 5 – 6 ÷ 2 + 7, the division is performed first and the expression becomes 5 – 3 + 7 = 9. If the convention is ignored and the operations are performed in order, the incorrect result, 6.5, is obtained.

outlier

An outlier is a data value that appears to stand out from the other members of the data set by being unusually high or low. The most effective way of identifying outliers in a data set is to graph the data.

For example, in the following list of ages of a group of 10 people, \(\left\{ {12,\,\,12,\,\,13,\,\,13,\,\,13,\,\,13,\,\,13,\,\,14,\,\,14,\,\,24}\right\}\), the 24 would be considered to be a possible outlier.

A dot plot diagram showing number of occurrences in each group 11-25 (age).

parabola

Definition 1

The graph of \(y = x^2\) is called a parabola. The point \(\left( {0,\,\,0} \right)\) is called the vertex of the parabola and the y-axis is the axis of symmetry of the parabola, called simply the axis.

The graph of y equals x squared is a parabola. The vertex is (0,0); y-axis is the axis of symmetry.

Some other parabolas are the graphs of \(y = ax^2 + bx + c\) where \(a \ne 0\).

More generally, every parabola is similar to the graph of \(y = x^2\).

Definition 2                                        

A parabola is the locus of all points P such that the distance from P to a fixed point F is equal to the distance from P to a fixed line l.

parallel box plots

Parallel box-and-whisker plots are used to visually compare the five-number summaries of two or more data sets.

For example, the box-and-whisker plots below can be used to compare the five-number summaries for the pulse rates of 19 students before and after gentle exercise.

Visual comparison of the 5-number summaries of 2 or more data sets.

Note that the box plot for pulse rates after exercise shows the pulse rate of 146 as a possible outlier (). This is because the distance of this data point above the upper quartile \(42 \, \left( 146-104 \right)\) is more than \(21 \, (1.5 \times IQR = 1.5 \times \left( 104 - 90 \right) = 1.5 \times 14 = 21)\).

The term 'parallel box-and-whisker plots' is commonly abbreviated to 'parallel box plots'.

parallelogram

A parallelogram is a quadrilateral whose opposite sides are parallel.

Thus the quadrilateral ABCD shown below is a parallelogram because AB || DC and DA || CB.

A parallelogram with vertices labelled A, B, C, D. Parallel sides are marked with matching arrows.

Properties of a parallelogram

  • The opposite angles of a parallelogram are equal.
  • The opposite sides of a parallelogram are equal.
  • The diagonals of a parallelogram bisect each other.
partitioning

Dividing a quantity into parts. In the early years it commonly refers to the ability to think about numbers as made up of two parts, for example, 10 is 8 and 2. In later years it refers to dividing both continuous and discrete quantities into equal parts.

percentage

A percentage is a fraction whose denominator is 100.

For example, 6 percent (written as 6%) is the percentage whose value is \(\frac{6}{{100}}\).

Similarly, 40 as a percentage of 250 is \(\frac{{40}}{{250}} \times 100\%  = 16\% \).

percentiles

Percentiles are the values that divide an ordered data set into 100 (approximately) equal parts. It is only possible to divide a data set into exactly 100 equal parts when the number of data values is a multiple of one hundred.

There are 99 percentiles. Within the above limitations, the first percentile divides off the lower 1% of data values. The second, the lower 2% and so on. In particular, the lower quartile is the 25th percentile, the median is the 50th percentile and the upper quartile is the 75th percentile.

perimeter

The perimeter of a plane figure is the length of its boundary.

pi

Pi is the name of the Greek letter \(\pi \) that is used to denote the ratio of the circumference of any circle to its diameter. The number \(\pi \) is irrational, but \(\frac{{22}}{7}\) is a rational approximation accurate to 2 decimal places. The decimal expansion of \(\pi \) begins \(\pi  = 3.141\,592\,653\,589\,79...\)

There is a very long history of attempts to estimate \(\pi \) accurately. One of the early successes was due to Archimedes (287–212 BC), who showed that \(3{\textstyle{{10} \over {71}}} < \pi  < 3{\textstyle{1 \over 7}}\).

The decimal expansion of \(\pi \) has now been calculated to at least the first \({10^{12}}\) places.

picture graphs

A picture graph is a statistical graph for organising and displaying categorical data.

Picture graph showing the number of students playing different sports. Key is included under the graph.

place value

The value of a digit as determined by its position in a number relative to the ones (or units) place. For integers the ones place is occupied by the rightmost digit in the number.

For example, in the number 2594.6 the 4 denotes 4 ones, the 9 denotes 90 ones or 9 tens, the 5 denotes 500 ones or 5 hundreds, the 2 denotes 2000 ones or 2 thousands, and the 6 denotes \(\frac{6}{{10}}\) of a one or 6 tenths.

point

A point marks a position, but has no size.

polygon

A polygon is a plane figure bounded by line segments.

The image shows a regular pentagon. The five interior angles are marked with arcs.

The figure shown above is a regular pentagon. It is a convex five-sided polygon. It is called a pentagon because it has five sides. It is called regular because all sides have equal length and all interior angles are equal.

polyhedron

A polyhedron is a solid figure bounded by plane polygonal faces. 

Two adjacent faces intersect at an edge and each edge joins two vertices.

A square pyramid with one vertex, one face and one edge labelled.

The polyhedron shown above is a pyramid with a square base. It has 5 vertices, 8 edges and 5 faces. It is a convex polyhedron.

A polyhedron with 6 rectangular side faces. The top face and bottom face are L-shaped.

The figure above is a non-convex polyhedron.

A convex polyhedron is a finite region bounded by planes, in the sense that the region lies entirely on one side of the plane.

polynomial

A polynomial in one variable x (simply called a polynomial) is a finite sum of terms of the form \(a{x^k}\) where a is a number and k is a non-negative integer.

A non-zero polynomial can be written in the form \({a_0} + {a_1}x + {a_2}{x^2} +\, ... \,+ \,{a_n}{x^n}\),
where n is a non-negative integer and \({a_n} \ne 0\).

population

A population is the complete set of individuals, objects, places, etc, that we want information about. A census is an attempt to collect information about the whole population.

prime factor

A prime factor of a natural number n is a factor of n that is a prime number.

For example, the prime factors of 330 are 2, 3, 5 and 11.

prime number

A prime number is a natural number greater than 1 that has no factor other than 1 and itself.

prism

A prism is a convex polyhedron that has two congruent and parallel faces and all its remaining faces are parallelograms.

A right prism is a convex polyhedron that has two congruent and parallel faces and all its remaining faces are rectangles. A prism that is not a right prism is often called an oblique prism.

Some examples of prisms are shown below.

3 different prisms – right rectangular prism, oblique rectangular prism, right triangular prism

probability

The probability of an event is a number between 0 and 1 that indicates the chance of something happening.

For example, the probability that the sun will come up tomorrow is 1, the probability that a fair coin will come up 'heads' when tossed is 0.5, while the probability of someone being physically present in Adelaide and Brisbane at exactly the same time is 0.

product

A product is the result of multiplying together two or more numbers or algebraic expressions. For example, 36 is the product of 9 and 4, and \({x^2} - {y^2}\)  is the product of \(x - y\) and \(x + y\).

proportion

Corresponding elements of two sets are in proportion if there is a constant ratio. For example, the circumference and diameter of a circle are in proportion because for any circle the ratio of their lengths is the constant \(\pi\).

pyramid

A pyramid is a convex polyhedron with a polygonal base and triangular sides that meet at a point called the vertex. The pyramid is named according to the shape of its base.

3 different pyramids - square-based pyramid, triangular-based pyramid, hexagonal-based pyramid.

Pythagoras' theorem

For a right-angled triangle

The square of the hypotenuse of a right-angled triangle equals the sum of the squares of the lengths of the other two sides.

In symbols, \({c^2} = {a^2} + {b^2}\).

The converse

If \({c^2} = {a^2} + {b^2}\) in a triangle ABC, then is a right angle.

quadratic equation

The general quadratic equation in one variable is \(a{x^2} + bx + c = 0\), where \(a \ne 0\).

The roots are given by the quadratic formula \(x = \frac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}\).

quadratic expression

A quadratic expression or function contains one or more of the terms in which the variable is raised to the second power, but no variable is raised to a higher power. Examples of quadratic expressions include \(3{x^2} + 7\) and \({x^2} + 2xy + {y^2} - 2x + y + 5\).

quartile

Quartiles are the values that divide an ordered data set into four (approximately) equal parts. It is only possible to divide a data set into exactly four equal parts when the number of data of values is a multiple of four.

There are three quartiles. The first, the lower quartile (\({Q_1}\)), divides off (approximately) the lower 25% of data values. The second quartile (\({Q_2}\)) is the median. The third quartile, the upper quartile (\({Q_3}\)), divides off (approximately) the upper 25% of data values.

quotient

A quotient is the result of dividing one number or algebraic expression by another. See also remainder.

random number

A random number is one whose value is governed by chance; for example, the number of dots showing when a fair die is tossed. The value of a random number cannot be predicted in advance.

range (statistics)

The range is the difference between the largest and smallest observations in a data set.

The range can be used as a measure of spread in a data set, but it is extremely sensitive to the presence of outliers and should only be used with care.

rate

A rate is a particular kind of ratio in which the two quantities are measured in different units. For example, the ratio of distance to time, known as speed, is a rate because distance and time are measured in different units (such as kilometres and hours). The value of the rate depends on the units in which the quantities are expressed.

ratio

A ratio is a quotient or proportion of two numbers, magnitudes or algebraic expressions. It is often used as a measure of the relative size of two objects. For example, the ratio of the length of a side of a square to the length of a diagonal is \(1 \! : \! \sqrt 2 \); that is, \(\frac{1}{{\sqrt 2 }}\).

rational number

A real number is rational if it can be expressed as a quotient of integers. It is irrational otherwise.

real numbers

The numbers generally used in mathematics, in scientific work and in everyday life are the real numbers. They can be pictured as points on a number line, with the integers evenly spaced along the line, and a real number b to the right of a real number a if \(a < b\).

A real number is either rational or irrational.

Every real number has a decimal expansion. Rational numbers are the ones whose decimal expansions are either terminating or recurring.

rectangle

A rectangle is a quadrilateral in which all angles are right angles.

The image shows a rectangle. Each right angle is marked with a square corner.

recurring decimal

A recurring decimal is a decimal that contains a pattern of digits that repeats indefinitely after a certain number of places.

For example, \(0.1\dot 0\dot 7 = 0.1070707...,\)

and this is the decimal expansion of the rational number

\(\frac{1}{{10}} + \frac{7}{{1000}} + \frac{7}{{100000}} + \frac{7}{{10000000}} +  \ldots  = \frac{1}{{10}} + \left( {\frac{{\frac{7}{{1000}}}}{{1-\frac{1}{{100}}}}} \right) = \frac{1}{{10}} + \frac{7}{{990}} = \frac{{106}}{{990}}\).

Every recurring decimal is the decimal expansion of a rational number.

reflection

To reflect the point \(A\) in an axis of reflection, a line has been drawn at right angles to the axis of reflection and the point \(A'\) is marked at the same distance from the axis of reflection as \(A\), but on the other side.

A line drawn at right angles to axis of reflection; point is marked at same distance on other side.

The point \(A'\) is called the reflection image of A.

A reflection is a transformation that moves each point to its reflection image.

regular shape

A regular shape can be a polygon. A polygon is regular if all of its sides are the same length and all of its angles have the same measure.

related denominators

Denominators are related when one is a multiple of the other. For example, the fractions \(\frac{1}{3}\) and \(\frac{5}{9}\) have related denominators because 9 is a multiple of 3.

Fractions with related denominators are more easily added and subtracted than fractions with unrelated denominators because only one needs to be renamed.

For example, to add \(\frac{1}{3}\) and \(\frac{5}{9}\) we can rename \(\frac{1}{3}\) as \(\frac{3}{9}\) and then compute \(\frac{3}{9} + \frac{5}{9} = \frac{8}{9}\).

relative frequency

Relative frequency is given by the ratio \(\frac{f}{n}\), where f is the frequency of occurrence of a particular data value or group of data values in a data set and n is the number of data values in the data set.

remainder

A remainder is the amount left over when one number or algebraic quantity is divided by another b. If a is divisible by b then the remainder is 0.

For example, when 68 is divided by 11, the remainder is 2, because 68 can be expressed as \(68 = 6 \times 11 + 2\).

revolution

A revolution is the amount of turning required to rotate a ray about its endpoint until it falls back onto itself. The size of 1 revolution is 360°.

A complete turn around a dot is represented as a circle; the angle is labelled 360°.

rhombus

A rhombus is a quadrilateral with all sides equal.

Two rhombuses shown in different orientations. Each side is marked with a pair of parallel lines.

right angle

Let AOB be a line, and let OX be a ray making equal angles with the ray OA and the ray OB. Then the equal angles AOX and BOX are called right angles.

A right angle is half of a straight angle, and so is equal to 90°.

The line OX runs perpendicular to AB. The angle XOB is marked with a right angle.

rotation

A rotation turns a figure about a fixed point, called the centre of rotation.

A rotation is specified by:

  • the centre of rotation O
  • the angle of rotation
  • the direction of rotation (clockwise or anticlockwise).

In the first diagram below, the point A is rotated through 120° clockwise about O. In the second diagram, it is rotated through 60° anticlockwise about O.

Image shows 2 different-sized angles of rotation - one clockwise; other anti-clockwise.

A rotation is a transformation that moves each point to its rotation image.

rotational symmetry

A plane figure F has rotational symmetry about a point O if there is a non-trivial rotation such that the image of F under the rotation is F itself.

Image shows an equilateral triangle. The centre of the triangle is shown by a dot and labelled O.

A rotation of 120° around moves the equilateral triangle onto itself.

rounding

The decimal expansion of a real number is rounded when it is approximated by a terminating decimal that has a given number of decimal digits to the right of the decimal point.

Rounding to n decimal places is achieved by removing all decimal digits beyond (to the right of) the \({n^{th}}\) digit to the right of the decimal place, and adjusting the remaining digits where necessary.

If the first digit removed (the \({\left( {n + 1} \right)^{th}}\) digit) is less than 5 the preceding digit is not changed.

For example, 4.02749 becomes 4.027 when rounded to 3 decimal places.

If the first digit removed is greater than 5, or 5 and some succeeding digit is non-zero, the preceding digit is increased by 1.

For example, 6.1234586 becomes 6.12346 when rounded to 5 decimal places.

sample

A sample is part of a population. It is a subset of the population, often randomly selected for the purpose of estimating the value of a characteristic of the population as a whole.

For instance, a randomly selected group of eight-year-old children (the sample) might be selected to estimate the incidence of tooth decay in eight-year-old children in Australia (the population).

sample space

A sample space is the set of all possible outcomes of a chance experiment. For example, the set of outcomes (also called sample points) from tossing two coins is \(\left\{ {\rm{HH, HT, TH, TT}} \right\}\), where \(H\) represents a 'head' and \(T\) a 'tail'.

scientific notation

A positive real number is expressed in scientific notation when it is written as the product of a power of 10 and a decimal that has just one digit to the left of the decimal point.

For example, the scientific notation for \(3459\) is \(3.459 \times {10^3}\), and the scientific notation for \(0.000004567\) is \(4.567 \times {10^{ - 6}}\).

Many electronic calculators will show these as \(3.459{\rm{E}}3\) and \({\rm{4.567E-6}}\).

secondary data set

Primary data is data collected by the user. Secondary data is data collected by others. Sources of secondary data include web-based data sets, the media, books, scientific papers, etc.

shape (statistics)

The shape of a numerical data distribution is most simply described as symmetric if it is roughly evenly spread around some central point, or skewed if it is not. If a distribution is skewed, it can be further described as positively skewed ('tailing-off' to the upper end of the distribution) or negatively skewed ('tailing-off' to the lower end of the distribution).

These three distribution shapes are illustrated in the parallel dot plot display below.

Image shows 3 distribution shapes, illustrated in parallel dot plot display.

Dot plots, histograms and stem plots can all be used to investigate the shape of a data distribution.

side-by-side column graph

A side-by-side column graph can be used to organise and display the data that arises when a group of individuals or things is categorised according to two or more criteria.

For example, the side-by-side column graph below displays the data obtained when 27 children are categorised according to hair type (straight or curly) and hair colour (red, brown, blonde, black). The legend indicates that dark grey columns represent children with straight hair and light grey columns children with curly hair.

A group of things is categorised by 2 criteria. A side-by-side column graph displays the data.

Side-by-side column graphs are frequently called side-by-side bar graphs or bar charts. In a bar graph or chart, the bars can be either vertical or horizontal.

similar

The four standard tests for two triangles to be similar

AAA: If two angles of one triangle are respectively equal to two angles of another triangle, then the two triangles are similar.

SAS: If the ratio of the lengths of two sides of one triangle is equal to the ratio of the lengths of two sides of another triangle, and the included angles are equal, then the two triangles are similar.

SSS: If we can match up the sides of one triangle with the sides of another so that the ratios of matching sides are equal, then the two triangles are similar.

RHS: If the ratio of the hypotenuse and one side of a right-angled triangle is equal to the ratio of the hypotenuse and one side of another right-angled triangle, then the two triangles are similar.

similarity

Two plane figures are called similar if an enlargement of one figure is congruent to the other.

That is, if one can be mapped to the other by a sequence of translations, rotations, reflections and enlargements.

Similar figures thus have the same shape, but not necessarily the same size.

simple interest

Simple interest is the interest accumulated when the interest payment in each period is a fixed fraction of the principal.

sine

In any right-angled triangle,

\(\sin \theta  = \frac{{{\rm{opposite}}}}{{{\rm{hypotenuse}}}}\), where \({0^ \circ } < \theta  < {90^ \circ }\).

Right-angled triangle with sides labelled hypotenuse, opposite and adjacent; 1 angle is marked theta.

sine rule

In any triangle ABC,

\(\frac{a}{{\sin A}} = \frac{b}{{\sin B}} = \frac{c}{{\sin C}}\).

The image shows a scalene triangle ABC. AB is labelled c, BC is labelled a and AC is labelled b.

In words it says:

Any side of a triangle over the sine of the opposite angle equals any other side of the triangle over the sine of its opposite angle.

square

A square is a quadrilateral that is both a rectangle and a rhombus.

Two squares are shown in different orientations. All right angles and parallel sides are marked.

A square thus has all the properties of a rectangle, and all the properties of a rhombus.

standard deviation

Standard deviation is a measure of the variability or spread of a data set. It gives an indication of the degree to which the individual data values are spread around their mean.

stem-and-leaf plot

A stem-and-leaf plot is a method of organising and displaying numerical data in which each data value is split into two parts, a 'stem' and a 'leaf'.

For example, the stem-and-leaf plot below displays the resting pulse rates of 19 students.

Numerical data is organised & displayed - each data value is split into 2 parts, a stem and a leaf.

In this plot, the stem unit is '10' and the leaf unit is '1'. Thus the top row in the plot 6 | 8 8 8 9 displays pulse rates of 68, 68, 68 and 69.

Stemplot is a synonym for stem-and-leaf plot.

straight angle

A straight angle is the angle formed by taking a ray and its opposite ray. A straight angle is half of a revolution, and so has size equal to 180°.

2 lines meet at a point marked by a dot. The straight angle is marked with an arc and labelled 180°.

subitising

Recognising the number of objects in a collection without consciously counting.

sum

A sum is the result of adding together two or more numbers or algebraic expressions.

supplementary angles

Two angles that add to 180° are called supplementary angles. For example, 45° and 135° are supplementary angles.

surd

A surd is a numerical expression involving one or more irrational roots of numbers. Examples of surds include \(\sqrt 2 \), \(\sqrt[3]{5}\) and \(4\sqrt 3  + 7\sqrt[3]{6}\).

tangent (plane geometry)

A tangent to a circle is a line that intersects a circle at just one point. It touches the circle at that point of contact, but does not pass inside it.

The image shows a circle. A tangent is drawn which touches the circumference at point T.

tangent (trigonometry)

In any right-angled triangle,

\(\tan \theta  = \frac{{{\rm{opposite}}}}{{{\rm{adjacent}}}}\), where \({0^ \circ } < \theta  < {90^ \circ }\).

Right-angled triangle with sides labelled hypotenuse, opposite and adjacent; 1 angle is marked theta

terminating decimal

A terminating decimal is a decimal that contains only finitely many decimal digits.

Every terminating decimal represents a rational number where the denominator is a power of 10. For example, 54.321 is the decimal expansion of the sum \(5 \times {10^1} + 4 \times {10^0} + 3 \times {10^{ - 1}} + 2 \times {10^{ - 2}} + 1 \times {10^{ - 3}} = \frac{{54321}}{{1000}}\).

transformation

The transformations included in this glossary are enlargements, reflections, rotations and translations.

translation

Shifting a figure in the plane without turning it is called translation. To describe a translation, it is enough to say how far left or right and how far up or down the figure is moved.

A translation is a transformation that moves each point to its translation image.

transversal

A transversal is a line that meets two or more other lines in a plane.

The image shows two non-parallel lines and a transversal.

trapezium

A trapezium is a quadrilateral with one pair of opposite sides parallel.

The image shows a trapezium. The 2 parallel sides are marked with arrows.

tree diagram

A tree diagram is a diagram that can be used to enumerate the outcomes of a multistep random experiment.

The diagram below shows a tree diagram that has been used to enumerate all of the possible outcomes when a coin is tossed twice. This is an example of a two-step random experiment.

A tree diagram showing the four possible outcomes when a coin is tossed twice.

triangular number

A triangular number is the number of dots required to make a triangular array of dots in which the top row consists of just one dot, and each of the other rows contains one more dot than the row above it. So the first triangular number is 1, the second is 3 (= 1 + 2), the third is 6 (= 1 + 2 + 3) and so on.

trigonometric ratios

Sine, cosine, tangent.

two-way table

A two-way table is commonly used for displaying the two-way frequency distribution that arises when a group of individuals or things is categorised according to two criteria.

For example, the two-way table below displays the two-way frequency distribution that arises when 27 children are categorised according to hair type (straight or curly) and hair colour (red, brown, blonde, black).

 

Hair colour

Hair type

 

Total

Straight

Curly

red

1

1

2

brown

8

4

12

blonde

1

3

4

black

7

2

9

Total

17

10

27

The information in a two-way table can also be displayed graphically using a side-by-side column graph.

unit fraction

A unit fraction is a simple fraction whose numerator is 1, that is, a fraction of the form \(\frac{1}{n}\) where \(n\) is a natural number.

univariate data

Univariate data is data relating to a single variable, for example, hair colour or the number of errors in a test.

variable (algebra)

A variable is a symbol, such as xor z, used to represent an unspecified member of some set. For example, the variable x could represent an unspecified real number.

variable (statistics)

A variable is something measurable or observable that is expected to change either over time or between individual observations.

Examples of variables in statistics include the age of students, their hair colour or a playing field's length or its shape.

Venn diagram

A Venn diagram is a graphical representation of the extent to which two or more events, for example \(A\) and \(B\), are mutually inclusive (overlap) or mutually exclusive (do not overlap).

The image shows a Venn diagram with 2 overlapping circles. The circles are labelled A and B.

vertically opposite angle

When two lines intersect, four angles are formed at the point of intersection. In the diagram, the angles marked AOX and BOY are called vertically opposite.

2 lines AB and XY intersect at O. The angles AOX and BOY are equal and marked with arcs.

volume

The volume of a solid region is a measure of the size of a region.

For a rectangular prism, \({\rm{Volume  =  Length }} \times {\rm{ Width }} \times {\rm{ Height}}\).

whole number

A whole number is a non-negative integer, that is, one of the numbers 0, 1, 2, 3, …

Sometimes it is taken to mean only a positive integer, or any integer.