Download Functional Maths and Numeracy Study Guide

Functional Maths and Numeracy study guide
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CONTENTS 1) THE FOUR RULES OF ARITHMETIC ‐
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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 ‐
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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) 6) ROUNDING ‐
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7) RATIO ‐
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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) ‐
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5) PERCENTAGES ‐
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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) Introduction & Worked Examples (page 67) Conversions & Proportion (page 71) Scale drawings (page 75) 8) NEGATIVE NUMBERS ‐
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Introduction (page 80) Addition and subtraction (page 81) 9) AVERAGES and RANGE ‐
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Range (page 87) Mean, Mode, Median (page 88) 10) REPRESENTING DATA ‐
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4) DECIMALS ‐
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Whole numbers (page 59) Decimals (page 62) Alternative method (page 65) to difficult limits (page 66) 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 ‐
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Introduction (page 105) BODMAS (page.106) 12) ALGEBRA AND FORMULAE ‐
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Introduction (page 108) BODMAS (page 109) August 2011. Kindly contributed by Shaun Bailey, Kirklees College. Search for Shaun on www.skillsworkshop.org
E3-L2 Functional Maths and adult numeracy. For related resources visit the download page for this resource at skillsworkshop
Page 1 of 110
L1-2 Functional Maths and Numeracy study guide
Addition
THE FOUR FUNDAMENTAL RULES OF ARITHMETIC 1: ADDITION Addition tells us the TOTAL of things. Q1: What is 23 + 140? 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 3 + 0. Write your answer DIRECTLY UNDERNEATH the column you’ve just added. Now move to the next column:
And finally the last column: Once you’ve done the last column you are finished! So, 23 + 140 = 163. August 2011. Kindly contributed by Shaun Bailey, Kirklees College. Search for Shaun on www.skillsworkshop.org
E3-L2 Functional Maths and adult numeracy. For related resources visit the download page for this resource at skillsworkshop
Page 2 of 110
L1-2 Functional Maths and Numeracy study guide
Addition
Q2: What is 123 + 189? 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 3 + 9 = 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 1 + 2 + 8. This is 11. Write “1” down and “carry over” the other 1 in the 11. Now you do the last column: 1 + 1 + the 1 you carried:
You’ve done the last column, so you have finished. So, 123 + 189 = 312.
August 2011. Kindly contributed by Shaun Bailey, Kirklees College. Search for Shaun on www.skillsworkshop.org
E3-L2 Functional Maths and adult numeracy. For related resources visit the download page for this resource at skillsworkshop
Page 3 of 110
L1-2 Functional Maths and Numeracy study guide
Addition of decimals
Q3: What is 5.5 + 0.98? 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 5 + 9 = 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, 5.5 + 0.98 = 6.48
August 2011. Kindly contributed by Shaun Bailey, Kirklees College. Search for Shaun on www.skillsworkshop.org
E3-L2 Functional Maths and adult numeracy. For related resources visit the download page for this resource at skillsworkshop
Page 4 of 110
L1-2 Functional Maths and Numeracy study guide
Subtraction
2: SUBTRACTION Subtraction is the reverse of addition! If you add something to a number, then take it back off, you’re back to where you started: 2 + 2 = 4  4 – 2 = 2. 7 + 31 = 38  38 – 31 = 7 Subtraction can tell you quite a few useful things: 
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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, 98 – 44 = 54. August 2011. Kindly contributed by Shaun Bailey, Kirklees College. Search for Shaun on www.skillsworkshop.org
E3-L2 Functional Maths and adult numeracy. For related resources visit the download page for this resource at skillsworkshop
Page 5 of 110
L1-2 Functional Maths and Numeracy study guide
Subtraction
Q2: What is 109 – 38? 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, 109 – 38 = 71. August 2011. Kindly contributed by Shaun Bailey, Kirklees College. Search for Shaun on www.skillsworkshop.org
E3-L2 Functional Maths and adult numeracy. For related resources visit the download page for this resource at skillsworkshop
Page 6 of 110
L1-2 Functional Maths and Numeracy study guide
Subtraction
Q3: What is 102 – 45? 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, 102 – 45 = 57
August 2011. Kindly contributed by Shaun Bailey, Kirklees College. Search for Shaun on www.skillsworkshop.org
E3-L2 Functional Maths and adult numeracy. For related resources visit the download page for this resource at skillsworkshop
Page 7 of 110
L1-2 Functional Maths and Numeracy study guide
Subtraction of decimals
Q4: What is 5 – 2.37? 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 5.00. 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 5.00 – 2.37: 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 9 – 3...
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, 5 – 2.37 (which is the same as 5.00 ‐ 2.37) = 2.63 August 2011. Kindly contributed by Shaun Bailey, Kirklees College. Search for Shaun on www.skillsworkshop.org
E3-L2 Functional Maths and adult numeracy. For related resources visit the download page for this resource at skillsworkshop
Page 8 of 110
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 512 – 149? 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 512 – 149 = 363, because 149 + 363 = 512 This method comes into its own when you may otherwise have to “borrow” from the other side again, and again, and again! August 2011. Kindly contributed by Shaun Bailey, Kirklees College. Search for Shaun on www.skillsworkshop.org
E3-L2 Functional Maths and adult numeracy. For related resources visit the download page for this resource at skillsworkshop
Page 9 of 110
L1-2 Functional Maths and Numeracy study guide
Subtraction – other methods
Q2: What is 50 001 – 49 999? 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 49 999 to get 50 000. You could ADD 1 to 50 000 to get 50 001. In total you just added 1 and 1, which is 2. So, 50 001 – 49 999 = 2. NB Both ways of subtraction will always work. Use whichever you feel most confident with. August 2011. Kindly contributed by Shaun Bailey, Kirklees College. Search for Shaun on www.skillsworkshop.org
E3-L2 Functional Maths and adult numeracy. For related resources visit the download page for this resource at skillsworkshop
Page 10 of 110
Functional Maths and Numeracy study guide
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: 9 + 9 + 9 + 9 + 9 + 9 + 9 + 9 (Slow)  9 x 8 (Quick!) 10 + 10 + 10 + 10 + 10 + 10 + 10 + 10 + 10 + 10 (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. August 2011. Kindly contributed by Shaun Bailey, Kirklees College. Search for Shaun on www.skillsworkshop.org
E3-L2 Functional Maths and adult numeracy. For related resources visit the download page for this resource at skillsworkshop
Page 11 of 110
Functional Maths and Numeracy study guide
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:
August 2011. Kindly contributed by Shaun Bailey, Kirklees College. Search for Shaun on www.skillsworkshop.org
E3-L2 Functional Maths and adult numeracy. For related resources visit the download page for this resource at skillsworkshop
Page 12 of 110
Functional Maths and Numeracy study guide
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. August 2011. Kindly contributed by Shaun Bailey, Kirklees College. Search for Shaun on www.skillsworkshop.org
E3-L2 Functional Maths and adult numeracy. For related resources visit the download page for this resource at skillsworkshop
Page 13 of 110
Functional Maths and Numeracy study guide
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 = 38 448. 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. August 2011. Kindly contributed by Shaun Bailey, Kirklees College. Search for Shaun on www.skillsworkshop.org
E3-L2 Functional Maths and adult numeracy. For related resources visit the download page for this resource at skillsworkshop
Page 14 of 110
Functional Maths and Numeracy study guide
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 10 ÷2 = 5 5x ? = 15 ? = 3 15÷5 = 3 10x ? = 20 ? = 2 20÷10 = 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. August 2011. Kindly contributed by Shaun Bailey, Kirklees College. Search for Shaun on www.skillsworkshop.org
E3-L2 Functional Maths and adult numeracy. For related resources visit the download page for this resource at skillsworkshop
Page 15 of 110
Functional Maths and Numeracy study guide
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 1 6 Too small 2
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Too small
4 24 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, 36... 1 2 3 4 5 6 Q1: What is 147 7? 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 = 140 + 7 = 147. So 147 7 = 21 21 7s 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, 147 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21
August 2011. Kindly contributed by Shaun Bailey, Kirklees College. Search for Shaun on www.skillsworkshop.org
E3-L2 Functional Maths and adult numeracy. For related resources visit the download page for this resource at skillsworkshop
Page 16 of 110
Functional Maths and Numeracy study guide
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. August 2011. Kindly contributed by Shaun Bailey, Kirklees College. Search for Shaun on www.skillsworkshop.org
E3-L2 Functional Maths and adult numeracy. For related resources visit the download page for this resource at skillsworkshop
Page 17 of 110
Functional Maths and Numeracy study guide
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, 468 ÷ 9 = 52. August 2011. Kindly contributed by Shaun Bailey, Kirklees College. Search for Shaun on www.skillsworkshop.org
E3-L2 Functional Maths and adult numeracy. For related resources visit the download page for this resource at skillsworkshop
Page 18 of 110
Functional Maths and Numeracy study guide
Division – long division
Q3: What is 5 000 750 ÷ 5? 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, 5 000 750 ÷ 5 = 1 000 150
August 2011. Kindly contributed by Shaun Bailey, Kirklees College. Search for Shaun on www.skillsworkshop.org
E3-L2 Functional Maths and adult numeracy. For related resources visit the download page for this resource at skillsworkshop
Page 19 of 110
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 = 6 3 3 2
2 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 = 3 6 3 3
‐ 6 3 = so six sweets divided between three groups = 2 6 2 2
2
You saw that 2 x 3 = 6, and that 6 2 = 3. Also, 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.
August 2011. Kindly contributed by Shaun Bailey, Kirklees College. Search for Shaun on www.skillsworkshop.org
E3-L2 Functional Maths and adult numeracy. For related resources visit the download page for this resource at skillsworkshop
Page 20 of 110
L1-2 Functional Maths and Numeracy study guide
Units of measure
UNITS OF MEASURE Length 1cm = 10mm 100cm = 1m 1 000m = 1km
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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 2
2
2
2
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mm, cm, m, km mm , cm , m , km
mm , cm3, m3, km3
Capacity 1000 ml = 1 l 1 l = 1000 cm3
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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
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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.
August 2011. Kindly contributed by Shaun Bailey, Kirklees College. Search for Shaun on www.skillsworkshop.org
E3-L2 Functional Maths and adult numeracy. For related resources visit the download page for this resource at skillsworkshop
Page 21 of 110
L1-2 Functional Maths and Numeracy study guide
Introduction to area, perimeter and volume
MEASURES, SHAPE AND SPACE August 2011. Kindly contributed by Shaun Bailey, Kirklees College. Search for Shaun on www.skillsworkshop.org
E3-L2 Functional Maths and adult numeracy. For related resources visit the download page for this resource at skillsworkshop
Page 22 of 110
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 cm2. How many centimetre squares can fit inside the rectangle? Count them. 7 lots of 6 = 7 x 6 = 42cm2 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 = 32cm2 Just in case you want to have a look:
32cm2 in total. August 2011. Kindly contributed by Shaun Bailey, Kirklees College. Search for Shaun on www.skillsworkshop.org
E3-L2 Functional Maths and adult numeracy. For related resources visit the download page for this resource at skillsworkshop
Page 23 of 110
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 = 54m2 (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 54m2 . 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. August 2011. Kindly contributed by Shaun Bailey, Kirklees College. Search for Shaun on www.skillsworkshop.org
E3-L2 Functional Maths and adult numeracy. For related resources visit the download page for this resource at skillsworkshop
Page 24 of 110
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. August 2011. Kindly contributed by Shaun Bailey, Kirklees College. Search for Shaun on www.skillsworkshop.org
E3-L2 Functional Maths and adult numeracy. For related resources visit the download page for this resource at skillsworkshop
Page 25 of 110
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 (m3). One cubic metre is 1m in length, width, and height: Volume = length x height x width = 5 x 3 x 3 = 45m3 45 cubic metres! August 2011. Kindly contributed by Shaun Bailey, Kirklees College. Search for Shaun on www.skillsworkshop.org
E3-L2 Functional Maths and adult numeracy. For related resources visit the download page for this resource at skillsworkshop
Page 26 of 110
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 (45m3) 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 August 2011. Kindly contributed by Shaun Bailey, Kirklees College. Search for Shaun on www.skillsworkshop.org
E3-L2 Functional Maths and adult numeracy. For related resources visit the download page for this resource at skillsworkshop
Page 27 of 110
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. August 2011. Kindly contributed by Shaun Bailey, Kirklees College. Search for Shaun on www.skillsworkshop.org
E3-L2 Functional Maths and adult numeracy. For related resources visit the download page for this resource at skillsworkshop
Page 28 of 110
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. August 2011. Kindly contributed by Shaun Bailey, Kirklees College. Search for Shaun on www.skillsworkshop.org
E3-L2 Functional Maths and adult numeracy. For related resources visit the download page for this resource at skillsworkshop
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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).
August 2011. Kindly contributed by Shaun Bailey, Kirklees College. Search for Shaun on www.skillsworkshop.org
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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 August 2011. Kindly contributed by Shaun Bailey, Kirklees College. Search for Shaun on www.skillsworkshop.org
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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: = August 2011. Kindly contributed by Shaun Bailey, Kirklees College. Search for Shaun on www.skillsworkshop.org
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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. August 2011. Kindly contributed by Shaun Bailey, Kirklees College. Search for Shaun on www.skillsworkshop.org
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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! 45
135
5⟶ 9
5 → 27
Can you simplify again? What fits into 9 AND 27? How about 9? ⟶
→
So
45
135
1
3
August 2011. Kindly contributed by Shaun Bailey, Kirklees College. Search for Shaun on www.skillsworkshop.org
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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, 40 ÷ 10 = 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 . August 2011. Kindly contributed by Shaun Bailey, Kirklees College. Search for Shaun on www.skillsworkshop.org
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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: + = = 1 (Because two halves make one whole). 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  = August 2011. Kindly contributed by Shaun Bailey, Kirklees College. Search for Shaun on www.skillsworkshop.org
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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 ‐ = ‐ = August 2011. Kindly contributed by Shaun Bailey, Kirklees College. Search for Shaun on www.skillsworkshop.org
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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. 100 5 = 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 = = 100 ÷ 5 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” August 2011. Kindly contributed by Shaun Bailey, Kirklees College. Search for Shaun on www.skillsworkshop.org
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. L1-2 Functional Maths and Numeracy study guide
Fractions – multiplying
Examples of 20 is the same as x 20. This means twenty quarters, . of 300 is the same as 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). £80 4 = £20 This tells you that one quarter = £20  However, you want 3 quarters, so multiply the answer by 3. 20 x 3 = £60  The answer to of £80 is therefore £60. August 2011. Kindly contributed by Shaun Bailey, Kirklees College. Search for Shaun on www.skillsworkshop.org
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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 2 + 2 = 4 halves. There is also an extra half (
2
= two AND a half), which is 5 HALVES. August 2011. Kindly contributed by Shaun Bailey, Kirklees College. Search for Shaun on www.skillsworkshop.org
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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: August 2011. Kindly contributed by Shaun Bailey, Kirklees College. Search for Shaun on www.skillsworkshop.org
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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 = 7.000 = 7.000 13 = 13.0 = 13.00 = 13.000 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 1.5700 = 1.57 ‐ these 0s can go since there’s nothing after them. The 0 in 1.206 CAN NOT be removed, because there is a 6 after the 0. 4.00230 = 4.0023 ‐ 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
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L1-2 Functional Maths and Numeracy study guide
Decimals – multiplying and dividing by 10, 100, 1000.
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 24.31  Move the decimal point to the left DECIMAL
24.31
 Move the decimal point to DECIMAL
24.31
the right 10
2.431
243.1
100 .2431 1000
.02431
2431.00 24310.00
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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 1.8  6 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 August 2011. Kindly contributed by Shaun Bailey, Kirklees College. Search for Shaun on www.skillsworkshop.org
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L1-2 Functional Maths and Numeracy study guide
Decimals – multiplying
Q2: What is 11 x 0.0015? 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. 0.0015 becomes 15. 11 stays the same – it is already a whole number. 11 x 0.0015  11 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 0.0015 is 0.0165 August 2011. Kindly contributed by Shaun Bailey, Kirklees College. Search for Shaun on www.skillsworkshop.org
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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 2 + 1 = 3 DIGITS. Now think of the numbers as whole numbers, without any decimals. 0.16 x 10.5 16 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 1.680 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 August 2011. Kindly contributed by Shaun Bailey, Kirklees College. Search for Shaun on www.skillsworkshop.org
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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 = 1.000 = 1.0000 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 4.2 ÷ 0.08 ? Something like 6 ÷ 2 is the same as (six halves, which is three.) So 4.2 ÷ 0.08 can be written as .
.
Then you can use equivalent fractions to get rid of awkward decimals. Multiplying by 10, 100, 1 000 and so on is an easy way to get rid of decimals. This tells you that 4.2 ÷ 0.08 is the same thing as 420 ÷ 8. August 2011. Kindly contributed by Shaun Bailey, Kirklees College. Search for Shaun on www.skillsworkshop.org
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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 4.2 ÷ 0.08 = 52.5 as a decimal. Q3: What is 8 ÷ 0.25 ? Similar to before, you can write 8 ÷ 0.25 as .
Then use the rules of equivalent fractions to change it into a “nicer” fraction: So 8 ÷ 0.25 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 August 2011. Kindly contributed by Shaun Bailey, Kirklees College. Search for Shaun on www.skillsworkshop.org
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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 2.0000 ... if you need to. This would go on forever! = 0.6666666... 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. August 2011. Kindly contributed by Shaun Bailey, Kirklees College. Search for Shaun on www.skillsworkshop.org
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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 August 2011. Kindly contributed by Shaun Bailey, Kirklees College. Search for Shaun on www.skillsworkshop.org
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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% = = August 2011. Kindly contributed by Shaun Bailey, Kirklees College. Search for Shaun on www.skillsworkshop.org
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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 =  divide top and bottom by 10  10% = 10 out of 100 = 5% = 5 out of 100 =  divide top and bottom by 5  1% = 1 out of 100 = 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 300 + of 300 of 300 = 300 ÷ 2 = 150 of 300 = 300 ÷ 4 = 75 So 75% of 300 = 150 + 75 = 225 You could have done this any number of ways. For example, you could have worked out 1% of 300 (=3) then multiplied by 75. 3 x 75 = 225 August 2011. Kindly contributed by Shaun Bailey, Kirklees College. Search for Shaun on www.skillsworkshop.org
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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. http://www.skillsworkshop.org/resources/percentage‐worker‐outer August 2011. Kindly contributed by Shaun Bailey, Kirklees College. Search for Shaun on www.skillsworkshop.org
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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 £500. 80% = = = . So, you need to find FOUR FIFTHS OF 500. of 500 = 500 ÷ 5 = 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, £500 + 80% = £500 + £400 = £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 £80. 25% = = so you need to find ONE QUARTER OF 80. of 80 = 80 ÷ 4 = 20. So the sale price = £80 – 25% = £80 ‐ £20 = £60.  Again, you started by working out 25% OF THE START PRICE. August 2011. Kindly contributed by Shaun Bailey, Kirklees College. Search for Shaun on www.skillsworkshop.org
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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. August 2011. Kindly contributed by Shaun Bailey, Kirklees College. Search for Shaun on www.skillsworkshop.org
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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 100. 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.
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Functional Maths and Numeracy study guide
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 0.125 = 12.5% August 2011. Kindly contributed by Shaun Bailey, Kirklees College. Search for Shaun on www.skillsworkshop.org
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Functional Maths and Numeracy study guide
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 0.125 = 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% = 0.084
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Functional Maths and Numeracy study guide
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 120 600” – 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, 10 000 AND SO ON When rounding to the nearest whole number, or a number ending in zero (10, 100, 1 000 and so on) think of the number in terms of “hundreds, tens, units”. Here is the number 9 605 872 written in this way: August 2011. Kindly contributed by Shaun Bailey, Kirklees College. Search for Shaun on www.skillsworkshop.org
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Functional Maths and Numeracy study guide
Rounding whole numbers
Q1: A total of 37 894 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, 37 894 rounded to the nearest hundred is 37 900 August 2011. Kindly contributed by Shaun Bailey, Kirklees College. Search for Shaun on www.skillsworkshop.org
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Functional Maths and Numeracy study guide
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.
August 2011. Kindly contributed by Shaun Bailey, Kirklees College. Search for Shaun on www.skillsworkshop.org
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Rounding decimals
Q4: What is 10.108 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, 10.108 to the nearest hundredth is 10.110 Because the last 0 is unnecessary, you must simplify this to just 10.11 August 2011. Kindly contributed by Shaun Bailey, Kirklees College. Search for Shaun on www.skillsworkshop.org
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Rounding decimals
Q5: A shop owner is calculating the price per unit of sweets in pounds. He uses his calculator and gets 0.177777777777… What is this to the nearest penny? Since this is in pounds, the calculator is saying £0.1777777777… The number TO THE RIGHT of the pennies column is 7, so you ROUND UP. The “7” becomes an “8”. So, £0.177777777… to the nearest penny is 18p. August 2011. Kindly contributed by Shaun Bailey, Kirklees College. Search for Shaun on www.skillsworkshop.org
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Rounding decimals
Q6 (HARD!): A jogger does a warm‐up run. He calculates that he has run 8.156 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 = 8.160 km (dividing by 1 000 to go back to kilometres) The last 0 is unnecessary. So IN KILOMETRES, 8.156km to the nearest 10m = 8.16km August 2011. Kindly contributed by Shaun Bailey, Kirklees College. Search for Shaun on www.skillsworkshop.org
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Rounding - alternative method
You can also think of rounding in an entirely different way… Q1: What is 3 088 to the nearest 100? You’re rounding the nearest 100. So, think of MULTIPLES OF 100. Which multiple of 100 is 3 088 closest to? Multiples of 100 are the 100 times table: 100, 200, 300, 400 … (fast forward a bit!) … 2 999, 3 000, 3 100. 3 088 is between 3 000 and 3 100. 3 088 is 88 AWAY from 3 000, but only 12 AWAY from 3100. So, 3 088 to the nearest 100 is 3 100. Q2: A care’s mileage reads “107 873”. How many miles is this to the nearest 10 000 miles? Here you’re rounding the nearest 10 000. So you need to think in MULTIPLES OF 10 000. 10 000, 20 000, 30 000, 40 000… …100 000, 110 000. 107 873 is between 100 000 and 110 000. 105 000 would be EXACTLY HALF‐WAY between 100 000 and 110 000. So, 107 783 is MORE THAN HALF WAY. So, 107 783 miles to the nearest 10 000 miles is 110 000 miles. August 2011. Kindly contributed by Shaun Bailey, Kirklees College. Search for Shaun on www.skillsworkshop.org
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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, 3.75. 3.68 is between 3.50 and 3.75. 3.68 is greater than 3.50 by 0.18, but it is only 0.07 smaller than 3.75. So, 3.68 is closer to 3.75 than 3.50 So 3.68 to the nearest quarter is 3.75 August 2011. Kindly contributed by Shaun Bailey, Kirklees College. Search for Shaun on www.skillsworkshop.org
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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 3 HARRY 2
 Add the ratios together to calculate how many parts the £20 is to be divided into altogether. 3 + 2 = 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. August 2011. Kindly contributed by Shaun Bailey, Kirklees College. Search for Shaun on www.skillsworkshop.org
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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 4. 500 x 4 = 2000ml  1000ml = 1 litre So 2000ml = 2 litres So you need 2000ml (= 2l) of white paint. August 2011. Kindly contributed by Shaun Bailey, Kirklees College. Search for Shaun on www.skillsworkshop.org
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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? Another way of thinking of this is that for every one lot of red, there are four lots of white. MIX 1 PART RED WITH 4 PARTS WHITE If you had two lots of red (1+1) we’d need eight lots of white(4+4): RED 1 5 500 WHITE 1+1+1+1 = 4 5+5+5+5 = 20 500+500+500+500 = 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 August 2011. Kindly contributed by Shaun Bailey, Kirklees College. Search for Shaun on www.skillsworkshop.org
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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 : 5  12 + 3 + 5 = 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
12 3
5
20 x by 3 x by 3
x by 3
x by 3 36 9
15
60 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…
August 2011. Kindly contributed by Shaun Bailey, Kirklees College. Search for Shaun on www.skillsworkshop.org
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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 1 2 4 10 cm 2.5 5 10 25 14 35 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 = 35 2.5 = .
= = 14 August 2011. Kindly contributed by Shaun Bailey, Kirklees College. Search for Shaun on www.skillsworkshop.org
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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 Length of model = the length of the theatre. = = = = 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. August 2011. Kindly contributed by Shaun Bailey, Kirklees College. Search for Shaun on www.skillsworkshop.org
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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). August 2011. Kindly contributed by Shaun Bailey, Kirklees College. Search for Shaun on www.skillsworkshop.org
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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. thof 320 = 320 8 = 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. August 2011. Kindly contributed by Shaun Bailey, Kirklees College. Search for Shaun on www.skillsworkshop.org
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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. August 2011. Kindly contributed by Shaun Bailey, Kirklees College. Search for Shaun on www.skillsworkshop.org
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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. August 2011. Kindly contributed by Shaun Bailey, Kirklees College. Search for Shaun on www.skillsworkshop.org
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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. August 2011. Kindly contributed by Shaun Bailey, Kirklees College. Search for Shaun on www.skillsworkshop.org
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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? August 2011. Kindly contributed by Shaun Bailey, Kirklees College. Search for Shaun on www.skillsworkshop.org
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Ratio – scale drawings
The only thing you have been given is the scale 1:25 000. You have to measure the scale drawing yourself. Measure the line from A to B:
This measures 5cm. You know the scale is 1: 25 000. The map will be smaller than the actual road! So, it MAKES SENSE that the “1” represents the map, and “25 000” the road. You can again use the fact that ratios are like fractions: = = So 5cm on the map represents an ACTUAL LENGTH of 125 000cm 100cm = 1m  125 000cm = (125 000 100) 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! August 2011. Kindly contributed by Shaun Bailey, Kirklees College. Search for Shaun on www.skillsworkshop.org
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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. August 2011. Kindly contributed by Shaun Bailey, Kirklees College. Search for Shaun on www.skillsworkshop.org
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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 + 3 = 7 4 + 2 = 6 4 + 1 = 5 4 + 0 = 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: 4 + 3 = 7 4 + 2 = 6 4 + 1 = 5 4 + 0 = 4 4 + ‐1= 3 (4 – 1 = 3) 4 +‐2 = 2 (4 – 2 = 2) 4 +‐3 = 1 (4 – 3 = 1) 4 +‐4 = 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 August 2011. Kindly contributed by Shaun Bailey, Kirklees College. Search for Shaun on www.skillsworkshop.org
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Negative numbers – addition and subtraction
Q1: What is 27 + ‐12? Using our rule, 27 +‐12 = 27 – 12 = 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 27 + 12?” 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, 27 + ‐12 = 27 – 12 = 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 August 2011. Kindly contributed by Shaun Bailey, Kirklees College. Search for Shaun on www.skillsworkshop.org
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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 = 1 4 – 2 = 2 4 – 1 = 3 4 – 0 = 4 4 ‐ ‐1 = 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. August 2011. Kindly contributed by Shaun Bailey, Kirklees College. Search for Shaun on www.skillsworkshop.org
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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 August 2011. Kindly contributed by Shaun Bailey, Kirklees College. Search for Shaun on www.skillsworkshop.org
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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? 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. August 2011. Kindly contributed by Shaun Bailey, Kirklees College. Search for Shaun on www.skillsworkshop.org
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Negative numbers – addition and subtraction
What is 1 ‐ 3? So, 1 – 3 = ‐2. What is ‐2+ 8?
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Range
AVERAGES and RANGE The 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: £30 ‐ £7.50 = £22.50 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. August 2011. Kindly contributed by Shaun Bailey, Kirklees College. Search for Shaun on www.skillsworkshop.org
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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. MEAN =  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 = = = = 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. August 2011. Kindly contributed by Shaun Bailey, Kirklees College. Search for Shaun on www.skillsworkshop.org
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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. August 2011. Kindly contributed by Shaun Bailey, Kirklees College. Search for Shaun on www.skillsworkshop.org
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Functional Maths and Numeracy study guide
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 6 + 8 = 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. August 2011. Kindly contributed by Shaun Bailey, Kirklees College. Search for Shaun on www.skillsworkshop.org
E3-L2 Functional Maths and adult numeracy. For related resources visit the download page for this resource at skillsworkshop
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Functional Maths and Numeracy study guide
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 93‐104 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. August 2011. Kindly contributed by Shaun Bailey, Kirklees College. Search for Shaun on www.skillsworkshop.org
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Functional Maths and Numeracy study guide
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: 
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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. August 2011. Kindly contributed by Shaun Bailey, Kirklees College. Search for Shaun on www.skillsworkshop.org
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Functional Maths and Numeracy study guide
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 August 2011. Kindly contributed by Shaun Bailey, Kirklees College. Search for Shaun on www.skillsworkshop.org
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Functional Maths and Numeracy study guide
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. August 2011. Kindly contributed by Shaun Bailey, Kirklees College. Search for Shaun on www.skillsworkshop.org
E3-L2 Functional Maths and adult numeracy. For related resources visit the download page for this resource at skillsworkshop
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Functional Maths and Numeracy study guide
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 August 2011. Kindly contributed by Shaun Bailey, Kirklees College. Search for Shaun on www.skillsworkshop.org
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Functional Maths and Numeracy study guide
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. August 2011. Kindly contributed by Shaun Bailey, Kirklees College. Search for Shaun on www.skillsworkshop.org
E3-L2 Functional Maths and adult numeracy. For related resources visit the download page for this resource at skillsworkshop
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Functional Maths and Numeracy study guide
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 August 2011. Kindly contributed by Shaun Bailey, Kirklees College. Search for Shaun on www.skillsworkshop.org
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Functional Maths and Numeracy study guide
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. August 2011. Kindly contributed by Shaun Bailey, Kirklees College. Search for Shaun on www.skillsworkshop.org
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Functional Maths and Numeracy study guide
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 August 2011. Kindly contributed by Shaun Bailey, Kirklees College. Search for Shaun on www.skillsworkshop.org
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Functional Maths and Numeracy study guide
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 21‐39 ‐ 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 August 2011. Kindly contributed by Shaun Bailey, Kirklees College. Search for Shaun on www.skillsworkshop.org
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Functional Maths and Numeracy study guide
Representing data – pictograms
PICTOGRAMS A Pictogram displays information through the use of symbols, or PICTURES, as the name suggests. Example 1 Example 2 August 2011. Kindly contributed by Shaun Bailey, Kirklees College. Search for Shaun on www.skillsworkshop.org
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Functional Maths and Numeracy study guide
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. August 2011. Kindly contributed by Shaun Bailey, Kirklees College. Search for Shaun on www.skillsworkshop.org
E3-L2 Functional Maths and adult numeracy. For related resources visit the download page for this resource at skillsworkshop
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Functional Maths and Numeracy study guide
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. August 2011. Kindly contributed by Shaun Bailey, Kirklees College. Search for Shaun on www.skillsworkshop.org
E3-L2 Functional Maths and adult numeracy. For related resources visit the download page for this resource at skillsworkshop
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Functional Maths and Numeracy study guide
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 16‐20 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. August 2011. Kindly contributed by Shaun Bailey, Kirklees College. Search for Shaun on www.skillsworkshop.org
E3-L2 Functional Maths and adult numeracy. For related resources visit the download page for this resource at skillsworkshop
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Functional Maths and Numeracy study guide
Order of calculations - introduction
ORDER OF CALCULATIONS Sometimes, it is not easy to know what to do first. Example What is 2 + 2 x 2? Do you work out what 2 + 2 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 3 + 6 x 2 ÷ 4 + 32 – 4? …NIGHTMARE! You need a rule so you know what to do first, what to do second, what to do third, and so on. August 2011. Kindly contributed by Shaun Bailey, Kirklees College. Search for Shaun on www.skillsworkshop.org
E3-L2 Functional Maths and adult numeracy. For related resources visit the download page for this resource at skillsworkshop
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Functional Maths and Numeracy study guide
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 42, 23. The LITTLE NUMBER tells us HOW MANY TIMES to multiply the BIG NUMBER by itself: 42 = 4 x 4 25 = 2 x 2 x 2 x 2 x 2 103 = 10 x 10 x 10 And so on. 3) THIRD, do the DIVISION and MULTIPLICATION (any order) 4) LAST, do the ADDITION and SUBTRACTION. August 2011. Kindly contributed by Shaun Bailey, Kirklees College. Search for Shaun on www.skillsworkshop.org
E3-L2 Functional Maths and adult numeracy. For related resources visit the download page for this resource at skillsworkshop
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Functional Maths and Numeracy study guide
Order of calculations - BODMAS
Example 1 What is 2 + 2 x 2? All you have here is Addition and Multiplication. “M” comes before “A” in BODMAS, so you do the multiplication first: 2 x 2 = 4 Then, do the addition: 2 + 4 = 6. So, 2 + 2 x 2 = 6 Example 2 What is 4 + 62 2 ? All you have is a POWER (the little floating 2), an ADDITION and a DIVISION. The 2 isn’t a +, ‐, x, ÷ or () …it’s something OTHER. So you have an OTHER, a DIVISION and an ADDITION. “O” comes before “D” and “A” in BODMAS, so you do the 62 first: 62 = 6 x 6 = 36. Then “D” comes next, so do the DIVISION: 36 ÷ 2 = 18. Finally, you do the addition: 4 + 18 = 22. So, 4 + 62 2 = 22 August 2011. Kindly contributed by Shaun Bailey, Kirklees College. Search for Shaun on www.skillsworkshop.org
E3-L2 Functional Maths and adult numeracy. For related resources visit the download page for this resource at skillsworkshop
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Functional Maths and Numeracy study guide
Algebra and formulae - introduction
ALGEBRA AND FORMULAE ALGEBRA IS THE USE OF LETTERS AND SYMBOLS TO REPRESENT NUMBERS.
Formulas – called formulae – often use algebra. Examples of formulae Circumference of circle = 2πr Volume of cuboid = lwh Perimeter of rectangle = 2(l + w) With formulae, if there is no “instruction” between one symbol and the next,
you ALWAYS assume to MULTIPLY. Example ABC = A x B x C 5(E + F) = 5 x (E + F) With formulae you will be given the values of the letters. You then solve the problem by TAKING THE LETTERS OUT, and PUTTING IN THE ACTUAL VALUES OF THOSE LETTERS. This is called SUBSTITUTION. August 2011. Kindly contributed by Shaun Bailey, Kirklees College. Search for Shaun on www.skillsworkshop.org
E3-L2 Functional Maths and adult numeracy. For related resources visit the download page for this resource at skillsworkshop
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Functional Maths and Numeracy study guide
Algebra and formulae – substitution
Q1: The area of a circle is given by the formula A = πr2 Where A = area of the circle = 3 r = radius of circle Work out the area of a circle with a radius of 5cm. You are working out A (the area). You know = 3 and r = 5. There is no instruction between and r2. So r2 = x r2 You can now SUBSTITUTE the values for the letters. A = π x r2 = 3 x 52 = 3 x 25 = 75cm2 So the area of a circle with radius 5cm is 75cm2 August 2011. Kindly contributed by Shaun Bailey, Kirklees College. Search for Shaun on www.skillsworkshop.org
E3-L2 Functional Maths and adult numeracy. For related resources visit the download page for this resource at skillsworkshop
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Functional Maths and Numeracy study guide
Algebra and formulae – substitution
Q2: A company uses a formula to calculate the amount (area) of wrapping paper, A, for its product: A = (L + H + 2)(2W + 2H + 2) Where: L = 11 cm W = 6 cm H = 4cm What is the area of the wrapping paper? There are two brackets, but no “instruction” between them. There is also no “instruction” between the 2 and W, or between the 2 and H. Where there is no instruction, it means you MULTIPLY. A = (L + H + 2)(2W + 2H + 2) A = (L + H + 2) x (2 x W + 2 x H + 2) You can now substitute in the values for L, W, and H: A = (11 + 4 + 2) x (2 x 6 + 2 x 4 + 2) = (17) x (12 + 8 + 2)  doing x before + because of BODMAS = (17) x (22) = 368cm2 August 2011. Kindly contributed by Shaun Bailey, Kirklees College. Search for Shaun on www.skillsworkshop.org
E3-L2 Functional Maths and adult numeracy. For related resources visit the download page for this resource at skillsworkshop
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