About this unit
How can we be sure a pattern will keep working?
Sections in this unit
Factual
- What makes a sequence arithmetic, geometric or neither?
- How do you find the general rule from two given terms?
- What do the first term, the common difference and the common ratio each tell you?
Conceptual
- Why do different-looking expressions give the same count?
- How can the same pattern be represented four ways, and what does each show that the others hide?
- What is the difference between verifying a rule and justifying it?
Debatable
- When is a rule proved rather than guessed?
- Is a pattern that holds for the first ten terms enough evidence that it holds for ever?
Skills
Students should be able to:
- Generalise from particular cases — move from counting a few examples to a rule that covers every case.
- Move between representations: read the same relationship as a diagram, a table, a graph and an algebraic rule, and convert between them.
- Verify a result by substituting a value back and checking it against an independent route.
- Justify why a rule must work, arguing from the structure it came from rather than asserting that it fits.
- Work backwards from a given result to an unknown condition.
How you are assessed
- Frogs Investigation, Criterion B
- Cumulative summative test, Criteria A and C
Investigation
Before you start
Students arrive from Year 9 with:
- The nth term of a linear sequence — finding it from a table, using it to generate terms, and testing whether a given number belongs to a sequence.
- Pattern spotting from visual and numerical starting points, and the square, triangular and Fibonacci-style sequences by name.
- Expanding single and double brackets, which the first homework deliberately recalls.
- Substitution, collecting like terms, and solving linear equations in one unknown.
Check what you remember
The nth term of a linear sequence
Expanding a single bracket
Expanding double brackets
Solving linear equations where the variable appears on one side of the equation only, including brackets
Solving linear equations where the variable appears on both sides of the equation