Commutative and non-commutative principles

Addition is commutative, meaning the order of operations doesn’t matter.

For example, $2+3=5$ and $3+2=5$ . Even if you flip around what you’re adding together, it doesn’t matter.

Multiplication is commutative too (so the order of operations doesn’t matter).

For example, $2*3=6$ and $3*2=6$ . Same principle as with addition.

Although addition and multiplication is commutative, subtraction and division are not, thus named non-commutative.

That is, $2-3=-1$ , whereas $3-2=1$ . Different answers.

Alternatively, $10/5=2$ , whereas $5/10=0.5$ . Different answers once again.

Introduction to order of operations

Numbers being added can be grouped in any order. Grouping occurs by using round brackets (parentheses), which show which operations to do first.

For example, $(1+2)+3=(3)+3=6$ . This matches $1+(2+3)=1+5=6$ .

Similarly, with multiplication, numbers can be grouped in any order as with addition.

For example, $(2*3)*4=(6)*4=24$ . This matches $2*(3*4)=2*(12)=24$ .

However, when you have addition and multiplication mixed together, do the multiplication first.

For example, $1+2*3=1+(2*3)=1+6=7$ .
The answer is NOT $1+2*3=(1+2)*3=(3)*3=9$ <== This is WRONG!

That’s why if you DO want the “1+2” to occur first, put them in brackets, i.e. $(1+2)*3=(3)*3=9$

Therefore, brackets occur before addition. In fact, brackets should be executed before addition, subtraction, multiplication and division, all of them!

The principle of BEDMAS

The principle behind “Order of operations” can be summarized in the acronym BEDMAS. This stands for:

Brackets
Exponent
Division
Multiplication
Addition
Subtraction

Forget about the Exponent for now. Note that (1) Brackets must be done first; then (2) Division and Multiplication; and then (4) Addition and Subtraction.

For example, $3*2+5-4/(2+2)$ . We must first do the Brackets. So we reduce down to $3*2+5-8/(4)$ . Then we do the multiplication and division. This means we can re-bracket what we need to do as $(3*2)+5-(8/4)=6+5-2$ . Now we can do the simple addition/subtraction, which gives us 9.

Order of operations on calculators

Not all calculators correctly execute order of operations.

For example, if given $1+2*3$ , the calculator may do the sum left-to-right, i.e. $(1+2)*3=3*3=9$ . This is WRONG mathematically.

Calculators which always work from left-to-right are called “arithmetic” calculators.

To do such calculations on such calculators, you must enter the operations in the correct order to get the correct answer, i.e. $2*3+1$ .

Better calculators, known as “scientific” or “algebraic” calculators know that multiplication should be done before addition, etc, and so will calculate the correct answer.

The calculator in Windows can be set to both arithmetic calculator (with view “standard”) or a scientific calculator (with view “scientific”).

Difficulty of calculations necessary by hand

By year 7, there are simple calculations you should be quite familiar with, including:

Addition and subtraction: Including when there are several items mixed together, for example $1+2-3=0$
Multiplication and division: Including when there are several items mixed together, for example $5*8/4=10$
Mixed addition/subtraction and multiplication/division: Remember that multiplication/division goes BEFORE addition/subtraction, for example $5*4-4/2=20-2=18$
With brackets: Remember due to “BEDMAS”, that brackets always come first. For example, $(5+4)*(3+2)=(9)*(5)=45$
Word questions: Where English words are used, and you need to first draw an equation out first, and THEN calculate it. This is perhaps the most complicated since it is essentially applied mathematics. For example, if Mandy owns 2 Phineas puppets, and each puppet has 2 jumpers, how many Phineas jumpers does Mandy have? $2*2=4$ .