Multiplication Patterns, Zeros and Estimates | Grade 5 Math | LevelKite

You already know basic facts such as $6 \times 7 = 42$ . A basic fact is a multiplication of two 1-digit numbers. In this lesson, one basic fact will help you find much bigger products.

Two words to know: the numbers you multiply are factors, and the answer is the product.

Start with $60 \times 7$ . The number $60$ is $6$ tens. So $60 \times 7$ is $6$ tens times $7$ . That is $42$ tens, just as $6$ ones times $7$ is $42$ ones. And $42$ tens is $420$ .

As a number sentence: $60 \times 7 = 42 \times 10 = 420$ .

The table shows what happens as the factors grow.

One basic fact, four products

Factors	In words	Power of ten	Product
$6 \times 7$	$6$ ones times $7$ ones is $42$ ones	none	$42$
$60 \times 7$	$6$ tens times $7$ ones is $42$ tens	$10 = 10^1$	$42 \times 10 = 420$
$60 \times 70$	$6$ tens times $7$ tens is $42$ hundreds	$10 \times 10 = 100 = 10^2$	$42 \times 100 = 4{,}200$
$600 \times 70$	$6$ hundreds times $7$ tens is $42$ thousands	$100 \times 10 = 1{,}000 = 10^3$	$42 \times 1{,}000 = 42{,}000$

Look down the last column. The digits $4$ and $2$ stay the same. Only their places change. Each time one factor becomes ten times as big and the other factor stays the same, the product becomes ten times as big.

Here is why. Remember that a power of ten is a number such as $10$ , $100$ or $1{,}000$ . You write these as $10^1$ , $10^2$ and $10^3$ . The small raised number is the exponent. It counts how many tens are multiplied to make the number.

A factor that ends in zeros splits into a nonzero part and a power of ten. The nonzero part is the digits in front of the ending zeros. So $60 = 6 \times 10$ and $70 = 7 \times 10$ .

You can multiply numbers in any order and group them in any way. The product stays the same. So you can put the nonzero parts together and the tens together:

60 \times 70 = (6 \times 7) \times (10 \times 10) = 42 \times 10^2 = 4{,}200

Now look at $42 \times 10^2$ . You know that multiplying a whole number by $10$ moves each digit one place to the left. The exponent $2$ means two tens are multiplied: $10^2 = 10 \times 10 = 100$ . So each digit of $42$ moves two places to the left.

The $4$ moves from the tens place to the thousands place. The $2$ moves from the ones place to the hundreds place. The tens place and the ones place are now empty, so you write a $0$ in each one. The product is $4{,}200$ .

When you multiply a whole number by a power of ten, the exponent tells you how many places each digit moves to the left. For $600 \times 70$ , the powers of ten are $100 \times 10 = 10^3$ . So the digits of $42$ move three places, and the product is $42{,}000$ .

Key concept

Nonzero parts first, then powers of ten

When whole-number factors end in zeros, you can build the product from two easy pieces.

Split each factor that ends in zeros into its nonzero part and a power of ten.
Multiply the nonzero parts.
Multiply the powers of ten.
Multiply the two results. The exponent of the power of ten tells you how many places each digit moves to the left.

Count the places the digits move. Do not just count zeros in the answer.

Example: $600 \times 70 = (6 \times 7) \times (100 \times 10) = 42 \times 10^3 = 42{,}000$ . Each digit of $42$ moves three places.

Worked example

A basic fact with a power of ten

What is $300 \times 40$ ?

Split each factor into its nonzero part and a power of ten. 300 is 3 hundreds, and 40 is 4 tens.
$300 = 3 \times 100 \quad \text{and} \quad 40 = 4 \times 10$
Multiply the nonzero parts. You may do this first because numbers can be multiplied in any order.
$3 \times 4 = 12$
Multiply the powers of ten. One hundred times ten is one thousand. That is three tens multiplied together, so the exponent is 3.
$100 \times 10 = 1{,}000 = 10^3$
Multiply the two results. The exponent 3 tells you that each digit of 12 moves three places to the left. The three empty places get zeros. In words, this is 12 thousands.
$12 \times 1{,}000 = 12{,}000$

Answer

The product of $300$ and $40$ is $12{,}000$ .

Worked example

When the basic fact ends in zero

What is $500 \times 60$ ?

Split each factor into its nonzero part and a power of ten. 500 is 5 hundreds, and 60 is 6 tens.
$500 = 5 \times 100 \quad \text{and} \quad 60 = 6 \times 10$
Multiply the nonzero parts. This basic fact ends in a zero of its own. Keep that zero, because it is part of the fact.
$5 \times 6 = 30$
Multiply the powers of ten. The exponent 3 means the digits will move three places.
$100 \times 10 = 1{,}000 = 10^3$
Multiply the two results. Move both digits of 30 three places to the left. The 3 lands in the ten-thousands place and the 0 lands in the thousands place. The three empty places get zeros.
$30 \times 1{,}000 = 30{,}000$

Answer

The product of $500$ and $60$ is $30{,}000$ . It ends in four zeros: one from $5 \times 6 = 30$ and three from moving the digits three places.

Most factors do not end in zeros. You can still find about how big a product is. An estimate is a number that is close to the exact answer but quicker to find.

One way to estimate a product is to round each factor to its greatest place. Rounding swaps a number for a nearby number that is easier to use. The greatest place is the place of the first digit on the left. That digit is worth the most. In $617$ , the greatest place is the hundreds place. In $38$ , it is the tens place.

To round $617$ to the hundreds place, pick the nearest hundred. $617$ is between $600$ and $700$ . It is nearer to $600$ , so it rounds to $600$ . In the same way, $38$ is nearer to $40$ than to $30$ , so it rounds to $40$ .

A quick check: 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. A number that is exactly halfway, such as $650$ , rounds up.

Each rounded factor is one nonzero digit followed by zeros. So you can multiply the rounded factors with a basic fact and powers of ten.

Worked example

Estimate a product

Estimate $73 \times 284$ by rounding each factor to its greatest place.

Round 73 to its greatest place, the tens place. The digit to the right is 3, which is less than 5. So 73 is nearer to 70 than to 80, and it rounds down.
$73 \text{ rounds to } 70$
Round 284 to its greatest place, the hundreds place. The digit to the right is 8, which is 5 or more. So 284 is nearer to 300 than to 200, and it rounds up.
$284 \text{ rounds to } 300$
Multiply the rounded factors. Split them as before: 70 is 7 tens and 300 is 3 hundreds. Multiply the nonzero parts, 7 and 3. Then multiply the powers of ten, 10 and 100.
$70 \times 300 = (7 \times 3) \times (10 \times 100) = 21 \times 1{,}000$
Move each digit of 21 three places to the left, because 1,000 is three tens multiplied together. This is 21 thousands.
$21 \times 1{,}000 = 21{,}000$

Answer

The product $73 \times 284$ is about $21{,}000$ . The exact product is $20{,}732$ , so the estimate is close.

More estimates. In the last row, the basic fact ends in a zero. Keep it.

Factors	Rounded factors	Nonzero parts and powers of ten	Estimate
$62 \times 81$	$60 \times 80$	$(6 \times 8) \times (10 \times 10) = 48 \times 100$	about $4{,}800$
$4{,}215 \times 67$	$4{,}000 \times 70$	$(4 \times 7) \times (1{,}000 \times 10) = 28 \times 10{,}000$	about $280{,}000$
$451 \times 78$	$500 \times 80$	$(5 \times 8) \times (100 \times 10) = 40 \times 1{,}000$	about $40{,}000$

The big idea: one basic fact can unlock many bigger products. Split each factor into its nonzero part and a power of ten. Multiply the nonzero parts, then multiply by the powers of ten. Count the places the digits move, and keep any zero that the basic fact already has.

When factors do not end in zeros, round each one to its greatest place first. Then the same steps give you an estimate.

Next, you will multiply two multi-digit numbers exactly. You will split each factor into its place values, multiply the parts and add the results. An estimate can help you check that an exact answer makes sense.