Problems using several operations

When you reach for it

Some stories cannot be answered in one step. The number the story asks for depends on a number the story never states, and that hidden number has to be found first.

Koldo’s class goes to the movie theater. The teacher buys 4 packs of movie tickets with 6 tickets in each pack, and 5 of the tickets go unused. How many tickets are used?

No single fact answers this. The story does not say how many tickets there were in all. That number is hidden inside “4 packs of 6”, and only once it is known can the 5 unused tickets be taken away. So the story has two steps, and the first step’s answer is the second step’s starting number.

Reach for this whenever a story’s question cannot be answered from the numbers it gives, but can be answered from a number that one of its own steps would produce. Two-step stories with adding and subtracting only are Two-step addition and subtraction problems. Here any of the four operations can appear in either step: multiplying and dividing within 100, and adding and subtracting within 1{,}000. The pieces are already in place. Equal group problems showed when a story multiplies or divides, and Using a letter for the unknown number wrote the number a story asks for as a letter.

The lesson takes four days, one part each. Part 1 writes each step as its own equation with a letter. Part 2 chooses the steps and their order from what happens in the story. Part 3 checks that an answer is reasonable by rounding or by mental math. Part 4 writes both steps as one expression and settles which operation goes first.

The procedure

Three habits carry every problem in this lesson, and each has a reason.

Picture the story and find the hidden number. A drawing, often a tape diagram, shows what the story knows and what it does not. The question asks for one unknown; the drawing shows which other unknown has to be found on the way. Naming that hidden number is the first step.

Write one equation for each step, with a letter for each unknown. Each equation tells one part of the story, in the order it happens. The letter in the first equation becomes a known number once it is found, and then it is used in the second equation. Saying what each letter stands for, in words, is what keeps the two steps from getting mixed up.

Check that the answer is reasonable, then answer in a sentence. Round the numbers and work the same steps with the rounded numbers, or work the story with friendlier numbers in your head. An exact answer far from the estimate means a step went wrong. Then say what the number counts.

Part 1: two steps, two equations

Back to the movie tickets. The hidden number is the number of tickets the teacher bought. Call it t. Four packs of 6 is an equal groups story with the total missing, so 4 \times 6 = t. Once t is known, the second step has a starting number: the 5 unused tickets come off it, and the rest, u, are used, so t - 5 = u. Example 1 works both steps.

The tape diagram holds both steps at once. The top bar is the packs: 4 equal parts of 6. The bottom bar is the same length, because it is the same tickets, now split into the ones used and the ones not used. Reading the top bar gives the first equation, and reading the bottom bar gives the second.

A story may need more than one number found before the last step. At an apple-picking farm, Pello fills 3 bags with 8 apples each and 2 bags with 7 apples each. The question, how many apples in all, needs both bag totals first. Example 2 finds each with a multiplication, then joins them.

Part 2: the order comes from the story

The steps of a story are not read off its words. They come from what happens.

On a zoo trip, 72 students ride in vans, 8 students to a van. Then 3 more vans join the trip for the teachers. How many vans go to the zoo?

The first thing that happens is that the students fill vans. The total is known (72) and the size of each group is known (8), so the number of vans is missing, and that is a division, 72 \div 8 = v. Only then can the 3 extra vans be added on, v + 3 = w. Example 3 works it through.

Notice the word “more” in this story. It does lead to an addition here, but the addition is the second step, and the first step is a division that no word in the story names. A single word does not choose the operation: “in all” ends both “3 bags of 8 apples, how many in all?” (a multiplication) and “29 apples and 34 apples, how many in all?” (an addition). What chooses the operation is the situation: equal groups being counted or filled call for multiplying or dividing, and amounts being joined, separated or compared call for adding or subtracting.

The order of the steps comes from the situation too. Sometimes the story tells the steps in the order they happen, as with the vans. Sometimes the question points at the last step and says nothing about the first. In Example 4, Xabier and Koldo pick apples and then pack them into bags of 7: the bags can only be counted once the two pickings are added together, so the addition comes first even though the question is about bags.

Part 3: is the answer reasonable?

A two-step answer can be worked out carefully and still be wrong, because a step was skipped, a step was done the wrong way round, or a fact slipped. A quick second answer, worked a different way, catches most of these.

The movie theater starts the weekend with 612 bags of popcorn. It sells 238 bags on Saturday and 197 bags on Sunday. How many bags are left?

Two subtractions, in the order the days happen:

where s is the number of bags left after Saturday and r the number left after Sunday. Working by place value (Subtracting three-digit numbers by place value, including across zero), 612 - 238 = 374, and then 374 - 197 = 177.

To check, round each number to the nearest ten and do the same two steps. The number 612 rounds to 610, 238 to 240, and 197 to 200. Then 610 - 240 = 370, and 370 - 200 = 170. The exact answer 177 is close to 170, so it is reasonable: it is about the right size, which is all an estimate can show.

An estimate cannot prove the exact answer right; it can only show whether it is in the right neighborhood. If the work had produced a number near 370, the estimate would show at once that the Sunday step was left out, since 370 is only one subtraction away from 610.

Mental math is the other check. For a story whose numbers are already small, a check can be done by working the steps in a friendlier way, or by working backward from the answer. Example 6 does both.

Part 4: two steps in one expression

Both steps of a story can be written as one expression, a row of numbers and operation signs that gives one number when it is worked out. The movie ticket story becomes

This expression must mean “find 4 \times 6, then take away 5”, which is 24 - 5 = 19. If it were worked the other way, 6 - 5 = 1 first and then 4 \times 1 = 4, it would give a different number that has nothing to do with the story. So everyone who reads an expression has to agree on which operation comes first. The agreement is:

• Work inside parentheses first.
• Then multiply and divide, before any adding or subtracting. When multiplying and dividing sit side by side, work them from left to right.
• Then add and subtract, from left to right.

By that rule, 4 \times 6 - 5 means the multiplication first, which matches the story. The apple bags of Example 2 become 3 \times 8 + 2 \times 7, and the rule does both multiplications first, 24 + 14, which again matches the story.

Parentheses are for a story whose adding or subtracting has to happen first. In the theater, each of 4 rows has 5 seats on one side of the aisle and 3 on the other. Each row holds 5 + 3 seats, and that sum has to be found before multiplying by the 4 rows. Parentheses say so:

Without the parentheses, 5 + 3 \times 4 means 5 + 12 = 17 by the rule: five seats and then four groups of three. That is a different story. The parentheses are what make the expression match the picture. This lesson uses only this much of the rule; the whole of it, and why it has to be agreed on, is Order of operations.

Worked examples

Examples 1 and 2 go with Part 1, 3 and 4 with Part 2, 5 and 6 with Part 3, and 7 and 8 with Part 4.

Boundaries

• Every division comes out even. Each story here divides a total that is a whole number of groups, such as 72 students in vans of 8. A story with something left over, such as 74 students in vans of 8, asks what to do with the leftover, which belongs to Remainders and what they mean.
• Within the facts and within 1{,}000. Multiplying and dividing stay within the facts up to 10 \times 10, and a one-digit number times a multiple of ten; adding and subtracting stay within 1{,}000. The same thinking works for larger numbers, with the methods that come later.
• An estimate checks size, not exactness. Rounding can show an answer is far off, but an answer near the estimate may still be wrong by a few. Working backward is the exact check.
• The order rule is an agreement about writing, not about stories. A story is always solved in the order its situation needs. The rule only says what an expression means, so that the expression written for a story matches it. When the story’s adding must come first, the expression needs parentheses.
• Not comparison by times. Stories in which one amount is a number of times another are Times as many, a different situation.

Where this goes next

The two-step habit of finding a hidden number and then using it is the start of every longer problem. Order of operations finishes the rule begun in Part 4 and says why an agreement is needed. Numerical expressions and Grouping symbols write longer expressions with several operations and several sets of parentheses. Remainders and what they mean takes on the division steps that do not come out even, and Times as many adds a new kind of multiplication story that can be one of the steps.

Exercises

Part 1: two steps, two equations

For 1 to 4, write an equation with a letter for each step, say what each letter stands for, and answer the question.

1. The movie theater snack stand has 7 boxes of granola bars with 8 bars in each box. It sells 29 bars. How many bars are left?
2. At the apple-picking farm, Pello fills 6 bags with 9 apples each, then picks 15 more apples that he carries loose. How many apples does he pick?
3. At the zoo, the keepers fill 4 baskets with 10 carrots each and 3 baskets with 5 carrots each. How many carrots are in the baskets?
4. Xabier has 48 apples and packs them in bags of 6. He gives 3 bags to his grandmother. How many bags does he keep?
5. Find each letter: 7 \times 4 = a, then a + 18 = b.
6. A class saves 3 rows of 9 seats at the movie theater, and 20 students sit in them. Draw a tape diagram and find how many saved seats are empty.

Part 2: the order comes from the story

7. At the zoo, 35 students split into groups of 5. Each group has 1 guide, and 3 more guides walk at the back. How many guides are there?
8. Koldo picks 29 apples and Pello picks 27. They share all the apples equally into 7 baskets. How many apples go in each basket?
9. At the zoo picnic area, 6 tables each seat 4 children, and 7 more children sit on the grass. How many children are there? Say which step comes first and why.
10. Xabier had 80 zoo raffle tickets. He sold 35, and then split the rest equally among 5 friends. How many tickets did each friend get?
11. Write a two-step story for the equations 9 \times 3 = p and p - 12 = q. Find p and q.
12. Koldo had some apples. He picked 4 bags of 7 more apples and now has 40 apples. How many apples did he have at the start?

Part 3: is the answer reasonable?

13. The movie theater had 846 cups. It used 312 cups on Friday and 289 on Saturday. How many cups are left? Check by rounding to the nearest ten.
14. At the apple orchard, 368 apples are picked on Monday and 257 on Tuesday, and 194 of them are sold. How many apples are not sold? Check by rounding.
15. The zoo had 503 visitors in the morning and 288 in the afternoon, and 176 of the visitors bought a map. Use rounding to decide whether 915 is a reasonable number of visitors who did not buy a map. Then find the exact number.
16. The zoo has 8 vans with 9 seats in each, and 58 people are on board. How many seats are empty? Check with mental math.
17. The zoo gift shop had 264 postcards. A delivery brought 350 more, and a second delivery brought 185. How many postcards does the shop have now? Check by rounding.

Part 4: two steps in one expression

18. Find the value of each expression. (a) 5 + 4 \times 6 (b) (5 + 4) \times 6 (c) 36 \div 4 - 2 (d) 36 \div (4 - 2)
19. Pello has 3 bags of 9 apples and 4 loose apples. Write one expression for the number of apples and find it.
20. A theater has 7 rows, each with 4 seats on the left of the aisle and 6 on the right. Write one expression with parentheses for the number of seats and find it.
21. “Xabier has 60 zoo stickers. He gives 5 friends 8 stickers each. How many stickers does he have left?” Which expression matches the story, (a) 60 - 5 \times 8 or (b) (60 - 5) \times 8? Find the answer.
22. Where can parentheses go in 3 \times 5 + 4 to make its value 27? What is its value with no parentheses?

Explaining and reasoning

23. The words “in all” end two stories: “4 boxes with 9 crayons each; how many in all?” and “36 red apples and 9 green apples; how many in all?” Explain why one is a multiplication and the other an addition, and what this says about choosing an operation.
24. Find 20 - 8 - 4 and 20 - (8 - 4). Explain with a story why they are different.
25. Explain why rounding can show that an answer is wrong but cannot show that it is exactly right.