An equation is a balance: do the same thing to both sides and it stays level. This unit starts with equations that need more than two steps, with brackets, fractions and the unknown on both sides, and then asks what happens when there are two unknowns and two facts about them. Two straight lines meet once, never, or everywhere, and three methods find where.
Every idea is something you can move: slide a value until a balance levels, multiply a fraction away, drag two lines until they cross, parallel or lie on top of each other, substitute, eliminate, and set a cost against an income to find where a small business breaks even. Contexts come from community businesses and fair trade, where the answer decides whether a good idea can keep going.
The lessons, in order
- Lesson 1 · Brackets and both sides Solving multi-step linear equations with brackets and the unknown on both sides; naming equations by degree. Slide x until the two sides of a bracket equation level, then read the expanded form. No calculator
- Lesson 2 · Equations with fractions Solving linear equations that contain fractions, including fractions with a bracket on top and the unknown on both sides. Multiply a fraction equation by 1 to 20 and see which multipliers clear every fraction. No calculator
- Lesson 3 · Where two lines meet A system of two linear equations; its solution as the point of intersection; solving by table and by graph. Move two lines until they cross, and read the pair of values at the crossing. No calculator
- Lesson 4 · How many solutions? A system of two linear equations has one, no or infinitely many solutions; predicting the case from gradients and intercepts. Change a second equation and watch a fixed line get met once, never, or everywhere. No calculator
- Lesson 5 · Substitution Solving a system of two linear equations by substitution, including rearranging first, and checking in both equations. Replace y by its expression and see how far the other equation is from its target as x changes. No calculator
- Lesson 6 · Elimination Solving a system by elimination: scale one or both equations, add or subtract, solve, find the other unknown, check. Scale two equations, add or subtract them, and watch one coefficient drop to zero. No calculator
- Lesson 7 · Break-even, and choosing a method Writing a cost and an income as a linear system, solving for the break-even point, interpreting it, and choosing a method. Change the price, the cost of each item and the fixed cost, and see where income catches up with cost. 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.