**< and >**

These symbols are referred to as the ‘greater than’ (>) and ‘less than’ (<) symbols. Children learn that they are used to show whether a number is bigger or smaller than another number (for example, 56 > 34 or 34 < 56).

**12-hour and 24-hour clock**

The 12-hour clock runs from 1am to 12 noon and then from 1pm to 12 midnight. The 24-hour clock uses the numbers 00:00 to 23:59 (midnight is 00:00).

**2D shapes**

2D shapes are two-dimensional, or ‘flat’. Examples of 2D shapes are squares, triangles, circles and rectangles.

**3D shapes**

3D shapes are three-dimensional, and have a volume. Examples of 3D shapes are cubes, cuboids, spheres, cylinders and prisms.

**Acute angle**

An angle that measures less than 90˚.

**Algebra**

An expression using letters to represent unknown values, e.g 2 (x + 3) = 16

**Angle**

There are three types of angle, depending on the number of degrees of the angle: acute, obtuse and reflex (see separate definitions).

**Analogue and digital clocks / time**

An analogue clock is a circular-faced clock with the numbers one to twelve around the outside and two hands, a shorter one to measure hours and a longer one to measure minutes. A digital clock is a clock which simply shows numbers to denote the time.

**Arc**

Part of the circumference of a circle.

**Area**

Area is the term used to define the amount of space taken up by a 2D shape or surface. We measure area in square units : cm² or m².

**Array**

An array is a pictorial representation, a picture of rows of dots, to help children understand multiplication and times tables.

**Arrow cards**

Arrow cards are a maths aid used to help children partition numbers (divide them unto units, tens, hundreds, etc) and understand place value.

**Ascending order**

When a group of numbers are given in ascending order, this means they are given in order from smallest to largest (ascending means ‘going up’). The opposite is descending order.

**Associative**

The associative law states that when we add or multiply numbers it doesn’t matter how we group them (we express this as (a + b) + c = a + (b + c) and (a × b) × c = a × (b × c) in formulae).

**Average**

In maths, there are three types of average- mean, mode and median (see separate definitions).

**Axes**

Axes are the horizontal and vertical lines used to frame a graph or chart.

**Bar chart**

A bar chart is a chart that displays information (data) by using rectangular bars of different heights, arranged on a vertical axis and a horizontal axis.

**Bar model**

This is used in Singapore maths (also known as Asian maths mastery) to help children visualise maths problems, first by handling actual objects, then by drawing pictures of the objects or cubes / dots representing them and finally by drawing one long bar and labelling it with numbers.

**Block graph**

A block graph (or block diagram) is a simple chart which shows numbers on the vertical axis and labels on the horizontal axis. Each unit is represented by one block.

**BODMAS**

BODMAS is an acronym used to help pupils remember the correct order to complete mathematical calculations in: Brackets, Orders, Division, Multiplication, Addition, Subtraction.

**Bridging through 10**

The “bridging through ten” method is a mental maths technique used to add numbers when the answer is larger than 10.

**Bus stop method**

The ‘bus stop’ method (also known as short-hand division or short division) is a division technique children are taught in primary school. Children are usually taught the bus stop method once they are confident with chunking (another division technique).

**Capacity**

Is the total amount of fluid that can be contained in a container. It is the word we use when we are measuring liquids (in litres or millilitres).

**Cardinal numbers**

Cardinal numbers allow us to count a set of objects and tell us about quantity (one, two, three, four, etc.).

**Carroll diagram**

A Carroll diagram is used to organise data and group it according to whether it fits certain criteria. The information is presented in rows and columns.

**Chord**

A straight line drawn between two points on the circumference of a circle.

**Chunking**

A method used for dividing large numbers. It involves using rough estimates of how many times a number will go into another number and then adjusting until the right answer is found (repeated subtraction of the divisor and multiples of the divisor – in other words, working out how many groups of a number fit into another number).

**Circle**

A 2D curved shape, every point of which is the same distance from a fixed point in the centre.

**Circumference**

The measurement all the way around the outside edge of a circle.

**Clockwise and anti-clockwise**

When something moves in a clockwise direction, it is moving in the same direction as the hands on the clock. When something moves in an anti-clockwise direction, it is moving in the opposite direction. Children are taught about the language of direction from KS1 onwards.

**Coordinates**

Coordinates are numbers that determine the position of a point or a shape in a particular space (a map or a graph). Points are marked by how far along they are on the x axis (the horizontal axis) and how far up they are on the y axis (the vertical axis).

**Column method**

The column method of addition and subtraction is the method where numbers are ‘carried’ and ‘borrowed’. The numbers to be added and subtracted are set up in columns of units, tens, hundreds, etc.

**Commutative**

The commutative law says that we can move numbers around within number sentences without affecting the results of our calculations; this can be expressed in algebra as a + b = b + a and a × b = b × a.

**Complementary addition (subtraction on a number line or the jump strategy)**

‘Complementary addition’ is a subtraction method that involves putting the smaller number at the start of a number line and then jumping up to the bigger number (it’s also known as the ‘jump strategy’). This makes the concept of subtraction being about finding the difference between two numbers very clear.

**Congruent**

Triangles are congruent if they are the same shape and size, eg two right-angled triangles with sides of 3cm, 4cm and 5cm would be ‘congruent’, even if one is the mirror image of the other. You can only create a congruent copy of a triangle by translation, reflection or rotation.

**Converting into same units**

Converting into the same units of measurement means understanding that the same length, capacity or weight can be expressed in different units of measurement. For example, weight can be shown in kilograms (kg) but also grams (g); there are 1000g in 1kg.

**Cube**

A solid shape with six square faces, e.g a dice.

**Cube numbers**

A cube number is a number that is the product of three numbers that are the same. In other words, if you multiply a number by itself and then by itself again, the result is a cube number. To write the mathematical formula for cube numbers we add a small 3 next to and above the number, for example: 23.

**Cube root**

A number that has to be multiplied by itself twice to make another number, eg 2 is the cube root of 8, as 2 x 2 x 2 = 8.

**Cuboid**: A solid shape with a rectangle for each of the six faces, eg a shoe box.

**Data handling**

The term used to refer to primary-school statistics. Children learn about how to collect, interpret and display data in pictograms, tally charts, block diagrams, bar charts, line graphs and pie charts.

**Decimal**

A decimal is a number expressed in the scale of tens. Commonly speaking we talk about decimals when numbers include a decimal point to represent a whole number plus a fraction of a whole number (tenths, hundredths, etc.).

**Degrees**

Degrees are the unit of measurement used to measure angles. A right angle is 90˚, a straight line angle is 180˚ and a full turn is 360˚. We use a protractor (an angle measurer) to measure degrees.

**Denominator**

The bottom number of a fraction. So in the fraction 3/8 the denominator is 8. In the fraction 5/6 the denominator is 6.

**Descending order**

When numbers are put in descending order, they are ordered from largest to smallest. The opposite is ascending order (from smallest to largest).

**Diagonal**

A diagonal is a straight line joining two nonadjacent vertices of a shape, that is, two corners of a shape that are not next to each other.

**Diameter**

The straight line going through the centre of a circle, connecting two points on the circumference.

**Distributive **

According to the distributive law, multiplication distributes over addition. This can be expressed in algebra as a × (b + c) = a × b + a × c.

**Division facts**

Division facts are the division number sentences related to times tables. For example, 30 ÷ 3 = 10, 27 ÷ 3 = 9 and 24 ÷ 3 = 8 are all division facts for the 3x table.

**Edge**

In 3D shapes, the edges are the lines where two faces meet.

**Equation**

An equation is a number sentence where one side equals the other, for example 4 + 4 = 10 – 2.

**Equilateral triangle**

A triangle that has three equal sides and three equal angles.

**Equivalent fractions and simplifying fractions**

When two fractions are equivalent, this means they are the same in terms of shape and size, but are expressed using different numbers (for example, 1/3 is equivalent to 2/6 or 3/9). Simplifying a fraction means finding an equivalent fraction where the numbers are reduced as much as possible.

**Estimate**

An estimate is sometimes called a ‘clever guess’. Estimating means roughly calculating or judging a number or value.

**Expanded notation**

Expanded notation is writing numbers or number sentences in which the numbers are partitioned (so 67 + 43 could be written as 60 + 7 + 40 + 3).

**Face**

In 3D shapes, the faces are the flat parts of the shape.

**Factor**

A factor is one of two or more numbers that divides a given number without a remainder. In the number sentence 4 x 5 = 20, both 5 and 4 are factors of 20.

**Fibonacci series**

A sequence of numbers created by adding the previous two numbers together to get the next one, eg 1, 1, 2, 3, 5, 8, 13…

**Finding the difference between two numbers**

Finding the difference between two numbers means subtracting the smaller number from the larger number. This concept is usually taught with complementary addition (subtraction on a number line).

**Formula**

A group of mathematical symbols and numbers that show how to work something out. Formulae children will learn in primary school include the formula for calculating the perimeter and area of 2D shapes and the formula for the volume for 3D shapes.

**Fraction**

A number that represents a part of a whole. It consists of a denominator (number at the bottom) and a numerator (number at the top). The numerator represents the number of parts of the whole and the denominator represents the total number of parts.

**Grid method**

A written technique used to teach children multiplication. It involves partitioning numbers into tens and units before they are multiplied, and placing them in a grid. The numbers are then multiplied two by two and the results are added together to give a total answer.

**Highest common factor (HCF)**

The highest common factor of two numbers is the largest whole number which is a factor of both. A factor is one of two or more numbers that divides a given number without a remainder.

**Horizontal**

A line that runs from right to left, like the horizon.

**Imperial units**

Imperial units of measurement were used in the UK before the adoption of the metric system.

**Improper fraction**

A fraction that is ‘top-heavy’ so the numerator is bigger than the denominator, for example 7/3.

**Investigation**

Carrying out a mathematical investigation means applying skills and knowledge to solving problems. Investigations differ from word problems because there isn’t always just one way of working them out and the solution might have to be found through trial and error. Often, there is not just one answer; there could be several.

**Integer**

A whole number. This can be a negative or positive number; 0 is also an integer.

**Inverse operation**

Inverse operations are opposite operations; one reverses the effect of the other. Subtraction is the inverse of addition and division is the inverse of multiplication.

**Isosceles triangle**

A triangle with two equal sides and two equal angles.

**Line graph**

A line graph is used to display information that changes over time. It is plotted on a graph as a series of points joined with straight lines.

**Long division**

Long division is a written method of dividing numbers (usually a three- or four-digit number by another large number).

**Long multiplication**

Long multiplication (or column multiplication) is a written method of multiplying numbers (usually a two- or three-digit number by another large number). As in column addition and column subtraction, the numbers are positioned in columns according to their place value.

**Lowest common denominator (LCD)**

The lowest common denominator (or least common denominator) is the smallest number that is exactly divisible by each denominator of a set of fractions.

**Lowest common multiple (LCM)**

The lowest common multiple of two numbers is the smallest whole number which is a multiple of both. A multiple is a number that can be divided by another number a certain number of times without a remainder.

**Mass**

Mass refers to the weight of an object. It is usually measured in grams and kilograms.

**Mean**

The total of all the values in a set of data, divided by the number of values.

**Median**

The middle number in a list of numbers, ordered from smallest to largest.

**Mirror line**

A central line that can be drawn on a symmetrical shape to show that both sides of the shape are exactly the same.

**Mixed number**

A number made up of a whole number and a fraction, for example 9 3/4.

**Mode**

The value that appears most often in a set of data. In this case, the mode is 9 seconds.

**Multiple**

A number that can be divided by another number a certain number of times without a remainder. In the number sentence 4 x 5 = 20, 20 is a multiple of 4 and a multiple of 5.

**Negative numbers**

A negative number is a number smaller than 0 (for example, -1, -3, -36).

**Net**

A net is what a 3D (three-dimensional) shape would look like if it were opened out flat.

**Number bonds**

The pairs of numbers that make up a given number (number bonds to 10 are 1 + 9, 2 + 8, 3 + 7, 4 + 6, 5 + 5; number bonds to 20 are 1 + 19, 2 + 18, 3 + 17, 4 + 16, 5 + 15).

**Number facts**

These are basic addition, subtraction, multiplication and division calculations that children should learn to recall instantly.

**Number line and number ladder**

A straight, horizontal line with numbers placed at even increments along the length. It’s not a ruler, so the space between each number doesn’t matter, but the numbers included on the line determine how it’s meant to be used. A number ladder is the vertical version of a number line.

**Number sentence**

An arrangement of numbers and symbols, such as the following: 6 + 7 = 13 (addition number sentence), 45 – 6 = 39 (subtraction number sentence), 8 x 9 = 72 (multiplication number sentence), 48 ÷ 8 = 6 (division number sentence).

**Number square**

A primary-school maths aid, a square filled with numbers (ordered sequentially). Younger children will often use a number square with numbers from one to 20. You’ll also commonly see number squares from one to 100.

**Numerator**

The top number of a fraction. So in the fraction 3/8 the numerator is 3. In the fraction 1/9 the numerator is 1. If a fraction has 1 as its numerator, it is called a unit fraction.

**Obtuse angle**

An angle that measures between 90˚ and 180˚.

**Odd and even numbers**

An even number is a number that can be divided into two equal groups. Even numbers always end in 2, 4, 6, 8 and 0. An odd number is a number that cannot be divided into two equal groups. Odd numbers end in 1, 3, 5, 7, 9.

**Operation**

The four mathematical operations are addition, subtraction, multiplication and division.

**Operator**

The sign telling you which mathematical operation to do. The most common ones are +, -, x and ÷.

**Order of operations **The sequence of doing basic mathematical sums when you have a mixture of, say, addition and multiplication. BIDMAS (or BODMAS) is a good way of remembering it, as it stands for:

*Brackets**Indices/Order (in other words, squares, cubes and so on)**Division**Multiplication**Addition**Subtraction*

Note that addition doesn’t come ‘before’ subtraction – these operations have to be done in the order in which they occur in the sum, and it makes a difference to the answer, eg 4 – 3 + 2 = 3 if you do the operations in order, which is correct, but you’d get the wrong answer of -1 if you did 3 + 2 first.

**Ordinal numbers**

Ordinal numbers (first, second, third, etc.) tell us the position of an item in a list. Dates are ordinal numbers.

**Parallel**

Parallel lines are straight lines that always stay the same distance from each other and never meet.

**Partitioning**

Partitioning means separating numbers into the tens, units, hundreds, thousands, etc. that make them up (so 2967 is 2000 + 900 + 60 + 7). Partitioning helps children understand place value.

**Percentage**

A number or ratio expressed as a fraction of 100. When we talk about percentages, we imagine that ‘a whole’ has been divided into 100 equal parts.

**Perimeter**

The distance around the edge of a 2D shape.

**Perpendicular**

When two lines are perpendicular, they are at right angles to each other.

**Pi **

A constant used to work out the circumference and area of circles, often shown as 22/7 or 3.14 although it’s actually an ‘irrational’ number, which means it goes on forever.

**Pictogram**

A chart that uses pictures to represent data. Pictograms are set out in the same way as bar charts, but instead of bars they use columns of pictures to show the numbers involved.

**Pie chart**

Circular charts divided up into sections (or ‘slices’) to represent values of different sizes.

**Place value**

The value of each digit in a number. It means understanding that 582 is made up of 500, 80 and 2, rather than 5, 8 and 2.

**Polygon**

These are 2D shapes with straight, fully closed sides. Polygons can have any number of sides. Common polygons are triangles, squares, pentagons, hexagons.

**Prime factors**

The lowest prime numbers that can be multiplied together to make a given number, eg the prime factors of 12 are 2² x 3.

**Prime number**

A number greater than 1 that cannot be divided evenly by any number other than itself or 1. For example: 13 is a prime number because you cannot divide it (without a remainder) by any number except 13 or 1.

**Prism**

A 3D shape with flat sides and two identical ends. Prims have the same cross-section all along the shape, from end to end.

**Probability, chance and likelihood**

Probability (or chance, or likelihood) tells us how likely something is to happen. It can be described in words, or more accurately in terms of fractions and percentages.

**Product**

The product of two numbers is the result you get when you multiply them together (for example, 12 is the product of 3 and 4 and 20 is the product of 4 and 5).

**Proportion**

Proportion tells us about a portion or part in relation to a whole.

**Pyramid**

A 3D shape with a polygon base and flat (triangular) sides that join at a common point (the apex).

**Quadrilateral**

A quadrilateral is a 2D shape with four sides. The following shapes are quadrilaterals: square, rectangle, rhombus, trapezium, parallelogram, kite. **Kite**: a quadrilateral with two pairs of equal sides next to each other (or ‘adjacent’ to each other).

**Parallelogram**: a quadrilateral with opposite sides parallel to each other.

**Rectangle**: a quadrilateral with two opposite pairs of equal sides and four right angles.

**Rhombus**: a quadrilateral with equal sides.

**Square**: a quadrilateral with equal sides and four right angles.

**Trapezium**: a quadrilateral with one pair of parallel sides. (Note: an isosceles trapezium is symmetrical.)

**Radius**

The distance from the centre of a circle to the edge of the circle.

**Range**

The difference between the lowest value and the highest value in a data set.

**Ratio**

A ratio compares values, telling us how much of one thing there is compared to another thing.

**Reflection of shapes**

Reflecting a shape in a mirror line means drawing a symmetrical copy of the shape, flipped over the line (exactly the same size but facing in the opposite direction).

**Reflective symmetry**

When a shape or pattern is reflected in a line of symmetry or a mirror line. The reflected shape will be exactly the same as the original, the same distance from the mirror line and the same size.

**Reflex angle**

An angle that measures between 180˚ and 360˚.

**Regular and irregular shapes**

Regular shapes have sides that are all equal and interior (inside) angles that are all equal. Irregular shapes have sides and angles of any length and size.

**Repeated addition**

A method of helping children understand multiplication. Children are asked to work out, for example, what 3 ‘lots of’ 5 are. They will be shown that this can be written as 5 + 5 + 5 (repeated addition number sentence) as well as 3 x 5 (multiplication number sentence).

**Right angle**

A right angle is an angle that measures 90˚. It is also known as a ‘quarter turn’ because it is a quarter of a full turn, which measures 360˚.

**Right-angled triangle**

A right-angled triangle is a 2D shape with three sides and an angle that measures 90º.

**Roman numerals**

The numbers that were used in ancient Rome, combinations of letters from the Latin alphabet (I, V, X, L, C, D and M).

**Rotation of shapes**

Rotating shapes means moving them around a fixed point (clockwise or anticlockwise, and by a certain number of degrees). The shape stays the same, but its position in the space will change.

**Rotational symmetry**

When a shape or pattern can be rotated or turned around a central point and remains the same.

**Rounding numbers**

Means adjusting the digits (up or down) to make rough calculations easier. We usually round numbers to the nearest 10, 100, and 1000.

**Scale factor**

When we talk about increasing the size of a 2D shape the size by which we make the shape larger is described by its scale factor.

**Scalene triangle**

A three-sided 2D shape in which the sides are all unequal.

**Sector**

A slice a circle between two radii.

**Segment**

A part of a circle is separated from the rest by a chord.

**Shapes**

The name of each shape depends on the number of sides. Here are the first 12 shapes: **Quadrilaterals** have four sides.**Pentagons** have five sides.**Hexagons** have six sides.**Heptagons** have seven sides.**Octagons** have eight sides.**Nonagons** have nine sides.**Decagons** have 10 sides.**Hendecagons** have 11 sides.**Dodecagons** have 12 sides.

**Shared between**

A term used in word problems to indicate division. Children learn that by sharing equally into groups they are dividing.

**Similar**

Triangles are similar if they are the same shape, but not necessarily the same size, eg a right-angled triangle with sides of 3cm, 4cm and 5cm is ‘similar’ to a right-angled triangle with sides of 6cm, 8cm and 10cm. (Note: similarity is the same as congruence, except that the triangles don’t have to be the same size.)

**Square numbers**

When you multiply a number by itself, the result is a square number. To write the mathematical formula for square numbers we add a small 2 next to and above the number, for example: 32.

**Square root ** A number that has to be multiplied by itself to make another number, eg 6 is the square root of 36, as 6 x 6 = 36.

**Standard and non-standard units**

Standard units are the units we usually use to measure the weight, length or capacity of objects (grams and kilograms, centimetres, metres and kilometres, millilitres and litres). Non-standard units are used by younger children to introduce them to the concept of measuring (for example: handspans, the length of a pencil).

**Sum**

The sum of two numbers is the answer you get when you add them both together (for example, the sum of 5 and 4 is 9).

**Symmetry**

Something is symmetrical when it is the same on both sides. A shape has symmetry if a central dividing line (the mirror line) can be drawn on it, to show that both sides of the shape are exactly the same.

**Tally chart**

A chart that uses marks representing numbers to collect data quickly and efficiently. One vertical mark is used to represent each unit; when five objects are counted the fifth line is crossed through the first four.

**Tangent ** Either a straight line that touches the circumference of a circle OR the length of the opposite side of a triangle divided by the length of the adjacent side

**Tessellation**

When shapes fit together exactly with no gaps. Common examples of tessellating shapes are floor tiles.

**Time intervals**

The length of time in between two given times.

**Translation of shapes**

In geometry, translation means moving a shape into a different position, without changing it in any way.

**Triangle**

A polygon with three sides and three angles. It is a 2D shape.

**Equilateral triangles** have all three sides the same length and all three angles the same.**Isosceles triangles** have two sides the same length and two angles the same.**Scalene triangles** have three sides of different lengths with three different angles.**Right-angled triangles** have one 90-degree angle.

**Triangular numbers**

A sequence of numbers generated by arranging a pattern of dots into equilateral triangles.

**Turns**

A quarter turn is a 90˚ movement, clockwise or anti-clockwise. A half-turn is 180˚ and a full turn is 360˚.

**Two-step and multi-step problems**

A word problem requiring two operations to solve it; a multi-step problem requires more than two operations before the solution can be found.

**Unit fractions**

A fraction where the numerator (top number) is 1 and the denominator (bottom number) is a whole number.

**Variable**

An unknown in algebra, e.g. x or y.

**Venn diagram**

A diagram that shows the relationship between a group of different things (a set) in a visual way, by sorting data into two or three circles that overlap in the middle. Each circle follows a certain rule, so any numbers or objects placed in the overlapping part (the intersection) follow both rules.

**Vertex / vertices**

In 3D shapes, the vertices are the points where two or more edges meet. The angles of a 2D shape are also sometimes referred to as ‘vertices’ (singular: vertex).

**Vertical**

A line that runs up and down the page, from top to bottom.

**Volume**

The amount of 3D space an object occupies or takes up.

**Word problem or story problem**

A ‘real-life’ scenario where a problem needs to be solved by way of a mathematical calculation.