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Unit 4: Math in Real World

4 Introduction

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. 

Practice Quiz

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!

Click here for the pre quiz.

4.0.1 What to Expect in this Unit

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

  • Repeated multiplication of the same number (exponents).
  • Order of calculation (PEMDAS).
  • The math problem solving cycle.

4.1 Exponents

[ 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 two multiplication two multiplication two multiplication two. In this case, 2 is being multiplied by itself four times.

In mathematics we write this as two super four 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 five cubed. This number would be read as “5 cubed.” This is mathematical shorthand for five multiplication five multiplication five. Thus, equation sequence five cubed equals five multiplication five multiplication five equals 25 multiplication five equals 125.

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.

4.1.1 Expressions with Exponents

Activity symbol Activity: Exponents

In your math notebook, determine the value of each of the given expressions and write out, in words, how you would say it.

(a) two cubed

Hint symbol
Comment

Rewrite the expression so that it is repeated multiplication.

Solution symbol
Answer

(a) equation sequence two cubed equals two multiplication two multiplication two equals four multiplication two equals eight; “2 cubed or 2 raised to the power 3””

(b) four squared

Solution symbol
Answer

(b) equation sequence four squared equals four multiplication four equals 16; “4 squared”

(c) three super four

Solution symbol
Answer

(c) equation sequence three super four equals three multiplication three multiplication three multiplication three equals nine multiplication nine equals 81; “3 raised to the power four”

(d) seven squared

Solution symbol
Answer

(d) equation sequence seven squared equals seven multiplication seven equals 49; “7 squared”

4.1.2 Calculator Exploration: Exponents

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.

Calculator symbol The calculator can be accessed in the left-hand side bar under Toolkit.

Suppose you want to calculate seven multiplication seven multiplication seven multiplication seven. 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 74, 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 74, 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 shift-6.

Mathematical operationCalculator buttonKeyboard key
Exponent
Shift-6 (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).

4.1 3 Further Investigation of Exponents on the Calculator

Calculator symbol The calculator can be accessed in the left-hand side bar under Toolkit.

Activity symbol Activity: Working with Exponents on the Calculator

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) 13 squared

Hint symbol
Comment

(a) This is 13 squared, or 13 times itself. Try using the key.

Solution symbol
Answer

(a)The key sequence is Key sequence showing 13 x squared equals and the answer is 169.

Calculator screen showing 13 squared = 169

If you used the Xy symbol key or the ^ key on your keyboard, followed by the number 2, that’s fine.

(b) four super six

Hint symbol
Comment

(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?

Solution symbol
Answer

(b) four super six equals 4096

(c) open 0.9 close super 10

Hint symbol
Comment

(c) Exponents work with decimal numbers, too. This is 0.9 to the power 10, or 0.9 multiplied by itself 10 times.

Solution symbol
Answer

(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.

Here’s what you have already learnt:

  • seven super four is a short way of writing seven multiplication seven multiplication seven multiplication seven, or 7 raised to the power 4.
  • To enter any exponent, use the key, or the ^ symbol on your keyboard.
  • You can use the key as a quicker way to find the square of a number.

4.2 Order of Operations

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!

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

Let’s look now at an everyday problem involving more than one operation.

4.2.1 An Everyday Problem

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: sum with, 4 , summands 15.99 plus 15.99 plus 15.99 plus 15.99 division six. 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 equation sequence sum with, 4 , summands 15.99 plus 15.99 plus 15.99 plus open 15.99 division six close equals 47.97 plus 2.665 equals 50.635, 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: open sum with, 4 , summands 15.99 plus 15.99 plus 15.99 plus 15.99 close.

Now you can divide (remember, P comes before D in PEMDAS). Then, you get open sum with, 4 , summands 15.99 plus 15.99 plus 15.99 plus 15.99 close division six equals 10.66 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.

4.2.2 Calculator Exploration: Understanding the Order of Operations

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.

Calculator Exploration: Understanding the Order of Operations

In the following exploration, you will explore the order in which mathematical operations are carried out.

Calculator symbol The calculator can be accessed in the left-hand side bar under Toolkit.

Suppose that you have the calculation three plus five multiplication two. Which do you do first, the addition or the multiplication? If you add first, you’ll get three plus five equals eight comma then multiply by two to get 16. If you do the multiplication first, then you get five multiplication two equals 10, 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.

Activity symbol Activity: Understanding the Order of Operations

Using the calculator, determine the answer to three plus five multiplication two.

Hint

Hint symbol

Comment

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.

Solution symbol

Answer

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 three plus five multiplication two, 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, equation sequence three plus five multiplication two equals three plus 10 equals 13.

4.2.3 Further Practice with PEMDAS

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.

Calculator symbol The calculator can be accessed in the left-hand side bar under Toolkit.

Activity symbol Activity: PEMDAS on Paper Versus the Calculator

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) open three plus four close multiplication two

Hint solutions

Comment

Remember PEMDAS. Are there parentheses? Then do the calculation inside those first. Next, look for exponents, then multiplication, division, addition, and subtraction.

Solution symbol
Answer

(a) open three plus four close multiplication two

Carry out the calculation in parentheses first: three plus four equals seven. Now multiply by two and the answer is 14.

(b) two plus three squared

Solution symbol
Answer

(b) two plus three squared

No parentheses, this time, so start with the exponent: three squared equals nine. Add the two and you get 11.

(c) open two plus three close squared

Solution symbol
Answer

(c) open two plus three close squared

This looks like part (b), but this time there are parentheses, so you must do the calculation inside the parentheses first: two plus three equals five. Now square the result, and the answer is 25.

(d) three squared plus four squared

Solution symbol
Answer

(d) three squared plus four squared

Work out the exponents first. three squared equals nine  and four squared equals 16. Finally add: nine plus 16 equals 25.

(e) two plus three multiplication four plus five . Take care with this one!

Solution symbol
Answer

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) two plus three multiplication four plus five

This time, you have addition and multiplication, so you must do the multiplication first: three multiplication four equals 12. So now, the calculation is sum with, 3 , summands two plus 12 plus five equals 19.

Using parentheses we would write two plus left parenthesis three multiplication four right parenthesis plus five.

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!

4.2.4 Moving Left to Right

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 five minus two plus four. There is a subtraction and an addition; these are at the same level, so work from left to right.

equation sequence five minus two plus four equals three plus four equals seven

Luckily the calculator knows about this aspect of the PEMDAS code and will automatically follow it.

Calculator symbol The calculator can be accessed in the left-hand side bar under Toolkit.

Activity symbol Activity: More on PEMDAS

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) open one plus two close multiplication open one plus two minus three close

Hint symbol
Comment

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.

Solution symbol
Answer

(a) open one plus two close multiplication open one plus two minus three close

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.

open one plus two close equals three and equation sequence open one plus two minus three close equals three minus three equals zero, so equation sequence open one plus two close multiplication open one plus two minus three close equals three multiplication zero equals zero

(b) open eight minus one plus three close multiplication two division five

Solution symbol
Answer

(b) open eight minus one plus three close multiplication two division five

Parentheses first: equation sequence open eight minus one plus three close equals seven plus three equals 10

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 equation sequence open eight minus one plus three close multiplication two division five equals open seven plus three close multiplication two division five equals 10 multiplication two division five

Now there is a multiplication and a division, so again, you’ll work from left to right:

equation sequence open eight minus one plus three close multiplication two division five equals 10 multiplication two division five equals 20 division five equals four

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!

4.2.5 PEMDAS Brainstretcher

Calculator symbolThe calculator can be accessed in the left-hand side bar under Toolkit.

Activity symbol Activity: Brainstretcher

Can you add parentheses to these calculations to make them correct?

(a) one plus three multiplication four equals 16

Hint symbol
Comment

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.

Solution symbol
Answer

(a) equation sequence open one plus three close multiplication four equals four multiplication four equals 16

(b) three plus seven minus two squared equals 28

Solution symbol
Answer

(b)equation sequence three plus open seven minus two close squared equals three plus five squared equals three plus 25 equals 28—not an easy one to spot! So well done if you did.

(c) two plus three multiplication four minus one equals 15

Solution symbol
Answer

(c) equation sequence open two plus three close multiplication open four minus one close equals five multiplication three equals 15

Here’s what you have already learnt in this section:

  • When you carry out a complicated calculation with more than one operator, follow the PEMDAS code:
  • Addition and subtraction are on the same level as each other as are multiplication and division.
  • Remember the addendum:
  • The calculator is programmed to follow this rule!

4.2.6 PEMDAS Performed By Hand

Now let’s practice performing operations by hand in the correct order.

Activity symbol Activity: PEMDAS

In your math notebook, simplify each expression by following the correct order of operations.

(a) three plus four multiplication two

Hint symbol
Comment

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.

Solution symbol
Answer

(a) three plus four multiplication two

P: There are no parentheses.

E: There are no exponents.

M/D: three plus eight(4 x 2 = 8)

A/S: 11

So, three plus four multiplication two equals 11.

(b) open eight minus three close squared plus 12 division four

Solution symbol
Answer

(b) open eight minus three close squared plus 12 division four

P: left parenthesis eight minus three equals five right parenthesis so now we have open five close squared plus 12 division four

E: (52 = 25) so now we have 25 plus 12 division four

M/D: (12 ÷ 4 = 3) so now we have 25 plus three

A/S: 28

So, open eight minus three close squared plus 12 division four equals 28.

(c) two multiplication open three close cubed plus six minus two multiplication open five minus two close plus one

Answer
Solution symbol

(c) two multiplication open three close cubed plus six minus two multiplication open five minus two close plus one

P: two multiplication open three close cubed plus six minus two multiplication three plus one

E: two multiplication 27 plus six minus two multiplication three plus one

M/D (from left to right): 54 plus six minus six plus one

A/S (from left to right): multiline equation line 1 60 minus six plus one equals 54 plus one equals 55

Thus, two multiplication open three close cubed plus six minus two multiplication open five minus two close plus one equals 55.

(d) 48 division six division two multiplication three squared

Solution symbol
Answer

(d) 48 division six division two multiplication three squared

P: There are none.

E: 48 division six division two multiplication nine

M/D (from left to right):

multiline equation line 1 eight division two multiplication nine equals four multiplication nine equals 36

Thus, 48 division six division two multiplication three squared equals 36.

4.3 Using Math in the Real World

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. Real-life 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!

4.3.1 Math Cycle

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:

  • Describe the problem concisely so that you are clear what you are trying to work out. This may involve discussing the problem with others.
  • Collect information and make assumptions so that you can use mathematics to solve it.
  • Decide what mathematical techniques to use, and carry out these calculations.
  • Work out what your results mean practically and check that they are reasonable. If they are not, you may need to refine the assumptions and go through the cycle again.

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!

4.3.2 Insulating the Attic

Try this example problem, using the steps of the mathematical modeling cycle.

Step 1: Describe the Problem

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.

Step 2: Collect Information, Make Assumptions, and Simplify the Problem

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:
  • The attic is approximately 312 inches by 332 inches.
  • Each joist is two inches wide.
  • A $400 grant is available.
  • The insulation comes in rolls that are 15 inches wide and 300 inches (25 feet) long.
  • Each roll costs $9, but there is a sale going on where I can buy three rolls for the price of two.
  • To find the cheaper option.
Note that all lengths have been converted to inches.

4.3.3 What Else is Needed?

Activity symbol Activity: Need More?

Do I need any further information? If so, what is it?

Hint Symbol

Comment

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.

Solution symbol
Answer

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?

4.3.4 Gathering Additional Information

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.

4.3.5 Tackling the Problem

Step 3: Tackling the Problem—Do the Math!

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.

  • How much would my father have to pay if he got the $400 grant and used the contractor?
  • How much would the “do-it-yourself” option cost?
    • How much insulation will I need?
    • How much will the materials cost?

Let’s focus on the first question.

4.3.6 Contractor Costs

Activity symbol Activity: The Contractor’s Costs

How much will it cost if the contractor installs the insulation and the grant is used?

Hint symbol

Comment

What are the fees and credits involved for contracting the job out?

Solution symbol
Answer

The contractor charges $650 and the grant is $400. So, the overall cost will be equation left hand side dollar times 650 minus dollar times 400 equals right hand side dollar times 250.

Now we must consider the do-it-yourself option.

4.3.7 Determining the Number of Rolls

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 332 minus four minus four inches = 324 inches. One roll between the joists is not quite enough to do the job—there is an extra 324 minus 300 equals 24 inches that need to be covered between any two joists.

Activity symbol Activity: How Many Rolls So I Need to Buy?

Let’s determine how many rolls of insulation I will need to properly insulate the attic.

Hint symbol

Comment

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 300-inch rolls will be needed to cover the gaps.

Solution symbol
Answer

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 24 multiplication 18 equals 432 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.

4.3.8 Overall Cost

Activity symbol Activity: The Price of Doing it Yourself

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?

Hint symbol

Comment

Remember:

  • Each roll costs $9.
  • There is a “three-for-two” sale.
  • I will also need to buy some safety equipment.
Solution symbol
Answer

Since I purchased 20 rolls of insulation, I will be able to take advantage of the “three-for-two” 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 equation left hand side dollar times nine multiplication two equals right hand side dollar times 18.

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 equation left hand side dollar times 18 multiplication six equals right hand side dollar times 108. Unfortunately, the two additional rolls won’t be on sale, so the insulation will cost equation left hand side sum with, 3 , summands dollar times 108 plus dollar times nine plus dollar times nine equals right hand side dollar times 126.

Don’t forget that the safety gear is $45. This puts the charges at equation left hand side dollar times 126 plus dollar times 45 equals right hand side dollar times 171.

4.3.9 Decision Time

Step 4: Interpret the Answer

Now you can help me make an informed decision.

Activity symbol Activity: Contractor or DIY?

Should I hire the contractor and use the grant for placing the insulation in the attic, or should I do it myself?

Hint symbol

Comment

Compare the costs for each option.

Solution symbol
Answer

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.

4.3.10 Reviewing the Process

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 first-hand 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!

4.4 Problem Solving Strategies

Activity symbol Activity: Stuck! What Do You Do?

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.

Hint symbol

Comment

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.”

Solution symbol

Answer

Some strategies are summarized below … but feel free to add your own!

  • Check to make sure you understand the problem.
  • Write down what you know—all the information you have, any techniques that might be helpful, and your ideas for tackling the problem.
  • Write down what you want to do or find.
  • Write down any questions you have.
  • Make a note of any ideas that occur to you, such as “I wonder if …” or “Maybe … might work …”
  • Break the problem down into smaller steps that you can tackle one at a time.
  • Have you seen something like this before? Could you use a similar approach?
  • Draw a diagram or use a physical model.
  • Can you solve a simpler problem first? For example, by temporarily replacing awkward numbers by smaller or simpler ones?
  • If it is an abstract problem, would looking at an example with numbers or a real context help?
  • Try out any ideas you’ve had and go through your work carefully writing down all your working—an arithmetic error or other mistake can stop you from moving ahead.
  • Always try to check your final answer for “reasonableness.” For instance, in the last example if your DIY cost was ten times the contractor’s estimate, you’d want to go back over your calculations.
  • Take a break, go for a walk, have a cup of coffee, or run some errands!
  • Discuss the problem with someone else—another student, a family member, or a friend.
  • Don’t panic, and try not to get too frustrated! It’s just math, after all.

4.5 Self-Check

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.

4.5.1 Self-Check on Exponents

Exercise symbolExercise 6: Exponents

Determine the value of each expression.

(a) two super five

Solution symbol
Answer

(a) equation sequence two super five equals two multiplication two multiplication two multiplication two multiplication two equals four multiplication four multiplication two equals 16 multiplication two equals 32

(b) four cubed

Solution symbol
Answer

(b) equation sequence four cubed equals four multiplication four multiplication four equals 16 multiplication four equals 64

4.5.2 Self-Check on Order of Operations

Exercise symbolExercise 7: PEMDAS

Find the correct value for each expression by following the correct order of operations.

(a) 20 minus two times open five minus two close squared plus 12

Solution symbol
Answer

multiline equation line 1 20 en dash two times open five en dash two close squared plus 12 line 2 equals 20 en dash two times left parenthesis cubed right parenthesis prefix plus of 12 line 3 equals 20 en dash two times open nine close plus 12 line 4 equals 20 en dash 18 plus 12 line 5 equation left hand side equals right hand side two plus 12 line 6 equals 14

(b) 27 minus 12 division three multiplication two squared

Solution symbol
Answer

(b) multiline equation line 1 27 minus 12 division three multiplication two squared line 2 equation left hand side equals right hand side 27 minus 12 division three multiplication four line 3 equation left hand side equals right hand side 27 minus four multiplication four line 4 equation left hand side equals right hand side 27 minus 16 line 5 equals 11

4.6 Quiz Time

Practice Quiz

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!

Click here for the post quiz.

4.7 Study Checklist

Study Checklist

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:

  • Work out everyday problems using addition, subtraction, multiplication, and division.
  • Carry out operations in the correct order (following the PEMDAS code).
  • Appreciate that strategies can be used to work out calculations on paper or in your head.
  • Use rounding and estimation and check results from your calculator.
  • Appreciate the steps in solving a problem mathematically.

Good start! You’re doing a great job.

Outlook on Unit 5

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!

4.8 Optional Activities

Extensions and Further Exploration

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!

4.8.1 Codes

According to Merriam-Webster, 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.

Activity symbol Activity: Codes

Using the Internet or personal knowledge, list some everyday items that use codes.

Hint symbol

Comment

Have you used your ATM card recently? Bought a book? Sent a letter through the post?

Solution symbol

Answer

There are many ways codes are used. Here are a few, although this is by no means a complete list:

  • Combination locks to protect your property.
  • ZIP/postal codes to identify a particular region of a country.
  • ISBN codes on books and other publications.
  • UPCs/barcodes on food, clothing, and almost any other products you buy.
  • Morse code for communication.
  • Banking codes to protect your money.
  • Credit card security codes.
  • IMEI numbers for mobile devices.

Videoclip symbol 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:

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

Reference

Merriam-Webster Online, s.v. “code,” http://www.merriam-webster.com/ dictionary/ code (accessed September 25, 2012).

4.8.2 Having Fun with Math—the Party Puzzle

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.

Activity symbol Activity: Party Puzzle

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:

  • For the first number, choose any number from the grid. In our example below, we chose 17.
  • Now cross out the numbers in the same row and column as the chosen number.
  • Choose the second number from the remaining numbers on the grid—our choice is 24. Then, cross out the numbers in the same row and column as the second number.
  • Continue in this way until you have chosen five numbers. For example, let’s choose 17, 24, 15, 6, and 3. For your fifth number, there will be only one number left that’s not crossed out.
  • Now add together the five selected numbers in your head or using pencil and paper. Check your answer with your calculator if you like. Here, the sum is sum with, 5 , summands 17 plus 24 plus 15 plus six plus three equals 65.

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?

Hint symbol

Comment

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.

Solution symbol
Answer

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 top-left to bottom-right because this gives one number in each row and column.

The sum is then sum with, 5 , summands one plus seven plus 13 plus 19 plus 25 equals 65. 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 sum with, 3 , summands open seven plus 13 close plus open one plus 19 close plus 25 and work out the sums in the parentheses first, we get sum with, 3 , summands 20 plus 20 plus 25, 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.

4.8.3 Understanding the Puzzle

Add up the numbers in the top row. This is sum with, 5 , summands one plus two plus three plus four plus five equals 15.

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.

  1. The number we pick from the first row is one of the five in the sum above. Suppose we pick 3.
  2. The number we pick from the second row is in the same column as one of the numbers above and is 5 more than it. For instance, if we choose the column headed by 2, then the number we get is two plus five equals seven.
  3. The number we pick from the third row is under a different one again of the five numbers above, and is 10 more than the one in the top row. Say we choose the column headed by 5 then the number we get is five plus 10 equals 15.
  4. The same goes for the fourth and fifth rows, with 15 and 20 more than in the first row, respectively.

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, sum with, 4 , summands five plus 10 plus 15 plus 20 equals 50, 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 50 plus 15 equals 65.

So whatever numbers you pick the sum will always be 65.

4.8.4 Here’s the Trick!

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!

4.8.5 Human Calculator

Now, let’s look at a quick way to attack division by nine.

Activity symbol Activity: The Human Calculator

Videoclip symbolWatch this short video to discover the power of your brain:

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

Now see how quickly you can perform these calculations in your head.

(a) 2154 division nine

Hint

Hint symbol

Comment

Write down the first number of the dividend, then use addition. Watch out for carries!

Solution symbol
Answer

(a) multiline equation line 1 two one five four prefix division of nine line 2 two three eight super one 12 cubed line 3 two three nine cap r times three

Thus, 2154 division nine equals 239 cap r times three.

(b) three one zero two one two prefix division of nine

Solution symbol

Answer

(b) multiline equation line 1 three one zero two one two prefix division of nine line 2 three four four six seven super one nine super zero line 3 three four four six eight

Thus, three one zero two one two prefix division of nine equals 34 comma 468.

(c)three two six four four one two division nine  (This one is tough—you might need to write the shortcut out!)

Solution symbol
Answer

multiline equation line 1 three two six four four one two prefix division of nine line 2 three five super one one super one five super one nine squared zero times super two 22 super four line 3 three six two six super one one two cap r times four line 4 three six two seven one two cap r times four

Thus, three two six four four one equation left hand side two division nine equals right hand side 362 comma 712 cap r times four.

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!