1.3+Addition

Section 1.3 - Addition
What's in this section?
 * Adding whole numbers
 * Multiple digit addition
 * Some rules for addition

What's __not__ in this section?
 * Adding negative numbers - Section 2.4
 * Adding fractions - Sections 4,7 and 4.8
 * Adding decimal numbers - Section 5.2

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There are actually many ways to describe the concept of adding numbers or adding to a number. the following:
 * Toni's budget was increased by $9,000.
 * Marco gained six yards on the last running play.
 * The stock market rose 2 points today.
 * Lacey added two new clients to her list.

Each of these examples uses a different word to describe the concept of increasing an amount or adding to a number. And yet, none of these examples gives us "an answer" or a total. We still don't know what Toni's budget is, or how many yards Marco has or where the ball is. In each case, we simply know that more has been added to what was previously there.

You might discover a lot of ways used to describe addition, but a few terms are important enough to get more attention.


 * Sum - The answer to an addition problem. ** The sum of six and four is ten.

Positive - Generally used to distinguish the numbers above zero, such as on a numberline. This is opposed to negative numbers, which are below zero.

Plus - Tricky one! Sometimes it just means "add", but other times it is used in place of "positive."

At its most basic sense, addition is like counting. If I count the number of snowmen in the picture, I get 12. But I stand in the middle and group them, say the 7 on my left and the 5 on my right, it becomes 7 + 5, which is still 12. Seen on the numberline, I move to the first number, which is 7. Then, adding 5 more, I find that the answer is 12.
 * Adding Whole Numbers**



Because we have to add large numbers, the numberline doesn't suffice, and counting from one number to the next would take forever. Taking advantage of our place-value system, we have a method for adding numbers of any size that is relatively quick and that doesn't overtax the brain.
 * Adding Multiple Digit Numbers**

The method is to line up the two numbers, one on top of the other, with the place values matching up.

Simply add the digits in the ones column, writing the result below the same column. Then repeat for each place value. If there is a missing digit in that column, consider it a zero, or ignore it.

If the sum of the digits in a column exceeds 9, we use a form of regrouping usually known as "carrying". The right-most digit is written, and the "tens" digit is added to the next column, to be included when that column is added. This may continue until the final column is added.



Here are two Khan Academy videos on addition and carrying. The first introduces the concept, the second is really good practice.

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Multiple numbers can be added at the same time, just by lining up the place values.




 * Commutative Property of Addition**

(Picture of Pizza guy with 5 or 6 pizzas.)

I've got 11 pizzas to deliver to the next house, and that's just a bit much for a single load, even for a muscleman like me. So, it'll take two loads. It doesn't matter if I take five pizzas the first trip and six the next, or six the first time and the last five on the second trip. Either way the total number of pizzas carried is 11. 6 + 5 = 5 + 6

This also works with more than two numbers. Essentially, as long as you are adding everything, it doesn't matter which order to you it in: 3 + 2 + 9 = 9 + 3 + 2


 * Associative Property of Addition**

Try to see the subtle difference between this math rule and the Commutative Property.

If you are adding a bunch of numbers, you can add some of the numbers together first, even if they aren't the first numbers in the list, and then add the rest of the numbers. In the example below, parentheses are used to indicate which numbers are being added first. As a general rule, do the math inside of parentheses first, even if it doesn't come first in the problem.

( 3 + 2 ) + 4 = 9 3 + ( 2 + 4 ) = 9 5 + 4 = 9 3 + 6 = 9

The Associative Property of Addition allows you to group (associate) numbers together and add them first -- even if there are no parentheses, but only if all of the numbers are being added (no subtraction, multiplication, etc.). When is it needed? Clearly it is needed when you are playing Blackjack. The dealer shows 8 4 6 and because your brain screams that 4 + 6 is 10, you naturally add those digits together first before adding the 8, for a total of 18.


 * Identity Property of Addition**

It's obvious, but still a property with a name, so it must be important to someone. If you add zero to a number, the answer is always the number you started with. Adding zero doesn't change the sum. Was it important enough to say two different ways?


 * Application of Addition**

Check out real-life problems involving perimeter (Section 9.1).

**Where to from here?**
1.1 Place Value 1.2 Rounding and Estimating Numbers 1.3 Addition 1.4 Subtraction 1.5 Multiplication 1.6 Division 1.7 Exponents 1.8 Order of Operations 1.9 Prime Numbers
 * Chapter One Review**
 * Chapter One Homework**