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

Repeating, Growing and Shrinking Patterns: Which Are Linear?

C1.1 A1About 35 minutes

Learning goals

I can identify and describe repeating, growing and shrinking patterns, including patterns in real life.
I can describe a growing or shrinking pattern by its first term and exactly how it changes.
I can tell whether a growing pattern is linear by checking that it grows by the same amount each time.
I can explain why the graph of a linear pattern makes a straight line.

Words to know

repeating pattern
A pattern with a core that repeats over and over.

growing pattern
A pattern whose terms get bigger by following a rule.

shrinking pattern
A pattern whose terms get smaller by following a rule.

jump (constant difference)
The amount you add or subtract to get from one term to the next. If it is the same every time, it is constant.

linear growing pattern
A growing pattern that adds the same amount every time. Its points on a graph line up in a straight line.

non-linear pattern
A growing pattern whose jumps are not all the same, like 1, 2, 4, 8. Its graph points do not make a straight line.

Remember from Grade 5

In Grade 5 you sorted patterns into repeating, growing and shrinking patterns. You described them exactly (start at 40 and subtract 2 each time), and you met patterns that multiply or divide, like 3, 6, 12, 24, and patterns whose jumps change, like 1, 3, 6, 10. You also found patterns in real life.

In Grade 6 you will do all of that again, and you will learn to say which growing patterns are linear.

Let's start with a story

It is November, and Amira's family is getting ready for winter. She notices three patterns:

The woodpile. On day 1 it has 10 logs. Her dad adds 6 logs every day: 10, 16, 22, 28, ...
The carnival news. Amira tells 1 friend about the school winter carnival. Every hour, the number of people who know doubles: 1, 2, 4, 8, 16, ...
The birdseed. The bird feeder bag holds 2000 g. The birds eat about 150 g a day: 2000, 1850, 1700, 1550, ...

The woodpile and the news both grow. The birdseed shrinks. But the woodpile and the news grow in very different ways.

A neat stack of split firewood beside a wooden shed at a snowy rural house, with a bird feeder hanging from a tree. — A woodpile like Amira's grows by the same number of logs every day, and a bird feeder slowly empties. Which of these patterns grows and which one shrinks? AI-generated image

Linear growing patterns

Look at the jumps (the amount added each time).

Woodpile: 10 → 16 → 22 → 28. The jumps are +6, +6, +6. Always the same!
News: 1 → 2 → 4 → 8 → 16. The jumps are +1, +2, +4, +8. They keep getting bigger.

A growing pattern that adds the same amount every time is called linear. The woodpile pattern is linear. The news pattern is not linear.

Why linear? Because when you graph a linear pattern, the points line up in a straight line.

Two graphs. Left: the linear pattern 3, 5, 7, 9, 11. Its five points line up in a straight line. Right: the doubling pattern 1, 2, 4, 8, 16. Its points curve upward more and more steeply, so they do not make a straight line.

Linear or not? Check every jump

To decide if a growing pattern is linear, find the jump between every pair of terms.

Pattern	Jumps	Linear?
10, 16, 22, 28, ...	+6, +6, +6	Yes: the same jump every time
5, 5.5, 6, 6.5, ...	+0.5, +0.5, +0.5	Yes: a decimal jump is fine, as long as it stays the same
1, 2, 4, 8, 16, ...	+1, +2, +4, +8	No: it doubles, so the jumps keep growing
1, 4, 9, 16, ...	+3, +5, +7	No: the jumps get 2 bigger each time
2, 3, 5, 8, 12, ...	+1, +2, +3, +4	No: the jumps get 1 bigger each time

Only the first two growing patterns are linear.

What about shrinking patterns?

A shrinking pattern that subtracts the same amount each time, like the birdseed (2000, 1850, 1700, ...), also makes points in a straight line, but the line goes down. In Grade 6 the word linear is used mostly for growing patterns, so that is what you will be asked to spot.

Worked example

Example 1: A linear pattern

Describe 4, 9, 14, 19, 24, ... Is it linear?

It gets bigger, so it is growing.
Jumps: 4 → 9 is +5, 9 → 14 is +5, 14 → 19 is +5, 19 → 24 is +5.
Every jump is +5, so the jump is constant.

Growing: start at 4 and add 5 each time. It is linear. The next term is 29.

Worked example

Example 2: Growing, but not linear

Describe 3, 6, 12, 24, ... Is it linear?

It is growing.
Jumps: +3, +6, +12. The jumps are not the same.
Try multiplying: 3 × 2 = 6, 6 × 2 = 12, 12 × 2 = 24. ✔ Each term is double the one before.

Growing: start at 3 and multiply by 2 each time. It is not linear, because the jumps keep getting bigger. The next term is 48.

A square tile pattern. Figure 1 is 1 tile. Figure 2 is a 2 by 2 square of 4 tiles. Figure 3 is a 3 by 3 square of 9 tiles.

Worked example

Example 3: A picture pattern

Look at the square tile pattern above. Describe it. Is it linear?

Count the tiles: 1, 4, 9. Figure 4 would be a 4 by 4 square: 16 tiles.
Jumps: 1 → 4 is +3, 4 → 9 is +5, 9 → 16 is +7.
The jumps are different, so the pattern is not linear. (The jumps grow by 2 each time.)

Growing. Figure n is an n by n square. It is not linear. You can also see it: each new square needs a whole new row and a whole new column, and those keep getting longer.

Worked example

Example 4: A pattern in real life

A pool is being filled with a hose. After 1 minute it has 60 L, after 2 minutes 120 L, after 3 minutes 180 L, after 4 minutes 240 L. Is this a linear pattern? How much water after 6 minutes?

Jumps: +60, +60, +60. The same each minute, so it is linear.
After 5 minutes: 240 + 60 = 300 L. After 6 minutes: 300 + 60 = 360 L.

Linear: the pool gains 60 L every minute. After 6 minutes it holds 360 L (if the hose keeps running at the same speed).

Is it linear? Three checks

Is it growing? (Repeating and shrinking patterns are not called linear growing patterns.)
Find every jump, not just the first one.
If all the jumps are the same, the pattern is linear. On a graph, a ruler would touch every point.

If you get stuck, don't give up. Write the jumps in a row under the pattern, then look again. Mistakes help you learn.

Just for fun

Funny but true: If Amira's carnival news really kept doubling every hour, about 1000 people would know after 10 hours (1024, to be exact), and more than a million would know after 20 hours. Doubling patterns aren't linear. They explode!

Hands-on activity

Linear or not? Build and graph

You need: about 40 coins, buttons or dried beans, grid paper (or lined paper), a pencil and a ruler.

Build a linear pattern. Make piles of 3, 5, 7, 9 coins. Write the jumps.
Build a doubling pattern. Make piles of 1, 2, 4, 8 coins. Write the jumps.
Graph both on grid paper. Put the pile number (1, 2, 3, 4) across the bottom and the number of coins up the side. Draw dots, don't join them.
Ruler test. Lay a ruler along the dots. For which pattern does the ruler touch every dot? (The linear one: 3, 5, 7, 9.)
Real-life hunt. With your grown-up, find one real pattern that is linear (for example, saving the same allowance every week) and one that is not.

What to look for: your child explains why a pattern is linear (same jump every time), not just that it grows.

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

Your turn

Is 8, 11, 14, 17, 20, ... linear? Explain.

Is 2, 4, 8, 16, ... linear? Explain.

Describe 50, 45, 40, 35, ... Is it growing or shrinking?

Is 1, 3, 6, 10, 15, ... linear?

What kind of pattern is ❄️⛄❄️⛄❄️⛄? What is its core?

A snowmobile club trail map shows markers at 2 km, 5 km, 8 km, 11 km. Describe the pattern. Is it linear? Where is the next marker?

Which pattern's graph would be a straight line: 2, 5, 8, 11 or 2, 4, 8, 16?

More practice: warm-up

Write the jumps in 7, 10, 13, 16.

Is 6, 12, 18, 24 linear?

Repeating, growing or shrinking: 30, 27, 24, 21?

What comes next: 5, 10, 20, 40, ...?

Finish the sentence: A growing pattern is linear when ...

More practice: on your own

Describe 2.5, 4, 5.5, 7, ... Is it linear? What is the next term?

Is 3, 4, 6, 9, 13, ... linear? What is the next term?

A tile staircase has 1 tile in Figure 1, 3 tiles in Figure 2 and 6 tiles in Figure 3 (each figure adds a new, taller column). Is it linear? How many tiles in Figure 4?

Kai has $25 and saves $8 every week. Write the amounts for the first 4 weeks. Is the pattern linear?

Jen says 1, 2, 4, 7, 11 is linear because it grows every time. Is she right?

Challenge

Make a linear growing pattern whose 1st term is 7 and whose 5th term is 31.

Two patterns both start 2, 6, ... Pattern A is linear. Pattern B multiplies by the same number each time. Write the next 3 terms of each.

A graph shows the points (1, 5), (2, 9), (3, 13), (4, 17). Then someone adds the point (5, 22). Is the pattern still linear? Explain.

Check your understanding

Quiz

Which growing pattern is linear?

On a graph, the points of a linear growing pattern ...

Which pattern is not linear?

How would you describe 40, 34, 28, 22?

Which real-life pattern is a linear growing pattern?

What is the next term in the linear pattern 12, 19, 26, 33, ...?

For parents and teachers

Big idea: Grade 6 students identify and describe repeating, growing and shrinking patterns, including real-life ones, and specify which growing patterns are linear (C1.1). A linear growing pattern adds the same amount each time, so its graph is a straight line.

Common difficulties:

Thinking every growing pattern is linear. Ask: "Is the jump the same every time?"
Checking only the first jump. 2, 6, ... could be +4 (linear) or ×3 (not linear). Ask for all the jumps.
Decimal jumps. 2.5, 4, 5.5 is still linear: the jump is always 1.5.

How to help: Look for real patterns together: a weekly allowance (linear), a pile of laundry, phone or streaming plans with a monthly fee, or a doubling story. Ask "Linear or not? How do you know?" Mathematical processes and social-emotional learning (A1) are woven in: your child reasons, explains and uses graphs to show their thinking. Praise careful checking more than speed.