Mathematics from zero
Algebra: build a pocket-money rule
Reading about algebra is not the same as turning a real situation into one. Take a small money problem from your own life, give the unknown a letter, and build it up into an expression, an equation, and an inequality — then solve each one by hand and check it. That round trip, from words to symbols and back, is the whole point of the unit.
Turn the unit’s four ideas into one connected piece of work: name a variable, write an expression for a total, set it equal to a target and solve the equation by the balance rule, then loosen it into an inequality and report the range — checking every answer by substituting back.
Take a real spending situation — saving pocket money, buying snacks at a fixed price, a phone plan with a fixed fee plus a per-unit cost — and model it as a variable, an expression, an equation, and an inequality, solving each by hand and proving every answer with a substitution check. Keep all numbers small and whole.
- A single page that moves cleanly from words to a variable, to an expression, to an equation, to an inequality — each step labelled and using the same letter consistently throughout.
- The equation is solved with every balance-rule move shown, and the solution is verified by a written substitution that makes both sides equal.
- The inequality answer is given as a range with the boundary correctly included or excluded, and is verified by one passing inside-value and one failing outside-value test.
- A two-sentence reflection naming where you used substitution, the balance rule, and the like-terms rule — and one mistake you caught by checking (for example treating 2n as 2 + n, or changing only one side).
- Add a second variable for a different choice (for example small snacks at one price and large snacks at another) and write a single expression that mixes both, then evaluate it for a sample basket.
- Find the exact spending amount where the equation and the inequality switch from true to false — the boundary — and explain in words what that boundary means back in the real situation.
- Rewrite your whole solution as a short 'rule' another person could follow for any target: state the expression once, then describe the steps to solve it for any total they pick.
- Build a small table of the expression's value for five different inputs, and use the table alone to estimate the equation's solution before solving it symbolically — then confirm the two agree.
This is the loop you will run on every word problem from here on: name the unknown with a letter, build an expression for the total, set it equal to a target and solve by the balance rule, loosen it to an inequality when the question is ‘at most’ rather than ‘exactly’, and check every answer by substituting back. Doing it once on a problem from your own life turns the four ideas of the unit — variable, expression, equation, inequality — into a single habit you can reuse.