Multiplication Properties, Facts and Multiples of Ten | Grade 5 Math | LevelKite

When you multiply, the numbers you multiply are called factors. The answer is called the product. In $3 \times 4 = 12$ , the factors are $3$ and $4$ , and the product is $12$ .

Picture $3$ rows with $4$ counters in each row. That is $3 \times 4$ , or $12$ counters. Now turn the picture on its side. You see $4$ rows with $3$ counters in each row, which is $4 \times 3$ . No counter was added or taken away, so there are still $12$ .

You just saw a property of multiplication. A property is a rule that works for any numbers you multiply. Properties let you rewrite a product in an easier form without changing its value.

The table shows five properties. The one you just saw is in the first row. In the table, parentheses ( ) tell you which part to work out first.

Five properties of multiplication

Property	In words	Number sentence	See the same amount
Commutative	You can change the order of the factors. The product stays the same.	$3 \times 4 = 4 \times 3$	$3$ rows of $4$ and $4$ rows of $3$ are both $12$ .
Associative	With three factors, you can change which two are grouped to multiply first. The product stays the same.	$(3 \times 2) \times 5 = 3 \times (2 \times 5)$	$6 \times 5$ and $3 \times 10$ are both $30$ .
Identity	When one factor is $1$ , the product is the other factor.	$8 \times 1 = 8$	$8$ groups of $1$ make $8$ .
Zero	When one factor is $0$ , the product is $0$ .	$8 \times 0 = 0$	$8$ groups of $0$ make $0$ .
Distributive	Break one factor into two parts that add up to it. Multiply each part by the other factor. Then add.	$6 \times 7 = 6 \times (5 + 2) = (6 \times 5) + (6 \times 2)$	$6$ groups of $7$ are $6$ groups of $5$ and $6$ groups of $2$ : $30 + 12 = 42$ .

Before you multiply, look at the factors. They often show you which property will help.

Is one of two factors $1$ ? The identity property says the product is the other factor: $1 \times 9 = 9$ .
Is any factor $0$ ? The zero property says the product is $0$ : $(7 \times 6) \times 0 = 0$ . You do not even need to find $7 \times 6$ .
Do two factors multiply to make $10$ , like $5$ and $2$ ? Use the commutative and associative properties to bring them together.
Is a product hard to recall? Use the distributive property to break one factor into parts that are easier.

Worked example

Regroup to make a ten

Find $(5 \times 9) \times 2$ in an easier way.

Look at the factors first. 5 and 2 multiply to make 10, but they are not side by side. Swap the 5 and the 9 inside the parentheses. The commutative property says that changing the order of the factors keeps the product the same.
$(5 \times 9) \times 2 = (9 \times 5) \times 2$
Now the 5 sits next to the 2. Move the parentheses so that 5 and 2 are grouped. The associative property says that changing the grouping keeps the product the same.
$(9 \times 5) \times 2 = 9 \times (5 \times 2)$
Work out the part in parentheses first. 5 × 2 = 10, so the product is now 9 × 10.
$9 \times (5 \times 2) = 9 \times 10$
Multiply. 9 × 10 means 9 groups of ten, which is 90.
$9 \times 10 = 90$
Check with the form you started from. 5 × 9 = 45, and 45 × 2 is 45 + 45, which is 90. Both forms have the same value. The new form was easier to work out.
$(5 \times 9) \times 2 = 45 \times 2 = 90$

Answer

The product $(5 \times 9) \times 2$ is $90$ .

A multiplication fact, or basic fact, tells you the product of two whole numbers from $0$ to $10$ , such as $6 \times 4 = 24$ . Aim to know every fact through $10 \times 10$ by heart. The larger products in this topic are all built from these facts.

When you are unsure of a fact, do not guess. Start from a fact you know and let a property take you the rest of the way.

Turn it around. Unsure of $8 \times 3$ ? The commutative property says it equals $3 \times 8$ . If you know that $3 \times 8 = 24$ , you know both.
Double a double. For $4 \times 6$ , think of $4$ as $2 \times 2$ . The associative property says $(2 \times 2) \times 6 = 2 \times (2 \times 6)$ . That is $2 \times 12$ , or $12 + 12 = 24$ .
Break it apart. Use the distributive property to split one factor into parts you know. The next example shows how.

Worked example

Build a fact from facts you know

You are unsure of $7 \times 8$ . Use facts you know and a property to work it out.

Think of 7 × 8 as 7 groups of 8. Break the 8 into 5 + 3, because facts with 5 and facts with 3 are easier to recall. Put the two parts in parentheses to show that together they still make 8.
$7 \times 8 = 7 \times (5 + 3)$
Use the distributive property. 7 groups of 8 is the same amount as 7 groups of 5 plus 7 groups of 3, so multiply each part by 7.
$7 \times (5 + 3) = (7 \times 5) + (7 \times 3)$
Use the two facts you know: 7 × 5 = 35 and 7 × 3 = 21.
$(7 \times 5) + (7 \times 3) = 35 + 21$
Add the two products. Together they count all 7 groups of 8.
$35 + 21 = 56$
Get a second fact for free. 8 × 7 has the same two factors in a different order, so the commutative property says it has the same product.
$8 \times 7 = 7 \times 8 = 56$

Answer

The product $7 \times 8$ is $56$ , and $8 \times 7$ is $56$ too.

Basic facts also let you multiply by much larger numbers.

A multiple of 10 is the product of $10$ and a whole number. For example, $70$ is a multiple of $10$ because $70 = 7 \times 10$ . In the same way, $700 = 7 \times 100$ is a multiple of $100$ , and $7{,}000 = 7 \times 1{,}000$ is a multiple of $1{,}000$ .

Place value is the value a digit gets from its place in a number. It tells you that $70$ is $7$ tens, $700$ is $7$ hundreds and $7{,}000$ is $7$ thousands.

So $6 \times 700$ means $6$ groups of $7$ hundreds. Count the hundreds with a basic fact. $6 \times 7 = 42$ , so you have $42$ hundreds. A whole number of hundreds has $0$ tens and $0$ ones, so $42$ hundreds is written $4{,}200$ .

The associative property shows why you may multiply $6 \times 7$ first. Watch the parentheses move in the line below.

6 \times 700 = 6 \times (7 \times 100) = (6 \times 7) \times 100 = 42 \times 100 = 4{,}200

The basic fact stays the same. Only the place value changes.

Product	In words	How many of that place	Number
$6 \times 7$	$6$ groups of $7$ ones	$42$ ones	$42$
$6 \times 70$	$6$ groups of $7$ tens	$42$ tens	$420$
$6 \times 700$	$6$ groups of $7$ hundreds	$42$ hundreds	$4{,}200$
$6 \times 7{,}000$	$6$ groups of $7$ thousands	$42$ thousands	$42{,}000$

Key concept

Same value, easier form

You can rewrite a product without changing its value, then work with the form that is easiest for you. The properties of multiplication let you change the order of the factors, change the grouping, or break a factor into parts. Place value lets you read $700$ as $7$ hundreds, so $6 \times 700$ becomes the basic fact $6 \times 7$ with a place name: $42$ hundreds. Each rewrite leads you back to basic facts you know.

Sometimes a close answer is all you need. An estimate is a number that is close to the exact answer and quick to find. It also helps you judge whether an exact answer makes sense.

To estimate a product, round the greater factor to its greatest place, then multiply. To round a number is to swap it for a nearby number that is easier to use. The greatest place is the place of the first digit on the left.

Try $6 \times 384$ . The greater factor is $384$ . Its first digit, $3$ , is in the hundreds place, so round $384$ to the nearest hundred. It lies between $300$ and $400$ . Look at the digit just to the right of the greatest place. If it is $5$ or more, round up. If it is less than $5$ , round down. Here that digit is $8$ , so $384$ rounds up to $400$ .

Now multiply. $6 \times 400$ is $6$ groups of $4$ hundreds, or $24$ hundreds. So $6 \times 384$ is about $2{,}400$ .

Rounding is not a property. A property keeps the value exactly the same. Rounding swaps in a nearby number, so an estimate tells you about how big the product is. Do not treat it as the exact product.

Worked example

Estimate with a four-digit factor

Estimate $8 \times 6{,}214$ .

Find the greater factor and its greatest place. The greater factor is 6,214. Its first digit on the left is 6, and that 6 is in the thousands place.
Round 6,214 to the nearest thousand. It lies between 6,000 and 7,000. The digit just to the right of the thousands place is 2. That is less than 5, so round down.
$6{,}214 \text{ rounds to } 6{,}000$
Multiply 8 by the rounded factor. 6,000 is 6 thousands, so use the basic fact 8 × 6 = 48 and name the place: 48 thousands.
$8 \times 6{,}000 = 48 \text{ thousands} = 48{,}000$
Think about what the estimate tells you. You rounded 6,214 down to 6,000, so 8 groups of 6,000 is less than 8 groups of 6,214. The exact product is more than 48,000, but close to it.

Answer

The product $8 \times 6{,}214$ is about $48{,}000$ .

Here is the idea to keep: a product can be written in more than one form, and a form you make with a property has the same value. So choose the form that is easiest. Change the order or the grouping of the factors. Break a factor into parts to reach a fact you are unsure of. When a factor is a multiple of $10$ , $100$ or $1{,}000$ , multiply the basic fact and name the place. When a close answer is enough, round the greater factor to its greatest place and multiply.

Next, you will use these ideas to multiply 2-digit, 3-digit and 4-digit numbers by a 1-digit number, and you will use an estimate to check each product.