Mental mathematics means working with numbers in your head using number relationships and useful strategies.
It does not mean trying to perform every written calculation from memory.
Instead, you look at the numbers, choose an efficient approach and check whether your answer makes sense.
Goal: use flexible mental strategies for common addition, subtraction, multiplication and division calculations.
Before you start
Review Practical Number Sense for Everyday Mathematics.
This lesson uses place value, decomposition, friendly numbers, estimation and inverse operations from that article.
You may use paper to record your thinking, but try each calculation mentally before writing a formal method.
Why is mental mathematics useful?
Mental calculation is useful when:
- estimating a shopping total
- checking change
- planning time
- comparing prices
- splitting a quantity equally
- checking a calculator result
- deciding whether an answer is reasonable
You do not always need an exact mental answer.
Sometimes a good estimate is enough to make a decision.
Choose a strategy instead of one fixed rule
Consider:
49 + 32
You could use several strategies.
One is to make 49 into the friendly number 50:
50 + 32 = 82
Then adjust by one:
82 less 1 = 81
So:
49 + 32 = 81
The strategy works because you changed one number temporarily and then corrected for that change.
Strategy 1: Make a ten
Numbers close to a multiple of ten are often easy to work with.
Calculate:
38 + 27
Move 2 from 27 to 38:
40 + 25 = 65
The total remains the same because you transferred 2 from one addend to the other.
Try it
Calculate mentally:
46 + 29
One solution
Think of 29 as 30 less 1.
46 + 30 = 76
Then:
76 less 1 = 75
Strategy 2: Separate tens and ones
Calculate:
54 + 23
Separate the tens:
50 + 20 = 70
Then the ones:
4 + 3 = 7
Combine them:
70 + 7 = 77
Another example
Calculate:
67 + 28
Separate:
60 + 20 = 80
7 + 8 = 15
Then:
80 + 15 = 95
Strategy 3: Add in convenient stages
For:
125 + 48
you might add 40 first:
125 + 40 = 165
Then add 8:
165 + 8 = 173
This can be easier than trying to combine every digit at once.
Subtraction can be thought of as difference
Suppose you want:
100 less 67
You can count upward from 67.
From 67 to 70 is 3.
From 70 to 100 is 30.
So the total difference is:
3 + 30 = 33
Therefore:
100 less 67 = 33
Everyday example: checking change
An item costs 74 units of money and you pay 100.
Think about the distance from 74 to 100.
From 74 to 80 is 6.
From 80 to 100 is 20.
So:
6 + 20 = 26
The change should be 26 units of money.
Strategy 4: Subtract a friendly amount and adjust
Calculate:
83 less 29
Subtract 30 first:
83 less 30 = 53
You subtracted one too much, so add one back:
53 + 1 = 54
Try another subtraction
Calculate:
152 less 48
You can subtract 50:
152 less 50 = 102
Because 48 is two less than 50, add 2:
102 + 2 = 104
Use known multiplication facts
Basic multiplication facts can become building blocks for larger calculations.
If you know:
7 × 8 = 56
you can use that fact in other situations.
For example:
70 × 8 = 560
Place value changes the size of the number, but the basic multiplication relationship is still useful.
Strategy 5: Break multiplication apart
Calculate:
6 × 24
Break 24 into 20 and 4:
6 × 20 = 120
6 × 4 = 24
Then:
120 + 24 = 144
Why this works
The calculation:
6 × 24
can be thought of as:
6 × (20 + 4)
which gives:
(6 × 20) + (6 × 4)
This is an example of using the structure of multiplication.
Strategy 6: Multiply by a friendly number and adjust
Calculate:
9 × 37
Multiplying by 10 is easy:
10 × 37 = 370
But you only need nine groups of 37, so remove one group:
370 less 37 = 333
Therefore:
9 × 37 = 333
Multiply by 5 using 10
Five is half of ten.
So for:
5 × 48
first calculate:
10 × 48 = 480
Then take half:
480 divided by 2 = 240
Therefore:
5 × 48 = 240
Multiply by 25 using quarters
Twenty five is one quarter of 100.
For:
8 × 25
you may already know that four groups of 25 make 100.
Eight groups therefore make:
200
Recognizing familiar number relationships can make calculations much faster.
Division is connected to multiplication
If you know:
8 × 6 = 48
then you also know:
48 divided by 6 = 8
and:
48 divided by 8 = 6
Related multiplication facts can therefore help with mental division.
Strategy 7: Split a division into easier parts
Calculate:
84 divided by 4
Break 84 into 80 and 4.
80 divided by 4 = 20
4 divided by 4 = 1
Combine:
20 + 1 = 21
Another division example
Calculate:
156 divided by 3
Break 156 into 150 and 6.
150 divided by 3 = 50
6 divided by 3 = 2
So:
156 divided by 3 = 52
Halving and doubling
Sometimes you can change a multiplication into an easier equivalent calculation by halving one factor and doubling the other.
Consider:
16 × 25
Double 25 and halve 16:
8 × 50
Do it again:
4 × 100 = 400
Therefore:
16 × 25 = 400
Use estimation before exact calculation
Suppose you need:
198 + 304
Before calculating exactly, estimate:
about 200 + about 300 = about 500
Now calculate:
198 + 304 = 502
The exact answer is close to the estimate.
Estimation can detect mistakes
Suppose a calculator shows:
198 + 304 = 5,020
Your estimate of about 500 immediately tells you that the displayed result is not reasonable.
The problem may be an input error.
Round only as much as you need
Suppose you are estimating:
61 + 38
You might use:
60 + 40 = about 100
That is enough to check the size of the answer.
You do not need unnecessary precision when the purpose is only estimation.
Use inverse operations to check
Suppose you calculate mentally:
67 + 28 = 95
You can check with subtraction:
95 less 28 = 67
This supports the original answer.
Check multiplication with division
Suppose:
7 × 18 = 126
You can check:
126 divided by 7 = 18
Using the inverse operation gives you another way to test your calculation.
Everyday example: estimating a shopping total
You plan to buy items costing:
39, 21 and 58
Before finding the exact total, estimate:
40 + 20 + 60 = about 120
The exact total is:
118
The estimate and exact result are close.
Everyday example: deciding whether a budget is enough
You have a budget of 200 units of money.
Three expected costs are about:
48, 72 and 61
Round to convenient values:
50 + 70 + 60 = about 180
The estimate suggests the costs may fit inside the budget.
You should still calculate the exact total before making a final decision.
Everyday example: sharing equally
A group has 96 items to divide equally among 8 people.
Think:
80 divided by 8 = 10
and:
16 divided by 8 = 2
So each person receives:
12 items
Everyday example: planning time
You have 120 minutes available.
Three tasks are expected to take about 28, 31 and 24 minutes.
Estimate:
30 + 30 + 25 = about 85 minutes
This suggests that 120 minutes should be enough, with some time remaining.
Exact answer or estimate?
Different situations require different levels of accuracy.
An estimate may be enough when:
- checking whether a budget is approximately sufficient
- judging whether a calculator result is sensible
- planning roughly how long several tasks will take
An exact answer may be needed when:
- paying a bill
- giving exact change
- recording an official measurement
- dividing a fixed quantity exactly
Good mathematical judgement includes knowing which kind of answer the situation requires.
Practice 1: Make a friendly number
Calculate mentally:
59 + 27
Suggested solution
Think:
60 + 27 = 87
Then:
87 less 1 = 86
Practice 2: Separate tens and ones
Calculate:
43 + 35
Suggested solution
40 + 30 = 70
3 + 5 = 8
70 + 8 = 78
Practice 3: Use difference
An item costs 63 and you pay 100.
How much change should you receive?
Answer
From 63 to 70 is 7.
From 70 to 100 is 30.
7 + 30 = 37
Practice 4: Adjust a subtraction
Calculate:
91 less 39
Suggested solution
Subtract 40:
91 less 40 = 51
Then add one:
51 + 1 = 52
Practice 5: Break multiplication apart
Calculate:
7 × 23
Suggested solution
7 × 20 = 140
7 × 3 = 21
140 + 21 = 161
Practice 6: Use a related multiplication fact
If:
9 × 7 = 63
what is:
63 divided by 7?
Answer
9
Practice 7: Split a division
Calculate:
92 divided by 4
Suggested solution
80 divided by 4 = 20
12 divided by 4 = 3
So:
92 divided by 4 = 23
Practice 8: Estimate first
Estimate:
297 + 405
One estimate
300 + 400 = about 700
The exact answer should therefore be close to 700.
Practice 9: Spot the unreasonable result
A calculator displays:
48 × 6 = 2,880
Why should you question the answer?
Answer
Fifty groups of 6 would equal only about 300.
So 2,880 is far too large.
A reasonable exact answer should be close to 300.
Practice 10: Choose the better method
You want to know whether four items costing about 24 each will fit within a budget of 100.
Which is most useful first?
- Estimate mentally
- Use a long written calculation immediately
Answer
Estimate mentally.
Think of each price as about 25:
4 × 25 = 100
This tells you quickly that the budget is very close to the expected total.
Activity: Build your own mental strategy
For each calculation below, decide which strategy you would use before calculating.
- 48 + 35
- 102 less 49
- 8 × 19
- 144 divided by 6
- 398 + 205
Possible strategies include:
- make a friendly number
- separate place values
- use difference
- break multiplication apart
- use a related multiplication fact
- split division into easier parts
- estimate first
There may be more than one good method.
Do not force mental calculation when another tool is better
Mental mathematics is one useful tool.
A written method, calculator or spreadsheet may be more appropriate for long, repetitive or highly precise calculations.
The important skill is choosing a suitable method and still using number sense to check the result.
Common mental mathematics mistakes
- trying to hold too many steps in memory at once
- adjusting a friendly number but forgetting to adjust the answer
- losing track of place value
- using an operation that does not match the problem
- accepting an unreasonable calculator result
- assuming there is only one correct mental strategy
- using an estimate when an exact answer is required
A mental calculation checklist
Before calculating, ask:
- Do I need an exact answer or only an estimate?
- Can I make one number friendlier?
- Can I separate the numbers using place value?
- Do I know a related multiplication or division fact?
- Can I use halving, doubling or an inverse operation?
- What approximate answer should I expect?
- Does my final answer fit that expectation?
Self check
- Why can 49 + 32 be changed temporarily to 50 + 32?
- How can a multiplication fact help with division?
- Why is estimation useful before using a calculator?
- When might an exact answer be more important than an estimate?
- Why can two people use different mental strategies and still both be correct?
Suggested answers
- You can use the friendly number 50 and then correct for the extra one that was added.
- Multiplication and division are inverse operations, so known multiplication facts can reveal related division facts.
- An estimate gives an expected size for the result and can reveal input or calculation errors.
- Examples include paying an exact bill, giving change or recording a required exact quantity.
- Number relationships can often be used in several valid ways, provided each method preserves the value of the calculation.
Lesson summary
Mental mathematics is flexible reasoning with numbers.
Useful strategies include making friendly numbers, separating place values, using difference, breaking multiplication apart, using known facts, splitting division and estimating before calculating.
The best strategy depends on the numbers and the purpose of the calculation.
Always use number sense to check whether your final answer is reasonable.
Continue learning
Review Practical Number Sense for Everyday Mathematics if you want more practice with estimation and number relationships.
You will use these mental strategies again when working with fractions and percentages.