Algebra and Patterns
Core ideas
Algebra expresses relationships generally and then manipulates them accurately. Candidates must substitute into expressions and formulas, collect like terms, expand and factorise, solve linear equations and inequalities, solve simultaneous equations, rearrange formulas to change the subject, use function notation, and describe sequences with an nth-term rule. Recognising arithmetic patterns (constant difference) and geometric patterns (constant ratio) is essential, as is interpreting straight-line graphs.
Traps
- Dropping a sign when subtracting a bracket: 5 - (x - 3) becomes 5 - x + 3, not 5 - x - 3.
- Treating 2x² as (2x)² - these are different expressions.
- Forgetting to reverse the inequality sign when multiplying or dividing by a negative number.
- Substituting n = 0 into an nth-term rule for a sequence that starts at n = 1.
- Assuming a relationship is linear when the differences between terms are not constant.
Worked example
Solve 5 - 2(x - 3) = 11. Expand the bracket: 5 - 2x + 6 = 11. Simplify: 11 - 2x = 11. Subtract 11 from both sides: -2x = 0, so x = 0. Check by substitution: 5 - 2(0 - 3) = 5 + 6 = 11. Correct.
Rule of thumb
For an arithmetic sequence with first term a and common difference d, the nth term is a + (n - 1)d. Always test a proposed formula with n = 1 and n = 2 before committing to it.