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 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.
Try calculating 29+38+12 in your head. Which two numbers did you add together first? If you added the 38 and 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.
This law doesn't apply to subtraction or division.
For example: 22-7-5 \neq 22-(7-5) and 24 \div 6 \div 2 \neq 24 \div (6 \div 2).
Consider 27+48+13.
Which pair of numbers will be easiest to add together first?
Find the missing value of the expression to complete the working out.
\displaystyle 27+48+13 | \displaystyle = | \displaystyle 27+⬚+⬚ | Apply the associative law |
\displaystyle = | \displaystyle ⬚+⬚ | Evaluate the first addition | |
\displaystyle = | \displaystyle ⬚ |
The associative law evaluates the operations in any order, if the operations are both addition or multiplication.
For example: 4+5+6=4+6+5.
Which of the following are true?
a) \,4+3=3+4 \quad \quadb) \,4-3=3-4 \quad \quadc) \,4\times 3=3\times 4 \quad \quadd) \,4\div 3=3\div 4
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 is demonstrated in the following applet.
The applet shows that the commutative law of multiplication holds, because even if we swap the numbers on either side of the multiplication the number of squares stays the same.
Consider 4\times13\times5.
Which pair of numbers will be easiest to multiply together first?
Complete the working out below.
\displaystyle 4\times13\times5 | \displaystyle = | \displaystyle 4\times⬚\times⬚ | Apply the commutative law |
\displaystyle = | \displaystyle ⬚\times⬚ | Evaluate the first multiplication | |
\displaystyle = | \displaystyle ⬚ |
The commutative law swaps the numbers on either side of the operation, if the operations are both addition or multiplication.
For example: 7\times8=8\times7.
This mental strategy is a similar to the commutative law but can be used when we have more than one of the same operation.
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 stays as the first number in the expression), unless the operation is also commutative (i.e. if it is either addition or multiplication).
Consider 52-24-12.
Which of the following expressions is equal to 52-24-12?
Which of the arithmetic rules can we apply to 52-24-12 to transform it into 52-12-24?
Reordering rearranges the order of the numbers in an expression.
For example: 13+18+7=13+7+18
The distributive law applies whenever there is multiplication outside some brackets and addition inside the brackets, such as the expression 3 \times (5+2). We apply the multiplication to each term inside the brackets individually, and add the results together:
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 \times 13 There is. The trick is find an "easy to multiply by" number close to 102. In this case we can use 100. So instead of trying to calculate 102 \times 13 we can instead calculate (100+2) \times 13. We can then use the distributive law to expand the brackets to get (100 \times 13) + (2 \times 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:
\begin{aligned} (24+6) \times 2 &= (24 \times 2)+ (6\times2) \\ 2 \times (24+6) &= (2\times 24) + (2 \times 6) \\ (24+6) \div 2 &= (24 \div 2) + (6 \div 2) \end{aligned}
We cannot apply the distributive law to division from the left side.
For example: 24 \div (6+2) \neq (24 \div 6) + (24 \div 2)
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.
We can use a similar trick to make division questions like 168 \div 14 easier. What is a number close to 168 that is "easy to divide by 14"? One way we can break up 168 is into 140 and 28, splitting a difficult to divide number into two easy to divide numbers.
Consider 168\div7.
Complete the statement:
168\div7 is the same as (140+⬚)\div7
Complete the statement:
(140+28)\div7 is the same as 140\div 7 + ⬚\div 7
Which arithmetic rule explains the equality between 168\div7 and 140\div 7+28\div7?
Find the missing value of the expression to complete the working out.
\displaystyle 168\div7 | \displaystyle = | \displaystyle (140+28)\div7 | Separate into two manageable components |
\displaystyle = | \displaystyle 140\div7+28\div7 | Distribute the parentheses using the distributive property | |
\displaystyle = | \displaystyle ⬚+⬚ | Evaluate the division | |
\displaystyle = | \displaystyle ⬚ | Evaluate |
The distributive law applies the operation outside the parentheses to the expression inside it in order to distribute the parentheses.
For example: 4\times(5+3)=(4\times5)+(4\times3).