Addition and Subtraction Story Problems

A story problem, or word problem, is a math story with a question. Read the whole story before you start counting. You can listen while a grown-up reads it, too.

Tell the story

“6 toy boats float in a tub. Nia takes out 2 boats. How many boats are still in the tub?”

Tell it in your own words: “There were boats in the water. Nia took some out. I need to find how many stayed.” Now say what you know:

• There were 6 boats at the start.
• Nia took out 2 boats.
• The number still in the tub is missing.

You can use 6 blocks to act out the boats. Move 2 blocks away. Count the 4 blocks still in the group.

Write, “There are 4 boats still in the tub.” The word “boats” is a label. It tells what your number counts.

Let the question guide you

A question can ask about different parts of a story. Try this one: “There were 6 boats in the tub. Nia took some out. Now 4 boats are still there. How many boats did Nia take out?”

This time, you know how many stayed. You need to find how many left. You can still act it out. Start with 6 blocks. Move blocks away until 4 remain. You moved 2, so Nia took out 2 boats.

Giving “4 boats” would answer a different question. Point to the words that tell what you need to find.

Notice what happens

Some stories have things joining or leaving a group. Others put groups together without anything moving. A compare story asks about two amounts, such as who has more. Picture the story to help you choose what to do.

A number can also be extra. “Nia has 6 toy boats and 3 dolls. She puts 2 boats away. How many boats are still out?” The dolls do not change the boat count. You still work out how many of the 6 boats remain after 2 leave.

For a new story, read, retell, and name the missing amount. Show it with objects or a drawing. Then solve it, write a labeled answer, and check that your answer fits what happened.

A model shows a math story. You can make one with blocks, counters, or marks on paper. A circle can stand for a dog, a book, or a child. Your marks only need to show the amounts clearly.

Draw things joining or leaving

“4 ladybugs sit on a leaf. 5 more ladybugs land. How many ladybugs are there now?” Draw 4 circles. Leave a little space, then draw 5 more. Count all 9 circles. Write “ladybugs” beside your drawing.

For things leaving, draw the starting group first. “There are 10 crackers. Eli eats 3. How many crackers are left?” Draw 10 circles and cross out 3. Count the circles that are still there: 7 crackers. Crossed-out circles show the crackers Eli ate.

Fill a ten frame

A ten frame has 10 spaces in two rows of 5. Put one counter in each space. A full frame shows 10 at a glance.

To show $7+5$, put 7 counters in the frame. There are 3 empty spaces. Use 3 of your 5 new counters to fill them. The other 2 go outside.

You used all 5 new counters: 3 inside and 2 outside. Moving them into the frame did not add any extra counters.

Hop on a number line

A number line shows numbers in order. To add, hop right. To take away, hop left.

For $10-3$, put your finger on 10. Make 3 hops left. You land on 9, then 8, then 7. Your landing place gives the answer.

Count the hops after you start moving. For $4+2$, start at 4. The first hop lands on 5. The second lands on 6. Saying “4, 5” counts the starting place and misses a hop.

Show tens with fewer marks

“20 paintbrushes are in one pot. 40 are in another.” Drawing 60 tiny circles would take a while. Draw one stick for each ten instead. Label it “1 stick = 10 paintbrushes.” Two sticks and four sticks make six sticks, or 60 paintbrushes.

Choose marks that are easy to read. Labels show what each group or stick means.

You can write a story with numbers and signs. This is a number sentence, also called an equation. A box marks the number you need to find.

Read the signs

The plus sign means add. The minus sign means subtract. The equal sign means “is the same as.”

“Luz has 5 markers. She gets 4 more. How many markers does she have now?”

The 5 stands for the markers Luz had. The 4 stands for the new markers. The box stands for how many she has now. Since 5 and 4 make 9, write 9 in the box.

The minus sign can show things leaving. It can also find a gap between amounts. For 9 markers when 4 are put away, $9-4=5$ finds what stays. When you compare 9 markers and 5 markers, $9-5=4$ finds how many more.

Both sides must match

Read $9=5+4$ aloud: “9 is the same as 5 plus 4.” It is true, even with the total on the left. You can also write:

Both sides make 9. The equal sign joins two amounts that match. It does not tell you where to put an answer.

To check a number sentence, work out both sides. $5+4=10$ is false: the left side makes 9, but the right side is 10.

Put the box where something is missing

“Luz had 5 markers. She got some more. Now she has 9.” The new markers are missing:

Try 4 in the box. $5+4=9$ is true, so Luz got 4 markers. You can find the same answer with $9-5=\mathrm{\square }$.

Now change the question: “Luz had some markers. She got 4 more. Now she has 9. How many did she have at first?” The starting amount is missing:

The box can go at the start, in the middle, or at the end. Some books use a question mark instead. Say what each number means before you solve. Writing $4+9$ would count the new markers twice, because they are already inside the final 9.

In an add to story, more things join a group. In a take from story, things leave. When you know the start and the change, you can find the end.

Some join

“8 children are at the art table. 6 more children come over. How many children are at the table now?”

The starting group has 8 children. The change is 6 children joining. Add to find the end:

You can make a ten. Split the 6 into 2 and 4. First add 2 to the 8. Then add the other 4.

There are 14 children at the table now. You could also count on 6 from 8. For a sum like $3+9$, starting at the bigger number, 9, gives you fewer counts.

Some leave

“16 crayons are in a box. Rosa takes out 7 crayons. How many crayons stay in the box?”

You can count back 7 from 16. Or use an addition fact: “7 and what make 16?” Since $7+9=16$, you know $16-7=9$. There are 9 crayons still in the box.

Check the size of the answer. Taking out crayons leaves fewer than 16. Adding the two counts would give more crayons, which does not fit what happened.

What if none join or leave?

“4 children are in a reading corner. No more children come.” Adding zero keeps the count at 4: $4+0=4$. If no children leave, $4-0=4$. Zero means there is no change in the count.

Use the same idea with cents, lengths, and tens

“Pia has 7 cents. She finds 5 more cents.” Adding gives $7+5=12$. Pia has 12 cents now.

“A strip of paper is 18 centimeters long. You cut 8 centimeters off one end.” Subtracting gives $18-8=10$. The piece that remains is 10 centimeters long.

For whole tens, count the tens. “There are 30 cups on a shelf. You put 40 more cups there.” Three tens plus four tens make seven tens: $30+40=70$. There are 70 cups on the shelf now.

The change tells how many things joined or left. Sometimes the story gives you the start and the end. Then you need to find what happened between them.

Find how many joined

“Tess had 6 beads. Her uncle gave her some more. Now she has 14 beads. How many beads did he give her?”

Write the story with a box for the new beads:

Draw 14 circles for the end amount. Circle 6 to show the beads Tess already had. The other 8 circles show the beads she got.

You can also count on from 6 to 14. Go from 6 to 10 first: 4 hops. Go from 10 to 14: 4 more hops. Together, that is 8 hops. Tess got 8 beads.

Subtracting finds the same missing piece:

Use a fact family

An addend is a number you add. The box in $6+\mathrm{\square }=14$ is a missing addend. Addition and subtraction use the same three amounts in a fact family:

The addition facts put the parts together. The subtraction facts take one part from the whole. A fact you know can help you find the missing change.

Find how many left

“17 oranges were in a basket. Some were eaten. Now 8 oranges are left. How many oranges were eaten?”

The 8 left and the eaten oranges together make the starting 17. Think, “8 and what make 17?” Since $8+9=17$, 9 oranges were eaten.

Check by following the story: $17-9=8$. There really are 8 left.

Answer the change question

In Tess’s story, 14 is how many beads she has now. It is not how many she got. Adding 6 and 14 would count her first 6 beads again. Look for the missing piece between the start and the end.

A savings story works the same way: “You had 4 cents. Now you have 10 cents after saving some more.” The new savings are $10-4=6$ cents. Say, “I saved 6 more cents.”

A story can begin with “some” because the starting number is missing. You know what happened next and how many there were at the end. Start with that end amount and undo the change.

Undo things joining

“Some frogs sat on a rock. 5 more frogs hopped onto it. Now there are 12 frogs. How many frogs were there at first?”

The box belongs at the start:

Draw 12 circles for all the frogs at the end. Mark 5 of them as the frogs that came later. Count the unmarked circles: 7 frogs were there at first.

You are taking the new frogs out of the final group:

Tell the story forward to check. Start with 7 frogs, then add the 5 that came. $7+5=12$ matches the end amount.

You could also think of $5+\mathrm{\square }=12$. Switching the two addends keeps the total the same. Count on from 5 to 12: 7 counts.

Undo things leaving

“Some books were on a cart. A librarian took away 6 books. Now 8 books are on the cart. How many were there at first?”

Make a group of 8 counters for the books left. Put back 6 counters for the books taken away. Together they show the starting group.

There were 14 books on the cart at first. Check forward: $14-6=8$.

Look past a clue word

The frog story says “more,” but you subtract to find the start. The book story says “took away,” but you add to rebuild the start. A word tells you what happened. The question tells you which amount is missing.

The frogs’ starting group must be smaller than 12 because 5 joined. The books’ starting group must be bigger than 8 because 6 left.

In a put together story, groups are there at the same time. Nothing needs to arrive or leave. You add the groups to find how many there are in all.

Two parts make a whole

“A shelf holds 8 animal books and 5 space books. How many books are on the shelf?”

The animal books are one part. The space books are another part. All the books together are the whole, or total.

There are 13 books on the shelf. The label “books” fits both groups. Calling the answer “13 animal books” would give the whole the wrong label.

You can draw a part-part-whole model, also called a number bond. Draw a circle for the whole. Join it with lines to two circles for the parts. Write 8 and 5 in the part circles. Put a question mark in the whole circle, then fill in 13 when you solve.

Choose an easier adding order

You can add the parts in either order:

Both sides make 13 books. Starting at 8 and counting on 5 takes fewer counts than starting at 5 and counting on 8. You are counting the same books either way.

Include a third part

“A bowl holds 3 peaches, 7 plums, and 4 pears. How many pieces of fruit are in the bowl?”

There are three parts. You can draw a whole circle linked to three part circles. Look for two parts that make ten:

There are 14 pieces of fruit. You can add three numbers in a different order and keep the same total. Choosing 3 and 7 first makes this sum easier.

Touch each part in the story as you use it. Stopping at 10 would leave out the pears. Adding the 4 pears twice would count fruit that is not there.

Put amounts of money together

“A card costs 6 cents. An envelope costs 9 cents. How many cents do they cost together?” The two costs are parts of the total cost.

The card and envelope cost 15 cents together. The numbers count cents, so the answer uses cents, too.

A take apart story gives you a whole. You know one part and need the other. The known part is already inside the whole.

Start with all of them

“A box holds 16 pencils. 7 are yellow and the rest are green. How many pencils are green?”

The whole is 16 pencils. One part is 7 yellow pencils. Draw 16 circles and mark 7 yellow. Count the unmarked circles: 9 are green.

You can also ask, “7 and what make 16?”

This is a missing-addend question. Counting on from 7 to 16 gives 9 counts. Both ways find the same green part.

Keep the whole and parts in their places

To subtract, start with the whole: 16. Take away the known part: 7. Adding 16 and 7 would count the yellow pencils twice. They are among the 16 already.

The missing part cannot be bigger than the whole. A guess of 23 green pencils would be too many for a box of 16 pencils. Check the actual answer with $7+9=16$.

Addition can swap the parts: $7+9=9+7$. Subtraction does not work in either order. Taking 7 from 16 is different from trying to take 16 from 7.

Number pairs you know help. “10 tiles are on a table. 6 are white and the rest are black.” Since 6 and 4 make 10, there are 4 black tiles.

Find all the ways to split a whole

“4 flags are each red or blue. How many of each color could there be?” One color can have no flags. Make an ordered list:

Each row still has 4 flags. As red goes up by 1, blue goes down by 1. This finds all 5 color splits without skipping any.

Find a part of a length

“A row of cubes is 15 centimeters long. The red part is 6 centimeters long. The rest is blue.” The blue part is $15-6=9$ centimeters long. The whole length includes both colors.

A tape diagram, or bar model, shows an amount with a long rectangle. The full bar is the whole. Pieces inside it show the parts.

Build a split bar

“5 toy trucks and 9 toy cars are in a bin. How many toy vehicles are there?”

1. Draw a long bar.
2. Split it into two pieces.
3. Label the shorter piece 5 trucks and the longer piece 9 cars.
4. Draw a line with downturned ends above the full bar. This bracket shows the whole.
5. Put a question mark above it for all the vehicles.

The 9-piece is longer because 9 is more than 5. You do not need to measure the pieces exactly. The bar shows that the two parts fit inside one whole.

The whole is missing, so add the parts:

There are 14 toy vehicles in the bin.

Put the question mark inside

“A bin holds 14 toy vehicles. 5 are trucks and the rest are cars. How many are cars?”

Draw the same split bar. This time, write 14 over the full bar. Write 5 inside the truck piece and ? inside the car piece. The whole is known. A part is missing.

There are 9 toy cars. Do not draw 14 as one piece beside 5. The 14 already counts every vehicle in the bin.

Use bars for things that happen

“7 children were in line. Some more joined. Now 12 children are in line.” The final 12 is the whole. Its parts are 7 and ?. Find the new children with $12-7=5$.

“Some children were in line. 4 left. Now 9 remain.” Here, the starting group is the whole. Its parts are 4 who left and 9 who stayed. Find the start with $4+9=13$.

Add a third piece

A bar can have three parts. For 2 green beads, 8 purple beads, and 6 white beads, draw three pieces. Write 2, 8, and 6 inside them. Put ? above the full bar. The total is $2+8+6=16$ beads.

A number bond and a split bar show the same parts and whole. Use the model that helps you see where the missing amount belongs.

A compare story asks how two amounts differ. You keep the groups separate and look for the gap between them. That gap is the difference.

Match one with one

“Ivy has 12 buttons. Max has 7 buttons. How many more buttons does Ivy have?”

Draw 12 circles in one row for Ivy. Draw 7 below them for Max. Begin both rows at the same place. Match one of Max’s circles to one of Ivy’s circles. When all 7 of Max’s circles have partners, 5 of Ivy’s do not.

Ivy has 5 more buttons than Max. Those 5 unmatched circles show the difference.

Draw two bars

You can draw a long bar for Ivy and a shorter bar for Max. Line up their left ends. Label the bars 12 and 7. Mark the extra piece at the right of Ivy’s bar with ?.

The extra piece is what you need to find. Subtract the smaller amount from the bigger amount:

Or ask how much you need to add to 7 to reach 12:

Counting on gives 5 counts. Both ways measure the same gap.

Turn the question around

“How many fewer buttons does Max have than Ivy?” still asks about the gap between 7 and 12. Max has 5 fewer buttons. Ivy has 5 more buttons.

The word “more” looks from the bigger side. The word “fewer” looks from the smaller side. Neither question asks for the two groups together. Adding $12+7$ gives 19 buttons in all, which is a different amount.

Be careful about what the difference counts. Ivy has 12 buttons, not 5. The 5 counts only her extra buttons after you match the groups.

Compare lengths

“One ribbon is 17 centimeters long. Another ribbon is 10 centimeters long. How much longer is the first ribbon?”

Imagine the ribbons lined up at one end. The extra length is $17-10=7$ centimeters. The first ribbon is 7 centimeters longer. The second is 7 centimeters shorter.

A difference cannot be bigger than the bigger amount when you compare these whole-number counts. Check that your answer fits the space between the two amounts.

Sometimes you know one amount and the difference. You need to find the other amount. First say who has more. That tells you which bar should be longer.

Find the bigger amount

“Jin has 6 cards. Ada has 5 more cards than Jin. How many cards does Ada have?”

Ada has more, so her bar is longer. It has a part matching Jin’s 6 and an extra part of 5. Add those parts:

Ada has 11 cards. The bigger amount is the smaller amount plus the difference.

Find the smaller amount

“Ada has 11 cards. Jin has 5 fewer cards than Ada. How many cards does Jin have?”

Jin has fewer, so his bar is shorter. Start with Ada’s 11 and take off the extra 5:

Jin has 6 cards. The smaller amount is the bigger amount minus the difference.

Follow the people, not just one word

“Ada has 11 cards. She has 5 more cards than Jin. How many cards does Jin have?”

This story says “more,” but Ada has more. You are finding Jin’s smaller amount. Subtract 5 from 11 to get 6.

Now read this: “Jin has 6 cards. He has 5 fewer cards than Ada. How many cards does Ada have?”

It says “fewer,” but Jin has fewer. Ada has the bigger amount, so add 5 to 6 to get 11. You can put the names beside two bars to keep track.

Check the gap

If Ada has 11 and Jin has 6, then $11-6=5$. The difference matches the 5 in the story. A guess of 16 for Jin would put him above Ada, even though she has more.

Compare ages and heights

“Bea is 6 years old. Her cousin is 4 years older. How old is her cousin?” Add to find the older child’s age: $6+4=10$. Her cousin is 10 years old.

“A tower is 15 cubes tall. A second tower is 7 cubes shorter.” Subtract to find the shorter tower: $15-7=8$. It is 8 cubes tall. “Older,” “taller,” and “shorter” compare amounts just as “more” and “fewer” do.

Checking means trying your answer in the story. A correct number sentence matters, and so does what the answer counts. You can explain both with a few words and a drawing.

Tell the story with your answer

“A box held 18 paper stars. You hung some up. Now 11 stars are in the box. How many did you hang up?”

Suppose you find 7. Put that number into the story: “There were 18 stars. I hung up 7. There were 11 left.”

It matches. The answer is “I hung up 7 paper stars.” Saying “11 stars” would name what stayed in the box.

Use the opposite operation

Addition and subtraction undo each other. You can check the star answer by putting the two parts together:

You get back to the starting amount. To check an addition such as $9+4=13$, subtract one part: $13-4=9$.

Think about size

Before working out an exact answer, think about how big it should be. “A sticker costs 6 cents. A tag costs 8 cents.” Both costs are more than 5 cents and less than 10 cents. Together they must cost more than 10 cents and less than 20 cents.

The total, 14 cents, fits. A total of 41 cents is too big. A rough size check can catch a mistake, but you still need the exact answer.

A part cannot be bigger than its whole. After some things leave, fewer remain. A difference is no bigger than the bigger amount. Use what happened to check the size you found.

Choose the right numbers and label

“Leo has 13 toy planes and 2 toy trains. He gives away 4 planes. How many planes does he have now?” The trains are extra information for this question.

An explanation could be, “I drew 13 planes and crossed out 4. There were 9 planes left.” The drawing, calculation, and label all tell the same story.

A clue word cannot do this thinking for you. A story can say “left” and still ask for the start. Read the question and picture the amounts. If counting on and subtracting give different answers, check the counts in both ways.

In a two-step story, you find one number first. Then you use it to find the final answer. Ask, “What do I need to find first?”

Follow two changes

“9 children are playing outside. 4 more children come. Then 6 children go inside. How many children are outside now?”

First find how many are outside after the new children come:

The 13 is a middle answer. It tells how many children are outside before anyone goes inside. Use it for the next step:

There are 7 children outside now. On a number line, start at 9. Hop right 4 to reach 13, then left 6 to reach 7. The two groups of hops show the two changes.

Stopping at 13 would miss the children going inside. Adding every number would treat the 6 leaving as if they joined. Follow what each group does.

Find one person’s amount, then the total

“Milo has 4 stamps. Noor has 3 more stamps than Milo. How many stamps do they have together?”

You know Milo’s count. First find Noor’s:

Now add the two children’s amounts:

They have 11 stamps together. The 3 is the gap between their amounts. Adding only 4 and 3 finds Noor’s stamps, not their total.

Keep track of three parts

“A reading basket has 6 animal books, 2 joke books, and 4 storybooks.” To find the total, all three parts must be included. You can choose an easy pair first:

There are 12 books. All three groups were there together; no group arrived or left. You used two calculations to add three parts.

Try whole tens

“70 sheets of paper are in a tray. You take out 30 sheets. Then you put 20 new sheets in. How many sheets are in the tray now?”

First, $70-30=40$. Then, $40+20=60$. There are 60 sheets in the tray now. Write down the first answer so you can use it in the next step.

You can turn a number sentence into a story for someone else to solve. Choose things to count, say what happens, and ask about the missing number.

Start with a number sentence

Try $7+2=\mathrm{\square }$. Here is one matching story:

“7 turtles rest beside a pond. 2 more turtles crawl over. How many turtles are there now?”

The 7 tells the start. The 2 tells how many join. The question asks for the total in the box. Solving gives 9 turtles.

Now try $13-6=\mathrm{\square }$. You can make different kinds of stories:

• Take from: “13 grapes are in a bowl. You eat 6. How many grapes are left?”
• Take apart: “13 socks are in a drawer. 6 are white and the rest are blue. How many are blue?”
• Compare: “You have 13 blocks. Your friend has 6 blocks. How many more blocks do you have?”

Each story fits the same subtraction. The answer has a different label in each one.

Start with a bar

Imagine a bar with 12 above the whole. One piece is 5 and the other is ?. A matching story could be:

“12 cups are on a tray. 5 are empty and the rest are full. How many cups are full?”

The whole is 12 cups. The parts are empty cups and full cups. Subtracting $12-5=7$ shows there are 7 full cups.

Choose something to find

Your question decides what is missing. “How many are there now?” asks for the end. “How many joined?” asks for the change. “How many were there at first?” asks for the start. “How many more?” asks for a difference.

If you write, “7 turtles rest by a pond. 2 more come. How many turtles came?”, you already told the answer: 2. Change the question to ask how many are there now.

Try solving your story

Make sure your numbers fit what happens. A child with 4 crackers cannot eat 6 of those crackers. Give enough information to find the missing amount, then solve your own story.

Ask a friend or grown-up to try it, too. They can draw a model and tell you how they solved it. Check that their answer matches the story you meant to tell.