Patterns & Algebra

Name the quantity that can change

A variable is a symbol representing a number or quantity whose value is not fixed in the current statement. An expression combines values and operations; an equation states that two expressions are equal. In 3n + 2, n is a variable and the whole is an expression. In 3n + 2 = 20, the equals sign makes an equation.

Define what the symbol represents and which values make sense. If n counts panels, it takes whole-number values in the model. A solution involving half a panel may be algebraically possible but outside a question that allows only complete panels.

Build an original rule

Imagine a fictional display containing n panels. Each panel contributes three tiles and the whole display has two extra marker tiles. The total is 3n + 2. For n = 1, 2 and 3, the totals are 5, 8 and 11. For ten panels, substitute n = 10: 3 × 10 + 2 = 32 tiles.

The extra two are added once for the whole display. The expression 3(n + 2) would add six extra tiles, because multiplication applies to everything inside the brackets. Explain the structure before relying on a memorised operation.

Use the relationship in reverse

If the total is 20 tiles, the equation is 3n + 2 = 20. Subtract 2 from both sides to get 3n = 18. Divide both sides by 3 to get n = 6. Check the result in the original relationship: 3 × 6 + 2 = 20.

Doing the same valid operation to both sides preserves equality. “Move the two across and change its sign” can hide that reason. A substitution check catches some errors and reconnects the answer with what the quantities mean.

A few terms do not determine every rule

The list 5, 8, 11 suggests adding three, but a finite list alone cannot establish a unique continuation. Here the tile-building description supplies the rule. Distinguish an observed pattern, a conjectured rule and a rule justified by a structure. That distinction makes a generalisation stronger than a guess.