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Numbers (Chapters 1 & 3) Name Ten Hundred Thousand Ten thousand Hundred thousand Million Billion Trillion
Number 10 100 1 000 10 000 100 000 1 000 000 1 000 000 000 1 000 000 000 000
Power of ten 101 102 103 104 105 106 109 1012
The number of zeros (also called noughts) present is written as the power. In large numbers digits are grouped in threes. Example: Consider the number 673 852. This number can be written in columns in the following way: Hundreds of Thousands 6
Tens of Thousands 7
Thousands 3
Hundreds 8
Tens 5
Units 2
So the number is made up of or can be split into: Six hundred thousand Seventy thousand Three thousand Eight hundred Fifty Two
600 000 70 000 3 000 800 50 + 2 673 852
And is said in words as: Six hundred seventy-three thousand eight hundred fifty-two. Formula One Maths, Exercise, Pg. 3 Multiplying and Dividing Numbers with Lots of Noughts Example: 330 × 200 000 This can also be written as 330 × 2 × 100 000 (because 200 000 = 2 × 100 000) First multiply 330 × 2 = 660
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Now multiply 660 × 100 000 = 66 000 000 (sixty-six million) Example: 6 300 ÷ 300 This can also be written as 6 300 ÷ 3 ÷ 100 First divide 6 300 ÷ 3 = 2 100 Now divide 2 100 ÷ 100 = 21 Summary: To divide by ten remove a nought To divide by hundred remove two noughts To divide by thousand remove three noughts
To multiply by ten add a nought To multiply by hundred add two noughts To multiply by thousand add three noughts
Formula One Maths, Exercise, Pg. 5 ST(P) Mathematics 1A, Exercise 2b, Pg. 14 ST(P) Mathematics 1A, Exercise 2g, Pg. 19 The Metric System Prefix Giga Mega Kilo Hecto Deca Basic unit Deci Centi Milli Micro Nano
Symbol G M k
c m µ η
Size × 109 (or 1 000 000 000) × 106 × 103 × 102 × 101 ×1 × 1/10 × 1/100 × 1/1 000 × 1/1 000 000 × 1/1 000 000 000
How the prefix is used
1 kilogram = 1 000 grams 1 hectare = 100 ares (1 are = 100 square metres) Metre, litre, gram, are 1 decilitre = 1/10 litre 1 centimetre = 1/100 metre 1 millimetre = 1/1 000 metre 1 micrometre = 1/1 000 000 metre 1 nanometre = 1/1 000 000 000 metre
Formula One Maths, Exercise, Pg. 7 Formula One Maths, Review Exercise, Pg. 8, 9 Addition and Subtraction ST(P) Mathematics 1A, Exercise 1b, Nos.41-45, Pg. 3 ST(P) Mathematics 1A, Exercise 1e, Nos.31-36, Pg. 6 ST(P) Mathematics 1A, Exercise 1g, Pg. 6
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ST(P) Mathematics 1A, Exercise 1h, Nos. 26-30, Pg. 8 ST(P) Mathematics 1A, Exercise 1i, Nos. 6-9, Pg. 8 Short Multiplication and Division ST(P) Mathematics 1A, Exercise 2a, Nos. 43-48, Pg. 13 ST(P) Mathematics 1A, Exercise 2f, Nos. 36-45, Pg. 18 Long Multiplication and More Division Example: The Patel family are taking a 16-day holiday costing €128 per day. How much is this altogether?
128 × 10 = 128 × 6 =
128 × 16 1280 + 768 € 2048
They look at another holiday. It is €910 for 14 days. Rita works out how much this is per day. 65 14 |910 140 – 770 700 – 70 28 – 42 42 0
Adding up all the multiples of 14 in the previous column: 10 × 14
10 + 50 + 2 + 3 = 65
50 × 14 2 × 14 3 × 14
ST(P) Mathematics 1A, Exercise 2c, Nos. 21-30, Pg. 15 ST(P) Mathematics 1A, Exercise 2e, Nos. 8-10, Pg. 16 ST(P) Mathematics 1A, Exercise 2h, Nos. 41-48, Pg. 20 Multiples and Divisibility Tests The multiple of a number is that number multiplied by any other number. All even numbers are divisible by 2. Even numbers end with multiples of 2 (0, 2, 4, 6 or 8). All numbers ending with a 5 or 0 are divisible by 5.
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All numbers ending with a 0 are divisible by 10. If a number is divisible by 3, you get a multiple of 3 when you add the single digits of that number. Example (Proof): Consider the number 3 852: 3 852 = 3 × 1 000 + 8 × 100 + 5 × 10 + 2 (every number is written in of powers of 10) = (3 × 999 + 3) + (8 × 99 + 8) + (5 × 9 + 5) + 2 (we can do this because 1 000 = 999 + 1, 100 = 99 + 1 and 10 = 9 + 1) = (3 × 999 + 8 × 99 + 5 × 9) + (3 + 8 + 5 +2) (999, 99 and 9 are all divisible by 3, so the first bracket is divisible by 3, the second bracket is the sum of the digits) Since 3 + 8 + 5 +2 = 18, which is divisible by 3, it follows that 3 852 is divisible by 3. If a number is divisible by 9, you get a multiple of 9 when you add the single digits of that number. A number can be divided by 4 exactly, if 4 goes exactly into the last two digits (4 will go into any number of hundreds since 100 is a multiple of 4, so we only need to know if 4 will go into the tens and units). A number can be divided by 8 exactly, if 8 goes exactly into the last three digits (8 will go into any number of thousands since 1 000 is a multiple of 8, so we only need to know if 8 will go into the hundreds, tens and units). If a number can be divided exactly by 2 AND by 3, then it can be divided exactly by 6. This is because 6 = 2 × 3. A number is divisible by 11 if the difference of the sums of alternating digits is divisible by 11. Example: 946 → (9 + 6) - 4 = 11 which is, of course, evenly divided by 11 so 946 es this divisibility test 10 813 → (1 + 8 + 3) - (0 + 1) = 12 - 1 = 11. 25 784 = (2 + 7 + 4) - (5 + 8) = 13 - 13 =0 . Yes, this does indeed work. In case you found this last bit confusing, that any number evenly divides 0. Think about it, how many 11's are there in 0? None, right. Well that means that 11 divides zero, zero times! 119 777 658 → (1 + 9 + 7 + 6 + 8) - (1 + 7 + 7 + 5) = 31 - 20 = 11
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Negative Numbers (Chapter 18) Numbers can be shown on a number line. Numbers like …, -3, -2, -1, 0, +1, +2, +3, …, are called integers. Example:
Brackets () mean × +×+=+ + × - = - and - × + = -×-=+
(+) (+) = + (+) (-) = - and (-) (+) = (-) (-) = +
Formula One Maths, Exercise, Pg. 157, Nos. 2,4,5 Subtraction of Negative Numbers So from the above list, 5 - (-1) = 6 because a negative sign (-) multiplied by another negative sign gives a positive sign (+). And 2 - 9 = -7. Formula One Maths, Exercise, Pg. 159 Multiplying and Dividing Negative Numbers Multiplication of division of negative numbers is done by first multiplying the numbers on their own, regardless of the sign in front of them. Then the signs are multiplied separately according to the above list and the resulting sign applied to the answer. Example:
(-4) × (-6) is worked out by first multiplying 4 × 6 = 24 and then multiplying (-) × (-) = +. So the answer is (-4) × (-6) = (+24) (-5) × (+6) is worked out by first multiplying 5 × 6 = 30 and then multiplying (-) × (+) = -. So the answer is (-6) × (+6) = (-30)
Formula One Maths, Exercise, Pg. 161 Formula One Maths, Review Exercise, Pg. 162, 163 5
Fractions (Chapter 9) A clock’s face is divided into 12 parts so the difference between the two times shown on the two clocks is of 1/12 of an hour.
There are 60 minutes in an hour:
Of means × Formula One Maths, Task, Pg. 72 Philip eats two pieces of cake at tea-time. This means he has eaten a quarter of the cake. So = The following fractions are the same or equivalent:
You can cancel as often as you like.
Whatever you do to the numerator (multiplication or division or cancelling) you must also do to the denominator! Formula One Maths, Exercise, Pg. 73 ST(P) Mathematics 1A, Exercise 3b, Nos. 16-19, Pg. 36 ST(P) Mathematics 1A, Exercise 3e, Nos. 25-30, Pg. 43 6
Adding and Subtracting Fractions Jared eats of a cake. He then goes back for ‘seconds’ and eats another of the cake. How much of the cake has he eaten altogether?
12 is called a common denominator, it is in fact the lowest common denominator (24 or 36 or any multiple of 12 could also have been used as the common denominator but this would involve bigger numbers and cancelling. Jacinta later eats of the cake. What fractions of the cake remains? The fraction left is
- . The lowest common denominator is 5 × 12 = 60.
You can now write: When you add and , the answer is . is called a top-heavy or improper fraction. You can write as
. Now it is a mixed number.
In order to perform multiplication and division on mixed numbers, first turn them into improper fractions. For addition and subtraction, the whole numbers can be worked out first and then the fractions. Formula One Maths, Exercise, Pg. 77 ST(P) Mathematics 1A, Exercise 3t, Pg. 58
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Multiplying Fractions Sally’s mother cuts a pizza into 2 halves. She then cuts each half piece into 3 slices. She has cut the whole pizza into 6 pieces, so each slice is one-sixth, or . In fact she did of = × =
The next day, Sally’s mother cuts a quiche into 3 equal pieces. She puts 2 of these pieces into Sally’s lunch-box. At school, Kevin has forgotten his lunch. Sally gives him one of her pieces of quiche. Kevin and Sally now each have one-third, or , of quiche.
Whatever you do to the numerator (multiplication or division or cancelling) you must also do to the denominator! This rule also applies to fractions which are being multiplied, but only one number can be cancelled at a time from the top and from the bottom. For example if we are dividing top and bottom by 7, we only divide one number from the top by 7 and not all the numbers at the top and we only divide one number from the bottom by 7 and not all the numbers at the bottom. Formula One Maths, Exercise, Pg. 75 8
ST(P) Mathematics 1A, Exercise 4b, Nos. 30-36, Pg. 61 ST(P) Mathematics 1A, Exercise 4c, Nos. 26-30, Pg. 62 A whole number can be written as a fraction with a denominator of 1. For instance 6 = . Doing this makes it easier to multiply a whole number by a fraction or a mixed number. Dividing by Fractions When we divide 6 by 3 we are finding how many threes there are in 6 and we say 6 ÷ 3 = 2. In the same way, when we divide 10 by we are finding how many halves there are in 10; we know that there are 20, so we say 10 ÷ = 20. But we also know that 10 × 2 = 20 so
and
This example shows that To divide by a fraction we turn that fraction upside down and multiply. ST(P) Mathematics 1A, Exercise 4f, Nos. 6-16, Pg. 65 ST(P) Mathematics 1A, Exercise 4g, Nos. 16-20, Pg. 67 Introduction to Percentages Percentage means out of 100 (from Italian; per cento) and is written as %. 20% means
.
Percentages can be written as fractions and cancelled in the same way to get them in their lowest .
You can find a percentage of a quantity by first changing the percentage into a fraction and then working it out normally:
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The whole amount is called 100%. Example: 82% of apples in a box are good, ripe apples. What percentage of the apples is not fit to eat? (100 – 82) % = 18% of the apples are not fit to eat. Formula One Maths, Exercise, Pg. 79 Ratio and Proportion Mortar is made up of 1 part cement to 4 parts sand, this can be written as the ratio 1 : 4 (reads as 1 is to 4). Order matters with ratios. Sand : cement = 4 : 1 But cement : sand = 1 : 4 If cement comes in 50 kg bags, how much sand should be mixed with it? If we decide that 1 part = 50 kg 4 parts = 4 × 50 = 200 kg. Therefore 200 kg of sand are needed. If 1250 kg of mortar are needed altogether we can say that 5 parts altogether ( 1 part + 4 parts) will be 1250 kg. So 1 part is
= 250 kg.
So 250 kg of cement are needed and 250 × 4 = 1000 kg of sand. Ratios can be cancelled down like fractions. Example: 2 : 8 = 1 : 4 (cancelling by 2) Example: 3 : 12 : 15 = 1 : 4 : 5 (cancelling by 3) Formula One Maths, Task, Pg. 80 Formula One Maths, Exercise, Pg. 81 Formula One Maths, Review Exercise, Pg. 82, 83
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Decimals (Chapter 7 & 13) Reading a scale with divisions:
Check to see in between which numbers the reading lies. Subtract the smaller of these two numbers from the biggest of these two numbers. Count the number of divisions (or intervals) between the two numbers. Divide the result obtained from the subtraction by the number of divisions between the two numbers. This will give you how much a division (or interval is worth).
Example: The reading lies in between the numbers 30 and 40. 40 - 30 = 10 The number of divisions between the two numbers is 5. 10 ÷ 5 = 2 So each division is worth 2. The dial reads 34 km/h. Example: The reading lies in between the numbers 69 and 70. 70 - 69 = 1 The number of divisions between the two numbers is 10. 1 ÷ 10 = 0.1 So each division is worth 0.1. The tape measure reads 69.3 cm. Formula One Maths, Exercise, Pg. 57, Nos. 2-5 Decimal Notation The decimal point system is a way of writing numbers with tenths, hundredths, thousandths, etc. This table shows how:
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Fraction 3/10 7/100 6 + 5/10 + 1/100 527 39/100
H hundreds
5
T tens
2
U Units
6 7
. . . .
1/10 tenths 3 5
1/100 1/1000 Decimal hundredths thousandths 0.3 7 0.07 1 6.51 39 527.039
To change a fraction to a decimal, always convert the fraction to one which has 10, 100 or 1000, etc. in the denominator. Formula One Maths, Exercise, Pg. 59 Addition and Subtraction of Decimals Addition and subtraction with decimals is just as easy as it is with whole numbers; all one has to do is to place the decimal points underneath each other when working! Example: Work out 7.53 + 420 + 0.086
Subtract 8.306 from 46.1
Zeros can be written down at the end of the number after the decimal point to help you work the sums out. Formula One Maths, Exercise, Pg. 61 Formula One Maths, Review Exercise, Pg. 62, 63 ST(P) Mathematics 1A, Exercise 5b, Nos. 9-12, Pg. 77, Nos. 21-24, Nos. 33-36, Pg.78 ST(P) Mathematics 1A, Exercise 5f, Nos. 16-28, Pg. 82-84
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Multiplying and Dividing Decimals (Chapter 13) To multiply decimals, first multiply the numbers as if there was no decimal point. Then count the numbers after the decimal point of each number and add them up. The number of numbers after the decimal point of the answer is the same number as the sum you obtained before. Example:
Formula One Maths, Exercise, Pg. 117, Nos. 1-6 Division of Decimals by a Whole Number Dividing decimals by whole numbers is done normally but the decimal point has to go down one underneath the other. You can add noughts in a s many decimal places as you need. Examples:
Sometimes when dividing one may obtain what is called a recurring number which is a number whose last digit or few digits are repeated for an infinite number of times. To indicate recurrence we write the recurring number only once and put a dot on it. When a group of digits recur, the dots are placed over the first and last recurring digits.
Formula One Maths, Exercise, Pg. 119 ST(P) Mathematics 1A, Exercise 6e, Nos. 1-12, Pg. 97 13
Division of Decimals (Short Division) In order to divide values, they must be in the same units. So if we are dividing amounts of money, both values must be in cents or both values must be in €. To divide a decimal by another decimal, multiply the decimals by 10 or by 100 or by 1000 (and so on) in such a way that the number you are dividing by is a whole number. Example:
Formula One Maths, Exercise, Pg. 121 Equivalent Percentages, Fractions and Decimals Percent (%) means out of 100. 80% means To change from fractions to percentages, multiply by 100.
To convert a decimal into a percentage, you multiply by 100.
Formula One Maths, Task, Pg. 122 Formula One Maths, Exercise, Pg. 123 ST(P) Mathematics 1A, Exercise 6h, Nos. 1-10, Pg. 101 Formula One Maths, Review Exercise, Pg. 124-125 ST(P) Mathematics 1A, Exercise 6k, Nos. 1-12, Pg. 104 14
Everyday Measures (Chapter 11) Length The metric system uses millimetres (mm), centimetres (cm), metres (m) and kilometres (km). Metric Measures 1 000 mm = 1 m 100 cm = 1 m
1 000 m = 1 km 10 mm = 1 cm
also
Measuring – To the Nearest cm and the Nearest mm The ruler shows that the length of the line is a bit more than 5 cm. The length of the line is 5 cm (to the nearest cm because the length is not equal to or does not exceed 5.5 cm) or 54 mm or 5.4 cm (to the nearest mm).
Formula One Maths, Exercise, Pg. 97, Nos. 2-5 Mass (or weight)
Volume
Metric Measures 1 000 milligrams (mg) = 1 g 1 000 grams (g) = 1 kilogram (kg) 1 000 kg = 1 tonne
Metric Measures 1 000 millilitres (ml) = 1 litre 100 centilitres (cl) = 1 litre
Formula One Maths, Exercise, Pg. 99, Nos. 1, 2, 3a, 4, 5, 6 Time
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60 seconds = 1 minute 60 minutes = 1 hour
Units of Time 24 hours = 1 day 365 days = 1 year (except in a leap year)
Timetables
Formula One Maths, Exercise, Pg. 101 Formula One Maths, Review Exercise, Pg. 102, Nos. 4-7
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Number Patterns (Chapter 10 &23) Factors 6 × 3 = 18 9 × 2 = 18 We say that 2, 3, 6 and 9 are factors of 18. Multiples 1×6=6 2 × 6 = 12 3 × 6 = 18 4 × 6 = 24 and so on. We say that 6, 12, 18, 24, … are multiples of 6. They are the answers to the 6 times table. The multiples of 2 are called even numbers Numbers which are not multiples of 2 are called odd numbers. Formula One Maths, Exercise, Pg. 85 Squares The numbers 1, 4, 9, 16, … are called square numbers. Square numbers are the answers when a number is multiplied by itself. 12 = 1 × 1 = 1 22 = 2 × 2 = 4 32 = 3 × 3 = 9 42 = 4 × 4 = 16 And so on. Square numbers make up squares.
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Square Roots 1 is called the square root of 1 2 is called the square root of 4 3 is called the square root of 9 4 is called the square root of 16 And so on. A quick way to write square root is to use the sign √ So: 5 × 5 = 25 √25 = 5 Formula One Maths, Exercise, Pg. 87, Nos. 1,2, 6, 7, 8 Primes Every whole number greater than 1 has at least two factors, the number itself and 1. A number which has exactly two factors is called a prime number. The smallest prime number is 2, and the next ones are 3, 5, 7 and 11. The number 180 is not prime since it has factors other than 180 and 1. For example 180 = 18 × 10. These factors have factors themselves: 18 = 2 × 9 = 2 × 3 × 3 and 10 = 2 × 5. So 180 = 2 × 3 × 3 × 2 × 5 = 22 × 32 × 5. We have written 180 as the product of prime factors. A number may have many sets of factors, but it has only one set of prime factors. Formula One Maths, Exercise, Pg. 89, No. 4 Cubes A centicube is a 1 cm by 1 cm by 1 cm cube. You can fit centicubes together to make cuboids. The first cuboid has 2 × 5 × 4 = 40 centicubes. A cube is a solid with all sides the same length. 18
If we make a cube out of these centicubes we will have used 3 × 3 × 3 = 27 centicubes and 13 centicubes will be left over. Now these 13 leftover centicubes can make up another cube of 2 × 2 × 2 = 8 centicubes and then only 5 centicubes will be leftover. The second cuboid has 3 × 6 × 2 = 36 centicubes. If we make a cube out of these centicubes we will have used 3 × 3 × 3 = 27 centicubes and 9 centicubes will be left over. Now these 9 leftover centicubes can make up another cube of 2 × 2 × 2 = 8 centicubes and then only 1 centicube will be leftover. Numbers like 1, 8, 27, 64, … are called cube numbers. 13 = 1 × 1 × 1 = 1 23 = 2 × 2 × 2 = 8 33 = 3 × 3 × 3 = 27 43 = 4 × 4 × 4 = 64 And so on. A cube may only be made by certain specific number of cubes. Cube numbers make up cubes. 1 is the cube root of 1 2 is the cube root of 8 3 is the cube root of 27 4 is the cube root of 64 Formula One Maths, Exercise, Pg. 91, Nos. 3, 4 Powers of 2 A certain type of bacteria grows by splitting into 2 every hour.
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These are the numbers of bacteria present each hour: 2, 4, 8, 16, 32, … This sequence of numbers has been obtained by doubling to get from one number to the next. The numbers are all powers of 2. The mathematical way of writing the powers of 2 is: 21 = 2 22 = 2 × 2 = 4 23 = 2 × 2 × 2 = 8 24 = 2 × 2 × 2 × 2 = 16 Formula One Maths, Review Exercise, Pg. 95-86
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Flow Diagrams and Number Patterns A decision tree diagram is a diagram which asks questions to help you sort information. Different shapes can be used so that one does different actions. Flow Charts and Number Patterns A flow chart is a list of instructions and questions which help you carry out a task. We can use flow charts to investigate number patterns. Flow charts can be used to make number chains.
Figure 1 flow chart for odd numbers
Figure 2 Flow Chart for even numbers
Working Systematically with Patterns One layer contains 1 ball Two layers contain 3 balls Three layers contain 6 balls Four layers contain 10 balls And so on. Add 2 to go from the first to the second layer. Then add 3 to go from the second to the third layer. Then add 4 to go from the third to the fourth layer.
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Growth Patterns The first three stages of a growth pattern are shown below:
Two matches are added every time to make a new stage. Table for growth pattern: Number of stages 1 2 3 4 5 6
Number of matches 3 5 7 9 11 13
Formula One Maths, Exercise, Pg. 205 Number Patterns – Making Predictions To build a fence one needs bars and posts. For one section, 2 posts and 2 bars are needed. For two sections, 3 posts and 4 bars are needed.
The following is the mapping for the number of posts and number of bars needed to build sections of the fence:
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Rules can be created from the mapping:
p stands for posts, s stands for sections and b stands for bars. Formula One Maths, Exercise, Pg. 207 Formula One Maths, Review Exercise, Pg. 208-209 ST(P) Mathematics 1A, Exercise 2l, Nos. 18, 20, 21, 25, 27, 29, 30 33, 34 35, Pg. 27-30
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Number Machines (Chapter 14) Making a phone call from a phone box costs 8cent per minute plus a 6cent connection charge. The cost of a call is calculated using a number machine like this:
The number of minutes that one is on the phone is called the input. The cost of the call is called the output. Given the output, the number machine can be reversed to find out what the input was.
The parts of the machine cannot be swapped.
Formula One Maths, Exercise, Pg. 127 Finding the Rule Following are the inputs and outputs of a number machine:
The rule chosen for the number machine must always give the correct output. The rule in this case is:
Formula One Maths, Exercise, Pg. 129
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Writing down the Rule Following are the inputs and outputs of a number machine:
The number machine can be drawn like this:
And can be written in words like this:
An easier way of writing this is:
The rule can be checked by working a few outputs for some inputs. Formula One Maths, Exercise, Pg. 131 Formula One Maths, Review Exercise, Pg. 132-133
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Averages (Chapter 16) The Mean To find the mean of a group of values; add up all the values and divide by the number of values. Formula One Maths, Exercise, Pg. 143 The Median To find the median, first write the values in order of size and then find the middle value. If there are two middle values, the median will be the mean of those two middle values Formula One Maths, Exercise, Pg. 145 The Mode The most popular value in the set of data is the mode (the one which occurs most often). When data is grouped, the group or class with the most items is called the modal class. In a bar chart or frequency chart, the modal class has the highest bar. The modal class in this bar chart is 11-15.
The Range The mean, median and mode are all ways of finding an average, or typical value. The range is the difference between the largest and smallest values of a set of data. The range is not an average. It tells you how spread out the data is. Formula One Maths, Exercise, Pg. 147 Formula One Maths, Review Exercise, Pg. 148-149
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Co-ordinates (Chapter 8) Positions on a grid can be referred to by two values separated by a comma and enclosed in brackets: (along, up). (along, up) is in alphabetical order. The first number tells you how many values you have to move along (horizontally) while the second value tells you how many values you have to move up (vertically). Formula One Maths, Exercise, Pg. 65 Using Co-ordinates in Mathematics In mathematics, co-ordinates are plotted on a pair of axes called the x and y axes. The horizontal axis is called the x axis while the vertical axis is called the y axis. The x co-ordinate is always given first, followed by the y co-ordinate. In general form this can be written as: (x, y) (x, y) is in alphabetical order. The point (0,0) has a special name. It is called the origin. Any point can be plotted using co-ordinates. Formula One Maths, Exercise, Pg. 67 Using Negative Numbers, Fractions and Decimals There are negative (minus) co-ordinates, to the left for x and down for y. Co-ordinates do not necessarily have to be whole numbers but can be fractions or decimals. Formula One Maths, Exercise, Pg. 69 Formula One Maths, Review Exercise, Pg. 70-71
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Scale (Chapter 15) Scales are used when a model or a picture is much smaller than the object in real life. They are written as: Scales can also be written as fraction in the form of: Formula One Maths, Exercise, Pg. 135 Drawing Plans If you want to fit the plan of a bedroom which is 4m long on a paper which is only 10 cm long you use a scale of 2 cm is to 1 m. This leads to a ratio of: 2 cm : 1 m 2 cm : 100 cm 2 : 100 1 : 50 If the bed is 1.8 m long, on the plan this is 2 × 1.8 cm = 3.6 cm. The ratio tells us that the real measurements are 50 times as big as the measurements on the plan. Formula One Maths, Exercise, Pg. 137 Map Scales 1 centimetre to the kilometre 1 : 25000
These are all examples of the way that a scale might be given on a map or in an atlas. A scale of 1 : 50 000 tells us that any length on the map is 50 000 times as big in real life. 1 cm on the map is equal to 50 000 cm in real life. 50 000 cm = 500 m = 0.5 km A scale of 1 : 50 000 is the same as 1 cm to 0.5 km.
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So if a length on the map measures 6.5 cm: 6.5 × 50 000 cm = 325 000 cm 325 000 cm = 3250 m = 3.25 km. If a place is less than 1 km away from another place: 1 km = 1000 m 1000 m = 100 000 cm 100 000 cm on the ground = 100 000 ÷ 50 000 = 2 cm on the map. Therefore the two places should be less than 2 cm away from each other on the map. Formula One Maths, Exercise, Pg. 139 Formula One Maths, Review Exercise, Pg. 140-141, Nos. 1-6.
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Displaying Data (Chapter 5) Data has to be displayed in an organized way so that one can make sense of it. Tally Charts One way of collecting data is to use a tally chart.
Frequency means the number of people. This is sometimes called a frequency table. Formula One Maths, Exercise, Pg. 39 Pictograms and Bar Charts A bar chart can also be drawn from a frequency table. It is important to always label the axes and give the chart a title.
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Pictograms can also be used; you can use any symbol you like for the picture.
Formula One Maths, Exercise, Pg. 41 Pie Charts
The angles of each section of a pie chart can tell us how much of the total that value is. Divide the value of that angle by 360° and reduce the fraction to its lowest to get the fraction.
Formula One Maths, Exercise, Pg. 43 Grouping Data Data can first be grouped into sets and then those sets can be plotted on one of the types of graphs mentioned above. Each set will consist of a range of values. Formula One Maths, Exercise, Pg. 45 Formula One Maths, Review Exercise, Pg. 46-47. 31
Angles (Chapter 4) Types of Angle
To measure angles you need a protractor.
It is important to be accurate when measuring angles.
The centre of the protractor must be on the centre of the angle. The start line must be lined up exactly over one of the angle lines. Your eyes must be directly over the protractor when reading the scale.
There are two sets of numbers round the protractor so you can measure angles on both sides. You use the set that is starting from zero in the angle you are measuring. There are 90° in a right angle. A straight line is 180°. A whole turn is 360°. You can use a protractor to measure an obtuse angle by first measuring up to 90° and making a mark and then measuring the rest of the angle from the mark. The value obtained should be added to the 90°. The internal angles of a triangle always make up 180°. Formula One Maths, Exercise, Pg. 29 32
Drawing Angles Protractors can be used to draw angles accurately. Always use a sharp pencil to angles and always keep your eyes over drawing. The following steps explain how to an angle of 57°.
very draw your draw
Formula One Maths, Exercise, Pg. 31 Angle Calculations As stated previously, there are 360° in a whole turn. It follows that a half-turn, or a straight line, is 180°. A right angle is 90°.
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A, B, C and D together make up 360°. Angles X and Y make up 180°. Two angles that add up to 180° are also called supplementary angles. Parallel Lines When two lines are always the same distance apart (equidistant), they are parallel. Railway lines are parallel. In the following diagrams, angles are made where lines intersect.
Formula One Maths, Exercise, Pg. 33 Angles in a Triangle The angles inside a shape are called interior angles. The interior angles of every triangle add up to exactly 180°. This statement can be proved as follows: Draw the line through one vertex (corner) of the triangle which is parallel to the opposite side. Then X = B and Y = C (*) by alternate angles But A + X + Y = 180° since these are adjacent angles (angles lying on a straight line). Therefore A + B + C = 180°.
Any shape with four straight sides is called a quadrilateral.
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If a diagonal line is drawn across a quadrilateral between two opposite corners, two triangles are formed. Hence the internal angles of a quadrilateral make up 180° + 180° = 360°. Formula One Maths, Exercise, Pg. 35 Formula One Maths, Review Exercise, Pg. 36-37 ST(P) Mathematics 1A, Exercise 9m, Pg. 145-146 ST(P) Mathematics 1A, Exercise 9n, Pg. 146-147 ST(P) Mathematics 1A, Exercise 15k, Pg. 234, Nos. 1-8
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Symmetry (Chapter 6) The Town Hall is symmetrical. Many buildings are designed like this. If you stand a mirror on the centre of the drawing, in line with the flagpole, the Town Hall will look exactly the same in the mirror as it does behind the mirror. This kind of symmetry is called reflection symmetry.
The green cross, seen on first aid kits, has four lines of symmetry. Two of them are diagonal, one is horizontal and one is vertical.
You can use squared or dotted paper to complete symmetrical shapes. Here is a shape with two lines of symmetry.
Formula One Maths, Exercise, Pg. 49 Rotation Symmetry Here is a drawing of a flag. There are two positions the flag can be in where it looks exactly the same. We say that the flag has rotational symmetry of order two. The point about which the flag rotates is the centre of rotation symmetry and is found at the centre of the flag itself.
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There are four positions where this square flag looks the same. This flag has rotational symmetry of order four. Its centre of rotation symmetry is found at the centre of the flag itself.
Some shapes have rotation symmetry and reflection symmetry. Formula One Maths, Exercise, Pg. 51 Paper Folding If you fold a paper once and cut out any shape, when you open up the paper, the shape will have reflection symmetry. But if you fold a paper once and cut out any shape, when you open up the paper, the shape will have reflection symmetry and rotation symmetry of order four. Formula One Maths, Exercise, Pg. 53, No. 4 Formula One Maths, Review Exercise, Pg. 54-55
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Flat Shapes (Chapter 12) Any shape made with straight sides is called a polygon. There are many different types of polygon. To describe them we give them special names.
Formula One Maths, Exercise, Pg. 105 A polygon has a number of sides. A point where two sides meet is called a vertex (plural vertices). In a quadrilateral, a diagonal s two opposite vertices. Classifying Triangles and Quadrilaterals
Formula One Maths, Exercise, Pg. 107 38
Circles A pair of comes are needed to draw a circle. The distance between the point and the pencil can be adjusted. The following steps show how to draw a circle.
The circle drawn in the steps above has a radius of length 6 cm. Its diameter is 12cm since the diameter is twice the radius.
Formula One Maths, Exercise, Pg. 109, Nos. 1-2.
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Constructing Triangles You need to use a pair of comes to make an accurate drawing of a triangle.
Making an accurate drawing like the one above is called a construction. The arcs and ‘extra bits’ of line are an important part of a construction. They must not be rubbed out! All the sides of an equilateral triangle are equal because the com was all the time open at the same length. All the angles of an equilateral triangle are also equal. Each angle is 60°. To draw an isosceles triangle, the same method can be used BUT the comes must be adjusted to a bigger length before starting Step 3. Formula One Maths, Exercise, Pg. 111, Nos. 1-6.
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Constructions with Angles
To construct an accurate drawing of this slide you need to follow the following steps.
The triangle constructed above has been drawn to a scale of 1 cm to 1m. Constructing an Angle of 90°
Formula One Maths, Exercise, Pg. 113, Nos. 1-6 Formula One Maths, Review Exercise, Pg. 114, Nos. 1-7 ST(P) Mathematics 1A, Exercise 11l, Pg. 178-179, Nos. 1-3 ST(P) Mathematics 1A, Exercise 11m, Pg. 179, Nos. 1-3 ST(P) Mathematics 1A, Exercise 11n, Pg. 180, Nos. 1-4
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Solid Shapes (Chapter 20) Here is a piece of card before it is folded to make a fruit pastilles packet. It is called a net. The shape of the packet will be a cuboid. The packet will have 6 faces, each face will have the shape of a rectangle. Where the card is folded there will be an edge of the packed. The packet will have 12 edges. 12 folds have to be made. The packet will have 8 vertices. This is an example of a mathematical net drawn on a centimetre squared paper. Flaps or tabs can be added to the net for sticking. The flaps do not necessarily have to be at the same locations. They can be arranged differently and one may have different numbers of flaps. This net should make a cuboid whose dimensions are 3 cm long by 2 cm wide by 2 cm high
Formula One Maths, Exercise, Pg. 173. Prisms Here is the card for a Toblerone packet.
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When folded and glued, it makes a triangular prism. A prism is called a polygon ‘stretched’ backwards to make a solid shape. A Toblerone backwards.
packet
is a
triangle
stretched
A triangular prism has 5 faces. 2 of them are triangles and the other 3 are rectangles. A triangular prism has 9 edges and 6 vertices. Shape Triangular prism Hexagonal prism Rectangular prism Pentagonal prism In general; V + F – E = 2
Number of faces (F) 5 8 6 7
Number of vertices (V) Number of edges (E) 6 9 12 18 8 12 10 15 This is called Euler’s Rule.
Further Solid Shapes There are many other solid shapes besides prisms. Here is a tetrahedron along with the net to make it. Tetra means four, hedron means face. The tetrahedron also obeys Euler’s Rule.
The five Platonic solids also obey Euler’s Rule. The mathematical name for the shape of a football is a sphere. Formula One Maths, Review Exercise, Pg. 178-179
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Measuring (Chapter 22) Perimeter The distance round the outside of a shape is called the perimeter. Area Area is the amount of flat space. The formula for the area of a rectangle is: A=L×B Where A is the area, L is the length and B is the breadth (the breadth is sometimes also called the width). Area is measured in square units e.g. m2. Volume Volume is the amount of solid space. The volume of a box or a cuboid can be found from the following equation: V=L×B×H Where V is the volume, L is the length, B is the breadth and H is the height. Volume is measured in cubic units e.g. cm3 Formula One Maths, Exercise, Pg. 189 More Areas Here is a rectangle with a triangle drawn inside it. The area of the rectangle is 8 cm × 5 cm = 40 cm2 The area of the triangle can be found by dividing the area of the rectangle by 2. The equation for finding the area of a triangle is:
Where b is the base and h is the height.
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If we want to find the area of an irregular shape such as the one shown, we have to divide the shape into simple areas for which we know how to find the areas.
This can be done in various ways as shown below:
All we have to do now is find the areas of the rectangles and then add them up. The surface area of a 3D object can be found by working out the area of each face and then adding up these areas. The surface area of the cuboid cardboard box is the total area of cardboard surrounding the volume.
Formula One Maths, Exercise, Pg. 191 Volume of Prisms Volume of prism = area of cross section × length
Formula One Maths, Exercise, Pg. 193 45
More Volumes The volume of a complicated shape can sometimes be found easily by taking the difference of the volumes of simple shapes. Formula One Maths, Exercise, Pg. 195 Formula One Maths, Review Exercise, Pg. 196-197 ST(P) Mathematics 1A, Exercise 14e, Pg. 209 ST(P) Mathematics 1A, Exercise 14f, Pg. 210-212
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Transformations (Chapter 26) An object is rotated when it moves along a circle. If it only moves along part of a circle, then it has moved along an arc. One point on the object remains fixed, the rest of the object rotates about this point. It is called the centre of rotation. The Plough is a constellation of stars. From midnight till 6 am, it rotated a quarter of a turn (90°) in an anticlockwise direction. To describe a rotation accurately, you must give the centre of rotation, the angle and the direction of turning. When an object rotates, its size and shape do not change. We say that the object remains congruent. Formula One Maths, Exercise, Pg. 229 Translation The movement of a shape in mathematics is called a transformation. Rotation is one kind of transformation. In translation, objects move without turning. When an object is translated, every point moves the same distance in a parallel direction. Object ABC is translated 5 squares to the right and 1 square up to become image A’B’C’. A maps to A’, B maps to B’ and C maps to C’. When an object is translated the shape and size remain the same. Formula One Maths, Exercise, Pg. 231
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Reflection Shapes can be transformed by reflection in a mirror. The figure below shows a shape before and after it has been reflected.
Objects are still congruent after having been reflected, that is, they retain their shape and size. To describe a reflection, you must say where the mirror line is. Formula One Maths, Exercise, Pg. 233 Formula One Maths, Review Exercise, Pg. 234-235
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Formulae (Chapter 17) You can give a formula in words or you can write it using letters. Examples of formulae:
To calculate the points for a team in a league: Points = 3 × W + 1 × D Where W stands for the number of wins and D stands for the number of draws. The cost to buy tickets: In words: The cost in euro is eight times the number of adults added to five times the number of children. In letters: P = 8 × a + 5 × c Where P is the cost in Euro, a is the number of adults and c is the number of children. It is common to leave the × signs out when you use letters. So the formula is P = 8a + 5c. This is called an expression. An expression has letters and usually some numbers too.
Example Find the value of 8a + 5c when a = 3 and c = 6. Solution: P = 8a + 5c P=8×3+5×6 P = 24 + 30 P = 54 Notice that you have to put the × signs in to work this out. When you write a formula in letters you are using algebra. The letters stand for the numbers. The numbers are not always the same. In the example above, a = 3, c = 6 is a party of 3 adults and 6 children. Formula One Maths, Exercise, Pg. 151 Formula One Maths, Exercise, Pg. 153 Formula One Maths, Review Exercise, Pg. 154-155
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Equations (Chapter 19) Balancing Equations Two identical packages are placed on some old-fashioned weighing scales. To know how much each package weighs one can use: The weight of two packages + 3 kg = 7 kg It would be quicker to write: 2 × p + 3 = 7 So: 2p + 3 = 7 This is called an equation – it only has one right answer. If 3 kg are removed from both sides of the scales, the scales still balance. When removing 3 kg from each side of 2p + 3 = 7, one can write: 2p = 4 One can work out how much one package weighs by dividing each side by two. Therefore: p = 2. Formula One Maths, Exercise, Pg. 165 Solving Equations
I think of a number and subtract 7. My answer is 4. To find out the number I thought about: Mystery number – 7 = 4 M–7=4 To find the mystery number add 7 to both sides of the equation: M – 7 + 7 = 4 + 7 So M = 11. I think of a number. I double it and add 15. My answer is 29. Doubling is the same as multiplying by 2. So one can write: 2n + 15 = 29. To undo the addition of 15, you can subtract 15. To undo the multiplication by 2, you can divide by 2. 2n + 15 – 15 = 29 – 15 2n = 14 2n ÷ 2 = 14 ÷ 2 n=7 It is important to subtract 15 FIRST and then the division by 2.
Formula One Maths, Exercise, Pg. 167
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Solving Problems with Equations The general plan is to: 1. 2. 3. 4.
Introduce a letter for what you want to find, with units. Form an equation. Solve the equation. State what the solution means.
Example The width of a rectangle is 4 cm less than its length, and the perimeter is 60 cm. Find the length. Solution: Let the length be L cm. The perimeter is L + (L – 4) + L + (L – 4) = 60 Collecting all the Ls together: 4L – 8 = 60 Adding 8 to each side of the equation: 4L – 8 + 8 = 60 + 8 Therefore 4L = 68 Dividing each side by 4: 4L ÷ 4 = 68 ÷ 4 Therefore L = 17. The length of the rectangle is 17 cm. Formula One Maths, Exercise, Pg. 169 Formula One Maths, Review Exercise, Pg. 170 – 171
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Graphs (Chapter 21) The Equation of a Line The co-ordinates of the points on Joe’s line are: (1, 1), (1, 2), (1, 3), (1, 4) The first co-ordinate is the x co-ordinate. The second co-ordinate is the y co-ordinate. Joe’s points are on the line called x = 1. This is called the equation of the line. The co-ordinates of the points on Sally’s line are: (1, 2), (2, 2), (3, 1), (4, 0) The equation of Sally’s line is x + y = 4. The equation of Oliver’s line is y = x. Formula One Maths, Exercise, Pg. 181 Drawing Lines Follow these steps to draw the graph of y = 3x. First of all make a table and choose some numbers for the x co-ordinates.
Then use the x co-ordinates to find the y co-ordinates. To know how big you need to draw your axes, you have to look at your table. Look again at the table. Draw your axes making sure that all your points will fit on it.
52
Plot all the points that you have found. up all your points with a straight line using a sharp pencil and a ruler. Look at the scales on the x-axis and y-axis. They are different. However, you must have the same scale for +x and –x, and the scale you use for –y must be the same as for +y. You can obtain the values of y at different values of x and vice versa from the graph. Formula One Maths, Exercise, Pg. 183 Interpreting Graphs Graphs can give us a lot of information. This line graph shows the temperatures at different times in two different towns. One can know when the greatest temperature difference occurred and when the temperature in both towns was the same. One small box on the temperature axis represents 2° and one small box on the time axis represents 15 minutes. The next graph is a conversion graph to convert between Fahrenheit and Celsius temperatures. One small box on the Fahrenheit axis represents 4° and one small box on the Celsius axis represents 1°.
Formula One Maths, Exercise, Pg. 185 Formula One Maths, Review Exercise, Pg. 186-187
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Operations in Algebra (Chapter 27) Brackets + and × are operations. - and ÷ are also operations. They must be carried out in the right order. You can use brackets to make the order clear. You work out inside the brackets first.
Brackets also come in pairs: ( opening and closing ). You can never have one on its own. You also use brackets in algebra.
Formula One Maths, Exercise, Pg. 237 BIDMAS You always work out inside brackets first. The word BIDMAS tells you the right order to carry out operations.
Formula One Maths, Exercise, Pg. 239
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Collecting Like To tidy up equations, you can collect all like together.
Sometimes there are numbers and minus .
Formula One Maths, Exercise, Pg. 241 Inverse Operations
The inverse operation of square is square root. Formula One Maths, Exercise, Pg. 243 Formula One Maths, Review Exercise, Pg. 244-245
55
Probability (Chapter 25)
56
Accuracy (Chapter 24)
57
Number 10 100 1 000 10 000 100 000 1 000 000 1 000 000 000 1 000 000 000 000
Power of ten 101 102 103 104 105 106 109 1012
The number of zeros (also called noughts) present is written as the power. In large numbers digits are grouped in threes. Example: Consider the number 673 852. This number can be written in columns in the following way: Hundreds of Thousands 6
Tens of Thousands 7
Thousands 3
Hundreds 8
Tens 5
Units 2
So the number is made up of or can be split into: Six hundred thousand Seventy thousand Three thousand Eight hundred Fifty Two
600 000 70 000 3 000 800 50 + 2 673 852
And is said in words as: Six hundred seventy-three thousand eight hundred fifty-two. Formula One Maths, Exercise, Pg. 3 Multiplying and Dividing Numbers with Lots of Noughts Example: 330 × 200 000 This can also be written as 330 × 2 × 100 000 (because 200 000 = 2 × 100 000) First multiply 330 × 2 = 660
1
Now multiply 660 × 100 000 = 66 000 000 (sixty-six million) Example: 6 300 ÷ 300 This can also be written as 6 300 ÷ 3 ÷ 100 First divide 6 300 ÷ 3 = 2 100 Now divide 2 100 ÷ 100 = 21 Summary: To divide by ten remove a nought To divide by hundred remove two noughts To divide by thousand remove three noughts
To multiply by ten add a nought To multiply by hundred add two noughts To multiply by thousand add three noughts
Formula One Maths, Exercise, Pg. 5 ST(P) Mathematics 1A, Exercise 2b, Pg. 14 ST(P) Mathematics 1A, Exercise 2g, Pg. 19 The Metric System Prefix Giga Mega Kilo Hecto Deca Basic unit Deci Centi Milli Micro Nano
Symbol G M k
c m µ η
Size × 109 (or 1 000 000 000) × 106 × 103 × 102 × 101 ×1 × 1/10 × 1/100 × 1/1 000 × 1/1 000 000 × 1/1 000 000 000
How the prefix is used
1 kilogram = 1 000 grams 1 hectare = 100 ares (1 are = 100 square metres) Metre, litre, gram, are 1 decilitre = 1/10 litre 1 centimetre = 1/100 metre 1 millimetre = 1/1 000 metre 1 micrometre = 1/1 000 000 metre 1 nanometre = 1/1 000 000 000 metre
Formula One Maths, Exercise, Pg. 7 Formula One Maths, Review Exercise, Pg. 8, 9 Addition and Subtraction ST(P) Mathematics 1A, Exercise 1b, Nos.41-45, Pg. 3 ST(P) Mathematics 1A, Exercise 1e, Nos.31-36, Pg. 6 ST(P) Mathematics 1A, Exercise 1g, Pg. 6
2
ST(P) Mathematics 1A, Exercise 1h, Nos. 26-30, Pg. 8 ST(P) Mathematics 1A, Exercise 1i, Nos. 6-9, Pg. 8 Short Multiplication and Division ST(P) Mathematics 1A, Exercise 2a, Nos. 43-48, Pg. 13 ST(P) Mathematics 1A, Exercise 2f, Nos. 36-45, Pg. 18 Long Multiplication and More Division Example: The Patel family are taking a 16-day holiday costing €128 per day. How much is this altogether?
128 × 10 = 128 × 6 =
128 × 16 1280 + 768 € 2048
They look at another holiday. It is €910 for 14 days. Rita works out how much this is per day. 65 14 |910 140 – 770 700 – 70 28 – 42 42 0
Adding up all the multiples of 14 in the previous column: 10 × 14
10 + 50 + 2 + 3 = 65
50 × 14 2 × 14 3 × 14
ST(P) Mathematics 1A, Exercise 2c, Nos. 21-30, Pg. 15 ST(P) Mathematics 1A, Exercise 2e, Nos. 8-10, Pg. 16 ST(P) Mathematics 1A, Exercise 2h, Nos. 41-48, Pg. 20 Multiples and Divisibility Tests The multiple of a number is that number multiplied by any other number. All even numbers are divisible by 2. Even numbers end with multiples of 2 (0, 2, 4, 6 or 8). All numbers ending with a 5 or 0 are divisible by 5.
3
All numbers ending with a 0 are divisible by 10. If a number is divisible by 3, you get a multiple of 3 when you add the single digits of that number. Example (Proof): Consider the number 3 852: 3 852 = 3 × 1 000 + 8 × 100 + 5 × 10 + 2 (every number is written in of powers of 10) = (3 × 999 + 3) + (8 × 99 + 8) + (5 × 9 + 5) + 2 (we can do this because 1 000 = 999 + 1, 100 = 99 + 1 and 10 = 9 + 1) = (3 × 999 + 8 × 99 + 5 × 9) + (3 + 8 + 5 +2) (999, 99 and 9 are all divisible by 3, so the first bracket is divisible by 3, the second bracket is the sum of the digits) Since 3 + 8 + 5 +2 = 18, which is divisible by 3, it follows that 3 852 is divisible by 3. If a number is divisible by 9, you get a multiple of 9 when you add the single digits of that number. A number can be divided by 4 exactly, if 4 goes exactly into the last two digits (4 will go into any number of hundreds since 100 is a multiple of 4, so we only need to know if 4 will go into the tens and units). A number can be divided by 8 exactly, if 8 goes exactly into the last three digits (8 will go into any number of thousands since 1 000 is a multiple of 8, so we only need to know if 8 will go into the hundreds, tens and units). If a number can be divided exactly by 2 AND by 3, then it can be divided exactly by 6. This is because 6 = 2 × 3. A number is divisible by 11 if the difference of the sums of alternating digits is divisible by 11. Example: 946 → (9 + 6) - 4 = 11 which is, of course, evenly divided by 11 so 946 es this divisibility test 10 813 → (1 + 8 + 3) - (0 + 1) = 12 - 1 = 11. 25 784 = (2 + 7 + 4) - (5 + 8) = 13 - 13 =0 . Yes, this does indeed work. In case you found this last bit confusing, that any number evenly divides 0. Think about it, how many 11's are there in 0? None, right. Well that means that 11 divides zero, zero times! 119 777 658 → (1 + 9 + 7 + 6 + 8) - (1 + 7 + 7 + 5) = 31 - 20 = 11
4
Negative Numbers (Chapter 18) Numbers can be shown on a number line. Numbers like …, -3, -2, -1, 0, +1, +2, +3, …, are called integers. Example:
Brackets () mean × +×+=+ + × - = - and - × + = -×-=+
(+) (+) = + (+) (-) = - and (-) (+) = (-) (-) = +
Formula One Maths, Exercise, Pg. 157, Nos. 2,4,5 Subtraction of Negative Numbers So from the above list, 5 - (-1) = 6 because a negative sign (-) multiplied by another negative sign gives a positive sign (+). And 2 - 9 = -7. Formula One Maths, Exercise, Pg. 159 Multiplying and Dividing Negative Numbers Multiplication of division of negative numbers is done by first multiplying the numbers on their own, regardless of the sign in front of them. Then the signs are multiplied separately according to the above list and the resulting sign applied to the answer. Example:
(-4) × (-6) is worked out by first multiplying 4 × 6 = 24 and then multiplying (-) × (-) = +. So the answer is (-4) × (-6) = (+24) (-5) × (+6) is worked out by first multiplying 5 × 6 = 30 and then multiplying (-) × (+) = -. So the answer is (-6) × (+6) = (-30)
Formula One Maths, Exercise, Pg. 161 Formula One Maths, Review Exercise, Pg. 162, 163 5
Fractions (Chapter 9) A clock’s face is divided into 12 parts so the difference between the two times shown on the two clocks is of 1/12 of an hour.
There are 60 minutes in an hour:
Of means × Formula One Maths, Task, Pg. 72 Philip eats two pieces of cake at tea-time. This means he has eaten a quarter of the cake. So = The following fractions are the same or equivalent:
You can cancel as often as you like.
Whatever you do to the numerator (multiplication or division or cancelling) you must also do to the denominator! Formula One Maths, Exercise, Pg. 73 ST(P) Mathematics 1A, Exercise 3b, Nos. 16-19, Pg. 36 ST(P) Mathematics 1A, Exercise 3e, Nos. 25-30, Pg. 43 6
Adding and Subtracting Fractions Jared eats of a cake. He then goes back for ‘seconds’ and eats another of the cake. How much of the cake has he eaten altogether?
12 is called a common denominator, it is in fact the lowest common denominator (24 or 36 or any multiple of 12 could also have been used as the common denominator but this would involve bigger numbers and cancelling. Jacinta later eats of the cake. What fractions of the cake remains? The fraction left is
- . The lowest common denominator is 5 × 12 = 60.
You can now write: When you add and , the answer is . is called a top-heavy or improper fraction. You can write as
. Now it is a mixed number.
In order to perform multiplication and division on mixed numbers, first turn them into improper fractions. For addition and subtraction, the whole numbers can be worked out first and then the fractions. Formula One Maths, Exercise, Pg. 77 ST(P) Mathematics 1A, Exercise 3t, Pg. 58
7
Multiplying Fractions Sally’s mother cuts a pizza into 2 halves. She then cuts each half piece into 3 slices. She has cut the whole pizza into 6 pieces, so each slice is one-sixth, or . In fact she did of = × =
The next day, Sally’s mother cuts a quiche into 3 equal pieces. She puts 2 of these pieces into Sally’s lunch-box. At school, Kevin has forgotten his lunch. Sally gives him one of her pieces of quiche. Kevin and Sally now each have one-third, or , of quiche.
Whatever you do to the numerator (multiplication or division or cancelling) you must also do to the denominator! This rule also applies to fractions which are being multiplied, but only one number can be cancelled at a time from the top and from the bottom. For example if we are dividing top and bottom by 7, we only divide one number from the top by 7 and not all the numbers at the top and we only divide one number from the bottom by 7 and not all the numbers at the bottom. Formula One Maths, Exercise, Pg. 75 8
ST(P) Mathematics 1A, Exercise 4b, Nos. 30-36, Pg. 61 ST(P) Mathematics 1A, Exercise 4c, Nos. 26-30, Pg. 62 A whole number can be written as a fraction with a denominator of 1. For instance 6 = . Doing this makes it easier to multiply a whole number by a fraction or a mixed number. Dividing by Fractions When we divide 6 by 3 we are finding how many threes there are in 6 and we say 6 ÷ 3 = 2. In the same way, when we divide 10 by we are finding how many halves there are in 10; we know that there are 20, so we say 10 ÷ = 20. But we also know that 10 × 2 = 20 so
and
This example shows that To divide by a fraction we turn that fraction upside down and multiply. ST(P) Mathematics 1A, Exercise 4f, Nos. 6-16, Pg. 65 ST(P) Mathematics 1A, Exercise 4g, Nos. 16-20, Pg. 67 Introduction to Percentages Percentage means out of 100 (from Italian; per cento) and is written as %. 20% means
.
Percentages can be written as fractions and cancelled in the same way to get them in their lowest .
You can find a percentage of a quantity by first changing the percentage into a fraction and then working it out normally:
9
The whole amount is called 100%. Example: 82% of apples in a box are good, ripe apples. What percentage of the apples is not fit to eat? (100 – 82) % = 18% of the apples are not fit to eat. Formula One Maths, Exercise, Pg. 79 Ratio and Proportion Mortar is made up of 1 part cement to 4 parts sand, this can be written as the ratio 1 : 4 (reads as 1 is to 4). Order matters with ratios. Sand : cement = 4 : 1 But cement : sand = 1 : 4 If cement comes in 50 kg bags, how much sand should be mixed with it? If we decide that 1 part = 50 kg 4 parts = 4 × 50 = 200 kg. Therefore 200 kg of sand are needed. If 1250 kg of mortar are needed altogether we can say that 5 parts altogether ( 1 part + 4 parts) will be 1250 kg. So 1 part is
= 250 kg.
So 250 kg of cement are needed and 250 × 4 = 1000 kg of sand. Ratios can be cancelled down like fractions. Example: 2 : 8 = 1 : 4 (cancelling by 2) Example: 3 : 12 : 15 = 1 : 4 : 5 (cancelling by 3) Formula One Maths, Task, Pg. 80 Formula One Maths, Exercise, Pg. 81 Formula One Maths, Review Exercise, Pg. 82, 83
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Decimals (Chapter 7 & 13) Reading a scale with divisions:
Check to see in between which numbers the reading lies. Subtract the smaller of these two numbers from the biggest of these two numbers. Count the number of divisions (or intervals) between the two numbers. Divide the result obtained from the subtraction by the number of divisions between the two numbers. This will give you how much a division (or interval is worth).
Example: The reading lies in between the numbers 30 and 40. 40 - 30 = 10 The number of divisions between the two numbers is 5. 10 ÷ 5 = 2 So each division is worth 2. The dial reads 34 km/h. Example: The reading lies in between the numbers 69 and 70. 70 - 69 = 1 The number of divisions between the two numbers is 10. 1 ÷ 10 = 0.1 So each division is worth 0.1. The tape measure reads 69.3 cm. Formula One Maths, Exercise, Pg. 57, Nos. 2-5 Decimal Notation The decimal point system is a way of writing numbers with tenths, hundredths, thousandths, etc. This table shows how:
11
Fraction 3/10 7/100 6 + 5/10 + 1/100 527 39/100
H hundreds
5
T tens
2
U Units
6 7
. . . .
1/10 tenths 3 5
1/100 1/1000 Decimal hundredths thousandths 0.3 7 0.07 1 6.51 39 527.039
To change a fraction to a decimal, always convert the fraction to one which has 10, 100 or 1000, etc. in the denominator. Formula One Maths, Exercise, Pg. 59 Addition and Subtraction of Decimals Addition and subtraction with decimals is just as easy as it is with whole numbers; all one has to do is to place the decimal points underneath each other when working! Example: Work out 7.53 + 420 + 0.086
Subtract 8.306 from 46.1
Zeros can be written down at the end of the number after the decimal point to help you work the sums out. Formula One Maths, Exercise, Pg. 61 Formula One Maths, Review Exercise, Pg. 62, 63 ST(P) Mathematics 1A, Exercise 5b, Nos. 9-12, Pg. 77, Nos. 21-24, Nos. 33-36, Pg.78 ST(P) Mathematics 1A, Exercise 5f, Nos. 16-28, Pg. 82-84
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Multiplying and Dividing Decimals (Chapter 13) To multiply decimals, first multiply the numbers as if there was no decimal point. Then count the numbers after the decimal point of each number and add them up. The number of numbers after the decimal point of the answer is the same number as the sum you obtained before. Example:
Formula One Maths, Exercise, Pg. 117, Nos. 1-6 Division of Decimals by a Whole Number Dividing decimals by whole numbers is done normally but the decimal point has to go down one underneath the other. You can add noughts in a s many decimal places as you need. Examples:
Sometimes when dividing one may obtain what is called a recurring number which is a number whose last digit or few digits are repeated for an infinite number of times. To indicate recurrence we write the recurring number only once and put a dot on it. When a group of digits recur, the dots are placed over the first and last recurring digits.
Formula One Maths, Exercise, Pg. 119 ST(P) Mathematics 1A, Exercise 6e, Nos. 1-12, Pg. 97 13
Division of Decimals (Short Division) In order to divide values, they must be in the same units. So if we are dividing amounts of money, both values must be in cents or both values must be in €. To divide a decimal by another decimal, multiply the decimals by 10 or by 100 or by 1000 (and so on) in such a way that the number you are dividing by is a whole number. Example:
Formula One Maths, Exercise, Pg. 121 Equivalent Percentages, Fractions and Decimals Percent (%) means out of 100. 80% means To change from fractions to percentages, multiply by 100.
To convert a decimal into a percentage, you multiply by 100.
Formula One Maths, Task, Pg. 122 Formula One Maths, Exercise, Pg. 123 ST(P) Mathematics 1A, Exercise 6h, Nos. 1-10, Pg. 101 Formula One Maths, Review Exercise, Pg. 124-125 ST(P) Mathematics 1A, Exercise 6k, Nos. 1-12, Pg. 104 14
Everyday Measures (Chapter 11) Length The metric system uses millimetres (mm), centimetres (cm), metres (m) and kilometres (km). Metric Measures 1 000 mm = 1 m 100 cm = 1 m
1 000 m = 1 km 10 mm = 1 cm
also
Measuring – To the Nearest cm and the Nearest mm The ruler shows that the length of the line is a bit more than 5 cm. The length of the line is 5 cm (to the nearest cm because the length is not equal to or does not exceed 5.5 cm) or 54 mm or 5.4 cm (to the nearest mm).
Formula One Maths, Exercise, Pg. 97, Nos. 2-5 Mass (or weight)
Volume
Metric Measures 1 000 milligrams (mg) = 1 g 1 000 grams (g) = 1 kilogram (kg) 1 000 kg = 1 tonne
Metric Measures 1 000 millilitres (ml) = 1 litre 100 centilitres (cl) = 1 litre
Formula One Maths, Exercise, Pg. 99, Nos. 1, 2, 3a, 4, 5, 6 Time
15
60 seconds = 1 minute 60 minutes = 1 hour
Units of Time 24 hours = 1 day 365 days = 1 year (except in a leap year)
Timetables
Formula One Maths, Exercise, Pg. 101 Formula One Maths, Review Exercise, Pg. 102, Nos. 4-7
16
Number Patterns (Chapter 10 &23) Factors 6 × 3 = 18 9 × 2 = 18 We say that 2, 3, 6 and 9 are factors of 18. Multiples 1×6=6 2 × 6 = 12 3 × 6 = 18 4 × 6 = 24 and so on. We say that 6, 12, 18, 24, … are multiples of 6. They are the answers to the 6 times table. The multiples of 2 are called even numbers Numbers which are not multiples of 2 are called odd numbers. Formula One Maths, Exercise, Pg. 85 Squares The numbers 1, 4, 9, 16, … are called square numbers. Square numbers are the answers when a number is multiplied by itself. 12 = 1 × 1 = 1 22 = 2 × 2 = 4 32 = 3 × 3 = 9 42 = 4 × 4 = 16 And so on. Square numbers make up squares.
17
Square Roots 1 is called the square root of 1 2 is called the square root of 4 3 is called the square root of 9 4 is called the square root of 16 And so on. A quick way to write square root is to use the sign √ So: 5 × 5 = 25 √25 = 5 Formula One Maths, Exercise, Pg. 87, Nos. 1,2, 6, 7, 8 Primes Every whole number greater than 1 has at least two factors, the number itself and 1. A number which has exactly two factors is called a prime number. The smallest prime number is 2, and the next ones are 3, 5, 7 and 11. The number 180 is not prime since it has factors other than 180 and 1. For example 180 = 18 × 10. These factors have factors themselves: 18 = 2 × 9 = 2 × 3 × 3 and 10 = 2 × 5. So 180 = 2 × 3 × 3 × 2 × 5 = 22 × 32 × 5. We have written 180 as the product of prime factors. A number may have many sets of factors, but it has only one set of prime factors. Formula One Maths, Exercise, Pg. 89, No. 4 Cubes A centicube is a 1 cm by 1 cm by 1 cm cube. You can fit centicubes together to make cuboids. The first cuboid has 2 × 5 × 4 = 40 centicubes. A cube is a solid with all sides the same length. 18
If we make a cube out of these centicubes we will have used 3 × 3 × 3 = 27 centicubes and 13 centicubes will be left over. Now these 13 leftover centicubes can make up another cube of 2 × 2 × 2 = 8 centicubes and then only 5 centicubes will be leftover. The second cuboid has 3 × 6 × 2 = 36 centicubes. If we make a cube out of these centicubes we will have used 3 × 3 × 3 = 27 centicubes and 9 centicubes will be left over. Now these 9 leftover centicubes can make up another cube of 2 × 2 × 2 = 8 centicubes and then only 1 centicube will be leftover. Numbers like 1, 8, 27, 64, … are called cube numbers. 13 = 1 × 1 × 1 = 1 23 = 2 × 2 × 2 = 8 33 = 3 × 3 × 3 = 27 43 = 4 × 4 × 4 = 64 And so on. A cube may only be made by certain specific number of cubes. Cube numbers make up cubes. 1 is the cube root of 1 2 is the cube root of 8 3 is the cube root of 27 4 is the cube root of 64 Formula One Maths, Exercise, Pg. 91, Nos. 3, 4 Powers of 2 A certain type of bacteria grows by splitting into 2 every hour.
19
These are the numbers of bacteria present each hour: 2, 4, 8, 16, 32, … This sequence of numbers has been obtained by doubling to get from one number to the next. The numbers are all powers of 2. The mathematical way of writing the powers of 2 is: 21 = 2 22 = 2 × 2 = 4 23 = 2 × 2 × 2 = 8 24 = 2 × 2 × 2 × 2 = 16 Formula One Maths, Review Exercise, Pg. 95-86
20
Flow Diagrams and Number Patterns A decision tree diagram is a diagram which asks questions to help you sort information. Different shapes can be used so that one does different actions. Flow Charts and Number Patterns A flow chart is a list of instructions and questions which help you carry out a task. We can use flow charts to investigate number patterns. Flow charts can be used to make number chains.
Figure 1 flow chart for odd numbers
Figure 2 Flow Chart for even numbers
Working Systematically with Patterns One layer contains 1 ball Two layers contain 3 balls Three layers contain 6 balls Four layers contain 10 balls And so on. Add 2 to go from the first to the second layer. Then add 3 to go from the second to the third layer. Then add 4 to go from the third to the fourth layer.
21
Growth Patterns The first three stages of a growth pattern are shown below:
Two matches are added every time to make a new stage. Table for growth pattern: Number of stages 1 2 3 4 5 6
Number of matches 3 5 7 9 11 13
Formula One Maths, Exercise, Pg. 205 Number Patterns – Making Predictions To build a fence one needs bars and posts. For one section, 2 posts and 2 bars are needed. For two sections, 3 posts and 4 bars are needed.
The following is the mapping for the number of posts and number of bars needed to build sections of the fence:
22
Rules can be created from the mapping:
p stands for posts, s stands for sections and b stands for bars. Formula One Maths, Exercise, Pg. 207 Formula One Maths, Review Exercise, Pg. 208-209 ST(P) Mathematics 1A, Exercise 2l, Nos. 18, 20, 21, 25, 27, 29, 30 33, 34 35, Pg. 27-30
23
Number Machines (Chapter 14) Making a phone call from a phone box costs 8cent per minute plus a 6cent connection charge. The cost of a call is calculated using a number machine like this:
The number of minutes that one is on the phone is called the input. The cost of the call is called the output. Given the output, the number machine can be reversed to find out what the input was.
The parts of the machine cannot be swapped.
Formula One Maths, Exercise, Pg. 127 Finding the Rule Following are the inputs and outputs of a number machine:
The rule chosen for the number machine must always give the correct output. The rule in this case is:
Formula One Maths, Exercise, Pg. 129
24
Writing down the Rule Following are the inputs and outputs of a number machine:
The number machine can be drawn like this:
And can be written in words like this:
An easier way of writing this is:
The rule can be checked by working a few outputs for some inputs. Formula One Maths, Exercise, Pg. 131 Formula One Maths, Review Exercise, Pg. 132-133
25
Averages (Chapter 16) The Mean To find the mean of a group of values; add up all the values and divide by the number of values. Formula One Maths, Exercise, Pg. 143 The Median To find the median, first write the values in order of size and then find the middle value. If there are two middle values, the median will be the mean of those two middle values Formula One Maths, Exercise, Pg. 145 The Mode The most popular value in the set of data is the mode (the one which occurs most often). When data is grouped, the group or class with the most items is called the modal class. In a bar chart or frequency chart, the modal class has the highest bar. The modal class in this bar chart is 11-15.
The Range The mean, median and mode are all ways of finding an average, or typical value. The range is the difference between the largest and smallest values of a set of data. The range is not an average. It tells you how spread out the data is. Formula One Maths, Exercise, Pg. 147 Formula One Maths, Review Exercise, Pg. 148-149
26
Co-ordinates (Chapter 8) Positions on a grid can be referred to by two values separated by a comma and enclosed in brackets: (along, up). (along, up) is in alphabetical order. The first number tells you how many values you have to move along (horizontally) while the second value tells you how many values you have to move up (vertically). Formula One Maths, Exercise, Pg. 65 Using Co-ordinates in Mathematics In mathematics, co-ordinates are plotted on a pair of axes called the x and y axes. The horizontal axis is called the x axis while the vertical axis is called the y axis. The x co-ordinate is always given first, followed by the y co-ordinate. In general form this can be written as: (x, y) (x, y) is in alphabetical order. The point (0,0) has a special name. It is called the origin. Any point can be plotted using co-ordinates. Formula One Maths, Exercise, Pg. 67 Using Negative Numbers, Fractions and Decimals There are negative (minus) co-ordinates, to the left for x and down for y. Co-ordinates do not necessarily have to be whole numbers but can be fractions or decimals. Formula One Maths, Exercise, Pg. 69 Formula One Maths, Review Exercise, Pg. 70-71
27
Scale (Chapter 15) Scales are used when a model or a picture is much smaller than the object in real life. They are written as: Scales can also be written as fraction in the form of: Formula One Maths, Exercise, Pg. 135 Drawing Plans If you want to fit the plan of a bedroom which is 4m long on a paper which is only 10 cm long you use a scale of 2 cm is to 1 m. This leads to a ratio of: 2 cm : 1 m 2 cm : 100 cm 2 : 100 1 : 50 If the bed is 1.8 m long, on the plan this is 2 × 1.8 cm = 3.6 cm. The ratio tells us that the real measurements are 50 times as big as the measurements on the plan. Formula One Maths, Exercise, Pg. 137 Map Scales 1 centimetre to the kilometre 1 : 25000
These are all examples of the way that a scale might be given on a map or in an atlas. A scale of 1 : 50 000 tells us that any length on the map is 50 000 times as big in real life. 1 cm on the map is equal to 50 000 cm in real life. 50 000 cm = 500 m = 0.5 km A scale of 1 : 50 000 is the same as 1 cm to 0.5 km.
28
So if a length on the map measures 6.5 cm: 6.5 × 50 000 cm = 325 000 cm 325 000 cm = 3250 m = 3.25 km. If a place is less than 1 km away from another place: 1 km = 1000 m 1000 m = 100 000 cm 100 000 cm on the ground = 100 000 ÷ 50 000 = 2 cm on the map. Therefore the two places should be less than 2 cm away from each other on the map. Formula One Maths, Exercise, Pg. 139 Formula One Maths, Review Exercise, Pg. 140-141, Nos. 1-6.
29
Displaying Data (Chapter 5) Data has to be displayed in an organized way so that one can make sense of it. Tally Charts One way of collecting data is to use a tally chart.
Frequency means the number of people. This is sometimes called a frequency table. Formula One Maths, Exercise, Pg. 39 Pictograms and Bar Charts A bar chart can also be drawn from a frequency table. It is important to always label the axes and give the chart a title.
30
Pictograms can also be used; you can use any symbol you like for the picture.
Formula One Maths, Exercise, Pg. 41 Pie Charts
The angles of each section of a pie chart can tell us how much of the total that value is. Divide the value of that angle by 360° and reduce the fraction to its lowest to get the fraction.
Formula One Maths, Exercise, Pg. 43 Grouping Data Data can first be grouped into sets and then those sets can be plotted on one of the types of graphs mentioned above. Each set will consist of a range of values. Formula One Maths, Exercise, Pg. 45 Formula One Maths, Review Exercise, Pg. 46-47. 31
Angles (Chapter 4) Types of Angle
To measure angles you need a protractor.
It is important to be accurate when measuring angles.
The centre of the protractor must be on the centre of the angle. The start line must be lined up exactly over one of the angle lines. Your eyes must be directly over the protractor when reading the scale.
There are two sets of numbers round the protractor so you can measure angles on both sides. You use the set that is starting from zero in the angle you are measuring. There are 90° in a right angle. A straight line is 180°. A whole turn is 360°. You can use a protractor to measure an obtuse angle by first measuring up to 90° and making a mark and then measuring the rest of the angle from the mark. The value obtained should be added to the 90°. The internal angles of a triangle always make up 180°. Formula One Maths, Exercise, Pg. 29 32
Drawing Angles Protractors can be used to draw angles accurately. Always use a sharp pencil to angles and always keep your eyes over drawing. The following steps explain how to an angle of 57°.
very draw your draw
Formula One Maths, Exercise, Pg. 31 Angle Calculations As stated previously, there are 360° in a whole turn. It follows that a half-turn, or a straight line, is 180°. A right angle is 90°.
33
A, B, C and D together make up 360°. Angles X and Y make up 180°. Two angles that add up to 180° are also called supplementary angles. Parallel Lines When two lines are always the same distance apart (equidistant), they are parallel. Railway lines are parallel. In the following diagrams, angles are made where lines intersect.
Formula One Maths, Exercise, Pg. 33 Angles in a Triangle The angles inside a shape are called interior angles. The interior angles of every triangle add up to exactly 180°. This statement can be proved as follows: Draw the line through one vertex (corner) of the triangle which is parallel to the opposite side. Then X = B and Y = C (*) by alternate angles But A + X + Y = 180° since these are adjacent angles (angles lying on a straight line). Therefore A + B + C = 180°.
Any shape with four straight sides is called a quadrilateral.
34
If a diagonal line is drawn across a quadrilateral between two opposite corners, two triangles are formed. Hence the internal angles of a quadrilateral make up 180° + 180° = 360°. Formula One Maths, Exercise, Pg. 35 Formula One Maths, Review Exercise, Pg. 36-37 ST(P) Mathematics 1A, Exercise 9m, Pg. 145-146 ST(P) Mathematics 1A, Exercise 9n, Pg. 146-147 ST(P) Mathematics 1A, Exercise 15k, Pg. 234, Nos. 1-8
35
Symmetry (Chapter 6) The Town Hall is symmetrical. Many buildings are designed like this. If you stand a mirror on the centre of the drawing, in line with the flagpole, the Town Hall will look exactly the same in the mirror as it does behind the mirror. This kind of symmetry is called reflection symmetry.
The green cross, seen on first aid kits, has four lines of symmetry. Two of them are diagonal, one is horizontal and one is vertical.
You can use squared or dotted paper to complete symmetrical shapes. Here is a shape with two lines of symmetry.
Formula One Maths, Exercise, Pg. 49 Rotation Symmetry Here is a drawing of a flag. There are two positions the flag can be in where it looks exactly the same. We say that the flag has rotational symmetry of order two. The point about which the flag rotates is the centre of rotation symmetry and is found at the centre of the flag itself.
36
There are four positions where this square flag looks the same. This flag has rotational symmetry of order four. Its centre of rotation symmetry is found at the centre of the flag itself.
Some shapes have rotation symmetry and reflection symmetry. Formula One Maths, Exercise, Pg. 51 Paper Folding If you fold a paper once and cut out any shape, when you open up the paper, the shape will have reflection symmetry. But if you fold a paper once and cut out any shape, when you open up the paper, the shape will have reflection symmetry and rotation symmetry of order four. Formula One Maths, Exercise, Pg. 53, No. 4 Formula One Maths, Review Exercise, Pg. 54-55
37
Flat Shapes (Chapter 12) Any shape made with straight sides is called a polygon. There are many different types of polygon. To describe them we give them special names.
Formula One Maths, Exercise, Pg. 105 A polygon has a number of sides. A point where two sides meet is called a vertex (plural vertices). In a quadrilateral, a diagonal s two opposite vertices. Classifying Triangles and Quadrilaterals
Formula One Maths, Exercise, Pg. 107 38
Circles A pair of comes are needed to draw a circle. The distance between the point and the pencil can be adjusted. The following steps show how to draw a circle.
The circle drawn in the steps above has a radius of length 6 cm. Its diameter is 12cm since the diameter is twice the radius.
Formula One Maths, Exercise, Pg. 109, Nos. 1-2.
39
Constructing Triangles You need to use a pair of comes to make an accurate drawing of a triangle.
Making an accurate drawing like the one above is called a construction. The arcs and ‘extra bits’ of line are an important part of a construction. They must not be rubbed out! All the sides of an equilateral triangle are equal because the com was all the time open at the same length. All the angles of an equilateral triangle are also equal. Each angle is 60°. To draw an isosceles triangle, the same method can be used BUT the comes must be adjusted to a bigger length before starting Step 3. Formula One Maths, Exercise, Pg. 111, Nos. 1-6.
40
Constructions with Angles
To construct an accurate drawing of this slide you need to follow the following steps.
The triangle constructed above has been drawn to a scale of 1 cm to 1m. Constructing an Angle of 90°
Formula One Maths, Exercise, Pg. 113, Nos. 1-6 Formula One Maths, Review Exercise, Pg. 114, Nos. 1-7 ST(P) Mathematics 1A, Exercise 11l, Pg. 178-179, Nos. 1-3 ST(P) Mathematics 1A, Exercise 11m, Pg. 179, Nos. 1-3 ST(P) Mathematics 1A, Exercise 11n, Pg. 180, Nos. 1-4
41
Solid Shapes (Chapter 20) Here is a piece of card before it is folded to make a fruit pastilles packet. It is called a net. The shape of the packet will be a cuboid. The packet will have 6 faces, each face will have the shape of a rectangle. Where the card is folded there will be an edge of the packed. The packet will have 12 edges. 12 folds have to be made. The packet will have 8 vertices. This is an example of a mathematical net drawn on a centimetre squared paper. Flaps or tabs can be added to the net for sticking. The flaps do not necessarily have to be at the same locations. They can be arranged differently and one may have different numbers of flaps. This net should make a cuboid whose dimensions are 3 cm long by 2 cm wide by 2 cm high
Formula One Maths, Exercise, Pg. 173. Prisms Here is the card for a Toblerone packet.
42
When folded and glued, it makes a triangular prism. A prism is called a polygon ‘stretched’ backwards to make a solid shape. A Toblerone backwards.
packet
is a
triangle
stretched
A triangular prism has 5 faces. 2 of them are triangles and the other 3 are rectangles. A triangular prism has 9 edges and 6 vertices. Shape Triangular prism Hexagonal prism Rectangular prism Pentagonal prism In general; V + F – E = 2
Number of faces (F) 5 8 6 7
Number of vertices (V) Number of edges (E) 6 9 12 18 8 12 10 15 This is called Euler’s Rule.
Further Solid Shapes There are many other solid shapes besides prisms. Here is a tetrahedron along with the net to make it. Tetra means four, hedron means face. The tetrahedron also obeys Euler’s Rule.
The five Platonic solids also obey Euler’s Rule. The mathematical name for the shape of a football is a sphere. Formula One Maths, Review Exercise, Pg. 178-179
43
Measuring (Chapter 22) Perimeter The distance round the outside of a shape is called the perimeter. Area Area is the amount of flat space. The formula for the area of a rectangle is: A=L×B Where A is the area, L is the length and B is the breadth (the breadth is sometimes also called the width). Area is measured in square units e.g. m2. Volume Volume is the amount of solid space. The volume of a box or a cuboid can be found from the following equation: V=L×B×H Where V is the volume, L is the length, B is the breadth and H is the height. Volume is measured in cubic units e.g. cm3 Formula One Maths, Exercise, Pg. 189 More Areas Here is a rectangle with a triangle drawn inside it. The area of the rectangle is 8 cm × 5 cm = 40 cm2 The area of the triangle can be found by dividing the area of the rectangle by 2. The equation for finding the area of a triangle is:
Where b is the base and h is the height.
44
If we want to find the area of an irregular shape such as the one shown, we have to divide the shape into simple areas for which we know how to find the areas.
This can be done in various ways as shown below:
All we have to do now is find the areas of the rectangles and then add them up. The surface area of a 3D object can be found by working out the area of each face and then adding up these areas. The surface area of the cuboid cardboard box is the total area of cardboard surrounding the volume.
Formula One Maths, Exercise, Pg. 191 Volume of Prisms Volume of prism = area of cross section × length
Formula One Maths, Exercise, Pg. 193 45
More Volumes The volume of a complicated shape can sometimes be found easily by taking the difference of the volumes of simple shapes. Formula One Maths, Exercise, Pg. 195 Formula One Maths, Review Exercise, Pg. 196-197 ST(P) Mathematics 1A, Exercise 14e, Pg. 209 ST(P) Mathematics 1A, Exercise 14f, Pg. 210-212
46
Transformations (Chapter 26) An object is rotated when it moves along a circle. If it only moves along part of a circle, then it has moved along an arc. One point on the object remains fixed, the rest of the object rotates about this point. It is called the centre of rotation. The Plough is a constellation of stars. From midnight till 6 am, it rotated a quarter of a turn (90°) in an anticlockwise direction. To describe a rotation accurately, you must give the centre of rotation, the angle and the direction of turning. When an object rotates, its size and shape do not change. We say that the object remains congruent. Formula One Maths, Exercise, Pg. 229 Translation The movement of a shape in mathematics is called a transformation. Rotation is one kind of transformation. In translation, objects move without turning. When an object is translated, every point moves the same distance in a parallel direction. Object ABC is translated 5 squares to the right and 1 square up to become image A’B’C’. A maps to A’, B maps to B’ and C maps to C’. When an object is translated the shape and size remain the same. Formula One Maths, Exercise, Pg. 231
47
Reflection Shapes can be transformed by reflection in a mirror. The figure below shows a shape before and after it has been reflected.
Objects are still congruent after having been reflected, that is, they retain their shape and size. To describe a reflection, you must say where the mirror line is. Formula One Maths, Exercise, Pg. 233 Formula One Maths, Review Exercise, Pg. 234-235
48
Formulae (Chapter 17) You can give a formula in words or you can write it using letters. Examples of formulae:
To calculate the points for a team in a league: Points = 3 × W + 1 × D Where W stands for the number of wins and D stands for the number of draws. The cost to buy tickets: In words: The cost in euro is eight times the number of adults added to five times the number of children. In letters: P = 8 × a + 5 × c Where P is the cost in Euro, a is the number of adults and c is the number of children. It is common to leave the × signs out when you use letters. So the formula is P = 8a + 5c. This is called an expression. An expression has letters and usually some numbers too.
Example Find the value of 8a + 5c when a = 3 and c = 6. Solution: P = 8a + 5c P=8×3+5×6 P = 24 + 30 P = 54 Notice that you have to put the × signs in to work this out. When you write a formula in letters you are using algebra. The letters stand for the numbers. The numbers are not always the same. In the example above, a = 3, c = 6 is a party of 3 adults and 6 children. Formula One Maths, Exercise, Pg. 151 Formula One Maths, Exercise, Pg. 153 Formula One Maths, Review Exercise, Pg. 154-155
49
Equations (Chapter 19) Balancing Equations Two identical packages are placed on some old-fashioned weighing scales. To know how much each package weighs one can use: The weight of two packages + 3 kg = 7 kg It would be quicker to write: 2 × p + 3 = 7 So: 2p + 3 = 7 This is called an equation – it only has one right answer. If 3 kg are removed from both sides of the scales, the scales still balance. When removing 3 kg from each side of 2p + 3 = 7, one can write: 2p = 4 One can work out how much one package weighs by dividing each side by two. Therefore: p = 2. Formula One Maths, Exercise, Pg. 165 Solving Equations
I think of a number and subtract 7. My answer is 4. To find out the number I thought about: Mystery number – 7 = 4 M–7=4 To find the mystery number add 7 to both sides of the equation: M – 7 + 7 = 4 + 7 So M = 11. I think of a number. I double it and add 15. My answer is 29. Doubling is the same as multiplying by 2. So one can write: 2n + 15 = 29. To undo the addition of 15, you can subtract 15. To undo the multiplication by 2, you can divide by 2. 2n + 15 – 15 = 29 – 15 2n = 14 2n ÷ 2 = 14 ÷ 2 n=7 It is important to subtract 15 FIRST and then the division by 2.
Formula One Maths, Exercise, Pg. 167
50
Solving Problems with Equations The general plan is to: 1. 2. 3. 4.
Introduce a letter for what you want to find, with units. Form an equation. Solve the equation. State what the solution means.
Example The width of a rectangle is 4 cm less than its length, and the perimeter is 60 cm. Find the length. Solution: Let the length be L cm. The perimeter is L + (L – 4) + L + (L – 4) = 60 Collecting all the Ls together: 4L – 8 = 60 Adding 8 to each side of the equation: 4L – 8 + 8 = 60 + 8 Therefore 4L = 68 Dividing each side by 4: 4L ÷ 4 = 68 ÷ 4 Therefore L = 17. The length of the rectangle is 17 cm. Formula One Maths, Exercise, Pg. 169 Formula One Maths, Review Exercise, Pg. 170 – 171
51
Graphs (Chapter 21) The Equation of a Line The co-ordinates of the points on Joe’s line are: (1, 1), (1, 2), (1, 3), (1, 4) The first co-ordinate is the x co-ordinate. The second co-ordinate is the y co-ordinate. Joe’s points are on the line called x = 1. This is called the equation of the line. The co-ordinates of the points on Sally’s line are: (1, 2), (2, 2), (3, 1), (4, 0) The equation of Sally’s line is x + y = 4. The equation of Oliver’s line is y = x. Formula One Maths, Exercise, Pg. 181 Drawing Lines Follow these steps to draw the graph of y = 3x. First of all make a table and choose some numbers for the x co-ordinates.
Then use the x co-ordinates to find the y co-ordinates. To know how big you need to draw your axes, you have to look at your table. Look again at the table. Draw your axes making sure that all your points will fit on it.
52
Plot all the points that you have found. up all your points with a straight line using a sharp pencil and a ruler. Look at the scales on the x-axis and y-axis. They are different. However, you must have the same scale for +x and –x, and the scale you use for –y must be the same as for +y. You can obtain the values of y at different values of x and vice versa from the graph. Formula One Maths, Exercise, Pg. 183 Interpreting Graphs Graphs can give us a lot of information. This line graph shows the temperatures at different times in two different towns. One can know when the greatest temperature difference occurred and when the temperature in both towns was the same. One small box on the temperature axis represents 2° and one small box on the time axis represents 15 minutes. The next graph is a conversion graph to convert between Fahrenheit and Celsius temperatures. One small box on the Fahrenheit axis represents 4° and one small box on the Celsius axis represents 1°.
Formula One Maths, Exercise, Pg. 185 Formula One Maths, Review Exercise, Pg. 186-187
53
Operations in Algebra (Chapter 27) Brackets + and × are operations. - and ÷ are also operations. They must be carried out in the right order. You can use brackets to make the order clear. You work out inside the brackets first.
Brackets also come in pairs: ( opening and closing ). You can never have one on its own. You also use brackets in algebra.
Formula One Maths, Exercise, Pg. 237 BIDMAS You always work out inside brackets first. The word BIDMAS tells you the right order to carry out operations.
Formula One Maths, Exercise, Pg. 239
54
Collecting Like To tidy up equations, you can collect all like together.
Sometimes there are numbers and minus .
Formula One Maths, Exercise, Pg. 241 Inverse Operations
The inverse operation of square is square root. Formula One Maths, Exercise, Pg. 243 Formula One Maths, Review Exercise, Pg. 244-245
55
Probability (Chapter 25)
56
Accuracy (Chapter 24)
57
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