The first three lessons are about exponential models: a function that multiplies by the same factor for every step, how to write one from a rate, a doubling time or two readings, and how to find when it reaches a target.
The next lesson is about choosing between linear, quadratic and exponential models from a table, fitting each by hand, and using residuals to say how good a fit is. Then a short taster of the sine rule and the cosine rule, which work in triangles with no right angle. The last lesson is a small exploration from a question to a tested model and its limits, a preview of the written exploration in the Diploma course.
A graphic display calculator (GDC) is optional all the way through. Lessons 3, 4 and 6 have a closed “With a GDC” section to open if you have one; everything else is done by hand.
The lessons, in order
- Lesson 1 · Exponential growth and decay The form y = a × bx, the y-intercept a, growth when b is above 1 and decay when b is between 0 and 1, evaluating a model, and comparing an exponential with a linear function. Set a, b and x to see the value and the curve, then race a doubling quantity against a line to see when the curve takes over. Scientific calculator
- Lesson 2 · Building an exponential model Writing y = a × bt from a start and a percentage rate, from a doubling or half-life time, and from two values one period apart. Move the half-life and the time to see the amount halve, then move the next reading to see the base and the percentage rate. Scientific calculator
- Lesson 3 · When does a model reach a target? Reading the first whole period at which a model reaches a target from a table, the crossing from a graph, whole versus exact time, doubling time and a comparison between an exponential and a line. Move a target line and the base to see the first whole period and the exact crossing, then move a growth rate to see the doubling year. GDC optional
- Lesson 4 · Choosing and fitting a model First differences, second differences and ratios to choose a family, fitting a line, a quadratic and an exponential by hand, residuals and the sum of squares, and when a model can be trusted. Switch between four data sets to see which row of differences or ratios is constant, then set a gradient and an intercept to make the squared residuals as small as you can. GDC optional
- Lesson 5 · A first look at the sine and cosine rules The sine rule a ÷ sin A = b ÷ sin B for a missing side, the cosine rule for the third side from two sides and the included angle, area = ½ ab sin C, and choosing between them. Build a triangle from two angles and see that every side divided by the sine of its opposite angle is the same, then slide an angle between two fixed sides to see the cosine rule. Scientific calculator
- Lesson 6 · A short mathematical exploration The stages of a short exploration, choosing a model from ratios, fitting by hand and with technology, testing with residuals and a new reading, judging a prediction outside the data, and evaluating a model honestly. Set a start and a factor to make the squared residuals small, then compare two models that agree on the data and disagree after it. GDC optional
Every lesson has a teacher page with the answers, the misconceptions each wrong answer reveals, discussion questions, differentiation and support for students learning in English, and extra questions in three tiers.