This lesson will introduce how to add and subtract with both positive and negative numbers. We will observe first how to add with integers using four scenarios. Then, rather than trying to remember more methods, we will discover how to transform any subtraction problem into an addition problem!

So without further ado, read through the slides below to get a feel for how to add and subtract integers and convert between addition and subtraction problems!

Phew, that’s a lot to take in. Once you’ve gone over this and found some practice problems to cement the idea in your head what integers are, you may move on to the next lesson below!

**Multiplying and Dividing Integers**

This lesson is a video generated from the fully animated PowerPoint I created that is available in my Teachers Pay Teachers store page. You may also head over to my YouTube channel to see the slideshows in video form there!

Just in case you’re into that sort of thing, here is a text outline of the main points of the lessons above!

- Integers
- Adding and subtracting integers
- Objectives
- By the end of this lesson you should feel comfortable:
- Adding and subtracting integers
- Recognizing the Commutitive Property of Addition
- Switching between addition and subtraction of integers
- Adding Positive integers
- Now that we have some new numbers to play with, of course the next step would be to try adding them together.
- Let’s start with the basics:
- Though this seems silly, remember, positive numbers are integers, too!
- So adding two positive numbers is something we are hopefully already used to.
- Or to help you can think of this as “a positive plus a positive equals a bigger positive.”
- Adding integers (positive and negative)
- So next we should try throwing a negative number into the mix!
- Let’s start small to see what is happening:
- Observe that there are two kinds of notation here.
- You may write your negative number in parentheses.
- Or you may write your negative sign smaller.
- Both denote a negative number.
- To keep from getting confused, it sometimes helps to write the negative in parentheses, so I will be doing so throughout these lessons.
- Adding integers (positive and negative)
- So, how do we go about solving this?
- Back to the number line!
- We start at our first number, 5.
- Normally, as we just saw, when you add, you get larger, which we know goes to the right on a number line.
- Negative numbers are opposites, however. (see it came back)
- This means we will move to the left instead of right!
- Adding integers (positive and negative)
- But wait…if we are moving to the left, isn’t that just subtracting?
- Yes! That is exactly what this problem is now like!
- You now have two ways to determine the solution:
- You may either know (hopefully) 5 – 3 = 2
- Or glance at the number line to see where we end up (2)
- Therefore, our original problem is solved: 5 + (-3) = 2
- Note
- It is possible when adding a positive and negative to get a negative answer if you move far enough to the left (negative is bigger than positive)
- Adding Integers (negative and Positive)
- Alright, so now what happens if we switch the order?
- Let’s take the same numbers but reversed:
- Well, yet again we have two ways to solve this.
- Way 1:
- First, let’s take a look at the number line:
- This time our starting point is -3.
- Now we are back in familiar territory. Since we add 5, we simply move 5 to the right.
- Adding Integers (Negative and positive)
- Way 2
- The
**commutative property of addition:** - As you saw from adding a positive and negative, the answer is the same.
- This is because you may reverse the order you add integers without affecting the outcome.
- This property is called the
*commutative property*. - Therefore, adding a negative and positive does not need to be thought of any differently than adding a positive and negative because they are the same problem regardless of the order.
- Adding Integers (negative and negative)
- Ok, one last scenario. We can also add two negative numbers together.
- You guessed it, let us once again return to the number line:
- At this point, you’ve probably already figured out where to start:
- Again we are adding a negative which means moving to the left because it is opposite.
- Adding integers (negative and negative)
- Wait!
- 2 + 3 = 5 right?
- So the easiest way to think of adding two negatives is to simply add the absolute values of the numbers (their positive values) and slap on your negative signs at the end!
- A negative plus a negative makes a bigger negative.
- Subtracting integers
- On to subtraction!
- First we need a problem to work with.
- Ah, yet again we see that positive numbers are integers, and most likely you are able to solve this already.
- However, let’s try observing this problem from another angle.
- To the number line!
- We start at 3 and take 2 away, so:
- This reminds me of something, though…..
- Subtracting integers
- Remember, when you add a negative, you move to the left because negatives
*negate*, or make opposite. - So both of these problems represent the same movement on the number line.
- While this is an interesting property, it is also useful.
- Rather than remember a whole new set of rules about subtraction with integers, we can just change them all into addition problems!
- Subtracting integers
- So without the number line, how can we do this with all our problems?
- When I first learned this method, it was affectionately called BAM! BAM!
- First, change the sign of your operation. BAM!
- Notice that the subtraction changed to addition.
- Then change the sign of your second number. BAM!
- Notice this time that the positive 2 became a negative 2.
- Now we can transform any subtraction into addition!
- Practice Transforming
- Now we know that we can change the two middle signs of any subtraction problem to make our lives a little easier, but it was silly to change an already easy problem.
- So instead let’s apply this technique to something a bit more annoying looking:
- Our two middle signs are the “minus” and “negative”, so BAM! BAM!
- Now we have “plus positive”
- Practice transforming
- From here you may either use the number line to travel 2 to the right from -3:
- Or think of taking 3 away from 2:
- Both methods will help you arrive at the same answer.
- -1
- Personally, I always think of subtracting two positive numbers like so:
- 2 – 3…can’t do that because 3 is bigger.
- 3 – 2 is equal to 1, though.
- Since I was taking something bigger from something smaller, tag on a negative…. -1 is the answer!
- The lemonade stand!
- I told you we would come back to it!
- What better way to solidify our new knowledge than by applying it to a real-life situation handling money?
- First we need to actually set up our problem.
- We have a negative amount to start off with, -10
- Then we add the $25 we made to this amount.
- Perfect! Now all we have to do is solve it.
- Starting at -10 we go 25 larger, or to the right, so we have…
- 15 as our final answer. Thus, our lemonade stand will earn us $15.
- conclusion
- Integers may be added four ways:
- Positive plus positive = bigger positive
- Positive plus negative where you move left along the number line (or subtract)
- Negative plus positive, which may be rewritten as positive plus negative
- Negative plus negative = bigger negative
- Subtracting integers may be rewritten as integer addition problems using BAM! BAM!
- You change both middle signs
- To solve real-world problems, you must first set up the number problem, then solve using either addition or subtraction.