Succeed with maths Part 1

SWMB_1 Succeed with maths Part 1 Copyright 2016 The Open University Contents Introduction and Guidance 3 Introduction 3 What is a badged course? 4 How to get a badge 4 6 Introduction 6 1 Getting started: puzzes and rea-word maths 7 2 Getting to know your cacuator 11 2.1 Another cacuator activity 12 3 Probem soving 14 3.1 New vehice 14 4 Understanding numbers 17 4.1 Parts of the whoe decima numbers 18 4.2 Parts of the whoe fractions and decimas 20 4.3 Rounding 22 5 This week's quiz 24 6 Summary 24 Everyday maths 25 Introduction 25 1 Addition and subtraction 26 1.1 Addition 26 1.2 Subtraction 28

1.2.1 Subtracting one number from another 29 2 Mutipication 30 2.1 Everyday mutipication 30 3 Division 33 4 Commutative properties of mutipication and division 35 5 Exponents or powers 37 5.1 Cacuator exporation: exponents 37 2 of 40 Thursday 23 November 2017

Introduction Introduction and Guidance Introduction Succeed with maths Part 1 is a free badged course which asts 8 weeks, with approximatey 3 hours' study time each week. You can work through the course at your own pace, so if you have more time one week there is no probem with pushing on to compete another week s study. You' start with the most famiiar territory of how numbers are put together and ook at addition, subtraction, division and mutipication. In the foowing weeks you' move on to ook at fractions, percentages and negative numbers. You use penty of rea-ife exampes to hep with this and give you penty of opportunities to practise your new understanding and skis. Part of this practice wi be the weeky interactive quizzes, of which Weeks 4 and 8 wi provide you an opportunity to earn a badge to demonstrate your new skis. You can read more on how to study the course and about badges in the next sections. After competing this course you wi be abe to: use maths in a variety of everyday situations understand how to use percentages, fractions and negative numbers in some everyday situations begin to deveop different probem soving skis use a cacuator effectivey. Moving around the course The easiest way to navigate around the course is through the 'My course progress' page. You can get back there at any time by cicking on 'Go to course progress' in the menu bar. From the quizzes cick on 'Return to Succeed with maths Part 1'. It's aso good practice, if you access a ink from within a course page, incuding inks to the quizzes, to open it in new window or tab. That way you can easiy return to where you've come from without having to use the back button on your browser. Viewing maths and equations As you view the different text styes and symbos used to dispay maths throughout the course, you may notice that some of the text appears cickabe or inked to something ese. This is part of the equation processing software used by The Open University. You can ignore these cicks or inks as they won't provide you with any usefu information, but if you do cick by accident you can just come back to where you were by using the back button. If you enjoy this course and want to practise more of your skis for maths, or if you find that you are aready famiiar with most of the materia in the course, you might ike to take a ook at Succeed with maths Part 2. 3 of 40 Thursday 23 November 2017

Introduction What is a badged course? Whie studying Succeed with maths Part 1 you have the option to work towards gaining a digita badge. Badged courses are a key part of The Open University s mission to promote the educationa we-being of the community. The courses aso provide another way of heping you to progress from informa to forma earning. To compete a course you need to be abe to find about 24 hours of study time, over a period of about 8 weeks. However, it is possibe to study them at any time, and at a pace to suit you. Badged courses are a avaiabe on The Open University's OpenLearn website and do not cost anything to study. They differ from Open University courses because you do not receive support from a tutor. But you do get usefu feedback from the interactive quizzes. What is a badge? Digita badges are a new way of demonstrating onine that you have gained a ski. Schoos, coeges and universities are working with empoyers and other organisations to deveop open badges that hep earners gain recognition for their skis, and support empoyers to identify the right candidate for a job. Badges demonstrate your work and achievement on the course. You can share your achievement with friends, famiy and empoyers, and on socia media. Badges are a great motivation, heping you to reach the end of the course. Gaining a badge often boosts confidence in the skis and abiities that underpin successfu study. So, competing this course shoud encourage you to think about taking other courses. How to get a badge Getting a badge is straightforward! Here's what you have to do: read each week of the course score 50% or more in the two badge quizzes in Week 4 and Week 8. For a the quizzes, you can have three attempts at most of the questions (for true or fase type questions you usuay ony get one attempt). If you get the answer right first time you wi get more marks than for a correct answer the second or third time. If one of your answers is incorrect you wi often receive hepfu feedback and suggestions about how to work out the correct answer. For the badge quizzes, if you're not successfu in getting 50% the first time, after 24 hours you can attempt the whoe quiz, and come back as many times as you ike. 4 of 40 Thursday 23 November 2017

Introduction We hope that as many peope as possibe wi gain an Open University badge so you shoud see getting a badge as an opportunity to refect on what you have earned rather than as a test. If you need more guidance on getting a badge and what you can do with it, take a ook at the OpenLearn FAQs. When you gain your badge you wi receive an emai to notify you and you wi be abe to view and manage a your badges in My OpenLearn!Warning! Tahoma not supportedwithin 24 hours of competing the criteria to gain a badge.. Get started with Week 1. 5 of 40 Thursday 23 November 2017

Week 1 Getting started Week 1 Getting started Introduction In this first week you wi be focusing on some introductory maths activities. Some of these probems may seem more ike puzzes than mathematics, and others wi ceary fa into the category of maths probems. To hep you there wi be hints for you to use, if you need them. A of these puzzes bring up points about how to engage in maths and what is usefu to know. You aso start to earn how to use your cacuator, and to understand how we put numbers together. In the foowing video, the course author Maria Townsend introduces you to Week 1: Video content is not avaiabe in this format. After this week, you shoud be abe to: tacke maths activities with more confidence use your cacuator for the four operations of +,, and use pace vaue and number-ines to understand numbers use rounding and estimation. 6 of 40 Thursday 23 November 2017

1 Getting started: puzzes and rea-word maths Before you start, The Open University woud reay appreciate a few minutes of your time to te us about yoursef and your expectations of the course. Your input wi hep to further improve the onine earning experience. If you d ike to hep, and if you haven't done so aready, pease fi in this optiona survey. 1 Getting started: puzzes and rea-word maths You probaby use maths every day without reaising, from working out how much change you might receive, to how ong a journey might take to soving puzzes. You re going to start by ooking at some Sudoku puzzes. In 2005, a puzze craze known as Sudoku swept across the word, starting in newspapers. These puzzes are cosey inked to a branch of mathematics known as graph theory. Sudoku invoves putting numbers on a square grid, and its creators caimed that soving it needed just a ogica mind no mathematics required. Most Sudoku puzzes are nine squares ong and nine squares wide, a nine-by-nine (9 9) ayout, but you are going to ook at a smaer exampe. You may be wondering what Sudoku has to do with soving rea probems. First of a, it wi introduce you to working ogicay, and that is an important genera technique that is used often in mathematics. Second, Sudoku puzzes are quite simiar to Latin squares, which are used extensivey in designing experiments such as finding out how different crops grow with different fertiiser treatments. Latin squares are square grids of numbers in which each number occurs just once in every row and coumn exacty the same as a Sudoku puzze! This happens quite often in mathematics. Very abstract mathematica topics, which seem to have no practica use whatsoever when they are discovered, ater turn out to be extremey important in science or technoogy. Just because it is difficut to see a use at the time does not mean that there wi not be some important practica deveopment ater! Activity 1 Do you do Sudoku? Aow approximatey 10 minutes Our arge square is composed of four bocks, each of which has four smaer squares within. The bocks are framed by thicker ines. There are aso four rows across and four coumns down, so this setup is a four-by-four (4 4) puzze. Draw this puzze for yoursef now. 7 of 40 Thursday 23 November 2017

1 Getting started: puzzes and rea-word maths Figure 1 Sudoku puzze The idea is to arrange the numbers so that each bock, row and coumn contains ony one of the numbers 1, 2, 3, and 4. For exampe, the bottom eft-hand bock of 4 aready has a 4 in it, so you need to put 1, 2 and 3 in the remaining ces in that bock. Make sure though that you do not end up with two numbers that are the same in any row or in any coumn. Try it now! Write your ideas directy on the copy of the diagram you drew. When you have either soved the puzze or spent about ten minutes on it, read on. Beow are some hints and tips if you fee you need more guidance. Cick on Revea comment to read these. Comment If you have never seen these puzzes before, you may find this quite tricky. There are many different ways that you can tacke this probem. The first step is to try to sort out exacty what you are being asked to do and to make sure you understand the probem. The next tactic to try is to make the probem simper by breaking it down into steps and concentrating on just one kind of number. You can choose the number that occurs most often and therefore is the number that you have most information about in this case, 2. Now, you need to find the number that makes most sense to fi in that does not vioate any of the rues given. Remember: each number (1, 2, 3 and 4) appears ony once in each bock, row and coumn. Keep the time imit of ten minutes in mind for the active part of this puzze. You can spend twice this time if you wish, but don t go beyond that. Try a soution now. If you need more guidance, review the next comment. Comment In this case, there are two 2s on the grid aready, so ony two more are needed, one in each of the bottom bocks. There is aready a 2 in the first coumn, so no more 2s can go in that coumn. The ony pace for the 2 in the bottom eft-hand bock is in the square under the 4. If you find this difficut to foow, draw a penci ine through the coumn and row that the given 2 is in to show that the 2s in the other bocks cannot be paced in this row or coumn. 8 of 40 Thursday 23 November 2017

1 Getting started: puzzes and rea-word maths Figure 2 Sudoku puzze with a number added Simiary, there is aready a 2 in the third coumn, so no more 2s can go in the third coumn. So the 2 in the bottom right-hand bock must go above the 3. Figure 3 Sudoku puzze with two numbers added Now, et s move on. You can see that the bottom row aready has a 3 in it, so the 3 in the bottom eft-hand bock must go next to the 4. 9 of 40 Thursday 23 November 2017

1 Getting started: puzzes and rea-word maths Figure 4 Sudoku puzze with four numbers added Now ook at the third row. Which number is missing? As more numbers are added, the puzze gets easier. If you have not finished the puzze for yoursef, try to do so now. Your fina grid shoud ook ike this: Figure 5 Competed Sudoku puzze Athough there is ony one correct answer to each Sudoku puzze, the approach described in the comment is not the ony way to get started on this puzze. If, for exampe, you focus on the number 4 in your first move, then you can reason that the bock in the ower right-hand corner must have a 4 paced beside the given 3. This woud ead to a different start as we as different steps to reason through, but it eads to the same answer. As there is ony one possibe arrangement of the numbers in the answer, if you don t reason your way through, reaching the correct answer is amost impossibe. As you can compete Sudoku with reason aone, you don t need to carry out any sums to find the soution. This means that there is no need for cacuations in your head or on 10 of 40 Thursday 23 November 2017

2 Getting to know your cacuator paper. For some of us that might be a bonus! But this might not aways be the case and athough it is a good idea to have some menta maths skis, there is nothing wrong with using a cacuator if you need to. So, before you move onto a maths puzze that invoves doing sums, you re going to take a quick ook at using a cacuator. 2 Getting to know your cacuator In the foowing activities you start to earn how to use your cacuator. You have penty of options when it comes to what cacuator to use. You may find it easier to use the one that you are most famiiar with, maybe a cacuator on your mobie phone, your computer or tabet, an onine cacuator, or a handhed cacuator. Let s start with a cacuation that you can check in your head to make sure your cacuator is working propery. Cick on or press the foowing keys: You shoud get the answer 7! If you didn t, have another go now, being carefu with the numbers that you enter. Once the cacuation is compete, it needs to be ceared before the next cacuation can be performed. You cear a cacuation by cicking on the button. Do this now; your ast cacuation shoud disappear. Now you re ready to do arithmetic on your cacuator in the next activity. You may find Tabe 1 hepfu in finding the correct button or key if you are using a computer keyboard. Tabe 1 Arithmetic on your cacuator Mathematica operation Cacuator button Computer keyboard key + Shift and = on the main keyboard or + on the number pad (the key to the eft of = ) on the main keyboard or on the number pad Shift and 8 on the main keyboard or * on the number pad / on the main keyboard or on the number pad = Enter Cear Backspace (as often as needed to cear a entries) 11 of 40 Thursday 23 November 2017

2 Getting to know your cacuator Activity 2 A few arithmetic probems Aow approximatey 10 minutes In this activity, you are going to do severa straightforward cacuations to make sure that you can get your cacuator to work propery. Check the answer in your head so that you know what your cacuator shoud show. Remember to cear the cacuation before you enter each new cacuation. (a) 5 + 20 (b) 100 99 (c) 2 4 (d) 10 2 Your cacuator shoud have given the foowing answers: (a) 25 (b) 1 (c) 8 (d) 5 How did that go? Hopefuy you found it fairy straightforward. There wi be penty of chances throughout the eight weeks of the course to become more confident using a cacuator. 2.1 Another cacuator activity This next number puzze, unike Sudoku, invoves doing cacuations. Activity 3 Cross-number puzze Aow approximatey 10 minutes Here s a cross-number puzze to give you some more practice using your cacuator. It s ike a crossword puzze, ony with numbers instead of words. You may want to downoad and print the puzze, and then use your cacuator to sove it. 12 of 40 Thursday 23 November 2017

2 Getting to know your cacuator Figure 6 Bank cross-number puzze Across Down 1. 21 47 1. 19 5 4. 1788 6 2. 8003 9 6. 497 105 + 12 3. 1234 1216 7. 1458 3 2 5. 1 + 11 + 111 29 9. 32 + 681 13 6. 517 11 7. 21 33 7 8. 49 + 49 48 Figure 7 Competed cross-number puzze We done, you ve competed two number puzzes aready and started to buid probemsoving skis as we! After a a probem is just another word for a puzze. So before you 13 of 40 Thursday 23 November 2017

3 Probem soving move on to thinking about how numbers are put together, you re going to spend some time thinking about probem soving and competing one more puzze (or probem). 3 Probem soving When you approach a probem there is every ikeihood that you may take a different route to the soution than somebody ese. That is absoutey fine! Some approaches may just be more efficient, and get you to answer more quicky than others. The more experience you have with probem soving the better you wi become at it. Depending on your previous experiences and how you initiay take in a particuar probem, the way you decide to begin to sove the probem wi vary. Soutions to everyday probems may just need the use of common sense and organisation to work your way through them. As you work through any probem, remember that there are usuay aternative methods for reaching the soution. If you get stuck using your initia approach, try a different one. Keep in mind that using pictures and staying eve-headed wi carry you far, and most ikey hep you finish soving an exercise. So try not to panic! With these thoughts in mind, try the next activity. 3.1 New vehice Just as with the Sudoku puzze, in the next probem you need to use ogic and reasoning to recover missing pieces of information. Activity 4 New vehice Aow approximatey 10 minutes Imagine you ve recenty been ooking for a new vehice. You don t know at the moment if you woud ike a motorbike or a car because you woud ike to get the best price possibe and have the most options avaiabe. When you ca the deaership to enquire about its stock, the assistant manager, Pau, jokingy tes you that they have 21 vehices avaiabe, with a tota of 54 whees. Just as you ask him how many of each type of vehice he has, the phone ca is inadvertenty disconnected. Can you work out how many motorcyces and how many cars the deaership currenty has? As with any probem there are different ways of approaching this probem, so when you ve got your answer take a ook at some other possibe methods. Remember, if you need a hint to hep you get started cick on Revea comment beow. Comment There are many ways to sove this probem. You might consider trying to use pictures (visuaisation can be very hepfu) or seect a starting point, such as assuming haf are motorbikes and haf are cars, and then revising your first guess. Method 1: Diagrams 14 of 40 Thursday 23 November 2017

3 Probem soving Suppose that a the vehices are motorbikes. Draw a diagram (you don t need to be an artist) that shows these 21 motorbikes. Be sure to use a representation that wi aow you to ceary distinguish between 2 and 4 whees. Figure 8 Picture representing 21 motorbikes Since there are 21 motorbikes with 2 whees each, your drawing shows a tota of 42 whees. Pau said that there were 54 whees, so you need an additiona 12 whees, because 54 42 = 12. In other words, some of the motorbikes drawn need to be turned into cars. You coud just start adding 2 whees to each motorbike unti you ve counted up to 12, or you coud be cever. If you change a motorbike into a car, you gain 2 additiona whees. Since you know you need 12 more whees to reach the required tota, you need to change 6 motorbikes to cars, because 12 2 = 6. Can you see how that worked? Figure 9 Picture representing 21 motorbikes atered to show the first 6 with 2 extra whees added to make 4 From the picture, you can observe that there are 6 cars and 15 motorbikes, for a tota of 21 vehices. You can check that the tota number of whees adds up to 54. Each car has 4 whees, which makes 6 4 = 24 whees. Each motorbike has 2 whees, which makes 15 2 = 30 whees. Together, this is 24 + 30 = 54 whees. Thus, the deaership has 15 motorbikes and 6 cars in stock. 15 of 40 Thursday 23 November 2017

3 Probem soving Method 2: Educated guess You can make any guess you think is reasonabe. For exampe, you might assume that about haf of the vehices were motorbikes say, ten of the vehices. To keep track of your guesses, a tabe is quite usefu. As you make adjustments to your guesses, remember that the number of motorbikes pus the number of cars must equa 21. Motorbikes Cars Tota number of whees 10 11 10 2 + 11 4 = 64 (too many whees need fewer cars) 12 9 12 2 + 9 4 = 60 (too many whees need fewer cars) 14 7 14 2 + 7 4 = 56 (too many whees need fewer cars) 15 6 15 2 + 6 4 = 54 This matches with what Pau tod you. Once again, the concusion is that there are 15 motorbikes and 6 cars at the deaership. Method 3: Pairs of whees Another way to sove this probem is to consider the pairs of whees. Because the assistant manager tod you that there are a tota of 54 whees, this means there are 27 pairs of whees, because 54 2 = 27. If a the vehices were motorbikes, you woud have 27 motorbikes, which is too many: remember, Pau tod you there are ony 21 vehices. Next, determine how many extra pairs of whees there are. Because 27 21 = 6, you know there are 6 extra pairs of whees. This indicates that 6 of the vehices have to have an extra pair of whees (beyond the pair for each vehice that you ve aready counted). In other words, 6 vehices must have 2 pairs of whees, or 4 whees each. Thus, there are 6 cars. To find the number of motorbikes, you need to take the cars away from the tota number of vehices: 21 6 = 15. Consequenty, there are 15 motorbikes and 6 cars. So, remember when you come across a probem that there wi usuay be more than one way to approach it. If you don t get anywhere with your first method, see if you can come at the probem a sighty different way. You are going to eave probem soving behind for a moment and move onto subjects that you might think fee ike proper maths! It is important to have a good understanding of how numbers are put together so that they make sense to us and what they represent. You wi turn your attention to this in the next section. 16 of 40 Thursday 23 November 2017

4 Understanding numbers 4 Understanding numbers To start your exporation of numbers you are going to concentrate on whoe numbers or what you might aso think about as the numbers that you count with. Every number that we see around us is made up of a series of digits, such as 1302 (one thousand, three hundred and two). The vaue that each digit has depends not ony on what that digit is but aso on its pace, or position, in a number. This aows you to represent any number that you want by using ony 10 digits (0, 1, 2, 3 etc.). Each pace in a number has a vaue of ten times the pace to its right, starting with ones or more propery caed units, then tens, hundreds, thousands and so on. So in 1302, the 2 is in the units pace, the 0 is in the tens pace, the 3 is in the hundreds pace and the 1 is in the thousands pace. This means that the 3 has a vaue of 300 and the 1 has a vaue of 1000. If you now rearrange the digits in your first number they wi have different vaues and you have a new number. So, 3120 has a the same digits but now the 3 is 3000, rather than 300 as it has moved one pace to the eft. Have a ook at this next exampe: Figure 10 Pace vaue You can see in the iustration above that the foowing is true: 1 is in the ten thousands pace 5 is in the thousands pace 7 is in the hundreds pace 6 is in the tens pace 4 is in the units pace. 17 of 40 Thursday 23 November 2017

4 Understanding numbers and the number is 15 764, which is spoken as fifteen thousand, seven hundred and sixty-four. Using this system you can write and understand any whoe number but as you are probaby aware numbers are not imited to ony whoe numbers. You aso need ways to represent parts of whoe numbers. The two ways that you do this is by using either decima numbers, ike 3.25, or fractions, such as. You start with a ook at decima numbers, before taking a very brief ook at fractions and how these to reate to decima numbers. The rea work on fractions wi be eft for Weeks 3 and 4, so this is just a gente introduction! 4.1 Parts of the whoe decima numbers An exampe of a decima number is 1.5, said as one point five. The point in the midde is caed the decima point and is used to separate the whoe numbers from decima parts of the number. The 5 in 1.5 represents the part of a whoe number. To understand how the decima part of a number works you can start by extending the pace vaues you have aready to the right, making each one 10 times smaer. This aows you to represent numbers that are parts, or have parts of whoe numbers. These are known as decimas numbers. This gives a pace vaue tabe that ooks ike this: Tabe 2 Extended pace vaue tabe The next pace vaue tabe shows a decima point added in, showing where the whoe numbers are separated from the parts of whoe numbers, and some exampes of decima numbers. Figure 11 A pace vaue tabe showing decima numbers The numbers shown in the tabe are: no units and two-tenths, written as 0.2 one unit, three-tenths and five hundredths, written as 1.35 one unit and eight-tenths, written as 1.8 four units, two tenths, zero hundredths and five thousandths, written as 4.205. 18 of 40 Thursday 23 November 2017

4 Understanding numbers As we as using a pace vaue tabe you can aso show decima numbers on a number ine. This may hep you to visuaise decima numbers. Take a ook at this number ine. The intervas between the whoe numbers (or units) have each been spit into ten equa intervas, each one is therefore 10 times smaer than a unit, so each of these is a tenth. If each tenth had then been spit into ten equa intervas, these new intervas wi be hundredths, since there wi be 100 of these intervas in a whoe unit. Figure 12 A number ine The number ine shows the foowing numbers expressed in decima form: two-tenths, written as 0.2 one unit and three-tenths, written as 1.3 one unit, three-tenths and five-hundredths, written as 1.35 one unit and four-tenths, written as 1.4 one unit and eight-tenths, written as 1.8. See how you get on with this next activity giving you practice at identifying the different pace vaues. Activity 5 Pace vaues Aow approximatey 10 minutes What digit is in the hundredths position in this number? 5.603 0 is in the hundredths position. What digit is in the thousands position? 32 657 2 is in the thousands position. What digit is in the thousandths position? 5.8943 4 is in the thousandths position. 19 of 40 Thursday 23 November 2017

4 Understanding numbers Now you ve seen how you can use decima numbers to represent numbers that incude parts of a whoe you turn your attention to the second way that you can do this fractions. But don t worry you re not going to get serious with fractions just yet that's for Weeks 3 and 4. Here, you re just going to think about a few everyday fractions and then see how you can reate fractions and decima numbers to each other. 4.2 Parts of the whoe fractions and decimas You use fractions every day even if you're not reay thinking about it. For exampe, if somebody asks you the time it coud be haf past 2 or a quarter to 6. The haf and the quarter are parts of an hour; in other words they are fractions of a whoe. The number at the bottom of the fraction tes you how many parts the whoe one has. So, has two parts in the whoe one and has 4. The number at the top tes you how many of these parts there are. Hence, is 1 ot of 2 parts. You be ooking at this again in more detai in Week 3. Since fractions are parts of a whoe, just as decimas are, this means that you can reate fractions and decimas. Here you wi ony be deaing with fractions that are easiy shown in a pace vaue tabe. So, say you needed to write out 2 and (three-tenths) as a decima. The whoe part is 2 and the fractiona part is. Remember in order to separate the whoe number from the fraction you use a decima point, with the part of the whoe number to the right. So, you woud write the 2 to the eft of the decima point and the fractiona part to the right of the decima point. Thus, it woud be written as 2.3. If you were deaing with on its own, you woud need show that there were no whoe parts by showing a zero to the eft of the decima point. is therefore written as 0.3. Now you ve seen a few exampes it s your turn to have a try yoursef in the next activity. You can use a pace vaue tabe if it heps you. Activity 6 Fractions to decimas Aow approximatey 10 minutes a) Rewrite each of the foowing fractions as a decima number. (i) Comment If the number does not have a whoe number part, a zero is written in the units pace. This makes the number easier to read (it s easy to overook the decima point). (i) (ii) 20 of 40 Thursday 23 November 2017

4 Understanding numbers (ii) (iii) (iii) (iv) (iv) b) Match the numbers beow to the etter shown on the number ine: Figure 13 Number ine exercise B C A D Now, you re going to turn your attention to rounding numbers, and making estimates. Understanding pace vaue wi hep with this but you aso investigate the idea of decima paces when rounding. 21 of 40 Thursday 23 November 2017

4 Understanding numbers 4.3 Rounding The word popuation in January 2013 was estimated to be 7.06 biion. That is not an exact vaue but it does give you an indication of the word s popuation at the time. This is an exampe of a rounded or estimated number. In newspapers and esewhere, numbers are often rounded so that peope can get a rough idea of the size, without getting ost in detais. You may use rounding yoursef, such as prices in a supermarket ( 2.99 is about 3) or distances (18.2 mies is about 20 mies). Using approximate vaues is aso usefu if you want to get a rough idea of an answer before you get out scrap paper or a cacuator. This estimation acts as a check on your cacuation and may hep you to catch any errors you may have. The genera rues for rounding are: Locate the pace vaue to be rounded. Look at the next digit to the right. If that digit is a 5 or greater, round up the previous pace vaue digit which means increase it by one. If that digit is a 4 or ess, eave the previous pace vaue digit unchanged. Repace a digits after the pace vaue digit with 0 but ony if these are to the eft of the decima point. This ast rue may be a itte harder to understand, so here s an exampe. If you are rounding 5423 to the nearest one hundred this woud be 5400 the tens and units paces being repaced by zeros. With decimas we often refer to the number of decima paces to round to rather than a digit s pace vaue. So, instead of asking you to round to the nearest tenth, you might see instructions to round to 1 decima pace (1 d.p.). The decima paces are numbered consecutivey from the right of the decima point. So, 1.54 has 2 decima paces, and 34.8942 has 4 decima paces. Whether you are rounding using pace vaue or decima paces the genera rues remain the same. Before you have a ook at some exampes you might ike to view this video on rounding. Here are a few more exampes to have a ook at before you have a go yoursef. Imagine you want to round the word popuation figure of 7.06 biion to the nearest biion. The pace vaue that you want to round in this case is the units as these represent the biions, which is a 7. The next number to the right of this is a 0, which is 4 or ess, so you don t need to round up and the 7 stays as it is. So 7.06 biion rounded to the nearest biion is 7 biion. Now you want to round 1.78 to 1 decima pace. The 7 is in the first decima pace and to the right of this is an 8. This is 5 or greater, so the 7 is rounded up to 8. So 1.78 rounded to 1 decima pace is 1.8. Now it s your turn. Activity 7 Rounding Aow approximatey 5 minutes Round each of the foowing numbers as stated. 22 of 40 Thursday 23 November 2017

4 Understanding numbers (a) Round 126.43 to the nearest tenth. (a) Locate the pace vaue to be rounded in this case, the number 4. Look at the next digit to the right in this case, it s 3. 3 is ess than 5, so you round down, and the answer is 126.4. (b) Round 0.015474 to the nearest thousandth. (b) Locate the pace vaue to be rounded in this case, the number 5. Look at the next digit to the right in this case, it s 4. Because 4 is ess than 5, you round down, and the answer is 0.015. (c) Round 1.5673 to 2 decima paces (2 d.p.). (c) Locate the decima pace to be rounded in this case the number is 6. Look at the next digit to the right in this case, it s 7. Because 7 is greater than 5, you round up, and the answer is 1.57. You have one fina activity on rounding and soving a puzze this week before it is time for the weeky quiz to check your progress, and prepare you for the Week 4 badged quiz. Activity 8 How much is it? Aow approximatey 10 minutes Suppose you want to buy a new pick-up truck. When you visit the deaership s website, it says the cost of the vehice you are ooking at is 19 748. When you ca the deaership on Thursday, the saes representative, Joe, tes you the cost is 19 750. However, the next day another saes representative, Tina, informs you that the vehice costs 19 700. Why did the quote you were given on the cost of the pick-up truck change? Comment Round the website cost to the nearest 10. Then round the website cost to the nearest 100. Here s how the variation in the quotes happened: Rounding to the nearest 10 (this is the same as rounding to the tens pace): The digit 4 is in the tens pace. Looking to the pace vaue to the right of the 4, the digit is an 8. Since 8 is arger than 5, you round up the 4 to 5 in the tens pace. Finay, you repace 23 of 40 Thursday 23 November 2017

5 This week's quiz the units digit with 0 (because this pace is to the eft of the decima point). The rounded number is 19 750. This expains Joe s quote. Rounding to the nearest 100 (this is the same as rounding to the hundreds): the digit 7 is in the hundreds pace. Looking to the tens pace (to the right of the 7), the digit is a 4. Since 4 is ess than 5, you eave the digit 7 unchanged and repace the digits after the 7 with zeros. The rounded number is 19 700. This expains Tina s quote. 5 This week's quiz We done, you ve just competed the ast of the activities in this week s study before the weeky quiz. Go to: Week 1 practice quiz. Open the quiz in a new tab or window (by hoding ctr [or cmd on a Mac] when you cick the ink). 6 Summary Congratuations on making it to the end of this week. You started by warming up your mathematica skis with some puzzes. By working your way ogicay through these puzzes to the answer, you have made a great start in honing probem-soving skis that you may not even have been aware that you had. As we as this, you ve begun your study of the foundations of maths. This new knowedge wi provide a soid base from which to move on to Week 2. In Week 2, you return to probem soving and aso the essentia maths operations (addition, subtraction, mutipication, division), using these in some everyday exampes. You shoud now fee that you can: tacke maths activities with more confidence use your cacuator for the four operations of +,, and use pace vaue and number-ines to understand numbers use rounding and estimation. You can now go to Week 2. 24 of 40 Thursday 23 November 2017

maths Everyday Everyday maths Introduction In this week you wi investigate the four basic operations used in maths, addition, subtraction, mutipication and division, as we as the cosey reated idea of powers or exponents. When using a of these, it is important to know what order to carry these out in, just ike you woud when making a piece of fat pack furniture. If you start in the wrong pace, you may not end up with a correcty finished wardrobe, or correct answer. So, you wi then ook at the correct order to carry out cacuations, before turning again to probem soving strategies. Maria introduces Week 2 in the foowing video: Video content is not avaiabe in this format. After this study week, you shoud be abe to: use addition, subtraction, mutipication and division in different situations use estimation to check your answers use and understand exponents (powers) carry out operations in the correct order appreciate the steps in soving a probem mathematicay. 25 of 40 Thursday 23 November 2017

maths Everyday 1 Addition and subtraction In Week 1 you earned how to write and round numbers, as we as making a start on probem-soving. You ve aready used some of the four basic operators, such as mutipication and addition in the new vehice probem but now you re going to ook at a these basic operators in turn, as they are a fundamenta to maths and everyday cacuations. 1.1 Addition So, you' start by considering addition. If you are working out a budget, checking a bi, caiming a benefit, assessing work expenses, determining the distance between two paces by car or any other everyday cacuations, you wi probaby need to add some numbers together. Consider the foowing questions. How much is the bi atogether? How much wi your hoiday cost now that the tota price of your airfare has increased by 50 due to fue surcharges? Five more peope want to come on the trip: What is the tota number of peope booked on the bus now? What is the cost pus VAT? A the questions above invove addition to work out the answer. The key or trigger words that indicate addition are in bod. The process of adding numbers together may aso be referred to more formay as finding the sum of a set of numbers. Something that you probaby take for granted is that when you add two numbers together, it does not matter which order you add them together, for exampe 2 + 3 gives the same answer as 3 + 2. In mathematica terms, we say that addition is commutative. Not every operation wi possess this property for exampe 5 3 does not equa 3 5. There are a few ways that you can carry out addition on paper, in our heads or using a cacuator for more compicated sums. You may aready fee comfortabe with adding on paper, which is great, but here s a quick overview just to make sure. To add numbers in the decima system, you write the numbers underneath each other so that the decima points and the corresponding coumns (or pace hoders) ine up. Then you add the numbers in each coumn, starting from the right. Remember that when you are adding, if the sum from one coumn is arger than 10, you wi need to carry into the next pace hoder. For exampe, adding 26.20, 408.75 and 0.07: 26 of 40 Thursday 23 November 2017

maths Everyday Figure 1 Adding decimas Now try this activity on adding up items for a bi. Activity 1 Checking a bi Aow approximatey 10 minutes Imagine you have bought three items costing 24.99, 16.99 and 37.25 from a mai order cataogue and the postage is 3.50. Start by rounding these prices to the nearest pound and working out an estimate for the tota bi in your head, or on paper. Figure 2 Adding up a bi Then check your estimate by writing out the probem without rounding and performing the cacuation again to give the accurate bi. You may have your own method of working things out in your head that works fine for you. If so, there is no need to change how you do this, but you might find the foowing method usefu to work through. The four rounded prices are 25, 17, 37 and 4. To find an estimate for the tota cost, add the four rounded prices together. This is easier to do if you spit each number apart from the first one, into tens and units, so 17 can be spit into 10 + 7 and 37 into 30 + 7. This makes the sum much easier to work out in your head and the sum then becomes 25 + 10 + 7 + 30 + 7 + 4. 27 of 40 Thursday 23 November 2017

maths Everyday Then working from the eft and adding each number in turn, you can say, 25 and another 10 makes 35; another 7 gives 42; another 30 gives 72; 7 more makes 79, and another 4 gives 83. The answer is 83, so the tota bi wi be approximatey 83. To check your estimate on paper, you wi need to ine the unrounded numbers up and then add them. Figure 3 Checking your estimate You can see that the estimate of 83 was very cose to the exact amount 82.73. Now you can be confident in your answer. The next section wi ook at subtraction. 1.2 Subtraction Subtraction foows naturay on from addition, because one way of viewing it is as the opposite of addition. For exampe, if you add 1 to 3, the answer is 4. If you subtract 1 from 4, the answer is 3, giving the origina vaue. Just as with addition there are trigger words in probems that te you that you wi need to subtract. Consider the foowing questions: What s the difference in distance between Miton Keynes and Edinburgh via the M6 or M1/A1? How much more money do you need to save? If you take away 45 of the pants for the front garden, how many wi be eft for the back garden? If a hoiday prices have been decreased (or reduced) by 20, how much is this one? A these questions invove the process of subtraction to find the answer a process that you often meet when deaing with money. You can see that the trigger words for subtraction are again in bod. Remembering that subtraction is the opposite of addition gives one way of tacking probems invoving subtraction using addition. A ot of peope find addition ot easier than subtraction, so it s a usefu tip to remember. 28 of 40 Thursday 23 November 2017

maths Everyday For exampe, instead of saying, 10 minus 7.85 eaves what? you coud say, What woud I have to add to 7.85 to get to 10?. Adding on 5 pence gives 7.90, another 10 pence gives 8 and another 2 wi give you a tota of 10. So, the tota amount to add on is 0.05 + 0.10 + 2.00 = 2.15. This is the same answer as the one obtained by subtraction: 10 7.85 = 2.15 This means that you can aso check an answer to a subtraction probem by using addition. You may need to carry out subtraction on paper though, so in the next section there is a quick reminder of how to do this. 1.2.1 Subtracting one number from another As with addition, you may fee comfortabe aready with how to perform subtraction on paper. You may have aso earned a different method to the one shown here and if this works for you it s fine to continue with your method. Here s a quick overview in case you fee that you need a reminder. Write out the cacuation so that the decima points and the corresponding coumns ine up, making sure the number you are subtracting is on the bottom. Then, subtract the bottom number from the top number in each coumn, starting at the right. Remembering that if you need to borrow from the next coumn, to change the vaue in that coumn. Figure 4 beow describes how you can do this. You wi ony need to do this if you don t have enough in one coumn to perform the subtraction. Figure 4 Subtracting one number from another Now it s time to put your knowedge to use in an activity using addition and subtraction when deaing with the sighty trickier situation of time. Activity 2 Time, addition and subtraction Aow approximatey 10 minutes Imagine you are traveing by train to attend a very important meeting. The train is eaving at 9:35 a.m. and arriving at 11:10 a.m. The ticket costs 48.30 but you have a refund voucher for 15.75, foowing the canceation of a train on an earier journey. How ong wi the journey take, and how much money wi you have to pay for the ticket if you use the entire refund voucher? Remember, if you need a hint, cick on revea comment. 29 of 40 Thursday 23 November 2017

maths Everyday Comment Start by taking out and isting a the information from the question this shoud hep to make the probem cearer. Working with times can be tricky because you have to remember that there are 60 minutes in an hour, so you are not working with the simpe decima system based on the number 10. One way to do this is by imagining a cock and saying, From 9:35 a.m. to 10:00 a.m. is 25 minutes; 10:00 a.m. to 11:00 a.m. is an hour and 11:00 a.m. to 11:10 a.m. is an extra ten minutes. So the journey time is 1 hour and 35 minutes, assuming the train runs on time. Now you have one more step: To find out how much you wi have to pay for the ticket, you need to subtract 15.75 from 48.30. There are different ways of doing this; one way is to work up from 15.75, adding on 25 pence to give you 16.00, then 32.00 to get to 48.00 and finay another 30 pence to reach 48.30. The tota to pay is 32.00 pus 25 pence pus 30 pence or 32.55. Now it s time to investigate the two other operations: mutipication and division, starting with mutipication. 2 Mutipication Mutipication is an extension of addition, and provides a quick way of carrying out repeated addition. For exampe, you know that 3 2 = 6. But why is this true? We, another way to think of 3 2 is as 3 ots of 2; which is the same as 2 and 2 and 2. So, 3 2 is teing us to add 2 to itsef 3 times: 2 + 2 + 2 = 6. We ve aready used one word that tes you to use mutipication, which is ots. But there are some others, as with addition, which mean the same thing. These are shown in bod beow: 5 times 6 On a cashpoint you might see the phrase Pease enter a mutipe of 10. A the words in bod, are again trigger words teing you to mutipy. Now you ve ooked at some of the anguage of mutipication you take a quick ook at carrying it out on paper. 2.1 Everyday mutipication Just as with addition and subtraction there are a number of ways that you can carry out mutipication, in your head, on paper or using a cacuator. You probaby find for more compicated exampes you wi need a cacuator but if you don t have one handy remembering how to do this on paper wi be a great hep. Of course, a good knowedge of the mutipication (times) tabes wi aso be a bonus! 30 of 40 Thursday 23 November 2017

maths Everyday To make sure that you are happy with mutipying on paper have a ook at a coupe of exampes. Suppose that you need to work out what 9 ots of 32.50 when you are out shopping. You need to set up the cacuation so that the smaer number is on the bottom, as shown beow (you have to do ess work then!). Working from right to eft mutipy each number in turn, writing the resut in ine with the appropriate coumn. If the resut is greater than 9, remember you wi need to carry to the next coumn. If you need to mutipy by a number that is 10 or more, then foow the same procedure as described above to set up the cacuation. Then mutipy each number in the top row by each digit in the ower number in turn, finay adding up the two resuts. When mutipying 3965 by 25, start with the 5 and then move onto the 2. But you are not reay mutipying by 2 but by 20 (2 is in the 10s pace). To take this into account, add a zero to the beginning of the 2nd ine of working, as shown beow. The next activity wi give you a chance to practise this. Here s a reminder of your times tabes. 31 of 40 Thursday 23 November 2017

maths Everyday Figure 5 Times tabe If you ike you can downoad and print a version of this tabe. Activity 3 Mutipication Aow approximatey 10 minutes On a piece of paper, perform the foowing operations by hand. (a) 348 37 Comment For mutipication, begin by writing the probem verticay and ining up the digits in the units pace. Thus, the answer is 12876. (This seems reasonabe since 300 40 = 12000.) (b) 560 23 32 of 40 Thursday 23 November 2017

maths Everyday Thus, the answer is 12880. (This seems reasonabe since 600 20 = 12000.) Now you ve ooked at mutipication, it s time for division, a cose reation of mutipication. 3 Division You know from the ast section that mutipication is repeated addition, simiary division can be thought of as repeated subtraction. For exampe, 20 divided by 4 is 5 (20 4 = 5). The answer is 5 because 5 is the number of times you subtract 4 from 20 to arrive at zero: Figure 6 Demonstrating division This aso eads to the reaisation that division is the opposite or undoes mutipication. If you add 4, five times to zero (repeated addition or mutipication) then you wi arrive back at 20. And just as with the other operators, there are words that te us you need to divide. Here are some exampes that you may recognise: How many times does 8 go into 72? How can you share 72 items equay among eight peope? How many ots of five are there in 20? Unike mutipication when there is ony one symbo that says mutipy, division has a whoe famiy of notation! If you want to divide 72 by 8 this can be written in the foowing ways: 72 8, or, or 72/8 or Not a division probems wi resut in a nice whoe number answer and you may have a number eft over. This number is known as the remainder. The common notation is to show this using the etter R. Now, you re going to ook at division on paper, just as you did with mutipication. You set up division on paper using the fina notation from our ist. Starting from the eft, in this case, divide into each number in turn, writing the resut above the number you have 33 of 40 Thursday 23 November 2017

maths Everyday divided into. In this case if you can t divide into a number move to the next and divide into both. Before the next activity you might ike to watch this video showing you how to carry out ong division. Now have a go at the next activity. If you wish you can view the times tabe again. Activity 4 Division Aow approximatey 10 minutes On a piece of paper, perform the foowing operations by hand. (a) 4968 24 The soution is 207. (b) 4035 15 The soution is 269. When ooking at subtraction and addition you thought about whether it mattered in what order these were carried out. You found that for addition the answer is the same whatever 34 of 40 Thursday 23 November 2017

maths Everyday the order but this is not the case for subtraction. What about mutipication and division? You find out in the next section. 4 Commutative properties of mutipication and division When you add two numbers together, the order does not matter the same as saying that addition is commutative; so 2 + 4 is the same as 4 + 2. But what about mutipication and division? Is 3 2 the same as 2 3? Is 4 2 the same as 2 4? To hep with this we use a diagram. On the eft this shows three rows of two dots (3 2), and on the right two rows of three dots (2 3). Figure 7 Order of mutipication The number of dots in both arrangements is the same, 6, and hence you can see that 3 2 = 2 3. However, you can t say the same for division, where order does matter. For exampe, if you divide 4.00 between two peope, each person gets 2.00. If instead you need to divide 2.00 among four peope, each person ony gets 0.50. Division is not commutative. Now, you ve ooked at the fundamentas of mutipication and division you are going to get a chance to appy these to a more everyday probem. The next activity uses both mutipication and division to sove a probem. Activity 5 How much paper? Aow approximatey 10 minutes Try doing these on paper and then check your answers using a cacuator. A coege bookshop buys pads of ega paper in buk to se to students in the aw department at a cheap rate. a) Each pack of paper contains 20 pads. If the shop wants 1500 pads for the term, how many packs shoud be ordered? a) You need to work out how many times 20 goes into 1500, so need to divide. Number of packs = 1500 20 = 75 Therefore, the bookshop needs to order 75 packs. 35 of 40 Thursday 23 November 2017