Teaching unit · Grade 3 (ages 8 to 9)

Two-step word problems

Solving problems that need two operations, using bar models and checking the answer is reasonable

About four lessons of 45 to 60 minutes

Start here · hook

A shopping trip is never just one sum

You go to the shop with a ten-dollar note to buy snacks for a picnic. First you pick up 3 packs of crackers at 2 dollars each, then you grab a 3-dollar tub of dip. How much change do you get back? Notice you cannot answer in one go. You have to work out the cost of the crackers first, then add the dip, then take that from your ten. That is a two-step problem: the answer to the first sum becomes a number you need for the second.

Most real problems are like this, a small chain of steps hidden inside a story. Today you will learn to read the whole story first, plan the steps in order, and draw a bar model so you can see which operation each step needs. Then you will check your answer is reasonable with a quick estimate, so a silly slip never slips through.

3 packs at $2, then a $3 dipstep 1: 3 x 2 = 6, step 2: 6 + 3 = $9 spent
Change from a $10 notestep 3: 10 - 9 = $1 change
4 bags of 6 apples, shared by 3step 1: 4 x 6 = 24, step 2: 24 / 3 = 8 each
Estimate to checkabout $6 plus $3 is $9, so change near $1 looks right

Learning objective

What students will be able to do

Students will solve two-step word problems using the four operations by reading the whole story, planning the two steps in order, drawing a part-whole or comparison bar model to choose each operation, solving with a labelled answer, and checking the answer is reasonable with an estimate found by rounding.

Success criteria

I can tell that a problem needs two steps and say what each step is.
I can draw a part-whole bar model to combine or find a missing part.
I can draw a comparison bar model for more than and times as many.
I can solve both steps in order and answer with the right label.
I can round to estimate and check my answer is reasonable.

Curriculum anchor

Standards this unit teaches

3.OA.D.8Common Core (US)
Two-step word problems
Solve two-step word problems using the four operations, represent them with an equation using a letter for the unknown quantity, and assess the reasonableness of answers using mental computation and estimation, including rounding.
3.NBT.A.1Common Core (US)
Round to estimate (foundation)
Use place value understanding to round whole numbers to the nearest 10 or 100. Rounding is how students form the estimate that judges whether a two-step answer is reasonable.
AC9M3N04Australian Curriculum v9 (ACARA)
Solve multiplication and division problems (Year 3)
Multiply and divide one- and two-digit numbers, representing problems using number sentences, diagrams and arrays, and using a variety of calculation strategies. Many two-step problems in this unit have a multiply or divide as one of their steps.
AC9M3N03Australian Curriculum v9 (ACARA)
Add and subtract to solve problems (Year 3)
Add and subtract two- and three-digit numbers by using place value to partition, rearrange and regroup, and by choosing efficient strategies. The combining and comparing steps here are exactly this add-and-subtract work.

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

Two-step problem: a problem that needs two calculations, where the first answer feeds the second
Bar model: a drawing of bars that shows how the amounts in a problem fit together
Part-whole: a bar where the parts underneath add up to the whole on top
Comparison: two bars on the same scale to show a difference or a multiple
Sum: the result of adding, the whole when parts are combined
Estimate: a close-enough answer found quickly by rounding, used to check

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 a two-step problem and planning

Concrete

A one-step problem hands you a single calculation. A two-step problem hides two, and you cannot reach the answer until you have worked out something in between. The first job is not to calculate, it is to read the whole story, underline the real question, and list the steps in order. For the picnic, the question is the change, and the plan is: step 1 find the cracker cost, step 2 add the dip, step 3 take it from ten.

Use a steady routine every time: read the whole problem once for the story, underline the question, then plan the steps before touching any numbers. Ask after each step, what have I found and what do I still need?

Planning first stops the most common error, grabbing the numbers and doing one calculation with them. The picnic is not solved after 3 x 2 = 6, because the dip has not been added and the change has not been found.

Check for understanding, ask

In the picnic problem, what is step 1 and what is step 2?
Why is the problem not finished after you work out 3 x 2 = 6?
What should you underline before you start calculating?

2. Part-whole bars: buy several, then find the total

Pictorial

A part-whole bar model turns each step into a picture. The whole sits on top and the parts underneath add up to it. For the shop, step 1 is a multiplication: 4 packs of stickers at 3 dollars each is 4 x 3 = 12 dollars. Then step 2 combines that with a 5-dollar folder in a part-whole bar: the two parts are the 12 and the 5, and the whole on top is the total spent, 12 + 5 = 17 dollars.

Drawing the bar makes the operation obvious: two known parts and an unknown whole means you add. If instead you knew the whole and one part and wanted the other, the same bar would call for subtraction.

The key move of a two-step problem is right here: the 12 from step 1 does not answer the question, it becomes a part in step 2. Always ask whether your first answer is the final answer or a stepping stone.

Step 2 as a part-whole bar: 12 dollars of stickers and a 5-dollar folder combine to the 17-dollar total. Two parts, unknown whole, so add.

Worked example

Priya buys 4 packs of stickers at $3 each and one folder for $5. How much does she spend in total?

Step 1, the sticker cost: 4 packs at $3 each is 4 x 3 = $12.
Step 2, add the folder: the parts are $12 and $5, so the whole is 12 + 5 = $17.
Answer the question asked, with a label: the total spent.

Answer: Priya spends $17 in total.

Check for understanding, ask

Which operation is step 1, and which is step 2?
In the part-whole bar, what is the whole and what are the parts?
Why is $12 not the final answer?

3. Two steps with subtraction: finding the change

Pictorial

Now finish the shopping trip. Priya spent 17 dollars and paid with a 20-dollar note. The change is the missing part of a part-whole bar: the whole is the 20 she handed over, one part is the 17 she spent, and the other part is the change. Whole minus a known part gives the other part, so 20 - 17 = 3 dollars change.

This is still a two-step problem: you needed the 17 from the last section before you could find the change. Many everyday problems are buy-then-change in exactly this shape.

The bar keeps you from subtracting the wrong way round. The 20 is the whole because it is the most, and the two smaller amounts, spent and change, must fit inside it.

The $20 note is the whole, split into the $17 spent and the $3 change. Whole minus a known part gives the other: 20 - 17 = 3.

Worked example

For the picnic, 3 packs of crackers cost $2 each and a tub of dip is $3. You pay with a $10 note. What is the change?

Step 1, the crackers: 3 x 2 = $6.
Step 2, total spent: 6 + 3 = $9.
Step 3, the change: whole $10 minus the $9 spent is 10 - 9 = $1.

The $10 note splits into $9 spent and $1 change: 10 - 9 = 1.

Answer: You get $1 change.

Check for understanding, ask

What are the whole and the two parts in the change bar?
Why is the $10 the whole and not one of the parts?
If you had paid with $20 instead, what would the change be?

4. Comparison bars: times as many, then altogether

Pictorial

Some two-step problems compare before they combine. Ben has 6 marbles. Sara has 3 times as many. How many do they have altogether? Step 1 is a comparison: times as many means multiply, so Sara has 3 x 6 = 18. Drawing Ben as one bar and Sara as three of those bars makes the multiple easy to see. Step 2 combines them: 6 + 18 = 24 marbles altogether.

Two phrases signal a comparison bar: more than, which is an add or subtract gap, and times as many, which is a multiplication. Draw the two bars on the same scale so the relationship is visible before you calculate.

Then read what the question actually wants. Altogether means combine both bars (add). How many more would mean the gap between them (subtract). The same picture answers either, so reread the question.

Ben's 6 and Sara's 18 on the same scale. Sara's bar is 3 times as long (3 x 6 = 18). Altogether is 6 + 18 = 24.

Worked example

Ben has 6 marbles. Sara has 3 times as many as Ben. How many marbles do they have altogether?

Step 1, times as many means multiply: Sara has 3 x 6 = 18 marbles.
Step 2, altogether means combine: 6 + 18 = 24.
Answer with a label: marbles altogether.

Answer: Together they have 24 marbles.

Check for understanding, ask

Which operation does times as many call for?
How would you find how many more Sara has than Ben?
Why do you add in step 2 rather than subtract?

5. Checking the answer is reasonable

Abstract

The last step of every two-step problem is to stand back and ask whether the answer makes sense. The quickest check is an estimate: round the numbers to friendly ones, work out the rough answer in your head, and see if the exact answer sits near it. If it does not, you have caught a slip before handing in a wrong answer.

For a trip that cost $17, estimate: about 4 x 3 is 12, plus about 5 is 17, so 17 is right in the ballpark. An answer of $170 or $2 would be a place-value or operation slip the estimate would flag at once.

Then reread the question to be sure you answered what was asked, with the right label. A bare number, 17, does not say 17 what. Two-step problems are easy to stop one step early, so the final check guards against answering the middle step.

Worked example

A class buys 6 boxes of pencils with 8 pencils in each, then hands out 40 pencils. How many are left, and is the answer reasonable?

Step 1, total pencils: 6 x 8 = 48.
Step 2, pencils left: 48 - 40 = 8.
Check with an estimate: about 6 x 8 is roughly 50, minus 40 is about 10, close to 8, so it is reasonable.

Answer: 8 pencils are left, and the estimate of about 10 confirms it is reasonable.

Check for understanding, ask

How would you estimate 6 x 8 to check the total quickly?
Why is a bare number like 8 not a complete answer?
What kind of mistake would an estimate of about 10 help you catch?

Watch for

Common misconceptions and how to address them

MisconceptionStop after the first step, giving the middle number as the answer.

Why it happens: The first calculation produces a number that looks like an answer, so students hand it in.

How to address it: After every step ask, does this answer the question, or is it a stepping stone? For the picnic, $6 for the crackers is a step, not the change. Reread the underlined question before stopping.

MisconceptionGrab all the numbers in the problem and do one calculation with them.

Why it happens: Students pull the numbers out without reading the whole story or planning the steps.

How to address it: Read the whole problem first and plan the steps in order. Not every number is used at once, and the order matters: you must find the total spent before you can find the change.

MisconceptionTrust a keyword, so altogether always means add and left always means subtract.

Why it happens: Keywords are taught as shortcuts, but they break on cleverly worded problems.

How to address it: Choose the operation from the meaning of the story, then check it against the bar model. Draw the bars and let the picture, not the keyword, decide add or subtract.

MisconceptionSubtract the wrong way round when finding change, doing 17 minus 20.

Why it happens: Students subtract in the order the numbers appear rather than from the whole.

How to address it: In the bar the note is the whole because it is the largest, and the spent and change are parts inside it. Take the part from the whole: 20 minus 17, never 17 minus 20.

The whole is the $20 note. Change is whole minus spent: 20 - 17 = 3.

MisconceptionAnswer with a bare number and no label.

Why it happens: Students give the numeral and forget the story asked for a quantity of something.

How to address it: Always label the answer: 8 pencils, $1 change, 24 marbles. The label is what turns a number back into an answer to the question.

MisconceptionSkip the reasonableness check, so a place-value slip goes unnoticed.

Why it happens: Once an answer is found students feel finished and do not look back.

How to address it: Make the estimate a fixed final step. Round, work out the rough answer, and compare. A quick estimate of about $10 catches an answer of $170 immediately.

Do it together

Guided practice (with answers)

1. 5 pens cost $2 each. You pay with a $20 note. What is the change?
The $20 splits into $10 spent and $10 change.
Answer: Step 1: 5 x 2 = $10. Step 2: 20 - 10 = $10 change.
2. A baker makes 4 trays of 6 buns, then sells 18. How many buns are left?
Answer: Step 1: 4 x 6 = 24. Step 2: 24 - 18 = 6 buns left.
3. Tom reads 15 pages on Monday and 3 times as many on Tuesday. How many pages altogether?
Tuesday is 3 times Monday: 3 x 15 = 45. Altogether 60.
Answer: Step 1: 3 x 15 = 45. Step 2: 15 + 45 = 60 pages.
4. 24 cards are shared equally among 4 children, then each child buys 2 more. How many cards does each child have?
Answer: Step 1: 24 / 4 = 6. Step 2: 6 + 2 = 8 cards each.
5. A shop has 30 apples. It sells 12, then a delivery of 20 arrives. How many apples now?
Answer: Step 1: 30 - 12 = 18. Step 2: 18 + 20 = 38 apples.
6. Estimate to check: 7 boxes of 9 pens, minus 50 pens given away. Is 13 left reasonable?
Answer: Yes. About 7 x 9 is roughly 63, minus 50 is about 13, and the exact answer 63 - 50 = 13 matches.

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 the two-step word-problem sets, then use the add-and-subtract problems to firm up the combining and change steps.

Grade 3 word problems worksheet (set A)two-step problems across the four operations Grade 3 word problems worksheet (set B)more two-step practice Grade 3 add and subtract word problems (set A)the combine and change steps Grade 3 estimation worksheet (set A)rounding to check an answer is reasonable All Grade 3 word problem worksheetsthe full set for extra practice

Reach every student

Differentiation

Support

Give a blank bar-model template so the student fills in the whole and parts before choosing the operation.
Break the problem into two written sub-questions the student answers one at a time, then combine.
Keep the numbers small and friendly at first so the arithmetic never hides the two-step structure.
Provide a sentence starter for the answer, such as 'The change is ___', so the label is not forgotten.

Extension

Write your own two-step shopping problem for a partner, with an answer key, and mark theirs.
Solve a problem where the two steps can be done in either order and show both routes give the same answer.
Add a hidden extra number that is not needed, and explain why it is not used.
Turn a two-step problem into an equation with a letter for the unknown, such as (4 x 3) + 5 = t.

Check it stuck

Assessment: exit ticket

A three-question exit ticket for the last five minutes. It samples a buy-then-change problem, a multiply-then-combine problem, and a reasonableness check.

1. 3 notebooks cost $4 each. You pay with a $20 note. What is the change?
Answer: Step 1: 3 x 4 = $12. Step 2: 20 - 12 = $8 change.
2. A team has 5 players. Each scores 6 points, then the team loses 4 points. What is the score?
Answer: Step 1: 5 x 6 = 30. Step 2: 30 - 4 = 26 points.
3. For 8 bags of 5 marbles, is a total of 45 reasonable? Estimate to decide.
Answer: No. About 8 x 5 is 40, and exactly 8 x 5 = 40, so 45 is not reasonable.

For the teacher

Teacher notes and timings

Rough timing across four lessons: Lesson 1 spotting and planning two-step problems (section 1), Lesson 2 part-whole bars to combine (section 2), Lesson 3 buy-then-change with subtraction (section 3), Lesson 4 comparison bars and the reasonableness check (sections 4 to 5) plus the exit ticket.
The pivotal idea is that the first answer is usually a stepping stone, not the final answer. Keep asking, does this answer the question or feed the next step? Most wrong answers here are one step short.
The bar model is the workhorse: it converts a story into a picture that chooses the operation. Insist on the bar before the arithmetic, so add versus subtract comes from the picture, not from a keyword.
Teach the reasonableness check as a non-negotiable final step. The standard names it explicitly, and a quick rounded estimate catches place-value and operation slips that a tired student would otherwise miss.
Money and sharing contexts keep the numbers small and the two steps clear. When you extend, raise the numbers before you raise the number of steps, so the structure stays visible.
US and AU alignment: the US names two-step problems with the four operations and a reasonableness check at Grade 3 (3.OA.D.8), with rounding drawn from 3.NBT.A.1. ACARA meets the same ground at Year 3 through solving multiplication and division problems (AC9M3N04) and add-and-subtract problem solving (AC9M3N03). The bar-model method here serves both.
Present mode and print both work: use the Print button for a clean handout, or project the bar models and build them live with the class straight from the diagrams.

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