TEK 6.3 C Learning Goal: I can represent integer operations with concrete models and connect the actions with the models to standardized algorithms.

TEK 6.3 D Learning Goal: I can add, subtract, multiply, and divide integers fluently.

Vocabulary

Integer: The set of whole numbers and their opposites (opposite meaning the negative version of a positive whole number), including (-2, -1, 0, 1, 2...).

Opposites/Additive Inverses: Numbers that are the same distance from zero, but on opposite sides of the number line.

Absolute Value: A number's distance from zero, no matter what side of the number line it is on.

Zero Pair: A positive counter paired with a negative counter, resulting in a value of zero.

Integer Operations Using Models

Adding, subtracting, multiplying, and dividing integers can all be represented with real life objects called counters. Positive counters represent one step to the right along the number line, while negative counters represent one step to the left. Watch this video to see how it works:

Now that you know how the counters system works, let's see how counters can be used to model the addition, subtraction, multiplication, and division of integers.

Addition

Subtraction

Multiplication

Division

Connecting Models to the Real World

You use integer operations all of the time in your everyday life. You even use models in the real world that are similar to the ones we just covered. One big example is measuring temperature. You know what a thermometer looks like:

Now that we have worked with the models, it may look different to you. A thermometer is basically a number line, where the red fluid represents where you are along the number line, and each mark on the side represents a step. You can see how you would be able to model the temperature increasing and decreasing throughout the day using counters and a number line. Integer operations are useful in all sorts of different applications, but this example clearly shows how they are useful since the numbers on a thermometer are similar to a number line.

Other cases may not be so obvious, but any time you are looking at a number that goes up and down across time - the height of a roller coaster, the amount of money in your bank account, your height as you snorkel in the ocean, your score in golf, etc. - it can be represented by adding and subtracting integers. You have to figure out what is zero based on the problem (sea level, zero dollars, the bottom of the ocean, zero points, etc.), and then you can describe changes in the data based on this zero point.

Any time you have a rate, which is a set amount of change in the data usually per amount of time, it is useful to use multiplication of integers. An example is a submarine that descends into the ocean at 90 feet per minute. That is a set amount of change (90 feet) per unit of time (minute in this case). If you wanted to find the position of the submarine after 7 minutes, you could just keep subtracting 90 feet seven times, but it would be easier just to multiply 7 times 90. This works because every minute, the submarine drops 90 feet. If there are 7 minutes, then the submarine has dropped 7 sets of 90 feet.

If you don't have the rate and you need to find it, division of integers is helpful. For example, if the temperature of a city starts off at 65°F, and it drops by 32°F over 4 days, you can divide 32 by 4, which gives you 8, to find the change in temperature per day. Division is basically repeated subtraction, so when you divide 32 by 4, you are basically saying that 8 gets subtracted from 65 four times, which ends up being a total of 32 by the end of the fourth time. Division of integers can be used in a lot of other cases that you probably already know about, like dividing food equally among people or finding out how many chairs need to go in each row of an auditorium.

Integer Operations - the Fast Version

You now know how doing operations on integers works and why it is useful - now you need to just focus on being able to do this math quickly.

You have been adding, subtracting, multiplying, and dividing positive integers for a long time now, so we don't need to worry about that. What we do need to pay attention to is when one or both of the integers are negative.

Multiplication/Division

Let's start with multiplying and dividing. Multiplying and dividing with negative integers is the same as if both integers were positive - except sometimes the answer will come out to be positive, and sometimes it will be negative. So, a quick way to multiply and divide integers is to do the math as if both numbers were positive and then figure out the sign of the answer afterwards. This chart can help you easily identify what sign the answer is supposed to have:

Drawing this chart is easy. You just split a circle into three parts and put a negative sign in two of them and a positive sign in the other. It doesn't matter whether the parts are exactly equal or not.

To use the chart for multiplication and division, you first need to identify what signs the two numbers in your problem have. See if they are both negative or if one is negative and one is positive. Next, you need to cover up the two parts of the circle that have those signs. If you are multiplying or dividing two negative numbers, both of the negative sections will be covered up. If you are multiplying/dividing a negative by a positive, one of the negative parts will be covered up (it doesn't matter which one), and the positive part will be covered up. The part that is left over after you have covered up two of the parts is the sign of the new number.

That was a lot of words just now, but if you look at some examples, you will get the hang of it.

Let's say you wanted to do this problem:

7 × -7

If we ignore all negative signs and do the problem, it comes out to 49. To find the sign of the final product, we use our chart. We cover up the positive part because one of our numbers (7) is positive, and we also cover up one of the negative parts because our other number (-7) is negative. This is what it would end up looking like:

As you can see, the only part left uncovered is one of the negative parts. So, the final answer is -49.

What if both of the sevens were negative:

-7 × -7

Well, you would cover up both negative parts of the chart like this:

The only part left is the positive part, so the answer would just be 49.

This chart works for both dividing and multiplying. Just do the problem as if both numbers were positive and then figure out what sign the final answer has by covering up the parts of the chart based on the signs of the numbers in the problem. Remember, only use this chart when one or both of the numbers in a problem are negative. If they are both positive, you just multiply and divide normally like you always have.

As you use this chart over and over again, you will start to notice a pattern: if one of the numbers is negative, the answer will be negative, and if both are negative, the answer will be positive. Once you start to remember this pattern, you won't have to draw out this chart and multiplying and dividing will be even quicker for you.

Addition/Subtraction

Adding and subtracting integers is actually less straightforward than multiplying and dividing them. We will use the same chart, but the main difference is that you can't do the problem as if both numbers were positive first, then figure out the sign. Instead, you use the chart to change the problem into a form that is easier to understand.

Using the Chart

The biggest thing you need to make sure you know is what to do if two operation signs (+ and -) are right next to each other, like with these two examples:

6 + (-2)

6 - (-2)

They are written with parenthesis to show that -2 is its own integer, but writing these problems with no parenthesis would be the same thing.

As you can see, these are not just simple addition or subtraction problems anymore. You have an operation sign, followed by another sign right after it. The good thing is, whenever you have two signs next to each other, you can always reduce them into one. To do that, you use the chart we used earlier:

If one of the signs is negative and one is positive, cover up the positive sign on the chart and one of the negative signs on the chart. If both signs are negative, cover up both negative signs on the chart. The sign left over is the new operation sign that can replace the original two.

Let's try this problem:

6 + (-2)

We have a plus sign (+) followed immediately by a minus sign (-). So, we will cover up one positive and one negative sign on the chart:

Since the only part left shows a negative sign, the two original signs can be turned into one negative sign, to form:

6 - 2

The problem is now a basic subtraction problem, and both integers are positive. You can easily do this problem now and see that the answer is 4.

Now let's try this one:

6 - (-2)

This time, the two operation signs that are next to each other are both negative signs. So, you will need to cover up both negative parts of the chart:

Now the positive section is the only part left on the chart. So, both of the original operation signs turn into a plus sign:

6 + 2

And the answer is 8 this time.

Moving Negative Integers to the Back

Another strategy to use when adding and subtracting integers is moving negative numbers that are at the front to the back. Let's say you had this problem:

-7 + 11

It can sometimes be hard to start with a negative number, especially if you are new to adding and subtracting integers or you are trying to do this quickly in your head. It's easily to understand if you just both the -7 to the back:

11 + (-7)

If you use the chart, you will see that a positive sign and a negative sign can be simplified to a negative sign, so you will have:

11 - 7

Now both integers in the problem are positive, and it is just a simple subtraction problem.

This strategy is only useful if you are adding the second number. If you are subtracting the second number, you will need to use the next strategy.

Subtracting a Positive Integer from a Negative Integer

Let's say you had a problem like this:

-9 + (-8)

If you use the chart, you know that the positive and negative signs that are next to each other in the middle of the problem can turn into a negative sign, which leaves:

-9 - 8

But you won't be able to move the negative integer to the back, because both integers are negative and it wouldn't help at all. Instead, what you can do is pretend that the first number is positive, and that you are adding except subtracting. Then, turn the number into its opposite, or the negative version, and that is the answer. Unlike multiplying and dividing, you always change the integer to negative in this case, and it is not based off of the chart.

After the first step, you will have:

9 + 8

You already know the answer is 17. Now all you have to do is turn 17 into its opposite. So, -17 is the answer.

This only works if you are subtracting a positive integer from a negative integer. Otherwise, don't do this!

As you have seen, multiple addition/subtraction strategies can be combined in the same problem. The most important one to know is what to do when there are two operation signs right next to each other. Once you know that, you can probably figure out the rest of the problem by just thinking about it. All of these strategies will become second nature to you after a little practice, and you will be able to add and subtract integers very quickly.

For practicing vocabulary with flashcards, go to:

https://quizlet.com/_6tpt33

Use these quizzes to make sure you fully understand what you have just learned. Click "view score" to see explanations for any wrong answers. Click "submit another response" to try again.