Teaching unit · Grade 4 (ages 9 to 10)

Multi-step word problems

Bar models for each step, choosing the operation, interpreting remainders, and checking with estimation

About four lessons of 45 to 60 minutes

Start here · hook

Planning the class trip budget

Your class is going to the zoo, and you have been put in charge of the money. There are 24 students, each ticket costs $7, and the bus is a flat $60. How much is the whole trip? You cannot answer that in one calculation, because you have to work out the ticket money first and then add the bus. That is a multi-step problem: the answer to the first step becomes a number you use in the second.

Real problems almost never hand you a single sum. They come as a small story with two or three calculations hidden inside, and the skill is planning the order and choosing the right operation at each step. Today you will use bar models to see each step clearly, keep track of what you have worked out, and check at the end that your answer actually makes sense.

24 tickets at $7 eachstep 1: 24 x 7 = $168 for all the tickets
A bus that costs $60step 2: add the bus, 168 + 60 = $228 in all
A budget of $300step 3: money left is 300 minus 228 = $72
A quick estimate to checkabout 24 x 7 is near 25 x 7 = 175, plus 60 is about 235, close to 228

Learning objective

What students will be able to do

Students will solve word problems that need two or more steps by planning the order of the calculations, drawing part-whole and comparison bar models to represent each step, choosing the right operation from the meaning of the problem, interpreting a remainder in context, and checking that the answer is reasonable with estimation.

Success criteria

I can read a problem and work out how many steps it needs.
I can draw a bar model to show the parts and the whole of a step.
I can choose the right operation from what the problem means, not from keywords.
I can decide what a remainder means in the story and round up or down.
I can estimate to check my answer is reasonable and answers the question.

Curriculum anchor

Standards this unit teaches

4.OA.A.3Common Core (US)
Multi-step word problems with the four operations
Solve multistep word problems posed with whole numbers and having whole-number answers using the four operations, including problems in which remainders must be interpreted. Represent these problems using equations with a letter standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding.
4.OA.A.1Common Core (US)
Multiplicative comparison
Interpret a multiplication equation as a comparison, e.g. interpret 35 = 5 x 7 as a statement that 35 is 5 times as many as 7. Represent verbal statements of multiplicative comparisons as multiplication equations. This is the reasoning behind a times-as-many comparison bar.
4.NBT.B.4Common Core (US)
Fluently add and subtract multi-digit numbers
Fluently add and subtract multi-digit whole numbers using the standard algorithm. Secure addition and subtraction is what lets students carry out the separate steps of a multi-step problem without the arithmetic getting in the way.
AC9M4N07Australian Curriculum v9 (ACARA)
Model money and number problems (Year 4)
Use mathematical modelling to solve financial and other practical problems, formulating the problem using number sentences, solving it and communicating the answer in the context of the problem.
AC9M4N05Australian Curriculum v9 (ACARA)
Efficient computation strategies (Year 4)
Develop efficient mental and written strategies and use appropriate digital tools for solving problems involving addition and subtraction, and multiplication and division where there is no remainder. These strategies are the tools each step of a multi-step problem calls on.

Before you start

Prior knowledge

This unit builds on skills students should already have met. Revisit any that are shaky first.

Key vocabulary

Words to teach and display

Word problem: a calculation hidden inside a real-life story you have to read and plan
Multi-step: needing two or more calculations, where one answer feeds the next
Sum: the answer to an addition
Difference: the answer to a subtraction
Product: the answer to a multiplication
Remainder: what is left over when a number does not divide exactly
Estimate: a rough answer from rounded numbers, used to check the real one

Teaching sequence

Teach it: concrete, pictorial, abstract

The lesson moves from things students can hold, to pictures and diagrams, to the written maths. The diagrams below are drawn from data, so they are accurate and print cleanly. Teach straight from them.

1. Spotting the steps

Concrete

A one-step problem hands you a single calculation: 24 tickets at $7 is just 24 x 7. A multi-step problem hides two or more, where you cannot reach the final answer until you have worked out something in between. The first job is not to calculate at all, it is to read the whole story and plan how many steps it needs and in what order.

Use a steady routine: read the whole problem once for the story, underline the actual question, then list the steps in order before doing any arithmetic. For the zoo trip the question is the total cost, and the plan is step 1 work out the ticket money, step 2 add the bus. Only then do you calculate.

Planning first stops the most common error, which is grabbing the numbers and doing one calculation with them. Ask at the end of every step: what have I found, and what do I still need? The trip is not solved after 24 x 7 = 168, because the bus has not been added yet.

Check for understanding, ask

In the zoo problem, what is step 1 and what is step 2?
Why is the trip not finished after you work out 24 x 7 = 168?
What should you underline before you start calculating?

2. Part-whole bars for combining steps

Pictorial

A bar model turns a step into a picture. In a part-whole bar, the whole sits along the top and the parts underneath add up to it. It answers two shapes of question: put the parts together to find the whole (add), or take one part from the whole to find the other (subtract). Drawing the bar makes the operation obvious.

For the zoo trip, step 1 is a multiplication: 24 students each paying $7 is 24 x 7 = $168 for tickets. That total becomes a part in step 2. Step 2 is a part-whole bar with two parts, the $168 tickets and the $60 bus, and the whole on top is the total cost, 168 + 60 = $228.

The picture makes clear why it is an addition: two known parts, an unknown whole, so you join them. If instead you knew the total and one part and wanted the other, the same bar would call for subtraction.

The whole trip cost splits into two parts: $168 of tickets and the $60 bus. Two known parts and an unknown whole means add, so the total is $228.

Worked example

There are 24 students going to the zoo. Each ticket is $7 and the bus is a flat $60. What is the total cost of the trip?

Step 1, the ticket money: 24 students at $7 each is 24 x 7 = $168.
Step 2, add the bus: the parts are $168 and $60, so the whole is 168 + 60 = $228.
Answer the question that was asked, with a label: the total cost.

Answer: The trip costs $228 in total.

Check for understanding, ask

Which operation does step 1 need, and which does step 2 need?
In the part-whole bar, what is the whole and what are the parts?
If you knew the total was $228 and the bus was $60, how would you find the ticket money?

3. Comparison bars for more than and times as many

Pictorial

Some problems compare two amounts rather than combine them. A comparison bar model draws the two quantities on the same scale, one above the other, so the difference or the multiple is easy to see. Two phrases signal this shape: more than (an addition or subtraction gap) and times as many (a multiplication).

Take a can drive. Room A collects 45 cans, and Room B collects 3 times as many. Drawing Room A as one bar and Room B as three of those bars shows Room B is 3 x 45 = 135 cans. That is the times-as-many meaning of the comparison.

Then a second step can ask how many more: the difference between the two bars is 135 minus 45 = 90 cans. The comparison bar shows both the multiple and the gap in one picture, so you can read off which operation each part of the question needs.

Room A's 45 cans and Room B's 135 on the same scale. Room B is 3 times as long (3 x 45 = 135) and 90 cans more (135 minus 45).

Worked example

Room A collects 45 cans. Room B collects 3 times as many as Room A. How many cans does Room B collect, and how many more than Room A?

Times as many means multiply: Room B is 3 x 45 = 135 cans.
How many more means the difference: 135 minus 45 = 90 cans.
Answer both parts of the question with labels.

The longer bar is 3 times the shorter one, and the extra length past Room A is the 90-can difference.

Answer: Room B collects 135 cans, which is 90 more than Room A.

Check for understanding, ask

Which operation does times as many call for?
How do you find how many more Room B has than Room A?
If Room B had 12 more than Room A instead of 3 times as many, would the first step change?

4. Interpreting a remainder

Abstract

When a step is a division that does not come out evenly, the leftover, the remainder, has to mean something in the story. The maths gives the same quotient and remainder every time, but what you do with the remainder depends on the question. Sometimes you round up, sometimes you drop it, and sometimes the remainder itself is the answer.

Suppose 94 students go on the trip and each bus holds 30. Dividing, 94 divided by 30 = 3 remainder 4. Three buses carry 90 students, but 4 are still standing on the pavement, so you round up to 4 buses. Dropping the remainder here would leave 4 children behind.

The same division answers a different question differently. If you asked how many students are on the fourth, not-full bus, the answer is the remainder itself, 4. And if you asked how many full buses there are, you drop the remainder and say 3. Always read the remainder back into the story.

Worked example

94 students are going on the trip. Each bus holds 30 students. How many buses are needed?

Divide to see how many full buses: 94 divided by 30 = 3 remainder 4.
3 buses hold 90 students, but 4 students are left over and still need to travel.
You cannot leave them behind, so round up: one more bus is needed.

Answer: 4 buses are needed. The remainder of 4 students forces an extra bus.

Check for understanding, ask

In the bus problem, why do you round up instead of down?
If the question asked how many students ride the last bus, what is the answer?
Name a problem where you would drop the remainder instead.

5. Checking the answer is reasonable

Abstract

The last step of every multi-step problem is to stand back and ask whether the answer makes sense. Two checks do most of the work: estimate with rounded numbers to see if the exact answer is in the right ballpark, and reread the question to be sure you answered what was asked, with the right label.

For the zoo trip total, estimate before trusting $228: about 24 x 7 is near 25 x 7 = 175, plus the $60 bus is about 235. The exact answer $228 sits close to $235, so it is reasonable. An answer of $28 or $2280 would be a place-value slip the estimate would catch.

Then reread the question. If it asked for the money left from a $300 budget, the total is only a middle step: the final part-whole bar has the $300 budget as the whole, the $228 spent as one part, and the change as the other, so 300 minus 228 = $72. Answering $228 there would be answering the wrong question.

Worked example

The trip costs $228 and the class budget is $300. How much money is left, and is the answer reasonable?

Money left is a part-whole bar: whole $300, one part $228 spent, other part the change.
Subtract: 300 minus 228 = $72.
Check with an estimate: about 300 minus 230 = 70, close to $72, so it is reasonable.

The $300 budget splits into the $228 spent and the $72 left. Whole minus a known part gives the other part.

Answer: $72 is left over, and the estimate of about $70 confirms it is reasonable.

Check for understanding, ask

How would you estimate 24 x 7 quickly to check $168?
Why is rereading the question part of checking your answer?
In the budget bar, what is the whole and what are the two parts?

Watch for

Common misconceptions and how to address them

MisconceptionThe numbers are grabbed and one calculation is done, so the problem is answered in a single step.

Why it happens: Students are used to problems that hand over one sum, so they stop after the first calculation.

How to address it: Plan the steps before calculating and ask after each one what is still needed. The zoo trip is not done after 24 x 7 = 168, because the bus is not yet added.

The bar shows two parts to combine. Stopping at the $168 tickets misses the $60 bus in the whole.

MisconceptionThe operation is chosen from a keyword rather than the meaning, so left always means subtract.

Why it happens: Keyword rules are taught as shortcuts, but cleverly worded problems break them.

How to address it: Choose the operation from what is happening in the story, then check it against a bar model. Total left over from a budget is a subtraction, but the word left is not what decides it, the part-whole picture is.

MisconceptionA remainder is ignored or always dropped, so 94 students in buses of 30 gives only 3 buses.

Why it happens: Division is practised as quotient and remainder without linking the remainder back to the situation.

How to address it: Read the remainder into the story. Four students left over cannot be left behind, so round up to 4 buses. The context, not a fixed rule, decides what happens to the remainder.

MisconceptionTimes as many is read as more than, so 3 times as many as 45 is worked out as 45 + 3.

Why it happens: The two comparison phrases sound similar and both describe one amount against another.

How to address it: Draw the comparison bar. Three times as many is three copies of the bar, 3 x 45 = 135, while 3 more would be almost the same length plus a sliver. The picture separates the two meanings.

Three times as many is three whole bars of 45, giving 135, not 45 with 3 added on.

MisconceptionThe final answer is a bare number with no label, or answers a middle step instead of the question.

Why it happens: Once a calculation is finished, students write the number and stop reading.

How to address it: Reread the underlined question and answer it in words with a unit. If the question asked for the change, $72 is the answer, not the $228 total from a middle step.

MisconceptionAnswers are not checked, so an unreasonable total is handed in.

Why it happens: The problem feels finished once the steps are done, with no habit of estimating.

How to address it: Round the numbers and estimate. About 25 x 7 = 175 plus 60 is about 235, so a total of $228 is reasonable but $28 or $2280 is not. The estimate catches place-value slips.

Do it together

Guided practice (with answers)

1. A shop sells 6 boxes of 8 pencils and then 5 loose pencils. How many pencils in all?
Answer: 53. Step 1: 6 x 8 = 48. Step 2: 48 + 5 = 53.
2. Mia reads 15 pages a day for 4 days, then 12 more pages. How many pages has she read?
Answer: 72. Step 1: 15 x 4 = 60. Step 2: 60 + 12 = 72.
3. Room A has 28 stickers. Room B has 12 more than Room A. How many do they have together?
Room B is 12 longer than Room A, so 40; together they make 68.
Answer: 68. Step 1: Room B is 28 + 12 = 40. Step 2: 28 + 40 = 68.
4. A baker makes 100 rolls and packs them in bags of 8. How many full bags, and how many rolls are left over?
Answer: 12 full bags with 4 rolls left over, because 100 divided by 8 = 12 remainder 4.
5. 78 children need to sit in rows of 10. How many rows are needed so everyone has a seat?
Answer: 8 rows. 78 divided by 10 = 7 remainder 8, and the 8 left over need one more row, so round up.
6. A class raises $85 and spends $37 on seeds. Estimate then find the money left.
Answer: About 90 minus 40 = 50. Exact: 85 minus 37 = $48, close to the estimate.

On their own

Independent practice worksheets

Set the matching ChalkBee worksheets for independent work. The answer keys are computed in code, so they are never wrong. Start with two-step problems, then move to problems that mix operations and interpret remainders.

Grade 4 word problems worksheet (Set A)two-step problems with the four operations Grade 4 word problems worksheet (Set B)more practice, including remainders and comparison All word problems worksheetsthe full set across grades 1 to 6 All mixed operations worksheetschoosing the operation, the core multi-step skill

Reach every student

Differentiation

Support

Give a blank part-whole and comparison bar template to fill in, so the model is a scaffold not a blank page.
Provide a two-line step planner (step 1, step 2) to write the plan before any arithmetic.
Start with two-step problems that use one operation, such as add then add, before mixing operations.
Keep the numbers small and friendly at first so the reading and planning, not the arithmetic, is the focus.

Extension

Move to three-step problems and to problems with a hidden extra step (a discount, then a total, then change).
Write equations with a letter for the unknown, such as 24 x 7 + 60 = c, to match the standard.
Have students invent a multi-step trip-budget problem and swap with a partner to solve.
Explore problems where the remainder is dropped, rounded up, or is the answer, and sort them by what the remainder means.

Check it stuck

Assessment: exit ticket

A three-question exit ticket for the last five minutes. It samples a two-step combine, a comparison, and a remainder to interpret.

1. A tray holds 4 rows of 6 muffins. 5 muffins are sold. How many are left?
Answer: 19. Step 1: 4 x 6 = 24. Step 2: 24 minus 5 = 19.
2. Sam has 16 marbles. Alex has 3 times as many. How many does Alex have, and how many more than Sam?
Answer: Alex has 48 (3 x 16), which is 32 more than Sam (48 minus 16).
3. 90 apples are packed into boxes of 12. How many boxes are needed for all the apples?
Answer: 8 boxes. 90 divided by 12 = 7 remainder 6, and the 6 left over need an eighth box, so round up.

For the teacher

Teacher notes and timings

Rough timing across four lessons: Lesson 1 spotting the steps and the routine (section 1), Lesson 2 part-whole bars (section 2), Lesson 3 comparison bars (section 3), Lesson 4 interpreting remainders and checking (sections 4 and 5) plus the exit ticket.
Language to keep saying: read then plan then solve then check, what have I found and what do I still need, choose the operation from the meaning, read the remainder back into the story. These pre-empt most of the misconceptions.
Keep bar-model templates out through the pictorial sections. Drawing the bar is what turns choosing the operation from a guess into a decision you can see.
The bar models use whole-dollar and whole-number values so the part widths stay clear. Comparison bars keep the longer bar within about three times the shorter, so the scale reads honestly on the page.
Curriculum note and a US and AU alignment: the US sets multi-step problems with interpreted remainders and an estimation check at Grade 4 (4.OA.A.3), leaning on multiplicative comparison (4.OA.A.1) and fluent addition and subtraction (4.NBT.B.4). ACARA meets the same work through Year 4 mathematical modelling of practical and money problems (AC9M4N07) and efficient strategies (AC9M4N05), so the US and Australian placements align closely at this level.
Present mode and print both work: use the Print button for a clean teacher copy or a student handout, and project the page to teach straight from the diagrams.

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