Grade by Grade
B. Number · Operations: Numbers to 100 000, Decimals, Fractions, Ratios and Rates

Solving Multi-Step Problems and Checking Answers

B2.1 B2.4 B2.6 B2.7 A1About 45 minutes
Learning goals
  • I can solve problems that need two or more operations, with whole numbers and decimals.
  • I can make a plan and draw a model (like a bar model) to show a problem.
  • I can check my answer with an estimate or by working backwards.

Words to know

multi-step problem
A problem that needs more than one calculation to solve.
bar model
A drawing with rectangles (bars) that shows how the numbers in a problem are related.
work backwards
Start from the answer and undo each step to check, or to find a starting number.

Let's start with a challenge

A school food drive has a goal of 1000 cans. 18 classes each bring 45 cans. How many more cans are needed?

This takes two steps:

  1. Find how many cans were collected: 18 × 45.
  2. Subtract from the goal: 1000 − that amount.

One calculation isn't enough. That's what makes it a multi-step problem.

goal: 1000 cans100018 × 45 = 810 collected?still needed
A bar model. The top bar is the goal of 1000 cans. The bottom bar shows 810 cans collected (18 × 45) and a small part marked ? that is still needed.

A four-step plan

  1. Understand: What do I know? What am I asked to find?
  2. Plan: What steps and operations will I use? Would a drawing help?
  3. Solve: Do each step. Write what each number means.
  4. Look back: Does my answer make sense? Check with an estimate or by working backwards.
Worked example

Example 1: The food drive

Goal 1000 cans. 18 classes bring 45 cans each. How many more cans are needed?

  1. Step 1: 18 × 45. Row 1: 18 × 5 = 90. Row 2: 18 × 40 = 720. 90 + 720 = 810 cans collected.
  2. Step 2: 1000 − 810 = 190.
  3. Look back: 20 classes × 45 = 900 would leave 100 needed. We have fewer classes, so we need a bit more than 100. 190 makes sense. ✓

190 more cans

Worked example

Example 2: Money with decimals

Sam buys a sandwich for $6.75 and two drinks for $1.85 each. He pays with a $20 bill. How much change does he get?

  1. Drinks: 1.85 + 1.85 = $3.70.
  2. Total: 6.75 + 3.70 = $10.45.
  3. Change: 20.00 − 10.45 = $9.55.
  4. Work backwards to check: 9.55 + 10.45 = 20.00. ✓

$9.55 in change

Sam's lunch: one problem, three stepsStep 1: 2 drinks1.85 + 1.85 = 3.70Step 2: total6.75 + 3.70 = 10.45Step 3: change20.00 − 10.45 = 9.55look back: 9.55 + 10.45 = 20.00 ✓
Each step uses the answer from the step before. The dashed arrow shows looking back: adding the change and the total gives back the $20.
Worked example

Example 3: Divide, then multiply

A store gets 432 water bottles packed in cases of 24. Each case costs $6. How much do all the cases cost?

  1. Step 1: number of cases. 432 ÷ 24: 24 × 10 = 240, 192 left. 24 × 8 = 192. So 18 cases.
  2. Step 2: cost. 18 × 6 = $108.
  3. Look back: 432 bottles is about 18 groups of 24 (24 × 18 = 432 ✓), and 18 × $6 is about 20 × $6 = $120. $108 is close. ✓

$108

Worked example

Example 4: Work backwards to find a number

I think of a number. I multiply it by 12, then subtract 15. I get 105. What's my number?

  1. Undo the last step first: before subtracting 15, I had 105 + 15 = 120.
  2. Undo the multiplying: 120 ÷ 12 = 10.
  3. Check: 10 × 12 = 120, and 120 − 15 = 105. ✓

My number is 10.

Problem-solving tips

  • Underline the question so you know what to find.
  • Label each step: "810 = cans collected".
  • If you're stuck, draw a bar model or try a simpler number first.
  • Mistakes are part of learning. If your check doesn't match, look for which step went wrong. That's what problem solvers do!
Just for fun

Silly what if: Skipping step 1 of a multi-step problem is like trying to eat a cake before you bake it. You can't ice what isn't there yet!

Hands-on activity

Plan a class party

You need: a grocery flyer (or made-up prices), paper and a pencil.

  1. Plan snacks for 24 students. Choose a snack that comes in packs (for example, juice boxes in packs of 8 or 10).
  2. Work out how many packs you need (round up!) and how much they cost.
  3. Add a second item, like fruit or crackers. Find the total cost.
  4. Your budget is $50. Are you under or over? By how much?
  5. Check your work with an estimate. Explain your plan to your grown-up.

Math talk: "Which step did you do first, and why?"

Your turn

1

3 packs of 24 juice boxes are bought for a class party. 15 are used. How many are left?

2

A family buys 12 hockey tickets at $25 each and 3 programs at $8 each. What is the total cost?

3

245 + 378 − 199 = ?

4

A road trip is 600 km. The family drives 238.5 km on day 1 and 197.25 km on day 2. How far is left?

5

15 boxes of 32 crayons are shared equally by 24 students. How many crayons does each student get?

6

Mia has $50. She buys 2 books at $12.95 each. How much money does she have left?

7

I think of a number. I multiply it by 12, then subtract 15. I get 105. What's my number?

More practice: warm-up

1

5 × 12 + 10 = ? (Multiply first.)

2

100 − 4 × 20 = ? (Multiply first.)

3

(36 + 14) ÷ 5 = ? (Brackets first.)

4

Juice costs $1.50. What do 3 juices cost?

5

You have 2 dozen eggs and use 5. How many are left?

More practice: on your own

1

A theatre has 24 rows of 35 seats. 712 tickets are sold. How many seats are empty?

2

Ski passes cost $45 for adults and $28 for kids. A family buys 2 adult and 3 kid passes and pays with $200. What is the change?

3

756 apples are packed in bags of 18. Each bag sells for $4. How much money do all the bags bring in?

4

Rain this week: 12.6 mm, 8.75 mm and 3.4 mm. How much less than 30 mm is that?

5

Ana says 16 × 25 − 150 = 350. Check her work. Is she right?

Challenge

1

312 students and 18 teachers go on a trip. Each bus holds 48 people and costs $275. What is the total bus cost?

2

A class sells 36 boxes of cookies at $12 each and 25 calendars at $15 each. Their goal is $900. Did they reach it? By how much are they over or under?

3

When you divide a mystery number by 14, you get 23 R5. What is the number?

Check your understanding

Quiz

1.

Jo buys 4 packs of 12 markers and loses 9. How many does she have?

2.

18 classes bring 45 cans each toward a goal of 1000. What is the first step to find how many more are needed?

3.

A sandwich costs $6.75 and two drinks cost $1.85 each. What is the change from $20?

4.

432 bottles come in cases of 24. Each case costs $6. What is the total cost?

5.

I multiply a number by 12 and subtract 15. I get 105. What's my number?

6.

Which is a good way to check a multi-step answer?

For parents and teachers

Big idea: B2.1 asks Grade 5 students to solve problems with whole numbers and decimals, "including those requiring more than one operation," and to check calculations. This lesson pulls together adding, subtracting, multiplying and dividing from the whole unit.

Common difficulties:

  • Stopping after one step. Ask "Did you answer the question that was asked?"
  • Using every number in the problem without thinking. Ask "What does each number mean?"
  • Mixing up which operation to use. Drawing a bar model often makes it clear.

How to help: Real planning makes great problems: budgeting for groceries, planning a trip, sharing costs. Let your child make a plan before calculating, and praise the checking step. Perseverance and reflecting on mistakes are part of A1.