Bar models for story problems

When you reach for it

A child who reads a story problem has to decide two things: which numbers belong together, and whether to add or to subtract. With small numbers they can act the story out with counters and watch the answer appear. With numbers like 38 and 81 that stops being practical. Nobody lays out 81 counters.

A bar model is a drawing that keeps the shape of the story and drops the counting. A long rectangle stands for an amount. Cut it into pieces and each piece is a part. A bracket over the whole bar holds the whole. A question mark goes wherever the story has a number missing. Once the bar is drawn, one question settles the calculation: is the missing number the whole, or a part? If the whole is missing, put the parts together and add. If a part is missing, take the known part away from the whole, or count up from the known part to the whole.

Children have already met bars inside single kinds of story: a whole and two parts in parts and the whole, the missing change in Problems with the start, change or result missing, and two bars side by side in How many more and how many fewer. This lesson puts the bar to work as one tool for every kind of adding and subtracting story within 100: putting together, taking apart, adding on, taking away, and comparing. It is taught over four days, one part a day. The arithmetic itself (adding and subtracting two-digit numbers) is done the ways the child already knows; the bar’s job is to decide which arithmetic to do.

Read every story aloud with the child, and let them draw the bar before any number sentence is written.

The procedure

Part 1: one bar, a whole and its parts

The plainest stories have two groups and the total of both. “38 bags were filled in the morning and some in the afternoon. 81 bags were filled in all.” The bags filled in the morning and the bags filled in the afternoon are the two parts. All the bags are the whole.

Draw one bar for the whole and cut it into two pieces, one for each part. Write each known number where it belongs: a part inside its piece, the whole over the bracket. Put a question mark where the missing number goes.

The bar says what a number bond says: the two parts together make the whole. So there are only two kinds of question, and each has its calculation.

Why does this work? Every bag is in exactly one of the two pieces, so counting all the bags is the same as counting one piece and carrying on through the other: that is adding. And if one piece is removed from the bar, what is left is exactly the other piece: that is subtracting. Counting up from the known part finds the same missing part, because it asks “this part and how many more make the whole?”, which is subtraction as a missing addend.

Drawing roughly to size. The bar does not need a square for every bag. It does help if the bigger number gets the longer piece, roughly. In the picture above the morning piece, 38, is a little less than half the bar, because 38 is a little less than half of 81. A bar drawn this way gives a free check: the missing piece looks a little more than half, so the answer should be a little more than 40. A sketch with the pieces the wrong way round still gives the right answer, but it loses that check.

Part 2: stories where something changes

Some stories have an event in them. Things are added on (“18 more were planted”) or taken away (“she gave 35 away”). These can be drawn with the same single bar, as long as the whole is put in the right place.

• Adding on. The start and the amount added are the two parts. The end is the whole, since everything there at the end was either there at the start or added.
• Taking away. The start is the whole. The amount taken and the amount left are the two parts, since every thing that was there at the start either went or stayed.

After that, the rule from Part 1 decides the calculation, whichever of the three numbers is missing. In an adding-on story with the change missing, the end (the whole) is known, so a part is missing and we subtract or count up, even though the story is about adding. In a taking-away story with the start missing, the whole is missing and we add, even though the story is about taking away. The action in the story does not choose the calculation; the place of the question mark on the bar does. Stories with the start missing have their own lesson, Problems with the starting amount missing.

Part 3: two bars for comparing

A comparing story has no joining and no taking away. Two amounts sit side by side and the story talks about the gap: “how many more”, “how many fewer”, “how much taller”. Here we draw two bars, one for each amount, lined up at the left edge, the bigger amount on top.

The longer bar then falls into two pieces: a piece as long as the shorter bar, and a piece that sticks out past its end. That second piece is the gap. So the longer bar is a whole, and its two parts are the smaller amount and the gap. The same rule applies:

• the bigger amount is the whole, so if it is missing, add the smaller amount and the gap;
• the smaller amount and the gap are parts, so if either is missing, subtract or count up.

Reading who has more from the sentence (“27 centimeters taller than”, “5 fewer than”) is the work of How many more and how many fewer. The bar adds nothing new there; it simply puts the comparing story into the same whole-and-parts shape as every other story.

Part 4: choosing the bar

Now every kind of story has been drawn, and they come down to the same picture. In each one, three numbers are tied together: a whole and two parts. The kinds differ only in where the whole is.

So drawing the bar is really one question asked of the story: which number is the whole? Find it, put it over the bracket, and the calculation follows. Two habits make the drawing reliable.

Only the numbers that belong. A story may give a number that is not about the same things. The bar is for one kind of thing, and a number about something else stays off it.

Check against the bar. When the answer is found, put it in its piece and check two things. The two parts must make the whole. And the pieces should look about right: a missing piece drawn much shorter than the known one cannot turn out much longer.

Going the other way is useful too. A bar with its numbers can be read as many different stories, because the bar keeps only the shape and the numbers, not the things or the action. That is why one fact answers many stories, and why it is worth having children write stories for a bar, not only bars for stories.

Worked examples

Each part of the lesson has two examples. Read each story aloud and let the child draw the bar first.

Boundaries

• The bar needs two of the three numbers. A whole and two parts are tied together, so any two decide the third. With only one, as in “Kapono filled some bags and Danica filled 20”, there is no single answer.
• A part cannot be bigger than the whole. With things that are counted, a piece of a bar is never longer than the bar. A story where the known part is bigger than the stated whole, such as “50 bags in all, 63 of them in the morning”, cannot happen, and no calculation will rescue it. A part can equal the whole only when the other part is 0.
• Everything on the bar counts the same kind of thing. Bags with bags, centimeters with centimeters. Shells do not go on a bar of bottle caps, and a length cannot be added to a number of plants.
• The bar shows one step. A whole and two parts give one addition or one subtraction. A story whose answer needs a first answer, such as the caps left in the bucket and then the shells added back in, needs two bars, one after the other. That is Two-step addition and subtraction problems.
• Roughly to size is a help, not a measurement. A sketch can show that one part is bigger or that a piece is about half; it cannot tell 43 from 44. The calculation finds the number; the bar only says which calculation, and whether the answer looks sensible.
• Two parts here, not more. A whole can be cut into three or more parts, and a bar can show that, but every story in this lesson ties exactly three numbers together: one whole and two parts.

Where this goes next

Bar models are one way into all of the story problems with adding and subtracting: every one-step story in that family, whatever its words, turns into a whole and two parts, and Problems with the total missing, Problems with the start, change or result missing, Problems with the starting amount missing and How many more and how many fewer are the same picture with the question mark in different places. Stacking two bars, the answer to the first becoming a number in the second, is how Two-step addition and subtraction problems are drawn. Each bar also gives a fact family: the beach bags give 38 + 43 = 81, 43 + 38 = 81, 81 - 38 = 43 and 81 - 43 = 38.

The bars here are always drawn for adding and subtracting. Later the same rectangle is cut into equal pieces, and then it shows sharing and groups. Years on, it becomes the tape diagram used for ratios and for setting up equations, where the habit of asking “which number is the whole?” still does the deciding.

Exercises

Part 1: one bar, a whole and its parts

1. At the beach cleanup, Abiola picked up 24 bottle caps and Danica picked up 35. How many bottle caps did they pick up in all?
2. The class garden has 62 plants. 37 of them are bean plants and the rest are tomato plants. How many tomato plants are there?
3. Kapono’s crew filled 17 bags before lunch and 26 bags after lunch. How many bags did the crew fill?
4. There are 50 bean poles in the garden. 28 are standing in the beds and the rest are in the shed. How many bean poles are in the shed?
5. The beach crew filled 74 bags, some with trash and some with recycling. 39 bags hold trash. How many bags hold recycling?

Part 2: stories where something changes

6. The class garden had 36 tomato plants. The class planted 25 more. How many tomato plants are there now?
7. There were 80 bottle caps in a pail. Kapono took 46 out to sort. How many bottle caps are still in the pail?
8. The truck held 29 bags of trash. The crew loaded more. Now the truck holds 57 bags. How many bags did the crew load?
9. A bag held 93 bean seeds. The class planted some. Now 58 seeds are left in the bag. How many seeds did the class plant?
10. Danica had some tomatoes in her basket. She picked 24 more. Now she has 71 tomatoes. How many tomatoes were in her basket at first?

Part 3: two bars for comparing

11. Kapono’s bean pole is 86 centimeters tall. Abiola’s bean pole is 59 centimeters tall. How much taller is Kapono’s bean pole?
12. Danica’s group filled 32 bags at the beach. Abiola’s group filled 19 more bags than Danica’s group. How many bags did Abiola’s group fill?
13. Kapono collected 70 bottle caps. He collected 24 more than Danica. How many bottle caps did Danica collect?
14. A tomato plant is 44 centimeters tall. It is 18 centimeters shorter than a bean plant. How tall is the bean plant?
15. The beach crew found 41 shells and 66 bottle caps. How many fewer shells than bottle caps did they find?

Part 4: choosing the bar

Use the bar models for Exercises 16 to 18. What number goes where the question mark is?

16. Bar model A.
17. Bar model B.
18. Bar model C.
19. Danica’s pail holds 54 bottle caps and 9 shells. She puts 30 of the bottle caps into a bag. How many bottle caps are in the pail now?
20. Write a story about the class garden that fits a bar with 75 over the bracket and pieces labeled 40 and 35, with the question mark on the 35 piece. Then write a comparing story with the same three numbers.

Explain

21. In a taking-away story, why is the starting amount the whole and not one of the parts?
22. A bar is drawn roughly to size, and the known part takes up more than half of it. Without working anything out, what can you say about the missing part?
23. A bar has 50 over the bracket and question marks in both pieces. Can you find the parts? What else would you need to know?
24. In a bar with 64 over the bracket, one piece is labeled 64. What goes in the other piece? Tell a story that fits.
25. In every bar model in this lesson, is the whole always the biggest of the three numbers? Why?