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# Unit 5: Numbers Everywhere

## 5 Introduction

Now that you’ve got the basics down, let’s look at some important characteristics and uses of numbers. People can use numbers to solve a wide variety of important problems such as in trading, building, and navigating, just to name a few. However, we will be concentrating on more everyday problems in this unit. It’s important that you have a working knowledge of various units of measurement, negative numbers, and reading and writing mathematics correctly so that you can successfully tackle real-world problems.

### 5.0.1 What to Expect in this Unit

This unit should take around 12 hours to complete. In this unit you will learn about:

• US and SI units of measurement.
• Negative numbers.

In the first section, we will look at the various units of measurement. In particular, we will focus on units used in the U.S., but will also consider those used by the scientific community.

In Sections 5.2 and 5.3, we will explore negative numbers. Whether you’re keeping track of your bank account or enjoying a round of golf, negative numbers might come into play!

In Section 5.4 , we will study the art of reading and writing mathematics. Remember that math is a language; reading and writing it takes practice, and is essential to your success in math.

Finally in Section 5.5 we will look at some extension work on negative numbers and tackle another home improvement problem.

#### Practice Quiz

Check your understanding of some basic techniques of measuring with numbers and using negative numbers before you start, by completing the Unit 5 pre quiz, then use the feedback to help you plan your study.

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 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!

## 5.1 Using Units of Measurement

Everyday problems often involve measurements, where you want to determine how much there is of some quantity. Things are measured to find out how big, how long, or how heavy they are. For these numbers to have any meaning, you need to specify what units the measurement is in. For example, there’s a big difference between 3 feet and 3 miles!

Although most of the scientific world has adopted an agreed-upon system of measurement, the United States still uses its own system, which is closely related to what is called the Imperial system. The internationally agreed units, and the rules for their use, form the Système Internationale (International System), known as SI, which uses metric units.

As you may encounter both systems of measurement in your daily life, we will look at each in this section.  Even if you don’t use the US system, the activities in the section are still good practice at converting between different units of measurement and problem solving—so give them go!

### 5.1.1 Length Measurements

[ The foot measurement is thought to be derived from the length of a man’s foot with shoes! (If your bare foot is a foot long, you have pretty big feet). It was first defined in law by Edward I of England in 1305. Most likely, the relationship among feet, yards, and miles were made to fit existing units. These units are also defined in terms of the metric system, so for example, 1 yard is equal to 0.9144 meters. ] The common units used for measurement of length in the U.S. are inches (abbreviated in), feet (abbreviated ft), yards (abbreviated yd), and miles (abbreviated mi). Unlike the SI, there is not an easy or constant relationship among these units. In short, there are 12 inches in one foot, 3 feet in one yard, and 1760 yards, or 5280 ft, in one mile. Whoa! That’s a lot of feet.

[ Perimeter and area will be discussed in greater detail in Unit 10. ] We measure items such as height, perimeter, and area using these common units. Perimeter is the total distance around an object. Area measures the size of the surface of an object.

Let’s suppose at your physical this year, your height was measured as 67 inches. How tall are you in feet and inches?

First, we see how many times 12 can divide into 67, because there are 12 inches in 1 foot: . We found that there are 5 feet in 67 inches, with 7 inches left over. So, you are 5 feet 7 inches tall. This is often written 5’7” in shorthand notation.

Here’s another example. If you have a photograph that measures 8 inches by 10 inches, what is the perimeter, in feet and inches?

The top and bottom of the photo have the same measurement, 8 inches. Similarly, the left side and right side share a measurement of 10 inches. Therefore, the perimeter, or the length of the border of the picture, is . Since , this means that 36 inches is the same as 3 feet. Thus, the perimeter of the picture is 3 feet.

#### Activity: Length

For each of the following scenarios, use the appropriate operation and unit to determine the answer.

(a) A three-year-old girl measures 34 inches in height. Over her lifetime, she grows another 34 inches. Fully grown, how tall is she in feet and inches?

##### Discussion

Try drawing pictures to help visualize each situation—you don’t need to be an artist!

(a) inches

(The calculation could also be carried out using addition: inches.)

Because , this means 68 inches is the same as 5 feet and 8 inches. Thus, the woman is 5’8”, which is rather tall for a woman.

(b) A rectangular garden measures 12 feet by 18 feet. What is the perimeter of the garden in feet? In yards?

##### Discussion

Make sure you know what each word means. For example, “perimeter” is a measurement of the border, or edges, of a structure.

(b) Because the garden is rectangular, this means that the opposite sides have the same length.

To determine the perimeter, we could just add up the sides of the garden: feet.

Alternatively, we could double each side:

. Since there are 3 feet in 1 yard, the number of yards is . Thus, the perimeter of the garden is 20 yards.

(c) During a road trip with your friends you drive 187 miles through Pennsylvania and then 225 miles in Kentucky. How many miles did you drive in total? How many feet is this?

(c) The total distance driven is .

To determine the number of feet, we need to remember that each mile contains 5,280 feet. Thus, the total distance can also be expressed as .

(d) A football field measures 120 yards in length. How many yards long are four football fields if they are arranged end-to-end?

(d) Four football fields each measuring 120 yards, will total .

### 5.1.2 Weight Measurements

[ The ounce was originally based on the weight of that very useful item, the barleycorn, with 480 barleycorns to the ounce (although it couldn't have been much fun counting 480 of them). Having 16 ounces to the pound was useful because 16 can be divided into halves (8, 4, 2, 1) very easily. ] To measure weight, ounces (abbreviated oz) and pounds (abbreviated lbs) are used. Once again, the relationship between these units may seem random; there are 16 ounces in one pound.

Let’s suppose you were moving boxes of electronic equipment. The first box weighs 14 pounds 11 ounces. The second one weighs 17 pounds 8 ounces. What is the total weight of both boxes?

First, we can add the pounds together: .

Next we add up the ounces: . Since there are 16 ounces in 1 pound, this means .

Thus, we have a total of . The boxes weigh a total of 32 pounds 3 ounces.

#### Activity: Weight

For each of the following scenarios, use the appropriate operation and unit to determine the answer.

(a) At birth, Samuel weighs 7 pounds 4 ounces. After one week, he has gained 13 ounces. How much does Samuel weigh now in pounds and ounces?

##### Discussion

Gaining weight would require addition. Losing weight would indicate the need to subtract. Multiplication is used to carry out repeated addition.

(a)

However, since there are 16 ounces in 1 pound then , the correct way to say this is that Samuel now weighs 8 pounds 1 ounce.

(b) When she started dieting, Beth weighed 203 pounds. She lost 37 pounds. How much does Beth now weigh?

(b) Beth now weighs .

(c) A recipe for a meal for one person requires 3 ounces of meat. How much meat would be needed to feed a party of 14 people? Give your answer in pounds and ounces.

(c) The recipe calls for a total of .

Because , this indicates that there would need to be 2 pounds 10 ounces of meat purchased for the party.

### 5.1.3 Volume Measurements

[ The gallon was originally based on a Roman vessel used for wine. Stout drinkers, those Romans. ] To measure volume, the system in the United States uses units like gallons and quarts. There are four quarts in one gallon.

#### Activity: Volume

For each of the following scenarios, use the appropriate operation and unit to determine the answer.

(a) You own an automotive garage. At the beginning of the month, you purchase 120 quarts of motor oil. Each week you use 22 quarts. How many quarts of oil will you have left over at the end of the month? What is this value in gallons?

##### Discussion

For part (a), you may assume that there are four weeks in a month. First find out how many quarts of oil were used during the month.

(a) If there are 22 quarts used each week, then there are a total of quarts used that month. Since we started with 120, there will be quarts left over.

There are 4 quarts in a gallon so 32 quarts in gallons is there are 8 gallons of oil remaining at the end of the month.

(b) On Monday, you pumped 14.5 gallons of gas into your car. The following Monday, you fill your tank with 15.7 gallons of gas. How much total gas did you put into your vehicle?

##### Discussion

Be sure to line the numbers up by the decimal point.

(b)

Thus, you pumped a total of 30.2 gallons of gas into your car.

### 5.1.4 Alternate Units of Measurement—the Système Internationale

There are many different units used throughout the world, but scientists and technologists of all nations have agreed to use a standard system of units called the Système Internationale (International System), known as SI. One major advantage to the SI is that everything you could ever want to measure can be measured using a few basic units, or combinations of them. [ Did you know that the U.S. actually began to convert to the metric system in the 1970s? As a result, there are quite a few cartoons out there that discuss and poke fun at these systems. ]

Although the U.S. has not adopted the SI, metric units are still in use. For example, perscription drugs will have  the dosage  stated in milligrams (mg), an SI measure of weight.

It’s really important not to confuse metric and Imperial units. In 1999, a NASA Mars Orbiter space craft was destroyed on arrival in the Martian atmosphere at a loss of $125m. An inquiry established that the flight system software on board the Orbiter was written to calculate thruster performance in metric Newtons (N), but mission control on Earth was inputting course corrections using the Imperial measure, pound-force (lbf). An expensive mistake … To show you how the SI system works, we shall look at the units of length. The SI base unit for length is the meter, symbol “m.” It is useful to know a rough idea of the size of a typical measurement in any given unit, so, for instance, the height of a typical domestic door is about 2 meters. The meter was originally intended to be one ten-millionth of the distance from the Earth’s equator to the North Pole (at sea level), but since 1983 it is defined as the length of the path travelled by light in vacuum in of a second. ### 5.1.5 The Metric System A meter is roughly equivalent to a yard. For longer lengths, such as the distance between two towns, the SI system uses a multiple of the base unit, a kilometer (km for short). [ The “kilo” in kilometer is an example of a prefix—a letter or word placed in front of another word. ] “Kilo” means “a thousand”. This means that 1 kilometer is the same distance as a thousand meters. There are other prefixes for smaller units. So, the width of your computer screen might be measured in centimeters (cm), where “centi” means “a hundredth”. So, there are 100 centimeters in 1 meter. Centimeters are further divided into millimeters—as the name suggests, there are 1000 mm in a meter, which means that 1 cm is equal to 10 mm. #### Length Conversion  kilometer (km) miles (mi) meter (m) feet (ft) centimeter (cm) inches (in) inches (in) centimeter (cm) yards (yd) meters (m) miles (m) kilometers (km) ### 5.1.6 Working with a Common Unit When you are using measurements in calculations, you often need to convert all the measurements into the same units before you start the calculation. This would be the case for example if you wanted to calculate the perimeter of something and you had some of the measurements in centimeters and others in meters. It does not make sense to add centimeters and meters so you would need to convert. For example, suppose you wanted to convert 2.3 meters (m) into centimeters (cm). We know that there are 100 cm in 1 m. So, to convert a measurement in m into cm, we need to multiply by 100. So, . You can see how to convert from one unit of length to another in the SI, in the diagram below. Notice that to convert from a big unit to a smaller unit, you multiply. This makes sense—you will need a lot more of the smaller units. On the other hand, to convert a measurement in a small unit to a bigger unit, you divide. The same logic works when converting in other measurement systems and it always worth pausing and asking yourself if you final answer should be bigger or smaller than the original value you have. ### 5.1.7 Length in the Metric System #### Activity: Units of Length (a) Write 327 centimeters as meters. ##### Discussion How many centimeters are in 1 meter? It’s okay to go back to the previous screen and look at the diagram at the bottom. ##### Answer (a) There are 100 cm in 1 m and we are converting from a small unit to a large unit, so we divide by 100 to change cm into m. Hence, . (b) Write 6.78 meters as centimeters. ##### Answer (b) There are 100 cm in 1 m. This time, we are converting from a large unit to a smaller unit, so we multiply by 100. So, . Want a quick reference? This website can help remind you how many of the metric units work. ### 5.1.8 Prefixes in the Metric System By now, you’ll probably have guessed the pattern of units. Working from liters, the two most common multiples are the centiliter (cl), which is one-hundredth of a liter (just as 1 centimeter is one-hundredth of a meter), and the milliliter (ml), which is one-thousandth of a liter. Try the next activity to see how these prefixes work. #### Activity: A Dose of Medicine A new bottle of medicine contains 14 cl. The dose is 5 ml. How many doses can be given from this bottle? ##### Discussion How would you describe, perhaps to a friend, how to convert between milliliters, centiliters, and liters? Use this diagram if it helps. ##### Answer To find the number of doses, the volume of the bottle and the dose need to be in the same units. There are 10 ml in 1 cl. Therefore, . So, the number of doses in the bottle is , or 28 doses. ## 5.2 Negative Numbers It took mathematicians a long time to come to terms with negative numbers, so do not be surprised if you need to work slowly and carefully through this section. In Unit 2 we looked at how to show numbers using number lines, which looked a little like this: Of course, the natural numbers, positive whole numbers, are not sufficient to describe the world around us. Think about going to the grocery store and the various prices you see. You have learned about zero, fractions, decimals, and maybe even before. , pronounced “pie,” is approximately 3.14. You will learn more about this interesting number in Unit 10 when studying circles. To put on a number line looks like this: Now, we extend the number line to the left, to add in the negative numbers, which are used when you want to describe a quantity that is less than zero. That number line looks like: For more on the number line and negative numbers take a look at this pencast. (Click on “View document.”) View document Can you think of a number line that you use that has negative numbers? How about a thermometer? For example, on a cold winter morning, your thermometer might read “4 ºF.” This number is read as “negative 4 degrees Fahrenheit” or “ minus 4 degrees Fahrenheit,” and it means 4 degrees below zero. Brrr! ### 5.2.1 Negative Numbers on a Thermometer Throughout our work, negative numbers will be denoted by using a raised negative () sign to distinguish it from the subtraction sign (−). (You will notice that in other textbooks and material, this distinction is not always made, and the same subtraction symbol is often used for both negative numbers and the subtraction operator.) Another common unit that is used to measure temperature is degrees Celsius (ºC). You can check the temperature of a fridge or freezer using a special thermometer which shows the temperature between about 30 ºC and 40 ºC. On the thermometer, the marks above each of the small divisions represents one degree C. This means that the temperature recorded is 4 ºC. You can see on the thermometer that the temperature increases to the right and decreases to the left. #### Activity: Freezer Stars On the thermometer above, what is the temperature range that corresponds to two stars and three stars? ##### Discussion You could draw the thermometer in your math notebook and mark the corresponding ranges. Remember that numbers to the left of zero have a negative sign, and become more negative as you move to the left. For example, 10 ºC is to the left of 4 ºC. ##### Answer The two stars symbol corresponds to a temperature range from 18 ºC to 8 ºC. The three stars mark from 30 ºC to 18 ºC. If you would like a little more practice getting comfortable with negative numbers, check out this website. ### 5.2.2 Uses of Negative Numbers Negative numbers are used in many other practical applications. Banks and other companies use negative numbers to represent overdrawn accounts or a debt. For example, if you had$50 in your account, but then withdrew $60, your account would be overdrawn by$10, and this might be recorded as $10 or ($10)—parentheses are sometimes used to show a negative balance.

You may also have seen negative numbers on maps, where they are used to record the depth below sea level, or on video recording equipment, where negative numbers are sometimes used to indicate position. If you’re an avid golfer, or enjoy following the Master’s on television, you’ve probably noticed that the scores are often negative (which is a good thing), representing the number of strokes below the average.

Negative numbers can be identified on the number line in a similar way to the temperatures on a thermometer. For example, the number line below shows the numbers 0.5, 1.8, and 2.4. These numbers could be read as “negative zero point five, negative one point eight, and negative two point four”.

[ Remember that the numbers increase to the right, so 1 is greater than 3 because it is to the right of 3. ]

### 5.2.3 The Number Line Revisited

#### Activity: Negative Numbers on the Number Line

Copy the scale below into your notebook and mark the numbers 2.5, 6, and 12 on it.

##### Discussion

The numbers become more negative as you move to theleft on the number line. Also, notice that the numbers go by fives. Between each value given on the number line, there are ten squares. Thus, every two squares represent one whole unit.

### 5.2.4 Determining if a Value is Positive or Negative

Let’s try one last exercise, now that you are an expert!

#### Activity: Positive or Negative?

Determine if each number in the given situation is negative or positive.

• 3 degrees below zero.
• 52 m below sea level.
• $1000 net gain. •$5000 withdrawal from ATM machine.
• $1000 deposit in savings account. • 3 kg weight loss. • 2 kg weight gain. • 80 m above sea level. • 37 above zero. •$2000 net loss.
##### Discussion

Look for key words, like “below”, “above”, or “loss” and “gain.”

The correct solutions have been sorted into a table below.

NumberPositiveNegative
3 degrees below zero
52 m below sea level52 m
$1000 net gain$1000
$5000 withdrawal from ATM machine$5000
$1000 deposit in savings account$1000
3 kg weight loss3 kg
2 kg weight gain2 kg
80 m above sea level80 m
37 above zero37°
$2000 net loss$2000

Now that we have looked at some examples of negative numbers and you have learnt how to represent them on a number line, we will move on to using negative numbers in calculations.

### 5.2.5 Addition and Subtraction on the Number Line

The number line can be used to show how calculations can be carried out with negative numbers. First check how this works with positive numbers by looking at the sum .

Start at 2 and move 4 units to the right (because you are adding a positive number) to get the answer 6.

The same method works if the starting number is negative. For example, suppose you are in debt by $5, so your account balance is recorded as$5. If you then pay $8 into your account,$5 of this will pay off your debt and $3 will be recorded as your new balance, so . On the number line, the starting number is 5, and moving 8 units to the right verifies the answer as 3. Now consider subtraction, for example . Here, the starting number is 6, but this time we move 4 units to the left to get to the answer, which is 2. Similarly, if your account is overdrawn by$2, and then you withdraw another $3, your account will be overdrawn by$5. This is represented on the number line by starting at 2, then moving 3 units to the left to get the answer 5. This is written as .

### 5.2.6 Working with Negative Numbers

#### Activity: Working with Negative Numbers

Let’s try the following examples. You can try this using pencil and paper, or click here for an interactive number line! Note that all these calculations involve adding or subtracting a positive number.

(a)

Start at 3, move 7 units to the left.

Thus, .

(b)

(b) Start at 2, move 5 units to the right.

Thus, .

(c)

(c) Start at 3, move 6 units to the left.

Thus, .

(d)

(d) Start at 4, move 2 units to the right.

Thus, .

Now let’s look at adding and subtracting negative numbers.

### 5.2.7 Adding and Subtracting Negative Numbers

We use a similar process when adding and subtracting negative numbers. Let’s consider a simple case first: . If you add a number to zero, you end up with the same number. For example, . Similarly, .

On the number line, we start at 0 and move 2 units to the left, which is the same operation as subtracting 2.

Adding a negative number is the same as subtracting the corresponding positive number. So for example:

and .

This is an important rule to remember, so make a note of it in your math notebook now.

Now, what happens when you subtract a negative number? This isn’t easy to visualize, so try thinking of subtraction in a different way. For example, for the calculation , instead of saying “30 take away 20 is …” you can say, “What do I have to add on to 20 to get 30?”

In the same way, can be interpreted as “What do I have to add on to (2) to get 0?” From the number line, you have to move 2 units to the right to arrive at zero, which is the same as adding on 2. Thus, .

Subtracting a negative number is the same as adding the corresponding positive number. For example, and .

Another important rule for your math notebook.

It’s not easy to see why subtracting a negative number is equivalent to adding a positive number in real life, so think of driving a car along a road, going in a forward direction which we’ll call “positive”. If you wanted to go back in the opposite direction, which we will call backward, or “negatively”, you have two choices. You could either reverse, or turn round. Either is kind of equivalent to a negative number. But, if the car did both at the same time (that is, turned themselves around, and then reversed), then they would continue in my original direction — that is, positively. So, subtraction of a negative number is addition of the positive number.

#### Activity: Adding and Subtracting Negative Numbers

Work out the following calculations in your math notebook.

(a)

##### Discussion

You could use a number line to help visualize each problem. Also, remember that subtracting a negative number is the same as adding the corresponding positive number, and adding a negative number is the same as subtracting the corresponding positive number.

(a)

(b)

(b)

(c)

(c)

(d)

(d)

If you’d like to see and hear these problems exercises explained, click on “View document” and let’s see how these work!

View document

### 5.2.8 Rules for Negatives

As you first learn how to work with negative numbers and develop your skills, a number line will be very helpful—but it can sometimes be tedious. Let’s create a few rules that can help us perform addition and subtraction when negative numbers are involved. Remember that you can always use a number line if you find it useful or if you’re unsure of your answer!

#### Rule 1:

If you are adding two numbers that have the same sign, like , momentarily ignore the signs, and add the numbers together. Then, place the sign in front of the sum.

We think of as , which equals 7. Since both numbers were negative, we place a negative sign in front of the answer. Thus, . You can verify this using a number line.

#### Rule 2:

If you are adding two numbers that have different signs, such as , momentarily ignore the signs. Then SUBTRACT the numbers and place the sign of the “larger” number (we are ignoring signs!) in front of the difference.

We ignore the signs in and subtract: , which equals 6. Since the “larger number” is 8, and it has a negative sign in front of it, the overall answer is also negative: .

#### Rule 3:

If you are subtracting a positive number, like , then rewrite it as addition of the negative number and then use the rules for addition.

So, . For help visualizing this example, look at the number line below.

#### Rule 4:

If you are subtracting a negative number, such as , then rewrite it as addition of the positive number and then use the rules for addition.

So, . Check out the number line below to see the addition.

Of course if you feel comfortable adding and subtracting negative without these rule there is no need to use them.

##### Activity: Adding and Subtracting Negative Numbers Using the Four Strategies

Try these questions again using the four strategies above.

(a)

###### Discussion

Look at rule 2.

(a) We are adding two numbers with different signs. Ignore the signs and subtract the numbers  20 – 5 and use the sign from the larger number in the answer. So

(b)

###### Discussion

Look at rule 1.

(b) We are adding two numbers with the same sign. Ignore the signs and add the numbers, 20 + 5 and put back the sign from the numbers. So

(c)

###### Discussion

Look at rule 4.

(c) Re-write as addition of a positive number, 20 + 5

(d)

(d) Re-write as addition of a positive number, -20 + 5. Then follow rule 2. Ignore the signs and subtract the numbers 20 – 5 and use the sign from the larger number in the answer.

If you’d like a little more explanation on subtracting negative numbers, you can check out this video:

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

### 5.2.9 Calculator Exploration—Negative Numbers

In this exploration, you will learn to use the calculator for negative numbers.

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

To enter a negative number on the calculator, use the key. This is between the 0 and the decimal point. Find this on the calculator now.

To enter a negative number like 3, you must use parentheses. The key sequence for 3 is:

Notice that the parentheses appear in the white window of the calculator but not in the black window; the calculator takes some short cuts in calculations with negative numbers, as you will see.

To work out calculations including negative numbers, follow exactly the same method as for positive numbers, but remember that negative numbers must always be enclosed by parentheses.

Here are some calculations for you to try. Work out the answer in your head (or using a number line) first to get some more practice, then check that it is correct on the calculator.

#### Activity: Adding and Subtracting Negative Numbers Using the Calculator

Remember to put negative numbers in parentheses

(a)

##### Discussion

Did you remember to include parentheses round each negative number?

(a) The key sequence is

When you have entered the calculation but before you click on “equals,” the calculator looks like this:

Notice that the full calculation is shown in the white window, but the calculator gives a shorter version in the black window. It is using the rule:

• Adding a negative number is the same as subtracting the corresponding positive number.

Watch the windows on the calculator as you enter the calculation and this should help you remember the rule.

(b)

(b)

(c)

(c) Entering the calculation, with parentheses round (5), the calculator looks like this:

This time the calculator is using the rule:

• Subtracting a negative number is the same as adding the corresponding positive number.

So, .

(d)

(d)

(e)

(e)

You may be wondering which key to use on your keyboard for negative numbers. There is no direct equivalent of the button, but the minus key performs the same task. Try this if you want to. Be careful, though—you must still enclose the negative number in parentheses.

Mathematical operationCalculator buttonKeyboard key
Negative number
The − key
Remember to enclose negative numbers in parentheses.

You know that practice makes perfect! Here is a link to a game that is fun and shows you more examples about adding and subtracting positive and negative numbers. Show how great your skills are and make the teacher walk the plank! (Click on each pirate to see which one is proposing the correct answer.)

Working with negative numbers can be quite confusing, so well done for getting this far. Get as much practice as you can to reinforce your learning. Try making up your own examples again or asking for some help from friends or family.

Here is what you’ve already learnt:

• Adding a negative number is the same as subtracting the corresponding positive number.
• Subtracting a negative number is the same as adding the corresponding positive number.
• Use parentheses to enter negative numbers on the calculator.
• To add the sign, use the key on the calculator, or the “minus” key on your keyboard.
• Calculations involving negative numbers are carried out as usual (remembering the parentheses).

Now let’s turn to multiplication and division with negative numbers.

## 5.3 Multiplication and Division with Negative Numbers

Multiplying and dividing with negative numbers is an important skill to possess. Let’s go through a calculator exploration to investigate what happens when you multiply by negative numbers.

### Calculator Exploration—Multiplication with Negative Numbers

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

#### Activity: Multiplying Negative Numbers

Copy the following tables into your notebook. Then use the calculator to find the answers and see if you can spot any patterns that emerge.

##### Discussion

Take your time, and remember to enclose each negative number in parentheses.

8
8
4
4
0
0
4
4
8
8
12
12

In the first solutions column, the numbers are going down by four each time, and in the second solutions column, the numbers are increasing by four each time.

#### Activity: What Happens Next?

Use the patterns in the tables to predict the answer to the next calculation in each sequence. Then, check your predictions using the calculator.

(a)

##### Discussion

What number comes next in the sequence of answers?

(a) The next number in the sequence 4, 8, 12 is 16 and on the calculator .

(b)

(b) The next number in the sequence 4, 8, 12 is 16 and .

Now we’ll see if we can write some rules to help us remember this pattern.

### 5.3.1 Investigating Multiplication by a Negative Number

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

Now it’s time to turn detective, and write some mathematical rules.

#### Activity: Being a Mathematical Detective

Use the results from the tables for the previous activity to complete the following statements:

If you multiply a negative number by a positive number, the answer is …

##### Discussion

Use the evidence from your tables that you wrote in your math notebook!

If you multiply a negative number by a positive number, the answer is negative.

gives .

You can extend these rules a little. The order of the numbers doesn’t matter, so multiplying a positive number by a negative number also gives a negative answer. And the rules work for division too.

If you multiply a negative number by a negative number, the answer is …

If you multiply a negative number by a negative number, the answer is positive. Again this rule also works for division.

gives .

Well done, you have discovered two other important mathematical rules about negative numbers!

### 5.3.2 Rules for Multiplication and Division of Signed Numbers

These rules for multiplication and division are really useful, so be sure to add them to your math notebook and take a little time to learn them.

1. or = Negative
2. or = Negative
3. = Positive
4. = Positive

Let’s think some more about what is happening here.

When you used your calculator, you found that , this was the same as four lots of , or , which gives an answer of 8.

You also found that . The product of any two negative numbers can be shown to be positive.

Now, consider . An alternative way of considering this problem is to say: “What do I have to multiply 2 by to get ...” ? Since 2 is positive, and 6 is a negative number, the answer must be negative. Because , then .

Similarly, for , since you would need to multiply by 3 to get , you can deduce that .

Now let’s get some practice without your calculator—you can do it! Look back at the rules if you need to.

### 5.3.3 Working with Negative Numbers by Hand

#### Activity: Multiplying and Dividing Negative Numbers

Work out the following calculations by hand in your math notebook.

After doing the work by hand, check your results on the calculator.

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

(a)

##### Discussion

Use the four rules for multiplying and dividing (Section 5.3.2) to determine the sign on your answer first, then carry out the operation without worrying about the signs.

(a) , since a negative number multiplied by a positive number gives a negative result.

(b)

(b) , since a negative number multiplied by a negative number gives a positive result.

(c)

(c) since a negative number divided by a positive number gives a negative result.

(d)

(d) , since a negative number divided by a negative number gives a positive result.

If you’d like to see these problems explained, click on “View document” and let’s see how these work!

View document

[ If you’d like some more explanation and extra practice with multiplication and negative numbers, visit this page. You can also create your own examples and check with the calculator. ]

You can also try using your multiplication skills with negative numbers in Grade or No Grade? to win the A+ from the “banker”.

### 5.3.4 Negative Numbers Used in Golf

Now that you’re a pro at working with negative numbers, let’s try one more activity!

#### Activity: Are You Ready for Some Golf?

On a lazy Sunday afternoon, you are watching golf on TV. There are two players that you are particularly interested in and their scores for the first nine holes of the round are shown below. What is the total score for each player and at this point in the game who is ahead?

Player 1Player 2
12
22
13
10
21
32
00
21
21

The score for each hole is the number of shots below or over the average for that hole, so the lower the score the better in this case!

##### Discussion

For each player count the number of times each different score occurs and use multiplication to calculate the overall scores.

Player 1:  three holes at one below average, four holes at two below average, one hole at three below average, and one hole at the average score.

So Player 1’s score = 3 × 1 + 4 × 2 + 1 × 3 + 1 × 0 = 3 + 8 + 3 + 0 = 14

Player 2: three holes at one below average, three holes at two below average, one hole at three below average, and two holes at the average score.

So Player 2’s score = 3 × 1 + 3 × 2 + 1 × 3 + 2 × 0 = 3 + 6 + 3 + 0 = 12

Because 14 is less than 12, this means that after nine holes, Player 1 is in the lead.

### 5.3.5 Wrapping Up Negative Numbers

You now know how to perform the basic operations with signed (negative) numbers by hand and using the calculator. Hopefully, your confidence has grown, and you will continue to utilize all of your resources as you extend your knowledge and develop new skills.

Here are the important rules to remember again:

• Adding a negative number is the same as subtracting the corresponding positive number.
• Subtracting a negative number is the same as adding the corresponding positive number.
• Negative × negative or Negative ÷ negative  = positive.
• Positive × positive or Positive ÷ positive = positive.
• Negative × positive or Negative ÷ positive  = negative.
• Positive × negative or Positive ÷ negative = negative.

If you’d like to see a few more examples of each operation involving negative numbers, watch this video:

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

Well done for reaching the end of this challenging section‡you’ve been working really hard! If you’d like to wrap up what we’ve learned about working with negative numbers try watching and listening to this this catchy song!

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

Now we’ve covered the basic operations in math, problem solving strategies, and how to handle negative numbers, it is time to learn about how you read and write the language of mathematics correctly.  This will mean that you can communicate what you have learnt so far to other people.

## 5.4 Reading and Writing Mathematics

Reading a piece of mathematics requires a more detailed and active approach than some other types of reading, so it is worth taking a few moments to think about strategies that you have used while working through this unit. Some useful tactics are summarized below, but you might want to add your own ideas to the list as well.

• Read carefully and check you understand any special terminology, symbols or abbreviations.
• Make sure you understand and what it is you have to do.
• Highlight (or underline) key pieces of information.
• Check that you have all the information you need at hand, including skills and techniques learned earlier.
• Add extra lines of working if that helps.
• Draw a diagram (to help you visualize the problem) and put the information you have on it.
• Mark the parts of the problems that you find difficult. You may want to come back to these or talk through the ideas with a friend or look on the Internet for some guidance.

### 5.4.1 Writing Mathematics

Working through the units, your writing fulfills different purposes. These include responding to the activities, making your own notes (as mentioned in Unit 1) when encountering a concept that is new to you, writing your assignments, working out self-checks, and explaining mathematical ideas whenever needed. These forms of writing use different styles. For example, something which may be perfectly acceptable as a quick note to remind yourself of a key point may not be suitable as part of your assignment for your tutor, or to resolve a problem during a self-check or for others to see.

You will need to make sure that whatever you write is going to be clear to anyone who needs to read it and that includes yourself in several weeks when you may come back to study what you wrote today. Asking someone else to read your work or imagining that you are writing for someone who has little knowledge of the topic and requires a full explanation can guide you to make sure you include sufficient detail.

Mathematical writing requires additional skills in using notation and specialist vocabulary—remember, math is a language! This skill does take time to develop. Practice is the key, and while you are working through the next units and sections, don’t forget to observe how the mathematics is set out, since this can serve as a model for the style that you can use to explain your mathematical reasoning. You will also find it helpful to try and use this style whenever you do your own calculations—the more you do this the more like second nature writing your math correctly will become and the easier it will be!

### 5.4.2 Importance of Proper Math Notation

Writing math correctly is important, especially for two reasons:

2. You can check your own work much more easily for mistakes.

The three most important things to remember are:

1. Explain each step in your working clearly.
2. Lay out your explanation clearly.
3. Use correct math—make sure that what you write is mathematically correct.

Let’s consider the following scenario.

You want to pave a patio that’s 18 feet long by 9 feet wide, with paving slabs that are two feet square. The paving slabs cost \$8 each. How many will you need, and what will they cost in total?

You go to your local contractor who says “Let me see …” and scribbles down the following calculation:

call it 44 to be safe, so = .

#### Activity: What’s Wrong with the Math?

Can you see the difficulty with his work and can you suggest any improvements?

##### Discussion

Remember that mathematics that has been written well is easy to follow and uses correct notation. List any problems or issues you see. Think about how you have seen math presented in the units so far.

• First of all, you’re going to have great difficulty in checking his work. This could be a problem—you really don’t want to buy more slabs than you need! Nor do you want to purchase too few to finish the job.
• Secondly it’s not clear what the answer is—what does that final figure of 352 represent? Slabs, or dollars?
• Third, it’s mathematically incorrect at a number of points. That could lead to some serious miscalculations—more or fewer slabs, and over- or under-charging. In particular, the contractor misuses the equal sign a few times. You can only use an equal sign if the expression on the left is equivalent, that is gives the same answer, to the expression on the right. If it’s not, this means you need another, separate step.

### 5.4.3 Understanding a Solution

Let’s take a closer look at some of the problems with the solution provided by the contractor:

[ Areas are measured in square units—this includes rectangles. You will look more closely at the concept of area in Unit 10. ]

Now let’s look at how the contractor should have written out your quote to be clear:

Remember the following steps as you are writing your math work:

1. Explain.
2. Layout.
3. Correct math.
4. Units.

### 5.4.4 Tips for Writing Mathematics

You may want to take a few minutes now, looking back over your own solutions you’ve written from activities you have worked on in Units 1–5. You may also find that it helps to read your work out loud.

Does it make sense and is each step explained? Did you use equal signs correctly? Did you skip any steps? Are there any improvements that you can make to your own work? If there are, you might like to identify and highlight these in the summary box below (or add them to the list) as a reminder for your future work.

Another reason for explaining your work carefully is that it could possibly help an instructor (in a future math class) to understand your thoughts and you may be awarded some partial credit for your ideas even if your final answer is incorrect.

#### Tips for Writing Mathematics

• Describe each step, explaining your work in words carefully, and checking that each sentence follows on logically from the previous one.
• Start each new sentence on a new line, and line up the equals signs underneath each other.  This helps you use equals signs correctly as well!
• Use notation correctly. Equals signs should only be used if two expressions are equal. They should not be used to link your solution together.
• Check your writing by reading it aloud. When you translate the symbols, it should still make sense.
• Use well-labeled graphs, charts, and tables to summarize data and results clearly.  We will look at this again in Unit 11.
• Include a concluding sentence which answers the question exactly.

If you’re interested in more help and practice in writing mathematics, here is a website about language, notation, and formulasthat will be useful. Note: There may be some topics that you haven’t studied yet, but the layout of the mathematics is what you want to pay attention to!

## 5.5 Extensions and Further Exploration

In this section, we will try to deepen our knowledge through the exploration of some of the concepts that were discussed throughout the unit. You might find some of these activities to be quite challenging. If you get stuck, feel free to discuss them with a friend. Don’t panic; just keep going.

The idea of this section is to expose you to some more math, and to get you thinking about the math that you run into on a daily basis. If you believe that you don’t have the time to spend further exploring these topics, this is a section that you could treat as optional.

Now get ready to investigate interior decoration, more about negative numbers and get some additional practice using the calculator!

### 5.5.1 Using the Number Line to Multiply

At the beginning of Unit 2, we learned about number lines. In Unit 5, we took advantage of this tool and used it to add and subtract, especially when negative numbers were involved. Did you know that you can also use the number line to multiply and divide? Let’s check it out!

Suppose you wanted to determine using the number line. You take the first factor, 2, and draw an arrow that is 2 units long. Because the 2 is positive, we will draw the arrows to the right. The second factor tells us how many arrows to draw. So, starting from zero, the arrow (of length 2) is drawn end-to-end 4 times, as shown below.

Thus, .

Suppose you wanted to determine using the number line. First, we will be drawing arrows that are 4 units long, but to the left since the 4 is negative. Starting at zero, we will draw an arrow that is 4 units long, 2 times, end to end, as shown below.

Thus, .

What happens if you need to determine ? The first factor tells us that the arrow will be 4 units long and go to the left, but we can’t draw the arrow a negative number of times. Unfortunately, our lovely number line model fails for this example.

Not to worry! In the calculator exploration earlier, we discovered that the reason a negative number times another negative number is positive is because it is following a pattern.

Let’s try a few more examples on the number line.

#### Activity: Multiplication on the Number Line

In your math notebook, determine the answer using a number line for each problem below.

(a)

##### Discussion

If you have a negative factor, write it first. Remember that both length and direction are given by the first factor.

(a) Our arrow will be 3 units long and go to the left. We will need to draw it three times.

Thus, .

(b)

(b) First, we must rewrite as , which is an application of the commutative property of multiplication. If you can’t quite remember what commutative means check your math notebook! So, our arrow will be two units long and go to the left. We will draw it five times.

Thus, .

### 5.5.2 Calculator Exploration—Exponents with Negative Numbers

In this exploration, you will explore exponents of negative numbers.

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

Here is an activity to help you understand exponents of negative numbers.

#### Activity: Taking Exponents of negative numbers

Copy the table below into your notebook and use the square or button to find the value of each of the following. Before you start think about what pattern you might see and make a note of it in your math notebook.

 left parenthesis super minus times one right parenthesis squared left parenthesis super minus times two right parenthesis squared left parenthesis super minus times one right parenthesis cubed left parenthesis super minus times two right parenthesis cubed left parenthesis super minus times one right parenthesis super four left parenthesis super minus times two right parenthesis super four left parenthesis super minus times one right parenthesis super five left parenthesis super minus times two right parenthesis super five left parenthesis super minus times one right parenthesis super six left parenthesis super minus times two right parenthesis super six
##### Discussion

Remember to enclose each negative number in parentheses, then follow it by the exponent button (or ^ and the exponent number on your keyboard).

The key sequence for finding is

 left parenthesis super minus times one right parenthesis squared 1 left parenthesis super minus times two right parenthesis squared 4 left parenthesis super minus times one right parenthesis cubed −1 left parenthesis super minus times two right parenthesis cubed −8 left parenthesis super minus times one right parenthesis super four 1 left parenthesis super minus times two right parenthesis super four 16 left parenthesis super minus times one right parenthesis super five −1 left parenthesis super minus times two right parenthesis super five −32 left parenthesis super minus times one right parenthesis super six 1 left parenthesis super minus times two right parenthesis super six 64

Look at the table you created in this activity. Can you see a pattern emerging? Was it what you expected to see?

#### Activity: Spotting the Pattern

See if you can complete the following sentences:

Raising a negative number to an even exponent gives a ______ number.

Raising a negative number to an odd exponent gives a ______ number.

##### Discussion

Remember that an even number divides exactly by 2 (for example, 2, 4, 6 …), while an odd number does not divide exactly by 2 (for example, 1, 3, 5 …).

Raising a negative number to an even exponent gives a positive number.

Raising a negative number to an odd exponent gives a negative number.

There are lots of patterns like these in mathematics, and it’s worth watching out for them!

### 5.5.3 Role of Parentheses

Now we’ll investigate why parentheses need to be placed around negative numbers.

Let’s check what happens if you square 5. When you multiply a negative number by a negative number, the answer is positive. So, if you want to find the square of 5, you calculate and the answer is 25.

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

#### Activity: Squaring −5

Use the calculator to find .

##### Discussion

Remember that negative numbers must be enclosed in parentheses.

You should indeed have found that . The calculator screen looks like this:

#### Activity: Forgotten Parentheses

Now let’s suppose that you forget the rule about enclosing negative numbers in parentheses and you enter with no parentheses. NOTE: You will have to do this using your keyboard because the calculator does not allow negative numbers created with the button to be entered without parentheses.

Use the keyboard and type in − 5 ^ 2, then hit the return key. What happens?

##### Discussion

Remember that you may have to clear the calculator by clicking on before you can begin typing. Also, do NOT use the button—use your keyboard.

The calculator screen should look like this:

You see that the answer given is 25. We obtained a negative value instead of a positive value!

### 5.5.4 Placement of Parentheses

Below are the screens from the two different calculations we carried out.

The first calculation was and the calculator shows exactly this with the answer 25.

The second calculation was . The calculator has added parentheses around 5, squared this to give 25, then made it negative. So, the answer is 25.

#### Activity: What’s Up with the Calculator?

Why has the calculator given the answer 25 this time, while it gave 25 for the earlier calculation?

##### Discussion

The calculator is using the PEMDAS code.

For :

In the PEMDAS code, parentheses are evaluated first. The cannot be simplified, so the calculator moves to the exponent, which operates on whatever is inside the parentheses. So, it carries out .

A negative number is the same thing as multiplying the corresponding positive number by 1. For example, . This is sometimes called finding the opposite.

In the second calculation, the calculator thinks you meant the “opposite of” five squared, since you did not place parentheses around the negative 5. Because of this, it places parentheses around the 5 only, so the calculator carries out , because exponents come before multiplication in the PEMDAS code.

Be careful when you work with negative numbers on the calculator, and remember to enclose them in parentheses to make sure that the calculator does what you want it to do.

#### To recap:

• Don’t try to take short cuts when you enter negative numbers on the calculator; negative numbers must be enclosed in parentheses.

### 5.5.5 Home Improvement Project

Let’s take a look at a real-world problem that pulls in many of the topics we have studied so far, like units of measurement, the basic operations, and rounding.

On a home improvement store’s website, the following instructions were given for calculating how many rolls of wallpaper are needed to decorate a room.

Calculate the number of rolls you will need, using the method outlined below.

• Step 1: A standard roll of wallpaper is approximately 10.5 m (33 ft) long and 530 mm (1 ft 9 in) wide. If you measure the height of the walls from the baseboard (skirting board) to the ceiling, you can determine how many strips of paper you can cut from a standard roll—four strips are about average.
• Step 2: Measure around the room (ignoring doors and windows) to work out how many roll widths you need to cover the walls. Divide this figure by the number of strips you can cut from one roll to calculate how many rolls you need to buy. Make a small allowance for waste.

(B&Q, 2005)

#### Activity: Following the Instructions

(a) Read through the directions above. Write down the important information. What did you do to make sense of these instructions?

##### Discussion

Try drawing a sketch, and think through the steps you would take to apply new wall paper in your kitchen. Thinking about the math cycle from Unit 4 may help as well.

(a) This is how Tony, a student, tackled the problem. His notes are given below.

• I read all the instructions through first of all to get an overall idea of what I would need to do. Then I went back and highlighted the key bits of information—the dimensions of the roll of wallpaper and the instructions for what I needed to measure.

I wasn’t sure what “ignoring doors and windows” meantin the instructions for measuring round the room. Did it mean you shouldn’t measure the part of the wall occupied by the door and window, or that you should measure the width and length of the room as though there wasn’t a door and window there? I discussed this with a friend and decided to measure as though there wasn’t a door and window, since that would overestimate the amount required and I didn’t want to end up with too few rolls of wallpaper.

• Then, I concentrated on the calculations. There are three calculations:
1. Working out how many strips of paper you can cut from a standard roll.
2. Working out how many roll widths there are round the room.
3. Calculating how many rolls are needed.
• I used the measurements to sketch out a diagram so that I could visualize the problem more easily as well. My room measured 3.2 m by 4 m, and the baseboard to ceiling height was 2.34 m. So, the distance round the room was .
• For the second calculation, I considered a simpler problem first. Say a wall that was 10 m long, and a roll of paper that was 0.5 m wide—that helped me to see that I needed to divide the length of the wall by the width of the roll, since the question was “how many widths are there along the wall?”.

(b) Using your own measurements for a room (or assume the room measures 3.2 m by 4 m and the height of the walls is 2.34 m), work out how many rolls will be needed.

(b) [ 10.5 m is equal to 1050 cm since there are 100 cm in 1 m. ] The first calculation is to work out how many lots of the height (2.34 m) you can get from a roll of wallpaper that is 10.5 m long (information provided at the website in Step 1).

Therefore, number of strips of wallpaper from one roll =

(Note that all the dimensions have been converted to the same units. The wavy equals sign means “approximately equals to.”)

You might be tempted to round this up to five, but in this real-world problem, we must round down to four, as the roll of wallpaper is not long enough to but cut into five strips each of length 234 cm.

So, you will only be able to get four strips from each roll.

Next, you need to work out how many strips are required to cover the room. If you already have a roll of wallpaper, you could count how many strips you’ll need by measuring along the wall with the roll of paper.

[ ] If you don’t have any wallpaper, you can calculate how many 53 cm (given at the website in Step 1) wide strips will fit in to 14.4 m. So, the total number of strips needed is . Rounding this up, to make sure we have enough paper, gives 28 strips altogether. As there are four strips in each roll, so rolls will be needed. This will allow for some wastage, because paper is not put over the doors or windows. However, if the paper has a large pattern that needs matching up, it might be worth buying an extra roll to be on the safe side. If money is tight, go with the seven rolls, and just be extra careful when hanging them up.

Well done for completing the problem—it had a lot of separate steps to get to the final answer.

Now it’s time to check back over the topics covered in this unit by completing the self-check section.

## 5.6 Self-Check

In this unit, you might have found certain activities very challenging, and that’s okay! We all get stuck on math problems. Keep in mind that there are plenty of resources that can provide guidance and advice, such as family, friends, teachers, old textbooks, and the Internet.

It’s important for you to mull over the material you have examined and consider the development of your skills. Use your math notebook to jot down some thoughts on this unit and anything you feel went particularly well or might need some more work before you do the self-check. Your confidence will continue to increase the more you practice. You can do this by working through the exercises for each section.

### 5.6.1 Self-Check on Units of Measurement

#### Exercise 1: Units of Measurement in the U.S.

For each of the following scenarios, use the appropriate operation and unit to determine the answer.

(a) At the start of soccer practice, the water jug contained 16 quarts of water. Five players each drank one quart, and two players drank two quarts each. How much water was left over at the end of practice?

(a) Quarts of water left =

(b) At birth, Peter weighs 7 pounds 9 ounces. During his first two days at the hospital, he loses 3 ounces each day. After he returns home, he gains 8 ounces. How much does Peter weigh now?

(b)

(b) So Peter’s weight now = 7 pounds + 9 ounces – 2 3 ounces + 8 ounces

= 7 pounds + 9 ounces – 6 ounces + 8 ounces

= 7 pounds + 3 ounces + 8 ounces

= 7 pounds + 11 ounces

Peter now weighs 7 pounds 11 ounces.

(c) A rectangular garden measures 12.4 feet by 18.3 feet. How much fencing will need to be purchased to enclose the garden?  How much is this in yards?