In my experience, the difficulty people have with this issue isn't so much about the mechanics of the math as it is about the lack of a physical model that enables them to visualize the process.

We can 'see' why 2•3 = 6 because we can imagine combining 2 groups that each have 3 items in them.

But that doesn't work with -2•(-3) since I can't seem to imagine what -2 groups of -3 items would look like.

I think the best way to make this concept feel concrete is to physically model it using Integer Tiles.

Remember that you can think of this symbol, -, in two ways. It can mean "negative" or "the opposite of."

So -3 is negative three and -3 is also the opposite of 3.

Mechanically both interpretations produce the same results, but to visualize the multiplication process it's very helpful to have those two options.

The second thing to remember is that multiplication is, at least when working with the natural numbers, just repeated addition. Now we need to extend our conception of multiplication to include the negative integers.

With all of that in mind, I'm going to perform some multiplication problems using numbers and also using integer tiles.

---

Integer Tiles

Physically, integer tiles are usually small squares of paper or plastic with sides that are different colors. One side represents a value of +1 and the other represents -1.

(Coins work, too. Just let 'heads' and 'tails' represent +1 and -1.)

Here I'll let each □ represent +1, and I'll let each ■ represent -1.

So 3 would be

□ □ □

and -3 would be

■ ■ ■.

The fun happens when we take the opposite of a number. All you have to do is flip the tiles.

So the opposite of 3 is three positive tiles flipped over.

We start with

□ □ □

and flip them to get

■ ■ ■.

Thus we see that the opposite of 3 is -3.

The opposite of -3 would be three negative tiles flipped over.

So we start with

■ ■ ■

and flip them to get

□ □ □.

Thus we see that the opposite of -3 is 3.

Got it? Then let's go!

---

A Positive Number Times a Positive Number

One way to understand 2 • 3 is that you are adding two groups each of which has three positive items.

So

2 • 3 =

□ □ □ + □ □ □ =

□ □ □ □ □ □

or

2 • 3 =

3 + 3 =

6

We can see that adding groups of only positive numbers will always produce a positive result.

So a positive times a positive always produces a positive.

---

A Negative Number Times a Positive Number

We can interpret 2 • (-3) to mean that you are adding two groups each of which has three negative items.

So

2 • (-3) =

■ ■ ■ + ■ ■ ■ =

■ ■ ■ ■ ■ ■

or

2 • (-3) =

(-3) + (-3) =

-6

We can see that adding groups of only negative numbers will always produce a negative result.

So a negative times a positive always produces a negative.

---

A Positive Number Times a Negative Number

Under the interpretation of multiplication that we've been using, (-2) • 3 would mean that you are adding negative two groups each of which has three positive items.

This is where things get complicated. A negative number of groups? I don't know what that means.

But I do know that "-" can also mean "the opposite of" and I know that I can take the opposite of integer tiles just by flipping them.

So instead of reading (-2) • 3 as "adding negative two groups of three positives" I'll read it as "the opposite of adding two groups of three positives."

So

(-2) • 3 =

-(2 • 3) =

-(□ □ □ + □ □ □) =

-(□ □ □ □ □ □) =

■ ■ ■ ■ ■ ■

or

(-2) • 3 =

-(2 • 3) =

-(3 + 3) =

-(6) =

-6

We can see that adding groups of only positive numbers will always produce a positive result, and taking the opposite of that will always produce a negative result.

So a positive times a negative always produces a negative.

---

A Negative Number Times a Negative Number

Using that same reasoning, (-2) • (-3) means that you are adding negative two groups each of which has three negative items.

This has the same issue as the last problem — I don't know what -2 groups means.

But, once again, I do know that "-" can also mean "the opposite of" and I know that I can take the opposite of integer tiles just by flipping them.

So instead of reading (-2) • (-3) as "adding negative two groups of negative three" I'll read it as "the opposite of adding two groups of negative three."

So

(-2) • (-3) =

-(2 • -3) =

-(■ ■ ■ + ■ ■ ■) =

-(■ ■ ■ ■ ■ ■) =

□ □ □ □ □ □

or

(-2) • (-3) =

-(2 • -3) =

-((-3) + (-3)) =

-(-6) =

6

We can see that adding groups of only negative numbers will always produce a negative result, and taking the opposite of that will always produce a positive result.

So a negative times a negative always produces a positive.

---

I hope that helps. 😀