# Glossary

This glossary contains the definitions in The Language of Mathematics in Science: A Guide for Teachers of 11-16 Science. It aims to promote a common understanding of these terms among teachers, publishers, awarding bodies and others; note that the definitions are not intended for pupils. Each of the chapters in the publication deals with clusters of key words selected from the glossary. In the definitions below,  italic terms are key words than can be found elsewhere in the glossary.

See equation.

#### anomaly

An anomaly (or anomalous value) is a measured value that appears not to fit the pattern of the other measurements, and is often (though not always) due to a mistake. For example, a value that is very different from the others in a set of repeated measurements, or a data point that lies far from a line of best fit. See also outlier.

#### approximation

A value that is not exact, but sufficiently close to the actual value for it to be useful.

#### area

A measure of the size of a surface (usually measured in square units, for example cm2 or m2).

#### area under the line (on a graph)

On a graph, the area under a straight line between two values on the horizontal axis may have a physical meaning. The area is found by multiplying two values: the mean of the values on the vertical axis, and the difference between the two values on the horizontal axis. For example, on a speed–time graph, the area represents the mean (or average) speed multiplied by the time interval: this gives the distance travelled. If a speed–time graph shows a curve rather than a straight line, the area under the curve also represents the distance travelled, though it is not so straightforward to calculate.

#### arithmetic mean

The sum of a set of values divided by the number of values in the set. Often referred to simply as a mean (though there are other types of mean, such as geometric mean).

#### average

A measure of the ‘typical value’ of a set of data. Sometimes used synonymously with mean (or arithmetic mean) even though there are other measures of average, such as median and mode.

#### axis

On a graph or a chart, an axis is a reference line along which distances may represent values of a variable. See also horizontal axis and vertical axis.

#### axis label

On the axis of a graph, the axis label shows the name of the variable and its unit where appropriate.

#### bar chart

A display for presenting data, in which bars of equal width represent the set of values. Each value is proportional to the length of the bar. The bars may be vertical or horizontal. See also grouped bar chart and stacked bar chart.

#### base unit

In the International System of Units (SI), there are seven base units. These are the units of the fundamental (and independent) quantities of length (metre), mass (kilogram), time (second), electric current (ampere), temperature (kelvin), chemical amount (mole) and light intensity (candela). See also derived unit.

#### batch

A set of values related to a single quantity or variable, for example repeated measurements of the time for a ball to drop a certain distance, or the heights of pupils in a school.

#### box plot

A diagram that represents the distribution of values in a batch of data. The central box represents the interquartile range, and the median is shown as a line within the box. Lines extend above and below the box to the highest and lowest values.

#### brackets

Symbols used to group numbers and letters in expressions, and indicating certain operations as having priority. See order of operations.

#### categorical

Categorical data are data that can be sorted into categories (e.g. different ‘eye colours’ or ‘food groups’) but cannot be ordered (since they are ‘labels’ that have no particular order). Categorical data are qualitative data.

#### circumference

The distance around a circle (its perimeter).

#### class interval

When drawing a histogram, the set of quantitative data is split into a number of classes (groups). The class interval is the range of values within each class.

#### coefficient

In mathematics, a coefficient is a factor of an algebraic term, though often it is used to mean a ‘numerical coefficient’. For example, in the term 4xy, 4 is the numerical coefficient of xy, but x is also the coefficient of 4y and y is the coefficient of 4x.

In science, the term is also used to apply to the properties of particular materials (e.g. the coefficient of expansion).

#### combined events

A combination of two or more events. The probability of a combined event can be calculated by multiplying together each of the probabilities of the separate events, but only if these are independent events.

#### compound measure

In mathematics, a compound measure is one that involves two (or more) other measures of different types. For example, speed (which can be calculated from distance / time) is a compound measure, and has units of metres per second.

#### constant

A number or quantity that does not vary. For example, in the equation y = 3x + 6, the 3 and 6 are constants, where x and y are variables. In the general equation for a straight line, y = mx+ c, m and c are constants for a specific line.

In science, the term may be used to refer to ‘universal constants’ (e.g. the speed of light or the Avogadro constant) or to values that are constant within a particular context (e.g. for a spring that follows Hooke’s Law, the value of the spring constant is constant for a specific spring).

#### constant of proportionality

In a directly proportional relationship between two variables, x and y, of the form y = kx, the constant k is referred to as the constant of proportionality.

#### continuous

Continuous data are a type of quantitative data (numerical data) for which the values can take on any value within a certain range (e.g. the heights of pupils or the temperatures of an object.). See also discrete.

#### control variable

In an investigation, the control variables are those that are kept constant by the investigator.

#### coordinate

On a graph, the coordinates determine the position of each data point in relation to the axes. See x-coordinate and y-coordinate.

#### correlation

A measure of the strength of the association between two variables. High correlation implies a close relationship and low correlation a less close one. If an increase in one variable corresponds to an increase in the other, the correlation is positive. If an increase in one variable corresponds to a decrease in the other, the correlation is negative.

#### cross-sectional area

The area of a cross-section of a three-dimensional object or geometrical figure. The cross-section is the surface that would be exposed by making a ‘straight cut’ through the object, often at right angles to an axis of symmetry.

#### cube

In geometry, a three-dimensional figure with six identical square faces.

In number (arithmetic) and algebra, the result of cubing a number or expression. For example, 23 (pronounced ‘two cubed’) is 2 × 2 × 2 = 8.

#### cube root

A value whose cube is equal to a given value. For example, the cube root of 8 is 2 (since 23 = 8), and this is represented as 3√8 = 2 or 81/3 = 2.

#### cuboid

A three-dimensional figure with six rectangular faces. (Some of the rectangular faces may be squares; a cube is a special cuboid in which all the faces are squares.)

#### data point

On a line graph or a scatter graph, a data point is represented by a symbol (e.g. × or +). Its position represents a pair of values for the two variables.

#### decimal

A term commonly used synonymously with ‘decimal fraction’, where the number of tenths, hundredths, thousandths, etc. are represented as digits following a decimal point.

#### decimal place

In a decimal, each column after the decimal point is a decimal place. For example, 5.275 is said to have three decimal places.

#### dependent variable

In an investigation, the dependent variables are those that are observed or measured by the investigator.

#### derived unit

All SI units, except for the seven base units, are derived units. They are produced by suitable multiplication or division involving two or more of the base units.

#### diameter

Any straight line segment joining two points on a circle or sphere that passes through the centre.

#### directly proportional

If the algebraic relation between two variables, x and y, is of the form y = kx (where k is a constant), y is directly proportional to x. It also follows that x is directly proportional to y (since x = 1/k y). Another way of expressing this (more common in mathematics than in science) is that x and y are in direct proportion.

If y is directly proportional to x, when x is doubled, y also doubles, and when x is multiplied by 10, y is also multiplied by 10.

The graphical representation of y = kx is a straight line through the origin, where k is the gradient of the line.

The word ‘directly’ is often dropped, and the term proportional is used to mean the same thing. Using the full term ‘directly proportional’, however, is helpful when it is being contrasted to inversely proportional.

#### discrete

Discrete data are a type of quantitative data (numerical data) for which the values can take on only certain values. These are often integer values produced by counting (e.g. the number of trees in a survey area). See also continuous.

#### displacement

The length and direction of the straight line from the initial position of an object to its position at a later time. Displacement is a vector quantity.

#### displacement–time graph

A graph showing how the displacement of an object changes over time.

#### distance

The length of the path along which an object has moved. Distance is a scalar quantity.

#### distance–time graph

A graph showing how the distance of an object changes over time.

#### distribution

For a set of data, the way in which values in the set are distributed (or spread out) between the highest and lowest values.

#### equation

A mathematical statement showing that two expressions are equal. The expressions are linked with the equals (=) sign. Also referred to as algebraic equations where the expressions contain variables. A formula is an equation that shows the relationship between real-world variables.

In science, the term ‘equation’ is also used to refer to a chemical equation.

#### estimate

A rough or approximate value, found by calculating with suitable approximations or using previous experience.

#### experiment

An investigation in which variables may be manipulated and data are collected by observing the effects of changing some of the variables. See also survey.

#### exponent

In index notation, the term ‘exponent’ is used synonymously with index.

#### exponential relationship

A relationship between two variables, x and y, of the form y = ax . For example, if a = 2 , each increase of 1 for x corresponds to a doubling of y.

#### expression

A mathematical form expressed symbolically, consisting of a combination of numbers and variables that may be evaluated. Expressions do not contain the equals (=) sign.

#### extrapolation

On a graph, extrapolation means estimating the value of one variable from a value of the other, using a line of best fit that is extended beyond the range of the available data. Care needs to be taken, since the relationship may not apply outside the data range. See also interpolation.

#### factor

In an investigation, an independent variable is often referred to as a factor, particularly when it is a categorical variable.

In mathematics, the term has an entirely different meaning: when a number can be expressed as the product of two or more numbers, these are factors of the first. For example, 2 and 3 are factors of 6.

#### formula

An equation that shows the relationship between real-world variables. By rearranging the formula, it is possible to make any of the other variables the subject of the formula.

In science, the term ‘formula’ is also used to refer to a chemical formula.

#### fraction

The result of dividing one integer by a second integer, which must be non-zero.

#### frequency

In statistics, the number of times an event occurs, or the number of individuals or objects with some specific property. (Although it is a very different context, in science, the frequency of a wave or an oscillation has a related meaning – the number of cycles per unit of time.)

#### frequency table

A table showing the frequencies of objects or events in different categories or class intervals.

On a graph, the gradient is a measure of the steepness of a line, and is calculated by dividing the vertical change by the corresponding horizontal change. It represents the rate at which the variable plotted on the vertical axis changes with the variable plotted on the horizontal axis.

#### grouped bar chart

A type of bar chart used to represent data categorised by two factors. Each group of bars represents one factor, and the bars within each group represent the other factor. (Also known as a clustered bar chart.)

#### grouped data

Discrete data and continuous data can be grouped into class intervals and counted to produce a frequency table. This is called grouped data.

#### histogram

In science, the term ‘histogram’ is used to refer to a representation of the distribution of data, in which the height of each bar is proportional to the frequency of values in each class: all of the class intervals are equal, and the bars are of equal width.

In mathematics and statistics, the class intervals may not all be equal, and so the bars may be of different widths. The area of each bar is proportional to the frequency of values in each class, and the height of each rectangle represents the ‘frequency density’ of the class.

#### horizontal axis

On a line graph or a scatter graph, the horizontal axis usually represents the independent variable. (See also x-axis.)

#### independent events

Two events are independent if the probability of the second event is not affected by the outcome of the first.

#### independent variable

In an investigation, the independent variables are those that are changed by the investigator.

#### index

In index notation, the superscript is called the index, for example in a4 the index is 4. (Note that the plural of index is indices.) It is also possible to have fractional and negative indices.

#### index notation

The notation in which a product such as a × a × a × a is recorded as a4.

#### integer

Any of the positive or negative whole numbers and zero (e.g. . . ., -2, -1, 0, +1, +2, . . .).

#### intercept

On a graph, the point at which a straight line or a curve crosses an axis is called an intercept. The term ‘intercept’ is typically used in relation to the vertical axis (y-axis), but also applies to the horizontal axis (x-axis).

#### interpolation

On a graph, interpolation means estimating the value of one variable from a value of the other, using a line of best fit that does not extend beyond the range of the data. See also extrapolation.

#### interquartile range

The difference between the upper and lower quartiles. It contains the middle half of the values in the ordered data set. It is a useful measure of spread since, unlike the range, it is not much affected by outliers.

#### inverse

Inverse operations are ‘opposite’ operations that ‘undo each other’. For example, subtraction is the inverse of addition, and -5 is the additive inverse of 5 since their sum is zero. Division is the inverse of multiplication, and ⅓ is the multiplicative inverse of 3 since their product is 1. (Sometimes, the term is used synonymously with reciprocal, for example ‘the inverse of 2 is ½’.)

#### inverse square relationship

A relationship between two variables, x and y, of the form y = a/x2 , where a is a constant.

#### inversely proportional

If the algebraic relation between two variables, x and y, is of the form y = k/x (where k is a constant), y is inversely proportional to x. It also follows that x is inversely proportional to y (since x = k/y ). Another way of expressing the equation is xy = k.

If y is inversely proportional to x then, for example, when x is doubled, y is halved, and when x is multiplied by 10, y is divided by 10.

#### line graph

In mathematics, a line graph is a graph in which adjacent data points are joined by straight-line segments. Such graphs are also used in science.

However, a ‘line graph’ in science more often refers to a graph where it is assumed that there is a simple relationship between the two variables, such that a line of best fit can be drawn that is very close to all the data points. In practice, not all the data points fit on this line because of measurement uncertainty.

#### line of best fit

A line drawn on a graph that passes through or as close as possible to the data points. It represents the best estimate of any underlying relationship between the variables. A ‘line of best fit’ often refers to a straight line but it may also be a curve.

#### linear

On a graph, a relationship is said to be linear if it is represented by a straight line. See also linear relationship.

#### linear dimension

A term often used in the context of scaling. A linear dimension refers to the distance between two points of a geometric figure. When comparing two similar geometric figures, the scale factor applies only to the linear dimensions (any two corresponding lengths), and not to the area or volume.

#### linear relationship

If the relationship between two variables, x and y, is linear, equal changes in x correspond to equal changes in y. For example, with a spring that follows Hooke’s Law, for each additional 100 g mass suspended from the spring, its length increases by the same amount as before.

The equation for a linear relationship can be expressed in the form y = mx + c. When represented as a graph, this is a straight line for which m is the gradient of the line and c is the intercept on the y-axis.

#### mass

A measure of the quantity of matter in an object. The SI base unit of mass is the kilogram (kg). In science, it is important to distinguish between mass and weight.

#### mean

The sum of a set of values divided by the number of values in the set. (More correctly called the arithmetic mean, as there are also other types of mean, such as geometric mean.) See also average.

#### median

The middle value in a set of data when all the values are arranged in order. An equal number of data values lie above and below the median. See also average.

#### mode

The most commonly occurring value in a set of discrete data. Some sets of data may have more than one mode. See also average.

#### non-linear

A non-linear relationship is one that is not linear and, on a graph, is represented by a curve and not by a straight line. See also linear relationship.

#### order of magnitude

The approximate size, often given as a power of 10. Orders of magnitude are particularly useful when comparing values of very different sizes. For example, 4763 is very roughly 1000 times larger than 3.8, i.e. 103 or ‘three orders of magnitude larger’.

#### order of operations

The order in which different mathematical operations are applied in a calculation. The convention is often encapsulated in the mnemonic BIDMAS (Brackets, Indices, Division/ Multiplication, Addition/Subtraction) or BODMAS (where O represents Order or ‘to the power Of’).

#### origin

On a graph, the origin is the point at which the values of both variables are zero (the x-coordinate and the y-coordinate are both zero).

#### outlier

A value in a set of data that is judged to be unusually large or unusually small in comparison with most of the other values, for whatever reason. In sampling a population, an outlier may indicate an individual with exceptional characteristics. By contrast, an outlier in a set of repeated measurements may indicate that a mistake has been made (see also anomaly).

#### percentage

A fraction expressed as the number of parts per hundred and recorded using the notation %.

#### percentile

When the values in a batch of data are arranged in order, the percentiles are the values that split the data into 100 groups containing (as far as possible) equal numbers of values. For example, 10% of the data values lie below the 10th percentile. See also quartile.

#### pie chart

A display for presenting data, in which the sectors (like ‘slices of a pie’) represent the proportions of each of the values. The size of each value is proportional to the angle at the centre of the circle.

#### population

In statistics, a population is the entire collection of objects or events of a similar nature that are of interest in a study, and about which data may be collected. This is usually done by selecting a sample.

#### power

In index notation, the term ‘power’ is often used synonymously with index. Using the term ‘power’ in its correct sense, the expression 34 can be described as ‘the fourth power of 3’.

#### power of 10

Any number of the form 10n is called a power of 10, where n is an integer (negative, zero, or positive), for example . . . 10−2, 10−1, 100, 101, 102, 103 (i.e. 0.01, 0.1, 1, 10, 100, 1000). Each number in the series is 10 times the previous number.

#### primary data

Data collected directly by the user. See also raw data and secondary data.

#### probability

The likelihood of an event happening. Probability is expressed on a scale from 0 to 1. Where an event cannot happen its probability is 0 and where it is certain its probability is 1.

#### proportional

The term ‘proportional’ is often used to mean the same as directly proportional. Using the full term ‘directly proportional’, however, is helpful when it is being contrasted to inversely proportional.

#### qualitative data

Data that are non-numerical (in contrast to quantitative data). See also categorical.

#### quantitative data

Data that are numerical (in contrast to qualitative data). See also continuous and discrete.

#### quantity

Any property that can be given a magnitude by measuring or counting.

#### quartile

When the values in a batch of data are arranged in order, the quartiles are the three values that split the data into four groups containing (as far as possible) equal numbers of values. They are called the first or lower quartile, the second quartile (or median), and the third or upper quartile. The difference between the upper and lower quartiles is the interquartile range.

The distance from the centre of a circle or sphere to any point on the circle or sphere.

#### random error

A component of measurement error due to measurements varying in unpredictable ways from one measurement to the next.

#### random sample

A sample from a population in which all the individuals in the population are selected at random and have an equal chance of being included in the sample.

#### range

For a measuring instrument, the range is the set of values that can be measured, describing its lower and upper limits.

In an experiment investigating the relationship between two quantitative variables, the range refers to the lowest and highest values of a variable. For the independent variable the range is chosen by the experimenter and for the dependent variable the range is determined by the results of the experiment.

On a graph, the range of an axis indicates the highest and lowest values on the axis.

For a distribution of data, the range is a measure of spread, and is the difference between the highest and lowest values. Note that in school science, the term ‘range’ is generally used to indicate both the lowest and highest values themselves, and not the difference between them.

#### rate

A measure of how quickly one variable changes in comparison with another variable. For example, speed is the rate of change of distance with time.

#### ratio

A ratio shows the relative sizes of two values, usually written in the form a : b (and pronounced ‘the ratio of a to b’). Since a ratio is a comparison of two similar quantities, it does not have units.

#### raw data

Data collected directly from experiments or surveys, before being processed. See also primary data.

#### reciprocal

The reciprocal of a value is 1 divided by the value; for example, the reciprocal of 2 is ½.

#### recurring decimal

A decimal with an infinitely repeating digit or group of digits (e.g. the fraction ⅓ is the decimal 0.33333…).

#### resolution

The resolution of a measuring instrument is the smallest change in the quantity being measured that gives a perceptible change in the indication on the instrument.

#### risk

Risk is related to the probability of harm occurring when exposed to a hazard. The actual value of a risk is often called the ‘absolute risk’, while a ‘relative risk’ may be used to compare the risks for two different situations or groups (e.g. in a clinical trial, to compare the risks for the control group and for the treatment group).

#### round

‘Rounding a number’ means expressing it as an approximation with fewer significant figures. For example, 5.432 rounded to the nearest 0.1 is 5.4 (from four to two significant figures).

#### sample

A subset of a population. In collecting data, a sample of observations may be made from which to draw inferences about a larger population.

#### scalar

A quantity that has a magnitude (size) but no direction, for example mass. See also vector.

#### scale

Used as a noun: a set of marks on a line with equal intervals. Applies to:

• an analogue measuring instrument
• the axis on a graph.

Used as a verb: to enlarge or reduce a number, quantity or measurement by a given amount (called a scale factor).

#### scale drawing

A representation of a physical object in which all lengths in the drawing are in the same ratio (the scale factor) to the corresponding lengths in the actual object.

#### scale factor

In a scale drawing, the ratio of any length in the drawing to the corresponding length in the physical object. More generally, the scale factor for two similar geometric figures is the ratio of any two corresponding lengths.

#### scatter graph

A graph on which paired values for two variables are plotted and which may indicate a relationship between the variables. On a scatter graph, it is not meaningful to join the data points with line segments, but a line of best fit may be drawn.

#### scientific notation

See standard form.

#### secondary data

Data obtained indirectly from sources such as books, articles or web pages. See also primary data.

#### significant figures

The number of digits that contribute information about the size of a value (related to the measurement uncertainty).

#### slope

Sometimes used as an informal alternative to gradient, although gradient is the preferred term.

#### speed

The rate of change of distance with time. Speed is a scalar quantity.

#### speed–time graph

A graph showing how the speed of an object changes over time.

For a batch of values, the term ‘spread’ refers to how close together or far apart the values are. Measures of spread include the range, interquartile range and standard deviation.

#### square

In geometry, a two-dimensional figure with four equal sides and four right angles.

In number (arithmetic) and algebra, the result of squaring a number or expression. For example, 52 (pronounced ‘five squared’) is 5 × 5 = 25.

#### square root

A value whose square is equal to a given value. For example, a square root of 25 is 5 (since 52 = 25), and this is recorded as √25 = 5. It also has a negative square root (-5), since (-5)2 = 25.

#### stacked bar chart

A type of bar chart used to represent data categorised by two factors. Each bar represents one factor, and the segments within each bar represent the other factor. (Also known as a compound bar chart.)

#### standard form

A form in which numbers are recorded as a number between 1 and 10 multiplied by a power of 10. For example, 193 in standard form is recorded as 1.93 × 102. It is also referred to as standard index form and scientific notation.

#### standard index form

See standard form.

#### stem-and-leaf diagram

A format for organising the values in a batch of data. The class intervals are represented on the vertical ‘stem’, and the values in each class interval are represented as horizontal rows forming the ‘leaves’.

#### subject of a formula

A formula is an equation that shows the relationship between real-world variables. It is conventionally written so that one of these variables is ‘on its own’ on the left of the equals sign – this variable is called the subject of the formula. By rearranging the formula, it is possible to make any of the other variables the subject of the formula.

#### surface area

The area of the surface of a three-dimensional object or geometric figure. 10.3 surface area : volume ratio The ratio of the surface area to the volume for a three-dimensional object or geometric figure.

#### survey

An investigation in which variables are hard to manipulate, and data are collected from samples of populations. See also experiment.

#### tangent

On a graph, a straight line that touches a curve at only one point. The line has the same gradient as the gradient of the curve at that point. (The term is also used in trigonometry: the tangent of an angle in a right-angled triangle is the ratio of the length of the opposite side to the length of the adjacent side.)

#### tick mark

On a graph or chart, the tick marks are the small lines along the axis at regular intervals, each representing a value on the scale.

#### tick mark label

On a graph or chart, the number next to a tick mark indicating the size of the value.

#### time series

A set of observations, generally measurements or counts, taken over time and usually at equally spaced intervals.

#### true value

The value that would be obtained in an ideal measurement.

#### two-way table

A frequency table in which the frequencies are categorised by two independent factors (categorical variables).

#### uncertainty

The interval within which the true value can be expected to lie, with a given level of confidence or probability.

#### unit

A unit of measurement is a standard used in measuring (e.g. the metre is a unit of length; the kilogram is a unit of mass).

#### unit prefix

The prefix used to form a decimal multiple or submultiple of an SI unit (e.g. ‘kilo’ or ‘milli’).

#### value

The value of a quantity or a variable may be a number, or may consist of a number and a unit.

#### variability

Variability in a set of data relates to how spread out or how close together the values are. It may arise due to measurement uncertainty or due to differences between the individuals in a population.

#### variable

In an investigation: a physical, chemical or biological quantity or characteristic that can differ from case to case.

In an algebraic equation: a quantity that can take on a range of values, often denoted by a letter (e.g. x, y, z, t).

#### vector

A quantity that has both a magnitude (size) and a direction, for example displacement. See also scalar.

#### velocity

The rate of change of displacement with time. Velocity is a vector quantity.

#### velocity–time graph

A graph showing how the velocity of an object changes over time.

#### vertical axis

On a line graph or a scatter graph, the vertical axis usually represents the dependent variable. (See also y-axis.)

#### volume

A measure of three-dimensional space (usually measured in cubic units, for example cm3, dm3 or m3).

#### weight

The weight of an object can be defined as the gravitational force exerted on the object. Its SI derived unit is the newton (N). In everyday language, it is common for ‘weight’ to be measured in units of mass, for example grams (g) or kilograms (kg). In science, however, it is important to distinguish between weight and mass.

#### x-axis

On a graph, the x-axis is the horizontal axis.

#### x-coordinate

On a graph, the x-coordinate of a data point is its distance along the x-axis.

#### y-axis

On a graph, the y-axis is the vertical axis.

#### y-coordinate

On a graph, the y-coordinate of a data point is its distance along the y-axis.