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Operations with Signed Numbers

This is something you’ll see all year long: operations with signed numbers.
Positive numbers, negative numbers, and everything in between show up everywhere: in equations, in word problems, in temperature changes, in bank accounts, in physics… everywhere.

Before going deeper into Math 8, it’s important to make sure that adding, subtracting, multiplying, and dividing signed numbers feels natural. Not memorized. Understood.

Here are the big ideas from this lesson and talk about the why behind each operation.

Adding Signed Numbers: Think in Terms of Cancellation

You can follow along with this topic in our teacher guided lesson video.

https://www.youtube.com/watch?v=TuBBxDIWkIQ&list=PLawHkYk9LuTPQXxvuP1f84v6Zz3xN-eU_

When you add a positive and a negative, the key idea is cancellation.

One positive cancels one negative.
That simple idea explains everything.

For example, in
-8 + 14

Picture eight negative ones and fourteen positive ones. Those eight positives cancel the eight negatives, and you’re left with six positives. That leaves 6.

When the signs differ, the larger “group” wins. If you have more negatives, your sum is negative. If you have more positives, your sum is positive.

When the signs are the same, like
-6 + -10
nothing cancels. You’re just adding more in the same direction.
So the result is
-16.

This is how signed addition works. Cancel what cancels. See what’s left.

Subtraction: Rewrite It as Addition

Subtraction isn’t as friendly as addition because it doesn’t play nicely with order or grouping. But here’s the cool part: every subtraction problem can be rewritten as addition.

For example:
10 minus 3 is the same as
10 + (-3).

You’re simply adding the opposite.

This helps especially when negatives are involved.
Look at:
6 minus (-2)

Subtracting a negative is the same as adding a positive, so this becomes
6 + 2
which is
8.

This “add the opposite” technique is something mathematics uses constantly, especially when expressions become longer and more complicated.

Using the Properties of Addition

Once everything is written with addition, you get access to the best tools in the toolbox:

Commutative Property: you can change the order of addition.
Associative Property: you can group the numbers however you want.

This means when you see something like:

7 minus 10 plus 2 minus 5 plus (-3) plus 11

you can rewrite it as:

7 + (-10) + 2 + (-5) + (-3) + 11

Then group the positives together and the negatives together.
The positives:
7 + 2 + 11 = 20
The negatives:
-10 + -5 + -3 = -18
Now add them:
20 + (-18) = 2.

This method reduces mistakes and helps you actually see how much positive and negative quantity you have.

Multiplying and Dividing Signed Numbers

Here’s the pattern. It’s beautifully consistent:

Same signs → positive result
Different signs → negative result

That’s it.

Positive times positive = positive
Negative times negative = positive
Positive times negative = negative
Negative divided by positive = negative
Negative divided by negative = positive

Then you simply multiply or divide the absolute values the way you always have.

For instance:
-4 × -7
The signs are the same, so the result is positive.
4 × 7 = 28
So the answer is
28.

Or:
48 ÷ -4
Different signs → negative result
48 ÷ 4 = 12
So
-12.

This rule works every time.

Bringing It All Together: Order of Operations

When addition, subtraction, multiplication, division, and exponents all appear together, order matters.

The order is:

1. Exponents
2. Multiplication and division (left to right)
3. Addition and subtraction (left to right)

For example:

-3 × 7 + (-5) × (-2)

Do the products first:
-3 × 7 = -21
-5 × -2 = 10

Then add:
-21 + 10 = -11.

Or another good one:

8 minus 2 × (-7)

Multiply first:
2 × (-7) = -14
So the expression becomes
8 − (-14)
which is the same as
8 + 14 = 22.

These examples show why understanding signed multiplication and rewriting subtraction matters. It keeps everything clean and consistent.

Fractions with Negatives: It’s Still Division

A fraction is a number, but it’s also division. This means signed fraction rules follow the same sign rules you just discussed.

For example:

-3 / 6
Different signs → negative result
Reduce 3/6 to 1/2
So this becomes
-1/2.

Or:

-24 / -3
Same sign → positive result
24 ÷ 3 = 8

Or:

20 / -16
Different signs → negative result
Reduce 20/16 to 5/4
So
-5/4.

When you simplify, just carry one negative. Not three. Not two. One.

Let’s Wrap It Up

This lesson is all about building confidence with signed numbers. None of the ideas are new, but it’s important to get comfortable with them. You’ll use positive and negative values constantly, and they should feel natural.

The big ideas:
• Cancelling positives and negatives when adding
• Changing subtraction into addition
• Grouping with the associative and commutative properties
• Same-sign products and quotients are positive
• Opposite-sign products and quotients are negative
• Fractions work just like division

As always, the more you practice, the better your instincts will get. So dive into practicing, try the problems without a calculator when you can, and don’t worry, these skills get smoother every time you use them.

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