\part{Addition}

Imagine that I have 4 apples and 3 bananas in a bag; what is the \emph{total} number of fruits altogether? Is there a way to put numbers together to find the total amount of things?

Let's try using counting! We know that I have 4 fruits from the apples, and 3 more fruits from the bananas. If I start at 4 and count 3 numbers up, I can find the total amount of fruits; 5,6,\emph{7}. I have 7 fruits!

This idea is called \emph{addition}, an operation which takes 2 numbers together to calculate what amount the numbers they make together. We calculate this by counting the first lot of objects, and then continuing from that number when counting the second lot of objects.


\begin{definition}
The \emph{addition operation} takes the first number and counts up as much as the second number.
\end{definition}

\section{Counting method}

The simplest way to add two numbers is to count up the second number, starting from the first.

\begin{example}
To solve 9+6, we count 6 up, starting from 9. So counting 6 numbers after 9, we have 10,11,12,13,14,\emph{15}. So 9+6=15!
\end{example}

Another way of thinking of this same idea is with a \emph{number line}.

\section{Basic rules}
- Commutative
\[7+5=5+7\]
- Associative
\[1+(2+3)=(1+2)+3\]


\section{Friends of 10}


These basic rules make addition a little easier, but to really get good at addition, it is useful to remember a few 'addition facts'. The most basic of these is the idea of 'friends of 10'.

This looks at all the single digit numbers that add together to get 10;
1+9=10
2+8=10
3+7=10
4+6=10
5+5=10

10 is a nice number to work with; adding 10 just adds 1 to the tens place.

\begin{example}
To solve 7+5, we remember that 7+3=10; so we know that after counting 3 more than 7 we have 10. We need to count 2 more numbers up to get our answer, which is the same as 10+2, which is 12. So we have calculated that 7+5=12.
\end{example}

\section{Addition algorithm}

Addition problems can get very large; how can we quicly calculate that 9237+2385=11622. This section will use the tricks we've learned to make a method of addition that we can use for any problem.