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

Repeating, Growing and Shrinking Patterns

C1.1 A1About 35 minutes

Learning goals

I can tell if a pattern is repeating, growing or shrinking.
I can describe a pattern by telling where it starts and exactly how it changes.
I can spot patterns that add, subtract, multiply or divide, and patterns whose jumps change.
I can find and describe patterns in real life.

Words to know

pattern
Something that repeats or changes by following a rule.

repeating pattern
A pattern with a core that repeats over and over. It never gets bigger or smaller.

core
The part of a repeating pattern that repeats.

growing pattern
A pattern where the terms get bigger by following a rule.

shrinking pattern
A pattern where the terms get smaller by following a rule.

term
One number or picture in a pattern. The term number tells its position: 1st, 2nd, 3rd, ...

element
One item in a pattern, like one shape or one picture.

Remember from Grade 4

In Grade 4 you sorted patterns into repeating patterns (a core repeats) and growing patterns (each term follows a rule, like start at 4 and add 3 each time). You also met shrinking patterns that subtract, and patterns where the jump changes, like 1, 3, 6, 10.

In Grade 5 you will describe patterns even more exactly. You will also meet patterns that multiply or divide, and you will look for patterns in the world around you.

Let's start with a story

It is a snowy Saturday in January. Noah notices patterns all day long:

His calendar: Monday, Tuesday, Wednesday, Thursday, Friday, Saturday, Sunday, then Monday again.
His piggy bank: he puts in $5 every week. It holds $5, $10, $15, $20, ...
His box of 40 granola bars: he packs 2 a day for school. 40, 38, 36, 34, ...

The calendar repeats. The piggy bank grows. The granola bars shrink. Every one of them follows a rule.

Repeating patterns

A repeating pattern has a core that repeats. To describe it, say what is in the core and how many elements it has.

🍁🍁❄️🍁🍁❄️🍁🍁❄️

The core is 🍁🍁❄️. It has 3 elements. The pattern repeats this core 3 times.

Growing and shrinking patterns

A growing pattern gets bigger. A shrinking pattern gets smaller. To describe one, tell two things:

Where it starts (the first term).
How it changes each time.

Here is a growing pattern made of square tiles.

An L-shaped tile pattern. Figure 1 has 3 tiles, Figure 2 has 5 tiles and Figure 3 has 7 tiles. Each new figure adds one tile to the top of the tall part and one tile to the end of the bottom row.

The pattern is 3, 5, 7, ... We say: start at 3 and add 2 each time. Figure 4 will have 9 tiles.

Four ways a pattern can change

Check the change from every term to the next. Patterns can change in different ways:

Pattern	How it changes	Kind
5, 10, 15, 20, ...	add 5 each time	growing
40, 38, 36, 34, ...	subtract 2 each time	shrinking
3, 6, 12, 24, ...	multiply by 2 (double) each time	growing
80, 40, 20, 10, ...	divide by 2 (halve) each time	shrinking
1, 3, 6, 10, 15, ...	add 2, add 3, add 4, add 5: the jump gets 1 bigger each time	growing

Growing and shrinking patterns can add, subtract, multiply or divide. Some have jumps that change.

Worked example

Example 1: A shrinking pattern

Describe the pattern 64, 32, 16, 8, ... and find the next three terms.

Is it getting bigger or smaller? Smaller, so it is shrinking.
Find the change. 64 − 32 = 32, but 32 − 16 = 16. The jumps are not the same, so it is not subtract the same number.
Try dividing: 64 ÷ 2 = 32, 32 ÷ 2 = 16, 16 ÷ 2 = 8. ✔ Every term is half the one before.
Next terms: 8 ÷ 2 = 4, 4 ÷ 2 = 2, 2 ÷ 2 = 1.

Shrinking pattern: start at 64 and divide by 2 each time. Next: 4, 2, 1.

Worked example

Example 2: The jumps change

Describe 2, 5, 9, 14, 20, ... and find the next term.

Find each jump: 2 → 5 is +3, 5 → 9 is +4, 9 → 14 is +5, 14 → 20 is +6.
The jump gets 1 bigger each time.
The next jump is +7: 20 + 7 = 27.

Growing pattern: start at 2, add 3, then add 1 more each time (+3, +4, +5, ...). Next term: 27.

The jumps between the terms are +3, +4, +5, +6. Each jump is 1 bigger than the one before, so the next jump is +7 and the next term is 27.

Worked example

Example 3: A pattern in real life

The city bus stops near the library at 7:15, 7:35, 7:55, 8:15, ... Describe the pattern. When is the next bus?

Find the change: 7:15 to 7:35 is 20 minutes. 7:35 to 7:55 is 20 minutes. 7:55 to 8:15 is 20 minutes.
The times get later by the same amount, so it is a growing pattern of times.
Next: 8:15 + 20 minutes = 8:35.

Start at 7:15 and add 20 minutes each time. The next bus is at 8:35.

Worked example

Example 4: A repeating pattern with more than one change

Find the core: 🔺🔵🔵🟦🔺🔵🔵🟦🔺🔵🔵🟦

Look for where the pattern starts again. After 🔺🔵🔵🟦 comes 🔺 again.
Check that the next part matches: 🔺🔵🔵🟦 ✔ and again 🔺🔵🔵🟦 ✔

The core is 🔺🔵🔵🟦. It has 4 elements: a triangle, two circles and a square. The shape changes, and the colour changes too: the triangle is red, and the circles and the square are blue.

Be a pattern detective

Check the change between every pair of terms, not just the first two.
If adding or subtracting doesn't fit, try multiplying or dividing.
If the jumps change, look for a pattern in the jumps.
Feeling stuck is normal. Take a breath, then try a different idea. Explaining your thinking to your grown-up helps you notice things.

Pattern riddle

What comes next?

J, F, M, A, M, J, J, ...

A! They are the first letters of the months: January, February, March, April, May, June, July, August. A repeating pattern you live inside every year!

Hands-on activity

Pattern hunt and coin patterns

You need: about 30 coins, buttons or dried beans, paper and a pencil.

Part 1: Pattern hunt (10 minutes). Walk around your home with a grown-up. Find 2 repeating patterns (tiles, a striped towel, a fence, a calendar) and 1 growing or shrinking pattern (stairs, stacked cups, the pages left in a book you are reading, money saved each week). Write each one down. Describe it: the core, or the start and the change.

Part 2: Build a shrinking pattern. Start with 25 coins in a pile. Take away 4 each time and write down how many are left: 25, 21, 17, ... Keep going until you can't take away 4 any more. How many are left at the end? (Answer: 1.)

Part 3: Build a doubling pattern. Make piles of 1, 2, 4, 8 coins. How many coins would the next pile need? (16.) Talk about how a doubling pattern grows faster and faster.

Grown-up note: keep small items like coins and beans away from babies and toddlers.

Your turn

What is the core of 🍁❄️❄️🍁❄️❄️🍁❄️❄️? How many elements does it have?

Describe 7, 12, 17, 22, ... and write the next two terms.

Describe 90, 81, 72, 63, ... and write the next two terms.

Describe 3, 6, 12, 24, ... and write the next two terms.

Describe 1, 2, 4, 7, 11, ... and write the next two terms.

Noah has a box of 40 granola bars. He uses 4 each day. Describe the pattern of bars left after each day. How many are left after day 5?

Is each pattern repeating, growing or shrinking?

a) the days of the week

b) your age on each birthday

c) the cookies left in a jar when you eat 2 a day

More practice: warm-up

Describe 5, 10, 15, 20, ...

What is the core of ☀️🌧️☀️🌧️☀️🌧️?

Write the next term: 50, 45, 40, 35, ...

Write the next two terms: 100, 200, 300, ...

Is 9, 7, 5, 3 growing or shrinking?

More practice: on your own

Describe 2, 6, 18, 54, ... and find the next term.

Describe 100, 95, 85, 70, 50, ... and find the next term.

Write elements 9 to 12 of 🔺🔺⬛🔵🔺🔺⬛🔵...

Maya says 4, 8, 12, 16 and 4, 8, 16, 32 are the same pattern, because both start 4, 8. Is she right?

During a snowstorm, the snow on the deck was 10 cm, 14 cm, 18 cm, 22 cm deep at 6, 7, 8 and 9 o'clock. Describe the pattern. If it keeps going, how deep at 10 o'clock?

Challenge

Make a shrinking pattern that starts at 81 and ends at 1, where every term is divided by the same number.

Describe 1, 4, 9, 16, 25, ... in two different ways, and find the next term.

A repeating pattern of beads has a core of 5 beads. A necklace has 32 beads. How many full cores are there, and how many beads are left over? Which beads are they?

Check your understanding

Quiz

Which pattern is shrinking?

What is the rule for 6, 11, 16, 21, ...?

What comes next: 2, 4, 8, 16, ...?

What comes next: 1, 3, 6, 10, 15, ...?

What is the core of 🍎🍎🍐🍎🍎🍐🍎🍎🍐?

Which real-life pattern is repeating?

For parents and teachers

Big idea: Grade 5 students identify and describe repeating, growing and shrinking patterns, including real-life ones (C1.1). The new step this year is describing the change precisely, including patterns that multiply or divide (like doubling) and patterns whose jumps change.

Common difficulties:

Checking only the first jump. For 4, 8, 16, 32 a child may say "add 4". Ask them to check every jump.
Saying "it goes up" instead of a rule. Push for a full description: "Start at ___ and ___ each time."
Mixing up the core with the whole pattern. Have your child put a finger where the pattern starts again.

How to help: Point out patterns in daily life: bus times, a weekly allowance, the pages left in a book, a striped scarf. Ask "Is it repeating, growing or shrinking? How do you know?" Social-emotional learning (A1) is woven in: when your child is stuck, encourage them to try another idea (dividing instead of subtracting) rather than giving up.