Solving Multistep Word Problems

Planning, Equations, Remainders, and Reasonableness

In this lesson:

• Plan and solve word problems that need two or more operations
• Write equations with a letter for the unknown at each step
• Interpret remainders and check your answers with estimation

Learning Objectives for Multistep Problem Solving

By the end of this lesson, you should be able to:

1. Solve problems needing two or more operations
2. Write equations with a letter at each step
3. Interpret remainders for the context
4. Estimate before and after solving
5. Plan by working backward from the question

One Step Becomes Two: The Muffin Problem

"A bakery makes 48 muffins. They pack them into boxes of 6. How many boxes?"

48 ÷ 6 = 8 boxes

Now: "48 muffins each morning for 5 mornings, boxes of 6. How many boxes?"

You know both operations — so why can't you just start computing?

Same Bar Diagram, New Job

The bar diagram from 4.OA.A.2 showed "times as many." Today it shows a total split into equal parts.

Same tool — new job: partition a total instead of copying a reference.

Work backward: what do you need last? Trace back to what's given.

Worked Example: Muffins in Boxes

Plan: find the total muffins, then divide into boxes.

Let m = total muffins. m = 48 × 5 = 240
Let b = number of boxes. b = 240 ÷ 6 = 40

Worked Example: Apples Packed in Bags

"A farmer picks 35 apples Monday and 45 Tuesday. She packs them into bags of 8. How many full bags?"

Let a = total apples. a = 35 + 45 = 80
Let g = number of bags. g = 80 ÷ 8 = 10

Worked Example: Water Bottles in Cases

"A store has 150 bottles. They sell 30 Saturday. The rest are packed equally into 4 cases. How many bottles per case?"

Let r = bottles remaining. r = 150 - 30 = 120
Let c = bottles per case. c = 120 ÷ 4 = 30

Plan First: Name the Operation Order

"A school orders 3 cartons of 60 pencils. They give 15 pencils to each of several classrooms. How many classrooms get pencils?"

Don't solve yet — just name the two operations, in order.

Always Define Your Letter First

Every equation in this lesson starts the same way:

"Let ______ = ______."

You've already seen this three times — "let m," "let a," "let r" — now it's a rule, not just something I did.

A letter with no definition is a computation, not a representation of the problem.

Practice: Three Two-Step Word Problems

Draw the bar diagram, define your letter, write the equation, then solve:

1. 5 boxes of 12 pens, split among 4 teachers. Find each share.
2. 90 stickers, minus 18 given away, packed in 8s. Find the packs.
3. 3 shelves of 14 books, plus 20 more. Find the total.

Challenge: One Three-Step Word Problem

"A store gets 6 crates of 15 apples each. They already had 40 apples. They pack everything into bags of 5. How many bags?"

Three operations this time — plan all three before computing any of them.

One Division, Three Different Questions

Let's all compute: 30 ÷ 7 = 4 remainder 2

Keep this on the board — three completely different questions are about to ask about it.

The division never changes. The question decides which piece of the answer you need.

Problem A: Drop the Remainder

"You have 30 stickers. Each album page holds 7. How many full pages can you fill?"

Is the answer 4, 5, or 2?

Answer: 4 pages. The leftover 2 stickers don't fill a page, so they don't count.

Problem B: Do You Round Up?

"30 students travel by van. Each van holds 7 students. How many vans are needed?"

Is the answer 4, 5, or 2? Predict before reading on.

Answer: 5 vans. 4 vans hold only 28 students — 2 would be left behind.

Problem C: The Remainder Is the Answer

"You have 30 cookies and put 7 in each bag. How many cookies are left over?"

Is the answer 4, 5, or 2?

Answer: 2 cookies. The question asks specifically about the leftovers.

Your Three-Question Routine for Remainders

Whenever a remainder appears, ask:

1. Full groups only? Drop the remainder.
2. Must everyone be covered? Round up.
3. Asking about the leftovers? The remainder is the answer.

Practice: Interpret Remainders in New Contexts

Same division for all three: 50 ÷ 8 = 6 remainder 2

1. 50 cupcakes, boxes of 8 — how many full boxes?
2. 50 people, vans of 8 — how many vans needed?
3. 50 buttons, bags of 8 — how many buttons left over?

Practice: A Multistep Remainder Problem

"A clerk has 6 boxes of 8 pens. She gives 7 pens to each customer. How many full sets of customers can she serve?"

The remainder interpretation is now buried inside a two-step problem.

Is 31 Notebooks a Reasonable Answer?

"4 classes of 28 students. Each student gets 3 notebooks. How many notebooks in all?"

A classmate's answer: 31 notebooks.

Before computing exactly — does that seem right?

Estimate Before Solving, Compare After

• Before solving: round the numbers, compute a rough estimate — sets your expectation
• After solving: compare your exact answer to the estimate — reasonable, or recheck?

Worked Example: T-Shirt Sales Estimate

"A store sold 47 t-shirts Friday and 53 Saturday. Each costs $6. Total collected?"

Estimate: 50 + 50 = 100 shirts, 100 × 6 = 600 dollars

Exact: 47 + 53 = 100, 100 × 6 = 600 dollars — matches exactly.

Worked Example: Eggs Packed Into Cartons

"A farmer has 295 eggs, packed into cartons of 6. How many full cartons?"

Estimate: 300 ÷ 6 = 50 cartons

Exact: 295 ÷ 6 = 49 remainder 1 — 49 full cartons

49 is close to 50 — reasonable, even though they're not equal.

Practice: Judge a Classmate's Answer

"A theater has 9 rows of 24 seats. Cleaning costs $5 per seat. Total cost?"

A classmate's answer: $38.

Using only estimation — no exact computation — is this reasonable? Why or why not?

Your Six-Step Multistep Problem Checklist

You already know all six pieces — here they are as one routine:

Estimate → Plan → Write equations → Solve → Interpret → Check

Nothing new — just the last three chunks combined into a repeatable sequence.

Model Problem: Planning the Relay Teams

"6 classes, 23 students each. Relay teams need 4. How many full teams?"

Estimate: 6 × 25 = 150, 150 ÷ 4 ≈ 37

Let s = students, s = 6 × 23 = 138
Let t = teams, t = 138 ÷ 4 = 34 R 2

Same Numbers, a Different Question

Same problem, new question: "How many students will NOT be on a team?"

Same computation: 138 ÷ 4 = 34 remainder 2

This time, the remainder itself is the answer: 2 students.

Class Practice: One More Integrated Problem

"A library has 8 shelves of 19 books. They box them for storage, 12 books per box. How many full boxes?"

Work through all six steps together: estimate, plan, write, solve, interpret, check.

Your Turn: The Full Checklist

"A school buys 340 pencils, then 60 more. They box them 8 per box, then give 3 boxes to each class. How many classes get pencils?"

Run the full checklist on your own — no scaffolding this time.

Four Errors Worth Watching For

1. Wrong order: read the question first — plan before you compute
2. Quotient-only thinking: ask what the question is really asking about
3. Skipping estimation: a far-off answer usually means the wrong operation
4. Undefined letters: always write "Let ___ = ___" before the equation

Key Takeaways: Your Six-Step Checklist

Estimate → Plan → Write equations → Solve → Interpret → Check

✓ A multistep problem is one problem, not several — each step feeds the next

✓ The same routine works no matter which four operations appear, or in what order

What's Next: Factors and Multiples

4.OA.B.4 — Factor Pairs, Prime, and Composite Numbers

• Uses the same "does it divide evenly?" reasoning from remainder interpretation
• Find every factor pair a number has, within 1 through 100
• Decide whether a number is prime or composite