Caveat Academy course · Grades 8 to 12

Mathematical Modeling

A semester course on turning real questions into math you can test: estimates, linear and exponential models, fitting data, optimization and chance. Six modules at your own pace, with a check in every lesson and a certificate at the end.

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$19one-time · every device you sign in on
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Included with the Caveat Academy pass and Whole-school licenses. Or get it in the STEM explorations set, $49 for 9 items.

6 modules · 18 lessonsabout 13 hours$19

Six modules, eighteen lessons

Each module ends with a practice set that tracks mastery, an explain-it-back prompt, a short written response and a module exam. The final opens when every module is proficient; passing it earns a certificate with a verification code.

  1. Module 1: Thinking like a modeler

    Apply the modeling cycle to a real question: state assumptions, name variables and parameters with units, make Fermi estimates and revise a model after validation.

    • The modeling cycle
    • Assumptions, variables and parameters
    • Estimation and Fermi problems
    about 2.2 h

  2. Module 2: Linear and piecewise models

    Build linear, piecewise and break-even models from measurements and compute residuals for a fitted line.

    • Rates of change and linear models
    • Fitting a line to data
    • Piecewise and break-even models
    about 2.2 h

  3. Module 3: Exponential growth and decay

    Compute exponential growth and decay, doubling times and half-lives, and compare linear with exponential models over the long run.

    • Growing and shrinking by a percent
    • Doubling time and half-life
    • Compound interest and comparing models
    about 2.3 h

  4. Module 4: Models from data

    Analyze data to choose a model's form from differences, ratios and residual plots, and evaluate correlation, causation and extrapolation claims.

    • Scatterplots, correlation and causation
    • Choosing the form of a model
    • Interpolation, extrapolation and limits
    about 2.2 h

  5. Module 5: Optimization

    Solve optimization problems with quadratic vertices, corner points of linear programs and weighted decision matrices, including a sensitivity check.

    • Quadratic models and the best value
    • Linear programming
    • Decisions with several goals
    about 2.2 h

  6. Module 6: Chance and change over time

    Compute expected values, design a simulation, run a logistic recurrence and write a modeling report with its limitations.

    • Expected value
    • Simulation
    • Step-by-step change and the modeling report
    about 2.3 h

Try it before you start

This is how every lesson works: step through a worked example at your own pace, then answer a check and get the reason right away. Nothing here is saved; the course keeps your progress.

Module 5 · Linear programming

Worked example: a bake sale plan

  1. x = trays of muffins, y = trays of bread. Each muffin tray earns $3 and each bread tray $4 (example figures).

A lesson check from each module

The checks need JavaScript turned on. The course itself works the same way.

What you get

Enrichment · STEM ExplorationsAlso for Academics · High School$19 · included with the Academy pass and Whole-school licenses

What you finish with
A finished semester of mathematical modeling: you can run the modeling cycle on a real question, make Fermi estimates, build linear, piecewise, exponential and logistic models, fit a line and read its residuals, choose a model from a table, solve optimization problems with vertices, corner points and decision matrices, and compute expected values and design simulations. Passing the final earns a Caveat certificate of completion.
Who it is for
Students in grades 8 to 12 who have finished or are taking first-year algebra and want to use it on real questions, preview a school modeling course or prepare for modeling competitions; homeschool families who want a complete semester unit. A calculator or spreadsheet helps.
Format
An online self-paced course: 6 modules and 18 lessons with worked examples you reveal step by step, 152 practice and exam items (multiple choice, numeric, fill-in and ordering), an explain-it-back prompt and a rubric-checked short response in every module, module exams with one retake, a 24-question final and a certificate with a verification code. About 13 hours in all; works on a phone or a computer.
Access
$19 one-time, yours on every device you sign in on; the course opens once you are signed in with it. Included with the Caveat Academy pass and Whole-school licenses. Also in the STEM explorations set ($49) and the Enrichment Suite pass ($119), and 1 more bundle. Progress is saved in this browser on this device, and the certificate page lets you download your record.
Where it lives
The STEM Explorations space in Caveat Enrichment; also listed for the High School space in Caveat Academics (the same course serves the high school Mathematical Modeling title).
Support
Access, a receipt, a refund or a correction to the course: team@caveatacademic.com.

How it works

  1. Open the course and start Module 1. Each lesson takes about 25 to 35 minutes; two or three a week finishes the course in about a semester.
  2. Answer the checks as you read. Every answer shows the reason, and a miss links you back to the lesson that covers it.
  3. Finish each module with its practice set and exam. Proficient on every module opens the final; 70 percent passes it.
  4. Print or download your certificate and record from the certificate page.

Start the course

Sources and accuracy

The lessons are original writing by Caveat Academic Services. Facts were checked against these free, open references, which are also good places to read further.

  1. SIAM and COMAP, Guidelines for Assessment and Instruction in Mathematical Modeling Education (GAIMME), free report: siam.org/publications/reports
  2. OpenStax, College Algebra 2e, exponential and logarithmic models (free textbook): openstax.org
  3. OpenStax, Introductory Statistics 2e, linear regression and correlation: openstax.org
  4. NIST/SEMATECH e-Handbook of Statistical Methods, residuals and model checking: itl.nist.gov/div898/handbook
  5. NIST Physical Measurement Laboratory, radionuclide half-life measurements: nist.gov/pml

Mathematical Modeling is a self-paced course from Caveat Academic Services. It follows the sequence of a typical one-semester course but does not replace a school course or award credit. Rates, prices and populations in worked examples are labeled as examples; the interest lessons are math practice, not financial advice. Support: team@caveatacademic.com.