Learn Mathematical Modeling unit by unit

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Units at a glance

Every unit in this course, what you will be able to do once you finish it, and roughly when it lands in a 12-week schedule.

Mathematical Modeling course map: 6 units
#	Unit	What you can do after it	Weeks
01	Units and estimation	You can carry units through a calculation and catch a wrong answer because the units do not match.	Weeks 1-2
02	Linear models	You can write a straight-line rule for a real situation and explain the slope and starting value in words.	Weeks 3-4
03	Growth and decay	You can decide whether a quantity changes by a fixed amount or a fixed percent, and pick the model that matches.	Weeks 5-6
04	Optimization	You can find the choice that gives the biggest or smallest result while staying inside a limit.	Weeks 7-8
05	Regression models	You can fit a line, a curve or an exponential to real data and pick the one that fits best.	Weeks 9-10
06	Model validation	You can test a model on data it has never seen and say exactly where it fails.	Weeks 11-12

Mathematical Modeling course units

Work in order for a complete course, or jump directly to the unit you need for an upcoming assessment.

Units and estimation

Weeks 1-2

You can carry units through a calculation and catch a wrong answer because the units do not match. You can also make a rough estimate first, so a bad answer stands out.

Convert 60 miles per hour into feet per second.
Estimate an answer to one digit before computing, then compare the two.
Cancel units through a chain of conversion fractions.

Worked example

Convert 60 miles per hour into feet per second

60 x 5280 = 316,800 feet per hour
Divide by 3600 seconds in an hour

Answer88 feet per second

Most common mistakeMultiplying by 3600 instead of dividing: reporting over a billion feet per second, faster than light, which the units check catches instantly.

Explain units and estimation Practice units and estimation

Linear models

Weeks 3-4

You can write a straight-line rule for a real situation and explain the slope and starting value in words. You can also say where the model stops making sense.

Turn 'a 30 fee plus 12 per hour' into C = 12h + 30.
Explain a slope of -4 as a loss of 4 units every week.
State the realistic domain, such as h being zero or more.

Worked example

A plumber charges a 45 call-out fee plus 60 per hour. Find the cost of a 3.5 hour job.

C = 60h + 45
C = 60(3.5) + 45 = 210 + 45

Answer255

Most common mistakeMultiplying the call-out fee by the hours too: computing (60 + 45)(3.5) = 367.50, which charges the one-off fee three and a half times.

Explain linear models Practice linear models

Growth and decay

Weeks 5-6

You can decide whether a quantity changes by a fixed amount or a fixed percent, and pick the model that matches. You can also find a doubling time or a half-life.

Compare a table's differences and its ratios to choose linear or exponential.
Write a 3% yearly decline as a base of 0.97.
Estimate a doubling time with the rule of 72.

Worked example

A town of 40,000 shrinks 3% each year. Find the population after 5 years.

Base = 1 - 0.03 = 0.97
P = 40,000 x 0.97 to the power 5 = 40,000 x 0.8587

AnswerAbout 34,350

Most common mistakeSubtracting 3% of the original five times: computing 40,000 - 5(1,200) = 34,000, which ignores that each year's 3% is taken from a smaller town.

Explain growth and decay Practice growth and decay

Optimization

Weeks 7-8

You can find the choice that gives the biggest or smallest result while staying inside a limit. You can also write those limits down as inequalities before you start.

Write a constraint such as 2L + 2W = 100 from a fencing budget.
Substitute the constraint into the goal so only one variable is left.
Find the maximum from the vertex of a downward parabola.

Worked example

You have 100 m of fencing for a rectangle. What shape gives the largest area?

2L + 2W = 100, so W = 50 - L
Area = L(50 - L) = 50L - L squared, with vertex at L = 25

Answer25 m by 25 m, giving 625 square m

Most common mistakeOptimising the perimeter rather than the area: a 49 m by 1 m rectangle also uses all 100 m of fence but encloses only 49 square m.

Explain optimization Practice optimization

Regression models

Weeks 9-10

You can fit a line, a curve or an exponential to real data and pick the one that fits best. You can also say how far past the data a prediction can be trusted.

Compare r squared across a linear fit and an exponential fit of the same data.
Read a residual plot for a bend that a straight line cannot handle.
Explain why predicting far beyond the data is risky.

Worked example

Data reads 2, 4, 8, 16, 32. Which model fits?

Differences are 2, 4, 8, 16, so they are not constant
Ratios are 2, 2, 2, 2, so those are constant

AnswerAn exponential model fits; a straight line does not

Most common mistakeAccepting a line because r = 0.92 sounds strong: the residual plot bends into a clear U shape, and the exponential fit predicts the next value far better.

Explain regression models Practice regression models

Model validation

Weeks 11-12

You can test a model on data it has never seen and say exactly where it fails. You can also list the assumptions the model is quietly making.

Hold back part of the data, then compare predictions with what really happened.
Compute the percent error between a prediction and the actual value.
Name an assumption, such as a constant rate, and say when it breaks down.

Worked example

A model predicts 480 units sold. The real number is 525. Find the percent error.

Difference = 525 - 480 = 45
Divide by the actual value: 45 / 525

AnswerAbout 8.6% too low

Most common mistakeDividing by the prediction instead of the real value: reporting 45/480 = 9.4%, which measures the error against a number that was itself wrong.

Explain model validation Practice model validation

Prepare for Mathematical Modeling practice

Start with the earliest uncertain prerequisite

Before timing yourself, check whether you can explain Units and estimation from a blank page. Then connect it to Linear models. If either explanation depends on copying a formula, review the unit first and complete two untimed examples.

Use tools to verify, not to choose the method for you

The Graphing calculator can test calculations and representations used in Mathematical Modeling. Make the setup yourself, predict the sign or scale, and compare the tool result with that prediction. Use the formula library to check conditions as well as notation.

Know when to move to the full test

Move from Mathematical Modeling practice to the complete course test after you can correct a missed problem without reopening the worked answer. Record the earliest wrong decision—not only the final score—so the next study session has a precise target.

Before and after the syllabus

Learn the ideas, then practise them

The unit list tells you what is covered. These two pages are where the method is explained and where you find out whether it stuck.

GuidesAlgebra explainedConcepts, worked methods and checks behind this course.

PracticeMixed Algebra I questionsAnswer review after every attempt.

ReferenceFormulas for this courseConditions and a worked example for each one.

Assess

Take the complete Mathematical Modeling test

Begin test →

Reference

Review essential formulas

Open library →

Calculate

Use the Graphing calculator

Open tool →

Plan

Prepare around your exam date

Build plan →

Questions about the Mathematical Modeling course

Where should I start?

Start with Units and estimation if you are following the full sequence. If that unit feels automatic, open the Mathematical Modeling practice page, choose mixed review, and let the first errors identify the earliest prerequisite to revisit.

How do I know I am ready for the course test?

You are ready when you can choose a method without a hint, show the governing steps, and explain why the result is reasonable. Use the complete Mathematical Modeling test only after you can correct practice errors from a blank page.

Which calculator supports this course?

The Graphing calculator supports the calculations and representations used in this course. Use it to test or visualize a result after making your own setup.

Learn Mathematical Modeling unit by unit

Units at a glance

Mathematical Modeling course units

Units and estimation

Worked example

Linear models

Worked example

Growth and decay

Worked example

Optimization

Worked example

Regression models

Worked example

Model validation

Worked example

Prepare for Mathematical Modeling practice

Start with the earliest uncertain prerequisite

Use tools to verify, not to choose the method for you

Know when to move to the full test

Learn the ideas, then practise them

Other high school courses

Pre-Algebra

Algebra I

High School Geometry

Take the complete Mathematical Modeling test

Review essential formulas

Use the Graphing calculator

Prepare around your exam date

Questions about the Mathematical Modeling course