B. Number · Operations: Numbers to 10 000, Decimal Tenths, Multiplying and Dividing

Rates: Multiplicative Relationships

B2.8About 40 minutes

Learning goals

I can describe a rate using words like "per", "each" and "for every".
I can use a table or a double number line to show a rate.
I can use multiplication and division to solve rate problems.

Words to know

rate
A comparison of two different kinds of things, like 8 legs per spider or $6 per hour.

per
For each one. "5 km per hour" means 5 km in each hour.

double number line
Two number lines lined up together, showing how two amounts grow at the same time.

multiplicative relationship
A relationship where one amount is always a certain number of times the other.

Remember from Grade 3

In Grade 3 you used ratios like "2 wheels for every 1 bike" and scaled them up: 5 bikes have 10 wheels. In Grade 4 you will use rates with any whole number, and solve problems going both ways.

What is a rate?

A rate compares two different kinds of things.

A spider has 8 legs per spider.
Lucas earns $6 per hour raking leaves.
A snowplow clears 3 km of road per hour.

The word per means "for each". A rate tells you how much for 1. Then you can multiply to find how much for any number.

An orange snowplow pushes snow off a country road past snowy farm fields. — A snowplow clears road at a rate, such as 3 km per hour. Multiply the rate by the number of hours to find the total. AI-generated image

Spiders	1	2	3	4	5
Legs	8	16	24	32	40

8 legs per spider. Each time you add a spider, you add 8 legs. The number of legs is always 8 times the number of spiders.

Seeing the pattern

In the table, the number of legs is always 8 times the number of spiders. This is a multiplicative relationship:

legs = 8 × spiders

So 10 spiders have 8 × 10 = 80 legs. You don't need to fill in the whole table!

Worked example

Example 1: Use a rate to multiply

A snowplow clears 3 km of road per hour. How much road does it clear in 5 hours?

The rate is 3 km for each 1 hour.
5 hours is 5 groups of 3 km.
5 × 3 = 15.

15 km

A double number line. Hours on top, kilometres below. Each hour lines up with 3 more km: 1 hour → 3 km, 5 hours → 15 km.

Worked example

Example 2: Use a rate to divide

Lucas earns $6 per hour raking leaves. How many hours must he work to earn $42?

Each hour gives $6. How many groups of 6 make 42?
42 ÷ 6 = 7. (6 × 7 = 42.)

7 hours

Worked example

Example 3: Bigger numbers

A bus travels 60 km per hour. How far does it go in 4 hours?

4 groups of 60 km.
4 × 60 = 4 × 6 tens = 24 tens = 240.

240 km

Worked example

Example 4: Comparing rates

Ali types 30 words per minute. Bea types 120 words in 5 minutes. Who types faster?

Find Bea's rate for 1 minute: 120 ÷ 5 = 24 words per minute.
Compare: 30 is more than 24.

Ali types faster.

A Canadian rate: maple syrup

In early spring, many families and communities in Ontario collect sap from maple trees and boil it down into syrup. First Nations peoples, such as the Anishinaabe and the Haudenosaunee, have made maple sugar and syrup for a very long time, and many communities carry on this tradition today. It takes about 40 L of sap to make 1 L of maple syrup. So 3 L of syrup takes about 3 × 40 = 120 L of sap!

The rate is about 40 L of sap for every 1 L of syrup. Three bottles of syrup need three buckets of sap: 3 × 40 = 120 L.

Rate clues

When you see "per", "each" or "for every", look for a rate. To find the total for many, multiply. To find how many groups, divide. Making a table can help you see the pattern.

Just for fun

A spider has 8 legs, so it would need 4 pairs of shoes. Ten spiders would need 40 pairs. Good thing spiders go barefoot!

Hands-on activity

Clap rate

You need: a timer (a phone or a clock with a second hand), paper and a pencil.

Your grown-up times 10 seconds. Clap as many times as you can (gently!) while they count.
Write your rate: "___ claps per 10 seconds".
There are 6 groups of 10 seconds in 1 minute. Make a table: 10 seconds, 20 seconds, 30 seconds ... up to 60 seconds. Fill it in by adding your number each time.
Multiply to check: your claps per 10 seconds × 6 = claps per minute.
Try another action, like jumping jacks or blinking. Which rate is fastest?

Example: 12 claps per 10 seconds → 24, 36, 48, 60, 72 claps per minute (12 × 6 = 72).

Your turn

A tricycle has 3 wheels. How many wheels do 7 tricycles have?

Pencils cost $2 each. How much for 9 pencils?

Juice boxes come 6 per pack. Complete the table for 1 to 5 packs.

A tap drips 5 mL per minute. How much water drips in 8 minutes?

Lucas earns $7 per hour. How many hours does he need to work to earn $35?

There are 7 days per week. How many days are in 6 weeks?

Mia bikes 9 km per hour. How far does she bike in 3 hours?

More practice: warm-up

Each person has 2 eyes. How many eyes do 5 people have?

10 dimes make $1. How many dimes make $4?

Each chair has 4 legs. How many legs on 3 chairs?

Apples come 5 per bag. How many apples in 2 bags?

There are 60 minutes per hour. How many minutes in 2 hours?

More practice: on your own

An octopus has 8 arms. How many arms do 6 octopuses have?

A train travels 70 km per hour. How far does it go in 3 hours?

One batch of cookies uses 4 eggs. How many batches can you make with 24 eggs?

Movie tickets cost $9 each. How many tickets can you buy with $63?

Fill in the double number line: 1 hour → 5 km. What goes with 2, 3 and 4 hours?

Challenge

It takes about 40 L of maple sap to make 1 L of syrup. About how much sap is needed for 6 L of syrup?

A plant is 5 cm tall. It grows 3 cm per week. How tall will it be after 6 weeks?

Store A sells 4 granola bars for $8. Store B sells 1 granola bar for $3. Which store has the better deal per bar?

Check your understanding

Quiz

An insect has 6 legs. How many legs do 5 insects have?

Ava earns $4 per hour. How much does she earn in 8 hours?

A family walks 3 km per hour. How long does it take to walk 15 km?

A table shows 1 → 7, 2 → 14, 3 → 21. What goes with 4?

Which one is a rate?

A box holds 9 muffins. How many boxes do you need for 54 muffins?

For parents and teachers

Big idea: The curriculum (B2.8) asks Grade 4 students to show simple multiplicative relationships involving whole-number rates, using tools and drawings. A rate compares two different quantities (km per hour, dollars per item). Tables and double number lines make the "times as many" pattern visible.

Common difficulties:

Adding instead of multiplying (5 insects have 6 + 5 = 11 legs). Build the table one row at a time so your child sees the jumps of 6.
Not knowing when to divide. Ask "Do we know the number of groups, or are we finding it?"
Mixing up the units in a double number line. Label both lines.

How to help: Point out rates around you: price per kilogram at the grocery store, km per hour on road signs, pages per day when reading. The maple syrup rate is an approximation; the amount of sap needed varies with the trees and the season.