6

What Is Multiplication?

Multiplication is an operation used to combine equal groups.

For example:

6 × 4 = 24

can mean:

6 groups of 4

or:

4 groups of 6

For larger numbers, the same idea applies.

For example:

23 × 14

means 23 multiplied by 14.

Multi-digit multiplication uses our understanding of:

• place value
• multiplication facts
• the distributive property
• addition
• estimation

Factors and Products

The numbers being multiplied are called factors.

The result is called the product.

For example:

24 × 15 = 360

The factors are:

24 and 15

The product is:

360

 

 

 

5

Place Value in Multiplication

Place value is extremely important when multiplying large numbers.

Consider:

34 × 20

Since:

20 = 2 tens

we can think:

34 × 2 = 68

then multiply by 10:

68 × 10 = 680

Therefore:

34 × 20 = 680

The zero is not simply "added to the answer." It represents the fact that we are multiplying by tens rather than ones.

Multiplying by Powers of Ten

Understanding powers of ten makes multi-digit multiplication easier.

37 × 10 = 370

37 × 100 = 3,700

37 × 1,000 = 37,000

Similarly:

24 × 30

can be thought of as:

24 × 3 × 10

= 72 × 10

= 720

Multiplying a Multi-Digit Number by One Digit

Consider:

243 × 4

Using place value:

243 = 200 + 40 + 3

Multiply each part:

4 × 200 = 800

4 × 40 = 160

4 × 3 = 12

Then add:

800 + 160 + 12 = 972

Therefore:

243 × 4 = 972

 

 

 

4

The Standard Multiplication Algorithm

The standard algorithm is an efficient written method for multiplication.

Consider:

243 × 4

Write:

  243
×   4
-----

Begin with the ones.

4 × 3 = 12

Write 2 and regroup 1 ten.

Then:

4 × 4 tens = 16 tens

plus the regrouped ten:

17 tens

Write 7 and regroup 1 hundred.

Then:

4 × 2 hundreds = 8 hundreds

plus the regrouped hundred:

9 hundreds

So:

  243
×   4
-----
  972

Therefore:

243 × 4 = 972

Why Regrouping Works

The standard algorithm is based on place value.

For:

4 × 243

we really calculated:

4(200 + 40 + 3)

Using the distributive property:

4 × 200 + 4 × 40 + 4 × 3

= 800 + 160 + 12

= 972

The standard algorithm records the same mathematics in a more compact form.

Worked Example 1

Calculate:

586 × 7

Ones:

7 × 6 = 42

Write 2 and regroup 4 tens.

Tens:

7 × 8 = 56

Add the regrouped 4:

56 + 4 = 60

Write 0 and regroup 6 hundreds.

Hundreds:

7 × 5 = 35

Add 6:

35 + 6 = 41

Therefore:

586 × 7 = 4,102

Multiplying Two Multi-Digit Numbers

Now consider:

23 × 14

We can break 14 into:

10 + 4

Therefore:

23 × 14

= 23(10 + 4)

= 23 × 10 + 23 × 4

= 230 + 92

= 322

This idea forms the basis of the standard algorithm for multiplying two multi-digit numbers.

 

 

 

Partial Products

A partial product is the result of multiplying one part of a number by another part.

For:

23 × 14

we have:

23 × 4 = 92

and:

23 × 10 = 230

These are the partial products.

Then:

92 + 230 = 322

So:

23 × 14 = 322

Standard Algorithm: Two-Digit by Two-Digit

Calculate:

23 × 14

Write:

   23
×  14
-----

First multiply 23 by the ones digit:

23 × 4 = 92

   23
×  14
-----
   92

Now multiply 23 by the tens digit.

The 1 in 14 represents:

10

Therefore:

23 × 10 = 230

   23
×  14
-----
   92
  230
-----
  322

Therefore:

23 × 14 = 322

Why the Second Row Shifts Left

In:

23 × 14

the second partial product comes from:

23 × 10

not:

23 × 1

That is why its digits are shifted one place to the left.

This represents multiplication by a ten.

Thinking about place value is better than simply memorizing "put a zero."

 

 

 

6

Worked Example 2

Calculate:

46 × 32

First:

46 × 2 = 92

Then:

46 × 30 = 1,380

Add:

92 + 1,380 = 1,472

Using the standard algorithm:

    46
×   32
------
    92
 1,380
------
 1,472

Therefore:

46 × 32 = 1,472

Worked Example 3

Calculate:

67 × 45

First partial product:

67 × 5 = 335

Second partial product:

67 × 40 = 2,680

Add:

335 + 2,680 = 3,015

Therefore:

67 × 45 = 3,015

Area Models

An area model represents multiplication using the area of a rectangle.

Suppose we want:

23 × 14

Split:

23 = 20 + 3

and:

14 = 10 + 4

 

 

 

4

This creates four smaller multiplication problems.

20 × 10 = 200

20 × 4 = 80

3 × 10 = 30

3 × 4 = 12

Add:

200 + 80 + 30 + 12 = 322

Therefore:

23 × 14 = 322

Why the Area Model Works

The area model uses the distributive property.

We are rewriting:

23 × 14

as:

(20 + 3)(10 + 4)

Then multiplying every part:

20 × 10

20 × 4

3 × 10

3 × 4

Finally, we add the partial products.

This makes the place-value structure of multiplication visible.

Area Model Example

Calculate:

34 × 26

Break apart:

34 = 30 + 4

26 = 20 + 6

Partial products:

30 × 20 = 600

30 × 6 = 180

4 × 20 = 80

4 × 6 = 24

Add:

600 + 180 + 80 + 24

= 884

Therefore:

34 × 26 = 884

 

 

 

Area Model and Standard Algorithm

The area model and standard algorithm are not separate kinds of multiplication.

They represent the same mathematics.

For:

34 × 26

the area model produces:

600 + 180 + 80 + 24

The standard algorithm combines these partial products more efficiently.

Both rely on:

• place value
• distributive property
• addition

The area model helps explain why the standard algorithm works.

Multiplying Three-Digit Numbers

The same method extends to larger numbers.

Consider:

326 × 24

First multiply by 4:

326 × 4 = 1,304

Then multiply by 20:

326 × 20 = 6,520

Add:

1,304 + 6,520 = 7,824

Therefore:

326 × 24 = 7,824

Worked Example 4

Calculate:

418 × 35

First:

418 × 5 = 2,090

Then:

418 × 30 = 12,540

Add:

2,090 + 12,540 = 14,630

Therefore:

418 × 35 = 14,630

Multiplication with Zeros

Consider:

305 × 24

Do not ignore the zero in 305.

First:

305 × 4 = 1,220

Then:

305 × 20 = 6,100

Add:

1,220 + 6,100 = 7,320

Therefore:

305 × 24 = 7,320

The zero is a placeholder showing that there are zero tens in 305.

Multiplying Numbers Ending in Zero

Consider:

240 × 30

Think:

24 × 3 = 72

The original factors contain:

two factors of 10 altogether

because:

240 = 24 × 10

and:

30 = 3 × 10

Therefore:

240 × 30

= 24 × 3 × 10 × 10

= 72 × 100

= 7,200

 

 

 

4

Estimating Products

Before calculating an exact product, it is often useful to estimate.

Estimation helps us:

• predict the approximate answer
• check whether an exact answer is reasonable
• detect place-value mistakes
• make quick decisions

Estimating by Rounding

Suppose:

47 × 31

Round:

47 ≈ 50

31 ≈ 30

Estimate:

50 × 30 = 1,500

Now calculate exactly:

47 × 31 = 1,457

Since:

1,457

is close to:

1,500

the answer appears reasonable.

 

 

 

Another Estimation Example

Calculate:

198 × 42

Estimate:

198 ≈ 200

42 ≈ 40

So:

200 × 40 = 8,000

Exact calculation:

198 × 42 = 8,316

The exact answer is close to:

8,000

Therefore, it is reasonable.

Choosing How Much to Round

For:

62 × 39

rounding to tens is useful:

60 × 40 = 2,400

For:

487 × 213

we might use:

500 × 200 = 100,000

The goal is not to produce a very precise estimate.

The goal is to understand the approximate size of the product.

Compatible Numbers

Sometimes it is easier to choose nearby numbers that multiply easily.

For:

49 × 21

think:

50 × 20 = 1,000

The exact product is:

49 × 21 = 1,029

The estimate gives us a useful benchmark.

Order of Magnitude

Estimation can help detect serious place-value mistakes.

Suppose someone claims:

298 × 41 = 1,221

Estimate:

300 × 40 = 12,000

The exact answer should therefore be somewhere around:

12,000

not:

1,200

The missing place value tells us the calculation is incorrect.

Checking Multiplication with Division

Multiplication and division are inverse operations.

If:

24 × 36 = 864

then:

864 ÷ 36 = 24

and:

864 ÷ 24 = 36

 

 

 

6

Division can therefore be used to check multiplication.

Worked Example 5: Check with Division

Suppose we calculate:

32 × 45 = 1,440

Check:

1,440 ÷ 45 = 32

Since the quotient returns the other factor, the multiplication is confirmed.

Checking with Estimation

Division is not the only way to check.

For:

32 × 45 = 1,440

estimate:

30 × 50 = 1,500

Since:

1,440

is close to:

1,500

the answer is reasonable.

Using both estimation and inverse operations gives an even stronger check.

Worked Example 6: Full Process

Calculate:

76 × 43

First estimate:

80 × 40 = 3,200

Now calculate.

76 × 3 = 228

76 × 40 = 3,040

Add:

228 + 3,040 = 3,268

Exact answer:

3,268

Compare with estimate:

3,268 ≈ 3,200

Check using division:

3,268 ÷ 43 = 76

Therefore:

76 × 43 = 3,268

A Reliable Multiplication Strategy

When using the standard algorithm:

Step 1: Write the factors with place values aligned.

Step 2: Multiply by the ones digit.

Step 3: Record any regrouping carefully.

Step 4: Multiply by the tens digit, remembering its place value.

Step 5: Continue for hundreds or larger place values if necessary.

Step 6: Add the partial products.

Step 7: Estimate to check reasonableness.

Step 8: If appropriate, check using division.

Multi-Digit by Multi-Digit Example

Calculate:

247 × 136

Break the second factor into:

100 + 30 + 6

Then:

247 × 6 = 1,482

247 × 30 = 7,410

247 × 100 = 24,700

Add:

1,482 + 7,410 + 24,700

= 33,592

Therefore:

247 × 136 = 33,592

 

 

 

5

Area Model for Larger Numbers

The area model can also represent:

247 × 136

Break:

247 = 200 + 40 + 7

and:

136 = 100 + 30 + 6

This produces nine partial products:

200 × 100

200 × 30

200 × 6

40 × 100

40 × 30

40 × 6

7 × 100

7 × 30

7 × 6

Adding all the partial products gives the same final answer.

The model becomes larger, which explains why the standard algorithm is often more efficient for large numbers.

Real-World Application: Shopping

A school buys:

28 calculators

at:

$37 each

Total cost:

28 × 37

Estimate:

30 × 40 = $1,200

Exact calculation:

37 × 8 = 296

37 × 20 = 740

Add:

296 + 740 = 1,036

Therefore:

total cost = $1,036

 

 

 

5

Real-World Application: Seating

A stadium section contains:

46 rows

with:

32 seats per row

Total seats:

46 × 32

46 × 2 = 92

46 × 30 = 1,380

Total:

1,472

Therefore:

1,472 seats

are in the section.

Real-World Application: Manufacturing

A factory produces:

245 components per day

for:

28 days

Total production:

245 × 28

Estimate:

250 × 30 = 7,500

Exact:

245 × 8 = 1,960

245 × 20 = 4,900

Total:

6,860

Therefore:

6,860 components

are produced.

The estimate confirms that the answer is reasonable.

Real-World Application: Distance

A delivery vehicle travels:

184 km per day

for:

23 days

Total distance:

184 × 23

184 × 3 = 552

184 × 20 = 3,680

Total:

552 + 3,680 = 4,232

Therefore:

4,232 km

are travelled.

 

 

 

Real-World Application: Area

A rectangular field measures:

125 m

by:

48 m

Area:

125 × 48

One efficient method is:

48 = 6 × 8

Since:

125 × 8 = 1,000

then:

1,000 × 6 = 6,000

Therefore:

Area = 6,000 m²

Multiplication is essential for calculating rectangular areas.

Real-World Application: Inventory

A warehouse has:

64 boxes

Each box contains:

125 items

Total:

64 × 125

Use a convenient strategy:

64 × 125

= 8 × 8 × 125

Since:

8 × 125 = 1,000

then:

8 × 1,000 = 8,000

Therefore:

8,000 items

are stored.

Real-World Application: Tickets

A theatre sells:

325 tickets

for each of:

16 performances

Total tickets:

325 × 16

325 × 6 = 1,950

325 × 10 = 3,250

Total:

5,200

Therefore:

5,200 tickets

are sold.

Recognizing Multiplication Problems

Multiplication is often appropriate when a problem contains:

• equal groups
• the same amount repeated
• price per item
• distance per day
• items per box
• seats per row
• people per group
• length × width
• rate × number of units

Do not rely only on keywords.

Think about the relationship between the quantities.

One-Step Practical Problem

A farmer plants:

38 rows

with:

47 plants in each row

How many plants are there?

The situation contains equal groups:

38 groups of 47

So:

38 × 47

Estimate:

40 × 50 = 2,000

Exact:

47 × 8 = 376

47 × 30 = 1,410

Add:

1,786

Therefore:

1,786 plants

are planted.

Multi-Step Practical Problem

A company packs:

36 boxes

with:

48 bottles in each box

It then sells:

275 bottles

First find the total:

36 × 48

= 1,728

Then subtract those sold:

1,728 − 275

= 1,453

Therefore:

1,453 bottles remain

 

 

 

4

Comparing Two Options

A store can purchase:

Option A: 24 boxes containing 36 items each

Option B: 18 boxes containing 50 items each

Option A:

24 × 36 = 864

Option B:

18 × 50 = 900

Therefore:

Option B contains 36 more items.

This type of problem requires multiplication followed by comparison.

Using Mental Math When Appropriate

The standard algorithm is useful, but it is not always the fastest strategy.

For:

25 × 40

mental math is easier:

25 × 4 × 10

= 100 × 10

= 1,000

For:

99 × 27

use compensation:

100 × 27 − 27

= 2,700 − 27

= 2,673

For:

348 × 67

the standard algorithm is likely more efficient.

A strong mathematician chooses the method that fits the numbers.

Common Mistakes

Mistake 1: Forgetting place value in the second partial product

In:

42 × 36

the 3 represents:

30

not:

3

Mistake 2: Forgetting regrouped values

If a multiplication produces a value greater than 9, record the regrouped value carefully.

Mistake 3: Adding partial products incorrectly

Multiplication may be correct but the final addition may still contain an error.

Check both stages.

Mistake 4: Misaligning partial products

Place-value columns must stay aligned.

Mistake 5: Ignoring zeros inside numbers

In:

304 × 27

the zero is an important place-value placeholder.

Mistake 6: Treating estimation as an exact calculation

An estimate provides an approximate value.

Use:

≈

when appropriate.

Mistake 7: Skipping the reasonableness check

An estimate can quickly reveal a missing zero or incorrect place value.

Error Analysis

Suppose a student calculates:

   48
×  23
-----
  144
   96
-----
  240

The first row:

48 × 3 = 144

is correct.

But the second row should represent:

48 × 20

not:

48 × 2

Therefore:

48 × 20 = 960

Correct calculation:

144 + 960 = 1,104

So:

48 × 23 = 1,104

 

 

 

4

Using Estimation to Detect the Error

For:

48 × 23

estimate:

50 × 20 = 1,000

The incorrect answer:

240

is nowhere near:

1,000

The estimate immediately tells us to check the calculation.

Area Models as an Error-Checking Tool

Area models are useful when a standard algorithm seems confusing.

For:

48 × 23

split:

48 = 40 + 8

23 = 20 + 3

Calculate:

40 × 20 = 800

40 × 3 = 120

8 × 20 = 160

8 × 3 = 24

Add:

800 + 120 + 160 + 24

= 1,104

The area model confirms the standard algorithm result.

A Complete Problem-Solving Process

For practical multiplication problems:

Step 1: Read the problem carefully.

Step 2: Identify the quantities and units.

Step 3: Decide whether multiplication is appropriate.

Step 4: Estimate the product.

Step 5: Choose a calculation strategy.

Step 6: Calculate the exact product.

Step 7: Compare the exact result with the estimate.

Step 8: Check using division when appropriate.

Step 9: State the answer with correct units and context.

Did You Know?

The standard multiplication algorithm is essentially a compressed version of an area model.

 

 

 

7

When we calculate:

34 × 26

the area model explicitly shows:

30 × 20

30 × 6

4 × 20

4 × 6

The standard algorithm combines these calculations into fewer written steps.

Both methods depend on the same mathematical ideas:

• place value
• distributive property
• partial products

Understanding the area model therefore helps explain why the standard algorithm works rather than simply memorizing a procedure.

Key Terms

• Multiplication: Operation used to combine equal groups or scale a quantity.
• Factor: Number being multiplied.
• Product: Result of multiplication.
• Place value: Value of a digit based on its position.
• Partial product: Product created by multiplying part of one factor by part of another.
• Standard algorithm: Organized written procedure for multiplication.
• Area model: Rectangle model used to represent multiplication through partial areas.
• Distributive property: Property allowing multiplication to be distributed across addition or subtraction.
• Regrouping: Rewriting a quantity using different place-value units.
• Estimate: Approximate value.
• Rounding: Replacing a number with a nearby convenient value.
• Compatible numbers: Nearby numbers that make calculations easier.
• Inverse operations: Operations that undo one another.
• Reasonableness: Whether an answer makes sense based on the original calculation.

Key Relationships

factor × factor = product

For example:

24 × 35 = 840

Using place value:

24 × 35

= 24 × (30 + 5)

= 720 + 120

= 840

Using an area model:

(20 + 4)(30 + 5)

= 600 + 100 + 120 + 20

= 840

Checking with division:

840 ÷ 35 = 24

Key Takeaways

• Multiplication combines equal groups and can also represent scaling, area, rates, and repeated quantities.
• The numbers being multiplied are called factors, and the answer is the product.
• Place value is essential when multiplying multi-digit numbers.
• The standard multiplication algorithm is based on place value and the distributive property.
• A multi-digit multiplication can be broken into partial products.
• When multiplying by a tens digit, the partial product represents multiplication by tens, not ones.
• Area models make the partial products visible and help explain why the standard algorithm works.
• The distributive property allows numbers to be broken into convenient place-value parts.
• Estimating before calculating provides a useful prediction of the approximate product.
• Rounding and compatible numbers are useful for estimating products.
• Comparing an exact product with an estimate helps check whether the result is reasonable.
• Multiplication and division are inverse operations, so division can be used to check a product.
• Errors in multi-digit multiplication often come from place-value mistakes, forgotten regrouping, or incorrect addition of partial products.
• Mental strategies can be more efficient when the factors have useful structures such as 10, 25, 50, 100, or numbers close to them.
• Standard algorithms are particularly useful for larger or less convenient factors.
• Multiplication is used in practical situations involving cost, inventory, seating, production, distance, area, packaging, and repeated quantities.
• A strong solution should include an estimate, an accurate calculation, a reasonableness check, and appropriate units or real-world context.