Algebra and Patterns
Algebra and Patterns tests whether you can see structure: extend a pattern, state its rule, write and evaluate expressions, and solve simple equations. The emphasis is on relationships, not just arithmetic.
Patterns and rules
- A sequence can change by a constant difference, as in 5, 9, 13, 17, where the difference is 4.
- When first differences change, look for a second-level pattern or a rule involving the term number, such as n² + 3.
- An input-output table pairs each input with exactly one output. Test a proposed rule on every row, not just the first.
Expressions, variables, and equations
- An expression has numbers, variables, and operations but no equals sign: 4n + 7.
- An equation states that two expressions are equal: 4n + 7 = 31.
- A variable stands for an unknown or a changing quantity.
- Evaluate by substituting: if n = 6, then 4n + 7 = 4(6) + 7 = 31.
- Solve one-step equations using inverse operations while keeping both sides balanced.
Equality and properties
- The equals sign means “same value,” not “compute.”
- Commutative property: a + b = b + a and ab = ba. Associative property: grouping changes but value does not. Distributive property: a(b + c) = ab + ac.
- Apply order of operations when evaluating any expression.
Worked example
Find the next term: 4, 7, 12, 19, 28, ___. The differences are 3, 5, 7, 9, so the next difference is 11. Then 28 + 11 = 39. The rule n² + 3 also fits: 1 + 3 = 4, 4 + 3 = 7, 9 + 3 = 12, 16 + 3 = 19, 25 + 3 = 28, and 36 + 3 = 39.
Traps that cost marks
- Assuming every sequence has a constant difference.
- Confusing an expression with an equation because both contain variables.
- Applying an inverse operation to only one side of an equation.
- Ignoring order of operations when substituting a value.
- Accepting a rule that fits one row of a table but fails on another.