This unit starts with a warning: a pattern that fits many cases can still be false. Students build counterexamples that meet the conditions of a claim, and see that a condition can turn a false claim into a theorem. Then letters take over: one letter for every free choice proves a pattern for every case, and the same idea with coordinates in letters proves a fact about every quadrilateral.
The middle of the unit asks how many solutions a system of equations can have, and what the coefficients say before anything is solved. The last two lessons turn an inequality into a region and a decision into a linear programming problem: the quantity to make largest is a line that slides across the region, and the best choice is at a corner.
A graphic display calculator (GDC) is optional all the way through. Lessons 1, 4, 5 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 · Evidence is not a proof Conjecture and theorem, inductive and deductive reasoning, building a counterexample that meets the conditions, why a long run of successes is not a proof, and adding a condition to rescue a claim. Join points on a circle and count the regions, then test a divisibility claim with two sliders. GDC optional
- Lesson 2 · Proving with the general case Representing any, even, odd and consecutive numbers with letters, expanding and collecting to reveal a factor, the number-grid box for any grid width and box size, and writing a proof as a chain that ends in a sentence. Move a box about a number grid and change its size, then grow a ring between two squares. No calculator
- Lesson 3 · Coordinate proof with letters Midpoint, squared distance and gradient in letters, parallel (equal gradients) and perpendicular (product of gradients minus 1), placing a shape on axes without losing generality, and the midpoint quadrilateral of any quadrilateral. Move two corners of a quadrilateral and watch what the midpoints of its sides always make. No calculator
- Lesson 4 · One, none or infinitely many The three cases for two lines, deciding the case from the coefficients of ax + by = c, solving by elimination, equivalence transformations, a parameter in a system, and a break-even model. Move a second line against a fixed one, then slide a plan's fee and rate to find where two costs meet. GDC optional
- Lesson 5 · Inequalities and regions a > b if and only if a − b > 0, adding and multiplying inequalities, reversing for a negative, double inequalities, solid and dashed boundaries, testing a point, and the region of a system of linear inequalities. Flip the sign of a coefficient and watch the solution ray turn, then move a boundary line and a test point. GDC optional
- Lesson 6 · Linear programming at the vertex Variables, constraints, the feasible region, the objective function, the optimum at a vertex, ties along an edge, minimizing, and interpreting a solution in context including whole-number limits. Slide the objective line across a region until it leaves, then compare the corner values. 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.