If you have not completed Units 2 and 3 yet, it is recommended that you go back and finish those sections first.
Click here to go back to Unit 2.
Click here to go back to Unit 3.
In this unit we will take a look at exponents (which follows on from multiplication) and the importance of carrying out calculations in the correct order when faced with a problem with several different operators involved, for example addition and division. Finally we will turn to a method of solving problems and how this can be used in our everyday lives.
To check your understanding of exponents and PEMDAS before you start, do the Unit 4 pre quiz, then use the feedback to help you plan your study.
The quiz checks most of the topics in the unit, and should give you a good idea of the areas you may need to spend most 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 for this unit!
This unit should take around six hours to complete. In this unit you will learn about:
[ When the exponent is 2, we say “squared.” When the exponent is 3, we say “cubed.” ] We know that multiplication is a way to represent and quickly calculate repeated addition. What if we have repeated multiplication? For example, say we have . In this case, 2 is being multiplied by itself four times.
In mathematics we write this as and say “2 raised to the power 4.” The 4 is superscripted (raised) and referred to as the exponent or power. This notation tells us to take the base, 2, and multiply it by itself four times.
Let’s suppose we have . This number would be read as “5 cubed.” This is mathematical shorthand for . Thus, .
There is some more new math language introduced in bold here. Add them to your glossary of terms in your math notebook now with your own definitions and examples.
In your math notebook, determine the value of each of the given expressions and write out, in words, how you would say it.
(a)
Rewrite the expression so that it is repeated multiplication.
(a) ; “2 cubed or 2 raised to the power 3””
(b)
(b) ; “4 squared”
(c)
(c) ; “3 raised to the power four”
(d)
(d) ; “7 squared”
Some exponential operations are easy to carry out in our heads or on paper, and you will start to remember the more common ones the more you use them; others will be more efficiently done with a calculator. In the following exploration, you will find out how to use the calculator to calculate exponential values.
The calculator can be accessed in the lefthand side bar under Toolkit.
Suppose you want to calculate . Do this on the calculator now; you should get 2401. This is fine, but as you have just seen, this calculation can be written more concisely as 7^{4}, and there is a quicker way of calculating it, too.
To work out an exponent on the calculator, we use the button (x raised to the power y). Find it on the calculator now. It is almost in the middle of the block of keys below the numbers.
To calculate 7^{4}, you need to click on the following keys:
Try it now, and check that you get the same answer, 2401.
When you have entered the numbers, but before you click on equals, the calculator looks like this:
You will see that the sign used for ‘exponent’ is ^. This is called a “caret,” or sometimes, a “hat.” You’ll find it above the number 6 on your keyboard. So, to enter an exponent using your keyboard, use shift6.
Mathematical operation  Calculator button  Keyboard key 

Exponent  Shift6 (which gives ^) 
Above the button is the button. This is a shortcut for finding the square of a number (the number times itself which is the same as the number raised to the power 2).
The calculator can be accessed in the lefthand side bar under Toolkit.
Now you know how to use the calculator to find exponents of numbers, so it’s time for some practice. Read each of the following out loud to make sure that you understand what it means. Then, use the calculator to work out the answer.
(a)
(a) This is 13 squared, or 13 times itself. Try using the key.
(a)The key sequence is and the answer is 169.
If you used the key or the ^ key on your keyboard, followed by the number 2, that’s fine.
(b)
(b) This is 4 raised to the power 6, or 4 multiplied by itself 6 times. Which key do you think you’ll need this time?
(c)
(c) Exponents work with decimal numbers, too. This is 0.9 to the power 10, or 0.9 multiplied by itself 10 times.
(c)
Notice that the question included parentheses around 0.9. This is simply to make it easier to read and understand. You don’t need to include parentheses in your calculation, though it is fine if you want to do so. You’ll get the same answer—try it!
Did you expect (0.9)^{10} to give you an answer that was smaller than the base number (0.9)? This can seem counterintuitive but think about what the operation means (0.9)^{10} is 0.9 of 0.9 of 0.9 and so on thus the answer will be smaller not larger than the base number.
Did you know that when an expression includes some combination of addition, subtraction, multiplication, and division, order matters? In other words, the order in which you perform the operations makes a difference in the answer you get. That’s right: Certain operations take priority over others. Since many everyday problems that we encounter require the use of more than one operation, we need to make sure we know how to correctly proceed, write, and carry out the calculation.
The five operations we have looked at so far are joined by parentheses when considering order of operation.
Parentheses indicate the highest priority, so you need to work out anything in parentheses first, followed by any exponents (powers). Then, carry out the multiplication and division, and finally any addition and subtraction. If part of the calculation involves only multiplication and division or only addition and subtraction, work through from left to right.
The correct order to carry out the operations can be summarized by using the mnemonic PEMDAS, where the letters stand for Parentheses, Exponents, Multiplication, Division, Addition, and Subtraction. A saying that may help you remember PEMDAS is: “Please excuse my dear Aunt Sally.”
Let’s watch this quick video, which puts the order of operations to a quick song that might help you remember better!
Let’s look now at an everyday problem involving more than one operation.
Let’s suppose you purchased four large pepperoni pizzas that cost $15.99 each, and you want to split the total cost among six people evenly. To determine how much each person needs to pay, both addition and division will be used.
Fortunately, you’ve brought your calculator along, so you type it in as follows: . You push the “=” button or enter and the result is $50.635. Your friends aren’t going to be too pleased about that! What went wrong?
Well, you soon realize that what your calculator has done is to calculate , because it follows the order of operations, PEMDAS. Your calculator has divided only the last $15.99, not the total, by six. What you need it to do is calculate the total first by putting it into parentheses: .
Now you can divide (remember, P comes before D in PEMDAS). Then, you get each. Each person owes $10.66, not more than $50. Phew—much relief all around!
So you can see now how important it is to get the correct order for calculations.
Many everyday problems can be solved by doing a little arithmetic. While there will be situations when you might want to work out the calculation quickly in your head or on a piece of paper, using a calculator will often be easier. However, if you do use a calculator, you need to be confident that your answer is correct. This involves understanding what the different keys on your calculator do, using them appropriately, and then checking that your answer is reasonable.
In the following exploration, you will explore the order in which mathematical operations are carried out.
The calculator can be accessed in the lefthand side bar under Toolkit.
Suppose that you have the calculation . Which do you do first, the addition or the multiplication? If you add first, you’ll get then multiply by two to get 16. If you do the multiplication first, then you get , and adding three gives 13.
Oh dear, we have different answers depending on which order we do the mathematical operations in! It is important that there are rules to say which order calculations like these should be carried out, and that we all follow the rules so that we get the same answer.
So let’s see what happens when we use the calculator for this.
Using the calculator, determine the answer to .
You can input the calculation exactly how it appears into the calculator using the buttons or your keyboard. Don’t be tempted to press the equals button part way through, otherwise you will break up the calculation and the order of operation as well.
The calculator gives the answer 13.
The calculator has performed the multiplication first. It is programmed to perform the calculations in the correct order.
Remember that to avoid any confusion, a code has been set for the order of mathematical operations, PEMDAS. This stands for
To work out a complicated calculation, follow this code. Work out any parentheses first, then exponents, followed by multiplication and division (from left to right), and finish with addition and subtraction (from left to right).
This tells you how to calculate , which we tried above. There are no parentheses or exponents, but we do have multiplication and addition. Multiplication comes higher in the PEMDAS list so you must do the multiplication first, then the addition.
So, .
Scientific and graphing calculators will follow the PEMDAS code, and the calculator used in this course certainly does. However, be careful, because a typical handheld calculator might not be programmed to perform operations in the correct order, as described earlier in the pizza example.
The calculator can be accessed in the lefthand side bar under Toolkit.
To make sure that you understand PEMDAS, try these calculations in your math notebook without using the calculator. Then, check your answer using the calculator.
(a)
Remember PEMDAS. Are there parentheses? Then do the calculation inside those first. Next, look for exponents, then multiplication, division, addition, and subtraction.
(a)
Carry out the calculation in parentheses first: . Now multiply by two and the answer is 14.
(b)
(b)
No parentheses, this time, so start with the exponent: . Add the two and you get 11.
(c)
(c)
This looks like part (b), but this time there are parentheses, so you must do the calculation inside the parentheses first: . Now square the result, and the answer is 25.
(d)
(d)
Work out the exponents first. and . Finally add: .
(e) . Take care with this one!
It can help to make calculations like this easier to read if you put parentheses around the part that you need to do first.
(e)
This time, you have addition and multiplication, so you must do the multiplication first: . So now, the calculation is .
Using parentheses we would write .
This will give you the same answer but just might make your job easier!
Did you get the same answers using the calculator? You should!
There is one other point to note about using the PEMDAS code. Division is the opposite of multiplication; it “undoes” multiplication. If you multiply by three and then divide by three, you end up where you began. So, although multiplication comes before division in the PEMDAS code, they are really on the same priority level. The same is true of addition and subtraction: Because subtraction is the opposite of addition, addition and subtraction are on the same level. Therefore, we have an addendum to the PEMDAS code:
Let’s see how this works.
Suppose that you are trying to calculate . There is a subtraction and an addition; these are at the same level, so work from left to right.
Luckily the calculator knows about this aspect of the PEMDAS code and will automatically follow it.
The calculator can be accessed in the lefthand side bar under Toolkit.
Here are a couple more calculations to make sure that you have things straight. Try them without the calculator, and then use the calculator to check your answer.
(a)
Start by thinking through PEMDAS code. If you find more than one calculation at the same priority level, work through them from left to right.
(a)
Parentheses come first, but in the second set of parentheses, there is an addition followed by a subtraction. These are on the same level so when you do that calculation, work from left to right.
and , so
(b)
(b)
Parentheses first:
Inside the parentheses, there is a subtraction operation followed by an addition operation. Because addition and subtraction have the same priority level, you will work from left to right, so
Now there is a multiplication and a division, so again, you’ll work from left to right:
Make sure that you have added PEMDAS to your math notebook so that you can refer back to it easily if you need a reminder.
Now you have the hang of the basics try the activity on the next page. It’s a brainstretcher so don’t feel bad if you need to use the hints for help!
The calculator can be accessed in the lefthand side bar under Toolkit.
Can you add parentheses to these calculations to make them correct?
(a)
None of the calculations are correct as they stand. Try adding parentheses so that the order of operations changes. Trial and error will get you there.
(a)
(b)
(b)—not an easy one to spot! So well done if you did.
(c)
(c)
Here’s what you have already learnt in this section:
Now let’s practice performing operations by hand in the correct order.
In your math notebook, simplify each expression by following the correct order of operations.
(a)
For each problem, first go through and work on the parentheses. Next, look for exponents. Multiplication and division should be carried out from left to right. Finally, perform addition and subtraction from left to right.
(a)
P: There are no parentheses.
E: There are no exponents.
M/D: (4 x 2 = 8)
A/S: 11
So, .
(b)
(b)
P: so now we have
E: (5^{2} = 25) so now we have
M/D: (12 ÷ 4 = 3) so now we have
A/S: 28
So, .
(c)
(c)
P:
E:
M/D (from left to right):
A/S (from left to right):
Thus, .
(d)
(d)
P: There are none.
E:
M/D (from left to right):
Thus, .
In the examples you have worked through so far, all the information you needed was given to you, and sometimes you were given a hint on how to tackle the problem, too. Using math in your own life isn’t like that.
Quite often, you will need to decide how to tackle the problem, as well as what information you need. That can be a lot more challenging than working through some examples in a textbook. Reallife math can be messy, and some of the decisions you make may depend on your priorities, such as cost or time.
Let’s work through an example that uses multiple operations. We will focus on various strategies that you can try if you get stuck while solving a problem. Keep in mind that this problem might have some tricky steps. Do the best you can and embrace this activity as a learning tool, because everyone gets stuck at some point when they are working on math. Discussing the problem with someone else, drawing a diagram, or taking a break can help—you may be surprised at how much work your brain can do on a problem without you even knowing!
Before we jump in, let’s talk about some of the main steps in solving a real problem mathematically. These steps can be summarized in the mathematical modeling cycle shown below.
There are four main steps in this cycle:
This sounds a little complicated, but with a little practice it’ll become second nature to you—you may well already be using the model but had just not realized it!
Try this example problem, using the steps of the mathematical modeling cycle.
My father has recently moved into an old townhouse to be closer to us, and it needs some serious renovating, including putting some insulation in the attic. I have volunteered to take care of this task. Although I think it’s going to cost less to put in the insulation myself, my dad can get a grant from local government if the insulation is installed by an approved contractor, which has the added advantage that the work would be guaranteed.
Which is the better option?
Now you should check whether you understand the problem. Can you explain it in your own words? Is there any vocabulary that you are unfamiliar with? In this problem, what do you think is meant by the “better” option? For the moment, concentrate on costs and interpret it as the “cheaper” option.
It is often helpful to write down what you know and what you want to find. This can include information and mathematical techniques that you think might be relevant.
After measuring the floor space and joists, visiting a home improvement store, and contacting the local government department about the grant, I wrote down the following lists, after converting my measurements to inches for easier calculations.
I know:  I want: 



Do I need any further information? If so, what is it?
Before I can determine which option is cheaper, I need to know more about contracting out the job. Also, if I opt to do the insulation myself, then I will need to know more about the layout of the attic.
Yes, I still need to get some quotes from approved contractors. I also need to check how to install the insulation. Will I need to buy any extra equipment? How many joists are there in the attic? How far apart are the joists?
After calling several contractors, I found that the cheapest quote came in at $650. For installing insulation, the instructions recommend wearing safety glasses, a face mask, and gloves (an additional $45 altogether).
The insulation has to be fitted between the joists, which connect to the frame of the roof that is four inches wide around the edges and will not be covered by insulation. Fortunately, a quick check in the attic shows that there are 17 joists and the insulation fits perfectly between them.
This is a complicated problem, so a useful strategy would be to break it down into more manageable chunks by concentrating on one piece at a time. There are two separate problems, and the second can be broken down more.
Let’s focus on the first question.
How much will it cost if the contractor installs the insulation and the grant is used?
What are the fees and credits involved for contracting the job out?
The contractor charges $650 and the grant is $400. So, the overall cost will be .
Now we must consider the doityourself option.
In order to determine how much the DIY choice will cost, I must determine how many rolls of insulation are needed in the attic. In this case, drawing a diagram might help. After counting how many joists there were in the attic, I sketched out a rough plan:
Note: the attic door is in the floor and will be covered by insulation. Then, the insulation will later be cut to allow the door to open.
As the insulation fits perfectly between the joists, as well as the frame and outer joists, the amount of insulation needed will be easy to calculate. Recall that one roll of insulation is 300 inches long. The length of the attic minus the frame at each end is inches = 324 inches. One roll between the joists is not quite enough to do the job—there is an extra inches that need to be covered between any two joists.
Let’s determine how many rolls of insulation I will need to properly insulate the attic.
How many gaps are there between the joists? Between the frame and outer joists? Determine how many inches, in total, will be left over, and then see how many lengths of 300inch rolls will be needed to cover the gaps.
First, we can count that we will need 16 rolls to lie between the joists and two rolls between the frame and outer joists. So far, we will need at least 18 rolls of insulation.
However, we know that each roll leaves an extra 24 inches that need to be covered. This happens 18 times, so we are short inches. Since one roll is only 300 inches, we will need to buy two more rolls of insulation. Thus, we need a total of 20 rolls of insulation.
Now that I’ve determined how many rolls of insulation I need to buy, what will the total cost be of doing the job myself?
Remember:
Since I purchased 20 rolls of insulation, I will be able to take advantage of the “threefortwo” sale. This means that for every three rolls I purchase, I only pay for two rolls. Recall that each roll costs $9. Thus, for every set of three rolls I purchase, I will be charged only .
To determine how many times I will get to take advantage of the sale, we can use a picture.
Since there are six sets of three rolls, it will cost . Unfortunately, the two additional rolls won’t be on sale, so the insulation will cost .
Don’t forget that the safety gear is $45. This puts the charges at .
Now you can help me make an informed decision.
Should I hire the contractor and use the grant for placing the insulation in the attic, or should I do it myself?
Compare the costs for each option.
The cost of using the contractor is $250, after the grant is applied. The price of installing the insulation myself is $171. So, just based on these values, I might choose the DIY option.
However, there are additional factors that should be considered. Depending on the size of my vehicle, I might need to make several trips, or even rent a small moving van. The amount of time required to purchase, transport, and install the insulation, would be substantial, and as they say, “time is priceless.” Perhaps hiring the contractor is worth the difference in cost, making it the better option after all.
Although the mathematical solution suggests the DIY option, within the context of the problem as a whole, I’ve decided that using the contractor is the better option for me and my dad. Using math has helped me to make the decision, but other considerations have also played a part.
In the next unit will you have another chance to complete a similar home improvement problem yourself.
With any problem, it is always worth thinking back over how you approached the problem and whether you could use any of the ideas for future problems. Breaking the problem down into small steps and using a diagram helped. It is also worth thinking how the problem ties in with what you already know. For this problem, perhaps you have firsthand knowledge about installing insulation.
Another important strategy is discussing the problem with someone else. By explaining in your own words how you have tried to tackle the problem, you might clarify your thinking sufficiently to see a way forward. Alternatively, the person may be able to suggest some of his or her own ideas which might help, or maybe someone will ask you a good, leading question that makes you think of something you hadn’t considered. Trying to teach someone else is a very good way of learning and sorting out your own ideas!
Think back over the occasions when you have used math in real life. Remember a specific situation where you got stuck. What did you try to do to resolve the situation? What strategies have you seen so far?
In your math notebook, make a list of the strategies you can use when approaching a problem.
You probably have several strategies from your own mathematical experiences already.
For example, a student nurse made the following comment:
“I’m still not sure about working out the more complicated drug doses, so I check the answer is similar to previous doses and then I discuss the calculation and result with the pharmacist before I proceed with the patient.”
Some strategies are summarized below … but feel free to add your own!
The more you practice, the more your skills improve. Below are some exercises that will help you continue to develop your ability and check to make sure you understand the concepts discussed in this unit. Be sure to write your work out in your math notebook so that you can refer to it later if necessary.
Determine the value of each expression.
(a)
(a)
(b)
(b)
Find the correct value for each expression by following the correct order of operations.
(a)
(b)
(b)
Now that you have taken the time to work through these sections, consider giving this short quiz a try! You may find that it will help you to monitor your progress, particularly if you took the quiz at the start of the unit as well.
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!
Read through the list below and think over all the work you have done in this unit. If there is a checkpoint that doesn’t seem familiar, skim your notes to jog your memory. Remember that your mathematical skills will develop and grow stronger over time. Just keep working at it!
You should feel confident to:
Good start! You’re doing a great job.
If you don’t have the time to do the optional activities in this unit, move onto Unit 5 now and continue to work with numbers. To be precise, you will work with various units of measurement and negative numbers. See you there and keep going!
In this section, we will try to extend our knowledge that was discussed throughout the unit. If you find a problem difficult, read back over it, briefly set it aside and revisit it later, or discuss it with a friend. But don’t panic; just keep going!
You should be starting to feel more confident about your math skills. The next section is optional and gives you the opportunity to practice all the skills you've already learnt in this unit, and begin to think about how you might apply them in your everyday life. If you choose to do it, this section will take you about an hour and a half more.
Let’s begin by taking a look at how numbers used in codes affect our daily lives!
According to MerriamWebster, a code is “a system of symbols (as letters or numbers) used to represent assigned and often secret meanings.” Codes are used for many things in the modern world, and most are based on a mathematical scheme.
Using the Internet or personal knowledge, list some everyday items that use codes.
Have you used your ATM card recently? Bought a book? Sent a letter through the post?
There are many ways codes are used. Here are a few, although this is by no means a complete list:
Never heard of IMEI numbers? IMEI numbers are intended to protect your cell phone from unintended use. If you’re interested, check out this short video that investigates IMEIs:
MerriamWebster Online, s.v. “code,” http://www.merriamwebster.com/ dictionary/ code (accessed September 25, 2012).
The next activity is a number puzzle, which introduces some more ways of adding numbers quickly in your head. You will see how to turn it into a party trick to amuse your friends.
Write down the numbers 1 to 25 in order, in a 5×5 grid, so that the first row reads 1, 2, 3, 4, 5; the next row begins with 6, and so on.
Now choose five numbers from the grid as follows:
Try the puzzle at least three more times, choosing different numbers each time. Work out the sums on paper or calculate in your head. What do you notice about the sums? Can you work out how to prove this?
In the end, you will have selected exactly one number from each row, but none of these numbers appear in the same column. Find an easy set of five numbers from the grid that have this property.
Whichever numbers you choose, you should find that the sum is always 65.
By specifying the way that the numbers must be chosen, exactly one number is chosen from each row and each column of the grid. It is then possible to prove that the sum will always be 65, without trying all the different choices of five numbers—that would be tedious!
In particular, you can choose the numbers on the diagonal from topleft to bottomright because this gives one number in each row and column.
The sum is then . Finding the total of the diagonal elements is a quick way of working out the sum for this puzzle. Here, if we rearrange the order of the numbers so that we have and work out the sums in the parentheses first, we get , which is 65 as before.
Next is a general proof for why this trick works, if you’re interested in exploring the math a little more.
Add up the numbers in the top row. This is .
Now, instead of picking the five numbers from the top row, we pick one from each of the rows in turn, selecting one from each of the columns.
So, compared with picking all five numbers from the top row, choosing one number from each row, with the constraint that each must also be in a different column, means that their sum will exceed the sum of the numbers in the first row by 5 (for the second row rather than the first) + 10 (for the third row rather than the first) + 15 (for the fourth row rather than the first) + 20 (for the fifth row rather than the first).
Now, , so it will be 50 more than 15 (the sum of the numbers in 1st row). Now to find the total we need to add back in the sum of the first row numbers and we have .
So whatever numbers you pick the sum will always be 65.
Now for the party trick! Create a large 5×5 grid. Write down the number 65 on a piece of paper and fold it up so that the number is not visible. Announce that you are going to predict the sum of the five numbers your friend chooses from the grid. Ask the friend to choose the numbers following the rules given above and also to work out the sum.
[ It is not a good idea to repeat this puzzle with the same audience—otherwise your trick may be discovered. However, you can extend these grids or even invent some new puzzles of your own! ]
Assuming no mistakes have been made, you can then dazzle your audience by producing the piece of paper with exactly the same sum written on it—ensuring your party is a success instead of a snooze!
Now, let’s look at a quick way to attack division by nine.
Watch this short video to discover the power of your brain:
Now see how quickly you can perform these calculations in your head.
(a)
Write down the first number of the dividend, then use addition. Watch out for carries!
(a)
Thus, .
(b)
(b)
Thus, .
(c) (This one is tough—you might need to write the shortcut out!)
Thus, .
Aren’t you impressed by the power of your brain? You should be! Of course, the real math is understanding why this trick works, but you’ll have to study much more math to find that out.
Now you have finished the optional activities as well it is time to move onto Unit 5 and continue to work with numbers. To be precise, you will work with various units of measurement and negative numbers. See you there and keep going!