You Gotta Have Rules

This is math, so you are getting used to the idea of rules. Math is about using logic, rules, and organization so that you can get the same answer every time you do a problem. If the rules changed, 1 + 1 would not always equal 2. That would get too confusing.

If you have studied science, you know about laws. There is the law of gravity and there are the laws of motion in physics. Math also has laws. They set up rules that always allow you to do certain things. You have already used three of those laws in the addition pages. Do you remember how you were able to rearrange numbers or group numbers when we were adding? We can do that because there are laws that say it is okay and the math will continue to work properly. You get a lot of the same rules in multiplication.

Rearrange the Numbers

When you rearrange numbers, you are using the Commutative Law of Multiplication. The word commutation may seem big, but it just means rearrange. If you want to, you could call it the Rearrangement Law of Multiplication. The law lets us move all of the factors around in any multiplication problem.

1 * 95 * 1,345 * 2 * 15 * 7 = ?
• This layout is a little weird to look at.
1 * 2 * 7 * 15 * 95 * 1,345 = ?
• When it is rearranged, it's a little easier to imagine multiplying the numbers.

You can't just rearrange anything. You still need to pay attention to parentheses and other operations. But for multiplication, you can reorder the values all you want. If you were a mathematician, you would use letters called variables to write out the idea. In math, we use letters to represent any number we want. You'll learn more about variables in algebra, but know that "a" and "b" can be any number you imagine. The official description of the commutative law is...

a * b = b * a

The order doesn't matter when looking at the multiplication operation.

Grouping the Numbers

Now you know that you can rearrange numbers in multiplication. You can also group them. You have already been doing that in many of your problems.

1 * 5 * 9 * 6 * 5 * 4 =?
• Rearrange the values and group them...
(1*9) * (5*5) * (6*4) = ?

Mathematicians know that grouping is helpful, so they made a law: the Associative Law of Multiplication. The law looks at the way numbers can associate with each other when you multiply. You might have a group of two and a group of three. The law lets you break up the groups and move things around.

(1*2*75) * (3*4*25) = (1*2*3*4) * (75*25) = (1*2) * (3*4) * (75*25)

See how we just shifted those parentheses around? We set up new groups of factors. As before, this only works with the multiplication operation. You need to pay attention to parentheses and other operations like subtraction or division. You also need to notice that we can group any number of factors. The associative law is there to make your life easier.

(75-1+2) * (3-4*25)
• You can't rearrange this or group them in other ways.
• You need to pay attention to the other symbols in the problem.
• Multiplication is special, not every operation in math is associative.

The official way to describe it using variables to represent any numbers would look like this...

(a*b) * c = a * (b*c) = a * b * c

Learn about more multiplication rules in part two.


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