CONTENTS. Functional Maths and Numeracy study guide Name 1) THE FOUR RULES OF ARITHMETIC 6) ROUNDING 2) MEASURES, SHAPE AND SPACE 8) NEGATIVE NUMBERS

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1 Name CONTENTS 1) THE FOUR RULES OF ARITHMETIC Addition (page 2) Addition of decimals (page 4) Subtraction (page 5) Subtraction of decimals (page 8) Subtraction other methods (page 9) Multiplication (page 11) Division (page 15) Long division (page 17) Number relationships (page 20) 2) MEASURES, SHAPE AND SPACE Units of Measure (page 21) Perimeter, Area, Volume (page 22) Area (page 23) Volume (page 26) 3) FRACTIONS Introduction (page 28) Equivalent fractions (page 32) Simplifying fractions (page 34) Comparing fractions (page 35) Adding and Subtracting (page 36) Multiplying with fractions (page 38) Mixed numbers (page 40) 4) DECIMALS Introduction and Ordering (page 42) Dividing and multiplying with powers of 10 (page 43) Multiplying with decimals (page 44) Converting fractions to decimals, division with decimals (Page 47) 5) PERCENTAGES Introduction, %s to fractions (page 51) Percentages of amounts (page 52) Percentage Worker Outer (page 53) Percentage changes (page 54) Converting between fractions, decimals and percentages (page 57) 6) ROUNDING Whole numbers (page 59) Decimals (page 62) Alternative method (page 65) to difficult limits (page 66) 7) RATIO Introduction & Worked Examples (page 67) Conversions & Proportion (page 71) Scale drawings (page 75) 8) NEGATIVE NUMBERS Introduction (page 80) Addition and subtraction (page 81) 9) AVERAGES and RANGE Range (page 87) Mean, Mode, Median (page 88) 10) REPRESENTING DATA Introduction (page 91) Discrete and Continuous Data (page 92) Bar Charts (page 93) Line Graphs (page 95) Scatter Graphs (page 97) Pie Charts (page 99) Pictograms (page 101) Tally Charts (page.103) Frequency Tables (page 104) 11) ORDER OF CALCULATIONS Introduction (page 105) BODMAS (page.106) 12) ALGEBRA AND FORMULAE Introduction (page 108) BODMAS (page 109) Page 1 of 110

2 L1-2 Functional Maths and Numeracy study guide Addition THE FOUR FUNDAMENTAL RULES OF ARITHMETIC Addition tells us the TOTAL of things. Q1: What is ? 1: ADDITION The best start is to correctly line up the digits in HUNDREDS, TENS, and UNITS. START ON THE RIGHT HAND SIDE and add up the numbers IN COLUMNS. The first column is 3 and 0, so your first addition is Write your answer DIRECTLY UNDERNEATH the column you ve just added. And finally the last column: Now move to the next column: So, = 163. Once you ve done the last column you are finished! Page 2 of 110

3 L1-2 Functional Maths and Numeracy study guide Addition Q2: What is ? Again, first of all line the digits up according to place value: And again, you start on the RIGHT HAND SIDE. The first column is 3 + 9, but = 12. There s only have room for one number underneath! So, you carry over the 1 in 12, and write the 2 from the 12 underneath: You now move on the next column. It used to be 2 and 8. Now though, you have 2, 8, and the 1 you carried over before. So now, you do This is 11. Write 1 down and carry over the other 1 in the 11. Now you do the last column: the 1 you carried: So, = 312. You ve done the last column, so you have finished. Page 3 of 110

4 L1-2 Functional Maths and Numeracy study guide Addition of decimals Q3: What is ? Don t let the decimals bother you! You still start in the same way by lining up the digits according to place value. The easiest way to do this is: MAKE SURE THE DECIMAL POINTS LINE UP! The first column is just 8 on its own. 8 plus... nothing! Now the second column is 5 + 9, and = 14. Carry over the 1, and write the 4 underneath. You must line up the decimal points, so put in another point directly beneath the others: Then you finish off by doing the last column as usual: So, = 6.48 Page 4 of 110

5 L1-2 Functional Maths and Numeracy study guide Subtraction Subtraction is the reverse of addition! 2: SUBTRACTION If you add something to a number, then take it back off, you re back to where you started: = = = = 7 Subtraction can tell you quite a few useful things: The difference between two numbers; The RANGE of different values (see later!); What I need to add to a smaller number to get to the bigger one; How much change I can expect back from my 10 note! So how do you do it? Q1: What is 98 44? What you must ALWAYS do is PUT THE NUMBER YOU ARE TAKING AWAY ON THE BOTTOM. You are taking away 44, so this goes on the bottom. You line up the digits according to place value just like you do with addition: Again, like addition, you start on the RIGHT HAND COLUMN. You do THE TOP NUMBER MINUS THE BOTTOM NUMBER, which here is 8 4: Then the same for next column, 9 4: That was the last column so you are finished. So, = 54. Page 5 of 110

6 L1-2 Functional Maths and Numeracy study guide Subtraction Q2: What is ? Again, you start in the same way. Line up the digits according to place value and put the number you are taking away on the bottom: Your first column is 9 8: Now the next column is 0 3. How can you do that? To make it possible, YOU BORROW 1 FROM THE LEFT AND MOVE IT OVER. The digit you borrow from becomes smaller by 1, and the 1 you take is placed just beside the number on the right: The 0 has become 10 after this borrowing. The 1 you borrowed from becomes 1 smaller, so it becomes 0. This now means the last column has gone altogether there s nothing there. So all you need to do is the next column, 10 3: You have now done all columns. So, = 71. Page 6 of 110

7 L1-2 Functional Maths and Numeracy study guide Subtraction Q3: What is ? Line up the digits according to place value and put the number you re taking away on the bottom: The first column is 2 5, which you can t do. Can you borrow? The next number across from 2 is 0, so there s NOTHING THERE TO BORROW! You need to borrow from the very end, then move the 1 across gradually: Now the 0 has become 10 so you CAN now borrow from the middle column. You borrow 1, which makes the 10 smaller by 1: Now the 2 has become 12. You can now carry on and subtract in the usual way. Start with the right column, 12 5: Then 9 4: And you are now finished because the 1 on the left was moved over. So, = 57 Page 7 of 110

8 L1-2 Functional Maths and Numeracy study guide Subtraction of decimals Q4: What is ? You set out the subtraction calculation in the same way as before. However, one number is a decimal and the other is not! If 5 had a decimal point, it would be 5.0, or even There are 0s after the decimal, because there is no decimal part to 5 it is a whole number. So, you can think of the question as : You can t do 0 7, so you need to borrow. You can t borrow from the next digit, 0, so you need to borrow from the 5: The middle 0 has become 10. Now you borrow AGAIN: You can now subtract column by column as normal. First column: 10 7 : Then Now you need to add a decimal point below, so all decimal points line up. Just like you had to do for addition with decimals: And finally, the last column, 4 2: So, (which is the same as ) = 2.63 Page 8 of 110

9 L1-2 Functional Maths and Numeracy study guide Subtraction other methods Other subtraction methods Like most maths, there are several methods for any one problem. You can use subtraction to find the difference between two numbers, or to find out what you need to add to the smaller number to get to the bigger one. Using that idea, you can calculate subtraction problems by using addition. Q1: What is ? Thinking of this in terms of adding, you could also ask WHAT DO I NEED TO ADD TO 149 TO GET TO 512? The trick now is to start at 149 and get to the next nice number to make it easy on yourself. You could ADD 1 to 149 to get 150. You could ADD 50 to 150 to get 200. You could ADD 300 to 200 to get 500. You could ADD 12 to 500 to get 512! In total, you added 1, 50, 300 and 12: This tells you that = 363, because = 512 This method comes into its own when you may otherwise have to borrow from the other side again, and again, and again! Page 9 of 110

10 L1-2 Functional Maths and Numeracy study guide Subtraction other methods Q2: What is ? The standard way would be tedious: You can t do 0 9, so you must borrow, but the next THREE digits are all 0! You would have to borrow 1 all the way from one end to the other. OR... You could ADD 1to to get You could ADD 1 to to get In total you just added 1 and 1, which is 2. So, = 2. NB Both ways of subtraction will always work. Use whichever you feel most confident with. Page 10 of 110

11 Multiplication 3: MULTIPLICATION Multiplication can tell you lots of things, including: The AREA of a 2D shape (See later!); The VOLUME of a 3D shape (See later!). In some cases, it is like a faster way of doing addition: (Slow) 9 x 8 (Quick!) (Slow) 10 x 10 (Quick!) Q1: What is 17 x 9? You put the smaller number on the bottom: Then multiply the bottom number by the numbers above, MOVING RIGHT TO LEFT. First, do 9 x 7. This equals 63. Like addition, you carry over the 6. Then do 9 x 1, and remember to add on your carried 6: 9 x 1 equals 9, plus the carried 6 gives 15: So, 17 x 9 = 153. NB With multiplication ALWAYS PUT THE FIRST NUMBER DIRECTLY BENEATH THE DIGIT YOU ARE MULTIPLYING WITH. Page 11 of 110

12 Multiplication This general method also works when larger numbers are on the bottom. Start with the bottom right digit, then multiply it by the digits above, moving FROM right TO left. When you re done with that number move on to the next number to the left. Here s some examples for the pattern you would follow when multiplying with larger numbers: Example 1: 15 x 15 NB With multiplication ALWAYS PUT THE FIRST NUMBER DIRECTLY BENEATH THE DIGIT YOU ARE MULTIPLYING WITH. With the first number 5 the first digit would go below the 5. With the second number 1 the first digit would go below the 1: Page 12 of 110

13 Multiplication Just in case, here is 15 x 15 done STEP BY STEP: Start in the bottom right, the 5, and multiply with the digits above moving FROM right TO left. First is 5 x 5, which is 25: Next you do 5 x 1, PLUS the carried 2: Now move to the next number, the 1. Again, you multiply from the top right and move left, so first calculation is 1 x 5. THE ANSWER IS PUT DIRECTLY BENEATH THE 1 WHICH YOU ARE MULTIPLYING: Next you do 1x1: And you have finished. THE FINAL STEP IS TO ADD UP THE MULTIPLICATION ANSWERS: So, 15 x 15 = 225. Page 13 of 110

14 Multiplication Example 2: 108 x 356 Again, it is exactly the same method. I would advise you try this one yourself, then check back here to see if you got it right! Then finally, add the totals together: So, 108 x 356 = NB There are other ways of doing multiplication (the Lattice Method for example). Use the way you are most comfortable with. This way is just a personal suggestion. Page 14 of 110

15 Division 4: DIVISION Division is the opposite of multiplication. Because of this, division can tell you what you need to multiply a number by to get another number: Example 6 x? = 18. What is? In plain English: 6 multiplied by SOMETHING equals 18. What is that something? (Hopefully) you know that 6 x 3 = 18, so that something MUST BE 3. You could have worked this using division: 18 6 = 3 Other Examples 2x? = 10? = = 5 5x? = 15? = = 3 10x? = 20? = = 2 see a pattern? These mystery numbers may have been quite easy to work out WITHOUT thinking of division, but... if the question was 7 x? = 1477, division comes into its own! Division can tell you: What you need to multiply one number by to get the bigger number; How much of something maybe money each person receives; The FRACTION of a value. And those are just a few. NB The link between division and multiplication is explained in greater detail later on. Page 15 of 110

16 Division Division with smaller numbers: mental methods For smaller numbers, division can often be done without using LONG DIVISION. Q1: What is 24 6? As long as you remember what division tells you, you will be okay! This is asking how many 6s fit inside 24? How many 6s? Value Comments Too small Too small Too small Got it! So, FOUR 6s fit inside 24, which means 24 6 = 4. This is just counting the Six Times Table and stopping at the answer: 6, 12, 18, 24, 30, Q1: What is 1477? You could do this again by doing the Seven Times Table, but you might be going for a long time! Another method is called CHUNKING. See if so many 7s are enough, and if not, try to see how many more will fit. 7 x 10 = 70 Not enough! 7 x 20 = 140 Still not enough! (7 x 20) + 7 = = 147. So = s fit inside 147. As a check, you can see with the 7x table that there ARE twenty one 7s in 147: 7,14,21,28,35,42,49,56,63, 70, 77, 84, 91, 98, 105, 112, 119, 126, 133, 140, Page 16 of 110

17 Division long division LONG DIVISION Mental methods may work for smaller numbers, but when dealing with much larger numbers, LONG DIVISION is often much quicker. Q1: What is 96 4? The most important first thing is knowing which number goes where! Here is how it would look: THE NUMBER YOU ARE DIVIDING GOES ON THE INSIDE. THE NUMBER YOU ARE DIVIDING BY GOES ON THE OUTSIDE. Now see how many times 4 can fit inside the first digit (the 9). 4 goes into 9 twice, with 1 remainder. The remainder moves over to the next digit: THE 6 HAS NOW BECOME 16 because of the 1 remainder. Next, how many times does 4 fit inside 16? The answer is FOUR times: Because there is no remainder, and you ve done the last digit, you can stop. So, 96 4 = 24. Page 17 of 110

18 Division long division Q2: What is 468 9? The 9 goes on the OUTSIDE, the 468 goes on the INSIDE: How many times does 9 go into 4? NONE! It s TOO BIG. You ve not used any of the 4, so ALL OF IT STILL REMAINS (remainder = 4): How many times does 9 go into 46? The answer is FIVE TIMES, 1 remainder: How many times does 9 go into 18? The answer is TWO TIMES, no remainder: Because there is no remainder, and you ve done the last digit, you can stop. So, = 52. Page 18 of 110

19 Division long division Q3: What is ? This example shows it is important to carry on right to the end of the number you are dividing: How many times does 5 fit inside 5? EXACTLY ONCE, no remainder: Even though there is no remainder, you still carry on because you have not done right to the end. How many times does 5 fit inside 0? It doesn t! No remainder. AGAIN, SAME QUESTION: How many times does 5 fit inside 0? ZERO! Same question again: How many times does 5 fit inside 7? And finally: So, = Page 19 of 110

20 L1-2 Functional Maths and Numeracy study guide Number relationships: multiplication and division NUMBER RELATIONSHIPS: MULTIPLICATION AND DIVISION When multiplying two numbers together the answer will be the same no matter what order you put the numbers in. i.e. 2 x 3 = 6 IS THE SAME AS 3 x 2 = (two groups of three) = 6 (three groups of two) = 6 Division: Using the same numbers as the multiplication sums above you can see their relationship to division sums. 6 2 = so six sweets divided between two groups = = so six sweets divided between three groups = You saw that 2 x 3 = 6, and that 6 2 = 3. Also, 2 You saw that 3 x 2 = 6, and that 6 3 = 2. Can you see the relationship? Division tells you how many times (=multiplication) one number fits inside another. Page 20 of 110

21 L1-2 Functional Maths and Numeracy study guide Units of measure Length UNITS OF MEASURE 1cm = 10mm 100cm = 1m 1 000m = 1km Imperial Approximations * 1 inch is about 2.5 centimetres (2 centimetres); 1 yard is roughly the same as 1 metre; 1 mile is about 1.6 kilometres (1 kilometres). Length Area Volume mm, cm, m, km mm 2, cm 2, m 2, km 2 mm 3, cm 3, m 3, km 3 Capacity 1000 ml = 1 l 1 l = 1000 cm 3 Imperial Approximations * 1 gallon is about 4.5 litres (4 litres); 1 pint is about 0.6 litres ( ths of a litre); Weight 1000mg = 1g 1000g = 1kg 1000kg = 1 tonne Imperial Approximations * 1 kilogram is about 2.2 pounds (2 pounds) You don t NEED to know anything marked with a star (*). These are just to allow you to relate metric units of measure to imperial units of measure. Page 21 of 110

22 L1-2 Functional Maths and Numeracy study guide Introduction to area, perimeter and volume MEASURES, SHAPE AND SPACE Page 22 of 110

23 L1-2 Functional Maths and Numeracy study guide Area AREA EXAMPLES Q1: Find the area of the following rectangles. a) b) Remember what area tells us: the amount of room inside a 2D shape. a) In this question the sides are given in cm. The corresponding unit of area will be cm 2. How many centimetre squares can fit inside the rectangle? Count them. 7 lots of 6 = 7 x 6 = 42cm 2 This is the same answer that you d get if you did length x width at the start. This proves that length x width IS a true, fast way of finding the room within a rectangle the AREA of a shape. b) Now you know it works, just do AREA = length x width = 8 x 4 = 32cm 2 Just in case you want to have a look: 32cm 2 in total. Page 23 of 110

24 L1-2 Functional Maths and Numeracy study guide Area Q2: A gardener has a lawn which is 9 metres long and 6 metres wide. He uses 20 grams of fertilizer per square metre on the lawn. How much fertilizer does he use? To begin you must calculate the area of the lawn. Area of lawn = length x width From the information in the question you can see the lawn is 9m in length and 6m in width. So 9 x 6 = 54m 2 (area) The question tells you that the gardener uses 20g of fertilizer per square metre of lawn. You know that the area of the lawn is 54m 2. Now calculate how much fertilizer he would use. 20 grams per square metre. The gardener has 54 metres to cover. So Amount of fertiliser per square metre x number of square metres = Total fertiliser used 54 x 20 = 1080 So 1080 grams of fertilizer is needed by the gardener. Page 24 of 110

25 L1-2 Functional Maths and Numeracy study guide Area Q3: A builder using square slabs 45cm long makes a path two slabs wide and 9m long. How many slabs are used? Convert the length of the path (9m) into the same unit of measurement as the slabs (cm). You know that 1m = 100 cm. So 9m = 900 cm. Now that you know the path is 900cm long, calculate how many 45cm slabs would fit into the full length. Calculate this by dividing 900cm by 45cm. This answer gives you the information that the path is 20 slabs long. How many slabs are needed for the whole path? 20 slabs on a 900cm path Look at the question again. It tells you the path is two slabs wide. So another row of 20 slabs is needed to complete the path so that it is wide enough. 2 rows of 20 slabs means that 40 slabs are needed to complete the path. Page 25 of 110

26 L1-2 Functional Maths and Numeracy study guide Volume VOLUME EXAMPLE Q1: What is the volume of this cuboid? Volume of Cuboid = Length x Height x Width Volume tells you the amount of space INSIDE a shape, or how many cubes can fit inside. The cuboid above has side lengths in METRES. The corresponding unit of volume will be CUBIC METRES (m 3 ). One cubic metre is 1m in length, width, and height: Volume = length x height x width = 5 x 3 x 3 = 45m 3 45 cubic metres! Page 26 of 110

27 L1-2 Functional Maths and Numeracy study guide Volume Or, ask the question, How many cubic metres fit inside the cuboid? You can fit 5 cubes lengthways, and 3 widthways. As a birdseye view: There are 15 cubes in total (count them!). However, the cuboid is 3m high, so there is space for 3 layers of 15 cubes: 3 layers of 15 cubic metres = 15 x 3 = 45 cubic metres (45m 3 ) Which is EXACTLY WHAT YOU GOT by just doing length x width x height. This shows that length x width x height DOES calculate the amount of room, or space, inside a cuboid. NB: It doesn t matter what order you do the multiplying, so don t worry! 3 x 3 x 5 = 45 3 x 5 x 3 = 45 5 x 3 x 3 = 45 Bottom line for cubes/cuboids: Length x Width x Height Page 27 of 110

28 L1-2 Functional Maths and Numeracy study guide Fractions - introduction FRACTIONS A number can be divided into fractions (or parts) In a fraction the bottom number (denominator) tells you how many parts the number is divided into. The top number (numerator) tells you the amount of those parts you are working with (shaded) Fractions are a good way of telling us how much of the total we have. If you get in a quiz it means you have got 6 correct out of the total of 10. The top number = how much you have, or the amount in question The bottom number = the total number of parts, or things. Page 28 of 110

29 L1-2 Functional Maths and Numeracy study guide Fractions - introduction Division shares, or splits a number into different parts. Fractions are the same. Examples means ONE HALF: 1 2 which means 1 whole split into 2 equal parts means ONE FIFTH: 1 5 which means 1 whole split into 5 equal parts means ONE EIGHTH: 1 8 which means 1 whole split into 8 equal parts The fractions above are called UNIT FRACTIONS. They all have 1 as the top number. The top number isn t always one! means THREE EIGHTHS. You don t just have one eighth, you have three of them: = 3 x In the same way: = 4 x = 2 x = 7 x and so on. Page 29 of 110

30 L1-2 Functional Maths and Numeracy study guide Fractions - introduction NB If the top and bottom number of the fraction are the same, the fraction will be equal to one whole (=1). Page 30 of 110

31 L1-2 Functional Maths and Numeracy study guide Fractions - introduction FRACTION QUESTIONS Q1: Look at the picture. How much of the shape has been shaded? First look at how many parts the shape has been divided into. The shape has been divided into four parts. The number of parts will become the bottom number of the fraction, 4 (the denominator). Now look at how many parts have been shaded = 1 This number is the top number of the fraction (numerator). So this fraction = ( of the circle is shaded; one part out of four). Picture examples Page 31 of 110

32 L1-2 Functional Maths and Numeracy study guide Fractions equivalent fractions EQUIVALENT FRACTIONS Equivalent fractions look different to each other but have the same value. looks different to but they BOTH represent the same fraction of the circle that is shaded. is the same size as. In other words: = Like the first example: looks different to but they BOTH represent the same fraction of the circle that is shaded. is the same size as In other words: = Page 32 of 110

33 L1-2 Functional Maths and Numeracy study guide Fractions equivalent fractions Equivalent Fractions are fractions that have the same value. is the same as is the same as You can go from one equivalent fraction to another by MULTIPLYING or DIVIDING BOTH the fraction s TOP and BOTTOM number by the same thing: Whatever you do to the top, you do to the bottom. And you can guess Whatever you do to the bottom, you do to the top. Page 33 of 110

34 L1-2 Functional Maths and Numeracy study guide Fractions equivalent fractions SIMPLIFYING FRACTIONS SIMPLIFYING a fraction means finding a simpler equivalent fraction. Q1: Simplify To simplify a fraction, you make the top and bottom as small as you can. Remember what you do to the top you have to do to the bottom. You want to make the numbers smaller. You will be DIVIDING. You need to think of a number you can divide both 45 AND 135 by. What fits into 45 AND 135? If a number ends in 5, you can divide by 5! Can you simplify again? What fits into 9 AND 27? How about 9? So Page 34 of 110

35 L1-2 Functional Maths and Numeracy study guide Fractions comparing fractions COMPARING FRACTIONS Comparing fractions is best done by using equivalent fractions to see which fraction is bigger than the other. Q1: Put these in order of size:,,. The easiest way is to rewrite the fractions as equivalent fractions with the same bottom number. 5, 10 and 8 are on the bottom what number can 5, 10 and 8 all fit into? One possible answer is 40. (40 5 = 8, = 4, 40 8 = 5) Now, make each bottom number equal to 40. Remember the rule: what you do to the bottom number you also do to the top number! 24 out of 40 is less than 25 out of 40 which is less than 28 out of 40! So you can see that the smallest fraction is, then and the biggest is. Page 35 of 110

36 L1-2 Functional Maths and Numeracy study guide Fractions adding and subtracting ADDING AND SUBTRACTING FRACTIONS When you add or subtract fractions it is important the bottom numbers (denominators) are the same. Q1: (Easy!) Find + This means you have one half plus one half. How many halves in total? TWO: make one whole). + = = 1 (Because two halves Q2: Find + This says Find two quarters plus one quarter. How many quarters in total? Two quarters + One quarter = Three Quarters. + =. Q3: Find + Here you have halves and quarters, which are not the same! First of all let s picture it. There are two different fractions of a whole. So, one half is actually the same as two quarters = Page 36 of 110

37 L1-2 Functional Maths and Numeracy study guide Fractions adding and subtracting So + = Remember equivalent fractions. To add or subtract fractions you need the bottom numbers to be the same. With +, how can you make the bottom numbers equal? You could multiply 2 by 2 to give us 4, so both bottom numbers would be 4. Remember: if you multiply the bottom by 2, you do the same thing to the top: So + = Q4: Find + = Again, you need the bottom numbers to be equal before you can proceed. You can multiply the bottom 3 by 8 to give 24, then the bottom numbers would be the same. If you multiply the bottom by 8, do the same to the top. So = = Page 37 of 110

38 L1-2 Functional Maths and Numeracy study guide Fractions multiplying MULTIPLYING WITH FRACTIONS A UNIT FRACTION is a fraction where the TOP NUMBER is 1. Fractions like are ALL UNIT FRACTIONS. Q1: Find of 100. This is asking you to find one fifth of 100. To find a fifth, you divide 100 into five fifths = 20 So of 100 = 20 Q2: What is x 100? As with any multiplication, x 100 means 100 lots of. One fifth + one fifth + one fifth +... etc...+ one fifth = one hundred fifths =. = So x 100 = 20 NB Look at Q1 and Q2 the answers are identical. Find a fraction OF a number means the same as that fraction MULTIPLIED by that number. When finding a FRACTION OF something, OF means x Page 38 of 110

39 L1-2 Functional Maths and Numeracy study guide Fractions multiplying Examples of 20 is the same as of 300 is the same as x 20. This means twenty quarters,. x 300. This means three hundred thirds,. It works the other way around: both x 100 and mean x 100 Q3: Find of 80. This question is asking you to divide the 80 into four quarters and to calculate the value of three quarters. To find a fraction of a number first divide the number given ( 80) by the number under the fraction line (in this question it is 4) = 20 This tells you that one quarter = 20 However, you want 3 quarters, so multiply the answer by x 3 = 60 The answer to of 80 is therefore 60. Page 39 of 110

40 L1-2 Functional Maths and Numeracy study guide Fractions mixed numbers MIXED NUMBERS A mixed number is a number with two parts: a whole number part, and a fraction part. Examples 2 means two and a quarter. 1 means one and two thirds. You can convert mixed numbers to fractions by looking at the fraction part. Q1: Write 2 as a fraction. HALVES are the fraction, so you have to think in halves. TWO HALVES make one whole. There are 2 wholes, which is = 4 halves. There is also an extra half (2 = two AND a half), which is 5 HALVES. Page 40 of 110

41 L1-2 Functional Maths and Numeracy study guide Fractions mixed numbers Q2: Write 4 as a fraction. Here you re dealing with thirds, so you must think in thirds. 4 means four AND two thirds. NB You might spot a pattern. Look at the above 2 examples. 2 = 4 = The top number, 5, equals (2 x 2) + 1. The top number, 14, equals (4 x 3) + 2. For any mixed number, where A, B and C are numbers: Page 41 of 110

42 L1-2 Functional Maths and Numeracy study guide Decimals introduction DECIMALS When a number is in decimal form it consists of whole numbers and a fraction part (parts of that number). The fraction part is to the right of the decimal point. If this is zero, it means that there is no fraction part. 7 = 7.0 = 7.00 = = = 13.0 = = and so on. The zero AFTER a decimal point can be knocked off, as long as there are NO NON ZERO NUMBERS after the zero. Examples = 1.57 these 0s can go since there s nothing after them. The 0 in CAN NOT be removed, because there is a 6 after the = There is a 2 and 3 after the first 0s, so those 0s stay. Q1. Put these decimal numbers into the correct order, from the lowest to the highest value. 7.6, 8.1, 6.9, 7.3, 7.9 Look at the whole numbers first: There is one decimal that begins with 6 so this will be the lowest value decimal 6.9 There are three decimals beginning with a 7 (7.6, 7.3 and 7.9). As there are several decimals beginning with a 7 you then need to look at the fraction part and put them into order from low to high. This will give you an order of 7.3, 7.6, 7.9 The final decimal in the list begins with an 8 and is therefore the highest decimal, 8.1. The answer to this question is therefore: 6.9, 7.3, 7.6, 7.9, 8.1 Page 42 of 110

43 L1-2 Functional Maths and Numeracy study guide Decimals multiplying and dividing by 10, 100, HOW TO DIVIDE AND MULTIPLY DECIMALS BY 10, 100 OR 1000 When multiplying decimal numbers the decimal point should be moved to the right to make the number bigger. When dividing decimal numbers the decimal point should be moved to the left to make the number smaller. You should move the decimal points as many times as there are zeros in the number you are multiplying by. So by 10 move once by 100 move twice by 1000 move three times Q1. Show how to divide and multiply the decimal Move the decimal point to the left DECIMAL Move the decimal point to the right DECIMAL Page 43 of 110

44 L1-2 Functional Maths and Numeracy study guide Decimals multiplying MULTIPLYING DECIMALS by other numbers (not 10, 100, 1000, etc.) Decimals are best treated as whole numbers when multiplying. Keep count of how many digits the decimal point(s) is in front of, then put the decimal back in at the end when you have your answer. Q1: What is 6 x 1.8? Here there is only one decimal 6 does not have a decimal. The decimal point is in front of ONE DIGIT the 8. Now, imagine any decimal numbers are WHOLE NUMBERS. 1.8 becomes 18. The question is transformed by doing this: 6 x x 18 You now do 6 x 18 in the usual way: Then you PUT THE DECIMAL POINT BACK IN. There was one decimal point which was in front of one digit, so in your final answer, you must put the decimal back in front of one digit: So 6 x 1.8 = 10.8 Page 44 of 110

45 L1-2 Functional Maths and Numeracy study guide Decimals multiplying Q2: What is 11 x ? Again there is only one decimal, but this time it is in front of FOUR DIGITS 0015 Again, you imagine any decimal numbers are whole numbers becomes stays the same it is already a whole number. 11 x x 15 Now do 11 x 15 in the usual way: Then, PUT THE DECIMAL POINT BACK IN. There was one decimal point which was in front of four digits, so in your final answer, you need to put the decimal back in front of four digits. The decimal jumps over the 5, the 6, the 1, then JUMPS OVER NOTHING for the fourth digit. Nothing in maths is written as 0. So, the answer to 11 x is Page 45 of 110

46 L1-2 Functional Maths and Numeracy study guide Decimals multiplying Q3: What is 0.16 x 10.5? There are two decimal numbers. One decimal is in front of TWO DIGITS (The 1 and 6 in 0.16) One decimal is in front of ONE DIGIT (The 5 in 10.5 ) IN TOTAL, THIS IS = 3 DIGITS. Now think of the numbers as whole numbers, without any decimals x x 105 Again, you do 16 x 105 in the usual way: The decimal points were in front of a total of three digits, so in your final answer, you must put the decimal point back in front of three digits. So the answer to 0.16 x 10.5 is Because the last digit is a 0 and it comes after the decimal, you can knock it off if you want to. The answer is 1.680, or if you like, 1.68 Page 46 of 110

47 L1-2 Functional Maths and Numeracy study guide Decimals dividing, and converting fractions to decimals FRACTIONS TO DECIMALS, AND DIVIDING WITH DECIMALS Fractions and divisions are closely linked. Q1: What is as a decimal? means 1 split into 4. It means 1 4. If the question was something like 4 2 it would be much easier! With 1 4, though, there will be remainders. You solve this by adding extra 0s to the number you are dividing: 1 = 1.0 = 1.00 = = and so on. Just as many 0s as you need! You then carry out the division, being sure to LINE UP THE DECIMAL POINTS: So (which is the same as 1 4) = 0.25 as a decimal. Q2. What is ? Something like 6 2 is the same as (six halves, which is three.) So can be written as.. Then you can use equivalent fractions to get rid of awkward decimals. Multiplying by 10, 100, and so on is an easy way to get rid of decimals. This tells you that is the same thing as Page 47 of 110

48 L1-2 Functional Maths and Numeracy study guide Decimals dividing, and converting fractions to decimals You can now do this division in the usual way. The bottom number goes on the outside: So = 52.5 as a decimal. Q3: What is ? Similar to before, you can write as. Then use the rules of equivalent fractions to change it into a nicer fraction: So is the same as 32 1, which is 32. It is much easier to divide by a whole number than a decimal. So, as a general rule: WHEN DIVIDING BY A DECIMAL, REWRITE THE DIVISION QUESTION AS A FRACTION, AND CHANGE IT INTO A FRACTION WITH WHOLE NUMBERS Page 48 of 110

49 L1-2 Functional Maths and Numeracy study guide Decimals dividing, and converting fractions to decimals Q4. What is two thirds as a decimal? Two thirds written as a fraction is. This means 2 3. Again, you can write 2 as if you need to. This would go on forever! = You can ROUND your answer to make a reasonable approximation. = 0.67 to the nearest hundredth. You round the 6 hundredths up to 7 hundredths, because the number to the right of it was 5 or bigger. Page 49 of 110

50 L1-2 Functional Maths and Numeracy study guide Decimals dividing, and converting fractions to decimals Q5: Find of 27. The best start would be to find of 27, which is the same as 27 5: So of 27 = 5.4 But you want ths which is four times as much, so you need to multiply by 4: 5.4 x 4 54 x 4 (Remember, first take the decimal point out, then replace it): There was only one decimal point which was in front of 1 number (the 4). So you now put the decimal point back in front of one number: So of 27 = 21.6 Page 50 of 110

51 L1-2 Functional Maths and Numeracy study guide Percentages introduction, converting percentages to fractions PERCENTAGES PERCENTAGE MEANS OUT OF ONE HUNDRED It is important to remember this. Examples = 80% = 3%. = 7.908% = 1 000% Using this fact, it is very easy to change from a percentage to a decimal: Q1: What is 40% as a fraction? 40% means 40 OUT OF 100, which is written as. You can now use what you know about simplifying fractions to make the fraction simpler: So, 40% = = Page 51 of 110

52 L1-2 Functional Maths and Numeracy study guide Percentages of amounts Percentages of Amounts Many questions will ask you to work out the percentage of a given value. The percentages of values can be broken down piece by piece.: Examples 100% is one whole, = 1 50% = half of a 100% = half of one whole = 25% = half of 50% = half of one half = 10% = 10 out of 100 = divide top and bottom by 10 5% = 5 out of 100 = 1% = 1 out of 100 = divide top and bottom by 5 Using these pieces you can find any percentage you want. Q1: What is 75% of 300? 75% = 50% + 25% From above, you know that 50% = and 25% =. So, 75% = + 75% of 300 = of of 300 of 300 = = 150 of 300 = = 75 So 75% of 300 = = 225 You could have done this any number of ways. For example, you could have worked out 1% of 300 (=3) then multiplied by x 75 = 225 Page 52 of 110

53 L1-2 Functional Maths and Numeracy study guide Percentages of amounts The same method can be used to calculate harder percentages: Source: John Thompson A (2008). The Percentage worker outer. worker outer Page 53 of 110

54 L1-2 Functional Maths and Numeracy study guide Percentage changes Percentage Changes These questions fall into two categories: 1) You are given a value and have to change it by a certain percentage. 2) A given value has ALREADY been changed and you have to work out the percentage change between the two values (start value and the end value). Q1: Last year a flight to Egypt for two people cost 500. This year the price has increased by 80%. What is the total price for two people THIS YEAR? This question is saying the price has increased by 80% OF THE START PRICE. You therefore need to find 80% of 500, and then add this to % = = =. So, you need to find FOUR FIFTHS OF 500. of 500 = = 100. ths of 500 will be four lots of this. So (=80%) of 500 is 400. Remember, though the question is asking what is 500 PLUS AN EXTRA 80%, NOT what is 80% of 500. So, % = = 900. Notice how you worked out 80% OF THE START PRICE. Always start with the first price, or first value. Q2: A Shop in town is having a sale. The banner reads 25% OFF ALL ITEMS!. Before the sale, a dress cost 80. How much will it cost in the sale? The question is saying the price has fallen by 25% OF THE START PRICE You therefore need to find 25% of 80 and then subtract this from % = = so you need to find ONE QUARTER OF 80. of 80 = 80 4 = 20. So the sale price = 80 25% = = 60. Again, you started by working out 25% OF THE START PRICE. Page 54 of 110

55 L1-2 Functional Maths and Numeracy study guide Percentage changes Q3: A gas cylinder weighs 5kg. When filled with gas it weighs 6kg. What is the percentage increase in the weight of the cylinder once the gas is added? You must calculate the PERCENTAGE CHANGE (in this case, the percentage increase). Like before, you start by using the START VALUE. For any question asking you to work out a % increase or % decrease: PERCENTAGE CHANGE = ( X 100 ) % In this question: START VALUE = 5kg FINAL VALUE = 6kg This change is therefore an INCREASE by 1kg. Using the formula above: PERCENTAGE INCREASE = ( X 100) % = ( X 100) % = 20% So the percentage increase in the weight of the cylinder is 20%. You can check this is correct by rethinking the question: The empty weight of the cylinder is 5kg. What if I Increase it by 20%? If my answer is right, this should make the weight 6kg as the question says. 20% = = = of 5 = 5 5 = 1 So 5kg + 20% = 5kg + 1kg = 6kg. This proves the answer is right. Page 55 of 110

56 L1-2 Functional Maths and Numeracy study guide Percentage changes Q4: Katie has been on a diet for 3 months. Her starting weight was 12 stone. She now weighs 10 stone. By what percentage has her weight decreased? Again, remember: PERCENTAGE CHANGE = ( X 100) % In this question: START VALUE = 12 STONE FINAL VALUE = 10 STONE This change is therefore a decrease by 1 stone, or 1.5 stone in decimals. PERCENTAGE DECREASE = ( X 100 ) % =. X Looks awkward but you can simplify it using equivalent fractions:. = = So % Decrease =. X 100 = X 100 = So the percentage decrease = 12.5%. Her weight has decreased by one eighth. Page 56 of 110

57 Converting between fractions, decimals and percentages CONVERTING BETWEEN FRACTIONS, DECIMALS AND PERCENTAGES FRACTIONS DECIMALS You can go from fractions to decimals by direct division: So = 0.75 FRACTIONS PERCENTAGES Once you have turned the FRACTION into a DECIMAL, you then MULTIPLY BY 100 to find out the percentage. This works because decimals and fractions show PARTS OF ONE WHOLE. A percentage tells us PARTS OF ONE HUNDRED. Multiply the decimal by 100 by moving the decimal point 2 DIGITS TO THE RIGHT: So = 0.75 = 75% DECIMALS PERCENTAGES Multiply the decimal by 100 to find the percentage So = 12.5% Page 57 of 110

58 Converting between fractions, decimals and percentages DECIMALS FRACTIONS Once the decimal is converted into a percentage, use the fact that % means OUT OF 100 to convert the percentage into a fraction. Then simplify that fraction, if possible: So = 12.5% = PERCENTAGES FRACTIONS Percentage means out of 100, so A PERCENTAGE CAN BE WRITTEN AS A FRACTION OUT OF 100. This fraction may then be simplified: So 88% = PERCENTAGES DECIMALS You DIVIDE BY 100 to go from a percentage to a decimal. This means moving the decimal point 2 DIGITS TO THE LEFT: So 8.4% = Page 58 of 110

59 Rounding whole numbers ROUNDING Rounding can be useful when making approximations (also called estimations). We round all the time without thinking about it. How many people live in the UK? I bet it s about 62 million this is rounding the nearest million. How many were at Wembley for that Concert? I think the news said this is rounding to the nearest hundred. When rounding, it is helpful to think of the number in Hundreds, Tens, Units form. ROUNDING TO THE NEAREST 1, 10, 100, 1 000, AND SO ON When rounding to the nearest whole number, or a number ending in zero (10, 100, and so on) think of the number in terms of hundreds, tens, units. Here is the number written in this way: Page 59 of 110

60 Rounding whole numbers Q1: A total of people competed in a Marathon. How many is this to the nearest hundred? You are rounding to the nearest 100. First of all write out the number as hundreds, tens, units and so on: The 100s column is underlined because you are rounding to the nearest 100. Now look at the digit TO THE RIGHT of what you re rounding to: This digit tells you how to round the hundreds column you re concerned with: IF THE DIGIT IS 5 OR MORE, ROUND UP BY 1 IF THE DIGIT IS LESS THAN 5, DON T ROUND UP! Since 9 is bigger than 5, you ROUND UP. The 8 is rounded up by 1, and becomes 9. ALL DIGITS TO THE LEFT OF IT BECOME ZEROS. So, rounded to the nearest hundred is Page 60 of 110

61 Rounding whole numbers Q2: What is 192 to the nearest 10? You can write out 192 in hundreds, tens and units. The 9 is how many tens you have, and TO THE RIGHT is the 2 : IF THE DIGIT IS 5 OR MORE, ROUND UP BY 1 IF THE DIGIT IS LESS THAN 5, DON T ROUND UP! 2 is less than 5, so you DO NOT change the 9. Any numbers to the left of the 9 become zero. So, 192 to the nearest 10 is 200. Q3: What is 195 to the nearest 10? IF THE DIGIT IS 5 OR MORE, ROUND UP BY 1 IF THE DIGIT IS LESS THAN 5, DON T ROUND UP! Because the number TO THE RIGHT of the tens column is 5, you ROUND UP. 9 is rounded up by 1 to 10. Because 10 is 2 digits, the rounding up is done by carrying the 1 over to the left. The 9 becomes 0 and the 1 has 1 added to it, to become 2. So, 195 to the nearest 10 is 200. Page 61 of 110

62 Rounding decimals Q4: What is to the nearest hundredth? Now you are being asked to round to a decimal amount. Again, though, you can use the same idea as before and put the number in hundreds, tens and units: The DECIMAL POINT separates the whole part from the fraction part of the number. To the left of the decimal point is a whole number, and to the right is a part of a whole. Underline the hundredths, and look at the digit TO THE RIGHT: The 8 is bigger than 5 so you ROUND UP. The 0 becomes a 1, and all numbers to the right of it become zero. So, to the nearest hundredth is Because the last 0 is unnecessary, you must simplify this to just Page 62 of 110

63 Rounding decimals Q5: A shop owner is calculating the price per unit of sweets in pounds. He uses his calculator and gets What is this to the nearest penny? Since this is in pounds, the calculator is saying The number TO THE RIGHT of the pennies column is 7, so you ROUND UP. The 7 becomes an 8. So, to the nearest penny is 18p. Page 63 of 110

64 Rounding decimals Q6 (HARD!): A jogger does a warm up run. He calculates that he has run km. How far is this to the nearest 10m? The units you are using are kilometres. 1km = 1000 m To go from km m, you multiply by 1000, so move the decimal place TO THE RIGHT by 3 DIGITS: So 8.156km = 8156m You can now round this to the nearest 10m in the usual way: The 6 is 5 or more, so the underlined 5 is ROUNDED UP to 6. So, 8 156m to the nearest 10m is 8 160m 8 160m = km (dividing by to go back to kilometres) The last 0 is unnecessary. So IN KILOMETRES, 8.156km to the nearest 10m = 8.16km Page 64 of 110

65 Rounding - alternative method You can also think of rounding in an entirely different way Q1: What is to the nearest 100? You re rounding the nearest 100. So, think of MULTIPLES OF 100. Which multiple of 100 is closest to? Multiples of 100 are the 100 times table: 100, 200, 300, 400 (fast forward a bit!) 2 999, 3 000, is between and is 88 AWAY from 3 000, but only 12 AWAY from So, to the nearest 100 is Q2: A care s mileage reads How many miles is this to the nearest miles? Here you re rounding the nearest So you need to think in MULTIPLES OF , , , , is between and would be EXACTLY HALF WAY between and So, is MORE THAN HALF WAY. So, miles to the nearest miles is miles. Page 65 of 110

66 Rounding to more difficult limits A big benefit of this method is that it can be used to round to more difficult limits, like quarters, halves, and so on. Q3: What is 3.68kg to the nearest quarter of a kilogram? = 1 4 = 0.25 So a quarter of a kilogram = 0.25 kg. Because you re rounding to the nearest quarter, think in MULTIPLES OF ONE QUARTER. 0.25, 0.50, 0.75, 1.00, 1.25, 1.50, 3.00, 3.25, 3.50, is between 3.50 and is greater than 3.50 by 0.18, but it is only 0.07 smaller than So, 3.68 is closer to 3.75 than 3.50 So 3.68 to the nearest quarter is 3.75 Page 66 of 110

67 Ratio RATIO When a number is divided into different parts it can be described as a ratio. Examples: Q1: If Tom and Harry are given 20 to divide in the ratio 3:2 how much do they each receive? Look at the order of the question Tom, Harry. Then 3:2 TOM HARRY 3 2 Add the ratios together to calculate how many parts the 20 is to be divided into altogether = 5 so 5 parts in total. 5 parts = 20. So how much is one part equal to? 20 5= 4 So each part represents 4 Now look again at the question to tell you how the money is to be divided. Tom gets 3 parts and Harry gets 2 parts. You know that 1 part = 4 so now calculate Tom and Harry s share: Tom = 4 X 3 = 12 Harry = 4 X 2 = 8 So to share 20 in the ratio of 3:2 Tom gets 12 and Harry gets 8. Page 67 of 110

68 Ratio Q2: Read the instructions on the paint tin. How much white paint should you mix with 500ml of red paint to get a tin of pink paint? Now look at the question in red above. How much white paint do you need to make a tin of pink paint when using 500ml of red paint? The 500ml of red paint represents 1 part of the total tin of pink paint. red 1:4 white Remember you need 5 parts to make a full tin so you now need an extra 4 parts of white paint. You know that 1 part equals 500ml so to calculate the other 4 parts you need to multiply the 500ml by x 4 = 2000ml So 1000ml = 1 litre 2000ml = 2 litres So you need 2000ml (= 2l) of white paint. Page 68 of 110

69 Ratio Another Method Q2: Read the instructions on the paint tin. How much white paint should you mix with 500ml of red paint to get a tin of pink paint? MIX 1 PART RED WITH 4 PARTS WHITE Another way of thinking of this is that for every one lot of red, there are four lots of white. If you had two lots of red (1+1) we d need eight lots of white(4+4): RED WHITE = = = 2000 So for 500ml red you need 2000ml (=2l) of white. NB Because ratios are like fractions, you could also treat this like a fraction problem: = = =. So 2000ml white to go with 500ml red Page 69 of 110

70 Ratio Q3: An artist takes 60 paintings to be displayed at an exhibition. He displays them in the following ratio: How many portrait paintings will he take? Here, you need to see that the TOTAL OF ALL THE RATIO IS IMPORTANT. In this case, the total represents the total number of paintings: 12 : 3 : = Total of 20 But the total number of paintings the artist is taking is 60. THIS IS THREE TIMES AS MANY AS 20: LANDSCAPE PORTRAIT STILL LIFE TOTAL x by 3 x by 3 x by 3 x by So he takes 36 landscapes, 9 PORTRAITS, and 15 still life paintings. Ratio is used in a wide variety of problems, including conversions and scale drawings Page 70 of 110

71 Ratio conversions CONVERSIONS Q1: A builder is measuring a panel. He needs it to be 35cm long. If 1 inch = 2.5 centimetres, how long will the panel be in inches? Method 1 For every 1 inch, there are 2.5 centimetres. As a ratio this is 1:2.5 So, if there were 2 inches, this would be 5 centimetres, and so on inches cm So 35cm = 14 Method 2 Remember, ratios are very much like FRACTIONS: =. x top & bottom by 2 x top & bottom by 7 So 35 centimetres = 14 inches. Method 3 The ratio of inches to centimetres is 1:2.5 So, the number of centimetres is found by multiplying the inches by 2.5. Therefore, THE AMOUNT OF INCHES IS FOUND BY DIVIDING THE CMs BY 2.5 So number of inches = =. = = 14 Page 71 of 110

72 Ratio conversions Q1: An architect is constructing a scale model of a theatre. The scale of the model is 1:50.The length of the theatre is 20m. What will be the length of the model? Obviously, any scale model will be smaller than the real thing! So, THE SMALLER NUMBER REPRESENTS THE MODEL, THE LARGER NUMBER REPRESENTS THE REAL THING. This tells you that the theatre will be 50 times bigger than the model, and the model will be th the size of the theatre. Method 1 The length of the model will be the length of the theatre. Length of model = = = = = 40cm Method 2 Remember, ratios are like FRACTIONS. To avoid nasty decimals, you can work with centimetres: = = = = So for a theatre length of 2 000cm (= 20m) the length of the model is 40cm. Page 72 of 110

73 Ratio conversions Q2: A plan of a house is drawn on paper. The scale of the plan is 1:25. On the plan, the chimney is 24cm above ground level. How much above ground level is the ACTUAL CHIMNEY? The scale this time is 1:25. Again, the plan of a house will be smaller than the actual house! This tells us the actual house will be 25 times bigger than the plan, and the plan will be the size of the house. Method 1 The chimney s height will be 25 times greater than its height on the plan: So the actual chimney s height is 600cm, or 6m. Method 2 Remember, ratios are like FRACTIONS. To avoid nasty decimals, you can work with centimetres: = = =. So a chimney height of 24cm on the plan, will have an actual height of 600cm (=6m). Page 73 of 110

74 Ratio conversions Q3. A family are travelling across Europe. One day in their car they cover a total distance of 320km. How far is this journey in miles? Take 5 miles to equal 8 kilometres. The information in the question tells you that the ratio of miles to kilometres is 5:8 Method 1 Using the fact that ratios are like fractions, =. This tells us that the number of miles is the number of kilometres. They have travelled 320km. th of 320 = = 40 So ths of 320 = 5 x 40 = 200 So 320 km = 200 miles. Method 2 Again, treating the ratio like a fraction, you can find out how many miles are travelled by using equivalent fractions: = = = = So a distance of 320 kilometres is the same as 200 miles. Page 74 of 110

75 Ratio scale drawings SCALE DRAWINGS A SCALE DRAWING is a drawing or diagram that is IDENTICAL IN PROPORTION to the thing it is representing, but is a DIFFERENT SIZE. This sounds more complicated than it actually is! Example Imagine a pitch, 20m long by 10m wide (so the length is DOUBLE the width). A SCALE DRAWING of the pitch could be a rectangle 20cm long by 10cm wide. The PROPORTIONS of the field AND the drawing of it are the same the length is DOUBLE the width in each case. However, they are DIFFERENT SIZES: 20m x 10m compared to 20cm x 10cm. 1m = 100cm, so the length of the ACTUAL FIELD is 100 TIMES BIGGER than the length of the plan. Because of this, you would say the SCALE OF THE PLAN is 1:100 On the DRAWING OF THE PLAN, you could instead have said something like, 1cm represents 1m So the 20cm length represents the ACTUAL LENGTH of 20m on the pitch and the 10cm width represents the ACTUAL WIDTH of 10m on the pitch. You will ALMOST ALWAYS be told the scale of the drawing next to the drawing. This is needed so you can work out the ACTUAL SIZE, or ACTUAL DISTANCE, of the real thing. Questions involving scale drawings MIGHT need a RULER. Page 75 of 110

76 Ratio scale drawings Q1: A builder is laying some square decking. He uses a scale drawing: What is the actual length of decking? Here is a scale drawing and the LENGTH HAS BEEN GIVEN TO YOU. You don t need to measure anything yourself. It says Scale: 2cm to 1m. The scale drawing (of anything!) will be smaller than the real thing, so the SMALLER VALUE (2cm) represents the DRAWING, and the LARGER VALUE (1m) represents the ACTUAL SQUARE DECKING. So, the ratio of the drawing to the decking is 2cm : 1m. Using the fact ratios are like fractions: = = So for a drawing length of 10cm, in the diagram above, represents an ACTUAL LENGTH OF DECKING of 5m. Page 76 of 110

77 Ratio scale drawings Q2: The diagram shows a plan of a surgery waiting area: What is the actual length of the waiting area? Because you have been told no lengths, and you need to find out the length, you need to MEASURE THE SCALE DRAWING WITH A RULER: The length of the PLAN is 12cm. You were told the scale is 2 centimetres = 1 metre (= 100 cm). So the ratio of plan to room is 2:100 = = The scale drawing (above) measured 12cm. This tells you that a drawing of 12cm represents an actual LENGTH OF 6m. Page 77 of 110

78 Ratio scale drawings Q3. The map below shows part of the route of a marathon race. The starting point is Point A. What is the actual distance from Point A to Point B on the road? Page 78 of 110

79 Ratio scale drawings The only thing you have been given is the scale 1: You have to measure the scale drawing yourself. Measure the line from A to B: This measures 5cm. You know the scale is 1: The map will be smaller than the actual road! So, it MAKES SENSE that the 1 represents the map, and the road. You can again use the fact that ratios are like fractions: = = So 5cm on the map represents an ACTUAL LENGTH of cm 100cm = 1m cm = ( ) m = 1250m So in more reasonable units, the actual distance is 1 250m. (This is just short of a mile). NB If your answer came to thousands of miles, would this be realistic? THINK! Page 79 of 110

80 Negative numbers - introduction NEGATIVE NUMBERS This number 8 is often mistaken for a Negative Number, because he can have a negative attitude. 8 isn t negative he s just outspoken. When we say Negative Number, we mean something quite different: The numbers in RED are known as POSITIVE NUMBERS. The numbers in BLUE are known as NEGATIVE NUMBERS. A negative number always has a MINUS SIGN in front of it. You see negative numbers all the time in life. On weather forecasts, cold temperatures are MINUS NUMBERS (Blue = cold!) 7 C, 2 C, 1 C All very cold! Minus numbers are also used with banking. If you have 10 in your account, and withdraw 15, you are 5 OVERDRAWN. This is the same as saying you have 5. Page 80 of 110

81 Negative numbers addition and subtraction Minus numbers are the EXACT OPPOSITES of their positive number counterparts 1 is opposite to 1 2 is opposite to 2 59 is opposite to 59 And so on. ADDING ON A NEGATIVE NUMBER First of all, let s see if you can spot a pattern = = = = 4 The number you are adding to 4 is getting smaller by 1 each time. As a result, the answer is becoming smaller by 1 each time. So FOLLOWING THE PATTERN what will come next? You need to add one less than 0. This is 1. The answer will be 1 smaller than the last answer. This is 3. So the pattern will continue: = = = = = 3 (4 1 = 3) = 2 (4 2 = 2) = 1 (4 3 = 1) = 0 (4 4 = 0) Notice how the answers in brackets are the same as the answers to the left. ADDING A MINUS NUMBER IS DONE AS A NORMAL SUBTRACTION Page 81 of 110

82 Negative numbers addition and subtraction Q1: What is ? Using our rule, = = 15 Further Explanation Remember: Minus numbers are the EXACT OPPOSITES of the normal numbers. So 12 is the opposite of 12. Use this to help. Imagine the question was What is ? You d just ADD THE 12. The answer is 39. BUT in Q1 above you re not adding 12. You re adding MINUS 12, which is the OPPOSITE. So, YOU DO THE OPPOSITE. The opposite of adding 12 is SUBTRACTING 12. So, = = 15. If you are ever confused with adding negative numbers, remember that: Minus numbers are the EXACT OPPOSITES of their positive number counterparts And as a result of this, adding a minus number has the OPPOSITE EFFECT of normal addition. The OPPOSITE EFFECT OF NORMAL ADDING is SUBTRACTING. Which is why ADDING A MINUS NUMBER IS DONE LIKE NORMAL SUBTRACTION Page 82 of 110

83 Negative numbers addition and subtraction TAKING AWAY A NEGATIVE NUMBER Subtraction tells you the DISTANCE BETWEEN TWO NUMBERS. What is the distance between 5 and 4? What is 5 4? The answer in both cases is 1. What is the distance between 21 and 15? What is 21 15? The answer in both cases is 6. So when you need to subtract, think what is the distance between the two numbers? Q1: What is 4 1? You can think What is the difference between 4 and 1? Using a NUMBER LINE makes it easier: There is a distance of 5, so 4 1 = 5. You can also see this by looking for a pattern as you subtract numbers from 4: 4 3 = = = = = 5 As the number you are subtracting from 4 gets SMALLER BY 1 each time, the answer follows the pattern of getting BIGGER BY 1 each time. So, continuing the pattern, 4 2 would be 6. Check it with a number line! The distance between 2 and 4 is 6. Page 83 of 110

84 Negative numbers addition and subtraction Q2: What is 1 5? Use a number line, and again consider the distance between 5 and 1: There is a distance of 6, so 1 5 = 6. Further explanation If ever confused with adding negative numbers, remember that: Minus numbers are the EXACT OPPOSITES of their positive number counterparts And as a result of this, subtracting a minus number has the OPPOSITE EFFECT of normal subtraction. The OPPOSITE EFFECT OF NORMAL SUBTRACTION is ADDING, which is why SUBTRACTING A MINUS NUMBER IS DONE LIKE NORMAL ADDITION Summary: ADDING A MINUS NUMBER = SUBTRACTION SUBTRACTING A MINUS NUMBER = ADDITION Page 84 of 110

85 Negative numbers addition and subtraction ADDING TO, OR SUBTRACTING FROM, A NEGATIVE NUMBER You can think of adding to/subtracting from ANY NUMBER on a number line. Let s start with something straight forward: What is 5 + 3? is obviously 8, but let s think about it on a number line: When you add, start at your first number then MOVE TO THE RIGHT by the amount of the number you are adding. When subtracting, you just move to the left! When you subtract, start at your first number then MOVE TO THE LEFT by the amount of the number you are subtracting. Page 85 of 110

86 Negative numbers addition and subtraction What is 1 3? What is 2+ 8? So, 1 3 = 2. Page 86 of 110

87 Range The RANGE AVERAGES and RANGE THE RANGE of a set of numbers tells you the DIFFERENCE between the SMALLEST VALUE and the LARGEST VALUE. It s not too different to real life, really! Example Here at Mobiles 4 U we have a HUGE RANGE of price plans on our phones, FROM 7.50/month standard plan TO 30/month with all you can eat browsing! What the salesman is getting at is the DIFFERENCE in prices of the phone contracts. To find the difference, you subtract the LOWEST from the HIGHEST: = And this is the RANGE of prices. For a set of numbers: RANGE = HIGHEST VALUE LOWEST VALUE Q1: Find the range of this set of lengths: 5cm, 8cm, 9cm, 8cm, 6cm, 2cm, 5cm, 7cm, 6cm, 8cm, 7cm, 8cm The LONGEST LENGTH is 9cm. The SHORTEST LENGTH is 2cm. So, the RANGE is 9 2 = 7cm. Page 87 of 110

88 Averages mean, mode and median CALCULATING AVERAGES There is more than one type of average: mean, mode, and median. The MEAN To find the mean add all the values together then divide the answer my how many numbers you have used. The mean can take a while to work out because it involves ADDING EVERYTHING UP: THE MEAN IS MEAN! Q2: The number of calls made by Sally on her mobile, per day, over one week is 3, 4, 0, 6, 8, 4, 3. What is the mean number of calls per day? Mean = MEAN = = = = 4 Therefore, the mean number of calls Sally made over the week is 4. The MODE is the most common value in a set of numbers Q3: Find the mode of this set of numbers: 1,3,2,1,3,1,0,1,4,2,3,4,1,2,3. 0 appears once 1 appears 5 times 2 appears 3 times 3 appears 4 times 4 appears twice The number 1 is the most common number. So the mode is 1 Things relating to the MODE are called MODAL. It s the same thing. NB It is possible for a set of data to have more than one mode. Page 88 of 110

89 Averages mean, mode and median The MEDIAN is the middle value in an ordered list. Q4: Find the median of this set of numbers: 1,3,2,1,3,1,0,1,4,2,3,4,1,2,3. First PUT THE LIST IN ORDER, LOWEST TO HIGHEST: 0,1,1,1,1,1,2,2,2,3,3,3,3,4,4. Count from either end of list towards the middle to find the middle value. Seven numbers can be counted from either end to find the middle value of 2. Therefore the median of this set of data is 2. Page 89 of 110

90 Averages mean, mode and median Q5: Find the median of this set of numbers: 8, 6, 4, 1, 1, 12, 9, 1, 9, 8, 9, 4 Again, PUT THE NUMBERS IN ORDER: 1, 1, 1, 4, 4, 6, 8, 8, 9, 9, 9, 12 Again, count inwards towards the middle from both ends: There are 2 middle numbers! If there are two middle numbers the median is EXACTLY HALFWAY between these two numbers This is the same as adding the 2 numbers up then halving = 14, then 14 2 = 7. 7 is half way between 6 and 8; The median is SEVEN If both middle numbers are the same, the median is the same number: Example Find the median of this set of numbers: 1, 3, 3, 4 There are two 3s in the middle. The Median is 3. Page 90 of 110

91 Representing data key vocabulary REPRESENTING DATA There are various ways to represent data, and some ways are better than others, depending on what you want to show. Pages give you a summary of the common ways to represent data, with some advantages and disadvantages to each. First though, a note about language and definitions with graphs and charts: Title tells you what the chart/graph is showing. Every graph should have a title. The title goes at the TOP OF THE GRAPH rarely at the bottom. Horizontal Axis is a LINE ACROSS the PAGE, running directly left to right. The axis should be labelled, so you know what it is measuring. Vertical Axis is a LINE GOING UP THE PAGE (think vertigo a fear of heights). Key tells you WHAT THINGS MEAN in the graph. For example, a key might tell you which line is for boys and which is for girls, or what each colour stands for. Page 91 of 110

92 Representing data discrete and continuous data DISCRETE AND CONTINOUS DATA Data can be thought of as being either discrete or continuous. Data has to be ONE OR THE OTHER IT CAN T BE BOTH. CONTINUOUS DATA CONTINUOUS DATA is data that can flow from one value to another. Data like this is also INFINITE it can take be an endless number of different values. Examples include: SPEED WEIGHT TIME HEIGHT ENERGY Height is continuous. A tree got to the height it is now by GRADUAL and CONTINUAL growth from something small to something big. Time is continuous. It FLOWS. You CAN NOT SAY FOR DEFINITE It s either 1 minute or 2 minutes what about 1½ minutes? Seconds? Milliseconds? Billionths of a second? Time has endless values. Continuous data is ALWAYS NUMERICAL, and for any 2 different continuous values, you can work out half way between them. DISCRETE DATA DISCRETE DATA is things that can be counted, with definite values. Examples: Day of week (Mon, Tue, Wed, Thur, Fri, Sat, Sun) Month (Jan, Feb, Mar, Apr, May, Jun, Jul, Aug, Sep, Oct, Nov, Dec) Sex of baby (Boy, Girl) Favourite team (Man Utd, Arsenal, Liverpool, Huddersfield Town) For any 2 different discrete values, it is often nonsense to work out what is half way between them. What is half way between Monday and Tuesday?!! It makes no sense! It s EITHER Monday, OR it s Tuesday. Page 92 of 110

93 Representing data bar charts BAR CHARTS A bar chart is made up of bars (like chocolate bars) with gaps between them: Example 1 Example 2 Page 93 of 110

94 Representing data bar charts Bar Chart Facts Has a HORIZONTAL AXIS AND VERTICAL AXIS. The horizontal ( bottom ) axis DOES NOT HAVE TO BE NUMERICAL. Are used to represent discrete data. Usually the horizontal axis is the type of item(s) being represented (male/female, year, month, type of TV show) and the vertical axis tells us how much/how many people/items. Bar charts are VERY CLEAR. It is easy to see if one bar is bigger than another, and by how much. THEY CAN SOMETIMES BE UPDATED. For example, if the horizontal axis is year (2009, 2010, 2011, and so on) then the next year s information can be added on by drawing another bar for that year. Bad Points Bar charts are only really useful if the data you are showing can be grouped into a handful of different groups. You don t want to have a chart with hundreds or thousands of bars! As a general rule, bar charts with more than 10 bars are rare. Page 94 of 110

95 Representing data line graphs LINE GRAPHS A line graph has both horizontal and vertical axes, like a bar chart. Instead of drawing bars, you draw points. The points are then connected WITH STRAIGHT LINES. Example 1 Example 2 Page 95 of 110

96 Representing data line graphs Line graphs do share a lot in common with bar charts: Has a HORIZONTAL AXIS AND VERTICAL AXIS. THEY CAN SOMETIMES BE UPDATED. For example, if the horizontal axis is year (2009, 2010, 2011, and so on) then the next year s information can be added on by drawing another point for that year. There are some differences, though. A line graph is used to plot continuous data. A line graph is intended to show PATTERNS or TRENDS, hence why the points are LINKED TOGETHER with straight lines. Data is only recorded where actual points are plotted. For example, in Example 2, there was no actual data recorded for 1965, although there is a line that goes through this date. Line graphs can have MORE THAN ONE LINE. In those cases, a KEY is needed so you can work out which line is for what. Bad Points Line Graphs are only really useful for a handful of different points. With lots and lots of different data values a line graph would look like a scribbly mess and the values would become hard to read. Page 96 of 110

97 Representing data scatter graphs SCATTER GRAPHS A scatter graph is a collection of different points SCATTERED across a grid. A scatter graph will have a horizontal and vertical axis. Example 1 Example 2 Page 97 of 110

98 Representing data scatter graphs Scatter Graph Facts Great for showing lots and lots of data values ideal for when you have too many things to show on a bar chart or line graph for example! Both the horizontal axis and the vertical axis are numerical. Because of this, Scatter Graphs are great for showing large amounts data where both groups are numerical. Examples could be: Length of newborn babies VS Weight of newborn babies; Number of police in an area VS Number of crimes in that area; Wealth of countries in dollars VS Population of countries The points can be plotted ANYWHERE, hence Scatter Graph. They don t need to be drawn in any order. On a line graph you may have a value for 1, value for 2, value for 3, value for 4, and so on, in a nice order. On a scatter graph, however, the points and numbers can be anything: 2.3, 19.4, 2.0, 6.9 Scatter graphs are great for showing a GENERAL RELATIONSHIP between two different things. In Example 1 you can see a clear relationship between the ages of husband and wife, but in Example 2 the relationship between the age of adults and their weights seems random (as you might expect). EVERY SINGLE PIECE OF DATA is displayed. Bad Points Individual points are not labelled, so you can t tell which point is for what person, or for what country, etc. Page 98 of 110

99 Representing data pie charts PIE CHARTS A pie chart is a circular representation of data. It gets its name from the food it looks like a pie cut up into different sized slices: Example 1 Example 2 Page 99 of 110

100 Representing data pie charts Pie Chart Facts The main purpose of pie charts is to CLEARY SHOW THE PROPORTION (ie fraction or percentage) OF THE TOTAL each group has. In example 1, about 50% (about half) of viewers are of age In example 2, just under a quarter is Lime. The actual pie chart picture on its own is no good you need either labelling to go with it (like Example 1) or a KEY (like Example 2). Pie charts are highly visual, and are therefore good for displays and presentations. Bad Points On their own merits, Pie Charts are USELESS for telling us the ACTUAL NUMBERS of things they only show the PROPORTION of things. If you want the actual numbers and actual totals, pie charts are not appropriate. Look at the examples one had 4 slices, the other had 3 slices. Pie Charts are only useful when there is a HANDFUL OF THINGS TO REPRESENT. Any more than that and the Pie Chart loses its visual edge all slices start to look thin and it can be hard to see which slices are bigger and which are smaller. They cannot be updated once drawn. In general, if you are wanting to show the PROPORTION, the PERCENTAGE, or the FRACTION of values from different groups PIE CHART Page 100 of 110

101 Representing data pictograms PICTOGRAMS A Pictogram displays information through the use of symbols, or PICTURES, as the name suggests. Example 1 Example 2 Page 101 of 110

102 Representing data pictograms Pictogram Facts Every Pictogram needs a KEY to tell us what each picture represents. If a pictogram is without a key, it is INCOMPLETE. Pictograms are probably THE CLEAREST WAY OF DISPLAYING DATA and as such are a popular choice for notice boards and public displays. Pictograms are EASILY UPDATED. In Example 2, the pictogram could easily be updated by drawing in an extra house every time the next 100 homes are sold. They are convenient for younger people, or people who might not be comfortable with bar charts and other types of displays Bad Points Pictograms don t DIRECTLY tell us the total numbers of things. This needs to be worked out from how many pictures it has, and how much each picture represents. Page 102 of 110

103 Representing data tally charts TALLY CHARTS A tally chart can be thought of as the most basic pictogram: Example This tells us that Hannah owns 8 shirts, Alice 7, Ian 10, and Ray 3. A Tally Chart is the only kind of pictogram without a key. You are just expected to know already that each line represents 1 item, and each gate represents 5 items. If it DID have a key, it might look like this: Aside from this, they are the same as any other pictogram. Often, a tally chart is typically the first kind of pictogram or any mathematical graph that people use. Page 103 of 110

104 Representing data frequency tables FREQUENCY TABLES Frequency means How Many. A frequency table is a very simple and clear way of presenting data: Example Car registration plates sold at a second hand dealership. This tells us 20 R plates were sold, 36 S plates, 41 T plates and 23 V plates. A GROUPED FREQUENCY TABLE is the same idea as a normal frequency table, but puts the data into groups. Example GROUPED FREQUENCY TABLES are good for arranging lots of data into more manageable groups. A benefit of this is if you want to find the MODE OF A GROUP (THE MODAL GROUP) instead of the MODAL VALUE. The MODAL GROUP in the example above is age because there are more in that group than any other group. This is often useful for businesses and corporations. For example, it might help Disney s marketing department to know that the MODAL AGE GROUP who watch Mickey Mouse is 5 10 year olds. Companies tend to market products at age groups. Page 104 of 110

105 Order of calculations - introduction ORDER OF CALCULATIONS Sometimes, it is not easy to know what to do first. Example What is x 2? Do you work out what is, and then multiply that by 2? [2 + 2 = 4, then 4 x 2 = 8] OR Do you start with 2, and then add on what 2 x 2 is? [2 + 4 = 6] Who knows? Example What is x ? NIGHTMARE! You need a rule so you know what to do first, what to do second, what to do third, and so on. Page 105 of 110

106 Order of calculations - BODMAS The rule is called BODMAS: BODMAS tells you what to do. As you read down the list: 1) FIRST, work out the value of what is inside BRACKETS. 2) SECOND, look for OTHER stuff that is NOT brackets, division signs, multiply signs, subtraction signs, addition signs. This is usually POWERS things like 4 2, 2 3. The LITTLE NUMBER tells us HOW MANY TIMES to multiply the BIG NUMBER by itself: 4 2 = 4 x = 2 x 2 x 2 x 2 x = 10 x 10 x 10 And so on. 3) THIRD, do the DIVISION and MULTIPLICATION (any order) 4) LAST, do the ADDITION and SUBTRACTION. Page 106 of 110