Mathematical thinking probably started with addition. Someone may have combined two piles of bricks, and wondered how many were in the single large pile. Addition is the mathematical term that describes “joining quantities together”.
Properties of Addition
Looking at the patterns that can occur when quantities are joined together, you might have noticed that it does not matter whether 2 bricks are added to a pile of 3, or 3 bricks are added to a pile of 2… either way, we still end up with a pile of 5. So, the order in which we add two quantities does not change the result. This probably makes intuitive sense to you when you visualize the situation above and the final pile that results.
After playing around with this a bit, you will see that any number of starting quantities can be re-arranged in any way you wish without changing the result of adding them all together:
This property is called the commutative property of addition. The quantities being joined together can “commute” from one side of the plus sign to the other (switch places) without changing the result. When faced with two piles of bricks to combine, it does not matter which pile you consider your starting point… you will still end up with the same size pile once both have been combined.
But what if you are combining more than two piles of bricks? As shown in the example above, not only does it not matter which pile you consider your starting point, but it also does not matter which pile you combine with it next… you will always end up with a final pile of the same size, no matter what sequence you combine the piles in, once all the piles have been joined together.
Using algebraic notation to describe this situation produces a much shorter description:
The parentheses indicate that the steps inside them should be carried out first. The property illustrated above is called the associative property of addition. When a situation involves only addition, it does not matter which two numbers you add first – you will always get the same end result once all numbers have been included in the sum. You may choose which numbers to “associate” with one another first, without affecting the final result.
Order of Operations Revisited
The associative property of addition shows us that addition is more flexible than indicated by the “order of operations”. If a problem involves only addition, you may combine the quantities in any order you wish… you do not have to combine them “left to right” when only addition is involved. This can be very useful when adding numbers in your head. For example, if you were asked to add:
It is often easier to perform the above addition in your head if you first use the associative property of addition to add 8+2, and then 6+4 to arrive at a total of 20 (so far). Now, invite the commutative property of addition to the party, and move the final 7 next to the first 3 to produce another sum of 10, for a grand total of three tens, or thirty.
Earlier posts in this series:
Algebra Intro 1: Numbers and Variables
Posts that continue this series:
Algebra Intro 3: Subtraction
Algebra Intro 4: Negative Numbers, Zero, Absolute Values, and Opposites
Algebra Intro 5: Addition, Subtraction, and Terms
Algebra Intro 6: Multiplication
Algebra Intro 7: Properties of Multiplication
Algebra Intro 8: Division
Algebra Intro 9: Fractions, Reciprocals, and Properties of Division
Algebra Intro 10: Fractions and Multiplication
Algebra Intro 11: Dividing Fractions, Equivalent Fractions
Algebra Intro 12: Adding and Subtracting Fractions