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

Patterns with Whole Numbers, Tenths and Hundredths

C1.4 A1About 40 minutes

Learning goals

I can read and write hundredths, like 0.01 and 0.35.
I can count forward and backward by tenths and hundredths, even across a whole.
I can make and describe patterns that show how whole numbers, tenths and hundredths are related.
I can use a whole-number fact to help with tenths and hundredths.

Words to know

tenth
One of 10 equal parts of a whole. Written 0.1.

hundredth
One of 100 equal parts of a whole. Written 0.01. Ten hundredths make one tenth.

decimal point
The dot that separates the whole number part from the parts of a whole: 3.25

place value
The value of a digit because of where it is: in 0.47, the 4 means 4 tenths and the 7 means 7 hundredths.

relationship
A connection between numbers, like 10 hundredths = 1 tenth.

Remember from Grade 4

In Grade 4 you made patterns with tenths, like 0.3, 0.6, 0.9, 1.2. You learned that after 0.9 comes 1.0, because ten tenths make one whole.

In Grade 5 you will go one place further, to hundredths.

Hundredths

Cut one whole into 100 equal parts. Each part is one hundredth, written 0.01.

A loonie is one dollar. One cent is one hundredth of a dollar: $0.01.
A metre stick has 100 centimetres. One centimetre is one hundredth of a metre: 0.01 m.

In the grids below, each big square is one whole.

Two hundred grids. The left one has 23 small squares shaded: 23 hundredths, or 0.23. The right one has 3 full columns shaded: 30 hundredths, which is the same as 3 tenths, 0.3.

Each column of 10 small squares is one tenth. So 10 hundredths = 1 tenth, and 100 hundredths = 1 whole.

Whole	Tenths	Hundredths
1	10 tenths	100 hundredths
0.1	1 tenth	10 hundredths
0.3	3 tenths	30 hundredths
0.35	3 tenths and 5 hundredths	35 hundredths
1.2	12 tenths	120 hundredths

The same amounts written in wholes, tenths and hundredths.

Pattern 1: Moving one place

Look at this shrinking pattern: 300, 30, 3, 0.3, 0.03

Each term is divided by 10. Every digit moves one place to the right: hundreds → tens → ones → tenths → hundredths.

Read it backward and it multiplies by 10 each time: 0.03, 0.3, 3, 30, 300.

Pattern 2: Counting across a tenth

0.07, 0.08, 0.09, 0.10, 0.11, 0.12

After 9 hundredths comes 10 hundredths, which is 1 tenth. We can write it 0.10 or 0.1. They are equal, just like $0.10 is one dime.

Pattern 3: Facts that grow and shrink together

Each time the number is divided by 10, the digit 3 moves one place to the right: hundreds, tens, ones, tenths, hundredths. The dot is the decimal point.

Whole numbers	Tenths	Hundredths
7 + 3 = 10	0.7 + 0.3 = 1	0.07 + 0.03 = 0.1
6 + 4 = 10	0.6 + 0.4 = 1	0.06 + 0.04 = 0.1
2 + 8 = 10	0.2 + 0.8 = 1	0.02 + 0.08 = 0.1

If you know 7 + 3 = 10, you also know 7 tenths + 3 tenths = 10 tenths = 1, and 7 hundredths + 3 hundredths = 10 hundredths = 0.1.

Worked example

Example 1: Counting nickels

Lily saves a nickel ($0.05) every day. Extend the pattern for 8 days: $0.05, $0.10, $0.15, ...

The rule is start at 0.05 and add 0.05 each time.
Think in hundredths: 5, 10, 15, 20, 25, 30, 35, 40 hundredths.
Write them as decimals.

$0.05, $0.10, $0.15, $0.20, $0.25, $0.30, $0.35, $0.40. After 8 days Lily has 40 cents.

Worked example

Example 2: A shrinking tenths pattern

Describe 1.0, 0.9, 0.8, 0.7, ... and find the next three terms.

Think in tenths: 10, 9, 8, 7 tenths. It subtracts 1 tenth (0.1) each time.
Next: 6, 5, 4 tenths.

Start at 1.0 and subtract 0.1 each time. Next: 0.6, 0.5, 0.4.

Worked example

Example 3: Use a pattern to find a missing number

Use the pattern in the table above to find 0.09 + ☐ = 0.1.

The whole-number fact is 9 + 1 = 10.
So 9 hundredths + 1 hundredth = 10 hundredths, and 10 hundredths is 0.1.

☐ = 0.01

Worked example

Example 4: Predict with quarters

A quarter is $0.25. Ben puts one quarter in his jar each day: $0.25, $0.50, $0.75, ... How much after 8 days?

Look for a pattern: 4 quarters make $1.00 (25 + 25 + 25 + 25 = 100 hundredths).
8 days is 2 groups of 4 quarters, so 2 × $1.00.
The pattern goes $0.25, $0.50, $0.75, $1.00, $1.25, $1.50, $1.75, $2.00.

After 8 days Ben has $2.00.

Say it in hundredths

If a decimal pattern feels tricky, say each number in hundredths: 0.35 is 35 hundredths, 0.5 is 50 hundredths, 1.2 is 120 hundredths. Then it is whole-number counting! Turn your answer back into a decimal at the end. Money helps too: 0.35 is 35 cents.

Just for fun

Why is the decimal point so good at giving directions?

Because it always knows its place!

Hands-on activity

Coin and metre-stick patterns

You need: some coins (nickels, dimes, quarters and a loonie if you have them, or paper coins), and a metre stick or tape measure.

Count by dimes: lay down dimes one at a time and say $0.10, $0.20, $0.30, ... How many dimes make $1.00? (10.)
Count by quarters to $2.00. How many quarters? (8.)
Count by nickels from $0.05 to $0.50. Write the pattern.
On the metre stick, find 0.25 m (25 cm), 0.50 m (50 cm) and 0.75 m (75 cm). Each is a quarter of a metre more. Where is 1.00 m?
Make up your own pattern with coins for your grown-up to continue. Tell them your rule.

What to look for: your child says the amounts as decimals ($0.10, not "10 dollars") and explains, for example, that 10 dimes make a dollar because 10 tenths make a whole.

Grown-up note: keep coins away from babies and toddlers.

Your turn

Continue the pattern: 0.02, 0.04, 0.06, ☐, ☐

Continue the pattern: 0.7, 0.8, 0.9, ☐, ☐

Continue the pattern: 0.95, 0.96, 0.97, ☐, ☐, ☐

Describe the pattern 5, 0.5, 0.05.

You know 4 + 6 = 10. What is 0.04 + 0.06?

Continue: $0.25, $0.50, $0.75, ☐, ☐

Continue: 3.00, 2.75, 2.50, ☐, ☐

More practice: warm-up

How many hundredths make 1 tenth?

Continue: 0.01, 0.02, 0.03, ☐

Write 47 hundredths as a decimal.

What is the rule for 0.1, 0.2, 0.3, 0.4?

0.3 + 0.7 = ?

More practice: on your own

Continue: 0.12, 0.24, 0.36, 0.48, ☐

The pattern 1.5, 1.4, 1.3, ... subtracts 0.1 each time. What is term 8?

Find the missing numbers: 0.6 + ☐ = 1 and 0.06 + ☐ = 0.1

Continue: 0.03, 0.3, 3, ☐. What is the rule?

Rope is cut into pieces 25 cm long. 25 cm is 0.25 m. How many pieces make 2 m?

Challenge

0.05, 0.15, 0.25, 0.35, ... What is term 10?

Continue the doubling pattern: 0.01, 0.02, 0.04, 0.08, ☐, ☐

Ava wrote this pattern: 0.8, 0.9, 0.10, 0.11. She says it adds 0.1 each time. Find her mistake and fix it.

Check your understanding

Quiz

What comes next: 0.97, 0.98, 0.99, ☐?

Which is equal to 0.4?

What is the rule for 200, 20, 2, 0.2?

You know 3 + 7 = 10. Which is true?

Continue: $0.25, $0.50, $0.75, ☐

What is the 6th term of 0.1, 0.2, 0.3, ...?

For parents and teachers

Big idea: Grade 5 students create and describe patterns that show the relationships among whole numbers, tenths and hundredths (C1.4). Hundredths are new this year. Patterns like 300, 30, 3, 0.3, 0.03 show that each place is 10 times the place to its right.

Common difficulties:

Thinking 0.10 comes after 0.9. Use money: 0.9 is 90 cents and 0.10 is 10 cents. 10 tenths is 1.0.
Thinking 0.25 is bigger than 0.3 because 25 > 3. Say both in hundredths: 25 hundredths and 30 hundredths.
Reading 0.05 as "five tenths." Point to the place each digit is in.

How to help: Coins and a metre stick are the best tools: cents are hundredths of a dollar, and centimetres are hundredths of a metre. Count by dimes, nickels and quarters together. Social-emotional learning (A1): decimals can feel confusing at first; remind your child that it's fine to go back to coins or grids.