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.
01
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.
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.
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.
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.
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.
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.
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.
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.
