Choose the Operation and Estimate an Answer | Grade 6 Math | LevelKite

A word problem tells a short story about quantities. A quantity is an amount you can count or measure, such as $24$ boxes or $3.5$ kilometers.

Your first job is not to compute. It is to understand the story well enough to pick an operation. An operation is an action you do with numbers. In this lesson, the operations are addition, subtraction, multiplication and division.

Two questions help you pick:

What is happening in the story? Do parts make a total? Are two amounts compared? Do groups of the same size make a total?
Which quantities do you know, and which one do you need to find?

The same story can call for different operations. It depends on which quantity is unknown, meaning the one you need to find.

Equal groups are groups that all have the same size. Picture a bookcase where every shelf holds $12$ books.

The bookcase has $8$ shelves. How many books does it hold? You know the number of groups and the size of each group. You need the total, so you multiply: $8 \times 12 = 96$ books.
The bookcase holds $96$ books. How many shelves does it have? You know the total and the size of each group. You need the number of groups, so you divide: $96 \div 12 = 8$ shelves.

The story did not change. The unknown quantity changed, and the operation changed with it. The table shows this for three kinds of stories.

The kind of story and the unknown quantity point to the operation

Kind of story	You need to find	Operation	Example
Two parts make a total	The total	Add the two parts	A jar has $15$ red marbles and $9$ blue marbles. In all: $15 + 9 = 24$ marbles
Two parts make a total	The missing part	Subtract the known part from the total	A jar has $24$ marbles, and $9$ are blue. Not blue: $24 - 9 = 15$ marbles
Two amounts are compared	The difference: how much larger one amount is than the other	Subtract the smaller amount from the larger amount	A jar has $15$ red marbles and $9$ blue marbles. More red than blue: $15 - 9 = 6$ marbles
Equal groups make a total	The total	Multiply the number of groups by the size of each group	There are $6$ bags with $4$ marbles in each bag. In all: $6 \times 4 = 24$ marbles
Equal groups make a total	The number of groups	Divide the total by the size of each group	You put $24$ marbles into bags of $4$ . Number of bags: $24 \div 4 = 6$
Equal groups make a total	The size of each group	Divide the total by the number of groups	You share $24$ marbles equally among $6$ bags. In each bag: $24 \div 6 = 4$ marbles

Key concept

The story chooses the operation

Decide what is happening to the quantities and which quantity is unknown. Those two facts, taken together, point to the operation. A single word in the story, such as more, cannot do that by itself.

When a story has more than one part, do this for each part. It works the same way whether you need an exact answer or only a close one.

Worked example

Choose the operation for equal groups

A baker has $432$ muffins. She packs them into boxes. Each box holds $12$ muffins, and every box she uses is full. How many boxes does she use?

Name the quantities. The total is 432 muffins. Each box holds 12 muffins. The unknown quantity is the number of boxes, so let the letter b stand for the number of boxes.
$b = \text{the number of boxes}$
Say what happens in the story. Equal groups of 12 muffins make the total of 432 muffins, so 12 times the number of boxes equals 432. Write that as an equation, which is a statement that two amounts are equal.
$12 \times b = 432$
Choose the operation. The story has equal groups and you need the number of groups, so you divide the total by the size of each group. If 12 times b is 432, then b is 432 divided by 12, because dividing by 12 undoes multiplying by 12. Both equations state the same relationship.
$b = 432 \div 12$
Divide in two parts. Filling 30 boxes uses 12 times 30, or 360 muffins. That leaves 432 minus 360, or 72 muffins. Filling 6 more boxes uses 12 times 6, or 72 muffins, so no muffins are left over.
$432 \div 12 = 360 \div 12 + 72 \div 12 = 30 + 6 = 36$
Check the result in the first equation. If 36 boxes is right, then 12 times 36 gives back the total of 432 muffins.
$12 \times 36 = 12 \times 30 + 12 \times 6 = 360 + 72 = 432$

Answer

The baker uses $36$ boxes.

Sometimes you do not need an exact answer. A question may ask about how many, or a close answer may be all you need. Then you can estimate. An estimate is a number close to the exact answer that you find with easier numbers.

To get easier numbers, you round. Rounding replaces a number with a nearby number that is simpler to compute with, such as the nearest ten or the nearest whole number.

Estimating does not change the operation. You still choose it from the story. Only the numbers change.

The symbol $\approx$ means is about equal to. Here is an estimate for $198 + 305$ , with each number rounded to the nearest hundred:

198 + 305 \approx 200 + 300 = 500

Worked example

Estimate a total of equal groups

A theater has $18$ rows of seats. Each row has $32$ seats. About how many seats does the theater have?

Choose the operation from the story. The 18 rows are equal groups with 32 seats in each group. The total number of seats is unknown. When equal groups make a total and you need that total, you multiply.
$18 \times 32$
Round each number to the nearest ten, because tens are easy to multiply. The number 18 is 2 away from 20, and 32 is 2 away from 30, so both rounded numbers stay close to the originals.
$18 \approx 20 \text{ and } 32 \approx 30$
Compute with the rounded numbers, using the operation you chose in the first step. Because 20 is 2 times 10 and 30 is 3 times 10, the result is 2 times 3 times 100.
$20 \times 30 = 2 \times 3 \times 100 = 600$
See how close the estimate is. You do not need this step when a problem asks only for an estimate. Split 32 into 30 and 2 to find the exact total, which is 576 seats. The estimate of 600 is only 24 seats higher.
$18 \times 32 = 18 \times 30 + 18 \times 2 = 540 + 36 = 576$

Answer

The theater has about $600$ seats.

Worked example

Estimate with decimals and two operations

A hiking trail has $4$ sections. Each section is $2.85$ kilometers long. After the last section, the hikers walk another $1.2$ kilometers to reach their bus. About how many kilometers do they walk in all?

Choose an operation for each part of the story. The 4 sections are equal groups of 2.85 kilometers, and you need their total, so multiply. The walk to the bus is one more part of the whole distance, so add it. Write both operations in one expression, which is numbers and operation signs written together.
$4 \times 2.85 + 1.2$
Round each decimal to the nearest whole number. The decimal 2.85 is only 0.15 away from 3, and 1.2 is only 0.2 away from 1. Keep the 4 as it is, because it is already easy to compute with.
$2.85 \approx 3 \text{ and } 1.2 \approx 1$
Compute with the rounded numbers. The expression has no parentheses, so the order of operations tells you to multiply before you add.
$4 \times 3 + 1 = 12 + 1 = 13$
See how close the estimate is. The exact distance is 12.6 kilometers, so the estimate of 13 kilometers is only 0.4 kilometers higher.
$4 \times 2.85 + 1.2 = 11.4 + 1.2 = 12.6$

Answer

The hikers walk about $13$ kilometers in all.

Takeaway: The story chooses the operation. Ask what is happening to the quantities and which quantity is unknown, and do not let a single word decide for you. When a close answer is enough, keep the same operation, round each number to a nearby value, and compute.

Next, you will use these skills to solve two-step and multi-step word problems and to decide whether an answer is reasonable.