C. Algebra · Linear and Non-Linear Relations

Relations in Real Life: Linear or Non-Linear?

C3.1 A1About 70 minutes

Learning goals

I can describe a relation between two variables and tell which variable is independent and which is dependent.
I can calculate a rate of change and explain what it means in a real situation.
I can tell from the shape of a graph whether a relation is linear or non-linear, and whether it is growing or shrinking.
I can use a graph to make predictions, and I can explain when a prediction is reasonable.

Words to know

relation
A relationship between two variables. It can be shown with words, a table of values, a graph or an equation.

independent variable
The variable you choose or control (often time). It goes on the horizontal axis and is usually called x.

dependent variable
The variable whose value depends on the independent variable. It goes on the vertical axis and is usually called y.

rate of change
How much the dependent variable changes for each 1 unit change in the independent variable: change in y ÷ change in x. It always has units, like dollars per week.

linear relation
A relation with a constant rate of change. Its graph is a straight line.

non-linear relation
A relation whose rate of change is not constant. Its graph is a curve (or a path that bends).

interpolate
To estimate a value between known data points.

extrapolate
To estimate a value beyond the known data points, by extending the pattern.

Remember from Grade 8

In Grade 8 you sorted patterns into linear and non-linear by looking at the differences between terms, and you described linear patterns by their constant rate and initial value. In Grade 9 you will use the same ideas with relations: real situations where one quantity depends on another. You will learn to read the shape of a graph like a story.

One winter afternoon, two graphs

Kai already has $20 and saves $10 every week for new skates. At the same time, his sister pours a mug of hot chocolate at 80 °C and measures its temperature every 5 minutes as it cools.

Look at the two graphs below. One is a straight line. The other is a curve. Before reading on, ask yourself: why do you think they look so different?

Left: Kai's savings grow by the same amount every week, so the points line up in a straight line. Right: the hot chocolate cools quickly at first and then more slowly (sample measurements), so the graph is a curve that flattens out.

Relations and variables

A relation connects two variables. In each graph above, one variable depends on the other:

The independent variable is the one that changes on its own or that you control. Here it is time. It goes on the horizontal axis (the x-axis).
The dependent variable depends on it. Here it is savings or temperature. It goes on the vertical axis (the y-axis).

A quick test: say the sentence "___ depends on ___." "Savings depend on time" makes sense. "Time depends on savings" does not. The second blank is the independent variable.

Rate of change

The rate of change tells you how fast the dependent variable changes:

rate of change = change in y ÷ change in x

Kai's savings: from week 0 to week 1 they go from $20 to $30. Rate = (30 − 20) ÷ (1 − 0) = $10 per week. Check any two weeks and you get $10 per week again. The rate is constant, so the relation is linear and the graph is a straight line.
The hot chocolate: in the first 5 minutes it drops from 80 °C to 62 °C. Rate = (62 − 80) ÷ 5 = −18 ÷ 5 = −3.6 °C per minute. In the last 5 minutes it drops from 29 °C to 26 °C. Rate = −3 ÷ 5 = −0.6 °C per minute. The rate changes, so the relation is non-linear and the graph is a curve.

A positive rate means the relation is growing (the graph rises from left to right). A negative rate means it is shrinking (the graph falls from left to right).

Four shapes to know. A straight graph means a constant rate of change. A curve that gets steeper means the changes are getting bigger; a curve that flattens means the changes are getting smaller.

What the graph looks like	What the rate of change is doing	Real example
Straight line going up	Positive and constant	Earning $16 per hour
Straight line going down	Negative and constant	A candle burning 2 cm per hour
Horizontal line	Zero (no change)	Parked car: distance from home stays the same
Curve going up, getting steeper	Positive and increasing	Folding paper: layers double each fold
Curve going down, flattening	Negative, getting closer to 0	Hot chocolate cooling toward room temperature

Reading a graph's shape as a story about its rate of change.

Worked example

Example 1: Linear or non-linear?

Decide whether each relation is linear or non-linear, and whether it is growing or shrinking. Explain.

(a) Snow 40 cm deep melts at 2 cm per day.

(b) The area of a square tile compared with its side length.

(a) The depth goes down by the same 2 cm every day, so the rate is a constant −2 cm per day. Linear and shrinking.
(b) Make a quick table. Side 1, 2, 3, 4 cm gives area 1, 4, 9, 16 cm². The increases are 3, 5, 7: not the same. Non-linear and growing (growing faster and faster).
(c) Each apple adds the same $0.80, so the rate is constant at $0.80 per apple. Linear and growing.

(a) linear, shrinking; (b) non-linear, growing; (c) linear, growing.

Worked example

Example 2: Make predictions from a graph

Use the two graphs at the top of the lesson.

(a) When will Kai have $150?

(b) Estimate the temperature of the hot chocolate at 12.5 minutes.

(a) Kai starts with $20 and adds $10 each week. He needs 150 − 20 = 130 more dollars. 130 ÷ 10 = 13 weeks. (You could also extend the straight line until it reaches $150.)
(b) At 10 minutes it is 49 °C and at 15 minutes it is 40 °C. 12.5 minutes is halfway, so the temperature is roughly halfway: about 44 °C to 45 °C. Because the curve is bending, this is only an estimate.
(c) Week 13 is beyond the data (weeks 0 to 6), so (a) is extrapolation. 12.5 minutes is between data points, so (b) is interpolation.

(a) After 13 weeks, if he keeps saving $10 a week. (b) About 44 °C. (c) (a) extrapolation, (b) interpolation.

Is the prediction reasonable?

Extrapolating assumes the pattern keeps going. That is often fine for a short distance, but be careful far beyond your data. The hot chocolate will not keep cooling to 0 °C: it levels off near room temperature (about 20 °C). Always ask: does this make sense in real life?

Hands-on activity

Fold it, stack it

You need: one sheet of scrap paper, 10 coins of the same kind (for example, 10 pennies or 10 dimes), a ruler with millimetres, a pencil.

Fold the paper in half and count the layers. Fold again and count. Keep going as far as you can. Record a table: folds (0, 1, 2, ...) and layers.
Stack coins one at a time. After each coin, measure the height of the stack in millimetres. Record a table: coins and height.
Graph both tables (or sketch them). Which graph is straight? Which curves upward?
Calculate the rate of change for the first step and the last step of each table.

What to look for: the paper doubles (1, 2, 4, 8, 16, 32, ...), so it is non-linear and the increases keep getting bigger. The coin stack grows by about the same height each time (about the thickness of one coin), so it is linear. Your measured heights may wobble a little; that is normal measurement error.

Folding paper: the number of layers doubles with each fold (1, 2, 4, 8, 16, 32). The jumps get bigger every time, so the graph curves upward: a non-linear, growing relation.

Just for fun

People used to say no sheet of paper can be folded in half more than 7 or 8 times. Around 2002, a high school student named Britney Gallivan folded a very long strip of paper in half 12 times. Thanks, non-linear growth!

Your turn

A candle 24 cm tall burns down 2 cm every hour. Is the relation between time and height linear or non-linear? Growing or shrinking? What is the rate of change?

Is this relation linear?

x: 0, 1, 2, 3, 4

y: 5, 8, 11, 14, 17

Is this relation linear?

x: 0, 1, 2, 3, 4

y: 1, 3, 9, 27, 81

A graph rises from left to right but gets flatter and flatter. Describe its rate of change.

Name the independent and dependent variables: the time it takes to shovel a driveway and the number of people shovelling.

Use Kai's savings ($20 to start, $10 per week). How much does he have after 9 weeks?

Predicting the hot chocolate's temperature at 60 minutes: interpolation or extrapolation? Would you predict 0 °C? Explain.

More practice: warm-up

Which axis is used for the independent variable?

"The cost of a pizza depends on the number of toppings." Which is the independent variable?

If a graph is a straight line, what do you know about its rate of change?

y-values 10, 7, 4, 1 for x = 0, 1, 2, 3. What is the rate of change?

y-values 2, 4, 8, 16 for x = 0, 1, 2, 3. Linear or non-linear?

More practice: on your own

A taxi charges $3.50 plus $2 per kilometre. Make a table for 0 to 4 km. Is the relation linear?

A ball is dropped. The distance it has fallen after 0, 1, 2 and 3 seconds is 0 m, 4.9 m, 19.6 m and 44.1 m. Find the rate of change during the 1st second and during the 3rd second. Is the relation linear?

A phone battery starts at 100% and drops 8% every hour. When will it be empty, if the rate stays the same?

A town's population grows by 3% each year. Is the relation between years and population linear? Explain.

Find the rate of change between the points (2, 15) and (6, 35).

Challenge

Is this relation linear?

x: 0, 2, 4, 6

y: 1, 5, 9, 13

What is its rate of change?

The x-values in this table are not evenly spaced. Is the relation linear?

x: 1, 2, 4, 5

y: 3, 5, 9, 11

A young maple tree is 1.2 m tall and grows about 0.4 m per year. Use a linear model to predict its height after 30 years. Why might this prediction be wrong?

Check your understanding

Quiz

A relation has a constant rate of change. What does its graph look like?

Which relation is non-linear?

What is the rate of change from (0, 20) to (4, 60)?

A graph goes down from left to right and flattens out. What is happening?

Estimating a value between known data points is called:

For x = 0, 1, 2, 3 the y-values are 4, 9, 14, 19. Which is true?

For parents and teachers

Big idea: Ontario expectation C3.1 asks students to compare the shapes of graphs of linear and non-linear relations to describe their rates of change, connect them to growing and shrinking patterns, and make predictions. This lesson builds the habit of reading a graph as a story: straight means a steady rate; a curve means the rate is changing.

Common difficulties:

Mixing up the axes. Use the sentence "y depends on x" to decide which variable goes where.
Forgetting units in a rate. "10" is incomplete; "$10 per week" is a rate.
Checking only the first two values to decide if a relation is linear. Every step must have the same rate.
Trusting extrapolation too far. Ask whether the pattern can really continue (cooling drinks stop at room temperature; trees slow down).

How to help: Point out relations in daily life (gas bill and kilometres driven, a cooling oven, a growing savings account) and ask: "Would this graph be straight or curved? Why?" The activity uses only paper and coins.