Problems with the start, change or result missing

When you reach for it

Most early stories give the start and the change and ask for the end: “5 crabs, 3 more come, how many now?” This lesson is about the stories that turn that around. The story tells the end, and the missing number is somewhere earlier.

• The change is missing. “There were 5 hermit crabs in the tank. Folau put some more in. Now there are 13. How many did he put in?”
• The start is missing. “Some fire trucks were at the station. 4 drove out to a fire. 9 are left. How many were there at first?”

A child reading these stories cannot just act them out from the beginning, because the beginning, or the middle, is the very thing they do not know. Two things help. A picture of the story (a part-part-whole bar, or counters in a ten frame) shows which number is the whole and which numbers are parts. And an equation with a box, \square, written in the order the story happens, holds the place of the missing number until it is found. Then the box is filled by counting up or by thinking of the related fact.

All the numbers here stay within 20. The lesson is taught over four days:

• Part 1: adding-to stories with the change missing, 5 + \square = 13.
• Part 2: taking-from stories with the change missing, 12 - \square = 7.
• Part 3: stories with the start missing, both \square + 6 = 14 and \square - 4 = 9. These are the hardest kind, because the number the child would act out first is the one that is gone.
• Part 4: mixed stories. The child decides which number is the whole before doing anything else, and meets stories told out of time order.

The stories where the end (the total) is the missing number belong to the next lesson, problems with the total missing.

The procedure

Part 1: something was added, but how much?

Read the lead story aloud: “There were 5 hermit crabs in the tank. Folau put some more in. Now there are 13. How many did he put in?”

First the child pictures what happened. The crabs that were in the tank and the crabs Folau put in end up in one tank together. Together they are all 13. So 13 is the whole, and 5 and the missing number are its two parts.

A part-part-whole bar shows this. The long bar on top is the whole, all 13 crabs. Under it the same length is split in two: 5 for the crabs already there, and a box for the crabs put in. The two pieces underneath together are exactly as long as the bar on top, which is the picture of “the parts make the whole”. The bar is a tidier drawing of what a number bond shows, with the added benefit that a bigger number gets a longer piece.

Next the child writes an equation that follows the story in order. There were 5, then some were added, and that made 13:

The box sits exactly where the story put its unknown: in the middle, as the thing added. Writing it in story order matters because it lets the child check the equation against the story, line by line. “There were 5”: the 5 comes first. “Folau put some more in”: plus a box. “Now there are 13”: equals 13.

To fill the box, the child can count up from the part they know to the whole. Starting at 5, it is 5 steps to 10 and then 3 more to 13. Five and three make eight, so the box holds 8. Stopping at 10 on the way is the same move as making ten. A child who knows the fact 5 + 8 = 13 can simply recall it instead. Either way, the answer goes back into the story: Folau put in 8 crabs.

The same story can also be written 13 - 5 = \square: the whole take away the part we know leaves the part we want. That is the idea of subtraction as finding a missing part. Both equations are true, and both give 8. The one with the box in the middle is the one that tells the story as it happened.

Part 2: something was taken away, but how much?

Now a taking-from story: “There were 12 fire trucks at the station. Some drove out to a fire. Now 7 are left. How many drove out?”

The whole is the trucks at the station before anything happened: all 12. When some drive out, the 12 are split into two parts, the trucks that left and the trucks still parked. So 12 goes on top of the bar, and 7 and the box go underneath.

In story order the equation is

The child could find the box by counting back from 12 until they reach 7, counting how many steps that takes. That works. But the bar shows a shorter path: 7 and the missing part make 12, so the question is also “7 and what make 12?” Counting up from 7 to 12 is a short count, and it gives the same number.

This is the main idea of Part 2. A taking-from story with the change missing is solved by thinking of addition. The words say “drove out”, yet the picture says “part and part make the whole”, and the related fact 7 + 5 = 12 answers it.

Part 3: the start is missing

The lead story for this part: “Some fire trucks were at the station. 4 drove out to a fire. 9 are left. How many were at the station at first?”

These stories are the hardest for young children, for a plain reason. With counters, a child acts a story out from its beginning, and here the beginning is the very thing that is missing. There is nothing to put on the table first.

The bar gets around this. The trucks at first are the whole, because the 4 that drove out and the 9 that stayed were all there at the start. So the box goes on top of the bar, and 4 and 9 go underneath. In story order:

Once the box is on top of the bar, the child sees that the answer is the two parts put back together: 9 + 4 = 13. A minus sign is in the equation, yet the child adds. That is not a contradiction; the story took trucks away, and finding where it began means putting them back.

The adding-to kind of start-unknown story behaves differently, and the bar shows why. “Tane had some hermit crabs. He got 6 more. Now he has 14.” Here the start is not the whole. The crabs he had and the crabs he got are the two parts, and the 14 he has now is the whole:

So in an adding-to story the start is a part, and in a taking-from story the start is the whole. The child does not need to memorize that sentence. Drawing the bar, and asking “which number is all of them?”, sorts it out every time.

Part 4: decide which number is the whole

By now the child has met the missing number in three places. The one question that settles every story in this lesson is: which number is all of them together? That number goes on top of the bar, or is the box on top if the story does not say it. The other two numbers, one of which may be the box, are the parts.

From the bar, two moves always work:

• If the whole is known and one part is missing, count up from the known part to the whole, or recall the fact that joins the known part to the whole.
• If the whole is missing, put the two parts together.

Some stories tell events out of order (“Now there are 11 trucks. Before, there were 7, and some came back”). The child can retell such a story in the order it happened before writing the equation, so the box lands where the story’s unknown really is.

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.

Boundaries

A part cannot be bigger than its whole. In a taking-from story the number left is never more than the number at the start, so 9 - \square = 12 has no answer among the whole numbers 0, 1, 2, 3 and so on: nothing taken from 9 leaves 12. In the same way, 14 + \square = 6 has none. Such a story has an error in it, and the child can say so. Much later, negative numbers give equations like these an answer, but not as stories about trucks and crabs.

The box goes where the story puts the unknown. 5 + \square = 13 and 13 - 5 = \square have the same answer, and both are correct sentences, but only the first follows the story’s order. In Example 8 the story was told out of time order, and the equation was written in the order things happened, not the order they were told.

Words like “more” and “left” do not choose the operation. In Example 5 the story said trucks “drove out”, and the answer came from adding. The bar, not a single word, decides whether to add or to subtract.

This lesson stays within 20. The same bars work for any numbers. Start-unknown stories can easily run past 20: \square - 6 = 15 starts at 21. Those numbers belong to later work with bigger numbers.

Exactly one number fits. Once the whole and one part are known, the other part is fixed, so every story here has one answer. Stories with two missing numbers (“some crabs were in the tank, some more came, now there are 10”) have many answers and are a different kind of question.

Where this goes next

The stories with the end missing are the next lesson, problems with the total missing. Stories that compare two amounts instead of changing one 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 and to stories with more parts. Start-unknown stories return with larger numbers in problems with the starting amount missing, and stories where one answer must be found before the next are two-step problems. Much later, the box becomes a letter, and 5 + \square = 13 becomes 5 + x = 13, a one-step equation.

Exercises

Part 1: something was added

Use the bars for Exercises 1 to 3. What number goes in the box? Write the addition sentence.

1. Bar A.
2. Bar B.
3. Bar C.

Find the number that goes in the box.

4. 6 + \square = 10
5. 9 + \square = 15

Write an equation with a box, then answer the story.

6. “Folau has 8 hermit crabs. He gets some more at the pet store. Now he has 12. How many did he get?”
7. “Nita’s tank has 2 hermit crabs. She puts in some more. Now there are 11. How many did she put in?”

Part 2: something was taken away

8. 10 - \square = 3
9. 14 - \square = 7
10. “There were 15 fire trucks at the station. Some went out to a fire. 6 are left. How many went out?”
11. “Tane has 12 hermit crabs. He gives some to Nita. Now he has 9. How many did he give her?”

Part 3: the start is missing

12. \square - 5 = 6
13. \square + 8 = 17
14. “Some fire trucks were at the station. 3 drove out. 9 are left. How many were at the station at first?”
15. “Nita had some hermit crabs. Folau gave her 5 more. Now she has 12. How many did she have at first?”
16. “Some fire trucks were at the station. 9 drove out to a big fire. 2 are left. How many were there at first?”

Part 4: which number is the whole?

For each story, say which number is the whole (or that the whole is the missing number), write an equation with a box, and answer the story.

17. “At the end of the day there are 17 hermit crabs in the tank. Tane put 9 of them in that morning; the rest were there already. How many were there already?”
18. “5 fire trucks drove out of the station. Then 10 were left. How many were there before?”
19. “Folau’s tank had 14 hermit crabs. Now it has 20. How many crabs were added?”
20. Tell a story about fire trucks that goes with \square - 3 = 11. Then find the box.

Explain

21. In the story “Some fire trucks were at the station. 6 drove out. 8 are left,” trucks drive away, yet the answer comes from adding 8 + 6. Why is adding right?
22. Is it always true that the start is the whole in a taking-from story, but only a part in an adding-to story? Use a bar to explain.
23. Can any number go in the box in 7 - \square = 10? How do you know?
24. For 15 - \square = 13, would you count up from 13 or count back from 15? Find the box.