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

Repeating, Growing and Shrinking Patterns in Real Life

C1.1 A1About 45 minutes

Learning goals

I can identify repeating, growing and shrinking patterns, including patterns in real life.
I can describe exactly how a pattern changes from one term to the next.
I can tell whether a growing or shrinking pattern is linear.
I can compare two patterns and explain which one changes faster.

Words to know

pattern
Things (numbers, shapes, sounds, actions) that follow a rule, so you can predict what comes next.

repeating pattern
A pattern with a core that repeats over and over, like spring, summer, fall, winter, spring, ...

growing pattern
A pattern whose terms get bigger, like 4, 7, 10, 13, ...

shrinking pattern
A pattern whose terms get smaller, like 80, 40, 20, 10, ...

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

linear pattern
A growing or shrinking pattern that changes by the same amount every time. Its graph is a straight line.

non-linear pattern
A growing or shrinking pattern whose change is not the same every time, like 2, 6, 18, 54 (×3 each time).

Remember from Grade 6

In Grade 6 you sorted patterns into repeating, growing and shrinking, and you learned that a growing pattern is linear when it adds the same amount every time. You wrote equations like t = 3n + 1 for linear patterns and used tables and graphs.

In Grade 7 you will look at an even wider variety of patterns, many from real life, and you will compare patterns: which one grows faster, which one starts higher, and when one catches up to another.

Let's start with a question

Amir and Lily each start a savings jar in January. The numbers show how much is in each jar at the end of each week.

Amir: $5, $10, $15, $20, ... (he adds $5 every week)
Lily: $0.01, $0.02, $0.04, $0.08, ... (the amount in her jar doubles every week)

After 4 weeks Amir is way ahead. Who do you think will be ahead after 15 weeks? Keep your guess in mind. We'll check it in this lesson.

Three kinds of patterns

A repeating pattern has a core that repeats. The days of the week, the seasons and the rhythm clap, snap, snap are repeating patterns.
A growing pattern gets bigger. A shrinking pattern gets smaller.

To describe a growing or shrinking pattern, find how you get from one term to the next. Write the change between each pair of terms.

If the change is the same every time (add 3, add 3, add 3...), the pattern is linear.
If the change is not the same (it keeps getting bigger or smaller, or you multiply or divide), the pattern is non-linear.

A shrinking pattern can be linear too: 50, 45, 40, 35 subtracts 5 every time.

Pattern A: an L shape of 3, 5 and 7 tiles. Pattern B: a staircase of 1, 3 and 6 tiles.

Look at the picture. Pattern A goes 3, 5, 7. Each figure adds 2 tiles: one on top and one on the right. The change is always 2, so Pattern A is linear.

Pattern B goes 1, 3, 6. The changes are +2, then +3. The next figure adds a column of 4, so the next term is 10. The change keeps growing, so Pattern B is non-linear.

Pattern	Kind	How it changes	Linear?
Monday, Tuesday, ..., Sunday, Monday, ...	repeating	core of 7 days	does not apply
4, 7, 10, 13, ...	growing	add 3 each time	yes
2, 6, 18, 54, ...	growing	multiply by 3 each time (+4, +12, +36)	no
24, 22.5, 21, 19.5, ...	shrinking	subtract 1.5 each time	yes
160, 80, 40, 20, ...	shrinking	divide by 2 each time (−80, −40, −20)	no
1, 3, 6, 10, ...	growing	add 2, then 3, then 4, ...	no

Describe each pattern by the change from term to term.

Write the jumps

Under a number pattern, draw little arcs between the terms and write each change on them. If every arc shows the same number, the pattern is linear. Always check all the jumps, not just the first one: 2, 6, ... could be +4, +4 (linear) or ×3 (non-linear).

Worked example

Example 1: Name and describe the pattern

Describe each pattern. Is it repeating, growing or shrinking? If it grows or shrinks, is it linear?

a) 1.2, 1.5, 1.8, 2.1, ...

b) 3, 6, 12, 24, ...

c) 🍁 ❄️ ❄️ 🍁 ❄️ ❄️ 🍁 ...

a) The jumps are +0.3, +0.3, +0.3. The pattern grows by the same amount each time, so it is linear. Rule: start at 1.2 and add 0.3 each time.
b) The jumps are +3, +6, +12. They are not the same. Each term is double the one before. It is a non-linear growing pattern. Rule: start at 3 and multiply by 2 each time.
c) The core is 🍁 ❄️ ❄️ (three items). It is a repeating pattern.

a) linear growing, +0.3 each time. b) non-linear growing, ×2 each time. c) repeating, core of 3.

Worked example

Example 2: Patterns in real life

a) A candle is 24 cm tall. It burns down 1.5 cm every hour. b) A ball is dropped from 160 cm. Each bounce reaches half the height of the bounce before. Describe each pattern for the first four values after the start.

a) Subtract 1.5 cm each hour: 24 → 22.5, 21, 19.5, 18. Same change every hour: a linear shrinking pattern.
b) Divide by 2 each bounce: 160 → 80, 40, 20, 10. The changes are −80, −40, −20, −10. They are not the same: a non-linear shrinking pattern.
Notice: the ball's heights shrink fast at first, then slowly. The candle shrinks at a steady rate.

The candle makes a linear shrinking pattern. The ball makes a non-linear shrinking pattern.

Worked example

Example 3: Compare two growing patterns

Pattern P: 5, 9, 13, 17, ... Pattern Q: 2, 4, 8, 16, ... Which term is bigger at term 4? At term 6?

P adds 4 each time (linear). Q doubles each time (non-linear).
Make a table. Term 4: P = 17, Q = 16. P is still ahead.
Term 5: P = 21, Q = 32. Term 6: P = 25, Q = 64.
Q started smaller, but doubling makes jumps that keep getting bigger. From term 5 on, Q is ahead and the gap keeps growing.

Term 4: P is bigger (17 vs 16). Term 6: Q is bigger (64 vs 25).

Worked example

Example 4: Back to Amir and Lily

Amir: $5 more every week. Lily: starts at $0.01 and the amount in her jar doubles every week. Who is ahead after 15 weeks?

Amir's pattern is linear: after week 15 he has 15 × $5 = $75.
Lily's pattern doubles: 0.01, 0.02, 0.04, 0.08, 0.16, 0.32, 0.64, 1.28, 2.56, 5.12, 10.24, 20.48, 40.96, 81.92, 163.84. That is week 15.
When does Lily pass Amir? Week 13: Lily $40.96, Amir $65 (not yet). Week 14: Lily $81.92, Amir $70. Lily passes Amir in week 14.

Lily is ahead after 15 weeks: $163.84 compared with $75. Doubling starts slowly but grows very fast. (In real life, nobody could keep doubling for long!)

Amir's savings climb in a straight line. Lily's barely move for 10 weeks, then shoot up: she passes Amir in week 14 and has $163.84 by week 15.

Compare like a mathematician

When you compare two patterns, ask:

Which one starts higher?
Which one changes faster?
Does the change stay the same (linear) or grow (non-linear)?
Will one catch up to the other? When?

Just for fun

Fold a sheet of paper in half and its thickness doubles. Fold again: doubled again! Most people can't fold an ordinary sheet more than about 7 times, because doubling grows so fast.

Hands-on activity

Pattern hunt at home

You need: paper, a pencil and about 30 small objects that are all the same (coins, dry pasta or buttons). Keep small objects away from little brothers and sisters.

Walk around your home with a grown-up. Find three real-life patterns: a repeating one (a tile floor, a calendar, a knitted scarf), a growing one (stair heights, page numbers in a book, ages on birthdays) and a shrinking one (a roll of paper towels getting thinner, a phone battery, days left until a holiday).
Write each one and say how it changes.
With your objects, build Pattern A from the lesson (3, 5, 7 objects in L shapes). Then build figure 4. How many objects? (9)
Build the staircase (1, 3, 6). How many objects in figure 4? (10)
Math talk: "Which pattern will need more objects by figure 6? How do you know?" (Staircase figure 6 needs 21, the L shape needs 13.)

Your turn

Describe the pattern and say whether it is linear: 7, 12, 17, 22, ...

Describe the pattern and say whether it is linear: 100, 90, 81, 73, ...

Describe the pattern: 2.4, 2.1, 1.8, 1.5, ... Give the next two terms.

What is the core of this pattern? winter, spring, summer, fall, winter, spring, ...

Describe: 1, 4, 16, 64, ... Is it linear?

A pool is being filled. It has 400 L, then 650 L, then 900 L, then 1150 L. Linear or non-linear?

Pattern X: 10, 13, 16, 19. Pattern Y: 1, 5, 9, 13. Which pattern changes faster?

More practice: warm-up

Is 6, 11, 16, 21 growing, shrinking or repeating?

What is the next term? 45, 40, 35, 30, ...

What is the core? A B B C A B B C A ...

Is 5, 10, 20, 40 linear?

Next term: 0.5, 1, 1.5, 2, ...

More practice: on your own

A plant is 3.5 cm tall and grows 0.8 cm each week. Write its height for the next 4 weeks. Linear or not?

Describe the pattern 729, 243, 81, 27, ... and give the next two terms.

Describe 2, 5, 10, 17, 26, ... and give the next term.

Pattern M: 40, 37, 34, 31. Pattern N: 40, 36, 32, 28. Which pattern shrinks faster? Which one reaches 20 first?

What is the 20th item in 🍁 ❄️ ❄️ 🍁 ❄️ ❄️ ...?

Challenge

Pattern P: 20, 30, 40, 50, ... (add 10). Pattern Q: 1, 2, 4, 8, ... (double). At which term does Q first become bigger than P? Show your table.

A pattern starts at 3. Make up a rule so the pattern is non-linear and still has 3, then 6 as its first two terms. Make up a different rule so it is linear.

Jaxon says, "Every shrinking pattern eventually reaches 0." Give one example that agrees and one that does not.

Check your understanding

Quiz

Which pattern is linear?

What is the next term? 64, 32, 16, 8, ...

A candle is 30 cm tall and burns 2 cm each hour. What kind of pattern do its heights make?

What is the core of 1, 2, 2, 3, 1, 2, 2, 3, 1, ...?

Pattern A: 2, 4, 6, 8, ... Pattern B: 2, 4, 8, 16, ... What is true?

Which real-life pattern is non-linear?

For parents and teachers

Big idea: Grade 7 students identify and compare a variety of repeating, growing and shrinking patterns, including patterns in real-life contexts (C1.1). The key skill is to describe exactly how a pattern changes and to decide whether the change is constant (linear) or not (non-linear). The next lesson compares linear patterns by their rate and starting value.

Common difficulties:

Checking only the first jump. 2, 6, ... could be +4 or ×3. Ask for at least three jumps.
Thinking only growing patterns can be linear. A steady shrinking pattern (30, 28, 26) is linear too.
Assuming the pattern that starts ahead stays ahead. The savings jar and Challenge 1 show how doubling overtakes adding.

How to help: Go on the pattern hunt together and talk about the changes out loud. Ask "How do you know?" and "Will it ever catch up?". Praise careful checking. This lesson builds A1 skills: being curious, persevering with a long table, and explaining reasoning clearly.