Estimation and Mental Methods Mathematics Worksheet

This is one of a series of worksheets designed to help you increase your confidence in handling Mathematics. In this worksheet you will find guidelines for improving your number competence to make you confident that you can operate numerically in a technological age. There are often different ways of doing things in Mathematics and the methods suggested in the worksheets may not be the ones you were taught. If you are successful and happy with the methods you use it may not be necessary for you to change them. If you have problems or need help in any part of the work then there are a number of ways you can get help. For students at the University of Hull  Ask your lecturers.  You can contact a Mathematics Tutor from the Skills Team on the email shown below.  Access more Maths Skills Guides and resources at the website below.  Look at one of the many textbooks in the library.

Web: http://libguides.hull.ac.uk/skills Email: [email protected]

The arrival of calculators as the latest calculating aid in a long line of such aids undoubtedly calls for a radical rethink of what we have, throughout this century, seen to be the basics of arithmetic needed by the populace as a whole. With calculators now in almost universal use in the general community there is no longer the need there previously was to learn the standard written arithmetical methods learned by generations of children over the last 100 years or so. Indeed knowledge of those methods can specifically impede competent calculator use, which requires as its basic accompaniment that users can estimate in order to prevent the acceptance of nonsense answers sometimes called up by pressing wrong buttons on the machine. And since the process of finding an estimate of necessity requires the skill of being able to calculate mentally, these two Estimating and Mental strategies for calculating become the new basics of arithmetic in the calculator age. This booklet will therefore give readers hints and guidelines for competent calculator use by encouraging the development of estimating and mental methods of calculating. In order to facilitate this development it is important to know about, and be able to use in specific ways, an important mathematical principle which pervades mathematics at all levels: the inverse principle which states that inverse operations undo each other. For this it is only necessary at this stage to consider the four basic arithmetical operations of addition, subtraction, multiplication and division. It is not difficult to see that adding and subtracting undo each other (try adding any two numbers and then undoing that by taking the second one away from the sum to arrive back at the other one) as do multiplication and division (multiply two numbers together and then divide the answer by one of them and you will find you arrive at the other.) There are, however, two specific examples of this principle that help with estimating and mental calculating. One is encompassed in the following × 10   10  which is an invaluable tool for calculating. Another specific example of this comes from the ‘tables' that most people learned at school. Tables deal with the result of multiplying two numbers. There is an inverse operation to this, the idea of ‘what goes into a given number', ‘what a given number can be divided by’, or, in more mathematical terms, ‘what are the factors of a given number’. Tables are often concentrated on at school and divisibility tends to be assumed from it. But this is really insufficient unless you can see, on looking at a number, what its factors are or what goes into it. When dealing with whole numbers, most people recognise that a number ending in 0 is divisible by ten, some realise that, if a number ends in 5 or 0, it is divisible by 5 but, for many people, that is as far as they can go apart from realising in a rather vague way that all even numbers are divisible by 2. Apart from knowing your tables up to 12  12 , it would also be helpful to know your tables up to at least 4  20 and, if possible for larger numbers than that. For example you should know that the 15 table goes 15, 30, 45, 60 and perhaps also see that you can then create later terms in it by looking at these results. 1

So how do we use these ideas to help us to estimate and to calculate mentally? Most people have some idea about estimating though some are stumped when asked to do so. Let us consider a specific example: 36 × 59. You might notice that 59 is nearly 60 so consider that instead of 59; if you look at 36 you might say that it is nearer 40 than 30 so you will consider 40  60 and your estimate will be 2400. Notice further that, in doing this, you didn't do anything like you would normally do if you didn't use your calculator but use the written process we call long multiplication. But this would certainly not be the best way to do this example. It is better to look at the numbers to see if they can help you to see an easier way to operate than going through the process of long multiplication. We will not do this question at the moment but will return to it later. Let us be systematic about the four operations, taking each in turn, after looking at some general guidelines for developing efficiency in the basic skills for a calculator age.

General Guidelines for calculating 1. Make an estimate to ensure you have an idea of the answer. 2. Use a mental method if you have one. 3. Make it easy, if you can, by using the inverse principle or some other device. 4. Use a calculator if appropriate (estimate essential in this case). 5. Check your answer against your estimate, or by using the inverse or doing the calculation in a different way. If you follow these guidelines for a while you will find that they will become part of your normal repertoire for approaching calculations and you will no longer have to think about the five steps, for you will do them automatically. ADDITION Two possibilities arise here: two items only or a list. Two items 236 + 598 1. Make an estimate: 200 + 600 = 800 2. Notice that, for your estimate you started with the left hand digits after noticing that 598 is almost 600. The traditional written method is to start with the unit digits, noting any carrying figures and going on to the tens and then the hundreds digits. But it is more effective to do as you did for an estimate, ie look at the left hand figures first, and proceed along from left to right instead of the normal right to left 2

written process. This way your estimate not only provides a check on your final answer but is also part of the calculation itself. Some people are daunted at the thought of such a radical change. If you can begin to effect this change, it is the experience of students who have attended Numeracy courses, that you eventually find it easier and more efficient to carry the necessary figures in your head while working from left to right. 200 and 500 30 and 90 6 and 8

is 700 is 120 is 14

total 820 total 834

3. But the estimate can be used in another way: the sum can be rewritten or thought about as 234 + 600 (adding the 2 to make 598 into 600, easier to add than 598, and taking away the same number from the other number). Or alternatively, adding 600 to the original 236 and taking off the 2 from the answer. Notice too that you were using the inverse principle to do this. 4. With these possibilities a calculator would not be a good way of doing it because, with practice, they are quicker than keying the numbers into your calculator. 5. Checked against your estimate of 800, it is a bit more but it is easy to see that, in the estimate more has been taken away from 236 than added to 598 so you would expect the answer to be more: 34 more in fact which is the difference between what was added and what was taken away. Such analysis becomes automatic as you practise it and helps you to gain confidence in your own calculating ability as well as encouraging observation of numbers and the meanings behind results. For a list

3027 856 2347 1483

1. Estimate: 6000 if you only look at the thousands, 7000 if you glance also at the hundreds. 2.

Using the same left to right procedure as for two items, you can proceed across, adjusting as you go until you get to the final units column: 3000+2000+1000 800+ 300+ 400 20+80+50+40 7+ 3+ 6+ 7

=6000 =1500 = 190 = 23

Total 7500 Total 7690 Total 7713

Notice collecting terms can make the addition easier: 20 + 50 + 40 + 80 = 20 + 80 + 50 + 40 = 100 + 90 = 190 3.

Checked against your estimate you notice that this is quite a bit more than your estimate but that is because you did not take the last two columns into account. Notice too that, working in this way, you are getting better and better approximations to the final answer. 3

Some people find it irksome to go through this procedure and prefer to go to their calculator for long lists. Others don't because they are more uneasy about pressing the wrong buttons and hence making the calculator method too tedious. It is a matter of judgement but you should try doing it in your head with jottings before considering using a calculator automatically. Note that, in both cases, you are working from the left and, with practice, you will find that this is more efficient than the school way of adding from the right. SUBTRACTION Let us consider the following: 532 - 286 1.

Estimate: 300 or, if you notice that 286 is not far from 300, 200

2.

There is a wide variety of mental methods you can use for this and other subtractions. Here are some of them: i) Counting on: 286 up to 300 is 14, 300 up to 532 is another 232 so the answer is 232 + 14 = 246 ii) Counting back: 532 - 200 is 332, take off another 100 to get 232 but this is 14 too many off so add 14 to get 246. The way that people organise their counting back can vary considerably and be different for different examples. iii) Using negatives: 200 from 500 is 300, 80 from 30 is -50 (Total 250), 6 from 2 is -4 (Total 246)

3.

You can make it easier by changing the second number into something easy to subtract, in this case 300 by adding 14 to 286 but in doing this you have taken off 14 too many so you need to add 14 to the other number giving you 546 - 300 (246)

4.

You never need a calculator for subtracting.

5.

Checking against your estimate you notice the answer lies between your two possible estimates and if you examine the numbers you can see why.

Most people already have a mental method for subtraction. If you have one, use it; if you haven't or if you prefer one of those given above, choose one of them and practise it until you are proficient mentally. Notice that for addition you can add to one number and take away from the other, while for subtraction, the inverse of addition, you have to add or subtract the same number for both. We will return to this. MULTIPLICATION 4

1.

We already looked at estimating for an example above: 36  59 for which our estimate was 40  60  2400 .

2.

It may not be easy to do this in your head totally but it should be possible with a minimum of writing down.

3.

There are several ways to make this easier, by using the inverse principle and in other ways. Always try them in preference to going to the long multiplication process. Here are some of them: i) Look at the numbers and see if there is a factor of one or both of them. In this case only 36 has factors: 6, 4, 9, 3, 2 and 12 all go into 36; let us choose one of those to use, say 6. Divide 36 by 6 and you have 6. But in doing so you have not multiplied by the 36 and, in order to counteract the effect of what you have done, you will need to multiply the other number by 6. Note this corresponds to what happens in addition where you have to add to one and take from the other while here you have to divide one and multiply the other to give 6  354 (straight multiplication of 59 by 6 or 360 - 6). You now have a `short multiplication' which you can do without difficulty, 2124. i.e. 36  59  6  (6  59)  6  354  2124 Checked against your estimate you notice it is less than 2400 and recognise that this is because, for your estimate, you made both numbers bigger so you would expect the final answer to be smaller. ii) Looking at the 59 you can simply go straight to 36  60 and then subtract 36. iii) You can break up the 59 into 50 + 5 + 4 all of which are easier to do than 59 itself. 36  50 36  5 36  4

= 1800 = 180 = 144

36  59

= 2124

half of 100 times 36 a tenth of 36 times 50 twice 36 is 72, twice 72 is 144, (an example of knowing up to four times higher numbers) by addition

iv) Egyptian way of multiplying: keep on doubling as far as you need to and then add appropriate totals. Here: 59  2 59  4 59  8 59  16 59  32

= 118 = 236 = 472 = 944 = 1888

32 + 4 = 36

so adding 59 × 4 59 × 32 59 × 36

236 1888 2124

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v) Russian peasant way of multiplying employs the idea of doubling one number and halving the other discarding any pieces left over thus: 36  59 18  118 9  236 4  472 2  944 1 1888

one lot of 236 lost

236

one lot of 1888

1888 2124

For other examples, using this method, you will have to add in an extra figure every time you get an odd number because, in halving it for the next line of the answer, you will have discarded one of the odd number of times. 4.

We have used the idea of inverse when we multiplied and divided by 6 as one way of doing the question. It was also used, in a very inventive way, in the Russian peasant method.

5.

You can use a calculator for multiplication (remember you must then have an estimate to make sure you haven't pressed the wrong keys) but if you practise some of the above methods you will probably end up not wanting to use a calculator because you have become good at doing without one.

6.

Check your answer against your estimate and try to see why it is either more or less than the answer you finally got. DIVISION At this point, we return to the processes we identified for addition and subtraction and for multiplication above because it should be possible to deduce from them something that would apply to division of numbers.

Addition: add to one number, take away from the other; Subtraction: add or subtract the same number for both; Multiplication: multiply one number, divide the other. What about division? Division is the inverse of multiplication just as subtraction is of addition. From this we can postulate that, for division, you can multiply or divide both numbers by the same number. Using this will give us a way to make division easier which is using the important inverse principle. Suppose we want to divide 2348 by 28. This would normally be a case for long division but, if we know what goes into both 2348 and 28 we can divide both numbers by that. 4 goes into both these numbers and, dividing them both by 4 we get 587 divided by 7, a simple division that should not cause anybody any difficulty. So, for this, our estimate might be 2400 divided by 30 which is 80. Now, having made the question easier by getting rid of the need for long division, we get 587 divided by 7 which is 83 remainder 6 or 83 76 or, taking it into decimals 83.857.... This is more than our estimate. Here we have made both our numbers bigger so, on the basis of our estimates for addition and multiplication, we might expect the answer to be less than the estimate. But division does not work in quite the same 6

way. If we increase the number to be divided we would expect the answer to be more while if we increase the dividing number we would expect the answer to be smaller so we have two opposing effects. Only extensive experience with such calculations can you begin to judge what to expect here. Division is different in another way too, for there are no other obvious ways of making division easier so, unless both numbers have a common factor which will enable the question to be reduced to one of short division, you may need to go to your calculator for division. There is an alternative to long division, however, which itself uses the inverse principle so those of you who are unhappy with long division might like to learn this way instead for cases when you cannot make it easier and do not want, or cannot, use a calculator. Alternative to long division 3342  17

Look at the number 3342 and notice that the answer will certainly be less than 200 because 200  17  3400 and more than 100 because 100  17  1700 , 3342 being between 1700 and 3400 so you can proceed as follows: Estimate: 3000 divided by 20 is 150 or 3200 divided by 16 which is 200. 3342 17 × 100 1700 1642 850 17 × 50 (half of 1700) 792 510 17 × 30 (10 × 51 which is 3 × 17) 282 17 × 10 170 112 85 17 × 5 27 17 17 × 1 10 10 . At this point the answer can be read off as 196 remainder 10 or 196 17 The leaflet on fractions, decimals and percentages shows how to express this number as a decimal or, alternatively, the procedure can continue to as many decimal places as is required by continuing as before. Remember that for the next figure for which you would use 8.5 it will be 17  0.5 by using the appropriate example of inverse given at the beginning of this leaflet. Note that the answer is between the two estimates and nearer to the second one. Think about why this might be. This way of working, once you have become accustomed to referring to and using the inverse principle in the specific ways that are appropriate to adding, subtracting, multiplying and dividing, will eventually give you greater number flexibility. It will also 7

enable you to judge when and when not to use a calculator and, if you do, how to judge whether the answer your calculator gives you is correct, or not correct because you have somehow pressed the wrong buttons. At this point YOU would never produce some of the ridiculous anomalies sometimes produced in gas bills etc when the person operating the computer which produces the answers has not noticed that those answers are impossible. This is a skill worth having but you will need to practise the ideas the leaflet gives you if you are to become numerate for a technological age.

We would appreciate your comments on this worksheet, especially if you’ve found any errors, so that we can improve it for future use. Please contact the Maths tutor by email at [email protected] updated 22nd June 2012

The information in this leaflet can be made available in an alternative format on request. Telephone 01482 466199

© 2009

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