What Is Mental Math?

Mental math means solving calculations mainly in your head rather than relying on a calculator or a written algorithm.

For example:

48 + 22

can be calculated mentally:

48 + 20 = 68

68 + 2 = 70

So:

48 + 22 = 70

Mental math is not simply about calculating quickly. It is about recognizing number relationships and choosing an efficient strategy.

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Why Mental Math Matters

Mental math helps you:

• estimate answers
• check calculator results
• work efficiently
• recognize number patterns
• understand place value
• solve everyday problems
• develop stronger number sense

Mental math is useful when working with:

• money
• time
• measurements
• distances
• quantities
• percentages
• estimates

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There Is Usually More Than One Strategy

Consider:

38 + 27

One person might calculate:

38 + 20 + 7

Another might think:

40 + 27 − 2

Another might split both numbers:

30 + 20 + 8 + 7

All can produce the correct answer:

65

The best strategy is usually the one that is:

• accurate
• efficient
• easy for you to understand

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Addition Strategy 1: Break Apart by Place Value

Numbers can be split into tens, hundreds, and ones.

For example:

46 + 32

Break apart 32:

32 = 30 + 2

Then:

46 + 30 = 76

76 + 2 = 78

Therefore:

46 + 32 = 78

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Addition Strategy 2: Make a Ten

Numbers that combine to make multiples of 10 are especially useful.

Consider:

27 + 8

27 needs:

3

to reach 30.

Break 8 into:

3 + 5

Then:

27 + 3 = 30

30 + 5 = 35

Therefore:

27 + 8 = 35

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Making the Next Hundred

The same idea works with larger numbers.

Consider:

187 + 36

187 needs:

13

to reach 200.

Break 36 into:

13 + 23

Then:

187 + 13 = 200

200 + 23 = 223

Therefore:

187 + 36 = 223

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Addition Strategy 3: Compensation

Compensation means changing a number to make the calculation easier and then correcting for the change.

Consider:

49 + 36

49 is close to 50.

Think:

50 + 36 = 86

But we added one too much.

So:

86 − 1 = 85

Therefore:

49 + 36 = 85

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Addition Strategy 4: Make Friendly Pairs

Consider:

37 + 18 + 13 + 22

Instead of adding from left to right, rearrange:

37 + 13 + 18 + 22

Now:

37 + 13 = 50

and:

18 + 22 = 40

Therefore:

50 + 40 = 90

This uses the commutative and associative properties.

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Addition Strategy 5: Doubles

Knowing doubles can make many calculations easier.

Examples:

6 + 6 = 12

15 + 15 = 30

25 + 25 = 50

50 + 50 = 100

These facts can also help with numbers that are close to doubles.

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Near Doubles

Consider:

26 + 27

We know:

26 + 26 = 52

So:

26 + 27 = 53

Or use:

27 + 27 = 54

then subtract 1:

54 − 1 = 53

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Addition Strategy 6: Left-to-Right Addition

Consider:

346 + 221

Start with the largest place values.

Hundreds:

300 + 200 = 500

Tens:

40 + 20 = 60

Ones:

6 + 1 = 7

Combine:

500 + 60 + 7 = 567

This is often easier mentally than working from right to left.

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Worked Addition Example

Calculate:

298 + 47

298 is close to:

300

Think:

300 + 47 = 347

We added 2 too much.

Therefore:

347 − 2 = 345

So:

298 + 47 = 345

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Subtraction Strategy 1: Break Apart

Consider:

74 − 32

Break 32 into:

30 + 2

Then:

74 − 30 = 44

44 − 2 = 42

Therefore:

74 − 32 = 42

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Subtraction Strategy 2: Count Up

Sometimes subtraction is easier if we think:

How much must I add?

Consider:

83 − 67

Start at 67.

67 → 70:

+3

70 → 80:

+10

80 → 83:

+3

Total:

3 + 10 + 3 = 16

Therefore:

83 − 67 = 16

This strategy is especially useful when the numbers are relatively close together.

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Subtraction Strategy 3: Compensation

Consider:

92 − 39

39 is close to:

40

Calculate:

92 − 40 = 52

But we subtracted one too much.

Add it back:

52 + 1 = 53

Therefore:

92 − 39 = 53

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Subtracting Numbers Close to 100

Consider:

245 − 98

Think:

245 − 100 = 145

Since 98 is 2 less than 100, add 2 back:

145 + 2 = 147

Therefore:

245 − 98 = 147

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Subtraction Strategy 4: Constant Difference

Consider:

83 − 48

Add 2 to both numbers:

85 − 50

The difference does not change.

Now:

85 − 50 = 35

Therefore:

83 − 48 = 35

This works because moving both numbers by the same amount keeps the distance between them unchanged.

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Subtraction Strategy 5: Place Value

Consider:

685 − 243

Subtract hundreds:

685 − 200 = 485

Subtract tens:

485 − 40 = 445

Subtract ones:

445 − 3 = 442

Therefore:

685 − 243 = 442

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Choosing a Subtraction Strategy

For:

1000 − 997

counting up is very efficient:

997 → 1000 = 3

For:

532 − 200

place value is simplest:

332

For:

174 − 99

compensation works well:

174 − 100 + 1 = 75

Different calculations suggest different strategies.

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Multiplication Strategy 1: Use Known Facts

Strong multiplication facts make mental calculations easier.

For example:

7 × 8 = 56

helps us calculate:

70 × 8 = 560

and:

700 × 8 = 5600

Place value extends familiar multiplication facts.

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Multiplication Strategy 2: Break Apart a Factor

Consider:

7 × 23

Break 23 into:

20 + 3

Then:

7 × 20 = 140

7 × 3 = 21

Add:

140 + 21 = 161

Therefore:

7 × 23 = 161

This uses the distributive property.

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Multiplication Strategy 3: Use a Nearby Friendly Number

Consider:

8 × 49

49 is close to 50.

Think:

8 × 50 = 400

But this represents one extra group of 8.

Subtract:

400 − 8 = 392

Therefore:

8 × 49 = 392

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Multiplying by 9

A useful strategy is:

multiply by 10, then subtract one group

For example:

9 × 34

Think:

10 × 34 = 340

Subtract one 34:

340 − 34 = 306

Therefore:

9 × 34 = 306

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Multiplying by 11

For many calculations:

11 × n = 10n + n

For example:

11 × 42

= 420 + 42

= 462

This is another use of the distributive property.

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Multiplication Strategy 4: Double and Halve

Suppose we need:

16 × 25

Double one factor and halve the other:

16 × 25

becomes:

8 × 50

which becomes:

4 × 100

Therefore:

16 × 25 = 400

The product stays the same.

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Another Double-and-Halve Example

Calculate:

14 × 50

Half 14:

7

Double 50:

100

Therefore:

14 × 50 = 7 × 100

= 700

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Multiplication Strategy 5: Multiply by 5

To multiply by 5:

multiply by 10 and divide by 2

For example:

48 × 5

First:

48 × 10 = 480

Then:

480 ÷ 2 = 240

Therefore:

48 × 5 = 240

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Multiplication Strategy 6: Multiply by 25

Because:

25 = 100 ÷ 4

we can often multiply by 100 and then divide by 4.

For example:

36 × 25

First:

36 × 100 = 3600

Then:

3600 ÷ 4 = 900

Therefore:

36 × 25 = 900

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Multiplication Strategy 7: Multiply by 50

Because:

50 = 100 ÷ 2

consider:

26 × 50

Calculate:

26 × 100 = 2600

Then divide by 2:

2600 ÷ 2 = 1300

Therefore:

26 × 50 = 1300

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Multiplication Strategy 8: Factor and Regroup

Consider:

25 × 12

Break:

12 = 4 × 3

Then:

25 × 4 × 3

Group:

(25 × 4) × 3

= 100 × 3

= 300

Recognizing friendly factor combinations makes calculations easier.

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Division Strategy 1: Think Multiplication

Division and multiplication are inverse operations.

For:

56 ÷ 8

ask:

8 × what = 56?

Since:

8 × 7 = 56

then:

56 ÷ 8 = 7

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Division Strategy 2: Break Apart the Dividend

Consider:

84 ÷ 4

Break 84 into convenient parts:

80 + 4

Then:

80 ÷ 4 = 20

4 ÷ 4 = 1

Therefore:

84 ÷ 4 = 21

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Another Division Example

Calculate:

156 ÷ 3

Break 156 into:

150 + 6

Then:

150 ÷ 3 = 50

6 ÷ 3 = 2

Therefore:

156 ÷ 3 = 52

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Division Strategy 3: Use Compatible Numbers

Compatible numbers are numbers that divide easily.

For example:

240 ÷ 8

We know:

24 ÷ 8 = 3

Therefore:

240 ÷ 8 = 30

Similarly:

420 ÷ 7

Since:

42 ÷ 7 = 6

then:

420 ÷ 7 = 60

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Division Strategy 4: Halving

Division by powers of 2 can often be done through repeated halving.

For example:

320 ÷ 8

Since:

8 = 2 × 2 × 2

halve three times:

320 ÷ 2 = 160

160 ÷ 2 = 80

80 ÷ 2 = 40

Therefore:

320 ÷ 8 = 40

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Dividing by 4

To divide by 4, halve twice.

For example:

196 ÷ 4

First halve:

196 ÷ 2 = 98

Halve again:

98 ÷ 2 = 49

Therefore:

196 ÷ 4 = 49

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Dividing by 5

One useful strategy for dividing by 5 is:

multiply by 2, then divide by 10

For example:

135 ÷ 5

Double:

135 × 2 = 270

Divide by 10:

270 ÷ 10 = 27

Therefore:

135 ÷ 5 = 27

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Dividing by 25

Because:

25 × 4 = 100

we can multiply by 4 and divide by 100.

For example:

700 ÷ 25

Multiply:

700 × 4 = 2800

Then:

2800 ÷ 100 = 28

Therefore:

700 ÷ 25 = 28

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Division Strategy 5: Simplify Both Numbers

Consider:

360 ÷ 40

Both numbers can be divided by 10:

36 ÷ 4

Now:

36 ÷ 4 = 9

Therefore:

360 ÷ 40 = 9

This is especially useful when both numbers end in zeros.

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Worked Example 1: Addition

Calculate mentally:

398 + 57

Round 398 to 400:

400 + 57 = 457

Correct for the extra 2:

457 − 2 = 455

Therefore:

398 + 57 = 455

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Worked Example 2: Subtraction

Calculate:

503 − 198

Think:

503 − 200 = 303

Add back 2:

303 + 2 = 305

Therefore:

503 − 198 = 305

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Worked Example 3: Multiplication

Calculate:

12 × 48

Think:

48 = 50 − 2

Then:

12 × 50 = 600

12 × 2 = 24

Therefore:

600 − 24 = 576

So:

12 × 48 = 576

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Worked Example 4: Division

Calculate:

144 ÷ 6

Break 144 into:

120 + 24

Then:

120 ÷ 6 = 20

24 ÷ 6 = 4

Therefore:

144 ÷ 6 = 24

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Worked Example 5: Choosing a Strategy

Calculate:

199 + 68

Because 199 is very close to 200, compensation is efficient.

200 + 68 = 268

Subtract the extra 1:

268 − 1 = 267

Therefore:

199 + 68 = 267

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Worked Example 6: Friendly Multiplication

Calculate:

24 × 25

Use:

25 × 4 = 100

Since:

24 = 6 × 4

we can write:

24 × 25

= 6 × 4 × 25

= 6 × 100

= 600

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Worked Example 7: Count Up

Calculate:

502 − 487

487 → 500:

+13

500 → 502:

+2

Total:

13 + 2 = 15

Therefore:

502 − 487 = 15

Counting up is much easier here than performing a long subtraction.

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Worked Example 8: Rearrange and Group

Calculate:

18 + 36 + 22 + 14

Rearrange:

18 + 22 + 36 + 14

Group:

40 + 50

Therefore:

90

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Worked Example 9: Double and Halve

Calculate:

32 × 125

Half 32 and double 125:

16 × 250

Again:

8 × 500

Again:

4 × 1000

Therefore:

32 × 125 = 4000

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Worked Example 10: Division Using Known Facts

Calculate:

630 ÷ 9

We know:

63 ÷ 9 = 7

Therefore:

630 ÷ 9 = 70

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Estimation as a Mental Math Strategy

Sometimes an exact answer is not necessary.

Suppose a store has:

49 boxes

with:

21 items per box

For an estimate:

49 ≈ 50

21 ≈ 20

Then:

50 × 20 = 1000

So there are approximately:

1000 items

The exact answer is:

49 × 21 = 1029

Our estimate was reasonably close.

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Estimation as a Check

Suppose a calculation gives:

198 × 31 = 6138

Estimate:

200 × 30 = 6000

The calculated answer:

6138

is close to:

6000

so it appears reasonable.

If the calculator had displayed:

61,380

the estimate would immediately suggest that something had gone wrong.

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Mental Math with Money

Mental math is especially useful when shopping.

Suppose an item costs:

$19

and you buy:

3

Estimate:

$20 × 3 = $60

Exact calculation:

$19 × 3

= $20 × 3 − $1 × 3

= $60 − $3

= $57

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Mental Math with Change

Suppose an item costs:

$17.60

and you pay:

$20

Count up:

$17.60 → $18.00 = $0.40

$18.00 → $20.00 = $2.00

Total change:

$2.40

Counting up is often easier than direct subtraction when calculating change.

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Mental Math with Time

Suppose a lesson begins at:

10:45

and lasts:

50 minutes

Think:

10:45 + 15 minutes = 11:00

There are:

35 minutes

remaining.

11:00 + 35 minutes = 11:35

Therefore, the lesson ends at:

11:35

Breaking calculations around convenient points is useful for time as well as numbers.

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Mental Math with Measurements

Suppose a runner completes:

4 laps

of:

400 m

Total distance:

4 × 400 = 1600 m

Since:

1000 m = 1 km

the runner completes:

1.6 km

Mental arithmetic helps us quickly interpret measurements.

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Using Number Properties

Mental math often depends on the number properties from the previous topic.

Commutative property

allows numbers to be rearranged.

Associative property

allows numbers to be regrouped.

Distributive property

allows numbers to be broken apart.

These properties explain why many mental-math strategies work.

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How to Choose an Efficient Strategy

Before calculating, look at the numbers.

Ask:

Are any numbers close to 10, 100, or 1000?

Use:

compensation

Can I make a friendly pair?

Rearrange and group.

Can I break a number into easier parts?

Use:

place value or distribution

Are the subtraction numbers close together?

Try:

counting up

Can one multiplication factor be doubled while the other is halved?

Try:

double and halve

Does the division connect to a known multiplication fact?

Think:

multiplication

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Strategy Comparison

Consider:

48 × 25

Several methods are possible.

Method 1: Distribution

25(40 + 8)

= 1000 + 200

= 1200

Method 2: Multiply by 100 and divide by 4

48 × 100 = 4800

4800 ÷ 4 = 1200

Method 3: Double and halve

48 × 25

= 24 × 50

= 12 × 100

= 1200

All are correct.

The important skill is selecting a strategy that is efficient for the numbers involved.

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Flexible Thinking

Strong mental mathematicians do not use exactly the same method for every problem.

For:

99 + 47

compensation is useful.

For:

32 + 68

making 100 is useful.

For:

75 × 4

known facts are useful.

For:

16 × 25

double and halve is useful.

For:

1002 − 997

counting up is useful.

For:

480 ÷ 8

known multiplication facts or partitioning are useful.

Mental math is about flexibility.

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When Written Methods Are Better

Mental math is not always the best choice.

A written method may be better when:

• numbers are very large
• calculations contain many steps
• exact records are required
• numbers are awkward
• there is a high risk of forgetting intermediate values

A calculator may be appropriate when:

• calculations are extremely complex
• high precision is required
• many repetitive calculations are needed

The goal is not to avoid calculators.

The goal is to know when mental mathematics is more efficient.

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Common Mistakes

Mistake 1: Trying to use the same strategy for every calculation

Different numbers suggest different strategies.

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Mistake 2: Rounding but forgetting to compensate

For:

49 + 28

if you replace 49 with 50, remember to subtract:

1

afterward.

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Mistake 3: Breaking apart subtraction incorrectly

For:

82 − 37

you can calculate:

82 − 30 − 7

but not:

82 − 30 + 7

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Mistake 4: Losing place value

For:

60 × 7

the answer is:

420

not:

42

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Mistake 5: Assuming mental math must be extremely fast

Accuracy and understanding matter more than speed.

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Mistake 6: Not checking reasonableness

Estimate before or after calculating.

An answer that is far from the estimate should be checked.

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Did You Know?

People have used mental calculation strategies for thousands of years, long before modern calculators existed.

Even today, mental math remains important because it develops number sense.

Number sense means understanding:

• how numbers relate
• how large numbers are
• how operations affect numbers
• which calculations are reasonable
• how numbers can be rearranged or decomposed

This is why mental mathematics is useful even when calculators are readily available.

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Key Terms

• Mental math: Performing calculations mainly without written algorithms or calculators.
• Number sense: Understanding numbers and relationships between them.
• Strategy: Planned method used to solve a problem.
• Place value: Value of a digit based on its position.
• Partition: Break a number into useful parts.
• Compensation: Adjust a number to simplify a calculation, then correct the adjustment.
• Friendly number: Number that makes a calculation easier, such as 10, 50, 100, or 1000.
• Compatible numbers: Numbers that work together easily in a calculation.
• Double: Multiply by 2.
• Halve: Divide by 2.
• Estimate: Approximate value used to judge the size of an answer.
• Inverse operations: Operations that undo each other, such as multiplication and division.
• Commutative property: Property allowing the order of numbers to change in addition or multiplication.
• Associative property: Property allowing numbers to be regrouped in addition or multiplication.
• Distributive property: Property allowing multiplication to be distributed across addition or subtraction.

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Key Strategies

Addition

• break apart by place value
• make 10, 100, or another friendly number
• compensation
• doubles and near doubles
• rearrange and group
• add from left to right

Subtraction

• break apart
• count up
• compensation
• constant difference
• subtract by place value

Multiplication

• use known facts
• break apart factors
• use the distributive property
• use nearby friendly numbers
• double and halve
• use shortcuts for ×5, ×25, ×50, ×9, and ×11
• factor and regroup

Division

• think multiplication
• break apart the dividend
• use compatible numbers
• use repeated halving
• simplify both numbers
• use useful relationships for ÷5 and ÷25

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Key Takeaways

• Mental math involves finding efficient ways to calculate without relying immediately on written algorithms or calculators.
• Good mental math depends on number sense, not just speed.
• Numbers can often be broken apart using place value.
• Addition can be simplified by making friendly numbers such as 10, 100, or 1000.
• Compensation is useful when numbers are close to convenient values.
• Doubles and near doubles can simplify addition.
• Counting up is often efficient for subtraction when two numbers are close.
• Constant difference can transform a subtraction problem into an easier equivalent calculation.
• Multiplication can be simplified by breaking factors apart using the distributive property.
• Doubling one factor and halving another can preserve a product while making it easier to calculate.
• Multiplication by 5, 25, and 50 can often be connected to multiplication by 10 or 100.
• Division can often be solved by thinking about related multiplication facts.
• Repeated halving is useful when dividing by 4, 8, and other powers of 2.
• Estimation is useful both for approximate answers and for checking exact calculations.
• Commutative, associative, and distributive properties explain why many mental-math strategies work.
• Everyday applications include shopping, calculating change, working with time, estimating quantities, and converting measurements.
• There is rarely only one correct mental strategy.
• An important mathematical skill is recognizing the structure of a calculation and choosing the most efficient strategy for the numbers involved.