Two-step addition and subtraction problems
A story changes an amount twice, or compares two amounts and then asks about both. The question cannot be answered from the numbers in the story directly. What has to be found first, and how does a child keep track of it?
Where you are 7 in Addition and subtraction problems in words
When you reach for it
A one-step story gives two numbers and asks about a third that can be found from them at once. A two-step story asks a question that the numbers in it cannot answer directly. Something else has to be found first, and the story never says what. Read this one aloud:
“The food pantry had 45 cans. 28 more cans were brought in. Then 37 cans went out to families. How many cans does the pantry have now?”
The question is about the end of the day. The 37 cans went out of a pile, but the story never says how big that pile was. Before anyone can take 37 away, they need the answer to a question the story does not ask out loud: how many cans were there after the 28 came in? That is the hidden question. Answer it, and the story becomes an ordinary one-step story with a known start.
The whole lesson turns on that idea. A child who has met stories with the start, change or result missing and stories that compare two amounts already has every single step. What is new is seeing that a story holds two of them, one after the other, and that the answer to the first is a number in the second. The arithmetic stays within 100 and uses the adding and subtracting the child already knows: an open number line, partial sums, and regrouping in addition and subtraction.
The lesson is taught over four days:
- Part 1: two changes, one after the other. Something is added and something is taken away.
- Part 2: two changes of the same kind, and three parts of one whole. Here the order of the steps can be chosen.
- Part 3: a comparison, then a total or a change. The first step finds an amount the story only describes (“17 more than Seble”).
- Part 4: the missing number sits inside the second step, and the child checks whether an answer fits the story.
The procedure
Part 1: two changes in a row
Read the pantry story aloud again and act it out slowly, one sentence at a time. After each sentence, stop and ask: “How many cans are on the shelves right now?”
- “The food pantry had 45 cans.” Right now: 45.
- “28 more cans were brought in.” Right now: we do not know yet, but we can find it.
- “Then 37 cans went out.” This needs the number from the line before.
That pause after each sentence is how the hidden question shows itself. The second sentence changes the amount, and the third sentence acts on the changed amount. So the first job is the first change:
After the delivery the pantry has 73 cans. Only now does the third sentence make sense as a step, and it is a plain take-away from a known start:
The pantry has 36 cans now.
Draw one step at a time. A picture that tries to show the whole story at once gets crowded. An open number line for each step keeps it clear. The first line starts at 45 and hops 28 to the right. Where it lands, 73, is copied to the second line, and the second line hops 37 to the left from there.
The number copied from one line to the next is the answer to the hidden question. Children often circle it, because it is the number the story never printed and the one everything after it depends on.
Two equations, or one. Each step gets its own equation, as above. The two can also be written as one, read left to right in the order the story happens:
Read from left to right, this means: start at 45, add 28, then take away 37. That is exactly the story. Two equations are easier to check line by line against the story; one equation shows the whole story at a glance. Both are correct, and a child may use either.
Does the answer fit? One quick look saves many slips. More cans went out (37) than came in (28), so the pantry must end with fewer than the 45 it started with. The answer 36 is fewer than 45, which fits. In fact 37 is 9 more than 28, so the pantry ends with 9 fewer than it started: 45 - 9 = 36. That second road to the same answer is a strong check.
Part 2: two changes of the same kind
Now a story where both changes take away: “Bojan had 50 carnival tickets. He used 17 on rides and 15 on games. How many tickets does he have left?”
There are two hidden questions to choose from, and both work.
- Follow the story. How many were left after the rides? 50 - 17 = 33. Then the games: 33 - 15 = 18.
- Put the two uses together first. How many did he use in all? 17 + 15 = 32. Then 50 - 32 = 18.
Both roads give 18. They have to, and a bar shows why. All 50 tickets are the whole. The tickets for rides, the tickets for games and the tickets left are its three parts, and nothing else happened to any ticket.
Taking away 17 and then 15 removes the same two pieces of the bar as taking away the 32 that the two pieces make together. What is left is the same piece either way. Written as numbers:
So taking away one amount and then another is the same as taking away both amounts together. The same is true of adding: adding 20 and then 8 is the same as adding 28. Choosing the order is a real choice, not a trick, and a child can pick whichever is easier. Here 17 + 15 = 32 is quick, and 50 - 32 is one short count up from 32, instead of two subtractions.
Three parts of one whole come up in stories that do not sound like change at all: “24 cans of soup, 31 cans of beans, and the rest corn”. The same picture applies. Two of the parts put together answer the hidden question “how many are not corn?”, and the whole take away that gives the third part. When all three parts are known and the whole is wanted, adding three numbers can be done in any order; moving them around is the commutative property, and choosing which two to add first is the associative property. A child who spots two numbers that make a ten, such as 26 and 4, can add those first.
Part 3: a comparison inside the story
Some stories hide the first step in a comparing phrase: “Seble sold 36 carnival tickets. Lanakila sold 17 more than Seble. How many tickets did they sell together?”
The question asks for a total, so it needs both amounts. Seble’s is printed. Lanakila’s is not; the story only says how it compares with Seble’s. That is the hidden question: how many did Lanakila sell? “Seventeen more than Seble” means Lanakila’s amount is Seble’s amount and 17 more.
The bars make this plain. Lanakila’s bar is as long as Seble’s, and then 17 longer. So the first step is
and only now can the question be answered:
The 17 is used only once, inside Lanakila’s amount. The answer is not 36 + 17; that is Lanakila’s tickets alone, and the question asked for both children.
The comparing step can also run the other way. “Fewer than” tells you an amount is smaller: if the pantry has 25 fewer cans of fruit than its 64 cans of soup, the fruit is 64 - 25. And a comparing step can come second: a story can change two amounts and then ask how far apart they are now. Either way the comparison itself is the work of How many more and how many fewer; what is new here is noticing that its answer is only a stepping stone.
Part 4: the box in the second step, and checking the answer
In Parts 1 to 3 the first step was all known and the box was the final answer. A story can also put the missing number inside the second step: “The pantry had 52 cans. 19 went out in the morning. Then some cans came in. At closing time the pantry had 61 cans. How many came in?”
The first step is fully known, so do it first: 52 - 19 = 33. Now the second step is a story with the change missing, the kind met in Problems with the start, change or result missing: 33 cans, some come in, and then there are 61. In story order,
Count up from 33: 7 to reach 40, then 21 more to reach 61. Together that is 28, so 28 cans came in. Run the story forward to check: 52 - 19 = 33, and 33 + 28 = 61, the closing number.
The same story as one equation is 52 - 19 + \square = 61, with the box where the story put its unknown.
Checking whether an answer fits is the last habit of the lesson, and it uses no new arithmetic. Three questions catch most slips:
- Should the answer be more or less than the start? If a story only takes things away, the end is less than the start. If more comes in than goes out, the end is more.
- Is every middle number possible? You cannot take 37 cans out of a pile of 30. If a middle number comes out impossible, the steps were set up in a way the story does not allow.
- Put the answer back. Run the story forward with the answer in place. If it lands on the number the story gives, it fits.
Worked examples
Examples 1 and 2 go with Part 1, Examples 3 and 4 with Part 2, Examples 5 and 6 with Part 3, and Examples 7 and 8 with Part 4.
Jars in, jars out
Read aloud: “The food pantry had 39 jars of peanut butter. 46 more jars were brought in. Then 58 jars went out to families. How many jars does the pantry have now?”
SolutionThe 58 jars went out of the pile that was there after the delivery, so the hidden question is “how many after the delivery?” Add the jars that came in. With partial sums, 39 + 46 is 30 + 40 = 70 and 9 + 6 = 15, and 70 + 15 = 85:
Now take away the jars that went out. On an open number line, as in Part 1, hop back 50 from 85 to 35, then 8 more to 27:
The pantry has 27 jars now.
Check. More went out than came in, so the end should be less than 39; 27 is. The difference of the changes is 58 - 46 = 12, and 39 - 12 = 27.
Tickets spent, tickets won
Read aloud: “Lanakila had 62 carnival tickets. He spent 34 at the ring toss. Then he won 19 more. How many tickets does he have now?”
SolutionThe tickets he won were added to what was left after the ring toss, so first find that: how many after the ring toss? Taking 34 from 62 needs a regroup, since 2 ones cannot give 4. Trade a ten: 62 is 5 tens and 12 ones, and 12 - 4 = 8, 5 - 3 = 2 tens.
Then the tickets he won join those 28:
Lanakila has 47 tickets now. As one equation, 62 - 34 + 19 = 47.
Check. He spent more (34) than he won (19), so he should end below 62. He spent 15 more than he won, and 62 - 15 = 47. Both roads agree.
Rides and games
Read aloud: “Seble had 72 carnival tickets. She used 27 on rides and 33 on games. How many tickets does she have left?”
SolutionDraw the bar as in Part 2: all 72 tickets are the whole, with three parts: rides, games and the tickets left. Two orders work.
Following the story: after the rides Seble had 72 - 27 = 45. After the games she had 45 - 33 = 12.
Putting the uses together first: the hidden question is “how many did she use in all?” That is 27 + 33 = 60, since 20 + 30 = 50 and 7 + 3 = 10. Then
Seble has 12 tickets left. The second order is easier here: the two uses make a whole number of tens, so it is one subtraction with no regrouping instead of two subtractions, the first of which needs a trade.
Check. The three parts must make the whole: 27 + 33 + 12 = 72.
Soup, beans and corn
Read aloud: “The food pantry got 80 cans on Saturday. 24 were soup, 31 were beans, and the rest were corn. How many cans of corn were there?”
SolutionThe 80 cans are the whole, split into three parts: soup, beans and corn. Two parts are known. The hidden question is “how many cans were not corn?”, so put the two known parts together:
The corn is what is left of the whole:
There were 25 cans of corn. Counting up works well for the second step: from 55, five more reach 60, and 20 more reach 80, so the gap is 25.
The other order also works: 80 - 24 = 56 cans were not soup, and 56 - 31 = 25 of those were not beans either.
Check. 24 + 31 + 25 = 80.
Eighteen more than Bojan
Read aloud: “Bojan sold 34 carnival tickets. Seble sold 18 more than Bojan. How many tickets did they sell together?”
SolutionThe question wants a total, and only Bojan’s amount is printed. The hidden question is “how many did Seble sell?” Drawn as in Part 3, Seble’s bar is Bojan’s bar and 18 more:
Now both amounts are known. Together they are
They sold 86 tickets together. Partial sums: 30 + 50 = 80 and 4 + 2 = 6.
Check. Seble sold more than Bojan, so together they sold more than two of Bojan’s amounts: 34 + 34 = 68, and 68 + 18 = 86. That is the same total found a second way.
Fewer cans of fruit
Read aloud: “The pantry had 64 cans of soup. It had 25 fewer cans of fruit than cans of soup. Then 18 cans of fruit came in. How many cans of fruit does the pantry have now?”
SolutionThe question is about fruit, and the fruit was never counted in the story. “25 fewer than the soup” is the hidden question: how many cans of fruit were there at first? Fewer means smaller, so
Then the 18 cans that came in join the fruit:
The pantry has 57 cans of fruit now. The 64 cans of soup were used only to find the fruit; the soup itself did not change.
Check. At first the fruit was 25 short of the soup. 18 cans came in, which closes 18 of that gap, so the fruit is now 25 - 18 = 7 short of the soup: 64 - 7 = 57.
The box in the first step
Read aloud: “Lanakila had 81 carnival tickets. He used some on rides. Then he won 16. Now he has 58. How many did he use on rides?”
SolutionThis time the box sits in the first step, and the step that is fully known comes second. Do the known step first, working back from the end: before he won 16, Lanakila had
So the rides took him from 81 down to 42. That is a change-unknown story:
Count up from 42: 8 to reach 50, then 31 more to reach 81. Together that is 39, so \square = 39. Lanakila used 39 tickets on rides.
Check. Run the story forward: 81 - 39 = 42, and 42 + 16 = 58, the number the story ends on. It also fits on sight: he ended 23 below where he started even after winning 16, so the rides took more than 23, and 39 is more than 23.
Won some, used some
Read aloud: “Bojan had 47 carnival tickets. He won 29 at the ring toss. Then he used 31 on the big slide. How many tickets does he have now?”
SolutionFollowing the story: after winning, Bojan had 47 + 29 = 76. After the slide he had
There is a second road, and it shows how the two changes work together. He won 29 and used 31. Those almost cancel: he used 2 more than he won. So the two steps together take away 2:
Bojan has 45 tickets. Either way the answer is 45, and finding it both ways is a check in itself. Comparing the two changes first works whenever one change adds and the other takes away: it is the same idea as the check in Example 1, used to find the answer rather than to check it.
Boundaries
Every middle number has to be possible. “The pantry had 20 cans. 35 went out. Then 40 came in.” As numbers, 20 + 40 - 35 = 25. As a story it cannot happen in that order: there were never 35 cans to send out. A story whose middle number would be less than zero has an error in it. Changing the order of the steps on paper does not repair the story itself.
Combining two changes works for two of the same kind. Taking away 17 and then 15 is the same as taking away 32, and adding 20 and then 8 is the same as adding 28. Taking away 17 and then adding 15 is not the same as taking away 32; it is the same as taking away 2, the amount by which the two changes differ (Example 8).
Only like amounts combine. Cans add to cans and tickets to tickets. In Example 6 the soup was used to find the fruit, but soup and fruit were never added, because the question was about fruit only. A story that asks “how many cans in all” may add soup and fruit, since both are cans.
A two-step story needs one unknown. Every story here leaves exactly one number for the child to find, with the middle number fixed by the first step. “The pantry had 40 cans, some came in, some went out, and now it has 50” has many answers: 10 in and none out, 20 in and 10 out, and so on.
The numbers here stay within 100. Every start, middle and end number in this lesson is 100 or less. The same story shapes work with any numbers, and later they return with numbers up to 1{,}000, and then with all four operations in problems using several operations.
Where this goes next
Every step in a two-step story is a one-step story from earlier lessons: the total missing, the change missing, the start missing or a comparison. The lesson’s real content is the joint between steps: the number one step makes and the next step uses. Bar models carry the same pictures to stories with more parts and bigger numbers. With multiplication and division added, the joint stays the same and the steps get richer: problems using several operations. Much later, one equation with a box, such as 52 - 19 + \square = 61, becomes an equation with a letter, and the habit of doing the known step first is how such equations are solved.
Exercises
Part 1: two changes in a row
- “The pantry had 38 cans. 24 cans came in. Then 41 cans went out. How many cans are there now?”
- “Seble had 70 carnival tickets. She used 26 on rides. Then she won 15. How many tickets does she have now?”
- 27 + 35 - 18 = \square
- 81 - 46 + 23 = \square
- “The pantry had 56 bags of rice. 29 bags went out on Monday. 33 bags came in on Tuesday. How many bags are there now?”
- “Lanakila had 44 tickets. He won 18, then used 25.” Draw each step on its own open number line and find how many tickets he has now.
Part 2: the same kind of change, and three parts
- “Bojan had 64 carnival tickets. He used 23 on rides and 19 on games. How many does he have left?” Find it two ways.
- “The pantry had 90 cans. It gave 35 to one shelter and 28 to another. How many cans are left?”
- 75 - 18 - 22 = \square. Which order is easier? Find the box.
- “A family bought 26 ride tickets, 17 game tickets and 4 food tickets. How many tickets did they buy?”
- “The pantry got 100 cans. 36 were peaches, 27 were pears, and the rest were applesauce. How many cans of applesauce were there?”
- “Seble had 83 tickets. She used 40 on rides and 9 on a game. How many does she have left?”
Part 3: a comparison inside the story
- “Seble has 28 tickets. Bojan has 15 more than Seble. How many tickets do they have together?”
- “The pantry got 47 cans on Monday and 19 fewer cans on Tuesday. How many cans did it get on the two days?”
- “Lanakila has 63 tickets. Seble has 24 fewer than Lanakila. Then Seble wins 12. How many tickets does Seble have now?”
- “The pantry has 32 boxes of cereal and 45 boxes of pasta. 16 boxes of pasta go out. Now are there more boxes of cereal or of pasta, and how many more?”
- “Bojan has 54 tickets. That is 18 more than Lanakila has. Then Lanakila wins 25. How many tickets does Lanakila have now?”
Part 4: the box in the second step
- “The pantry had 43 cans. 26 came in. Then some went out. Now there are 38. How many went out?”
- “Seble had 60 tickets. She used some on rides. Then she won 14. Now she has 49. How many did she use on rides?”
- Write one equation with a box for this story, then find the box: “Bojan had 37 tickets. He won 45, then used 28.”
- Tell a food pantry story that goes with 64 - 27 + 30 = \square. Then find the box.
Explain
- Why is taking away 26 and then 14 the same as taking away 40? Use a bar.
- Lanakila had some tickets. He won 25 and then used 25. Without knowing how many he had, what can you say about how many he has now?
- “The pantry had 45 cans. 30 came in. Then 80 went out.” Can this story be true? How do you know?
- “Seble had 52 tickets. She used 18, and then she used 23 more.” Without working it out, could the answer be 91? Then find the answer.
Answers
- 21 cans. Hidden question: how many after the delivery? 38 + 24 = 62. Then 62 - 41 = 21.
- 59 tickets. After the rides: 70 - 26 = 44. Then 44 + 15 = 59.
- 44. Left to right: 27 + 35 = 62, then 62 - 18 = 44.
- 58. Left to right: 81 - 46 = 35, then 35 + 23 = 58.
- 60 bags. After Monday: 56 - 29 = 27. After Tuesday: 27 + 33 = 60.
- 37 tickets. Step 1: start at 44 and hop 18 to the right to 62, so 44 + 18 = 62. Step 2: start at 62 and hop 25 to the left to 37, so 62 - 25 = 37.
- 22 tickets. Following the story: 64 - 23 = 41, then 41 - 19 = 22. Uses first: 23 + 19 = 42, then 64 - 42 = 22.
- 27 cans. Given away in all: 35 + 28 = 63. Left: 90 - 63 = 27.
- 35. Putting the two together first is easier, because 18 + 22 = 40 is a whole number of tens. Then 75 - 40 = 35. Following left to right also works: 75 - 18 = 57, 57 - 22 = 35.
- 47 tickets. Add 26 and 4 first, since they make a ten: 26 + 4 = 30. Then 30 + 17 = 47. Adding in any order gives the same total.
- 37 cans. Not applesauce: 36 + 27 = 63. Applesauce: 100 - 63 = 37.
- 34 tickets. 83 - 40 = 43, then 43 - 9 = 34. Or uses first: 40 + 9 = 49, and 83 - 49 = 34.
- 71 tickets. Hidden question, how many does Bojan have? 28 + 15 = 43. Together: 28 + 43 = 71.
- 75 cans. Tuesday: 47 - 19 = 28. Both days: 47 + 28 = 75.
- 51 tickets. Seble at first: 63 - 24 = 39. After winning: 39 + 12 = 51.
- Cereal, by 3 boxes. Pasta after some went out: 45 - 16 = 29. Cereal is unchanged at 32, and 32 - 29 = 3.
- 61 tickets. Bojan has 18 more, so Lanakila has 18 fewer than Bojan: 54 - 18 = 36. After winning: 36 + 25 = 61.
- 31 cans. After the delivery: 43 + 26 = 69. Then 69 - \square = 38; counting up from 38 to 69 is 31, so 69 - 31 = 38.
- 25 tickets. Work back from the end: before she won 14 she had 49 - 14 = 35. So 60 - \square = 35, and 60 - 25 = 35. Check: 60 - 25 = 35, 35 + 14 = 49.
- 37 + 45 - 28 = \square. Left to right: 37 + 45 = 82, then 82 - 28 = 54. The box is 54.
- 67. One story: “The pantry had 64 cans. 27 went out to families. Then 30 came in. How many are there now?” 64 - 27 = 37, then 37 + 30 = 67.
- Draw a bar for the whole. Taking away 26 cuts off one piece, and taking away 14 cuts off a second piece next to it. The two pieces together are 26 + 14 = 40 long, so the piece left is the same as if one piece of 40 had been cut off at once.
- He has the same number he started with. Winning 25 and then using 25 adds and takes away the same amount, so the two steps together change nothing.
- No. After the delivery the pantry had 45 + 30 = 75 cans, and 80 cannot go out of a pile of 75. A middle number of less than zero is not possible in a story about cans.
- No. She only used tickets and never got any, so she must end with fewer than 52, and 91 is more. The answer is 11: 52 - 18 = 34, then 34 - 23 = 11. Or uses first: 18 + 23 = 41, and 52 - 41 = 11.