Lesson 9 of 10 · Grades 3 to 4
Word problems: choose the operation
Word problems hide the same two ideas under a story. One story tells you the group size, and asks for the total. Another tells you the total, and asks how many groups there are. Read the question twice, then decide.
Same story, two questions
Here are two stories built from the same numbers. Story one: a baker fills 6 boxes with 8 rolls in each box. How many rolls does she pack? Story two: a baker packs 48 rolls with 8 rolls in each box. How many boxes does she fill? The first story multiplies; the second divides.
The numbers are the same, but the missing piece is different. In story one the group size is known and the total is missing. In story two the group size is known and the number of groups is missing. Before you calculate, work out which piece of the picture the question is asking for.
Look for equal groups
Both operations live in stories with equal groups. Look for a phrase that gives the size of one group, such as "each bag holds 9 apples" or "6 pencils in every box". Once you spot it, ask what the story does not tell you.
If the total is missing, multiply the number of groups by the group size. If the number of groups is missing, divide the total by the group size. If the group size is missing, divide the total by the number of groups. Every equal-group story is one of those three questions.
A bar model helps
Draw the groups as a row of equal boxes. Fill in the numbers you know and put a question mark in the box you need. A row with one question mark where the total should be is a multiplication. A row where you know the total but not how many boxes there are is a division.
The drawing stops you from grabbing the first operation that comes to mind. It also shows why the two stories use the same numbers in a different order.
Useful tip
Check the size of your answer. If the story asks how many groups, the answer should be smaller than the total. If it asks for the total, the answer should be bigger than both numbers in the story. A quick sanity check catches many wrong choices.
Worked example
A teacher has 6 boxes with 8 pencils in each box. She also packs 48 loose pencils into boxes of 6. How many pencils are in the boxes, and how many boxes does she use for the loose pencils?
Step 1: the first question gives the group size and asks for the total, so multiply: 6 × 8 = 48 pencils.
Step 2: the second question gives the total and the group size, and asks for the number of groups, so divide: 48 ÷ 6 = 8 boxes.
Step 3: check both answers with an inverse operation. 48 ÷ 6 = 8 and 6 × 8 = 48, so both answers match the story.
Answer: 48 pencils in the first six boxes, and 8 boxes for the loose pencils.
Remember
- Equal-group stories use multiplication or division.
- Total missing: multiply. Number of groups or group size missing: divide.
- Draw a row of boxes with a question mark for the missing piece.
- Check whether the answer should be bigger or smaller than the numbers in the story.
Try it
Answer the five questions below. Work them out first, then open each answer to check.
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There are 6 bags with 9 apples in each bag. How many apples are there in all?
Show answer
54 apples. 6 × 9 = 54.
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54 apples are packed 9 to a bag. How many bags are needed?
Show answer
6 bags. 54 ÷ 9 = 6.
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A box holds 12 eggs. How many eggs are in 7 boxes?
Show answer
84 eggs. 7 × 12 = 84.
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84 eggs are packed into boxes of 12. How many boxes are filled?
Show answer
7 boxes. 84 ÷ 12 = 7.
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Maya reads 8 pages a day for 5 days. How many pages does she read?
Show answer
40 pages. 5 × 8 = 40.
Puzzle · Sports store
Puzzle time
A coach has 48 tennis balls. Each basket holds 6 balls. How many baskets can he fill completely?
Show a hint
The total and the group size are known. Divide the total by the group size.
Show the answer
Answer: 8. 48 ÷ 6 = 8 baskets. Check with multiplication: 8 × 6 = 48. The answer 6 would only hold 36 balls.