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What is multiplication?

We know that multiplication is a procedure in math, and we know that we need to memorize our times tables, but what is multiplication?

Well, multiplication is adding the same number, multiple times.

So, instead of writing something like:

We can instead, count the number of 7’s we have here:

And simplify this long equation to just:

7(10) = 70

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Rules/Ways to multiply numbers

So how do we actually do the multiplication?

Well, there are a few ways to do it, so here are some ways to multiply:

x 12

Multiplying the bottom right number by the top right number: 2(5)

1. When dealing with two or more digit integers, we can multiply them by first:

Multiplying the bottom right number by the top left number: 2(20)

Multiplying the bottom left number by the top right number:10(5)

Multiplying the bottom left number by the top right number:10(20)

200

+_____

300

2. When dealing with two or more digit integers, we can also multiply them by:

72�x 28

Breaking the numbers into their one and tens digits like so:

72 = 70 + 2

28 = 20 + 8

Making a box, and putting those numbers in the box:

X	70	2
20
8

Now we multiply each number on the left of the box, by the number on the top:

20 x 70 = 1400

1400

560

Lastly, we add all the numbers together:

1400

560

+ 16

2016

3. When dealing with two or more digit integers, we can also multiply them by:

x 27

Multiplying the bottom right number by the top right number, then carrying the tens digit (like adding): 7(3) = 21 (but we carry the 2)

Multiplying the bottom right number by the top left number: 7(4) then adding the left over 2: (28 + 2) = 30

70 x 8 = 560

20 x 2 = 40

8 x 2 = 16

Now, place a 0 in the ones place, and then multiply the bottom left number by the top right number: 2(3) = 6

Now, multiply the bottom left number by the top left number: 2(4) = 8

And finally, add everything together

+_____

1161

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Multiplying positive integers

So now that we know how to multiply, let’s look at multiplying positive numbers together to see what we get:

So, when we multiply two positive numbers like so:

12�x 7

What we get should be another positive number, since the two numbers are positive:

But why?

Why is it, that when we multiply two positive numbers together, we get another positive number?

Well, it goes back to the definition of multiplication.

Remember?

Multiplication is nothing more than adding the same number multiple times.

So, our multiplication problem is an easier way of writing:

And we know, that when we’re adding positive numbers together continuously,

we’re going to get another positive number.

Also, for those who are wondering, our equation doesn’t necessarily have to be just 7 added 12 times.

We could write it as 12 added 7 times:

And we would still get the same answer:

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Multiplying negative integers

So then, what would multiplying a negative integer with a positive integer look like?

Well, this is a little more complicated, but let’s go back to the definition of multiplication with a new problem:

(-13)� x (-9)

Now, we know that multiplication is adding the same number multiple times.

However, in this equation, we have two negative numbers.

So what does that tell us?

What it actually tells us, is that we are to subtract the same number multiple times.

And, of course, we know how to do that right?

Even though this is ugly, we know how to change this right?

We are subtracting, so we need to add its opposite.

So our subtraction problem will become:

So, I’m sure you’re wondering why the first -9 became a positive.

The reason why is because we are subtracting all of the -9’s from themselves.

So, in this rare instance, the first -9 is included and changes sign.

And again, we don’t have to subtract (-9) 13 times, we can also subtract (-13) nine times:

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Multiplying positive and negative integers

So now that we know how to multiply positive with positive integers together, and negative with negative integers together, what about positive and negative integers?

So, let’s start with an example and see what happens:

19� x (-3)

So, remember, multiplication is adding the same number multiple times

Unless we have a negative.

Then it’s subtracting the same number multiple times.

So in this case, written out, we would have:

But again, we know how to do that!

We just did it!

When we subtract, we have to make sure to add its opposite:

Again, I know you may be wondering why the first 19 is negative,

But this is a rare instance since we need to subtract all of the 19’s together.

So the first 19 would become a negative.

And yet again, we can see that if we switch this, and instead add (-3) 19 times,

(since 19 is positive, so we can add instead of subtract)

That we will get:

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So, what’s the general rule then?

Basically, when you are multiplying (and also dividing):

A positive times a positive will always give you a positive
A negative times a negative will always give you a positive
A positive times a negative will always give you a negative

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So what is a fraction?

So before we even begin looking at adding fractions, let’s look at what a fraction actually is.

So when you see a fraction, what mathematical procedure comes to mind?

Division

Whenever you see a fraction think division.

So why did you have such a hard time dealing with fractions?

Because you’ve been doing two procedures at once, that’s why.

So, we will go over fractions, but

Give yourself a break

We’ll get through this.

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Adding Fractions

So let’s look at what a fraction is:

A fraction of something is only a portion of that thing.

Or in other words, like said before, a division of a thing.

So how do we add fractions?

Well, we have to change the fractions some.

So, let’s go over that.

First, let’s look at something.

Let’s say we have something like this:

Now, what do we know that this reduces to?

But, the question becomes, did we change anything?

I mean, if we have 3 third slices of pizza, do we still have a full pizza?

So then, we can see that 3/3 is actually 1 in disguise.

Why bring this up?

Because what is the one number we can multiply any number by, and not change anything?

That’s right, 1.

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Finding the Least Common Multiple

So here’s how we add fractions.

Let’s start off with an example:

We can’t just add these right?

We have no idea how to add them.

However, what if we changed them to something we can add?

But, we can’t change the equation right?

So, how do we change the numbers, without changing the amount?

Well, we can try to multiply the fractions by 1, but what kind of one?

This is where the Least Common Multiple comes in.

So, to start, let’s see which multiple the two denominators share.

To find it, let’s list out the multiples of 2:

2 4 6 8 10 12 14 16 18 20

Now, let’s list out the multiples of 3:

3 6 9 12 15 18 21 24 27 30

Now, which number do they both share?

Now we have to find out how to make 3 into 6.

We multiply it by 2!

Now we have to find out how to make 2 into 6.

We multiply it by 3!

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THE CHEATING WAY

So, there is actually another way of creating a common denominator instead of finding the least common multiple

But, it’s not the best way, however, it will work every single time.

We multiply by opposite denominators.

Sounds weird, but let’s try it:

So, what we do, is look at:

( )

= 35

Now, we multiply each side by the opposite number to get 35:

Again, this works every time, but it can take more work.

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That’s all there is to it!

That’s how we add fractions!

We need to find the least common multiple, figure out what to multiply the fractions by, then add them.

This is also how we subtract fractions.

So, let’s look at a few more examples:

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EXAMPLE 1:

Let’s say we have:

Again, we can’t just add these right?

We need to find the least common multiple first!

So, to start, let’s see which multiple the two denominators share.

To find it, let’s list out the multiples of 5:

5 10 15 20 25 30 35 40 45

Now, let’s list out the multiples of 4:

4 8 12 16 20 24 28 32 36 40

Now, which number do they both share?

Now we have to find out how to make 5 into 20.

We multiply it by 4!

Now we have to find out how to make 4 into 20.

We multiply it by 5!

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EXAMPLE 2:

Let’s say we have:

Again, we can’t just add these right?

We need to find the least common multiple first!

So, to start, let’s see which multiple the two denominators share.

To find it, let’s list out the multiples of 7:

7 14 21 28 35 42 49 56 63

Now, let’s list out the multiples of 3:

3 6 9 12 15 18 21 24 27 30

Now, which number do they both share?

Now we have to find out how to make 7 into 21.

We multiply it by 3!

Now we have to find out how to make 3 into 21.

We multiply it by 7!

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EXAMPLE 3:

Let’s say we have:

Again, we can’t just add these right?

We need to find the least common multiple first!

So, to start, let’s see which multiple the two denominators share.

To find it, let’s list out the multiples of 11:

11 22 33 44 55 66 77 88 99

Now, let’s list out the multiples of 3:

3 6 9 12 15 18 21 24 27 30 33

Now, which number do they both share?

Now we have to find out how to make 11 into 33.

We multiply it by 3!

Now we have to find out how to make 3 into 33.

We multiply it by 11!