Questions Dictionary of Maths

Maths Dictionary By Kev Delaney, Adrian Pinel &. Derek Smith

Questions Publishing Company Birmingham

© 1996, 1997 The Questions Publishing Company 27 Frederick Street, Hockley, Birmingham Bl 3HH First published by Watts Books, London

ISBN 1-898149-70-4

Illustrated by Martin Lealan Typeset by The Questions Publishing Company Ltd Printed in Great Britain

To the teacher All areas of knowledge require language, and sometimes they use words which have a special meaning. In mathematics, there are special words, and also everyday words which are used in a particular way. This dictionary uses a symbol at the start of each entry to show what kind of word is being defined. There are cross-references at the head of some of the entries, where mathematical ideas are linked, and a word in BOLD CAPITALS is defined elsewhere in the dictionary. The purpose of this book is not only to help children to understand the meaning of mathematical words and to use them correctly; it also takes the opportunity to clarify and extend ideas, to help children to consolidate understanding, and even encounter new mathematics. It makes extensive use of examples and diagrams to show the mathematical ideas that the words represent. For example, this entry for SQUARE shows both the numerical and geometrical meanings.

The content has been chosen to include words within the capabilities of children in primary years, so some ideas are included that are not actually in the normal primary curriculum. More difficult aspects have been deliberately omitted. The dictionary is intended to be available in the classroom for children to use, either independently or together with the teacher, as the circumstances require. Made available as a book, or as individual entries, photocopied and fixed to AS cards, the dictionary will provide an extra opportunity for children to develop their skills in using reference material. It can also be a rich field for browsing during odd moments, giving as it does the opportunity to meet new mathematics in a friendly format. All mathematics is most clearly and easily understood when it is actively experienced. Children should be encouraged to try for themselves the ideas that they meet here. Repeating the examples the entry contains, to check that it works for them, or trying the ideas out in other cases, will help their understanding and knowledge to grow. In the Pupil's introduction, they are encouraged to have a pencil and paper handy for this purpose. Mathematics is very enjoyable when it is experienced in the right way. Wherever possible, the entries in this book have been presented in a manner that is fun as well as informative. Kev Delaney Adrian Pinel Derek Smith

To the pupil This book is not just a list of words and their meanings. It also has a great deal of mathematics that you will find interesting and useful. It will show you how the words are used as well as what they mean. There are different kinds of mathematical words. Some stand for a mathematical idea, some for a thing, and some are everyday words that are used in a special way in mathematics. You will find all kinds on this dictionary, and these symbols show what they are. A whole branch of mathematics (for example, ALGEBRA)

A word which is part of everyday language but takes on a special meaning when it is used in mathematics (for example, TRANSLATION) A technical word that is used mainly in mathematics (for example, ANGLE)

The name of a mathematical object (for example, CUBE)

If a word is in BOLD CAPITAL LETTERS, you can look it up in the dictionary. You can use this dictionary in many ways. If you come across a word in mathematics that you are not sure about or don't understand, you can look up what it means. You can also just pick any word and start finding out about interesting mathematical ideas. The dictionary can help you understand better what you already know or to learn some new mathematics. It is by doing mathematics that you understand it best, so why not use a pencil and paper to try out ideas as you look up a word? Above all, there should be a lot of fun in mathematics, so we have written about the words in a way that we hope you will enjoy. Kev De/oney Adrian Pinel Derek Smith

Abacus An abacus has counters that are moved to ADD, SUBTRACT, MULTIPLY or DIVIDE numbers. Here are some kinds of abacus.

All these abacuses show the number 52.

Above and below The seed is planted below the ground; then the plant grows above the ground and below the ground.

What is above and below you changes according to your position.

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Accurate (also see APPROXIMATE, BETWEEN and ESTIMATE) Being accurate when calculating is not making a mistake. If you start at 7 and COUNT on 9, you will get to 16 if you are accurate.

Some people check if their calculations are accurate by doing them another way.You could start at 9 and count on 7 to check their calculation. When MEASURING, being accurate means measuring as carefully as possible.

Being completely accurate when measuring is impossible.

Adding (also see SUBTRACTING) When you add two NUMBERS, you COUNT on from one of them by the amount of the other.

Adding numbers is called 'addition'. The sign for addition is + and this is called 'plus',

'Adding' is sometimes used to mean putting two or more amounts of objects (for example, counters) together. A better word for this is 'aggregating'.

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CTRATEGIES

Algebra

(also see FUNCTION and PATTERN)

Algebra is about PATTERNS and properties of NUMBERS.

Algebra (continued) Algebra is about FUNCTIONS too.

It is also about organising FUNCTIONS into TABLES and GRAPHS.

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Angle

(also see BEARING)

An angle can be the sharpness of a corner, or the amount of turn needed to move around a corner.

Angles are MEASURED in complete turns, right angles and degrees.

Angle (continued) Angles of less than 90° are called 'acute'. Angles of more than 90° but less than 180° are called 'obtuse'. Angles of more than 180° but less than 360° are called 'reflex'.

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CTRATEGIES

Approximate (also see ESTIMATE) When you MEASURE anything, the measure is always approximate.This is because anything measured will come to an end BETWEEN the marks on the SCALE, however close those marks are.

When you are giving the approximate measurement of something, you may be quite close to being exact (for example, between CENTIMETRE marks), or much closer to being exact (for example, between MILLIMETRE marks). The same is true when you are calculating. 'About' is a word to use when you are approximating. The pencil is about 62 cm long.' 'Seven sevens are about fifty.'

Area

(also see UNIT)

All shapes have an area.Their area is the amount of surface or surfaces you find they have when you MEASURE them. For flat shapes, any shape that TESSELLATES can be used as a measuring UNIT. Mostly, the shapes used to measure surfaces are SQUARES or equilateral TRIANGLES.

STRATEGI[S

9

Arithmetic

(also see ADDING, DIVIDING and MULTIPLYING)

Arithmetic is calculating with NUMBERS by ADDING, SUBTRACTING, MULTIPLYING or DIVIDING. It can be be done in your head, on paper and with a CALCULATOR. How many months have you been alive?

Arrangements Things can be arranged in DIFFERENT orders. For example, in the diagrams below, different shaped counters are arranged in different orders.

Each order is called an 'arrangement'. Below are five of the six possible arrangements of three SHAPES.

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STRATEGIES

Arrow When you want to link two things that are related in some way, you can use an arrow. When this is done several times together, the result is an arrow diagram. Here are four examples of arrow diagrams.

Average

(also see MEAN)

There are three different kinds of average.

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Bar chart, bar graph, blOCk grap Information such as how many medals countries won in athletics championships can be given on a bar chart, a bar graph or a block graph. The bar chart here shows the information with countries in alphabetical order, which helps you to find any country quickly. However, the countries could have been put on the graph in any order. A bar chart has bars all the same width.The bars also have gaps of the same width between them.

Bar chart, bar graph, blOCk gra The bar graph below presents the same information.The countries have been put in order of which won most medals, but they could have been put in any order.

The block graph below shows the same information with countries in order of the size of their population, but they could have been put in any order.

A bar graph has much thinner bars (or sometimes just lines), which can be horizontal (as above) or vertical.

A block graph is similar to a bar chart but its bars are divided up into blocks and each block stands for something. Here, each block in each line stands for 2 medals.

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sr

Base (1 - NUMBER) Objects can be grouped to make them easier to COUNT.These drums are hard to count when scattered

but easy to count when grouped in tens.

There are four tens or four-ty drums, written as 'forty' or '40'. Counting with tens as the base continues with five tens or five-ty (fifty, 50), sixty (60), seventy (70), eight-ty (eighty, 80) and ninety (90). Instead of ten-ty, we have one hundred, written '100'. TEN is the base we usually count in, which is why we only need ten DIGITS to record numbers: 0 I 2 3 4 5 6 7 8 9 . Other bases are used less often. In computing, base sixteen (called 'hex') is more convenient and ACCURATE.

Base (2 - GEOMETRY) Three dimensions A base is any flat face on which a shape is resting.

A cylinder has two possible bases. A cone has just one, and a sphere has none.The pyramid and cuboid each have several possible bases. Two dimensions A base is a side which is in a horizontal position at the bottom of a shape.The TRIANGLE below has three possible bases. It depends on how you turn it.

STRATEGY

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Bearing

(also see ANGLE)

To describe a bearing, you need to know where north is. The bearing of a place from where you are is the amount of turn from north that you would need to make to face towards that place. There are eight compass bearings in common use.

Sometimes we need to be more ACCURATE.

However, the most accurate bearings of all are MEASURED CLOCKWISE in degrees from the north line.

A bearing always has 3 DIGITS, e.g. 45° becomes 045°.

Between

(also see ESTIMATE)

When estimating a MEASURE, you can choose a NUMBER, knowing it is close, but not quite right. Or you can decide to describe the measure as being between two numbers. You are making a really good ESTIMATE if you say that the measure is between two numbers that are not far apart, and you get these right.

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Billion A billion is a thousand millions. It is written or

We need words like billion to answer questions such as: How far away is Saturn? About one billion miles away. How old is the Earth? About four and a half billion years old. How many people are there in the world? About six billion people. How many stars are there in our galaxy? About one hundred billion stars.

In some books, a billion means a million millions, but a million millions is more often called a 'trillion'. A trillion is written or

Brackets

These two CALCULATORS give different answers to 3 + 4 x 5. One calculator works out 3 + 4 , which is 7, and then 7 x 5 = 35. The other calculator works out 4 x 5 , which is 20, and then 3 + 20 = 23. To show which part of the calculation to work out first, we use brackets. (3 + 4) x 5 = 35

3 + (4 x 5) = 23

The RULE is then to work out the part of the sum in the brackets first. If there are no brackets, the rule is that MULTIPLYING and DIVIDING come before ADDING and SUBTRACTING, so 3 + 4 x 5 = 23. STRATEGY

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Calculator A calculator is a machine that helps people to find answers to NUMBER problems by doing ARITHMETIC. Electronic calculators are small and do arithmetic very fast. Calculators are useful for problems which people cannot work out in their heads, and which they don't want to work out with a pen and paper or in some practical way.They are often used in jobs where lots of calculations must be done ACCURATELY and in a short time. When you want to solve a problem by using a calculator, you need to know exactly what kind of sum you are trying to do.

Capacity

(also see VOLUME)

The capacity of a container is the amount of space inside it. If the container is a bottle, you can find out its capacity by MEASURING the amount of liquid it can hold. If the container is a box, you find out its capacity by measuring the amount of sand or other dry, flowing material it can hold.

Capacity is often measured in litres.

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Centimetre

(also see LENGTH)

A centimetre (or 'cm1 for short) is a HUNDREDTH of a METRE and is a handy UNIT for MEASURING many everyday objects. The rectangular frame around this dictionary entry is about 12cm by 19cm, so its PERIMETER is about 62 centimetres. In the top right hand corner of the frame, there are some square centimetres, cm2, which are used to measure AREAS. Centimetre cubes, cm3, are used to measure VOLUME.

Centre There are different kinds of centre.The centre of this kfce can be the POINT where its DIAGONALS cross each other, or the centre of the CIRCLE jnto which it just fits.

Shapes (2D or 3D) which are very symmetrical have centres that are easier to find. The centre of the oblong frame around this dictionary entry is where the diagpnals meet and cross.This is also the centre of the circle surreiinding the frame. Mere are two more symmetrical shapes with their centres. ,.:••-*'"

STRATEGIES

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Centre

(continued)

It is not so easy to find the centre of some shapes.

These shapes share the same centre, and so they are called 'concentric'.

Chance

(also see PROBABILITY)

When something may or may not happen, you need to decide what chance there is that it will happen. You may reckon that there is a good chance that something will happen, which means that it is more likely to happen than not to happen. You may reckon that there is a fifty-fifty chance that it will happen, which means that it is equally likely to happen or not to happen. You may reckon that there is a poor chance that it will happen, which means that it is less likely to happen than not to happen.

If something is sure to happen, then it is certain. If it has no chance of happening, then it is called 'impossible'.

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STRATEGIES

Circle

(also see Pi)

If you swing a conker on a string, the conker traces out a circle and the end you are holding is the CENTRE of the circle. The distance from the centre to the conker is the RADIUS of the circle.

The CIRCUMFERENCE of a circle is its PERIMETER (the distance all around its edge).

v-rirCUmrC alseose caiucale deimeter and RADIUS) The two ends of this length of plastic tube can be joined so that the tube forms a CIRCULAR hoop.The circumference is the PERIMETER of the hoop - the distance all the way round the outside of the circle.The widest part of the circle is its DIAMETER.

The diameter of a circle will fit into its circumference a certain NUMBER of times.This number is very nearly 31/? and is called 'PI*.

STRATEC1ES

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Classes People are different heights, so a chart of heights might look like this.

If grouped together into classes, the same heights look like this.

Heights that have been grouped together are in the same class interval. A SURVEY of shops in a town included dress shops, shoe shops and hat shops. In the survey results, these shops were all put together in the class of clothes shops.

Clockwise

(also see ROTATION)

Things can turn or spin in two different ways. To push a screw into wood, you need to turn the screwdriver clockwise. To put a lid on a jar, you need to turn it clockwise, and to take the lid off, you turn it anticlockwise.

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Column Below are a table of NUMBERS and a bus timetable. In each table, a column has been shaded to pick out some information. A Greek column

The 6 times table.

This bus starts its journey at Handsworth at 6.15. It gets to the City Centre at 6.42.

Commute If you commute, you travel the same distance both ways. Some number statements give the same answer either way round.

Above, 7 and 4 commute for ADDITION, and 6 and 4 commute for MULTIPLICATION. Any pairs of NUMBERS commute for ADDITION and multiplication, but this is not true for SUBTRACTION and DIVISION. CTRATEGIES

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Complement These pairs are sometimes called the 'complements' of ten because one NUMBER in the pair makes the other up to 10:

Here are the whole NUMBER complements of 7:

Complements of smaller numbers help you to deal with larger numbers.

And they can help us to ADD up:

Shop assistants may also use complementary addition to make up the change from the price of the article we buy to the amount we give.

Congruent Congruent shapes are exactly the same size and shape as each other. Two shapes are congruent if you can fit one on top of the other with no overlaps by using one of these movements:

3D objects like these gloves can also be congruent.

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Consecutive Consecutive things or NUMBERS are things or numbers that are next to each other. These are consecutive days of the week:Tuesday, Wednesday,Thursday. These are consecutive COUNTING numbers: 4 5 6 7 8 . These are consecutive ODD numbers: 5 7 9 II. These numbers are not consecutive: 6 7 1 0 . We use consecutive numbers for: COUNTING I 2 3 4 5 ...

labelling things, for example houses, so we know which is which.

Co-ordinates Co-ordinate systems give a code for the position of a space or a POINT. If we use a map co-ordinate system, we can show that some treasure is buried on the island somewhere in the SQUARE coded C4.This is not a very precise code. If we wanted to give a more ACCURATE code, we could make the squares smaller. If we use a Cartesian co-ordinate system, we can show that some treasure is buried on the island at the point coded (3.4). In Cartesian co-ordinate systems, the NUMBERS along the bottom are read first, and the numbers up the side are read second.You will not find the treasure at the point (4.3)! You can make Cartesian co-ordinate systems even more precise by dividing both sets of numbers into DECIMAL FRACTIONS.

STRATEGIES

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Co-ordinates

(continued)

In a polar co-ordinate system, we describe a point as being so many UNITS away from a fixed point and at an ANGLE of so many degrees (anticlockwise) from a fixed BASE line running through the fixed point. If we use a polar co-ordinate system, we can show that the treasure is buried on the island 5 units away from a fixed point and at an angle of 53° (anticlockwise) from the fixed base line.

Count

(also see NUMBER)

When we count, we are saying how many (the NUMBER) of something there are. One, two, three, four, etc. are called the 'counting numbers'. To count the buttons below, we can touch each one in turn, moving along the line and saying the next number up as we touch the next button.The number we say as we touch the last button tells us how many there are.

When you are counting, you need to understand why the numbers have the names they do. For example, when you say * 163', you need to understand that this stands for one HUNDRED, six TENS ('-ty' = 'tens') and three.

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CTRATECIES

Cube (also see NET and POLYHEDRON) A cube is a 3D (3 DIMENSIONAL) shape with 6 SQUARE faces. This picture of a cube looks like a hexagon. If you are looking directly at just one face of the cube, it looks like a square. 3D shapes made with 6 rectangular faces are called 'cuboids'.

This is called a 'cube NET'.You can imagine it being folded into a cube. There are several different ways of opening out a cube to rnake a cube net.

27 is called a 'cube NUMBER' because 27 identical cubes can be put together to make a larger cube. Cube numbers start with 1, 8, 27 ...

Curve Any line that is not straight can be called a 'curve'. Some curves are given special names.

It is possible to use straight lines to give the impression of curved surfaces.

3D shapes can have curved surfaces as well as flat ones.

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Data Facts that give you information about something can be called 'data'. Data may be given in NUMBERS, in words or even in pictures. If you wanted to find out as much as you could about everyone in a class, you could collect many kinds of data. The data for what each person likes to eat best might be a list of food words next to a list of names, or it could be a list of little pictures: NAME

FAVOURITE FOOD

Susan

ice cream

Siobhan

Mars Bar

Ranjit

chips

or

For the children's ages, the data would be in numbers. A 'spreadsheet' is a table of data produced by a computer.

Decimal The word 'decimal' comes from the Latin word 'decem' which means 'TEN'. The way we write NUMBERS is called a 'decimal system' because it is based on ten. Each time a DIGIT moves from 9 to 10, we make the number longer by placing a I on its left hand side, for example, 9-10, 99-100,999-1,000. The people who started our system of numbers probably used a decimal system because we have 10 fingers.

All numbers in the decimal system are made up from ten symbols:

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CTRATECIES

Decimal

(continued)

A decimal system for MEASUREMENT (LENGTH, weight, etc.) makes TEN of each UNIT equal to the next, larger unit. For example, in measuring length using the metric system 10 millimetres = I CENTIMETRE 10 centimetres = I decimetre 10 decimetres = I METRE, and so on. Once length was measured in yards, feet and inches. It took 12 inch units to make I foot unit and 3 foot units to make one yard unit, so this was not a decimal system of measurement.

With the decimal system, whole numbers are broken into decimal FRACTIONS (or decimals, for short) by using a dot called a 'decimal point' to show where the fraction starts. The way to show 1/4 in the decimal system is 0.25(4x 1 4 = I and 4 x 0 . 2 5 = I)

Diagonal

(also see CENTRE,)

Any flat shape that has four or more sides can have diagonals drawn across it. The diagonals are lines that DIVIDE the shape by joining two of its corners. For a line to be a diagonal, the corners it joins must not be next to each other.

A hexagon has nine diagonals.

This octagon has two of its diagonals shown.

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L^ldlTlC Lvrr (also s The tips of this aeroplane propeller sweep out in a CIRCLE when the propeller is ROTATING. The bolt marks the CENTRE of this circle.

The DIAMETER is the line from one tip to the other, and it passes through the centre of the circle. It is twice as long as the RADIUS.

If you don't know where the centre of something circular is, you can still find its widest part, or diameter, by using callipers.

Difference For NUMBERS, the difference is how much bigger one number is than the other.You SUBTRACT the smaller of two numbers from the larger number to find the difference. 5 is the difference between 8 and 13. For the numbers 27 and 20, the difference is 7. The difference between these rods is 3cm.

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Differences One of the ways we can look for the PATTERN in a NUMBER SEQUENCE is to look at how each number is different from the one before it. The sequence of TRIANGULAR numbers begins I 3 6 10 15 21 .. The differences between the numbers are 2 3 4 5 6 ... It is not very easy to see the pattern in this number sequence. 0 I 5 14 3 0 . . . but by looking at the differences I 4 9 16... we can see that they form a sequence of SQUARE numbers. By continuing the sequence of square numbers (25, 36), we can continue the first sequence (55, 91). 0 0 4 18 48 100 For some sequences, it is helpful to go on to find the 0 4 14 30 52 difference between the differences, and so on. 4 10 16 22 This creates a difference table. 6 6 6

Digit We write NUMBERS over 9 by putting together the digits 0, 1,2, 3,4, 5, 6, 7, 8 and 9. Three hundred and eighty-two is written as 382. The digit 3 stands for 300. The digit 8 stands for 80. The digit 2 just stands for 2. This sum has been made into a puzzle by hiding some digits.

There may be more than one answer to the puzzle. When we DOUBLE a number made up of 2 or more digits, then we double the new number, and so on, there is a pattern in the last digits: 12, 24, 48, 96, 192, 384, 768, 1536 ... 23, 46, 92, 184, 368, 736, 1472, 2944 ... S™nc,Es

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Digital A digital root is found by ADDING the DIGITS of a NUMBER. If there is still more than one digit, these must be added again. So, for 34: 3 + 4 = 7 the digital root is 7 and for 478:4+ 7 + 8 = 19 and I +9= the digital root is I.

10 and I + 0

A digital watch shows the time using digits. In a digital recording, the pitch and loudness of musical notes are measured by whole numbers.The digits that code the pitch and loudness are what is recorded on the tape or disc.

Dimension The MEASUREMENTS that tell us how big something is are its dimensions. We talk about things having one, two or three dimensions. To find the distance along a line from one POINT to another point - for example, how long the lead connecting a computer to an electric socket is - we only need one measurement, the LENGTH.We are measuring one dimension. If we are looking at how big a surface is, e.g. a computer screen, we measure its width as well as its length. The screen surface is two-dimensional. To know how big the whole computer monitor is, we measure its length and its width and its depth. We are measuring three dimensions. 30

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Dividing

(also see MULTIPLYING)

To divide something is to split it into parts. This hexagon has been divided into three parts and they are all different. It could have been divided into equal parts. When NUMBERS are divided into EQUAL parts, we use the division sign:

This diagram shows 12 divided into 4 parts and each part has 3. \ We write this as: 12 •*• 4 = 3. .' The same diagram shows how many 3s there are in 12. We can write this as: 12 •*• 3 = 4. The 12 objects can be seen in several different ways, such as 1 2 - ^ 2 = 6 and 3 x 4 = 12 and 12 = 2 x 6 . The opposite of dividing is MULTIPLYING.

Double The word 'double' is used when there are two of something. A double-yolked egg. A double-five domino has ten spots altogether because double 5 is 10. A double-headed monster.

If a sheet is folded in two, it is doubled over, as it is twice the thickness. If we start at I and keep doubling, we get this SEQUENCE: I 2 4 8 16 3 2 6 4 . . . The opposite of doubling a number is HALVING it. swere

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Equal Four threes are the same as 12. Four threes equal 12. 3 x 4 = 12

12 is the same as four threes. 12 equals four threes. 12 = 3x4

The two equations above show the same thing in different ways. The equation can be written either way because the NUMBERS on each side of the equals sign have the same answer. Sometimes we use the equals sign when we mean 'makes', as in five times seven makes thirty-five written 5 x 7 = 35. On a simple CALCULATOR, the = button is always used in this way.

LSrirYldTC An estimate is a reasonable guess that has not been COUNTED, MEASURED OT CALCULATED ACCURATELY

You could count this flock of birds, but you could also estimate how many there are. For example, you could count five birds and then guess how many fives there are.

If you know that the height of this panel is about 12 CENTIMETRES, you can estimate its width. Before working out 1 9 x 2 1 , you could estimate that the answer will be close to the answer for 20 x 20 = 400. Estimation is an important skill in all MATHEMATICS. 32

sm

Even Here are two children sharing out a NUMBER of CUBES. If the number of cubes is even, they can end up with the same number each. If the number is not even, there will be one cube left. Here are some PATTERNS of counters

These numbers make pairs with none left over, so they are called even numbers.

Even

These numbers make pairs, but have one left over, so they are ODD numbers.

(continued)

If you start from 0 and COUNT on in twos, you will make a list of even NUMBERS:

2 4 6 8 10 12 14 16 18 20 22 2 4 . . . 1 1 2 114 1 1 6 1 1 8 . . Notice how the even numbers begin with the two times table. All even numbers end in 0, 2, 4, 6 or 8. If you count your steps as you walk, one of your feet will count out the even numbers and the other foot will count out the ODD numbers.

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Factor

(also see MULTIPLYING)

5 is a factor of 30, because six 5s equal 30. 6 is also a factor of 30, because five 6s equal 30. So 5 and 6 are called a pair of factors of 30. Other pairs of factors of 30 are 3 and 10,2 and 15, and I and 30. 30 has four pairs of factors, and eight factors altogether. Starting with 30, we can grow a factor tree. In factor trees, we cannot use the number I. First, choose a pair of factors. 3 and 10

Factor

(continued)

Here is the completed tree. The leaves 2,3 and 5 are the prime factors of the tree that grows out of 30.

Factorial

I x 2 x 3 is called 'three factorial'. All the whole NUMBERS up to 3 are MULTIPLIED together.

The symbol for factorial is ! 3! = 6 and 4! = 24.

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STRATLGIELS

Fibonacci Most NUMBER PATTERNS have a RULE that you can use to continue them. You look at the last number and use the rule to find the next one. Fibonacci number patterns are different; you need to look at the last two numbers and use the rule. Leonardo of Pisa, nicknamed Fibonacci, discovered this pattern, which is called the Fibonacci sequence:

1

I 2 3 5 8

13 21 34 55 89 ...

This and other Fibonacci-type patterns have some fascinating properties. For example, choose any of the numbers above and make a note of its two neighbours. SQUARE your number and MULTIPLY its two neighbours together and compare the two results.Then try another number. You can create your own Fibonacci-type pattern by choosing any two numbers as its start, e.g. choosing 2 and 10 would give:

2

10

12 22 34 56 90 ...

Notice how quickly the numbers become quite similar to Fibonacci's own SEQUENCE.

Formula A formula is a recipe or RULE. It may be written in short form, using SYMBOLS to stand for words, e.g. In words: To find the PERIMETER of any oblong, MEASURE the longer and then the shorter side, then ADD these two measures and DOUBLE this. In short form: For any oblong, P = 2 x (L + S)

So a formula, whether it is symbols or words, gives instructions about what to measure or find out, and how to combine this information. In the case above, L and S need to be measured to find P. A formula given in symbols needs a key to explain what each symbol means: Key: L is the longer side; S is the shorter side; P is the perimeter. STRATEGY

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Fraction

(also see NUMBER LINE)

In this array of twelve dots, some are black and some white. Eight of the twelve are black, but you might also notice that two in every three are black. As a fraction, % (SYMBOL) of the dots are black. two-thirds (words) of the dots are black. 1 Also, /s (one-third) of the dots are white. Notice that 4/,I20 and 1/3 are two fractions that mean the same amount. They are equivalent fractions. 8 / |2 and % are another pair of equivalent fractions. 14 is a quarter. A quarter of the 12 dots would be 3 dots. Finding 14 of a NUMBER has the same effect as DIVIDING it by 4.

Fraction

(continued)

Fractions are sometimes used to think about how something should be DIVIDED before the division is done. If this chocolate bar with 10 pieces had to be shared EQUALLY between 3 people, each could have 3 pieces; but the last piece would need to be broken into smaller pieces, into 3 thirds. 1 0 - 3 = lo/3 = 31/3 Fractions are used in all kinds of MEASURES. Here are some examples: A line whose LENGTH has been split into quarters. A 2-dimensional shape whose AREA has been split into 2/s and %. A 3-dimensional shape whose volume has been split into HALVES.

36

STRATEGY

Frequency When making a SURVEY of the colours of cars passing their school gate, some children made this tally chart. There were 12 red cars, so 12 is the frequency of red cars during the time the survey covers. The results of the survey were put into a frequency table.They were also put into a GRAPH called a 'frequency diagram', which shows frequencies against a NUMBER LINE at the side. FREQUENCY TABLE Colour Frequency Red 12 Orange 3 Yellow 5 Green 9 10 Blue

FREQUENCY DIAGRAM

Function A function is an instruction to do something, or several things, in a definite order.You can get an overall idea of a function by using it lots of times with different starting NUMBERS.These starting numbers can be put into a function table and made into a function GRAPH. Here are 2 simple functions. Simple functions can also be combined. ADD THREE input output 0 3 I 4 2 5 3 6 4 7 5 8

DOUBLE input output

0 I 2 3 4 5

0 2 4 6 8 10

ADD THREE,THEN DOUBLE input output 0 6 I 8

2 3 4 5

10 12 14 16

Functions can be given as key presses on a CALCULATOR.

ST—

37

Generalise You may have noticed that a TRIANGLE has 3 sides and also 3 corners, and that generally POLYGONS have the same NUMBER of sides as they have corners. When we find a RULE that always works, we can generalise. These lines of crosses are made from sticks.

We can generalise by saying that the number of sticks needed is always 3 times the number of crosses, plus one. Often, such rules are written in SYMBOLS. Using 'C for the number of crosses, and 'S' for the number of sticks, the general rule is: S = 3 x C + I Here are two other generalisations to check on: 1. Any straight LINE drawn through the exact CENTRE of an oblong will cut it into two EQUAL pieces. 2. Neither 2 nor 7 is ever the last digit of a SQUARE number or of a TRIANGULAR number.

Graph

(also see BAR CHART, BLOCK GRAPH and CO-ORDINATES)

A graph is a picture that can make DATA easier to understand than when it is just NUMBERS in a table. There are many different kinds of graph, each with its own uses. For example, in this dictionary you will find BAR CHARTS, BAR GRAPHS, BLOCK GRAPHS and FREQUENCY diagrams, which are all kinds of graph. Pie charts are also kinds of graph.They are used to show how the total amount of something is shared out. MATHEMATICAL RULES and FUNCTIONS make connections between 2 numbers, so each rule can give us lots of pairs of numbers. Used as COORDINATES, these pairs can be made into a picture of the function or rule. Sometimes, it makes sense to join these co-ordinate points with a straight LINE or a CURVE. Imagine a journey of 60 kilometres where you travel steadily at the same speed all the way.The speed and the time the journey takes are a pair of numbers that can become co-ordinates on a graph. 38

CTRATEGIES

Half

(also see FRACTION)

Half of something must be one of 2 EQUAL parts.

Half of these counters are shaded.

Half of this square is shaded,

but the shaded part of this square is not a half.

Here are some other ways to halve a square.In each square, the line cuts the AREA into 2 EQUAL pieces. A half is also a NUMBER, written as 1/2 or 0.5 and this number is on the NUMBER LINE mid way between 0 and I.

To get half of a number, we DIVIDE it by 2. Half of 12 is 6. We use the V SYMBOL for 'of and write 1/2 x 12 = 6

Hundred

(also see TEN)

Our way of writing NUMBERS uses TEN as a BASE. It is a 'place value' system: the meaning of a DIGIT depends on its place. One hundred is written 100, with the I in the third place from the right. The number of hundreds is given in this place. In the number 872, the 8 means 8 hundreds. In the number 527,384 the 3 is in the hundreds place and so is worth 300. The other digits are as follows: 4 is in the UNITS place and is worth 4. 8 is in the tens place and is worth 80. 7 is in the thousands place and is worth 7,000. 2 is in the tens of thousands place and is worth 20,000. 5 is in the hundreds of thousands place and is worth 500,000. A number like 1,200 has thousand and 2 hundreds.We sometimes ca that 'twelve hundred'.

CTRATECIES

39

Infinite When something can be imagined as going on forever, it is called 'infinite'. COUNTING NUMBERS are infinite because, however far you have counted, you can always count on one more. This small L-shape can be repeated to make a bigger L-shape. By repeating this process, a larger and larger L-shape could be made. In your imagination, this could go on forever; so it would become infinite.

Inverse If you ADD 2 to a NUMBER, you can find the original number again by SUBTRACTING 2.This works no matter what number you start with, because 'Add 2' and 'Subtract 2' are inverses. In general, addition and subtraction are inverse operations. DOUBLING and HALVING numbers are inverse changes too. MULTIPLICATION and DIVISION are inverse operations, since (for example) x7 can be undone by using +7.This works for other numbers - except if you use xO, which has no invers Turning left' and 'turning right' are inverses, as long as you turn the same amount If you turn through one right angle, you can either turn back through one right angle or carry on turning through three more right angles, as either will be an inverse move to the first one.

40

ST~

Length

(also see UNIT)

The length of something is how far it is from one end to the other. The length of this pencil is 5cm.

Tools for MEASURING length include a ruler and a tape measure. Some UNITS for measuring length are: millimetre inch CENTIMETRE

foot

METRE kilometre

yard mile light year

Line symmetry These drawings are normally found on playing cards. Imagine folding the 'heart' along the dotted line. It would fold together exactly without overlapping.The two HALVES on either side of the line are mirror images.The heart has a line of symmetry. Check that each of the other drawings has a line of symmetry. One of the drawings has two lines of symmetry which are at right ANGLES.

Some shapes which have several lines of symmetry.

A CIRCLE has an INFINITE number of lines of symmetry: any straight line through its CENTRE.

STRATEGY

41

Mapping

(also see FUNCTION)

On a geographical map, each POINT represents a point on the ground. (See SCALE.) A MATHEMATICAL mapping shows how things are connected by a particular RULE. When the rule has to do with NUMBERS, these are sometimes shown on a NUMBER LINE. Often the connections are shown by ARROWS. + 2mapping

Mapping

(continued)

•* 2 mapping

A FACTOR mapping for the set of numbers from 2 to 10

42

CTRATECIE:

Mathematics

(also see PATTERN)

Mathematics happens when people find problems that involve NUMBER or size and try to solve them through reasoning and thinking logically. Mathematical problems are usually written down using: numbers or other SYMBOLS or shapes or moves in space. When there are several mathematics problems of the same kind, people try to find a METHOD or FORMULA that can help to solve them all.

Mean

(also see AVERAGE)

A mean is a kind of AVERAGE. Three children have these amounts of pocket money: If they shared their money equally, they would each have 50p.

They would have equal shares if John gave Fiona 20p and Ann I Op.

When we say what amount each child would have if the money were shared out EQUALLY, that average amount is called the mean. The mean amount of pocket money is 50p.

The mean height of this family is not actually the height of any one of its members. STRATEGY

43

Measure

(also see UNIT)

Measures are used to answer questions like these: How high is that tree? How big a surface has the desk top? How much cereal will fit into the box? How heavy is the fridge? How much space does the fridge take up? In each case, you need a UNIT to measure with. This changes the sort of questions given above into questions that begin with 'How many?' How many SQUARE CENTIMETRES is the desk top? How many kilograms is the fridge? How many CUBIC CENTIMETRES will it take up?

Method A method is a way of solving a problem or finding an answer to a question. Methods can be mental - the answer is worked out in someone's head written - the answer is worked out using pencil and paper mechanical - the answer is worked out using a machine such as a CALCULATOR.

44

STRATEGY

Metre

(also see LENGTH)

We use metres to say how long something is. When a tall person walks with big steps, each step is about a metre. The length and breadth of a room can be MEASURED in metres. For longer things we need a larger UNIT.To say how far it is from one town to another we can use kilometres. One kilometre is 1,000 metres. Many everyday objects are much smaller and so we use CENTIMETRES. 100 centimetres makes I metre.

The SYMBOL for metre is 'm'.

The measurements of the floor of this room are shown in metres.

Multiplying

(also see DIVIDING)

When there are several equal groups of objects, for example six groups of four, then the 4 has to be multiplied by 6 to get the total NUMBER. This is written as 4 x 6. 4 x 6 = 24, so 24 is a multiple of 4. 6 x 4 = 24, so 24 is also a multiple of 6. 24 is also a multiple of I, 2, 3, 8, 12 and 24. There are other ways of describing multiplications:

STRATEGIES

45

Negative numbers

(also see NUMBER)

NUMBERS less than ZERO are called 'negative numbers'. They are the answers to problems like these: 3-5 7-17 221-300 The answer to this SUBTRACTION, minus two, is a negative number.

Negative numbers can describe positions on a LINE that are below zero, that is they are on the other side of zero to positive numbers. Below are some negative numbers, with zero, and then some positive numbers.

Negative numbers also describe positions when used as CO-ORDINATES.

Net

(also see POLYHEDRON)

These flat shapes all fold into a CUBE. Each is a different net for a cube.

Any POLYHEDRON has several different nets.

If you buy things in boxes, you can open the boxes out to make nets.

46

CTRATEGIES

Network A network is made up of LINES joined at one or both of their ends. Here are some networks.

The lines in this drawing are a network.

This tree diagram is a network.

A network of main roads joins these places in the Isle of Wight.

Number There are many different kinds of numbers. The COUNTING numbers are I, 2, 3,4, 5, 6 .

Seven is a counting number.There are 7 buttons here. FRACTIONS are numbers for parts of things: The slice is 1 4of the pizza.

CTRATECIES

47

Number

(continued)

Numbers below ZERO are called NEGATIVE NUMBERS. Sometimes we need to show that something is less than ZERO. For example, we need to show that the temperature is less than zero when it is very cold. The thermometer on the right shows a negative temperature.

Numbers can also be used to tell us the position of something. At the swimming baths, we know we will find locker 214 between lockers 213 and 215.

Number line To help us think about NUMBERS, we can use a line of numbers. A number track A number line

Any FRACTION can be placed on a number line. On this number line, some fractions are shown in their places.

We can extend a number line to show NEGATIVE NUMBERS too.

48

CTRATECIES

Odd

(also see EVEN)

When an odd NUMBER of things is DIVIDED into twos, there is always an odd one left over.

Seven is odd.

If you add two odd numbers, the answer will always be EVEN.

If you add an odd and an even number, the answer will always be odd.

Parallel Two straight lines that are always the same distance apart are called parallel lines. Some POLYGONS have parallel sides: a TRAPEZIUM has one pair of parallel sides a PARALLELOGRAM has two pairs of parallel sides this 'parahexagon' has three pairs of parallel sides.

The faces of a POLYHEDRON can also be parallel. The CUBE has three pairs of parallel faces. STRATEC,ES

49

PcircUlclOgrcim fa/so see QUADRILATERAL, PARALLEL, RHOMBUS, RECTANGLE, SQUARE and PLANE SHAPES)

A parallelogram is a QUADRILATERAL with two pairs of PARALLEL sides. This RHOMBUS is a parallelogram.

This parallelogram is a SQUARE.

This RECTANGLE JS

a parallelogram.

All parallelograms have ROTATIONAL SYMMETRY of at least order 2. Most parallelograms do not have a LINE OF SYMMETRY.

P(also seePREDICT, SYMMETRY and TRANSLATION) A pattern is continued either by repeating exactly what is already there or by following a RULE.

This frieze pattern repeats in a line. This NUMBER pattern uses a DOUBLING rule to find the next number in the SEQUENCE. 3 6 12 24 48 .,

Here is another number pattern: 2 5 8 II

4 6 8

7 10 9 12 II 14

14 13 16 15 18 17 20

The vertical rule in the pattern above is different from the horizontal rule. 50

This pattern of a TESSELLATING shape repeats in several directions. CTRATEGIES

Percentage

(also see FRACTION)

Comparing, ADDING or SUBTRACTING FRACTIONS can be awkward when the bottom numbers are different It is not easy to say which is the largest and which is the smallest of these fractions:

Percentages are fractions written so that the bottom number is always 100. Instead of writing the ' 100', we write 'per cent' or put a % sign.

7/,o = 7°/mo = 7° per cent = 70% '8/25 = 72/,00 = 72 Per cer|t = 72% 33

It is now easy to see which is the largest.

/so = 66/,oo = 66 per cent = 66%

DECIMALS are easier than fractions to write as percentages, because the parts of a whole are already COUNTED using a BASE of TEN. 0.25 = 25 HUNDREDTHS = 25/mn 100 = 25% 0.2 = 20 hundredths = 20/IOO =20%

1.2 =120 hundredths = I20/I00 =120%

Perimeter The perimeter of a shape is the length of its boundary.

This RECTANGLE is 5cm wide and 3cm high so it has a perimeter of 16cm.

These two shapes have the same AREA, but different perimeters.

These three shapes have the same perimeter, but different areas. CTRATEO

51

Pi(7i) Pi is a NUMBER which comes from MEASURING CIRCLES. In any circle, the DIAMETER can be fitted 3 times round the CIRCUMFERENCE and there is still a small gap. The number of times the diameter fits into the circumference is very nearly 31/? and is called pi. Pi is written TT. It is a letter from the Greek alphabet. Circumference (of circle) = TT x diameter We are not able to write the number TT exactly using our number SYMBOLS, but 31/7 and 3.142 are very close to it. The number TT is also needed in CALCULATING the AREAS of circles and the VOLUMES of spheres.

plane (also see PLANE SHAPES) A plane is a surface that is completely flat. In MATHEMATICS a plane goes on and on in all directions without end, so we usually work with just part of the plane.

Plane shapes are shapes that can be drawn on a flat surface, such as a piece of paper.

The roof on this house is not a plane shape, but one side of the roof is a plane shape. The other side is another plane shape. Each wall is a plane shape too. 52

CTRATEGIES

Plane shapes

(also see POLYGON)

A plane shape is one that is entirely flat. Here are some plane shapes:

A CUBE is not a plane shape, but each of its faces is the plane shape that we call a SQUARE.

There is also a plane shape that can be found by cutting through the mid-points of six of the edges of a cube. It is a hexagon and so it is a plane shape.

plane symmetry (also see LINE SYMMETRY and SYMMETRY)

This cup has one plane of symmetry.

A solid has plane symmetry when it can be cut into HALVES by a plane so that each half is a REFLECTION of the other in that plane. Some solid shapes have more than one plane of symmetry.This cup has two.

This is a pyramid cut in half by one of its planes of symmetry. STRATEGIES

53

Polygon A polygon is a flat, closed shape made with straight lines. Each polygon has a name that describes how many sides it has. For example: A A A A

polygon polygon polygon polygon

with three sides is called a 'TRIANGLE*. with four sides is called a 'QUADRILATERAL*. with five sides is called a 'pentagon'. with six sides is called a 'hexagon'.

A regular polygon has all its sides the same length and all its angles equal.The hexagon shown above is regular, but the other polygons are not.The polygons below are also not regular.

Polyhedron A polyhedron is a solid shape which has several faces and each face is a POLYGON.

All polyhedrons can also be called a name that describes how many faces they have. This is a polyhedron which has eight faces. It is called an 'octahedron'. The polygons on the top and bottom are hexagons. The polygons on the other six faces are RECTANGLES (QUADRILATERALS). The faces of this octahedron are regular polygons, but there are two different kinds of these, so this is not a regular polyhedron.

This octahedron has a special name. It is called a 'hexagonal prism'.

54

S™:

Polyhedron (continued) This polyhedron also has eight faces, so it too is an octahedron. Its faces are isosceles TRIANGLES. The faces of this octahedron are polygons that are not regular, so this is not a regular polyhedron.

This octahedron has a special name. It is called a 'square dipyramid'.

This polyhedron has six faces. It is called a 'CUBE*. The polygons on each of its six faces are SQUARES (quadrilaterals). The faces of a cube are all the same and are all regular polygons, so this is a regular polyhedron.

Point A point is a single dot. Sometimes, we mark a point by using a small cross, with the 2 lines cutting each other at the point. These points are all on the same straight line.

These points all lie on a CIRCLE.

There are two points where these two circles cut each other.

A corner that sticks out is sometimes called a point. This star has 8 points.

STRATEGIES

The point on this shape is at the top. 55

Predict

(also see PATTERN)

Look at these NUMBERS:

4 7 10 13 16 From the PATTERN, you can say that the next number will be 19, and the

one after that 22. We predict that these will be the next 2 numbers. One of the most powerful things about MATHEMATICS is the way that, by using it, we can predict things. If you understand the pattern, then you can say what will happen in other cases.

Predict

(continued)

These PATTERNS show the first 4 TRIANGULAR NUMBERS:

We can see that they are right by COUNTING. We can also CALCULATE them like this:

(I x 2 ) - 2 = I (2x3)-2 = 3

(3x4)-2 = 6 ( 4 x 5 ) - 2 = 10

The pattern in these sums shows us how to find a triangular number without counting dots. The 100th triangular number can be calculated like this:

(100 x 1 0 1 ) - 2 = 5050 We have predicted the answer from the pattern. Nobody wants to COUNT that number of dots!

56

STRATECIES

Probability

(also see CHANCE)

Many things may or may not happen, for example you may or may not get a 4 when you throw a die; you may or may not win a raffle. Sometimes we can give a MEASURE of how likely something is to happen, and this measure is called the 'probability' of it happening. On the probability SCALE, we use 0 for impossible and I for certain. Things that might happen have a probability that is a FRACTION between Oand I. There are six faces on a die, and you are EQUALLY likely to throw any of the six, so the probability of getting a 4 is '/6. If 500 raffle tickets are sold and 7 of them will win prizes, the probability that your ticket will win a prize is 7/soo. You cannot have a probability that is greater than I, or a probability that is negative.

Quadrilateral

(also see POLYGON)

A quadrilateral is a POLYGON with four sides. Different kinds of quadrilaterals have special names

A kite has two pairs of sides EQUAL in LENGTH.

A trapezium has one pair of PARALLEL sides.

A PARALLELOGRAM has two pairs of parallel sides.

CTRATEOIIS

57

Quadrilateral

(continued)

These parallelograms have special names: A parallelogram with four right angles is a RECTANGLE.

A rectangle with longer and shorter sides is an oblong.

A rectangle with all sides equal is a SQUARE.

A RHOMBUS is a parallelogram that has all sides equal but is not a rectangle.

(also see CIRCLE and DIAMETER) radiu r

If you keep the string tight, this peg will mark out a CIRCLE on the ground as you ROTATE it about the stake in the CENTRE. Viewed from the side, this circle will look like an oval. The distance from the centre of the circle to a point on the outside is called the RADIUS and is half the size of the DIAMETER. The distance travelled by the peg, above, when it goes once round, is the CIRCUMFERENCE of the circle.

58

CTRATEG

Range The range of a set of NUMBERS is the DIFFERENCE between the smallest and the largest number. These numbers

40 45 31 41 30 36 34 32 47 range from 30 to 47. We also say that these numbers have a range of 17, because 4 7 - 3 0 = 17.

IvCC^Ccir liJ (also see QUADRILATERAL, PARALLEL,

PARALLELOGRAM, SQUARE and PLANE SHAPES)

A rectangle is a QUADRILATERAL with right angles at all four corners.

This rectangle is a SQUARE, because all its sides are the same length. Rectangles that are not squares are called 'oblongs'. All rectangles have at least two LINES OF SYMMETRY, and have ROTATIONAL SYMMETRY of Order 2.

This rectangle is called a 'domino', because it is made from two squares. The panel surrounding this definition is rectangular.

surui

59

IvGrlGCTlOr

(also see LINE SYMM

Reflections are seen in mirrors, puddles and shiny surfaces.

The striped POLYGON on the right is a reflection of the spotted polygon on the left.They are the same shape, but each is the flip side of the other. Although these two shapes made from CUBES are reflections of each other, you cannot move one to look exactly like the other.

These polygons have line symmetry.You can find lines that divide them in half, and each half is a reflection of the other half.

rhombus (also see QUADRILATERAL, PARALLEL, PARALLELOGRAM, SQUARE and PLANE SHAPES) A rhombus is a QUADRILATERAL with all four sides the same length.

This rhombus is a SQUARE, because its corners are right angles.

All rhombuses have at least two LINES OF SYMMETRY, and have ROTATIONAL SYMMETRY of order 2. This rhombus is a diamond, because it is made from two equilateral TRIANGLES. 60

CTRATEGIES

Rotation Rotation means turning about a CENTRAL POINT or a central LINE. Shapes or objects can rotate in different directions. In this picture, the sails of the windmill appear to us to rotate in a CLOCKWISE direction.To a person standing behind the windmill, they would be seen to rotate in an anticlockwise direction. Some objects, like a steering wheel, rotate about the centre. Other objects rotate about other places. For example, the hands of a clock rotate about the point where their ends meet. Other objects, like the pages of a book, rotate about a straight line.

Rotational symmetry

(also see LINE SYMMETRY and SYMMETRY

A SQUARE cut from a piece of plastic will fit back into its hole in four different ways:

The square has a rotational symmetry of order 4. If you turn this three-legged design, it will look the same as it does now in three different positions, so it has a rotational symmetry of order 3. These 2D and 3D shapes have different orders of rotational symmetry.

The cone will look the same in an INFINITE NUMBER of positions.

swrad

61

Rule A rule tells you how things or NUMBERS are connected: These numbers are connected by the rule 'ADD 4 to get the next number': 3 7 II 15 19 23. The rule connecting the ANGLES of a TRIANGLE is that they must add up to 180 degrees. A rule tells you how to do something: A rule for finding the CENTRE of a PARALLELOGRAM is: Draw the DIAGONAL LINES that join up the opposite corners, and where the diagonal lines cross is the centre.

Scale

(also see NUMBER LINE)

Scale can mean two things in MATHEMATICS: A series of regular marks on something that enables the user to take MEASUREMENTS is called a scale. For example, you can see a scale on a rule, on a measuring jug and on the vertical line at the side of a GRAPH. When two or more objects have the same shape but are different sizes, this difference between the objects is one of scale. The flag on the right has been enlarged from the one on the left by a scale factor of 3. A toy model car or train is the same shape as the full size car or train, but has been scaled down. On this map, 5cm represents I km at the place shown on the map. The map is said to have a scale of 5cm to I km.

62

STRATEGIES

Sequence

(also see PATTERN)

In a sequence, a set of NUMBERS or things is put in order according to a RULE. The numbers below have been put in a sequence according to the rule 'ADD 2 to get the next ODD number': I 3 5 7 9 II 13 15 17 19 21 Below we have a sequence of TRIANGULAR numbers.

This is a sequence of connections between POINTS.

Square (1 - Shape) A square is a QUADRILATERAL which has four EQUAL sides.

All the corners of a square are right ANGLES.

A DIAGONAL going across a square DIVIDES it into two SYMMETRICAL parts.

A set square is a drawing aid usually in the shape of a right-angled TRIANGLE. It is used for drawing right angles.

smi

63

Square (2 - Number) The square of a NUMBER is the answer you get when you MULTIPLY that number by itself. The square of 5 = 5 x 5 = 25 We say'Five squared is twenty-five' and write this as 52 = 25.

A square CENTIMETRE (cm2), a square METRE (m2) and a square kilometre (km2) are UNITS used to MEASURE AREA. A surface that measures 25km2 has the same area as a square with sides of 5km.

Square root The square root of a NUMBER is the number that must be squared to make it. Four squared is 16. So, the square root of 16 is 4. 2

4 = 4 x 4 = 16

The square root of the AREA of a square is the length of one side.

The SYMBOL for square root is / , so we can write /I6 = 4 This SQUARE is made out of four TRIANGLES and each is HALF a square CENTIMETRE. The area of the square is 2cm2.The side is /2, but /2 is not a whole number. A simple CALCULATOR gives /2 = 1.41421356, but if you square this DECIMAL it gives 1.99999999. 64

CTRATEGIES

^UDrrrtCTinS (also

see ADDING and DIFFERE

Subtracting is the opposite of ADDING. You COUNT on to add, and you count back to subtract.

1 0 + 6 = 16 so 16 - 6 = 10

Subtracting numbers is called 'subtraction'. The sign for subtraction is • and this is called 'minus'.

Subtracting is often used to find the DIFFERENCE between two amounts. At other times, it is used when an amount is taken away. Sometimes it is a good idea to use adding on to do a subtraction. People often work out the amount of money to give in change by adding on.

Standard units

(also see UNIT)

The UNITS we MEASURE things in must be the same every time they are used, otherwise we cannot be sure how big something is. At different times in history, people have used different units from those that we use now. For example, LENGTH was once measured in cubits and amounts of liquid were measured in firkins. Units that everyone measures in at a time are called the standard units for that time. Here are some standard units we use today: length

AREA

kilometre METRE

km

millimetre

square metre

m

square centimetre

m

mm

I km = 1,000m

weight

VOLUME 2

cubic metre m cm2

3

tonne cm3

cubic centimetre litre

2

2

Im = 10,000cm

kilogram gram

3

I litre = 1,000cm

kg

g

I tonne = 1,000kg

I m = 1,000mm The CENTIMETRE (cm)

is also used. I m = 100cm

STRATECIES

65

Survey (also see DATA) A survey looks at the whole of something in order to collect information about various parts of it. You could do a survey of all the cars that passed the school gate for an hour to find out how many of each colour there were. You could do a survey of the pets that the pupils in your class have, to find out how many have a cat, how many a dog, etc. The information collected in a survey may be recorded in several ways, for example stored in a computer database or shown on a GRAPH.

Symbol A symbol is a short, easy way of writing something. Numerals are symbols for NUMBERS, for example 5 is the symbol for five 20 is the symbol for twenty. There are plenty of other symbols used in MATHEMATICS, for example +

is the symbol for ADD

X

is the symbol for MULTIPLY

=

is the symbol for EQUALS

J

is the symbol for SQUARE ROOT.

If one hundred and three plus five equals one hundred and eight is written in symbols, it becomes 103 + 5 = 108

66

STRATEGY

Symmetry

(also see LINE SYMMETRY, PLANE SYMMETRY and ROTATIONAL SYMMETRY)

A shape has reflection symmetry when one half of the shape is a reflection of the other half.

This PLANE shape has

This solid shape has

LINE SYMMETRY.

PLANE SYMMETRY.

A shape has ROTATIONAL SYMMETRY if it looks exactly the same in different positions when rotated about a POINT or about a straight LINE.

Ten

(also see HUNDRED)

Ten is the BASE we usually COUNT in, so ten is the first counting NUMBER that does not have its own SYMBOL. Because ten is the base of our number system, if a DIGIT is moved one place to the left, it becomes worth ten times as much as it was worth in its original place. The 3 in 1320 is worth ten times as much as the 3 in 132. Because ten is the base number, MULTIPLYING or DIVIDING by 10 is easy to do. For xlO, move figures one place to the left.

STRATEGIES

For + 10, move figures one place to the right.

67

Tessellate

(also see PATTERN and TILING)

A shape tessellates if a NUMBER of exact copies of the shape will fit together, with no gaps or overlaps, to make a PATTERN that is INFINITE.

Parts of some tessellations of single shapes

It is possible to have two or more different shapes that tessellate together.

Parts of some tessellations

Shapes that tessellate together are often used for tiling surfaces.

Translation

(also see ROTATION and PATTERN)

A translation moves a shape from one place to another, just by sliding it. When this cat is moved, it is translated. It is not turned. It does not change its size. Translation is often used in PATTERNS. A basic design is translated to make the pattern.

68

CTRATEGIE:

trapeziu (also seeQUADRILATERAL, PARALLEL,

parra;;;eprpam

A trapezium is a QUADRILATERAL with a pair of PARALLEL sides.

This is a right-angled trapezium.

This trapezium is a SQUARE.

This has a LINE OF SYMMETRY, so it is an isoceles trapezium,

This RECTANGLE JS

a trapezium. The name for the trapeze that is used in circus acts comes from trapezium.

Tree diagram

(also see NETWORK)

A tree diagram, like a tree, has branches. The branches keep splitting, and cannot join up again. A tree diagram is used to show what may happen when there are a number of possibilities. Each branch splits into further branches, according to how many possibilities there are. By looking at the tree diagram, you can see how many different combinations of possibilities there are.This example is a binary (two-way branching) tree, with two new possibilities from each question. So there are four possible combinations altogether from the two questions. If another binary (Yes or No) question was now asked, the new tree would have eight combinations. CTRATEGILS

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Trial (and improvement) Trial (and improvement) is a way of solving problems. It works like this: First, choose a solution to your problem and try it. If your first solution isn't right, use what you have found out to choose another. Go on trying solutions until you get one that is correct, or close enough. We could use trial and improvement to solve this problem about a RECTANGLE: A rectangle has a perimeter of 20cm and an area of 20 cm2.What are its length and breadth? To have a PERIMETER of 20cm, the LENGTH and breadth must add up to I Ocm. Try length 6cm and breadth 4cm.The AREA is 24cm2.This area is too large. Try length 7cm and breadth 3cm.The area is 21 cm2.This area is still too large, but much closer. Try length 8cm and breadth 2cm.The area is 16cm2. Now the area is too small. So the length needs to be between 7cm and 8cm, but nearer to 7cm. A good size for the next trial would be 7.2cm. With just a few more trials, we can find quite a good answer to the problem.

Triangle

(also see POLYGON)

A triangle is a POLYGON with three sides. Different kinds of triangles have special names.

A scalene triangle has three sides of different lengths.

An isosceles triangle has at least two sides of the same length. An equilateral triangle has all three sides the same length and all three ANGLES are 60°. It is therefore a regular polygon. A right angled triangle has one angle of 90°. It can be isosceles or scalene, like these.

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Unit

(also see AREA, CAPACITY and VOLUME)

A unit means one of something. When we say that the LENGTH of the line above is 8 CENTIMETRES, the unit we are using is km. We could also say that the length is 80 millimetres by using a unit of I mm. Whenever we MEASURE, it is important to say what unit we have chosen as our measurement of one.

In 243 the 2 is two HUNDREDS, the 4 is four TENS and the 3 is three ones. The 3 is called the 'units figure'.

We use different kinds of units to measure: AREA

VALUE

VOLUME

CAPACITY

Value The value of something is what it is worth. In MATHEMATICS, value is numerical: it relates to a quantity of something. The NUMBERS 271, 337 and 718 all have a 7 in them - but it has a value of seventy in 271, seven in 337 and seven hundred in 718.

Prices on goods show numerical values. If 2 + A = 6 then we know that the value of A is 4. CTRATEGIES

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Volume

(also see CAPACITY)

The volume of a solid shape is the amount of space it takes up. Volume is often MEASURED using CENTIMETRE CUBES, which have a volume of one cubic centimetre (km3). Bigger volumes can be measured in cubic METRES (m3). Shapes that cannot be measured using centimetre cubes are still measured in cubic centimetres and cubic metres. This shape is made up of 40 centimetre cubes. Its volume is 40 cubic centimetres. It can be seen as two layers which each have 4 x 5 or 20 centimetre cubes.

The space taken up by I cubic centimetre of water (at room temperature) is I millilitre (or thousandth of a litre).We use litres as a measure of CAPACITY.

Zero (also see NUMBER) Zero is another name for nothing or nought. On a NUMBER LINE it is the point where numbers change from positive to negative.

The zero, or nought, in 407 shows that there are no TENS. The zero in 30 shows that there are no UNITS. The zero in 1,037 shows that there are no HUNDREDS. When you ADD zero to another number, it does not change that number and if you SUBTRACT a number from itself, the result is zero. If you MULTIPLY any number by zero, the answer will be zero. The rules of ARITHMETIC do not allow you to DIVIDE by zero.

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