 # Comparing Operations

You've been introduced to basic arithmetic operations such as addition, subtraction, multiplication, and division. We have full sections on how to use those operations. We just wanted to take a moment to talk about how those operations are related. Addition and subtraction are very close when you look at their concepts. Let's start by imagining a pile of nuts on a table. When you take a few away, you are subtracting. When you put a few in the pile, you are adding. Some out and some in. Multiplication is a fast way to do a lot of addition. We're going to talk about operations in more detail and you need to watch for patterns when we go through the examples.

We've talked about this idea a few times on previous pages. Starting with addition and subtraction, we find two opposite ideas. Addition puts things together and subtraction takes things away. Word problems might have you putting nuts in a bucket just as quickly as they will have you take them out of that bucket. Those actions are addition and subtraction. Every person (EVERY person) in the world uses addition and subtraction every day. It might not always be super-tough addition or subtraction, but the operations are there.

Example:
7 = 5 + 2 (Addition)
7 - 5 = 2 (Subtraction)

Addition and multiplication are related because they put things together. Multiplication is like a shortcut for long addition problems. Who wants to write out 3+3+3+3+3+3 when it is so much faster to write 6x3? Multiplying saves time when you write it out. It also gives you the chance to think about groups. When we wrote it the long way, we had six groups of three. If you convert that into math-speak you get 6 groups of 3 equals 6x3. By the way, the answer is 18.

Multiplication is about putting two or more groups together to make a larger group (the product). If we write it in words, addition might take one group of six, add it to another group of six, add that to another group of six, and finish off with another group of six (6+6+6+6). Multiplication takes the shortcut by saying, "Make an amount equal to four groups of six." They both get the same answer, but multiplication is much faster.

Example:
2 + 2 + 2 + 2 + 2 = 10 (Addition)
2 x 5 = 10 (Multiplication)

# Multiplication and Division

Multiplication and division are closely related to each other. You know that multiplication puts groups of objects together. How many stones do you have if there are five (5) groups of seven (7) stones? You get thirty-five (35) stones for an answer. Division is like the reverse action of multiplication. It takes a larger group and breaks it down into smaller groups. Using the same values, a division problem would start with thirty-five (35) stones and break them into smaller groups of seven (7). How many smaller groups would there be? That division problem would give you an answer of five (5). You should see a pattern when you look at the details of the example. It's almost as if we just switched the symbols around.

Example:
5 x 7 = 35 (Multiplication)
35 ÷ 7 = 5 (Division)

Write it a different way...
5 x 7 = 35
5 = 35/7
7 = 35/5

See how the last two examples take one number and switch it to the other side of the equals sign? We divided both sides by seven for the first example and both sides by five in the second example. We'll work out many more problems in the division section.

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