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

Patterns with Whole Numbers and Decimals

C1.4 A1About 40 minutes

Learning goals

I can create and extend patterns that count by tenths, hundredths and thousandths.
I can describe patterns made by multiplying and dividing by 10.
I can use a pattern to show how decimals and whole numbers are related, like 4 × 0.25 = 1.
I can find missing numbers in decimal patterns.

Words to know

decimal number
A number with a decimal point, like 2.5 or 0.125. Digits after the point show tenths, hundredths and thousandths.

tenth (0.1)
One part when a whole is split into 10 equal parts.

hundredth (0.01)
One part when a whole is split into 100 equal parts.

thousandth (0.001)
One part when a whole is split into 1000 equal parts. 1 mm is 0.001 m.

place value
The value of a digit depends on its place. Each place is 10 times the place to its right.

Remember from Grade 5

In Grade 5 you made patterns with tenths and hundredths, like counting nickels ($0.05, $0.10, $0.15, ...) and the pattern 300, 30, 3, 0.3, 0.03, where each number is a tenth of the one before.

In Grade 6 you will go one place further, to thousandths, and use patterns to show how whole numbers and decimals are connected.

Let's start at the store

Ethan is buying granola bars that cost $0.75 each. He makes a pattern of the cost:

$0.75, $1.50, $2.25, $3.00, $3.75, $4.50, $5.25, $6.00, ...

He notices something: every 4th term is a whole number of dollars ($3.00, $6.00). That's because 4 × $0.75 = $3.00. The pattern shows a relationship between the decimal 0.75 and the whole number 3.

A number line from 0 to 2 marked every 0.25: 0, 0.25, 0.5, 0.75, 1, 1.25, 1.5, 1.75, 2. Equal jumps of 0.25 connect the marks. The whole numbers 0, 1 and 2 are larger dots.

Counting by a decimal

Counting by 0.25 goes 0.25, 0.5, 0.75, 1, 1.25, 1.5, 1.75, 2, ... It takes 4 jumps to make 1, because 4 × 0.25 = 1.

Counting by 0.2 takes 5 jumps to make 1 (5 × 0.2 = 1). Counting by 0.125 takes 8 jumps (8 × 0.125 = 1).

Multiplying and dividing by 10

When you divide by 10 again and again, each digit moves one place to the right. When you multiply by 10, each digit moves one place to the left.

Number	In words	Pattern
4000	four thousand	start
400	four hundred	÷ 10
40	forty	÷ 10
4	four	÷ 10
0.4	four tenths	÷ 10
0.04	four hundredths	÷ 10
0.004	four thousandths	÷ 10

A dividing-by-10 pattern. Read it from the bottom up and it is a multiplying-by-10 pattern.

Each ÷ 10 slides the digit 4 one place to the right: from 4 thousands down to 4 thousandths. Read upward and every step is × 10.

Worked example

Example 1: Counting by tenths

Count by 0.2 from 0 to 2. Which terms are whole numbers?

0.2, 0.4, 0.6, 0.8, 1.0, 1.2, 1.4, 1.6, 1.8, 2.0.
The 5th term is 1 and the 10th term is 2.
That's because 5 × 0.2 = 1.

Every 5th term is a whole number: 1, then 2.

Worked example

Example 2: A dividing-by-10 pattern

Continue the pattern 2500, 250, 25, ... for three more terms. Describe it.

250 is 2500 ÷ 10, and 25 is 250 ÷ 10.
25 ÷ 10 = 2.5. 2.5 ÷ 10 = 0.25. 0.25 ÷ 10 = 0.025.
The digits 2 and 5 stay the same. They move one place to the right each time.

2500, 250, 25, 2.5, 0.25, 0.025. Shrinking: divide by 10 each time.

Worked example

Example 3: A multiplying-by-10 pattern

Continue 0.006, 0.06, ... until you reach a number greater than 10.

0.006 × 10 = 0.06 (6 thousandths become 6 hundredths).
0.06 × 10 = 0.6. 0.6 × 10 = 6. 6 × 10 = 60.

0.006, 0.06, 0.6, 6, 60. Growing: multiply by 10 each time.

Worked example

Example 4: Thousandths that make a whole

Count by 0.125. How many terms until you reach 1?

0.125, 0.25, 0.375, 0.5, ...
Each jump adds 125 thousandths. 0.5 is 500 thousandths, so after 4 jumps we have half.
Keep going: 0.625, 0.75, 0.875, 1.

8 terms. So 8 × 0.125 = 1, and 0.125 is one-eighth of a whole.

Worked example

Example 5: A shrinking decimal pattern

The pattern is 5, 4.75, 4.5, 4.25, ... Which term is 3?

The rule: start at 5 and subtract 0.25 each time.
From 5 down to 3 is 2. Each jump is 0.25, and 4 jumps make 1, so 8 jumps make 2.
8 jumps after the 1st term is the 9th term.

The 9th term is 3. (Check: 5, 4.75, 4.5, 4.25, 4, 3.75, 3.5, 3.25, 3.)

Worked example

Example 6: Missing numbers

Find the missing terms: 1.2, 1.5, ☐, 2.1, ☐

1.5 − 1.2 = 0.3. Check: from 1.5 to 2.1 is 0.6, which is 2 jumps of 0.3 ✔.
1.5 + 0.3 = 1.8. 2.1 + 0.3 = 2.4.

1.2, 1.5, 1.8, 2.1, 2.4. Add 0.3 each time.

Decimal pattern tips

Line up the decimal points when you add or subtract.
Say it in the smallest unit: 0.375 is 375 thousandths, 0.5 is 500 thousandths. Then it's like whole numbers.
0.5, 0.50 and 0.500 are all the same amount.
Money and metres help: $0.25 is a quarter, 0.01 m is 1 cm, and 0.001 m is 1 mm.

Just for fun

Joke: Why did the decimal point keep moving to the right?

It was tired of being divided by 10 all the time. (Shh, don't tell it: it's the digits that really move!)

Hands-on activity

Coins and a measuring tape

You need: some coins (or play money), a metric measuring tape or metre stick, paper and a pencil.

Part 1: Coins. Count by quarters ($0.25) out loud to $3.00. How many quarters did you use? (12.) Then count by dimes ($0.10) to $1.00. How many dimes? (10.) Write each pattern.

Part 2: The ÷ 10 pattern on a tape. Find 1 m (100 cm). Then find 0.1 m (10 cm), 0.01 m (1 cm) and 0.001 m (1 mm). Each one is one-tenth of the one before. Write the pattern 1, 0.1, 0.01, 0.001.

Part 3: Count by eighths of a metre. 0.125 m is 12.5 cm. Put your finger on 12.5 cm, 25 cm, 37.5 cm, ... How many jumps to reach 100 cm (1 m)? (8.)

What to look for: your child connects the decimal to the real amount (0.001 m is the tiny 1 mm mark) and explains why 8 jumps of 0.125 make 1.

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

Your turn

Write the next three terms: 0.4, 0.8, 1.2, 1.6, ... Which terms in the pattern are whole numbers?

Continue: 7000, 700, 70, 7, ☐, ☐, ☐

Continue: 0.003, 0.03, ☐, ☐

Find the missing terms: 2.5, 2.75, ☐, 3.25, ☐

If you count by 0.05, how many terms does it take to reach 1?

Describe 10, 9.5, 9, 8.5, ... and write the next two terms.

In the 0.125 pattern, what is the 4th term? What does that tell you?

More practice: warm-up

What comes next: 0.1, 0.2, 0.3, ...?

What is 0.25 + 0.25?

Continue: 5000, 500, 50, ☐

Count by 0.5 from 0 to 2.

Which is greater, 0.3 or 0.25?

More practice: on your own

Describe 1.5, 2.7, 3.9, 5.1, ... Is it linear? What is the next term?

Look at 0.9, 0.99, 0.999. How far is each number from 1? What pattern do you see?

Continue: 36, 3.6, ☐, ☐. What is the rule?

Muffins cost $1.25 each. Write the cost of 1, 2, 3 and 4 muffins. How much do 8 muffins cost?

Find the missing terms: 0.006, 0.06, ☐, 6, ☐

Challenge

Make a pattern that starts at 0 and reaches exactly 3 in 8 equal jumps.

In the pattern 0.3, 0.6, 0.9, 1.2, ..., which term is 6? Explain.

Describe 1, 0.5, 0.25, 0.125, ... Is it shrinking by the same amount each time?

Check your understanding

Quiz

What comes next: 0.25, 0.5, 0.75, ...?

What is 3.7 × 10?

Find the missing term: 0.08, 0.8, ☐, 80

How many jumps of 0.2 make 1?

What is the rule for 4.5, 4.2, 3.9, 3.6, ...?

Which is a divide-by-10 pattern?

For parents and teachers

Big idea: Grade 6 students create and describe patterns that show relationships among whole numbers and decimal numbers (C1.4), now down to thousandths. Patterns like 0.25, 0.5, 0.75, 1 show that 4 quarters make a whole, and 4000, 400, ..., 0.004 shows that each place is 10 times the place to its right.

Common difficulties:

Thinking 0.25 > 0.3 because 25 > 3. Write 0.3 as 0.30, or use money: 30 cents beats 25 cents.
Counting 0.8, 0.9, 0.10. After 0.9 comes 1.0, because 10 tenths make 1.
Moving the decimal point the wrong way when multiplying or dividing by 10. Ask: "Should the answer be bigger or smaller?"

How to help: Use money and a metric measuring tape. Ask questions like "How many dimes make a dollar?" and "How many millimetres in a metre?" A calculator can be used to check a pattern, but have your child predict first. Mathematical processes (A1): your child is making connections between decimals, money and measurement.