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# Unit 1: Math and You

## 1 Introduction

This unit is centered on introductory activities and opportunities to try solving them. (Don’t worry: There will be hints for you to use if you get stuck.) Some of these problems may seem more like puzzles than mathematics, and others will clearly fall into the category of math problems. But all of them bring up points about how to engage in math and what is useful to know before going on to the next units. In Unit 1, you’ll also start to learn how to use the web calculator—a useful tool when you are working online.

This unit includes a look at how best to organize yourself to be successful in learning math. This includes using a math notebook and making a schedule.

First of all, you are invited to meet a few people who will explain how they use math in their jobs. Succeed with Math would not exist were it not for people like these folks and you, who agree that knowing math is important and useful. So let’s get started!

### 1.0.1 What to Expect in this Unit

This unit should take around five hours to complete. In this unit you will learn about:

• How to learn online.
• The online calculator.
• Math puzzles and problem solving strategies.

## 1.1 Everyone Uses Math

Math is the basis of many things we do or use every day. We need it in jobs or business, and have to understand mathematical concepts to excel in certain aspects of our lives. In the short videos below you will meet people who use math in their work every day.

Media Production Specialist

Click on the white arrow in the center of the black screen to start the video.  Make sure that you have the sound turned on. If you are having problems, ask someone for help.

Skip transcript

### Math on the Job

I use math on the job every day. For instance, when I go out on a video shoot there are two things we look at on the camera; one is the F-stop, the other is the focus. F-stop is a measurement of how much light you are allowing to go into camera. So for example; if you are in a really brightly lit area, day lit area, you’re going to close down your F-stop to an F16. When you’re in a really dimly lit area, or a dark room, you’re going open up your camera iris to a 1.8 or 2.4.

Another way we use math out on a video shoot is when we are focusing the camera. Um, subject for instance might be ten feet away, well you’re going to check your focus rein and make sure it was set for ten feet, so at ten feet your subject will be in focus.

We also use math back in the editing room. When we are capturing video we will capture it at different aspects ratios depending on what our project calls for. Sometimes we will use a 16×9 aspect ratio gives you a wide screen shoot, or we will use a 4×3 or standard depth ratio. Another way we use math in video editing is when we are working with time code. Time code is both a measurement of time but also shows us things like frames per second. Most video is shot at 30 frames per second, and then this will transition back when we go back into video editing. In a sense that If you are trying to create a transition, like a cross fade or a dissolve, you’re going to what to adjust that depending on the mood that you’re trying to create. Sometimes you want a really long transition, so that might be, I don’t know, 60 frames, so it would be about two seconds long. Sometimes you’re going to want something that is a lot quicker and that might be 15 frames, but you have to do that and kind of gage it with what your project is.

And those are a couple of different ways we use math here when we are going video production on the job.

End transcript

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Potter

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Firefighter

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Did you realize that these professions involve so much math? You can probably think of many more math applications on the job besides the ones you have just seen. So keep in mind that most professionals are in need of some mathematical skills to perform their job.

And this does not even include the professions that usually come to mind when we talk about jobs heavily based on mathematics—engineers, computer programmers, architects, statisticians, and many more.

Thinking about where math is used in the real world may help motivate you to start your mathematical journey as well.

### 1.1.1 Math Notebook

Keeping all of your own math notes in one place will be a big help. To learn math, you need to try problems on your own and write things down. Prepare a notebook to keep all the math notes you create. It can be a bound or spiral notebook, or even a binder. Use it when you do an activity in the main portion or a problem in the self-check section. In addition, make a few notes from the screens that explain a concept that is new to you or that you previously found unclear.

It is also a good idea to have a section set aside in your notebook to write down any new math terminology you run across, and its definition. Think of it as a vocabulary section. Looking back at the definition of a word can clarify a task or an explanation—and if you keep these words in your math notebook, you may actually learn them better and be able to look them up less and less frequently.

Write down the section a problem or fact came from (for example, “Section 2.1”) and give each entry a specific, meaningful title. This is important and will make it easier to find the original place the item came from.

Thinking about how you are getting on with your learning is also very helpful and you can use your math notebook to do this as well. Consider things like if you have found the best time to study, if you are taking enough notes, and if you have understood all the concepts covered in a unit? If you do this and spend some time reviewing your thoughts at the end of the course, you may well surprise yourself in how far you have come from the beginning. Think of it as your learning diary that is just for you to look at, so you can say just what you want!

Other than that, be as creative as you like. Make your math notebook uniquely yours: Individualize it. Just don’t take too much time to make things look nice.

### 1.1.2 Sudoku

#### Activity: Do You Do Sudoku?

[ These puzzles are closely linked to a branch of mathematics known as Graph Theory. ] In 2005, a puzzle craze known as sudoku swept across the world, starting in newspapers. Sudoku involves putting numbers on a square grid, and its creators claimed that solving it needed just a logical mind—no mathematics required. Most sudoku puzzles are nine squares long and nine squares wide, which is called a nine-by-nine (9×9) layout, but we are going to look at a smaller example shown below.

Our large square is composed of four blocks, each of which has four smaller squares within. The blocks are framed by thicker lines. There are also four rows across and four columns down, so this setup is a four-by-four (4×4) puzzle. Draw this puzzle in your notebook.

The idea is to arrange the numbers 1, 2, 3, and 4 in each block, so that each row and each column contains only one of the numbers 1, 2, 3, and 4. For example, the bottom left-hand block already has a 4 in it, so you’ll need to put 1, 2, and 3 in the remaining cells in that block. Make sure that you do not end up with two numbers that are the same in any row or in any column, though. Try it! Write your ideas directly on the copy of the diagram you drew. When you have either solved the puzzle or spent about ten minutes on it, read on. You can reveal the hint at any time that you feel you want more guidance. Just click on “Reveal discussion.”

##### Discussion

If you have never seen these puzzles before, you may find this one quite tricky. There are many different ways that you can tackle this problem. The first step is to try to sort out exactly what you are being asked to do and to make sure you understand the problem. You may find it helpful to use a highlighter pen to mark rows, columns, or blocks as you work.

Then you might like to get a feel for the problem by putting in a few numbers just by guesswork. Unless you have been lucky, you will probably realize fairly soon that this does not work very well, but it will have given you a better idea of what is involved. Starting to get a feel for what might be involved is a very important first stage for any problem.

The next tactic to try is to make the problem simpler by breaking it down into steps concentrating on just one kind of number. You can choose the number that occurs most often and therefore is the number which you have most information about—in this case, 2. Begin by focusing on the number you chose to start with.

Now, you need to find the number that makes most sense to fill in that does not violate any of the rules given. Remember: Each number 1, 2, 3, and 4 appears only once in each block, in each row, and in each column. Keep the time limit of ten minutes in mind for the active part of this puzzle. You can spend twice this time if you wish, but don’t go beyond that.

Try a solution now. If you need more guidance, there is another hint to reveal below.

##### Discussion

In this case, there are two 2s on the grid already, so you only need to add two more, one in each of the bottom blocks. There is already a 2 in the first column, so no more 2s can go in that column and so the only place for the 2 in the bottom left-hand block is in the square under the 4.

If you find this difficult to follow, draw a pencil line through the column and row that the given 2 is in to show that the 2s in the other blocks cannot be placed in this row or column. (Here’s a helpful tip: Write your numbers in pen and make the lines in pencil, so you can erase the pencil lines after each thought.)

Similarly, there is already a 2 in the third column, so no more 2s can go in the third column. So the 2 in the bottom right-hand block must go above the 3.

Now, let’s move on. You can see that the bottom row already has a 3 in it, so the 3 in the bottom left-hand block must go next to the 4.

Now look at the third row. Which number is missing?

As more numbers are added, the puzzle gets easier. If you have not finished the puzzle for yourself, try to do so now.

Your final grid should look like this:

Although there is only one correct answer to each sudoku puzzle, the approach described in the hint is not the only way to get started on this puzzle. If, for example, you focus on the number 4 in your first move, then you can reason that the block in the lower right-hand corner must have a 4 placed beside the given 3. This would lead to a different start as well as different steps to reason through, but it leads to the same answer.

Each sudoku puzzle has only one possible arrangement of the numbers in the answer. If you don’t reason your way through, reaching the correct answer is almost impossible.

### 1.1.3 Reflection on Sudoku

An important part of learning is the time spent thinking about what you have done and how effective your methods have been so that you can improve your work. The next activity will help you to do this.

#### Activity: How Did This Work for You?

Spend a few minutes thinking about how you felt about doing the previous activity. If you were asked to solve a similar puzzle, would you approach it differently? What have you learned from tackling the activity about your studying or how you approach problems? Write down a few ideas summarizing your thoughts in your math notebook.

##### Discussion

You can think about the strategies that worked for you that you would use again, and the ones that lead to a violation of the rules. Was it possible to fix a mistake easily? How confident did you feel in your approach? Can you think of a reason why a sudoku puzzle would be part of a math lesson?

You may have felt slightly apprehensive about being asked to do an activity so early in the unit—that’s a natural feeling when you are not quite sure what to expect! Or you might have been ready to try, or you might even have felt very confident if you already enjoy sudoku puzzles.

Your comments will obviously be rather personal, but some things you might have noted are the hints on tackling the problem: Crossing out rows and columns and concentrating on the smaller blocks, or looking along the rows to see what numbers were missing. You may not have used quite the same strategy for solving the puzzle, and that’s okay, as there is usually more than one way to solve a problem … even if sometimes one way is much more efficient than another.

Logical thinking and reasoning are important skills when engaging in math. Finding strategies that work is the goal, but going wrong often will be part of the process. When you watch somebody with lots of practice do math, you may be fooled into thinking that this person just knows instinctively what to do next. But it is much more likely that this person has practiced this type of problem and eliminated the wrong turns along the way.

Learners make mistakes! It’s natural and even beneficial. Unfortunately, you may have noticed with this puzzle that making a mistake early on may lead to more mistakes, and since you cannot be sure where the first one occurred, fixing these is almost impossible. Starting over and ensuring that each number that you put down is allowed to go where you put it may be the only way out.

### 1.1.4 Further Reflection on Sudoku

#### Did You Find the Activity Difficult?

If you get stuck with an activity, it is fine to use part or all of the hint and see if you can then understand how to work the problem. For example, in the sudoku puzzle, the idea to cross out the column and row might have been sufficient for you to tackle the rest of the problem yourself. Or you might have needed to read through more of the hint and solution comments to fully understand what is going on. That is okay, too—it is a good way to learn. You can then try the similar problem in the self-check section to make sure that you have fully understood the ideas.

If you are still puzzled by it, try discussing the problem with someone else. Talking about a problem, particularly to someone who is not familiar with it, will help you to see it in a different light, and that might be sufficient for you to find the way forward yourself. Alternatively, a friend who has tried sudokus in the past or is following your introduction closely might be able to suggest a different approach to help you get started.

It can also be very helpful to take a break from a task and come back to it another time. Often an idea will come to you when you are doing something completely unrelated to your math study.

### 1.1.5 Mistakes

#### What If You Make a Mistake?

Making mistakes is part of the learning process. Mistakes tend to be of two different kinds: Small or careless errors, and more serious misunderstandings. If you have made a small error, it is worth trying to work more carefully in the future and taking time to check the intermediate steps and your answer. One thing that will help with this is making sure that you write down all of your math working, however obvious to you it may appear in your math notebook. If, however, you have misunderstood some aspect, then you might need to look back over the topic—or, if you are seriously stuck, ask somebody else for help.

Sometimes you may think you do not understand something, but in reality, you may just have made a minor error early on that causes a problem later. So, if you do find you are stuck with a problem, check back over your work first to see if you have made a mistake. Whatever has happened, the mistake will have been useful in deepening your understanding. Making mistakes is a bit like falling down when learning to walk. Everyone makes mistakes, and it is how we handle them and what they teach us that counts.

Just learn from the experience and move forward! Have you ever tried to learn a new language? It takes a lot of practice and error correction to become proficient. The same is true for math skills—we just seem to think that learning math should require less of a time commitment. Well, think of it as an investment in your future.

If you have time and you’d like to read more about the value of mistakes, here’s an interesting article on the topic.

A brief video on making mistakes in math

Although you are not working through a book, the advice for online material and books is very much the same.

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### 1.1.6 More Reflection on Sudoku

#### Was the Activity Easy for You?

Any sudoku expert will have solved this particular puzzle quite quickly. After you have solved the puzzle and analyzed your approach, challenge yourself by asking questions. For example, what makes some sudoku puzzles easy while others are more difficult? Is there only one solution? How can you tell if a starting position will give a solution at all?

Whether or not you found it easy, as you work through the units, you will probably find topics that you do already understand and can therefore spend less time on. For most units, there will be extensions that encourage you to think about more challenging problems or ideas. Give them a try if you have some spare time. However, do any self-check sections or quizzes for all units. These will help you to see how you are getting on and if you might need to review a particular concept before moving on or seek some further help from a friend or member of your family.

### 1.1.7 Puzzles and Real-World Math

[ “Latin squares” are square grids of numbers in which each number occurs just once in every row and column. ] One theme of this course is the use of mathematics in practical situations. You may be wondering what this particular puzzle has to do with solving real problems. Well, first of all, it has introduced you to working logically, and that is an important general technique that is used often in mathematics. Second, sudoku puzzles are quite similar to Latin squares, which are used extensively in designing experiments such as finding out how different crops grow with different fertiliser treatments.

This happens quite often in mathematics. Very abstract mathematical topics, which seem to have no practical use whatsoever when they are discovered, later turn out to be extremely important in science or technology. Just because it is difficult to see a use at the time does not mean that there will not be some important practical development later!

## 1.2 Getting to Know the Calculator

In the following activities, you’ll learn how to use the web calculator that is easily accessible through your computer.

The calculator can be accessed on the left-hand side bar under Toolkit. You might need to scroll down the screen to find it!

Look for the calculator picture on the left-hand side bar. Clicking on the link will open the calculator. If you would like to move the calculator to one side of the screen away from the main text, you can do this by clicking on the calculator with the left mouse button and holding this down. You will then be able to move the calculator where you would like.

Note that if you click away from the calculator it may appear to disappear from your screen. Don’t worry, it’s still there—but will be hiding behind the main text!

We will not be using most of the buttons toward the bottom of the calculator yet. Everything you will need at first is in the upper half. Don’t worry about looking at all of the features. If you need a new button, you will be introduced to it through the text.

The calculator works like most handheld calculators, though you can enter numbers and calculations in two ways, either by clicking on the buttons on the calculator itself, or by typing in numbers and mathematical signs directly in the white entry window.

First, use the calculator buttons to create the entries. Although you are not typing on your keyboard, you still need to make sure your cursor is positioned in the white window before you start. Watch the two screens at the top of the calculator (one black and one white) as you click on the following keys: .

The black screen shows the complete calculation with the answer. The white screen shows the calculation as you are doing it, and after you press the equals sign, it shows only the answer.

Once the calculation is complete, it needs to be cleared before the next calculation can be performed. You clear a calculation by clicking on the button. Do this now; your last calculation should disappear.

Try the same calculation by entering it using your keyboard. Type the numbers and mathematical signs either on the main keyboard or, if you have one, on the number pad. Type 2 + 5, then hit the Enter key. The calculator should display the same calculation as before.

Note that the flashing cursor must be in the white calculator window before you can begin typing. If you can’t see it, click in the window, or click on .

### Correcting Mistakes

Suppose you want to calculate 37 − 19, but you mistype and enter 37 − 29 by mistake. The easiest way to correct this is to use the left arrow key on your keyboard until the cursor is flashing after the number you want to correct, then hit the Backspace key on your keyboard and type in the correct number. You can of course also erase more with backspace and retype as needed.

An alternative way is to use the little left arrow on the right side of the white entry window. It works like the Backspace key on your keyboard. Typing errors can be corrected by clicking on it.

Try it! Type 37 − 29 and correct it to 37 − 19.

### Arithmetic on the Calculator

Now you’re ready to do arithmetic on this calculator. You can use either the buttons on the calculator or the ones on your keyboard, or a combination of the two—whichever suits you best. You may find the following table helpful in finding the correct button or key. You may find it useful to copy this table into your math notebook if you are not familiar with some of the keyboard alternatives.

In this activity, you are doing several straightforward calculations to make sure that you can get the calculator to work properly. Check the answer in your head so that you know what the calculator should show. Remember to clear the screen after each calculation, before you enter each new calculation.

### Activity: A Few Arithmetic Problems

• (a)
• (b)
• (c)
• (d)

#### Discussion

Watch the screens at the top of the calculator to make sure you are entering the calculation correctly, and correct any mistakes as you go along. Remember to clear the screen each time before you start the next calculation.

The calculator should look like this:

 (a) (b) (c) (d)

### Activity: Making the Calculator Do the Work

Now for some calculations that you can’t do in your head so easily; use the calculator to find the answer to these.

• (a)
• (b)

Remember that although large numbers are sometimes written with commas to make reading the number easier (7,094 for example), commas (or spaces) are not entered on the calculator.

#### Discussion

Don’t be distracted by the size or complexity of the numbers. Enter each one carefully watching the screen to make sure that it is correct.

 (a) 363 plus 905 en dash 189 equals 1079The calculator shows: (b) 7094 multiplication 2758 division 394 equals 49658Before you click or Enter, the calculator looks like this: After pressing , it looks like this:

### 1.2.1 Calculator Activities

Now that you see how the calculator works, you are ready to investigate some number puzzles and patterns.

#### Activity: Doubling Up

Enter any three-digit number into the calculator. Multiply that value by 11 and then by 91. What do you notice about the answer? Now try a different three-digit number. Do you think this always works?

##### Discussion

Look at your six-digit number output. Then compare to the number that you put in.

Try at least one other number. Does it show the same pattern?

and .

Multiplying by 11 and then by 91 seems to give you a number with the original three digits repeated.

There’s further explanation of this concept in the following pencast (click on “View document”).

View document

#### Activity: Following Up on Your Discovery

##### Discussion

What is 11 multiplied by 91? How does a multiplication by this combined product create an answer?

Multiplying 11 by 91 gives 1001, so multiplying by 11 and then by 91 is the same as multiplying by 1001. Write out 139 × 1001 as a long multiplication, and see if this helps you explain why the trick works.  Don’t worry if you’re not comfortable with this just yet: that’s why you’re doing this course! Just accept it as a party trick, and consider showing it to your friends.

If you are interested in having the long multiplication explained have a look at this pencast by clicking on view document below.  You will need the volume turned up on your computer and the newest version of Adobe Reader, Adobe X or higher.  If you didn’t download this earlier, do so now.

#### Activity: All at Elevens

Type the following calculations with the calculator and write out the answers in your notebook:

Can you predict the answer for ? And for ? Check it on the calculator.

##### Discussion

Look at the pattern the digits in each answer create. In the next two problems you have five and six 1s respectively.

Note: For the last calculation, the answer is too long to fit on the top black screen, but if you look below, the full answer is shown in the white screen. Keep an eye out for this when you have very long calculations or answers.

#### Activity: Another Number Pattern

Here’s another number pattern to try.

Work out the first two answers in your head and the next two on the calculator, and see if you can spot the pattern. What do you think the next line in the pattern will be?

##### Discussion

Try to describe the pattern in the answers. Then go back and look at each calculation in turn. Can you see a pattern in the first number on each line? And a pattern in the last number of the problem? How about a pattern in each answer?

The lines start 1, 12, 123, 1234, so the next line will start 12345. Multiply that number by 8. The number that is added forms a pattern, too: 1, 2, 3, 4 …, so the next line will be the multiplication part +5. The answers form a pattern as well: 9, 98, 987, 9876 …, so the answer to the next line should be 98765.

The next line reads . Check that it’s correct on your calculator.

If you’re feeling brave write down the next line as well and check it with the calculator.

The calculator can be accessed on the left-hand side bar under Toolkit.

### 1.2.2 More Calculator Activities

#### Activity: Cross-Number Puzzle

Here’s a cross-number puzzle to give you some more practice using the calculator. It’s like a crossword puzzle, only with numbers instead of words. Draw the puzzle in your math notebook and then use the calculator to solve it. Do a few of the calculations in your head or on notebook paper, if you feel up to it. That makes great practice.

The calculator can be accessed on the left-hand side bar under Toolkit.

 Across Down 1. 21 multiplication 47 1. 19 multiplication five 4. 1788 division six 2. 8003 multiplication nine 6. 497 en dash 105 plus 12 3. 1234 en dash 1216 7. 1458 division three multiplication two 5. sum with, 3 , summands one plus 11 plus 111 en dash 29 9. 32 plus 681 en dash 13 6. 517 division 11 7. 21 multiplication 33 division seven 8. 49 plus 49 en dash 48

##### Discussion

One great thing about cross-number puzzles is that hints are built into the puzzle. Most digits have to be correct in both directions, across and down.

Well done! You now know how to use the calculator and have done more math problems along the way, some containing patterns. Patterns play an important role in mathematics. You will discover more details about this in Unit 8.

To recap:

• You can use the buttons on the calculator or the keys on your keyboard to enter numbers and mathematical signs.
• Remember to clear each calculation before you begin the next using C.
• Watch the screen as you enter the calculation to make sure that it is correct.
• If a calculation or answer is very long, you may have to look for the complete answer in the white window, underneath the cropped number that shows in the black window.

## 1.3 Online Learning

### The Online Learning Environment

Learning online or taking online classes is different from learning in a classroom. It can be hard to stay focused with so many distractions that can occur in an online environment, especially when you link out to other sites. Remind yourself to complete just the suggested task and don’t get sidetracked. Be mindful of your time when you are on the computer. If you find yourself venturing away from the math, stop and remind yourself of the goals you are trying to achieve. It can be very helpful to set a five-minute timer whenever you go to outside websites. When the timer rings, you will know that you should return to the screen you were working on.

It is also advisable not to spend too long working at your computer screen.  Give yourself regular breaks away from it.  This will also help you study more effectively.

On the other hand, learning online can be fun and engaging. Reading a book is usually not as exciting as spending study time in a multimedia environment. Enjoy the opportunities it brings.

Don’t isolate yourself. Talking to family and friends about the math you study, and maybe discussing details of problems that you find tricky, are important experiences and can help tremendously. Everybody has been exposed to math. Don’t be shy to talk about it.

### 1.3.1 Where Did the Time Go?

You know that we never seem to have enough time to do all the things we want to do in a day. Using your time wisely is a part of becoming a successful student. It does not matter if you have half-hour segments or several-hour segments set aside for working on your math. The truth is that you have to put the time in your daily/weekly schedule or you won’t be able to accomplish your goals. “I’ll get to it when I get to it” is not going to work. So make choices that support your goals. If learning math is important to you, give it the right place in your schedule.

Here are three more tips:

• Planning. You could try to write out a schedule. It might be helpful to plan out your lessons on paper (maybe even in your math notebook), or in a computer document that you then print and post where you’ll see it. You could map out your day in hourly slots. Put in all responsibilities, then times for eating and sleeping and some relaxation, and then plan the rest that has not been accounted for. (That’s where the math goes.) Then comes the hard part: Stick to this schedule as best you can. Revise your schedule periodically or as needed to keep it realistic. Being realistic about how much you can get done in a given time is very important—if you are too ambitious you may end up feeling deflated if things don’t go to plan.
• Making lists. One of the difficult things about being a student is all the stuff that competes for your time and attention alongside your studies. All these uncompleted tasks create distraction and stress just by being there. The best way to get rid of that stress is to get the stuff out of your head and onto a list.
• Now here is the useful part. You don’t necessarily have to complete one item on your list before moving on to the next. When you have a little time you just take some small actions on any of the things on the list that will move them forward. The satisfaction you get from crossing something off your list will give you a sense of achievement that will be very motivating!
• Managing procrastination. Do you have a tendency to put things off? Well, you are not alone … practically all of us do! Reports suggest that nearly 80% of students procrastinate about their studies from time to time. Psychologists suggest you can manage your tendency to procrastinate by:
• Setting realistic goals.
• Planning (see above).
• “Making molehills out of mountains”—that is, breaking down a big task into a set of smaller ones.
• Figuring out why you procrastinate (perfectionism, fear of failure, fear of disappointing others) and trying to relax about these obstacles.
• Tell your friends and family your goals—they could help to keep you motivated.

If you tend to put things off, maybe one or more of these ideas will help.

### 1.3.2 Study Time

One of the most difficult aspects of being an adult student is fitting in your studying with everything else in your life. It is important both to find enough time to study, and to try to make the most effective use of your time. Finding enough time can be quite a challenge! It often means giving up some activities you currently enjoy or perhaps negotiating with your family and friends to pass on some of the daily chores or to allow you some time to yourself. You could try the 4D method:

• Defer. There may be things you think you must do but that you can put off for a while.
• Delegate. There may be things you do outside your classwork that you can get someone else to do, giving you more time to devote to your studies.
• Downgrade. There may be things you don't have to do quite so perfectly, and maybe you can spend a bit less time doing those.
• Decommit. This is a fancy way of saying there may be things you don’t have to do at all.

Even a ten-minute slot in your schedule can be used for recapping previous work, sorting out paperwork, planning future work, or working through an example/activity or two. So any time you have is valuable time!

One more thought: Having found some time, it is also worth thinking about whether this is the best time for you to study. Consider the times of day and the lengths of sessions where you work most productively. For example, if you know you are going to lose concentration after a half-hour or so, and also that you are just too tired to study in the evenings, it is probably a good idea to schedule your study time for new topics in half-hour slots in the morning, and use the evenings for other chores.

A timer might be useful when you’re studying. Some people like to work in 25-minute bursts and then take a five-minute break.

If you want to try this, there are many online timers available, such as this one.

### 1.3.3 Taking Notes

Keep your learning active. Consider using colored pens or highlighters to mark up different aspects in your notebook; for example, blue for important definitions, yellow for useful techniques, pink for questions that you want to come back to, and so on. This will make things easier to find if you need to scan back over your work later. Alternately, you may find a traffic light system works well for you—red for things you do not understand, yellow for areas that need more work, and green if you do understand the topic—and can apply it.

Some students like to use a fairly big notebook with one page for the mathematics and the facing page for notes and comments on how a concept works. Writing things down in your own words will help your understanding. This will give you a handy review of the main ideas that you can refer to later. People learn in different ways, and it is important for you to experiment to find out which methods work best for you. Your math notebook is yours!

Now back to the math …

### 1.3.4 New Vehicle

It’s time for another math activity. If we are given only partial information, then sometimes our logic and reasoning skills can help us recover the missing pieces. In other words, we often rely, perhaps unknowingly, on mathematics to fill in the gaps. Let’s consider an example where we can retrieve information despite not having the entire picture.

#### Activity: New Vehicle

You’ve recently been shopping for a new vehicle. You don’t know at the moment if you would like a motorbike or a car. Because you would like to get the best price possible and have the most options available, you would like to start by visiting the dealership that has the greatest number of vehicles of each type.

When you call the dealership to inquire about its inventory, the assistant manager, Paul, jokingly tells you that they have 21 vehicles available, with a total of 54 wheels. Just as you ask him how many of each type of vehicle he has, the phone call is inadvertently disconnected.

Do you have enough information to answer your own question? Can you figure out how many motorcycles and how many cars the dealership currently has?

##### Discussion

There are many ways to solve this problem. You might consider trying using pictures (visualization can be very helpful) or select a starting point, such as assuming half are motorbikes and half are cars, and then revising your first guess.

Method 1: Diagrams

Since there are different methods that can be used to tackle this problem, it’s possible you might have tried a different (but perfectly valid!) approach than the ones presented here.

Let’s suppose we have all motorbikes. Draw a picture (you don’t need to be an artist) that shows 21 motorbikes. Be sure to use a representation that will allow you to clearly distinguish between two and four wheels.

Since there are 21 motorbikes with two wheels each, your drawing shows a total of 42 wheels. Paul said that there were 54 wheels, so we need an additional 12 wheels, because . In other words, some of the motorbikes we drew need to be turned into cars.

We could just start adding two wheels to each motorbike until we’ve counted up to 12, or we could be clever. If we change a motorbike into a car, we gain two additional wheels. Since we know we need 12 more wheels to reach the required total, we need to change six motorbikes over to cars, because . Can you see how that worked?

From our picture, we observe that there are six cars and 15 motorbikes, for a total of 21 vehicles. We can check that the total number of wheels adds up to 54.

• Each car has four wheels, which makes wheels.
• Each motorbike has two wheels, which makes wheels.
• Together, this is wheels.

Thus, the dealership has 15 motorbikes and six cars in stock.

Method 2: Educated Guess

You can make any guess you think is reasonable. For example, you might assume that about half of the vehicles were motorbikes—say, ten of the vehicles. To keep track of your guesses, a table is quite useful. As you make adjustments to your guesses, remember that the number of motorbikes plus the number of cars must equal 21.

MotorbikesCarsTotal Number of Wheels
1011
(too many wheels → need fewer cars)
129
(too many wheels → need fewer cars)
147
(too many wheels → need fewer cars)
156
This matches with what Paul told you.

Once again, we come to the conclusion that there are 15 motorbikes and six cars at the dealership.

Method 3: Pairs of Wheels

Another way to solve this problem is to consider the pairs of wheels. Because the assistant manager told you that there are a total of 54 wheels, this means there are 27 pairs of wheels, because . If all of the vehicles were motorbikes, you would have 27 motorbikes, wthis hich is too many: remember, Paul told you there are only 21 vehicles.

Next, determine how many extra pairs of wheels there are. Because , we know there are six extra pairs of wheels. This indicates that six of the vehicles have to have an extra pair of wheels (beyond the pair for each vehicle that you’ve already counted). In other words, six vehicles must have two pairs of wheels, or four wheels each. Thus, there are six cars. To find the number of motorbikes, we need to take the cars away from the total number of vehicles: .

Consequently, there are 15 motorbikes and six cars.

### 1.3.5 Problem Solving

Did you use a different method to approach this problem? It’s likely that you did. You might have used a similar method, but it probably wasn’t identical to any of the three solutions shown.

Depending on your previous experiences and how you initially take in a particular problem, the way you decide to begin will vary. For this activity, it would have been possible to solve it using algebra with a system of two equations in two variables.  Don’t worry if these ideas are new to you at the moment—remember you are here to learn. The methods presented here required no knowledge of formal mathematical techniques. Instead, the solutions discussed demanded the use of common sense and organization.

As you work through any problem, remember that there are usually alternative methods for reaching the solution. If you get stuck using your initial approach, try a different one. Keep in mind that using pictures and staying level-headed will carry you far, and most likely help you finish solving an exercise. So try not to panic!

### 1.3.6 Learning from the Activity

When you have worked through an activity and read any comments, it is worth thinking back on how it worked for you. Do you need more practice? Could you have tackled the problem in a different way than the one you tried? How does this fit in with what you already know, and your own experiences? For example, are there any areas in your life where you could use the techniques that you have just learned in a practical way? This will help you to develop a firm basis for approaching further problems confidently.

You might also want to make a few notes on techniques you can use in the future, either for solving problems or for improving your learning. For example, you may decide to try out one of the different ideas of marking your notes or use a diagram.

Have you heard that there is only one way to solve a math problem or that men are more talented in math than women? These are two of a multitude of misconceptions about math that persist in people’s minds … and they are completely wrong. Unfortunately, such incorrect statements are made every day, and many students believe them.

### More Myths

• You need a special math gene to be good at math.
• False. We're all hard-wired at birth to do math. (It may be that dyscalculia—the numeric counterpart to dyslexia—affects up to 5% of the population. But if you’ve gotten this far, you’re not one of them.)
• Using a calculator means that you don’t learn basics.
• False. People may actually learn better by getting the mechanics of arithmetic out of the way.

And yes, research has shown that women can be just as good at math as men.

## 1.5 Vision

### Where Can Studying Math Take You?

Now you are on the journey to Succeed with Math. Dream big. Challenge yourself. You saw that many professions use math, and you are aware that the more math you learn, the farther it can take you in your career and help in your everyday life as well.

There is no limit to what you can achieve if you become a lifelong learner who is motivated and persistent and takes challenges one at a time.

We Use Math

This video is longer than the ones you have seen so far, but it is worth watching.

Interactive feature not available in single page view (see it in standard view).

## 1.6 Self-Check for Unit 1

(a) The website you will go to has many sudoku puzzles. Pick just one or two to solve. Be mindful of your time and don’t spend more than 20 minutes on this activity, even if you enjoy solving sudoku puzzles. If you cross out lines and columns, use your math notebook as you do the steps. Then type your answers directly into the puzzle. The sudokus to select from are called “mini sudokus,” and they are in the third row of this sudoku collection. Try the 4×4. When you enter the website, scroll down to access it.

(a) The sudoku puzzles from this page tell you if you filled them in correctly. When you have solved the 4 × 4 sudoku successfully, you can go on to the 6 × 6 version right next to it. The site saves your correct solution, so unfortunately you cannot ask another person to try this one from the same computer if you already filled it in correctly.

(b) In a dog park, there are 24 creatures (dogs plus people) present. Together, they have 80 legs. How many people and how many dogs were in the dog park?

(b) Eight people and 16 dogs make 24 creatures and legs.

You can go about this in different ways (and you may even have another way than the ones listed here).

##### Solution Idea 1: Draw a Picture

If 24 creatures have two legs each, we are using up 48 legs. There are 32 legs left over, which still have to be placed in the following (incomplete) picture:

Put two more legs with each of your bodies until you run out of legs:

This last picture now shows 16 dogs and eight people.

##### Solution Idea 2: Make an Educated Guess and Revise as Necessary

Make an educated guess: What if half of the creatures are people and half are dogs.

• 12 people have 24 legs
• 12 dogs have 48 legs.
• Together that makes 72 legs, which leaves us eight legs short.
• These eight extra legs make four two-legged creatures into four four-legged creatures. So we have four fewer people, but four more dogs.
• 12 people minus four people makes eight people.
• 12 dogs plus four dogs makes 16 dogs.
• There must have been eight people and 16 dogs

(You could have also continued your educated guesses as shown in the “New Vehicle” activity. For the revision you then use more dogs, since this will bring up the number of legs.)

##### Solution Idea 3: Use Pairs of Legs
• 80 legs make 40 pairs of legs, as .
• We are using 24 pairs of legs to give every creature two legs.
• 40 pairs of legs minus 24 pairs of legs leaves 16 pairs of legs left over.
• These 16 pairs are put with one creature each, which means each of these 16 creatures is a dog. 24 minus 16 is eight, so eight creatures only had one pair of legs, which is true for a person.

The answer is 16 dogs and eight people.

## 1.7 Quiz Time

Now that you have revisited the skills you have learned in this unit, give this short quiz a try! If you do get stuck on any of the questions, don’t worry—just go back and have another look at the material in this unit and then try again.

### Practice Quiz

Now that you have taken the time to work through these sections, do this short quiz it will help you to monitor your progress

The quiz does not check all the topics in the unit, but it should give you some idea of the areas you may need to spend more time on. Remember, it doesn’t matter if you get some, or even all of the questions wrong—it just indicates how much time you may need to come back and review this unit! Or you may need to think about how you have been studying so far as well.  Maybe you were rushed at some points or needed to take more notes?  It is always worth thinking about your study techniques and if you may need to tweak these.

## 1.8 Study Checklist

### Study Checklist

You should now be able to:

• Make a plan to organize your study time.
• Tackle math activities with more confidence.
• Understand that making mistakes is part of learning.
• Use your calculator for the four operations of +, –, ×, and ÷, and correct input mistakes.
• Start to take some useful notes.
• Appreciate that people use different strategies to solve problems.
• Appreciate that math can be used in many different situations.

### Outlook on Unit 2

In the next unit, you will look at numbers. The topics you are visiting will include the history of numbers, the place value system, rounding, decimals, and estimating. Enjoy your mathematical journey through the rest of the units!