C. Algebra · Linear and Non-Linear Relations

Modelling Real Data: Choosing a Linear or Non-Linear Model to Make Predictions

C3.1 C3.2 C4.1 A1 A2About 75 minutes

Learning goals

I can collect data from an experiment and decide whether a linear or non-linear model fits it better.
I can draw a line of good fit, find its equation and use it to make predictions.
I can describe a non-linear pattern, such as a constant percent decrease, and use it to predict.
I can judge whether a prediction is reasonable and explain the limits of a model.

Words to know

mathematical model
An equation, graph or rule that describes a real situation well enough to make useful predictions.

scatter plot
A graph of separate data points, used to look for a relationship between two variables.

line of good fit
A straight line drawn through the middle of a set of points that are roughly linear, with about as many points above it as below it.

measurement error
The small, unavoidable differences between measured values and the true values.

From experiment to equation

Scientists, engineers, kinesiologists, economists and data analysts all do the same thing: they measure something, look for a pattern, choose a model and use it to predict. Real measurements are never perfect, so the points do not land exactly on a line or curve. Your job is to see the overall shape and decide which model fits.

In this lesson you will model two home experiments: an elastic band stretched by washers, and a ball bouncing on the floor.

The modelling steps

Collect data and record it in a table.
Plot a scatter plot.
Look at the shape and the rates of change. Roughly constant first differences and a straight-looking plot suggest a linear model. Steadily changing differences or a curved plot suggest a non-linear model.
Build the model. For a linear model, draw a line of good fit and find its equation. For a non-linear model, describe the pattern (for example, "each value is about 70% of the one before").
Predict, then check whether the prediction makes sense in real life.

Washers, w	0	1	2	3	4	5	6
Length, L (cm)	10.0	11.4	13.1	14.4	16.0	17.6	19.0
First difference		1.4	1.7	1.3	1.6	1.6	1.4

Sample data: an elastic band with washers hanging from it. The first differences are not exactly equal, but they all stay close to 1.5.

Worked example

Example 1: A linear model for the elastic band

Use the table above. (a) Is a linear model reasonable? (b) Find an equation for a line of good fit. (c) Predict the length with 8 washers, and the number of washers for a length of 25 cm.

(a) The first differences are all between 1.3 and 1.7: roughly constant. The points lie close to a straight line. Yes, a linear model is reasonable. The small wobbles are measurement error.
(b) Draw a line through the middle of the points. It passes close to (0, 10) and (6, 19). Slope: (19 − 10) ÷ (6 − 0) = 1.5. y-intercept: 10. L = 1.5w + 10.
Meaning: the band stretches about 1.5 cm per washer, and its unstretched length is about 10 cm.
(c) 8 washers: L = 1.5(8) + 10 = 22 cm (extrapolation, just beyond the data). Length 25 cm: 1.5w + 10 = 25, so 1.5w = 15 and w = 10 washers.

L = 1.5w + 10. About 22 cm for 8 washers; about 10 washers for 25 cm.

The data points (dots) are close to, but not exactly on, the line of good fit L = 1.5w + 10 (dashed). The open circle is the prediction for 8 washers.

How far can you trust a model?

The elastic model predicts 1.5(40) + 10 = 70 cm for 40 washers. That is risky: an elastic band stretches less evenly when pulled hard, and it might snap. Predictions close to your data are more reliable than predictions far beyond it.

A non-linear model: the bouncing ball

Drop a ball from 100 cm and measure the height of each bounce (sample data):

Bounce number: 0, 1, 2, 3, 4, 5

Height (cm): 100, 70, 49, 34, 24, 17

First differences: −30, −21, −15, −10, −7. They get smaller, so the relation is non-linear. Look at ratios instead: 70 ÷ 100 = 0.70, 49 ÷ 70 = 0.70, 34 ÷ 49 ≈ 0.69, 24 ÷ 34 ≈ 0.71, 17 ÷ 24 ≈ 0.71. Each bounce is about 70% of the one before.

The bounce heights (dots) follow a curve that flattens out: each bounce is about 70% of the last one. A straight line (dashed) through the first two points would predict negative heights after bounce 3, which is impossible.

Worked example

Example 2: Predict with the bounce model

Use the bounce data. (a) Predict the height of bounce 6. (b) Why is a linear model a poor choice here?

(a) 70% of 17 cm: 0.7 × 17 = 11.9. About 12 cm.
(b) A line through the first two points has slope −30 and would give 100 − 30(4) = −20 cm at bounce 4: a negative height. The real heights shrink by a percent, not a fixed amount, so they get closer and closer to 0 without going below it.

(a) About 12 cm. (b) The decrease is not constant; a straight line would predict impossible negative heights.

Worked example

Example 3: Choose a model before you have data

Would you expect a linear or non-linear model? (a) Total cost of gas at $1.50 per litre. (b) Value of a car that loses 15% of its value each year. (c) Length of a shadow over a whole afternoon. (d) Pay for hours worked at $17 per hour.

(a) Each litre adds the same $1.50: linear (direct variation).
(b) 15% of a smaller value is a smaller amount, so the drops shrink each year: non-linear.
(c) Shadows grow slowly at first and then quickly as the sun gets low: non-linear.
(d) Each hour adds the same $17: linear.

(a) linear, (b) non-linear, (c) non-linear, (d) linear.

Just for fun

Tennis balls for official play are tested with a bounce: a standard ball dropped from 254 cm onto concrete must bounce back up somewhere between 135 cm and 147 cm. A tennis ball that is too bouncy is not allowed in official play!

Hands-on activity

Bounce lab

You need: a ball that bounces well (a tennis ball or rubber ball), a tape measure or metre stick taped to a wall, a hard floor, a helper, and (if you have one) a phone that records slow-motion video.

Part A (linear): drop the ball from 40, 60, 80, 100 and 120 cm. Measure the height of the first bounce each time. Plot drop height (x) against bounce height (y). Draw a line of good fit and find its equation. Does the line pass close to (0, 0)? Why would it?

Part B (non-linear): drop the ball from 100 cm and measure the first four or five bounce heights. Find the ratio between bounces. Predict the next bounce, then test it.

What to look for: in Part A the points are close to a line through the origin (direct variation: dropping from 0 cm gives a bounce of 0 cm). In Part B each bounce is about the same percent of the one before. Slow-motion video makes the top of each bounce much easier to read. Keep the area clear so nobody trips while chasing the ball.

Your turn

x: 1, 2, 3, 4, 5 and y: 3.1, 4.9, 7.0, 9.1, 10.9. Is a linear model reasonable? Find an equation for a line of good fit using the points (1, 3) and (5, 11).

Use L = 1.5w + 10. Predict the length with 4 washers and compare it with the measured value 16.0 cm.

Use L = 1.5w + 10. How many washers give a length of 22 cm?

Use the bounce model (70% each time). Predict the bounce after a 24 cm bounce.

Linear or non-linear: the value of a phone that loses 20% of its value every year?

Linear or non-linear: the total cost of gas at $1.50 per litre? Direct or partial variation?

Why should you be careful using L = 1.5w + 10 to predict the length with 50 washers?

More practice: warm-up

Where should a line of good fit go?

The points on a scatter plot lie almost in a straight line. Which kind of model should you use?

The points curve upward more and more steeply. Which kind of model?

What is predicting beyond the data called?

In L = 1.5w + 10, what does the 10 represent?

More practice: on your own

A bean plant's height (cm) on days 0, 5, 10, 15, 20 is 2.0, 4.6, 7.1, 9.4, 12.1. Write a linear model and predict the height on day 30. Is the prediction reliable?

A cup of hot water is left on the counter and its temperature is measured every 5 minutes. Should you expect a linear or a non-linear model? Explain.

A model y = 2.5x + 8 predicts y for x = 6. The measured value was 21.5. How far off is the model? Is that reasonable?

Bounce heights: 120 cm, 84 cm, 59 cm. Find the ratio and predict the next bounce.

Explain why a line of good fit does not usually pass through every point.

Challenge

Another student used the data points (1, 11.4) and (6, 19.0) for the elastic band. Find their equation and compare its prediction for 10 washers with L = 1.5w + 10.

A ball is dropped from 100 cm and each bounce is 70% of the one before. On which bounce is the height first less than 5 cm?

Data: x = 0, 1, 2, 3, 4 and y = 1, 2, 4, 8, 16. A student draws a line through (0, 1) and (4, 16). Show why this linear model is a poor choice.

Check your understanding

Quiz

First differences of a data set are 2.9, 3.1, 3.0, 2.8, 3.2. Which model fits best?

Bounce heights 80, 60, 45, 34 (cm). What is the pattern?

A line of good fit passes through (0, 4) and (5, 14). What is its equation?

Which prediction is most reliable for data collected from x = 0 to x = 10?

Which situation is best modelled by a non-linear relation?

Using L = 1.5w + 10, what length is predicted for 6 washers?

For parents and teachers

Big idea: Overall expectation C3 is to represent and compare linear and non-linear relations that model real-life situations and use these representations to make predictions. This lesson brings the unit together with real measurements, which are never perfect. Students learn to choose a model by looking at the shape and the rates of change, and to judge whether a prediction is reasonable.

Common difficulties:

Expecting points to fall exactly on a line. Real data wobbles; the line shows the trend.
Connecting the dots instead of drawing one straight line of good fit.
Using a linear model for percent changes (bounces, depreciation).
Over-trusting extrapolation.

How to help: Do the bounce lab together; it takes about 20 minutes. Slow-motion phone video makes the measurements much easier. Talk about careers that build models from data (engineering, sports science, economics, environmental science).