A linear relationship has one defining feature: equal steps in one quantity always change the other by the same amount. This unit starts from that idea, names the constant change as a gradient, and then follows a straight line through every way of writing it: y = mx + c, a point and a gradient, two points, and general form.
Parallel and perpendicular lines come out of the gradient, and the last lesson uses a line as a model: the cost of a plan, the point where two plans cost the same, and where a model stops being believable. Every idea is something you can move: slide a table into a graph, drag two points, change m and c, test two gradients against each other. The unit's real contexts are about how people's decisions add up: recycling, water, materials and energy.
The lessons, in order
- Lesson 1 · Constant change and straight lines Tables of values, the change in y for equal steps in x, and the test for a linear relationship, including falling and flat lines. Pick one of six relationships and see its table, the changes between rows and its points against a straight guide line. No calculator
- Lesson 2 · Gradient: rise over run Gradient as rise over run from two points, its sign, zero and undefined gradients, and gradient as a rate of change in context. Set two points with four sliders and read the rise, the run and the gradient; then slide two points along one fixed line. No calculator
- Lesson 3 · y = mx + c: gradient and intercept y = mx + c, the meaning of m and c, the x-intercept, graphing from the gradient and y-intercept, and horizontal and vertical lines. Set m and c with two sliders, then build a line from its y-intercept by repeating an across-and-up step. No calculator
- Lesson 4 · Writing the equation of a line Point-slope form from a point and a gradient, the equation through two points, converting to y = mx + c, and checking with the second point. Set a gradient and a point to see point-slope form become y = mx + c, then pick two points and follow the gradient and c. No calculator
- Lesson 5 · General form and intercepts General form ax + by = d, intercepts by setting each variable to zero, conversion to y = mx + c, and vertical lines. Set a, b and d with three sliders and read the intercepts and gradient straight from the general form. No calculator
- Lesson 6 · Parallel and perpendicular lines Parallel lines have equal gradients; perpendicular lines have gradients whose product is negative one; equations of parallel and perpendicular lines through a point. Set two gradients and read the product and the verdict, then build the perpendicular to a line through a point. No calculator
- Lesson 7 · Linear models and break-even Building y = mx + c from a context, interpreting m and c, break-even between two plans, a two-rate (piecewise) model, and the limits of a model. Compare two plans against the number of uses and move the price of the flask, then run a growth model past the days it was measured. No calculator
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.