Check Multi-Step Answers and Fix Mistakes | Grade 5 Math | LevelKite

You already know how to solve a multi-step problem, which is a word problem that needs more than one calculation. You can record the calculations in an expression, a line of numbers and signs such as $6 \times 4 - 5$ .

Good problem solvers do one more thing after they get an answer: they check it. One wrong step can give an answer that makes no sense. In this lesson, you learn how to tell whether an answer makes sense. You also learn how to find and fix a mistake in the written calculations, which we call the work.

An estimate is a quick answer that shows about how big the exact answer should be. To make one, you round: you swap each number for a nearby number that is easy to work with, such as the nearest ten or hundred. So $38$ becomes $40$ , and $420$ becomes $400$ . Then you do the same steps as the story with the rounded numbers.

The rounded numbers are near the real ones, so the estimate is a guide to the size of the answer. It is not the exact answer.

A computed answer is the answer you get from the exact calculations. It is reasonable when it is about the same size as an estimate made from the story. Reasonable means believable. It does not mean proven correct. In this lesson, comparing an answer with an estimate is called the size check.

Worked example

Is the answer reasonable?

A farm packs $38$ crates of apples with $52$ apples in each crate. Then it sells $420$ apples. How many apples are left? Mia calculates $38 \times 52 = 1{,}976$ and then $1{,}976 - 420 = 1{,}556$ . She says $1{,}556$ apples are left. Use an estimate to decide whether her answer is reasonable.

Write the steps of the story as an expression. 38 crates with 52 apples in each are 38 equal groups, so multiply to find the apples packed. Selling 420 apples takes them away, so subtract.
$38 \times 52 - 420$
Round each number to a nearby number that is easy to work with. Each amount stays about the same size: 38 crates is about 40 crates, 52 apples is about 50 apples, and 420 apples sold is about 400 apples.
$40 \times 50 - 400$
Do the first step with the rounded numbers. About 40 crates with about 50 apples in each is about 2,000 apples packed.
$40 \times 50 = 2{,}000$
Do the second step. Take away about 400 apples sold from about 2,000 apples packed. The estimate is about 1,600 apples left.
$2{,}000 - 400 = 1{,}600$
Compare. Mia's answer is 1,556 and the estimate is 1,600. They are only 44 apart, which is a small gap for numbers this big. The two are about the same size, so her answer is reasonable.
$1{,}600 - 1{,}556 = 44$

Answer

Mia's answer of $1{,}556$ apples is reasonable, because it is about the same size as the estimate of $1{,}600$ apples.

The size check can tell you that an answer looks wrong. It does not tell you where the mistake is. For that, you use a story check.

Read the story one part at a time. For each part, ask what happens and which operation fits. An operation is one of the four things you do with numbers: add, subtract, multiply or divide. Then find the line of the work that goes with that part. Last, read the question again. The final number must be the thing the question asks for.

The table below shows the story check on Mia's apple work. Each part of the story has a matching line with the operation that fits, and the final number answers the question. So her work passes the story check.

The story check on Mia's apple work

Part of the story	What happens	Line in Mia's work
$38$ crates with $52$ apples in each	Equal groups, so multiply	$38 \times 52 = 1{,}976$ apples packed
$420$ apples are sold	Apples are taken away, so subtract	$1{,}976 - 420 = 1{,}556$ apples left
The question: how many apples are left?	The final number must tell the apples left	$1{,}556$ is the number of apples left

Key concept

Two checks for every multi-step answer

Size check. Make an estimate from the story. A reasonable answer is about the same size as that estimate.

Story check. Match the work to the story, one part at a time. Then make sure the final number is the thing the question asks for.

Do both checks. An answer can pass the size check and still be wrong, because an estimate is only a rough guide. When the work fails the story check, you have found a mistake to fix.

Three mistakes the story check can find

Mistake	What goes wrong	Question to ask
Skipped step	A fact the question needs is never used in the work.	Can I find every needed fact from the story in the work?
Wrong operation	A step adds, subtracts, multiplies or divides when the story needs a different operation.	Does each step do what happens in the story?
Answer to a different question	The final number is not the thing the question asks for.	What does the question ask for? Is my final number that thing?

Worked example

Find and fix a mistake: bracelets

A craft club has $21$ packs of beads with $38$ beads in each pack. It uses $19$ beads to make each bracelet. How many bracelets can the club make? Kai writes $21 \times 38 = 798$ and then $798 \times 19 = 15{,}162$ . He says the club can make $15{,}162$ bracelets. Find and fix his mistake.

Start with the size check. Estimate from the story, not from Kai's lines. About 20 packs with about 40 beads in each is about 800 beads. With about 20 beads in each bracelet, that makes about 40 bracelets. Kai's answer, 15,162, is hundreds of times as big as 40, so it is not reasonable. A club with about 800 beads cannot make thousands of bracelets.
$20 \times 40 = 800 \qquad 800 \div 20 = 40$
Now do the story check to find the mistake. Kai's first line matches the story: 21 packs with 38 beads in each are equal groups, and multiplying gives 798 beads. His second line does not match. Making bracelets splits the 798 beads into equal groups of 19, and splitting into equal groups is division. Kai multiplied, so this step has the wrong operation.
$\text{Kai: } 798 \times 19 \qquad \text{Story: } 798 \div 19$
Fix the step by dividing 798 by 19. Use multiplication to help. 40 groups of 19 use 760 beads. That leaves 38 beads still to be grouped.
$19 \times 40 = 760 \qquad 798 - 760 = 38$
The 38 beads that are left make 2 more groups of 19. So there are 40 + 2 = 42 groups in all, and each group makes one bracelet.
$19 \times 2 = 38 \qquad 798 \div 19 = 42$
Check the new answer. 42 is about the same size as the estimate of 40, and it is a number of bracelets, which is what the question asks for.

Answer

Kai used the wrong operation in his second step: he multiplied where the story needs division. The club can make $42$ bracelets.

Worked example

Find and fix a mistake: library shelves

A library gets $42$ cartons of books with $39$ books in each carton. It puts $210$ of the books on display tables. It puts the rest on shelves, with $21$ books on each shelf. How many shelves does it fill? Leo writes $42 \times 39 = 1{,}638$ and then $1{,}638 - 210 = 1{,}428$ . His answer is $1{,}428$ . Find and fix his mistake.

Size check. Estimate from the story: about 40 cartons with about 40 books in each is about 1,600 books. Take away about 200 display books to leave about 1,400. Put about 20 on each shelf to fill about 70 shelves. Leo's answer, 1,428, is about 20 times as big as 70, so it is not reasonable.
$(40 \times 40 - 200) \div 20 = 70$
Story check. Leo's two lines match the first two parts of the story: 1,638 books arrive, and 1,428 books are left for the shelves. But the fact that each shelf holds 21 books is never used. Leo skipped the last step. So his number answers a different question: how many books go on the shelves, not how many shelves are filled.
Add the missing step. Putting 21 books on each shelf splits the 1,428 books into equal groups of 21, so divide. 60 groups of 21 use 1,260 books. That leaves 168 books, which is 8 more groups of 21. So there are 60 + 8 = 68 groups, and each group fills one shelf.
$1{,}428 \div 21 = 60 + 8 = 68$
Check the new answer. 68 is about the same size as the estimate of 70, and it is a number of shelves, which is what the question asks for. One expression now records all three steps of the story.
$(42 \times 39 - 210) \div 21 = 68$

Answer

Leo skipped the last step, so his answer was a number of books and not a number of shelves. The library fills $68$ shelves.

The takeaway: check every multi-step answer in two ways. First, estimate from the story and see whether the answer is about that size. Second, match the work to the story. Make sure every needed fact is used, each step has the right operation, and the final number answers the question that was asked. When a check fails, find the step that is wrong or missing, fix it, and check the new answer again.