B. Number · Operations: Divisibility, Decimals, Fractions, Ratios and Percents

Solving Multi-Step Problems

B2.1 B2.4 B2.7 B2.9 B2.12 A1About 45 minutes

Learning goals

I can use a plan to solve problems with more than one step.
I can write one expression with brackets for a multi-step problem.
I can solve problems that mix whole numbers, decimals, fractions and percents.
I can check that my answer makes sense.

Words to know

multi-step problem
A problem that needs two or more calculations to solve.

expression
Numbers and operation signs that show a calculation, like 30 × 12.50 + 345.

reasonable
Makes sense. A reasonable answer is close to your estimate and fits the situation.

bar model
A drawing of bars that shows the parts and the whole in a problem.

Let's start with a field trip

30 students are going to the science centre. Tickets cost $12.50 each, and the bus costs $345. The cost is shared equally. How much does each student pay?

One calculation isn't enough. You need a plan.

A yellow school bus parked outside a large glass science centre on a sunny day. — A field trip has more than one cost: tickets for every student plus the bus. Finding each student's share takes several steps. AI-generated image

A four-step plan

Understand. What do you know? What are you asked to find? Draw a picture or bar model.
Plan. Which operations do you need, and in what order?
Solve. Do each step. Writing one expression with brackets can help.
Look back. Is the answer reasonable? Check with an estimate or the inverse operation. Did you answer the question that was asked?

Worked example

Example 1: The field trip

30 students. Tickets $12.50 each. Bus $345. The total is shared equally. How much does each student pay?

Understand: Find the total cost, then share it by 30.
Plan: (30 × 12.50 + 345) ÷ 30.
Solve: Tickets: 30 × 12.50 = 375. Total: 375 + 345 = 720. Each: 720 ÷ 30 = 24.
Look back: Each student pays their own ticket ($12.50) plus a share of the bus: 345 ÷ 30 = 11.50. 12.50 + 11.50 = 24. ✓

Each student pays $24.

A bar model: the whole bar is the total cost ($720). Cut it into 30 equal shares and each student pays $24.

Worked example

Example 2: A percent and a fraction

A snowboard costs $240. It goes on sale for 15% off. Kai has saved 3/4 of the sale price. How much more does he need?

Discount: 10% of 240 = 24, and 5% = 12. 15% = 36.
Sale price: 240 − 36 = 204.
Kai's savings: 3/4 of 204. 204 ÷ 4 = 51, and 51 × 3 = 153.
Still needed: 204 − 153 = 51. (That's the last 1/4 of the sale price!)
Look back: 153 + 51 = 204. ✓

Kai needs $51 more.

Take 15% ($36) off the $240 price to get $204. Cut $204 into quarters of $51: Kai has 3 of them ($153), so he needs $51 more.

Worked example

Example 3: Write one expression

You buy 3 notebooks at $2.45 and 2 pens at $1.30. You pay with a $20 bill. Write one expression for your change and find its value.

Total cost: 3 × 2.45 + 2 × 1.30. It must be subtracted from 20 as a group, so use brackets.
Expression: 20 − (3 × 2.45 + 2 × 1.30).
Brackets first: 3 × 2.45 = 7.35 and 2 × 1.30 = 2.60. 7.35 + 2.60 = 9.95.
Then 20 − 9.95 = 10.05.

Your change is $10.05.

Stuck? Take one step

When a problem feels big, don't give up. Break it into small questions. Ask: "What can I find first with the numbers I have?" Then: "What does that let me find next?" Each small answer is a step closer.

Just for fun

Multi-step problems are like making a sandwich. Do the steps in the wrong order and you end up with cheese on the outside of the bread.

Hands-on activity

Plan a Thanksgiving dinner budget

You need: a grocery flyer (paper or online), paper and a pencil. A calculator is OK for checking.

Plan a meal for your family (or 8 people). Choose at least 5 items from the flyer. Write their prices.
Some items need more than one: "2 bags of potatoes". Multiply.
Find the total. Pretend one item is 25% off. Find the new total.
Find the cost per person: divide the total by the number of people.
Write your whole plan as one expression with brackets.

Math talk: "Which step did you do first? Why?" (You can use any family meal or celebration instead of Thanksgiving.)

Your turn

A teacher buys 4 packs of 12 juice boxes for 50 students. Is that enough? Explain.

Movie tickets cost $14.50 for adults and $9.75 for children. How much for 2 adults and 3 children?

Four friends have a $40 restaurant bill. They add a 15% tip and split the total equally. How much does each pay?

A recipe uses 2 1/4 cups of flour. You make 3 batches. A bag holds 10 cups. How much flour is left?

120 − 4 × (15 + 8) = ?

Mia runs 2.4 km a day for 5 days, then 3.75 km on Saturday. How far does she run in all?

More practice: warm-up

5 × 4 + 3 = ?

You have $10. You buy 2 snacks at $3.50. How much is left?

What is half of 3/4 of 16?

Find 25% of 80, then add 5.

2 × 0.6 + 0.8 = ?

More practice: on your own

A hockey team buys 15 jerseys at $42.80 each. A sponsor pays 50%. How much does the team pay?

A garden is a rectangle 12.5 m by 8 m. Fencing costs $9 per metre. How much to fence all around it?

360 people come to a winter fair. 5/8 are adults. Adults pay $6 and children pay $3.50. How much money is collected?

A school raises $1 250. 40% goes to the food bank. The rest is split equally among 3 other charities. How much does each of those charities get?

Ella says 4 items at $19.95 cost about $8. Is that reasonable? Explain.

Challenge

A family collects 320 L of maple sap. Their ratio is 40 L of sap for 1 L of syrup. They fill 0.5 L bottles and sell them for $12.50 each. How much money do they make?

A book has 240 pages. Elena reads 3/8 of it on Monday and 25% of it on Tuesday. How many pages are left?

Write one expression for this problem and find its value: A class of 28 shares 7 pizzas. Each pizza is cut into 8 slices. How many slices does each student get?

Check your understanding

Quiz

You buy 6 tickets at $8.50 and pay $5 for parking. What is the total?

What is 50 − 3 × (4 + 6)?

35 students. 2/5 walk to school. How many do not walk?

Which is the best estimate for 4 × $24.95 + $9.90?

A $45 game is 20% off. You pay with a $50 bill. What is your change?

In the four-step plan, what does look back mean?

For parents and teachers

Big idea: B2.1 asks Grade 6 students to solve problems involving whole numbers, decimals, fractions, ratios, rates and whole-number percents, including those requiring multiple steps or multiple operations. This lesson pulls the whole unit together and practises a problem-solving plan.

Common difficulties:

Stopping after one step, or answering a middle step (the discount instead of the sale price).
Using every number in the problem without thinking about whether it is needed.
Not checking. An estimate catches many mistakes.

How to help: Real planning tasks are perfect: a grocery budget, a trip, a fundraiser. Ask "What do you need to find first?" rather than giving the next step. Praise perseverance and the habit of looking back; these are key A1 skills (problem solving, reasoning and reflecting).