C. Algebra · Patterns, Equations, Inequalities, Coding and Modelling

Patterns with Whole Numbers and Decimal Tenths

C1.4 A1About 40 minutes

Learning goals

I can count forward and backward by tenths.
I can create and describe patterns with decimal tenths.
I can use patterns to show that 10 tenths make 1 whole.
I can use whole-number facts to add tenths.

Words to know

tenth
One of 10 equal parts of a whole. One tenth is written 0.1.

decimal point
The dot that separates the whole number part from the tenths: 2.3 means 2 wholes and 3 tenths.

decimal tenths
Numbers like 0.4, 1.7 and 3.0 that show wholes and tenths.

relationship
How two sets of numbers are connected, like 1 whole = 10 tenths.

Tenths: a quick reminder

When one whole is cut into 10 equal parts, each part is one tenth. We write one tenth as 0.1.

The strip below is one whole. 3 of its 10 parts are shaded, so the shaded part is 3 tenths, or 0.3.

One whole cut into 10 equal parts. 3 parts are shaded: 0.3.

Let's start with a story

Jonah and his aunt walk the path around the park near their home. One trip around the path is 0.3 km (3 tenths of a kilometre). Jonah wants to know how far they walk after 1, 2, 3, ... trips. That's a pattern with tenths!

Each trip adds 0.3 km: 0.3, 0.6, 0.9, 1.2, 1.5, 1.8.

Count in tenths: 3 tenths, 6 tenths, 9 tenths, 12 tenths... But 10 tenths make 1 whole. So 12 tenths is 1 whole and 2 tenths: 1.2. The pattern is start at 0.3 and add 0.3 each time.

Trips (term number)	Distance in tenths	Distance in km
1	3 tenths	0.3
2	6 tenths	0.6
3	9 tenths	0.9
4	12 tenths	1.2
5	15 tenths	1.5
6	18 tenths	1.8

The tenths pattern looks just like counting by 3s: 3, 6, 9, 12, 15, 18.

Pattern 1: Counting by tenths across a whole

0.7, 0.8, 0.9, 1.0, 1.1, 1.2

After 0.9 comes 1.0, not 0.10! Ten tenths make one whole, just like ten ones make one ten.

Pattern 2: Wholes and tenths

Every whole is 10 tenths. Look at the pattern:

Wholes	Tenths
1	10
2	20
3	30
4	40
5	50

Each time we add 1 whole, we add 10 tenths. Number of tenths = number of wholes × 10.

So 2.3 is 2 wholes and 3 tenths = 20 tenths + 3 tenths = 23 tenths.

Pattern 3: Whole-number facts help with tenths

If you know 3 + 4 = 7, then you know 3 tenths + 4 tenths = 7 tenths, so 0.3 + 0.4 = 0.7.

Look at this pattern of pairs that make 10 and pairs of tenths that make 1:

Whole numbers that make 10	Tenths that make 1
1 + 9 = 10	0.1 + 0.9 = 1.0
2 + 8 = 10	0.2 + 0.8 = 1.0
3 + 7 = 10	0.3 + 0.7 = 1.0
4 + 6 = 10	0.4 + 0.6 = 1.0
5 + 5 = 10	0.5 + 0.5 = 1.0

As the first number goes up by 1 tenth, the second goes down by 1 tenth. The total stays 1.

Worked example

Example 1: Extend a tenths pattern

Extend: 0.6, 0.8, 1.0, 1.2, ?, ?

Find the jump: 0.8 − 0.6 = 0.2 (2 tenths). Each jump is +0.2.
1.2 + 0.2 = 1.4.
1.4 + 0.2 = 1.6.

1.4, 1.6. The rule is start at 0.6 and add 0.2 (2 tenths) each time.

Worked example

Example 2: A shrinking pattern

Create a pattern that starts at 2.5 and subtracts 0.5 each time, down to 0.

0.5 is 5 tenths. 2.5 is 25 tenths.
Count back by 5 tenths: 25, 20, 15, 10, 5, 0 tenths.
Write them as decimals: 2.5, 2.0, 1.5, 1.0, 0.5, 0.

2.5, 2.0, 1.5, 1.0, 0.5, 0. It is like counting back by 5s: 25, 20, 15, 10, 5, 0.

Worked example

Example 3: Predict with a tenths pattern

Jonah walks around the park path (0.3 km) 10 times over a week. How far does he walk?

Each trip is 3 tenths. 10 trips are 10 × 3 = 30 tenths.
10 tenths make 1 whole, so 30 tenths make 3 wholes.

3.0 km (3 km). Justify: the pattern goes 3, 6, 9, ... tenths, so the 10th term is 30 tenths, which is 3 wholes.

Worked example

Example 4: Describe a relationship

Describe the pattern: 4 + 5 = 9, 0.4 + 0.5 = 0.9 and 8 − 3 = 5, 0.8 − 0.3 = 0.5

In each pair, the second fact uses the same digits as the first, but in tenths.
4 tenths + 5 tenths = 9 tenths, just like 4 + 5 = 9.
8 tenths − 3 tenths = 5 tenths, just like 8 − 3 = 5.

When you add or subtract tenths, you can use the whole-number fact and then write the answer in tenths.

Say it in tenths

If a decimal pattern feels tricky, say the numbers in tenths: 1.2 is 12 tenths, 0.8 is 8 tenths. Then it is just counting with whole numbers. Turn the answer back into a decimal at the end.

Just for fun

Why was 0.9 so excited?

It was just one tenth away from becoming a whole new number!

Hands-on activity

Tenths with paper strips

You need: 3 strips of paper the same length, a ruler, a pencil and crayons.

With a grown-up, mark each strip into 10 equal parts (if your strip is 20 cm long, make a mark every 2 cm). Each part is 0.1 of the strip.
Colour parts to show this pattern: 0.2, 0.4, 0.6, 0.8, 1.0, 1.2. When you fill one strip, keep going on the next one!
Write the pattern and its rule.
Now use the strips to show the pairs that make 1: colour 0.3 on a strip. How many tenths are left? (0.7) Find all the pairs.
Make up your own tenths pattern and ask your grown-up to extend it.

What to look for: your child says 1.0 (not 0.10) after 0.8 + 0.2, and explains that 10 tenths make a whole.

Your turn

Count on by tenths: 1.6, 1.7, 1.8, ?, ?, ?

Extend: 0.4, 0.8, 1.2, 1.6, ?

How many tenths are in 4 wholes?

You know 6 + 2 = 8. What is 0.6 + 0.2?

What goes with 0.7 to make 1?

How many tenths is 3.5?

Describe the pattern 5.0, 4.5, 4.0, 3.5, 3.0.

More practice: warm-up

Count on by tenths: 0.1, 0.2, 0.3, ?, ?

What comes after 0.9 when you count by tenths?

Write 7 tenths as a decimal.

What is 0.5 + 0.5?

Count back by tenths: 2.3, 2.2, 2.1, ?

More practice: on your own

Find the missing term: 1.1, 1.4, ☐, 2.0, 2.3

Make a table: 1 whole → 10 tenths, 2 → 20, ... What is the rule? Use it for 9 wholes.

Use 9 − 4 = 5 to solve 0.9 − 0.4.

A plant grows 0.2 cm each day. It starts at 3.0 cm. How tall is it after 5 days? Show the pattern.

Ben counts by tenths: 0.7, 0.8, 0.9, 0.10, 0.11. What is his mistake?

Challenge

A pattern starts at 0.5 and adds 0.5. What is the 10th term? Justify.

Find all the pairs of decimal tenths (like 0.4 + 0.8) that make 1.2, using tenths from 0.1 to 1.1.

Two patterns: A is 0.2, 0.4, 0.6, ... B is 1.0, 1.1, 1.2, ... At which term are they equal?

Check your understanding

Quiz

What comes next: 1.7, 1.8, 1.9, ?

How many tenths are in 2 wholes?

What is the rule for 0.3, 0.6, 0.9, 1.2?

Which pair makes 1?

You know 5 + 3 = 8. What is 0.5 + 0.3?

A path is 0.4 km long. Mira walks it 5 times. How far does she walk?

For parents and teachers

Big idea: The Ontario curriculum (C1.4) asks Grade 4 students to create and describe patterns to illustrate relationships among whole numbers and decimal tenths. Patterns show children that tenths behave like whole numbers (3 tenths + 4 tenths = 7 tenths) and that 10 tenths make 1 whole.

Common difficulties:

Writing 0.10 after 0.9. This is a very common mistake. Use paper strips or a metre stick (10 cm is 0.1 m) to show that ten tenths fill a whole.
Reading 1.2 as "one point twelve" or thinking 0.12 is bigger than 0.9. Have your child say decimals in tenths: "12 tenths".
Lining up digits wrongly when adding. Say the numbers in tenths first, add, then write the decimal.

How to help: Look for tenths in real life: kilometres on road signs or a running app (2.5 km), centimetres on a ruler (0.1 cm is 1 mm), a dime as one tenth of a dollar. Count by tenths together on walks: "0.1, 0.2, 0.3 ...". Encourage your child to explain using the words tenths and wholes (Strand A: communicating).