Problems with the total missing
A story gives two numbers and asks how many there are in all, or how many are left at the end. How does a child decide whether to add or subtract, and know the answer fits the story?
Where you are 7 in Addition and subtraction problems in words
When you reach for it
Many first-grade story problems ask one of two plain questions: how many in all? or how many are left? The two numbers in the story are both given, and the missing number is the last one, the total or the end.
- Putting together, total missing. Nothing moves. Two groups are simply there, and the story asks about both at once: “Asta’s bowl has 7 red cherry tomatoes and 9 yellow cherry tomatoes. How many cherry tomatoes are in the bowl?”
- Adding to, result missing. Something happens: more arrive. “Trine has 8 carrot sticks in her lunch. Arnfinn gives her 6 more. How many does she have now?”
- Taking from, result missing. Something happens: some go away. “Asta picked 16 cherry tomatoes. She ate 7 of them. How many does she have left?”
These look like the easiest stories, and in a way they are: a child can act them out from the beginning, because the beginning is given. The work of the lesson is in three habits that make the harder stories later possible. First, decide which number is the whole (all of them together) and whether the story gives it or asks for it. Second, write an equation with a box for the missing number, in the order the story goes. Third, check the answer against the story: put it back in and see that the story still makes sense.
All the numbers stay within 20. The stories where the change or the start is missing came in the lesson before, problems with the start, change or result missing; here the box always sits at the end of the equation. The lesson is taught over four days:
- Part 1: putting together, with both parts given and the total missing, 7 + 9 = \square.
- Part 2: adding to, with the result missing, 8 + 6 = \square.
- Part 3: taking from, with the result missing, 16 - 7 = \square.
- Part 4: mixed stories. The child decides first whether the whole or a part is missing, and checks every answer against the story.
The procedure
Part 1: two groups, how many in all?
Read the lead story aloud: “Asta’s bowl has 7 red cherry tomatoes and 9 yellow cherry tomatoes. How many cherry tomatoes are in the bowl?”
Nobody adds or takes anything here. The red tomatoes and the yellow tomatoes are in the bowl the whole time, and the question asks about all of them together. This is the meaning of addition as putting together: two parts, seen as one whole.
So 7 is a part, 9 is a part, and the whole is the missing number. A part-part-whole bar shows it. The long bar on top is all the tomatoes in the bowl, and it holds the box. Under it, the same length is split into a piece for the red tomatoes and a piece for the yellow ones. The two pieces underneath are exactly as long as the bar on top, which is the picture of “the parts make the whole”. It is the same idea as a number bond, drawn so that a bigger number gets a longer piece.
The equation says what the bar says. The parts are joined, and the box is the whole:
To fill the box, the child can count all sixteen tomatoes, and at first that is fine. A quicker path uses ten. Line up the 9 yellow tomatoes in a ten frame: one space is empty. Move one red tomato in to fill it. Now there is a full ten, and 6 red tomatoes are still out:
This is making ten. A child who knows that 8 + 8 = 16 can also see 7 + 9 as one less and one more than that double; that is the move in doubles and near doubles. Either way the bowl has 16 cherry tomatoes.
Because nothing happens in a put-together story, there is no first and second group. The story could name the yellow tomatoes first, and the total would not change: 9 + 7 = 16 as well. That is the commutative property, and it lets the child start from whichever part is easier.
Part 2: more arrive, how many now?
Read aloud: “Trine has 8 carrot sticks in her lunch. Arnfinn gives her 6 more. How many carrot sticks does she have now?”
This story has an action and a time order. First there were 8, then 6 joined them, and the question is about the end. Once the 6 are in the lunch, the 8 and the 6 are together, so the end is the whole. The start and the change are its two parts.
In story order, the equation is
The box sits at the end, just where the story put its question: “how many now?” A child can act this out with counters. Put down 8, push in 6 more, count them all. Or they can count on: say “eight” and count six more, 9, 10, 11, 12, 13, 14. A number line with a stop at 10 makes the count shorter (Example 3).
The bar for an adding-to story looks exactly like the bar for a put-together story: the whole on top, two parts below. The bar does not show time, and it does not need to. What decides the operation is that the whole is missing and both parts are known, so the parts are joined.
Part 3: some go away, how many are left?
Read the lead story aloud: “Asta picked 16 cherry tomatoes. She ate 7 of them. How many does she have left?”
Now the first number is the whole. All 16 tomatoes were there before anything happened. When Asta eats 7, the 16 are split into two parts: the tomatoes eaten and the tomatoes left. So 16 goes on top of the bar, and 7 and the box go underneath.
In story order:
This is subtraction as taking away, now with a start bigger than ten. There are two good ways to fill the box.
Count back, stopping at 10. Taking 7 from 16 all at once is a long count back. Split the 7 so the first piece lands on 10: 16 take away 6 is 10, and 10 take away the last 1 is 9. This is bridging back through ten.
Think of addition. The bar says 7 and the missing part make 16. So the question is also “7 and what make 16?” From 7, three steps reach 10, and six more reach 16. Three and six make nine. This is subtraction as finding a missing part, and it is often the shorter count when a lot is taken away.
Both ways give 9. Asta has 9 cherry tomatoes left.
Part 4: is the whole missing, or a part?
By now the child has seen the box at the end of three kinds of story. The question that sorts every one of them is the same question as in the lesson before: which number is all of them together, and does the story tell it?
- If the story gives both parts and asks for the whole (put together, or added to), join the parts: add.
- If the story gives the whole and one part and asks for the other part (taken from, or “the rest are”), find the missing part: subtract, or count up from the part you know.
Single words do not settle it. “Left” can appear in a story whose answer comes from adding (see Boundaries), and in the lesson before, trucks that “drove out” were found by adding. The bar settles it.
Then every answer goes back into the story. In a put-together or adding-to story the total must be at least as big as each part. In a taking-from story the number left must be no bigger than the start. And the answer, put in place of the box, must make the story true: if Asta has 9 left after eating 7, then 9 and 7 must make the 16 she picked, and they do.
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.
Tomatoes in the bowl
Read aloud: “Asta’s bowl has 7 red cherry tomatoes and 9 yellow cherry tomatoes. How many cherry tomatoes are in the bowl?”
SolutionThe red and the yellow tomatoes are both in the bowl, and the question asks about all of them, so the whole is missing and 7 and 9 are the parts (first figure above). The equation is
Put the 9 yellow tomatoes in a ten frame. One space is empty, so move 1 red tomato in. That makes a full 10, and 7 - 1 = 6 red tomatoes are still out (second figure above).
There are 16 cherry tomatoes in the bowl.
Check. The answer is the whole, so it should be bigger than each part: 16 is bigger than 7 and than 9. Counting by ones gives 16 too.
Teddy bears on the shelf
Read aloud: “Arnfinn’s shelf has 5 big teddy bears and 8 small teddy bears. How many teddy bears are on the shelf?”
SolutionThe big bears and the small bears sit on the shelf together. Both parts are given, and the total is missing:
The story names the 5 first, but in a put-together story the order does not matter, so the child may start from the bigger part, 8, and count on the smaller one: “eight”, then 9, 10, 11, 12, 13. Five numbers were said after eight, so
There are 13 teddy bears on the shelf. Counting on from 8 took five counts; counting on from 5 would have taken eight. Starting from the bigger part is the shorter way.
Check. Take the 5 big bears off the shelf again: 13 - 5 = 8 small bears are still there, which is what the story said.
Carrot sticks in the lunch
Read aloud: “Trine has 8 carrot sticks in her lunch. Arnfinn gives her 6 more. How many carrot sticks does she have now?”
SolutionFirst there were 8, then 6 more joined them. The question asks about the end, when all the carrot sticks are together, so the end is the whole and it is missing:
Start at 8 on the number line. The 6 is split so the first hop lands on 10: two steps reach 10, because 8 and 2 make ten. That leaves 6 - 2 = 4 more, and four steps from 10 reach 14.
Trine has 14 carrot sticks now.
Check. If Arnfinn took his 6 back, Trine would have 14 - 6 = 8, which is where the story began.
More tomatoes for the bowl
Read aloud: “There were 12 cherry tomatoes in the bowl. Asta picked 5 more and put them in. How many cherry tomatoes are in the bowl now?”
SolutionThe 12 were there first, and 5 more joined them. The end is the whole:
No bridge through ten is needed here, because 12 is already past ten. Think of 12 as one ten and 2 ones. The 5 new tomatoes join the ones: 2 + 5 = 7. The ten is still there:
There are 17 cherry tomatoes in the bowl now. Seeing 12 as a ten and some ones makes this sum as easy as 2 + 5, a fact the child already knows. That move is part of adding and subtracting within 20.
Check. 17 is more than the 12 at the start, as it must be after tomatoes are added.
Tomatoes eaten
Read aloud: “Asta picked 16 cherry tomatoes. She ate 7 of them. How many does she have left?”
SolutionAll 16 were there at the start, so 16 is the whole. The 7 eaten and the tomatoes left are the parts (the bar in Part 3). In story order:
Take the 7 away in two hops back. Six gets from 16 down to 10; the last one gets from 10 to 9. Six and one make the seven that was eaten.
Asta has 9 cherry tomatoes left.
Check. The tomatoes left and the tomatoes eaten must make all she picked: 9 + 7 = 16. They do.
Bears go to the playroom
Read aloud: “There were 15 teddy bears on the shelf. Trine took 9 of them to the playroom. How many teddy bears are still on the shelf?”
SolutionThe 15 bears on the shelf at first are the whole; the 9 taken and the bears still there are its parts:
Counting back nine from 15 is long. Nine is a lot to take, and the part left is small, so thinking of addition is shorter: “9 and what make 15?” From 9, one step reaches 10, and five more reach 15. One and five make six.
There are 6 teddy bears still on the shelf.
Check. The answer is a part, so it must be smaller than the 15 at the start, and it is. Putting the 9 bears back on the shelf with the 6 makes 15 again.
Same numbers, three stories
Read aloud each story, then decide whether the whole or a part is missing, write an equation, and answer it.
- “Arnfinn has 11 carrot sticks. He eats 5. How many are left?”
- “Arnfinn has 5 carrot sticks. Trine gives him 6 more. How many does he have now?”
- “Arnfinn has 5 carrot sticks. Trine gives him some more. Now he has 11. How many did she give him?”
In each story, ask first which number is all of the carrot sticks together.
| Story | Whole | Equation | Box |
|---|---|---|---|
| 1 | 11 | 11 - 5 = \square | 6 |
| 2 | \square | 5 + 6 = \square | 11 |
| 3 | 11 | 5 + \square = 11 | 6 |
In Stories 1 and 3 a part is missing; in Story 2 the whole is.
Story 1 starts with all 11 and takes 5 away, so the whole is known and the end is a part: from 5, five steps reach 10 and one more reaches 11, so 6 are left. Story 2 gives both parts and asks for the end, which is the whole: 5 + 6 = 11, one more than the double 5 + 5. Story 3 gives the whole and asks for the change; it is the kind of story taught in the lesson before, and it is answered the same way as Story 1.
All three stories draw one bar: 11 on top, 5 and 6 below. Only the place of the box moves. Stories 1 and 2 both have the box at the end of the equation, yet one subtracts and one adds. What decides it is not where the box sits but whether the box is the whole.
Brown bears and white bears
Read aloud: “There are 14 teddy bears on the shelf. 8 of them are brown and the rest are white. How many are white?”
SolutionNothing moves in this story; it is a put-together story. But it is not a total-missing one. The story gives all the bears, 14, which is the whole, and one part, the 8 brown bears. The missing number is the other part, the white bears. The bar has 14 on top and 8 and a box below:
The whole is known and a part is missing, so count up from the part we know. From 8, two steps reach 10 and four more reach 14. Two and four make six.
There are 6 white teddy bears.
Check. Two things. The answer is a part, so it must be smaller than 14, and it is. And 6 white bears with 8 brown bears make 14 bears, the number the story began with. Had the child added 14 and 8, the white bears alone would be more than all the bears on the shelf, which the story cannot allow. That is why the first question is always whether the whole is given.
Boundaries
A total is at least as big as each of its parts. When both parts are counted, the whole is bigger than either one, unless a part is 0. “Trine has 8 carrot sticks and Arnfinn gives her 0” leaves her with 8, the same as the part. A total smaller than one of its parts means the wrong operation was used.
Nothing can be taken away that is not there. A taking-from story cannot take 9 from 7: 7 - 9 = \square has no answer among the whole numbers 0, 1, 2, 3 and so on. The number left is never more than the start, and it is 0 when everything is taken. Much later, negative numbers give 7 - 9 a value, but not as tomatoes left in a bowl.
Only things of one kind are joined into a total. “7 cherry tomatoes and 9 carrot sticks” make 16 pieces of food, but not 16 cherry tomatoes. The question decides what the whole is, and both parts must belong to it.
The words do not choose the operation. “Left” usually comes with taking away, yet a story like “Asta ate 3 tomatoes and has 9 left; how many did she have?” asks for the whole, and the answer comes from adding. The bar, not a single word, decides.
Two parts, within 20. Every story here joins or separates two groups, with numbers up to 20. Stories with three groups at once (“red, yellow and orange tomatoes”) are adding three numbers. Bigger numbers use the same bars and the same thinking.
Where this goes next
Stories that compare two amounts instead of joining or taking are how many more and how many fewer. The part-part-whole bar grows into bar models for story problems, which carry the same picture to bigger numbers. When one answer has to be found before the next question can be asked, the stories are two-step problems, and the start-missing kind returns with larger numbers in problems with the starting amount missing. All of these together are addition and subtraction problems in words. Much later the box becomes a letter, and 16 - 7 = \square becomes 16 - 7 = x, the simplest kind of one-step equation.
Exercises
Part 1: two groups, how many in all?
Use the bars for Exercises 1 and 2. What number goes in the box? Write the addition sentence.
- Bar A.
- Bar B.
Find the number that goes in the box.
- 9 + 5 = \square
Write an equation with a box, then answer the story.
- “Asta’s bowl has 7 red cherry tomatoes and 7 orange cherry tomatoes. How many cherry tomatoes are in the bowl?”
- “On the shelf are 10 teddy bears with bows and 7 teddy bears without bows. How many teddy bears are on the shelf?”
- “Trine’s lunch has 9 carrot sticks and Arnfinn’s lunch has 9 carrot sticks. How many carrot sticks are in the two lunches?”
Part 2: more arrive, how many now?
- 13 + 5 = \square
- 4 + 8 = \square
- “Arnfinn put 8 teddy bears on the shelf. Then Asta put 3 more on it. How many teddy bears are on the shelf now?”
- “Trine picked 9 cherry tomatoes. Then she picked 8 more. How many has she picked in all?”
- “Asta has 4 carrot sticks in her lunch box. Arnfinn puts 10 more in. How many carrot sticks are in the lunch box now?”
Part 3: some go away, how many are left?
- Bar C. What number goes in the box? Write the subtraction sentence.
- 14 - 5 = \square
- 12 - 8 = \square
- “There were 13 teddy bears on the shelf. Trine took 5 down to play with. How many are on the shelf now?”
- “Arnfinn had 11 carrot sticks. He ate 2. How many does he have left?”
- “Asta picked 20 cherry tomatoes. She gave 10 to Trine. How many does Asta have now?”
Part 4: is the whole missing, or a part?
For each story, say whether the whole or a part is missing, write an equation with a box, and answer the story.
- “Trine had 12 carrot sticks. She gave some to Asta. Now she has 7. How many did she give Asta?”
- “Asta’s bowl holds 13 cherry tomatoes. 9 are red and the rest are yellow. How many are yellow?”
- “Asta has 8 red cherry tomatoes and 8 yellow cherry tomatoes. How many cherry tomatoes does she have?”
- Tell a taking-from story about teddy bears that goes with 11 - 3 = \square. Then find the box.
Explain
- In “Asta’s bowl has 7 red and 9 yellow cherry tomatoes,” nothing is added to the bowl. Why is the answer still found by adding?
- Is the answer to a total-missing story always bigger than each of the two numbers in it? Give an example to explain.
- In a taking-from story with the end missing, can the answer be bigger than the number at the start? Why or why not?
- A story uses the numbers 9 and 6. Tell one story whose answer is 15 and one whose answer is 3. Say which number is the whole in each.
Answers
- 12. Both parts are given and the whole is missing: 5 + 7 = \square. From 7, three steps reach 10 and two more reach 12, so 5 + 7 = 12.
- 13. Start from the bigger part: 9 and 1 make ten, and 3 more make 13, so 4 + 9 = 13.
- 14. 9 and 1 make ten; the other 4 make 14: 9 + 5 = 14.
- 7 + 7 = \square, and the box is 14. It is a double: 7 + 7 = 14. There are 14 cherry tomatoes in the bowl.
- 10 + 7 = \square, and the box is 17. One ten and 7 ones: 10 + 7 = 17 teddy bears.
- 9 + 9 = \square, and the box is 18. A double: 9 + 9 = 18 carrot sticks.
- 18. 13 is a ten and 3 ones; 3 + 5 = 8, so 13 + 5 = 18.
- 12. Count on from the bigger number: 8, then 9, 10, 11, 12. So 4 + 8 = 12.
- 8 + 3 = \square, and the box is 11. Count on: 9, 10, 11, so 8 + 3 = 11 teddy bears.
- 9 + 8 = \square, and the box is 17. 9 and 1 make ten, and 7 more make 17: 9 + 8 = 17 tomatoes.
- 4 + 10 = \square, and the box is 14. Ten and four is 14: 4 + 10 = 14 carrot sticks.
- 7. The whole is 15 and one part is 8: 15 - 8 = \square. From 8, two steps reach 10 and five more reach 15, so 8 + 7 = 15 and 15 - 8 = 7.
- 9. Take 4 to reach 10, then 1 more: 14 - 5 = 9.
- 4. Think “8 and what make 12?”: two to 10, two more to 12, so 8 + 4 = 12 and 12 - 8 = 4.
- 13 - 5 = \square, and the box is 8. Take 3 to reach 10, then 2 more: 13 - 5 = 8 teddy bears.
- 11 - 2 = \square, and the box is 9. Count back two: 10, 9. So 11 - 2 = 9 carrot sticks.
- 20 - 10 = \square, and the box is 10. Two tens take away one ten leaves one ten: 20 - 10 = 10 tomatoes.
- A part is missing (the change). The whole is 12: 12 - \square = 7. From 7, three steps reach 10 and two more reach 12, so 7 + 5 = 12 and the box is 5. Trine gave Asta 5 carrot sticks.
- A part is missing. The whole is 13 and the red tomatoes are one part: 9 + \square = 13. From 9, one step reaches 10 and three more reach 13, so 9 + 4 = 13. There are 4 yellow tomatoes.
- The whole is missing: 8 + 8 = \square. A double: 8 + 8 = 16 cherry tomatoes.
- One story: “There were 11 teddy bears on the shelf. Trine took 3 of them to bed. How many are on the shelf now?” The whole is 11 and the box is the bears left: count back 10, 9, 8, so 11 - 3 = 8. Check: 8 + 3 = 11.
- The question asks how many tomatoes are in the bowl, which means all of them together. The red ones and the yellow ones are the two parts of that whole, and putting two parts together into one whole is what adding means, whether or not anything moves.
- Almost always. The whole is made of both parts, so it is bigger than each part when both parts have something in them. The one exception is a part of 0: “Asta has 6 tomatoes and picks 0 more” gives 6 + 0 = 6, the same as the first part, not bigger.
- No. Everything left was there at the start, so the number left is a part of the start, and a part cannot be bigger than its whole. If nothing is taken, the end equals the start; otherwise it is smaller.
- One pair: “Arnfinn has 9 carrot sticks and Trine gives him 6 more. How many now?” The whole is missing, and 9 + 6 = 15. “Arnfinn has 9 carrot sticks and eats 6. How many are left?” The whole is 9, and 9 - 6 = 3.