B. Number · Operations: Adding and Subtracting to 1000, Multiplying and Dividing

Ratios: Scaling Up by 2, 5 and 10

B2.9 B2.2About 35 minutes

Learning goals

I can describe a ratio using "for every".
I can use the ratios 1 to 2, 1 to 5 and 1 to 10 to scale up numbers.
I can make a ratio table.
I can solve problems with ratios using my 2, 5 and 10 facts.

Words to know

ratio
A way to compare two amounts. "For every 1 bike there are 2 wheels" is a ratio of 1 to 2.

for every
Words we use to say a ratio: for every 1 of this, there are 2 of that.

scale up
Make both amounts in a ratio bigger by multiplying them by the same number.

ratio table
A table that shows how two amounts grow together.

pattern
Something that repeats or grows in a regular way.

Let's start with a question

Every bike has 2 wheels. How many wheels are on 5 bikes?

For every 1 bike, there are 2 wheels. That is a ratio of 1 to 2. To find the wheels on 5 bikes, we scale up: 5 bikes, 5 × 2 = 10 wheels.

Six children's bicycles in a metal bike rack outside a brick school in spring. — For every 1 bike there are 2 wheels. Count the bikes, then scale up by 2 to find how many wheels are in the rack. AI-generated image

For every 1 square, there are 2 circles. 4 squares go with 8 circles. The ratio is 1 to 2.

What is a ratio?

A ratio compares two amounts. We say it with the words "for every":

1 to 2: For every 1 bike, there are 2 wheels. For every 1 pair of mittens, there are 2 mittens.
1 to 5: For every 1 hand, there are 5 fingers. For every 1 nickel, there are 5 cents.
1 to 10: For every 1 dime, there are 10 cents. For every 1 ten rod, there are 10 ones cubes.

The order matters. 1 to 2 (bikes to wheels) is not the same as 2 to 1.

Ratio tables

A ratio table shows how two amounts grow together. Each time you add 1 more bike, you add 2 more wheels.

Bikes	1	2	3	4	5	6
Wheels	2	4	6	8	10	12

A ratio table for 1 to 2. The wheels are the 2 facts!

Scaling up

To scale up a ratio, multiply both amounts by the same number.

1 bike to 2 wheels. Times 6: 6 bikes to 12 wheels.
1 hand to 5 fingers. Times 4: 4 hands to 20 fingers.
1 dime to 10 cents. Times 7: 7 dimes to 70 cents.

Your facts of 2, 5 and 10 make this quick!

Worked example

Example 1: Ratio of 1 to 5

One nickel is worth 5 cents. How many cents are 8 nickels worth?

The ratio is 1 nickel to 5 cents (1 to 5).
Scale up by 8: 8 nickels to 8 × 5 cents.
8 × 5 = 40.

8 nickels are worth 40¢.

Nickels	1	2	3	4	5	6	7	8
Cents	5	10	15	20	25	30	35	40

A ratio table for 1 to 5.

Worked example

Example 2: A recipe

To cook rice, use 2 cups of water for every 1 cup of rice. How much water do you need for 4 cups of rice?

The ratio of rice to water is 1 to 2.
Scale up by 4: 4 cups of rice to 4 × 2 cups of water.
4 × 2 = 8.

8 cups of water.

For every 1 cup of rice, add 2 cups of water. Each new row adds 1 cup of rice and 2 cups of water, so 4 cups of rice need 8 cups of water.

Worked example

Example 3: Going backwards

Mira has 70 cents in dimes. How many dimes does she have?

The ratio is 1 dime to 10 cents (1 to 10).
Ask: 10 times what makes 70? That is 70 ÷ 10.
10 × 7 = 70, so there are 7 dimes.

7 dimes

Dimes	1	2	3	4	5	6	7
Cents	10	20	30	40	50	60	70

A ratio table for 1 to 10.

Worked example

Example 4: Find the missing numbers

Fill in the ratio table for 1 to 5: 1 → 5, 3 → □, □ → 30.

3 groups of 5: 3 × 5 = 15.
What times 5 makes 30? 6 × 5 = 30, so the missing number is 6.

3 → 15 and 6 → 30.

Look for patterns

In a 1 to 2 table, the second row goes up by 2 each time. In a 1 to 5 table, it goes up by 5. In a 1 to 10 table, it goes up by 10. You can skip count to fill in a ratio table, or use your multiplication facts to jump straight to the answer.

Just for fun

A spider has 8 legs. For every 1 spider, there are 8 legs. So 10 spiders have 80 legs. Imagine buying shoes for all of them!

Hands-on activity

Ratio hunt and table

You need: paper, a pencil, and a ruler to draw a table.

Hunt around your home for things that come in a 1 to 2 ratio (a pair of socks, a pair of shoes), a 1 to 5 ratio (a hand and its fingers, a 5-pack of juice boxes), and a 1 to 10 ratio (a 10-pack of pencils, toes on a person).
Pick one and make a ratio table with 6 columns. For example: People: 1, 2, 3, 4, 5, 6 and Toes: 10, 20, 30, 40, 50, 60.
Ask a question you can answer with your table, like "How many toes are in our family?"
Make one more table with a different ratio.

Math talk: "What pattern do you see in the second row? Which multiplication facts are hiding in your table?"

Your turn

For every 1 bike there are 2 wheels. How many wheels on 7 bikes?

For every 1 hand there are 5 fingers. How many fingers on 6 hands?

For every 1 dime there are 10 cents. How many cents in 9 dimes?

Fill in the table for 1 to 2: 1 → 2, 2 → 4, 3 → □, 4 → □.

Say this ratio using "for every": 1 to 10, packs to pencils.

35 cents in nickels is how many nickels?

A recipe uses 2 eggs for every 1 cake. How many eggs for 5 cakes?

More practice: warm-up

For every 1 pair of socks there are 2 socks. How many socks in 3 pairs?

How many cents are 4 dimes?

How many fingers on 2 hands?

Is the ratio of hands to fingers 1 to 5 or 1 to 2?

Count by 10s to fill in: 1 → 10, 2 → 20, 3 → □

More practice: on your own

Fill in the ratio table for 1 to 5: 2 → □, 4 → □, □ → 45.

Juice boxes come in packs of 5. How many juice boxes are in 8 packs?

For every 1 canoe there are 2 paddles. How many paddles for 9 canoes?

Pencils come in boxes of 10. The class has 60 pencils, all in full boxes. How many boxes?

For every 1 cup of rice you use 2 cups of water. You used 12 cups of water. How much rice did you cook?

Challenge

1 dime is worth the same as 2 nickels. How many nickels are worth the same as 8 dimes?

Make a ratio table for 1 to 5 that ends at 50. How many columns does it have?

Ava has 3 dimes. Noah has 3 nickels. Who has more money? How much more?

Check your understanding

Quiz

For every 1 bike there are 2 wheels. How many wheels on 4 bikes?

Which describes a ratio of 1 to 5?

How many cents are 6 dimes worth?

In a 1 to 5 ratio table, what goes with 7?

50 cents in nickels. How many nickels?

For every 1 cup of rice, use 2 cups of water. How much water for 3 cups of rice?

For parents and teachers

Big idea: Grade 3 students use the ratios 1 to 2, 1 to 5 and 1 to 10 to scale up numbers and solve problems (B2.9). This links directly to the multiplication facts of 2, 5 and 10 (B2.2) and to money (nickels and dimes). It is an early step toward proportional reasoning.

Common difficulties:

Adding instead of multiplying (6 bikes have 6 + 2 = 8 wheels). Draw the bikes, or build the ratio table one column at a time.
Reversing the ratio (thinking 1 bike to 2 wheels means 2 bikes and 1 wheel). Always say it with "for every".
Going backwards (70¢ in dimes). Ask: "10 times what makes 70?"

How to help: Ratios are everywhere: pairs of shoes, packs of juice boxes, coins, recipes. Ask questions like "For every person we need 2 slices. How many slices for our family?"