Patterns and Relations: Rules, Expressions, and Equations
What you must understand
This strand asks you to describe, extend, and generalize patterns, and to move between words, tables, expressions, and equations. A numeric pattern may grow by a constant difference (arithmetic) or by a constant multiplier (geometric). For a table of paired values, you should find the rule linking the input to the output, often written as an expression such as 3n + 2. You must also evaluate expressions by substituting values, solve one-step and simple two-step equations, and use a variable to represent an unknown quantity.
Traps to watch for
- Confusing the step with the starting value: in 3n + 2, the multiplier 3 is how much the output grows per input, while 2 is the value when n = 0, not when n = 1.
- Off-by-one term numbers: check whether the pattern starts at term 1 or term 0 before writing the rule.
- Order of operations in substitution: in 3n + 2 with n = 4, multiply first (12) then add 2, giving 14, not 18.
- Solving equations: whatever you do to one side you must do to the other; undo addition before undoing multiplication.
- Preserving equality: an equation is a balance, so a solution must make both sides equal when substituted back.
Worked example
A pattern of tile figures uses 5 tiles for figure 1, 8 for figure 2, and 11 for figure 3. The outputs rise by 3 each time, so the rule has the form 3n + b. Using n = 1: 3(1) + b = 5, so b = 2. The rule is 3n + 2. Check with n = 3: 3(3) + 2 = 11, which matches. For figure 10 the count is 3(10) + 2 = 32 tiles.
Rule of thumb
Always test a proposed rule on at least two table entries, and substitute your solution back into the original equation to confirm it balances.