C. Algebra · Algebra: Patterns, Rational Numbers, Equations, Inequalities, Coding and Modelling

Patterns in Real Life: Linear or Not?

C1.1 A1About 45 minutes

Learning goals

I can identify repeating, growing and shrinking patterns in real-life situations.
I can use the differences between terms to decide whether a pattern is linear or non-linear, even with fractions, decimals and negative numbers.
I can compare patterns using words, tables and graphs.

Words to know

repeating pattern
A pattern with a core (a group of terms) that repeats over and over, like Monday to Sunday.

growing pattern
A pattern whose terms get larger.

shrinking pattern
A pattern whose terms get smaller.

term number
The position of a term in a pattern: 1st, 2nd, 3rd and so on. We often call it n.

difference
What you add to one term to get the next term. For 5, 8, 11 the differences are +3 and +3.

linear pattern
A growing or shrinking pattern where the difference between terms is always the same. Its points on a graph lie on a straight line.

non-linear pattern
A growing or shrinking pattern where the differences change. Its points on a graph do not lie on a straight line.

Remember from Grade 7

In Grade 7 you found repeating, growing and shrinking patterns in real life. You learned that a linear pattern adds or subtracts the same amount each time, and that a pattern like 1, 2, 4, 8 (doubling) is non-linear. You also compared linear patterns using their constant rate and initial value.

In Grade 8 the patterns can include fractions, decimals and negative numbers. These are all rational numbers. You'll use the differences between terms as a quick, reliable test for whether a pattern is linear.

Let's start on a winter morning

At 6 a.m. it is −12°C. The sun comes up and the temperature rises 1.5°C every hour until noon. The temperatures are:

−12, −10.5, −9, −7.5, −6, −4.5, −3

That's a growing pattern, even though all the numbers are negative! Each one is 1.5 more than the last.

Now think about a phone tree for a snow day. The principal calls 2 families, each of those calls 2 more, and so on: 2, 4, 8, 16, 32. That is also growing, but in a very different way. Today you'll learn how to tell these two kinds of patterns apart.

Three kinds of patterns

A repeating pattern has a core that repeats: the days of the week, the order of a traffic light, or garbage and recycling weeks that take turns.
A growing pattern gets larger: −12, −10.5, −9, ...
A shrinking pattern gets smaller: the height of a candle as it burns, or the money left on a transit card.

The difference test

To check whether a growing or shrinking pattern is linear, find the difference from each term to the next (next term minus this term).

If the differences are all the same, the pattern is linear.
If the differences change, the pattern is non-linear.

Check at least three differences. Two can match by accident.

Pattern	Differences	Type
5, 8, 11, 14	+3, +3, +3	linear, growing
20, 16.5, 13, 9.5	−3.5, −3.5, −3.5	linear, shrinking
−12, −10.5, −9, −7.5	+1.5, +1.5, +1.5	linear, growing
1, 4, 9, 16	+3, +5, +7	non-linear, growing
80, 40, 20, 10	−40, −20, −10	non-linear, shrinking

Constant differences mean linear. Changing differences mean non-linear.

The difference test: 5, 8, 11, 14 adds +3 every time, so it is linear. 1, 4, 9, 16 adds +3, then +5, then +7, so it is non-linear.

The linear pattern 2, 5, 8, 11, 14 (circles) lies on a straight line. The square numbers 1, 4, 9, 16, 25 (squares, dashed) curve upward, because the differences keep getting bigger.

Watch out!

A pattern can be growing and still be non-linear. "Growing" tells you the direction. "Linear" tells you whether the change is the same every time.

Worked example

Example 1: A decimal pattern

Is the pattern 2.4, 3.1, 3.8, 4.5, 5.2 linear?

Find the differences: 3.1 − 2.4 = 0.7, 3.8 − 3.1 = 0.7, 4.5 − 3.8 = 0.7, 5.2 − 4.5 = 0.7.
All four differences are +0.7.

Yes, it is a linear growing pattern. It goes up 0.7 each time.

Worked example

Example 2: A burning candle

A candle is 24 cm tall. Each hour it is measured: 24, 22.5, 21, 19.5 (cm). Is this linear? If the pattern continues, when will the candle be used up?

Differences: −1.5, −1.5, −1.5. The pattern is linear and shrinking.
The candle loses 1.5 cm each hour. 24 ÷ 1.5 = 16.

Linear. The candle would last 16 hours. (A real candle might not burn at exactly the same speed the whole time, so this is a prediction.)

Worked example

Example 3: A bouncing ball

A ball is dropped from 200 cm. Its bounces reach 200, 100, 50, 25 (cm). Is this linear?

Differences: 100 − 200 = −100, 50 − 100 = −50, 25 − 50 = −25.
The differences change. Each height is half of the one before.

Non-linear, shrinking. It halves each time instead of subtracting the same amount.

Worked example

Example 4: A repeating pattern

A town collects garbage and recycling on alternating weeks: week 1 garbage, week 2 recycling, week 3 garbage, and so on. What is collected in week 27?

The core is (garbage, recycling). It has 2 terms.
Odd weeks are garbage and even weeks are recycling. 27 is odd.

Garbage is collected in week 27.

Hands-on activity

Cups and bounces

You need: 5 paper or plastic cups that stack, a ruler or tape measure, a small ball, a wall, a pencil and paper.

Part 1: Stacking cups. Measure the height of 1 cup. Stack a 2nd cup inside and measure again. Keep going to 5 cups. Make a table and find the differences.

Part 2: Bouncing ball. Tape a tape measure to a wall (or have a helper hold it). Drop the ball from 100 cm and watch how high it bounces. Then drop it from the bounce height you saw, and so on, for 3 or 4 drops. Record the heights and the differences.

What to look for: the cup heights grow by about the same amount each time (just the rim sticks out), so they are close to linear. The bounce heights shrink by smaller and smaller amounts, so they are non-linear. Real measurements are a little messy, so "about the same" differences count as linear.

Math talk: "Why is the first cup's height different from the amount each new cup adds?"

Just for fun: a doubling surprise

If you could fold a 0.1 mm sheet of paper in half 42 times, doubling its thickness each fold, the stack would be about 440 000 km thick: farther than the Moon! That's non-linear growth for you.

Don't try to beat the record, though: real paper gets too thick to fold after about 12 or 13 folds.

Your turn

Is 3, 7, 11, 15 linear or non-linear? Growing or shrinking?

Is 100, 91, 82, 73 linear or non-linear? Growing or shrinking?

Is 2, 6, 18, 54 linear?

Is 1.5, 2.25, 3, 3.75 linear?

Is 0, 1, 3, 6, 10 linear?

Is −6, −4, −2, 0, 2 linear? Growing or shrinking?

School runs Monday to Friday. Day 1 of a unit is a Monday. What weekday is school day 100 of the unit, if there are no holidays?

More practice: warm-up

Write the next two terms: 4, 9, 14, 19, ...

Is 50, 45, 40, 35 growing, shrinking or repeating?

Find the differences for 10, 13, 16, 19.

Is 1, 2, 4, 8 linear?

What is the core of the pattern ▲ ● ● ▲ ● ● ▲ ● ●?

More practice: on your own

After a thaw, the snow depth in a backyard is measured each day: 60, 54.5, 49, 43.5 (cm). Is this linear? If it continues, what is the next depth?

Which pattern is linear? A: 3, 6, 12, 24 B: 3, 6, 9, 12 C: 3, 4, 6, 9

The points (1, 2), (2, 5), (3, 10), (4, 17) are plotted on a graph. Will they lie on a straight line? Explain.

A traffic light is green for 40 s, then yellow for 4 s, then red for 36 s, and then repeats. What colour is it 250 s after it turns green?

Is 7, 7.25, 7.75, 8.5 linear? Explain.

Challenge

A pattern starts 2, 5, ... Give a third term that makes it linear, and a different third term that makes it non-linear. Describe each rule.

The pattern 1, 3, 6, 10, 15, ... has differences +2, +3, +4, +5. What is the 10th term?

Lena saves $5 every week, starting from $0, so she has $5 after week 1. Omar has 1 cent after week 1, and his amount doubles every week. In which week does Omar first have more money than Lena?

Check your understanding

Quiz

Which pattern is linear?

What are the differences in 12, 9.5, 7, 4.5?

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

Which pattern is non-linear and shrinking?

In the repeating pattern A, B, B, C, A, B, B, C, ... what is the 30th letter?

What are the differences in 1, 4, 9, 16?

For parents and teachers

Big idea: The Ontario curriculum (C1.1) asks Grade 8 students to identify and compare repeating, growing and shrinking patterns, including real-life ones. The key tool in this lesson is the difference test: constant differences mean a linear pattern.

Common difficulties:

Thinking a pattern with negative numbers can't be growing. −12, −10.5, −9 is growing because each term is larger.
Subtracting in the wrong order. Always do next term minus this term, so shrinking patterns get negative differences.
Checking only one or two differences. Ask for at least three.
Expecting real measurements to be perfect. Measured data (cups, bounces) is close to a pattern, not exact. That's normal.

How to help: Look for patterns on a walk or in the kitchen: stacked bowls, a transit card balance, temperatures in a weather app. Ask "Is it growing by the same amount each time?" This lesson also builds A1 skills: making conjectures, checking them, and explaining reasoning.