To my knowledge:

[math]

3 \cdot 5 = 5 + 5 + 5

[/math]

 

However

[math]

3 \, \mathrm{m} \cdot 5 \,\mathrm{m} = 15 \, \mathrm{m}^2 \neq 5 \,\mathrm{m} + 5 \,\mathrm{m} + 5 \,\mathrm{m}

[/math]

 

What is wrong?

Simply that you've not done the multiplication right.

 

[math]

3 \, \mathrm{m} \cdot 5 \,\mathrm{m} = (3\cdot 5) \mathrm{m}\cdot \mathrm{m} = (3\cdot 5) \mathrm{m}^2

[/math]

 

and the rest follows.

I don't think I have made my self clear.

The problem is that if multiplication is repeated addition, how do I add the units to make them square?

Well, crudely speaking:

[math]3m \cdot 5m = (3\cdot 5) (m \cdot m) = \underbrace{(5 + 5 + 5)}_{ 3 \; \mbox{times} } \underbrace{(m + m \dots + m)}_{ m \; \mbox{times} } = 15m^2 [/math]

 

Like Dave said, it's mostly just a case of doing the multiplication right.

Ah yes. Of course. Thank you both.