1.08 Number operation properties

One of the hardest things to do in mathematics is to not use your calculator. It's just so much easier than working things out in your head! But what if you don't have a calculator or simply aren't allowed one? Is it even possible to solve something like $24\times13+78\times13$24×13+78×13 in your head?

Yes. By using mental arithmetic strategies we can make questions easier by changing the way we approach them.

But we don't have the tools to solve something like this yet. Let's start with some simple mental strategies to build up our tool kit.

The associative law

Try calculating $29+38+12$29+38+12 in your head. Which two numbers did you add together first?
If you added the $38$38 and $12$12 together first, you were using the associative law!
The associative law lets us evaluate the operations in any order, so long as the operations are all addition or all multiplication. Mathematically, this looks like we are adding or moving brackets to change which pair of numbers we add or multiply first.

The Associative Law for Addition

The commutative law

Which of the following are true?

If you answered a) and c), you're correct!

These two show applications of the commutative law which lets us swap the numbers on either side of the operation. Notice that since only a) and c) are true, the commutative law can be used for addition and multiplication, but not subtraction and division.

The Commutative Law for Multiplication

Reordering

This mental strategy is a similar to the commutative law but can be used when we have more than one of the same operation.
When calculating $13+18+7$13+18+7 which number would it be easier to add first, $18$18 or $7$7? Unless you are a calculator it is usually easier to add $7$7 from $13$13 first and then add $18$18 to their sum. This is an example of using reordering.

Reordering for Addition

Reordering lets us rearrange the numbers in the expression to make the calculations easier when solving from left to right. The rules for using reordering are that all the operations must be same and the first number cannot be moved (notice that $13$13 stays as the first number in the expression), unless the operation is also commutative (i.e. if it is either addition or multiplication).

The distributive law

The distributive law applies whenever there is multiplication outside some brackets and addition inside the brackets, such as the expression $3\times\left(5+2\right)$3×(5+2). We apply the multiplication to each term inside the brackets individually, and add the results together:

The Distributive Law for Multiplication

This law will be very useful when we study algebra, but for now it is still a clever strategy for mental arithmetic.

Is there an easy way to solve $102\times13$102×13?
There is! The trick is find an "easy to multiply by" number close to $102$102. In this case we can use $100$100. So instead of trying to calculate $102\times13$102×13 we can instead calculate $\left(100+2\right)\times13$(100+2)×13. We can then use the distributive law to expand the brackets to get $\left(100\times13\right)+\left(2\times13\right)$(100×13)+(2×13), which is much easier to multiply and add.

The distributive law can be used whenever there are brackets with addition (or subtraction) inside, and multiplication on the outside (on either side). It can also be used when the brackets are being divided from the right side.

For example:

We can show the distributive law visually as well through area. We can either break up the total area into two simpler areas or subtract some excess area from an approximate total.

Finding Areas using the Distributive Law

We can use a similar trick to make division questions like $168\div14$168÷​14 easier. What is a number close to $168$168 that is "easy to divide by $14$14"? One way we can break up $168$168 is into $140$140 and $28$28, splitting a difficult to divide number into two easy to divide numbers.

The Distributive Law for Division

Now that we're a bit more familiar with these strategies, let's see them in action.

Worked example

Remember the question from the start of this lesson?
Let's solve $24\times13+78\times13$24×13+78×13.

Think: It will be difficult to solve both these multiplications individually so instead we need to find an easier way to solve the problem. Notice that both expressions involve multiplication by $13$13. Does this remind you of any of the laws? Yes, this is the result of the distributive law!

Do: If we can use the distributive law to expand brackets, we can also use it to bring them back. This is called 'factorising' and will give us:

$24\times13+78\times13=(24+78)\times13$24×13+78×13=(24+78)×13

But why did we do this? We did this because $24$24 and $78$78 are not easy numbers to multiply by, so instead let's find some nicer numbers to work with. We can first add $24$24 and $78$78 together:

$(102)\times13$(102)×13

How can we break up $102$102 into some easy to multiply numbers?
Let's break up $102$102 into $100$100 and $2$2 since they are very easy to multiply by. This will give us:

$(100+2)\times13$(100+2)×13

Now we can use the distributive law to expand the brackets into easy multiplication:

$(100\times13)+(2\times13)$(100×13)+(2×13)

Then we can evaluate the multiplication:

$(1300)+(26)$(1300)+(26)

And finally we can evaluate the addition to get the final solution:

$1326$1326

Reflect: Every step involves writing the expression in a way that is easier to solve mentally by using one of the mental arithmetic strategies. It is worth noting that a lot of the steps in the working out do not need to be written down and can be done in your head. With enough practice we can sometimes skip all the way from the second step to the solution without writing down anything; it's not called "mental arithmetic" for nothing!
 

As we can see, mental arithmetic strategies can make difficult questions much easier by applying a few clever techniques. While it is true that this leads to more steps of working than what a calculator would need, the advantage of these techniques is that there are no difficult calculations and we can use them wherever we are, whenever we want, even if there isn't a calculator in sight.

Practice questions

Question 1

Question 2

Question 3

Question 4