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Addition

Addition is a basic operation in math. So think about it for a second what types of numbers can you add?

Sources: Strawberry - Merlinul/iStock/Thinkstock; Pizza - vitalssss/iStock/Getty Images Plus; Thermometer - Milena_Vuckovic/iStock/Thinkstock

In school, you learn about many models to show addition including objects or pizzas; we can even use temperature.

What we want to do today is extend our knowledge of addition to include any type of number.

To do this, number lines can model the addition of any type of number.

Lesson Goals

• Review the operation of addition.
• Model addition on the number line.
• Learn how to add integers, including negative integers.

Try This!

One spring morning, the outside temperature is \(3^\circ\)C below zero.  By noon, the temperature has increased \(10^\circ\)C.  

1. What is the temperature at noon?
2. How can we represent this scenario using integer addition?  

Think about this problem, then move on to the next part of the lesson.

Negative Integers

Integers on a Thermometer

Recall the Try This problem: One spring, morning the outside temperature is \(3^\circ\)C below zero.  By noon, the temperature has increased \(10^\circ\)C.  

We have to pay special attention to phrases such as, below \(0\). Because they tell us where this temperature is located relative to \(0\) on the thermometer. 

So if we look at the thermometer for a second, we can locate \(0\).

 

Then locate \(3^\circ\)C below.

You're going to notice that on the thermometer this is denoted by the integer, \(-3\).

Now recall that the temperature scale is how we measure and describe temperature. The ultimate goal is typically to compare different temperatures.

So let's ask ourselves the following question:

To make sense of this comparison let's look to a thermometer.

If we compare \(3\) and \(+3\), we can see that they are the exact same distance from \(0\), but lie on opposite sides of \(0\).

Each of these integers represents a change of exactly \(3^\circ\)C from \(0^\circ\)C. \(+3\) represents an increase, and \(-3\) represents a decrease.

Source: Thermometer - _human/iStock/Getty Images Plus

Example 1

We're now going to look at how we can use the thermometer to represent changes in temperature.

Solution

We can represent an increase of \(6^\circ\)C using an arrow that's \(6\) units long.

Since our starting temperature is \(1^\circ\)C, the arrow will start at \(1\).

Then increase \(6\) units, and end at \(+7\).

As a result, the final temperature would be \(7^\circ\)C.

Solution

It makes sense that the arrow we use to represent a change of \(+6\) would be the same as the example we just did. So what needs to change here?

Well, our starting point. Instead, the tail of the arrow starts at \(-4\), and increases \(6\) units from there.

What we see is that an increase of \(6\) units from \(-4^\circ\)C gives us a final temperature of \(2^\circ\)C.

Source: Thermometer - _human/iStock/Getty Images Plus

So even though we're getting different answers here, what we really want to understand is that adding positive \(6\) to a number is the same concept regardless of whether that number is a positive integer, or a negative integer.

Example 2

So let's spin this problem a little bit.

Solution

Our starting point is the same as the first example we did, so we know that the arrow is going to start at \(+1\).

But we're not increasing our temperature this time. Instead, we're decreasing. So instead of moving the arrow up, it makes sense to do the opposite, and move the arrow downward. 

To show this addition, we would start the arrow at \(+1\), and move it \(5\) units down.

As a result, if we start at \(+1^\circ\)C,, and our temperature decreases by \(5^\circ\)C, our final temperature is \(-4^\circ\)C.

Source: Thermometer - _human/iStock/Getty Images Plus

What we see is that when we change temperature, we could end up with a negative answer.

Opposite Integers

Now, we can think of a thermometer as a number line that's been written vertically.

To understand quantity when it comes to negative numbers, it makes sense that we should actually start looking at the number line.

Each negative number on the number line is opposite to a positive number. 
For example, \(-5\) is opposite to \(+5\).

\(-10\) is opposite to \(+10\).

\(-30\) is opposite to \(+30\).

Source: Thermometer - _human/iStock/Getty Images Plus

So we've given you three examples of integers that are opposite. However, any integer has an opposite. So we can formalize this idea in a definition.

So the only integer left to consider is \(0\).

This is a special case of our definition. Since \(0\) is the point at which we're referencing to define opposite integers, \(0\) is its own opposite.

Check Your Understanding 1

Question

Determine the integer that is opposite to \(3\) and plot it on the following number line.

Answer

The integer that is opposite to \(3\) is \(-3\) and it is \(3\) ticks to the left of \(0\) on the number line.

Created with GeoGebra. Author: University of Waterloo. CC BY-NC-SA 4.0. https://ggbm.at/tK648BKT

Feedback

Recall that an integer is opposite to \(3\) if it is the same distance away from zero but on the opposite side of zero on the number line.

Addition on the Number Line

Example 3

Take a moment to try this problem on your own.

Solution

Now, we are already familiar with plotting positive integers. So \(+4\) is simply \(4\) units to the right of \(0\).

\(-4\) is the opposite of \(+4\), so it will be the same distance away from \(0\) as \(+4\) is, but it will be in the opposite direction.

\(-7\) is opposite to \(+7\), which means it's going to be to the left of \(0\). So we locate \(-7\), \(7\) units to the left of \(0\).

\(+1\) is \(1\) unit to the right of \(0\).

\(-3\) is \(3\) units to the left of the \(0\).

Now that we have all the numbers plotted, recall that the number line orders numbers from least to greatest when read from left to right.

So we see that in order of least to greatest, we have

\(\large -7,~-4,~-3,~+1,~+4\)

The number line also reinforces the relative size differences between integers.

For example, the difference between \(-7\) and \(-4\) is greater than the difference between \(-4\) and \(-3\). 

On the number line, we see that because the space on the number line between \(-7\) and \(-4\) is larger than the space between \(-4\) and \(-3\).

Check Your Understanding 2

Question

Plot \(-8\), \(-1), and \(2\) on the following number line.

Answer

Summary

Let's bring this all together for a moment.What we are seeing is that the number line has two sides:

• Positive integers are found to the right of \(0\). 
• Negative integers are found to the left of \(0\). 

Going forward, I'd like you to keep one thing in mind: you already know how to add integers on the number line when they're to the right of \(0\). Now what we've done is we've put in this negative side. But keep in mind that the process does not change.

Addition as Movement on the Number Line

One way to view addition is as movement along the number line.

For example, consider the addition statement, \(6+4\).

Remember from the previous example when we increased our temperature by \(6^\circ\)C we drew an arrow moving up \(6\) units. Our number line is like a thermometer.

To increase our sum by \(6\) units on the number line, we need to draw an arrow that is \(6\) units long and moves to the right.

From there, we want to increase our sum by another \(4\). So we draw an arrow that is \(4\) units long. And again, it moves to the right.

What we see from our number line is the answer is

\(6+4=10\)

How might we include negative numbers?

Example 4

Solution

Let's go back to the thermometer example. We can start by showing each integer on its own thermometer.

\(-3\) is a decrease in \(3^\circ\)C from \(0\), which we can represent with an arrow that starts at \(0\), and moves downwards ending at \(-3\). Similarly, \(8\) is an increase in \(8^\circ\)C from \(0\), which we can represent with an arrow that starts at \(0\), and moves upwards, ending at \(+8\).

So what does it look like when we add these two integers? 

Essentially, all we have to do is copy our arrows onto the same thermometer, making sure that they start and end in the appropriate places.

So currently, both the arrows we have start at \(0\). Does this make sense, when we're trying to find the sum of two integers?

Hopefully, you agree that the answer is no, because we're supposed to be adding \(+8\) to \(-3\), we're not adding both to \(0\).

So to show that we are adding \(+8\) to \((-3)\), the arrow representing \(+8\) should start where the arrow representing \((-3)\) ends.

Source: Thermometer - _human/iStock/Getty Images Plus

This shows that we are increasing the temperature by \(+8^\circ\)C after it has been decreased by \(3^\circ\)C first.

When you're showing integer addition, the arrows should line up from head to tail, to head, to tail. It should be a path starting at \(0\), and moving until you reach the final answer.

We conclude that

\((-3)+8 = +5\)

Adding Integers on the Number Line

Example 5

Take a moment to try this problem on your own.

Solution

Remember that number lines are very similar to thermometers. So while we're working on the number line, it's sometimes helpful to recall the similarities to thermometers.

First, let's consider the two quantities on their own number line. What does a change of \(-7\) look like?

Think about it for a second. On a thermometer, \(-7\) was a decrease. On the number line, a decrease is represented by moving left. So we can represent \(-7\) using an arrow that moves \(7\) units to the left.

So now, what does a change of \(+3\) look like?

Again, think about it in terms of a thermometer. On a thermometer, \(+3\) represents an increase. On the number line, an increase is represented by moving right. We can represent \(+3\) using an arrow that moves \(3\) units to the right.

Now, we got to think about it. What does it mean to add these two quantities?

Let's start with the first integer, \(-7\). All we need to do is copy this first integer to our final number line, which we're going to use to represent addition.

Currently, we're located at \(-7\). From this location, we want to increase our sum by \(3\) units. So we copy the arrow that we used to represent \(+3\), making sure that the arrow starts where the arrow for \(-7\) ended. 

To add quantities means to combine them.  We arrange the arrows from head to tail.

Again, you're going to notice that starting at \(0\), we have a path of arrows that we can follow. The final location is the answer to our addition problem.

Therefore, \((-7)+3 = -4\).

Check Your Understanding 3

Question

Calculate \((\class{hl3}{-14})+\class{hl1}{6}\) using the number line.

Answer

We can represent \((\class{hl3}{-14})\) using an arrow that moves \(14\) units to the left.

We can represent \(+ \class{hl1}{6}\) using an arrow that moves \(6\) units to the right. To add quantities means to combine them.  We arrange the arrows from head to tail and the final location is at \((-8)\). 

Created with GeoGebra. Author: University of Waterloo. CC BY-NC-SA 4.0. https://ggbm.at/P2Y7Cxuj

Therefore, \((\class{hl3}{-14})+\class{hl1}{6}= -8\).

Feedback

Use the sign of each integer to determine whether your arrow should move left (negative) or right (positive). Your final answer is where the path of arrows ends.

Use the counters along the bottom of the number line to determine where the last arrow head ends. This is the answer to the addition problem.

Example 6

As you do this, keep in mind that you can always think about these numbers in terms of temperature if you need to visualize what's going on.

Take a moment and try this problem on your own.

Solution

So what happens when two negatives are added together? Again, let's look at the number line so we can actually visualize what's going on.

As we did before, we're going to first consider the two quantities that we want to add on their own.

What does a change of \(-4\) look like? 

We can represent \(-4\) using an arrow that moves \(4\) units to the left.

What does a change of \(-5\) look like?

We represent the change of \(-5\) using an arrow that moves \(5\) units to the left.

So again we ask ourselves, what does it mean to add these quantities? We start with the first integer and copy that arrow to represent \(-4\). Again, we're now \(4\) units to the left of \(0\).

To add quantities means to combine them. We arrange the arrows from head to tail. To this number we want to add \(-5\). So we copy the arrow that was representing \(-5\), making sure that that arrow starts where the arrow for \(-4\) ended.

We follow the path, and we see that \((-4)+(-5) = (-9)\).

Now there's something interesting happening here. Let's take a moment and compare the representations for \((-4)+(-5)\) and \(4+5\) on the number line. What do you notice right away?

Can you see that the two representations are actually mirror images of one another?

This tells us that these two calculations are very similar.

The only difference is that the arrows point in opposite directions.

We will use this observation to help us with mental arithmetic.

Check Your Understanding 4

Question

Calculate \((\class{hl3}{-5}) + (\class{hl1}{-6})\) using the number line.

Answer

We can represent \((\class{hl3}{-5})\) using an arrow that moves \(5\) units to the left.

We can represent \((\class{hl1}{-6})\) using an arrow that moves \(6\) units to the right. To add quantities means to combine them. We arrange the arrows from head to tail and the final location is at \((-8)\).

Created with GeoGebra. Author: University of Waterloo. CC BY-NC-SA 4.0. https://ggbm.at/BF4Y6Bhe

Therefore, \((\class{hl3}{-5}) + (\class{hl1}{-6})=-11\).

Feedback

Use the sign of each integer to determine whether your arrow should move left (negative) or right (positive). Your final answer is where the path of arrows ends.

Use the counters along the bottom of the number line to determine where the last arrowhead ends. This is the answer to the addition problem.

Addition is Commutative

Example 7

Solution

On the number line, we move \(8\) units to the right, and we show this with an arrow.

And from there, we move \(3\) units to the left. Again, using an arrow to show the movement.

Our final answer is

\(\class{hl1}{(+8)}+\class{hl2}{(-3)}=5\)

Now we did an earlier example that felt very, very similar to this. In fact, the only difference between the two examples that we did is the order in which we added the two integers.

\(\class{hl2}{(-3)}+\class{hl1}{(+8)}=5\)

Source: Thermometer - _human/iStock/Getty Images Plus

 

What happens when we change the order in which we add two integers?

How does positive \((+8) + (-3)\) compare to \((-3) +(+8)\)?

 \(\class{hl2}{(-3)}+\class{hl1}{(+8)}=5\)

What we can see is that because the integers are added in the opposite order, the arrows appear in opposite order as well.

The important thing is that both show a final answer of \(5\).

The Commutative Property

This leads us to a very important fact about addition.

\( \class{hl2}{(-3)} +\class{hl1}{5} = \class{hl1}{5} + \class{hl2}{(-3)}\)

Example 8

Take a moment and try this problem on your own.

Solution

We can represent both expressions on the number line. 

\(\class{hl2}{(-11)}+\class{hl1}{4}\)

\(\class{hl1}{4}+\class{hl2}{(-11)}\)

Again, we see that they give the same result as they did in the previous example with thermometers. The number lines look different, because the arrows are appearing in different order.

In both cases, we get a final result of \(-7\).

Explore This 1

Description

The following investigation is intended to let you explore many examples to convince yourself that you can add numbers in any order to obtain the same result.

Example 1 

Example 2

Example 3

Created with GeoGebra. Author: University of Waterloo. CC BY-NC-SA 4.0. https://ggbm.at/azGCEEdG

By completing the investigation, hopefully you have convinced yourself that the order in which you add integers doesn't matter.

Online Version

https://ggbm.at/azGCEEdG

Take It With You

Number lines can model integer addition.

For example, the following number line shows that

\(\class{hl1}{4}+\class{hl2}{(-6)}=-2\)

What other tools (besides the number line) could you use to model integer addition?