Multiplication and Division with 2, 5 and 10

Look at three plates with five strawberries on each. Every plate has the same number. These are equal groups. You can count all the strawberries without touching each one.

Count one group at a time

Add the strawberries on each plate:

You added 5 three times. This is repeated addition. Multiplication writes those equal groups in a shorter way:

Read it aloud: “Three times five equals fifteen.” The sign × means “times” or “multiplied by.” Here, 3 counts the plates. The 5 tells how many strawberries are on each plate. The 15 counts all the strawberries.

The numbers you multiply are factors. The answer is the product. In this sentence, 3 and 5 are factors. The product is 15.

Use counting steps or jumps

“There are four pairs of socks. How many socks are there?” Each pair has two socks. Count one pair at a time: “2, 4, 6, 8.” Four pairs make eight socks, so $4\times 2=8$.

Put up a finger for each pair as you count. Your fourth finger goes up when you say “8.” Counting just the pairs would give 4, but you want the number of socks.

A number line shows the same counting. Start at 0. Make four jumps of 2, landing on 2, 4, 6, then 8. The number of jumps tells how many groups. Each jump's size tells how many are in one group.

Read the whole number sentence

The equal sign means “is the same amount as.” You can also write $15=3\times 5$. Both sides still show fifteen strawberries.

Adding the two factors gives a different amount. For the plates, $3+5=8$ does not count the strawberries. You need three copies of five, which make 15.

Check the groups before multiplying. Bags holding 2, 3, and 5 marbles have different amounts. Add their actual amounts: $2+3+5=10$. One group size cannot describe all three bags.

An array lines things up in equal rows and columns. Rows go across. Columns go up and down. The objects make a rectangle with no gaps.

Read an array across

Draw three rows with five dots in each row:

● ● ● ● ●
● ● ● ● ●
● ● ● ● ●

There are three rows, so there are three groups. Each row holds five dots. Count “5, 10, 15.” The array shows $3\times 5=15$.

You can draw an array when you forget a fact. For four groups of five, draw four equal rows. Count each row once to find the total.

Read the same array down

Now look down the columns of the three-row array. There are five columns with three dots in each. Those columns show $5\times 3=15$.

No dots were added or taken away. You counted the same dots in a different way. Turning the picture a quarter turn makes the old columns into rows.

This shows the turn-around rule: changing the order of the factors keeps the product the same. You may also hear “commutative property of multiplication.”

Use a fact you know

If you know $5\times 10=50$, you also know $10\times 5=50$. Five rows of ten and ten rows of five have the same total.

For $7\times 2$, you can think of $2\times 7$. That is two sevens, or double seven: $7+7=14$. The rows may look different, but both products are 14.

Check the rows

Two rows of ten have twenty objects. Adding two and ten gives twelve, which counts neither the whole first row nor the whole second row. Multiply the row count by the number in each row: $2\times 10=20$.

Rows of 5, 5, and 4 do not make a complete array. One row is short. They hold $5+5+4=14$ dots. Saying “three rows of five” would count a dot that is missing.

An array can also help you divide. Put twenty dots into rows of five. You can make four full rows. This shows $20÷5=4$: four groups of five fit into twenty.

You have twelve cookies and three friends. You want each friend to get the same number. You need to share equally.

Deal them out

Draw three circles, one for each friend. Put one cookie in each circle. Keep going around, giving one cookie to each friend each time.

After four rounds, all twelve cookies are gone. Count one friend's share: four cookies. Every friend has four.

Two things must be true. All the cookies have been shared, and every share is the same size. Shares of 5, 4, and 3 use twelve cookies, but they are not equal.

Write the sharing

Division writes equal sharing with the sign ÷. Read this sentence as “Twelve divided by three equals four”:

The 12 is the total you start with. The 3 tells how many equal shares you make. The 4 tells how many cookies each friend gets. The answer to a division is called the quotient.

The question asks how many cookies one friend gets. Answer “four cookies,” not “three friends.” Three tells you how many shares to make; it is already known.

Check the shares

Put the three shares back together in your mind:

You can also write $3\times 4=12$. The shares rebuild the starting total, so the division is right.

Share a larger amount

“Thirty pencils are shared equally among ten cups. How many pencils go in each cup?” Draw ten cups. Give each cup one pencil at a time. After three rounds, you have used all thirty.

You can use a multiplication fact too. Ask, “Ten times what makes thirty?” Since $10\times 3=30$, each cup gets three pencils. Write $30÷10=3$.

Sharing between two finds half. Fourteen grapes shared equally between two children give seven grapes each. The two sevens make fourteen, so $14÷2=7$.

You have fifteen apples. Each bag holds five apples. How many bags can you fill? You know the group size. You need to find the number of groups.

Make one full group at a time

Put five apples in the first bag. Put five in the next bag, then five in another. No apples remain. You filled three bags:

This way of dividing is called grouping. Count the finished bags, because the question asks how many bags you need. The answer is “three bags,” not “three apples.”

Take away equal amounts

Each full bag takes five apples from the pile. You can show this with repeated subtraction:

You took away five three times. Those three subtractions match the three bags. Subtracting five just once tells how many apples remain after one bag. Keep going until you reach zero.

On a number line, start at fifteen. Jump back five to ten, then to five, then to zero. Count the jumps: three. Count the spaces you jumped, not the four landing numbers including the start.

Count forward instead

For thirty-five buttons in groups of five, count “5, 10, 15, 20, 25, 30, 35.” Hold up one finger for each group. You reach thirty-five at the seventh group:

Seven groups of five use all thirty-five buttons. Check with $7\times 5=35$.

For forty blocks in groups of ten, count all the way to forty: “10, 20, 30, 40.” That makes four groups. Stopping at thirty would leave ten blocks unused.

Rows can show grouping too. Place twenty counters in rows of five. Four full rows use every counter, so $20÷5=4$. The row size is known; the division finds how many rows fit.

Two stories can use the same division but ask different questions. Look at what you know and what you need to find.

Find the size of a share

“Eighteen stickers are shared equally between two children. How many stickers does each child get?”

You know the total: eighteen stickers. You also know the number of shares: two children. Draw two circles and share all the stickers. Each circle gets nine:

This is sharing. The answer tells how many stickers are in one child's share. Say, “Each child gets nine stickers.”

Find the number of groups

“Eighteen stickers go on pages. Each page holds two stickers. How many pages do you need?”

This time, two tells you the group size. Put two stickers on each page. Nine full pages use all eighteen:

This is grouping. The answer tells how many pages you need. Say, “You need nine pages.”

Keep track of three parts

Every equal-group division story has a total, a number of groups, and a group size. You know the total and one group part. Division finds the missing group part.

In the sharing story, the two children are known. In the grouping story, the two stickers per page are known. The written numbers match, but the answer words are different.

Draw the known parts first. For sharing, draw the empty groups before dealing things out. For grouping, draw one group of the given size, then keep making more.

Start with the total

Both stories start with eighteen, so eighteen comes first in the division sentence. Writing two divided by eighteen starts with only two stickers. That does not match either story.

Read the whole question before choosing. “Each” can tell you a group size in a grouping story, or ask for a share size in a sharing story. To check either story, use $9\times 2=18$. Then read your answer with its word: stickers for each child, or pages.

A pair has two things. Shoes, socks, and bicycle wheels often come in pairs. Counting pairs uses the 2 times table.

Learn the twos

Each new group adds two. Read down the facts:

Say the totals: “2, 4, 6, 8, 10, 12, 14, 16, 18, 20.” Put up one finger for each pair. Six children wearing two shoes each have $6\times 2=12$ shoes.

Multiplying by two is also doubling. Seven pairs give the same total as two sevens: $7\times 2=7+7=14$. You can use a double you know to remember a twos fact.

Make partners

An even number of things can all have partners. Eight socks make four pairs with none left without a partner. The twos facts all have even answers.

An odd number leaves one without a partner. With seven socks, you can make three pairs, but one sock has no partner.

Odd and even numbers take turns as you count. One is odd, two is even, three is odd, four is even. Adding one changes which kind comes next.

Look at the last digit

The ones digit is the last digit in a whole number. Even numbers end in 0, 2, 4, 6, or 8. Odd numbers end in 1, 3, 5, 7, or 9.

Thirty-four is even because its last digit is 4. Forty-three is odd because its last digit is 3. A 4 elsewhere in the number does not make it even.

The counting keeps going past twenty. The next two facts are $11\times 2=22$ and $12\times 2=24$.

A box holds ten crayons. Eight boxes hold eight tens. You can count the tens to find how many crayons there are.

Learn the tens

Say “10, 20, 30, 40, 50, 60, 70, 80.” You said eight totals, one for each box. Eight boxes hold eighty crayons: $8\times 10=80$.

Each new group gives ten more. All the totals end in 0 because there are no extra ones outside the full tens.

Think about tens and ones

In 4, the digit 4 means four ones. In 40, it means four tens. This is place value: a digit's place tells what it is worth.

For $4\times 10=40$, four ones become four tens. The digit 4 moves into the tens place. A zero fills the ones place.

Adding zero does a different job. $4+0=4$ leaves four ones. Multiplying by ten makes four groups of ten, which is forty. Saying “add zero” can mix up these two actions.

Use the table both ways

Six tens make sixty, so $6\times 10=60$. The turn-around rule also gives $10\times 6=60$. Ten groups of six and six groups of ten have the same total.

At the end of the first ten facts, ten tens make one hundred: $10\times 10=100$. You can picture ten bundles, each holding ten sticks. Together they hold a hundred sticks.

Two more steps give $11\times 10=110$ and $12\times 10=120$. Each step still adds one full ten.

Each hand has five fingers. Count one hand at a time and you count in fives. Every new hand adds five fingers.

Learn the fives

The totals go “5, 10, 15, 20, 25, 30, 35, 40, 45, 50.” Their last digits take turns: 5, then 0, then 5, then 0. An odd number of fives ends in 5. An even number of fives ends in 0.

If you say “5, 10, 15, 25,” you skipped twenty. From fifteen, add five to reach twenty. The jump must stay the same size each time.

Let tens help

A five is half a ten. So six groups of five give half as many as six groups of ten.

Start with $6\times 10=60$. Half of sixty is thirty. That gives $6\times 5=30$.

Keep the number of groups the same. For $8\times 5$, first find eight tens: $8\times 10=80$. Halve eighty to get forty. Eight fives make forty.

You can picture each group of ten as two rows of five. Taking one row from every group keeps half the total.

Count minutes in fives

A clock face has a five-minute step from one number to the next. Count from twelve, where the minutes start at zero. At one, five minutes have passed; at two, ten minutes have passed.

When the minute hand reaches four, four steps of five make twenty minutes past the hour. At nine, nine steps of five make forty-five minutes past.

The fives continue after fifty. Two more facts are $11\times 5=55$ and $12\times 5=60$.

The answers you get when multiplying by a whole number are called multiples. You can keep counting on to find more. The multiples of five go past fifty to fifty-five, sixty, and onward.

Check the ones digit

The last digit of a whole number helps you spot multiples:

Forty-six ends in 6, so it is a multiple of two. Forty-seven ends in 7, so it is not.

Eighty-five ends in 5, so it is a multiple of five. Fifty-eight ends in 8. It is not a multiple of five, even though it has a 5 at the front. You need the ones digit, not just any digit.

Seventy is a multiple of ten because it ends in 0. Seventy-five ends in 5, so it is a multiple of five but not ten.

A number can belong in several lists

Sort these numbers: 15, 20, 24, 35, and 40.

• Multiples of two: 20, 24, 40.
• Multiples of five: 15, 20, 35, 40.
• Multiples of ten: 20, 40.

Twenty and forty belong in all three lists. Their last digit is 0. You can make whole groups of two, five, or ten from either amount.

Twenty-four belongs in the twos list only. Thirty-five belongs in the fives list only. Checking each rule lets you decide where a number goes.

Connect the fives and tens

Every full ten can be split into two fives. A total made of full tens can therefore also be made of full fives. Tens can also be split into pairs. This is why a number ending in 0 fits all three lists.

For the same number of groups, tens give double the fives total. Three fives make fifteen. Double fifteen is thirty, which is three tens:

A multiple of two need not be a multiple of five. Twelve makes pairs but does not end in 0 or 5. A multiple of five need not be a multiple of ten: fifteen ends in 5.

To double a number, make two equal copies. Add the number to itself or multiply it by two. Double eight is $8+8=16$.

Use small doubles

The doubles of one through ten are 2, 4, 6, 8, 10, 12, 14, 16, 18, and 20. These match the twos facts.

“Sam has six marbles. Ana has twice as many. How many does Ana have?” Twice as many means double. Two copies of six give $6+6=12$ marbles.

Doubling does not mean adding two. Double seven is $7+7=14$. Adding two gives nine, which is only two more than seven.

Double tens

Think of thirty as three tens. Double three tens is six tens, so double thirty is sixty. Double fifty is a hundred because five tens and five tens make ten tens.

Split a two-digit number

To double twenty-three, split it into twenty and three. Double both parts, then add the results:

1. Double twenty: $20+20=40$.
2. Double three: $3+3=6$.
3. Join the doubled parts: $40+6=46$.

So double twenty-three is forty-six. You made a second copy of all twenty-three, including the ones.

When the ones make a ten

For double seventeen, split seventeen into ten and seven. Double ten is twenty. Double seven is fourteen. Then $20+14=34$.

The fourteen contains a new ten. Keep all fourteen when you join the parts; do not keep just its last digit.

Doubling only the tens leaves part of the number unchanged. For double twenty-four, forty-four is too small. You need forty and eight: $40+8=48$.

Doubling can link your tables too. Seven fives make thirty-five. Double thirty-five is seventy, which is seven tens. Every group of five has become a group of ten.

To halve a number, split it into two equal parts. One half is one of those parts. Halving is the same as dividing by two.

Use a double you know

Double nine is eighteen. So half of eighteen is nine. Two nines fit back together to make the whole amount.

Subtracting two does not find half. Half of twenty is ten, because two tens make twenty. Taking two from twenty leaves eighteen.

Halve tens

Sixty is six tens. Split six tens into two equal groups. Each has three tens, so half of sixty is thirty.

Thirty has an odd number of tens. Use a helpful split: $30=20+10$. Half of twenty is ten, and half of ten is five. Put those halves together: $10+5=15$. Each half of thirty is fifteen.

This also works for fifty and seventy. Half of fifty is twenty-five. Half of seventy is thirty-five.

Halve a two-digit number

To halve forty-six, work with forty and six:

1. Half of forty is twenty.
2. Half of six is three.
3. Add those halves: $20+3=23$.

For thirty-four, use thirty and four. Half of thirty is fifteen. Half of four is two. Add $15+2=17$. Check by doubling: $17+17=34$.

You must halve every part. Halving just forty in forty-eight leaves the eight too large. Half of forty is twenty, and half of eight is four. The whole answer is twenty-four.

Whole things and equal halves

Fourteen marbles split into two equal groups of seven whole marbles. Thirteen whole marbles cannot make two equal groups without splitting a marble. One would be left without a partner.

Multiplication puts equal groups together. Division splits the total back into equal groups. They are inverse operations: each undoes the other.

Use three linked numbers

Four groups of five make twenty. The numbers 4, 5, and 20 make a fact family:

The first two facts find the total. The last two start with that total and find a group part.

See the family in an array

Draw four rows of five dots. There are twenty dots in all. Count across the rows to see four fives. Count down the columns to see five fours.

If you know the twenty dots make four equal rows, each row has five. If you know each row has five, there are four rows. The array connects all four sentences.

Keep the total first when dividing

The product is twenty, so twenty comes first in both division facts. Changing which factor you divide by gives the other factor.

You may turn around the factors in multiplication. You cannot use that rule to turn around a division. Twenty apples shared among five children give four each. Starting with five apples and sharing among twenty children cannot give four each; you do not have enough apples.

The two jobs have changed. Keep the starting total in the right place.

When the factors match

For five groups of five, turning the factors around repeats the same sentence: $5\times 5=25$. Both division facts repeat $25÷5=5$. This family has just two different facts.

The family for $2\times 2=4$ works the same way. Its other different fact is $4÷2=2$.

A multiplication-and-division family uses its linked factors and product. Adding four and five gives nine, so $4+5=9$ is not part of the family for 4, 5, and 20.

For $35÷5$, ask, “How many fives make thirty-five?” A multiplication fact can answer that question.

Find the missing factor

You know $7\times 5=35$. Seven fives make thirty-five, so $35÷5=7$. The answer is the factor that was missing.

If you forget the fact, count “5, 10, 15, 20, 25, 30, 35.” Count the steps on your fingers. You reach thirty-five at seven steps.

Giving five as the answer repeats the group size. Giving thirty-five repeats the total. You need the number of fives that fit.

Use the three tables

Dividing by two finds half. For $18÷2$, think of the double that makes eighteen. Since $9\times 2=18$, the answer is nine. The twos division facts run from $2÷2=1$ to $20÷2=10$.

Dividing by ten counts tens. Seventy is seven tens, so $70÷10=7$. The tens division facts run from $10÷10=1$ to $100÷10=10$.

Dividing by five counts fives. Forty-five has nine fives, so $45÷5=9$. The fives division facts run from $5÷5=1$ to $50÷5=10$.

Check by rebuilding the total

To check $60÷10=6$, multiply the answer by ten: $6\times 10=60$. You get the starting total. Here, ten is the divisor, the number you divide by.

Dividing a number greater than zero by itself gives one. Five objects make exactly one group of five, so $5÷5=1$. Two objects make one pair; ten objects make one group of ten.

Same total, different group sizes

Thirty objects fill three groups of ten: $30÷10=3$. They fill six groups of five: $30÷5=6$.

Each ten splits into two fives. Groups half the size fit twice as many times into the same total. Keep the total at thirty while you compare the groups.

A question mark or box can stand for a missing number. The number you choose must make the whole sentence true.

A missing factor

For $6\times ?=30$, ask, “Six times what makes thirty?” Six fives make thirty, so the missing number is five. Put five back into the sentence to check.

For $?\times 10=80$, ask how many tens make eighty. Eight tens do, so the missing number is eight.

The answer can be on the left: $25=5\times ?$ still needs five. The equal sign asks for the same amount on both sides.

A missing number in division

In $40÷?=4$, the total is forty and there are four groups. Find the group size. Since $4\times 10=40$, the missing number is ten. Check: $40÷10=4$.

In $?÷2=9$, the starting total is missing. Put nine pairs together: $9\times 2=18$. Check by writing $18÷2=9$.

The position of the question mark changes what you need to find. Subtracting two from nine would not rebuild the missing total.

Equal amounts on both sides

For $3\times 10=?\times 3$, the left side makes thirty. The turn-around rule gives $10\times 3=30$, so the missing number is ten.

Test a missing sign

In “10 ? 2 = 20,” the question mark stands for a sign. Try addition, subtraction, multiplication, and division. Ten plus two is twelve; ten minus two is eight; ten times two is twenty; ten divided by two is five. The sign is ×.

With the same starting numbers, changing the answer changes the sign. “10 ? 2 = 5” needs ÷, “10 ? 2 = 12” needs +, and “10 ? 2 = 8” needs −.

Sometimes two signs work. In “2 ? 2 = 4,” both + and × make a true sentence. Test each allowed sign before saying you found them all.

Read what the story tells you. Then ask what you need to find. The missing part helps you choose the action.

Do you need the total?

“There are five vases with ten flowers in each vase. How many flowers are there?”

You know how many groups: five vases. You know the size of each group: ten flowers. Multiply to find the total: $5\times 10=50$ flowers.

Rows work the same way. Four rows with five chairs in every row make $4\times 5=20$ chairs. An array can show the equal groups.

Do you know the total?

“There are fifty flowers. Each vase holds ten flowers. How many vases do you need?”

Now fifty is known, and the vase count is missing. Divide: $50÷10=5$ vases. The word “each” appears in both flower stories. You need the whole story to choose the operation.

When a total is shared among a known number of groups, divide to find how much goes in each. Both kinds of division start with the total.

Work with lengths

“A ribbon thirty centimeters long is cut into pieces five centimeters long. How many pieces are there?” Each piece is one group of five centimeters. There are $30÷5=6$ pieces.

“Six strips, each ten centimeters long, are laid end to end. How long is the line?” You need the total length of six equal strips: $6\times 10=60$ centimeters.

Listen for double or half

“Twice as many” means double, so multiply by two. “Half as many” means divide by two. If Ben has sixteen cards and Lily has half as many, Lily has $16÷2=8$ cards.

Some stories need addition

“Five flowers are red and ten flowers are yellow. How many flowers are there?” You are joining two amounts, not making five groups of ten. Add $5+10=15$ flowers.

Use the given amounts and the question together. A single word cannot tell you which action will solve every story.

A drawing can help you see what a story asks. Then a number sentence records the same idea with numbers and signs.

Show what you know

“There are six bikes. Each bike has two wheels. How many wheels are there?”

You know there are six groups with two in each. Draw six circles with two dots inside each. Write $6\times 2=?$. The question mark stands for the wheel count.

Count the groups of two: “2, 4, 6, 8, 10, 12.” Finish with the answer word: “There are twelve wheels.” The question asks for wheels, so “six bikes” does not answer it.

Check by undoing the multiplication: $12÷2=6$. Twelve wheels make the six pairs you started with.

Draw the groups you need to find

“Mia puts thirty-five stickers in rows of five. How many rows does she make?”

Draw rows of five dots until you have thirty-five dots. You need the row count, so write $35÷5=?$. Seven fives make thirty-five. Answer, “Mia makes seven rows.”

Check with $7\times 5=35$. Your rows hold all the stickers.

Pick a helpful picture

Use circles and dots for groups, a rectangle of dots for an array, or equal jumps on a number line. A bar model, also called a tape diagram, shows the whole amount as one bar split into equal parts.

For forty shared into four equal parts, draw a bar labeled 40. Split it into four equal boxes. Each box is ten, since $4\times 10=40$. The drawing shows $40÷4=10$.

Read your answer back into the story

Does your answer name what the question asks for? Does your check rebuild the starting amount?

For two or more equal groups, the total is bigger than one group. Three bags of ten apples must hold more than ten apples. An answer of thirteen comes from adding the group count to the group size. Three tens make thirty.

When sharing among two or more, each share is smaller than the total. Use this size check alongside your multiplication or division check.