Problems with the starting amount missing
A story tells what happened and how things ended, but not how they began. With numbers in the tens, how does a child go back to the start without guessing?
Where you are 7 in Addition and subtraction problems in words
When you reach for it
Some stories hide their beginning. “Some toys were in the toy drive bin. 26 more were given. Now there are 63. How many toys were in the bin before?” The story tells the change and the end, and asks for the start.
The child has met these stories already, within 20, in problems with the start, change or result missing. The one-line recap: in an adding-to story the start is a part and the end is the whole, and in a taking-from story the start is the whole, so there the child adds even though the story took things away.
What changes now is the size of the numbers. With 9 trucks left and 4 driven out, a child can try 12, then 13, and land on the answer by trying. With 63 toys and 26 given, trying starts does not work: there are too many to try, and each try is a two-digit sum. The child needs a reason for which way to go, and a sure way to get there.
The reason is a simple one. A story runs forward in time. To find its start, run it backward. Whatever the story did, undo it. If toys were given, take them back off. If fish were let go, put them back in the bucket. Then run the story forward again from the number found, to check that it really ends where the story says.
This lesson takes three days. Day 1 is adding-to stories, Day 2 is taking-from stories, and Day 3 is choosing between the two moves when the numbers are big and the story is told in a mixed-up order.
The procedure
Day 1: something was added, so take it back off
The lead story for the day: “Some toys were in the toy drive bin. 26 more were given. Now there are 63. How many toys were in the bin before?”
Write the story in the order it happened, with a box for the missing start:
Now run the story backward. The last thing that happened was that 26 toys went in. Before that, the bin held 26 fewer. So the start is what is left when the 26 come back out:
This is the same fact read the other way, which is the idea of subtraction as an unknown addend: taking away 26 from 63 and finding what joins 26 to make 63 are one question.
There are two good ways to find the number, both on an open number line.
Take it back off. Start at 63, the end of the story, and jump back by the 26 that came in. Jump back the tens first, then the ones. A jump back of 20 lands on 43. The 6 ones are split so the line stops at a friendly ten: back 3 to 40, then back 3 more to 37. The first worked example below draws these jumps.
Count up from the change. Start at the 26 toys that were given and count up to the 63 in the bin now. The jumps add up to the start. This is counting up to find a difference, and it is often easier when the change is close to the end.
Either way, the bin held 37 toys before. Run the story forward to check: 37 toys, then 26 more given. From 37, jump 20 to 57, then 6 to 63. The story ends at 63, just as it says.
One thing the child can say before doing any arithmetic: in an adding-to story the start is less than the end, because things were added to it. The start of \square + 26 = 63 has to be below 63. That is a quick way to catch a wrong move later on.
Day 2: something was taken away, so put it back
The lead story for the day: “Some fish were in a bucket on the fishing pier. 18 were let go. 47 are left. How many fish were in the bucket at first?”
In story order:
Run it backward. The last thing that happened was that 18 fish went back into the water. Before that, the bucket held those 18 as well as the 47 that stayed. So put them back:
The story has a minus sign, and the answer comes from adding. That is not a trick; it is what undoing a take-away means.
Put it back on the number line. Start at 47, the fish left, and jump forward by the 18 that were let go: 10 to 57, then the 8 ones split at the friendly ten, 3 to 60 and 5 to 65. The third worked example below draws these jumps.
Put it back with tens and ones. The 47 fish are 4 tens and 7 ones; the 18 fish are 1 ten and 8 ones. Together that is 5 tens and 15 ones. Fifteen ones make one more ten and 5 ones, so the bucket held 6 tens and 5 ones. This is regrouping in addition, and the child can do it with base ten blocks, with a drawing of sticks and dots, or in writing.
There were 65 fish at first. Run the story forward to check: 65 fish, and 18 are let go. Back 10 to 55, back 5 to 50, back 3 to 47. The story ends with 47 left, as it says.
The size check for this kind of story is the other way round. In a taking-from story the start is more than the end, because things were taken from it. The start of \square - 18 = 47 has to be above 47.
Day 3: choosing the move when a guess will not do
On Day 3 the two kinds of story are mixed, and the child has to choose between “take it back off” and “put it back” from the story itself.
The choice comes from one question: what happened to the start?
- If things were added to the start (given, bought, caught, came in), undo it by taking them back off: subtract the change from the end.
- If things were taken from the start (let go, given away, sold, left), undo it by putting them back: add the change to the end.
A word alone does not decide this. “Given” can mean toys given to the bin, which made it grow, or toys given away, which made it shrink. The child pictures what the story did to the pile and undoes that.
Two habits make the choice safe with big numbers.
Say the size first. Before working anything out, say whether the start is more or less than the end. Added to: the start is less. Taken from: the start is more. A number found on the wrong side means the wrong move was made.
Run it forward last. Put the number found back into the story, in the order the story happened, and see whether it ends where the story ends. With small numbers, a child may check by counting objects. With numbers like these, running the story forward is the check that always works.
Some stories tell the end first: “At the end of the day there were 71 toys in the bin. During the day, 45 were given. How many were there in the morning?” Before choosing a move, retell it in the order it happened: some toys in the morning, 45 given, 71 at the end. Then the box lands at the start, where the story’s unknown really is: \square + 45 = 71.
Worked examples
Read each story aloud to the child, and let the child draw it or act it out before writing any numbers.
Day 1: The toy drive bin
“Some toys were in the toy drive bin. 26 more were given. Now there are 63. How many toys were in the bin before?”
SolutionIn story order, a box for the start:
The toys were added, so the start is less than 63. To run the story backward, take the 26 toys back off the 63. On an open line, start at 63 and jump back: 20 to 43, 3 to 40, and 3 more to 37. The three jumps back make 26 in all, because 20 + 3 + 3 = 26.
There were 37 toys in the bin before. Check: 37 + 26. Jump 20 from 37 to 57, then 6 to 63. The story ends at 63.
Day 1: Board games, counting up
“Jelani looked in the toy drive bin. Some board games were in it. Then 19 more board games were given. Now there are 54. How many board games were in the bin before?”
SolutionThe games were added, so the start is less than 54. Here the change, 19, is close to a ten, so counting up from it is easy. Adding in either order gives the same total (the commutative property), so the start is the number that joins 19 to make 54.
From 19, jump 1 to the friendly ten, 20. From 20, jump 30 to 50. From 50, jump 4 to 54. The jumps were 1, 30 and 4, and 1 + 30 + 4 = 35.
There were 35 board games in the bin before. Check: 35 games and 19 more. 35 + 20 = 55, and one fewer than that is 54. The story ends at 54.
Day 2: The fish bucket
“Some fish were in a bucket on the fishing pier. 18 were let go. 47 are left. How many fish were in the bucket at first?”
SolutionFish were taken from the bucket, so the start is more than 47. To run the story backward, put the 18 fish back. Start at 47 and jump forward: 10 to 57, 3 to the friendly ten 60, and 5 to 65.
There were 65 fish at first. Check: start the story at 65 and let 18 go. Back 10 to 55, back 5 to 50, back 3 to 47. The bucket ends with 47, as the story says.
Day 2: Nohea’s bucket, tens and ones
“Nohea had some fish in her bucket. She let 26 of them go. 38 fish are still in the bucket. How many fish did she have at first?”
SolutionFish were let go, so the start is more than 38. Put the 26 back with base ten blocks: 38 is 3 tens and 8 ones, and 26 is 2 tens and 6 ones.
Put the tens together: 3 tens and 2 tens make 5 tens. Put the ones together: 8 ones and 6 ones make 14 ones. Fourteen ones are too many for the ones place, so trade ten of them for one more ten: 14 ones are 1 ten and 4 ones. Now there are 6 tens and 4 ones.
Nohea had 64 fish at first. Check: 64 - 26. Back 20 from 64 to 44, then back 6 to 38. That is the number still in the bucket.
Day 3: Same numbers, two stories
Read both, then find each start.
“Some toys were in the toy drive bin. 34 more were given. Now there are 52. How many were in the bin before?”
“Some fish were in Svetla’s bucket. She let 34 go. 52 are left. How many fish were in the bucket at first?”
SolutionThe numbers are the same, so the stories must be read for what happened to the start.
In the toy story, toys were added. The start is less than 52, and the move is to take the 34 back off:
From 52, back 30 to 22, back 2 to 20, back 2 to 18. There were 18 toys before. Check: 18 + 34: 18 + 30 = 48, and 4 more is 52.
In the fish story, fish were taken away. The start is more than 52, and the move is to put the 34 back:
5 tens and 3 tens make 8 tens; 2 ones and 4 ones make 6 ones. There were 86 fish at first. Check: 86 - 34: back 30 to 56, back 4 to 52.
Same three numbers on the page, and the two starts are 18 and 86. The size check said this before any arithmetic: one start had to be below 52 and the other above it.
Day 3: The end told first
“At the end of the day, Jelani counted 71 toys in the toy drive bin. During the day, 45 toys were given. How many toys were in the bin in the morning?”
SolutionThe story is told end first. Retell it in the order it happened: in the morning there were some toys; during the day 45 were given; at the end there were 71.
Toys were added during the day, so the morning count is less than 71. Take the 45 back off. From 71, back 40 to 31, then back 1 to 30 and back 4 more to 26.
There were 26 toys in the morning. Check: run the day forward. 26 + 45: 26 + 40 = 66, and 5 more is 71. That is what Jelani counted.
Day 3: A story for an equation
Tell a fishing pier story that \square - 48 = 39 fits, then find the start.
SolutionThe box comes first, so the story begins with an unknown amount. Then 48 are taken away, and 39 remain. One story: “Svetla had some fish in her bucket. She let 48 of them go. 39 were left. How many did she have at first?”
Something was taken from the start, so the start is more than 39, and the move is to put the 48 back:
From 39, jump 1 to the friendly ten 40. From 40, jump 40 to 80. That is 41 of the 48 put back, so 7 more remain: 80 and 7 is 87.
The bucket held 87 fish at first. Check: 87 - 48. Back 40 to 47, then back 8 to 39. The story ends with 39 left.
Any story works as long as it starts with the unknown amount, takes 48 away, and ends with 39: fish let go, toys taken out of the bin, all fit the same equation.
Boundaries
The end and the change decide the start. Once the story says what changed and how it ended, exactly one start fits it. A story that leaves two of the three numbers out (“some toys were in the bin, some more came, now there are 50”) has many answers and is a different kind of question.
A take-away cannot leave more than the start, and an add-to cannot end below it. A taking-from story whose numbers make the start smaller than the end has an error in it. In \square - 18 = 47 the start is 47 + 18, which is always more than 47. Among the whole numbers 0, 1, 2, 3 and so on, an adding-to story such as \square + 70 = 50 has no start at all: nothing with 70 added to it ends at 50. The child can say so.
The numbers stay within 100. A taking-from story can push the start past 100 even when every number printed is smaller: \square - 45 = 70 starts at 115. The same moves work, and those numbers belong to adding and subtracting within 1{,}000.
One change, one step. Every story here has a single change between the start and the end. A story with two changes, “some fish, 12 let go, then 9 more caught, now 30”, has to be undone one change at a time, last change first, and belongs to two-step problems.
The box goes where the story puts the unknown. \square + 45 = 71 and 71 - 45 = \square have the same answer, but only the first tells the story in the order it happened. The child writes the story order first, then the move that undoes it.
Where this goes next
Stories that compare two amounts rather than change one are how many more and how many fewer. The part-whole picture behind every start-unknown story is drawn to size in bar models for story problems, and stories that need one answer found before the next are two-step problems. Much later, the box becomes a letter, and \square - 18 = 47 becomes x - 18 = 47, a one-step equation. Undoing the last thing done is still how it is solved.
Exercises
Day 1: something was added
- \square + 34 = 76. Find the start, then run it forward to check.
- Some toys were in the toy drive bin. 27 more were given. Now there are 81. How many toys were in the bin before?
- \square + 48 = 92
- Some puzzles were in the toy drive bin. Nohea added 16 puzzles. Now there are 50. How many puzzles were in the bin before?
- \square + 65 = 93. Find the start by counting up from 65. Name each number you land on.
- To find the start of \square + 29 = 75, Svetla jumps back 30 from 75 and then forward 1. Where does each jump land, and what is the start?
Day 2: something was taken away
- \square - 23 = 45
- Some fish were in a bucket on the fishing pier. 17 were let go. 56 are left. How many fish were in the bucket at first?
- \square - 38 = 54. Put the 38 back on an open number line.
- Jelani had some fish in his bucket. He let 9 go. 66 are left. How many did he have at first?
- \square - 46 = 37. Find the start with tens and ones, and say where you trade ten ones for a ten.
- \square - 50 = 28
Day 3: choosing the move
- Find the start of each story. (a) Some toys were in the toy drive bin. 25 more were given. Now there are 61. (b) Some fish were in a bucket on the pier. 25 were let go. 61 are left.
- Before working anything out, say whether the start is more or less than 70. Then find it. (a) \square + 18 = 70 (b) \square - 18 = 70
- At closing time the toy drive bin held 84 toys. During the day, 39 toys were given. How many toys were in the bin when it opened?
- When Svetla packed up, 27 fish were in her bucket. Earlier that day she had let 44 fish go. How many fish were in the bucket before she let any go?
- Some fish were in Nohea’s bucket. She let 33 go. 33 are left. How many fish were in the bucket at first?
- Tell a toy drive story that \square + 57 = 90 fits, then find the start.
- Some toys were in the toy drive bin. 45 more were given. Now there are 100. How many toys were in the bin before?
Explain
- In an adding-to story with the start missing, can the start ever be more than the end? Explain using the toy drive bin.
- Explain, using the fish bucket, why putting the fish back gives the start of a story where fish were let go.
- Do \square + 35 = 80 and 35 + \square = 80 have the same missing number? Find it, and say why.
- For \square - 24 = 51, run the story forward from 75 and from 27. Which one is the start, and how does running it forward show that?
Answers
- 42. Take 34 back off 76: back 30 to 46, back 4 to 42. Check: 42 + 34 = 76.
- 54 toys. \square + 27 = 81, so take the 27 back off: 81 - 20 = 61, 61 - 1 = 60, 60 - 6 = 54. Check: 54 + 27 = 81.
- 44. 92 - 40 = 52, 52 - 2 = 50, 50 - 6 = 44. Check: 44 + 48 = 92.
- 34 puzzles. \square + 16 = 50, so 50 - 16: back 10 to 40, back 6 to 34. Check: 34 + 16 = 50.
- 28. From 65: +5 to 70, +20 to 90, +3 to 93. The jumps make 5 + 20 + 3 = 28. Check: 28 + 65 = 93.
- 46. Back 30 from 75 lands on 45; that took off one too many, so forward 1 lands on 46. Check: 46 + 29 = 75.
- 68. Put the 23 back: 45 + 20 = 65, 65 + 3 = 68. Check: 68 - 23 = 45.
- 73 fish. \square - 17 = 56, so put the 17 back: 56 + 10 = 66, 66 + 4 = 70, 70 + 3 = 73. Check: 73 - 17 = 56.
- 92. From 54: +30 to 84, +6 to 90, +2 to 92. Check: 92 - 38 = 54.
- 75 fish. \square - 9 = 66, so 66 + 9: +4 to 70, +5 to 75. Check: 75 - 9 = 66.
- 83. 37 is 3 tens 7 ones and 46 is 4 tens 6 ones. Together: 7 tens and 13 ones. Trade ten of the 13 ones for a ten: 8 tens 3 ones. Check: 83 - 46 = 37.
- 78. Put back 5 tens: 28 + 50 = 78. Check: 78 - 50 = 28.
- (a) 36 toys. Added, so take off: 61 - 25 = 36. Check: 36 + 25 = 61. (b) 86 fish. Let go, so put back: 61 + 25 = 86. Check: 86 - 25 = 61.
- (a) Less than 70, because 18 was added to the start: 70 - 18 = 52. Check: 52 + 18 = 70. (b) More than 70, because 18 was taken from it: 70 + 18 = 88. Check: 88 - 18 = 70.
- 45 toys. In time order: \square + 39 = 84. Take the 39 back off: 84 - 40 = 44, and one back, 45. Check: 45 + 39 = 84.
- 71 fish. In time order: \square - 44 = 27. Put the 44 back: 27 + 44 = 71. Check: 71 - 44 = 27.
- 66 fish. \square - 33 = 33, so 33 + 33 = 66. Check: 66 - 33 = 33.
- 33. One story: “Some toys were in the bin. 57 more were given. Now there are 90.” Take the 57 back off: 90 - 50 = 40, 40 - 7 = 33. Check: 33 + 57 = 90.
- 55 toys. \square + 45 = 100, so 100 - 45 = 55. Check: 55 + 45 = 100.
- No. The toys at the start are still in the bin at the end, with the new toys as well, so the end is the start and some more. The start can equal the end only if 0 toys were given.
- The fish left in the bucket and the fish let go were all in the bucket at the start. Putting the let-go fish back joins the two groups again, so the bucket is as it was before anything happened: the start.
- Yes, 45 both times. Adding in either order gives the same total, so both ask what joins 35 to make 80: 80 - 35 = 45.
- 75. From 75, letting 24 go leaves 75 - 24 = 51, which is the story’s end. From 27, letting 24 go leaves 27 - 24 = 3, not 51. Only the true start runs forward to the end the story gives.