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Course 218 · Life course 40

Model Exponential Growth and Decay

A population and cooling model sheet comparing repeated addition with repeated multiplication and checking predictions against supplied observations. Six introductory lessons use supplied practice cases.

Before you start

Use paper or a free text document; every core input is supplied in the lessons.

Read and Build Linear Functions See everything you’ll need

Free download · Original practice data · No external account required. Download the starter pack

What you’ll leave with

A population and cooling model sheet comparing repeated addition with repeated multiplication and checking predictions against supplied observations.

Start course Start without an account. Sign in to save lesson progress across devices.

Your project

Start with the practice files. Build your project one lesson at a time.

Practice files are free without an account. Sign in to save your project record in My Proof: a description, checklist, and optional work link. Keep the original files in your own app or notebook; saving a project record does not upload them.

  1. Get your materialsDownload practice filesSupplied practice cases, a blank working record, complete lesson instructions, and annotated worked examples.
  2. Make your projectView the project checklist Work in your own notebook, document, or software. Use the supplied practice examples when you prefer.
  3. Keep a private recordSave a project record Sign in to save a description and checklist, then return to update them.
Course handbook

Know the route before you begin.

Here’s what you’ll need and what you’ll make. You can return to this guide during any lesson.

Who this is for

Learners aged 14+ building subject foundations and practical independence. Adults can use the same supplied cases.

Prepare before lesson one
  • Use paper or a free text document; every core input is supplied in the lessons.
  • A basic calculator is optional; show the method, units and checks.
  • Suggested earlier course: Read and Build Linear Functions
Your final project

Finish with proof you can use.

A population and cooling model sheet comparing repeated addition with repeated multiplication and checking predictions against supplied observations.

Save a project record in My Proof
See project checklist and review criteria

Keep your working files in your own notebook or software. Save a private description, checklist, and work link in My Proof.

What to include

  • Compare a linear and an exponential sequence.
  • Show repeated-factor tables and direct formulas.
  • Check two growth and two decay calculations.
  • Explain a doubling or halving interval from supplied data.
  • Compare a model with observations and retain its mismatch.

Review and revise

Use these criteria to check your work and choose what to improve.

  • Method: applies the course-specific rules to the supplied cases with visible intermediate work.
  • Accuracy: results agree with the stated facts, units, boundaries and independent checks.
  • Reasoning: explains why the result follows and identifies a limitation or unresolved question.
  • Transfer: completes changed cases and revises the final project when a relevant assumption changes.
Three ideas to help you along the way
Three useful ideas

The rules behind the steps.

01

Name the quantities before calculating

Linear change adds the same amount in equal intervals; exponential change multiplies by the same factor. “Fast” alone does not identify an exponential process.

02

Use the correct relationship

A percent change of r per interval uses factor 1+r for growth and 1−r for decay. The factor multiplies the current amount, so later absolute changes can differ.

03

Keep the check and boundary visible

A model should be chosen by its mechanism and evidence, not by the larger predicted value. Compare observed and predicted values at the same inputs, and state whether a finite table can establish a universal rule.

What do these words mean?
Key words

Key terms worth knowing.

Factor
A multiplier applied each interval.
Initial value
The modeled output at zero elapsed intervals.
Growth
Increase by a repeated factor greater than 1.
Decay
Decrease by a factor between 0 and 1.
Absolute error
The nonnegative magnitude of observed minus predicted value.
Your route

6 lessons.
One useful project.

Work in order the first time. Each lesson makes something the next lesson can use.

  1. 1
    Up next

    Distinguish additive and multiplicative change

    Distinguish constant addition from constant multiplication in equal-interval sequences.

  2. 2
    10 min

    Build a repeated-factor table

    Construct an exponential table by applying a repeated factor once per interval.

  3. 3
    10 min

    Write and use an exponential rule

    Convert a percent change into a factor and apply it to the changing quantity.

  4. 4
    10 min

    Compare growth and decay

    Compare exponential growth and decay while tracking the remaining quantity.

  5. 5
    9 min

    Find doubling and halving intervals

    Identify a doubling or halving interval from supplied times and values.

  6. 6
    13 min

    Evaluate competing predictions

    Compare observed values with linear and exponential predictions and retain their mismatches.

Use the tools

Tools to help you practice.

See all tools
ZIP · practice files

Model Exponential Growth and Decay practice files

Supplied practice cases, a blank working record, complete lesson instructions, and annotated worked examples.

Text worksheet

Your course project worksheet

A downloadable text worksheet with each lesson’s working output, space for your answers, and the shared final-project checklist. Save your own copy.

Source transparency

Sources and further study.

Explore the references behind the lessons and take your learning further. Source notes explain access and reuse.

OpenStax: exponential functions
Source & access notes · checked 2026-10-01

Web open/search showed the official page, its topic heading and relevant concept description. This is a content/source check, not a provider endorsement. Exact topic chapters were opened in web result turn 325view 0–14.

Access: Official page retrieved with the web reader on 2026-10-01; interactive tools and videos were not certified by retrieval.

Reuse: Original fictional teaching cases, calculations, questions and examples; link and short factual paraphrase only. No copied source exercises or illustrations.

OpenStax: exponential graphs
Source & access notes · checked 2026-10-01

Web open/search showed the official page, its topic heading and relevant concept description. This is a content/source check, not a provider endorsement.

Access: Official page retrieved with the web reader on 2026-10-01; interactive tools and videos were not certified by retrieval.

Reuse: Original fictional teaching cases, calculations, questions and examples; link and short factual paraphrase only. No copied source exercises or illustrations.

OpenStax: exponential models
Source & access notes · checked 2026-10-01

Web open/search showed the official page, its topic heading and relevant concept description. This is a content/source check, not a provider endorsement.

Access: Official page retrieved with the web reader on 2026-10-01; interactive tools and videos were not certified by retrieval.

Reuse: Original fictional teaching cases, calculations, questions and examples; link and short factual paraphrase only. No copied source exercises or illustrations.

Ready when you are

Start with the first useful move.

You’ll always know what to do, what to make, and where to go next.

Start lesson one