Grade by Grade
B. Exponential Functions · Exponential Functions

Collecting and Modelling Exponential Data

B3.1 B3.2About 75 minutes
Learning goals
  • I can collect data that behave exponentially using coins, number cubes or measurements.
  • I can graph data and decide whether an exponential model is reasonable using ratios.
  • I can fit a model y = a(bˣ) by hand using the starting value and the mean ratio.
  • I can describe how technology finds an exponential curve of best fit.

Words to know

primary data
Data you collect yourself, for example by experiment or measurement.
secondary data
Data collected by someone else, such as tables on the Statistics Canada website.
scatter plot
A graph of data points, used to see the pattern before choosing a model.
curve of best fit
A smooth curve that follows the general pattern of the data points.
exponential regression
A technology tool (Desmos, a graphing calculator, a spreadsheet) that finds the exponential equation that best fits the data.
discrete data
Data that can only take separate values, such as whole numbers of rounds or coins.

Where exponential models come from

Nobody hands a biologist the equation of a bacteria population. They measure, plot, look at the pattern, and then choose a model. In this lesson you will collect data that behaves exponentially, graph it, and fit a model of the form y = a(bˣ).

Data can come from primary sources (you collect it with coins, number cubes, a thermometer, a ruler, or electronic probes and sensors) or secondary sources (someone else collected it; for example, Statistics Canada publishes population and economic tables on its website).

The coin-decay experiment

Start with 60 coins. Shake them in a cup and spill them on a table. Remove every coin showing heads. Count the coins left. Repeat with the coins that remain until none are left.

Each coin has a 1/2 chance of being removed each round, so each round should leave about half of the coins. The predicted model is y = 60(0.5)ⁿ, where n is the round number. Real results wobble around the prediction because of chance.

round 060round 131round 216round 37Each round, flip every coin and remove the heads: about half are left.
One sample run of the coin experiment: 60, 31, 16, 7, … coins. The number left is roughly halved each round.
Round n012345
Coins left (sample run)603116742
Ratio to previous0.520.520.440.570.50
Model 60(0.5)ⁿ6030157.53.751.9
Data from one sample run of the coin experiment, with ratios rounded to two decimal places.
roundcoins left01234560102030405060model y = 60(0.5)ⁿ (dashed)dots: one sample run
The sample data (dots) follow the exponential model y = 60(0.5)ⁿ (dashed) closely, but not perfectly. Chance makes real data scatter around the model.

Fitting a model by hand

  1. Plot the data and look at the shape. Does it curve toward a horizontal asymptote? Are the ratios roughly constant (for equal steps)?
  2. Choose a: the starting value (the y-value at x = 0).
  3. Choose b: the average ratio of consecutive y-values. For the coin data the ratios are 0.517, 0.516, 0.438, 0.571 and 0.5, with a mean of about 0.51.
  4. Write the model y = a(bˣ) and check it against a few data points.

Fitting a model with technology

In Desmos, enter the data in a table (columns x₁ and y₁), then type y₁ ~ a·b^x₁. Desmos finds the a and b that fit best (it may choose values slightly different from your hand fit). Graphing calculators call this ExpReg, and spreadsheets call it an exponential trendline.

Worked example

Example 1: Fitting the coin data

Use the sample coin data above to write a model by hand. Compare it with the theoretical model y = 60(0.5)ⁿ.

  1. a = 60 (60 coins at round 0).
  2. Mean ratio: (0.517 + 0.516 + 0.438 + 0.571 + 0.5) ÷ 5 ≈ 0.508, so b ≈ 0.51.
  3. Hand model: y = 60(0.51)ⁿ. At n = 3 it predicts 60(0.51)³ ≈ 8.0 coins; the data had 7 and the theory 7.5.
  4. The fitted base 0.51 is very close to the theoretical 0.5. With more coins or more runs, the average gets even closer to 0.5.

y ≈ 60(0.51)ⁿ, close to the theoretical y = 60(0.5)ⁿ.

Worked example

Example 2: Number cubes

Hamid rolls 60 number cubes, removes every 6, and repeats. His counts are 60, 49, 41, 35, 29, 24, 20. Predict the theoretical model, fit a model to his data, and predict the count after 10 rolls.

  1. Each cube has a 5/6 chance of staying, so the theoretical model is y = 60(5/6)ⁿ ≈ 60(0.833)ⁿ.
  2. His ratios: 49/60 ≈ 0.817, 41/49 ≈ 0.837, 35/41 ≈ 0.854, 29/35 ≈ 0.829, 24/29 ≈ 0.828, 20/24 ≈ 0.833. Mean ≈ 0.833.
  3. Fitted model: y = 60(0.833)ⁿ, matching the theory.
  4. After 10 rolls: 60(0.833)¹⁰ ≈ 9.6, so about 10 cubes.

y ≈ 60(0.833)ⁿ; about 10 cubes after 10 rolls.

drop100 cmbounce 172 cmbounce 252 cmbounce 337 cmbounce 427 cm
Sample data: a ball dropped from 100 cm rebounds to about 72, 52, 37 and 27 cm. Each rebound is about 72% of the one before. (Heights drawn to scale.)
Worked example

Example 3: Bouncing-ball data

A ball is dropped from 100 cm. The rebound heights (sample data) are 72, 52, 37 and 27 cm. Is an exponential model reasonable? Find one and predict the 6th rebound.

  1. Ratios: 72/100 = 0.72, 52/72 ≈ 0.722, 37/52 ≈ 0.712, 27/37 ≈ 0.730. Nearly constant, mean ≈ 0.72: an exponential model fits.
  2. Model: h = 100(0.72)ⁿ, where n is the bounce number and h is in centimetres.
  3. 6th rebound: 100(0.72)⁶ ≈ 13.9 cm.
  4. The number of the bounce is a whole number, so this model is discrete: we plot points, not a continuous curve, though a dashed curve can guide the eye.

h = 100(0.72)ⁿ; about 14 cm on the 6th rebound.

Worked example

Example 4: Secondary data (practice set)

A small town's population (made-up practice data) was 8000 in 2016, 8400 in 2017, 8820 in 2018 and 9261 in 2019. Is the growth linear or exponential? Write a model.

  1. First differences: 400, 420, 441: not constant, so not linear.
  2. Ratios: 8400/8000 = 1.05, 8820/8400 = 1.05, 9261/8820 = 1.05: constant. Exponential.
  3. Let t = years since 2016: P = 8000(1.05)ᵗ, which is 5% growth per year.
  4. With real Statistics Canada tables, the ratios will not be exactly equal; use their average.

P = 8000(1.05)ᵗ

Just for fun

Why are coins so good at the decay experiment?

Because when it's time to leave, they don't argue. Heads, you're out.

Hands-on activity

Run your own decay experiment

You need: 50 to 100 small identical objects with two different sides (coins, or squares of paper marked with an X on one side), a cup or box, graph paper and a calculator. Number cubes work too: remove the 6s.

  1. Count your objects (round 0). Shake them and spill them out. Remove every one that lands heads (or X up). Count what is left. Record round number and count.
  2. Repeat until none remain (or for 8 rounds).
  3. Compute the ratio of each count to the one before it, and the mean ratio.
  4. Plot the data on graph paper (round on the horizontal axis, count on the vertical axis).
  5. Write your fitted model y = a(bˣ) and the theoretical model y = a(0.5)ˣ (or a(5/6)ˣ for number cubes). Sketch both on your graph.
  6. If you can, enter the data in Desmos and use y₁ ~ a·b^x₁ to compare.

What to look for: your mean ratio should be close to 0.5 for coins (or 0.83 for number cubes), but probably not exact. Later rounds wobble most, because a few coins more or less is a big percentage of a small number. With more objects or by combining several runs, the fit improves. The data are discrete: you cannot have round 2.5.

Your turn

1

A data set has y = 200, 100, 51, 24, 13 for x = 0 to 4. Is an exponential model reasonable? Why?

2

Data: x = 0, 1, 2, 3; y = 5, 8, 13, 20. Find the mean ratio and an exponential model.

3

Name two primary-source ways and one secondary source for collecting exponential data.

4

Why is the number of coins left after each round discrete data?

5

In an experiment with 40 number cubes where 6s are removed, what model do you expect?

6

Use y = 100(0.72)ⁿ to predict the 3rd rebound height of the ball in Example 3.

More practice: warm-up

1

What does a constant ratio in data suggest?

2

In y = a(bˣ), what does a represent in an experiment?

3

Coins: about what fraction should remain each round?

4

Is Statistics Canada a primary or secondary source?

5

Ratios 3.1, 2.9, 3.0, 3.0: what base would you use?

More practice: on your own

1

Mug of water data (temperature above room temperature, °C): 50, 42, 35, 30, 25 at t = 0, 5, 10, 15, 20 min. Find the ratio per 5 minutes and a model with t in 5-minute steps.

2

Coin data from 80 coins: 80, 38, 21, 9, 5. Find a fitted model and compare with the theory.

3

Why do later rounds of the coin experiment usually fit the model worse?

4

Rebound heights: 150, 105, 74, 52 cm. Find a model and predict when the rebound first drops below 20 cm.

5

A student fits y = 60(0.52)ⁿ to coin data. Predict the count at n = 4 and comment.

Challenge

1

Explain why removing the coins that show heads gives exponential decay, but removing exactly 5 coins each round would give linear decay.

2

Two classes do the coin experiment: one with 20 coins, one with 200 coins. Whose mean ratio will probably be closer to 0.5? Why?

3

A modelling app gives y = 98.6(0.715)ⁿ for bouncing-ball data that started at 100 cm. Why might a ≠ 100 exactly, and is the model still sensible?

Check your understanding

Quiz

1.

Which data source is primary?

2.

In the coin experiment, the theoretical model for 80 coins is:

3.

Data ratios are 0.81, 0.84, 0.82, 0.85. A good base for the model is about:

4.

Removing 6s from number cubes each roll gives which base?

5.

Why don't experimental data fit the model exactly?

6.

In Desmos, which entry fits an exponential model to table data x₁, y₁?

For parents and teachers

Big idea: Expectation B3.1: collect data that can be modelled exponentially, from primary sources (coins, number cubes, probes) or secondary sources (such as Statistics Canada), and graph the data. The examples also preview B3.2 (identifying exponential models in context).

How to help: Do the coin experiment together; it takes about 10 minutes. Ask "what fraction is left each time?" and "why isn't it exactly half?"

Common difficulties:

  • Using differences instead of ratios to test for exponential data.
  • Expecting experimental data to match the theory exactly.
  • Forgetting that a is the value at x = 0, not at x = 1.
  • Joining discrete data with a solid curve as if every in-between value were meaningful.