Two students describe the same pattern (2, 5, 8, 11, 14). Student A says 'add 3 each time.' Student B says 'each term is 3 times its position minus 1.' Are both rules correct?
AOnly Student A is correct — you can see the +3 in the gaps
BOnly Student B is correct — position rules are always better
CBoth are correct — they describe the same pattern from different perspectives
DNeither is correct — the rule is 'start at 2'
Both rules produce the same pattern. Student A gives a term-to-term rule (recursive): to get the next term, add 3 to the current one. Student B gives a position rule: term = 3 x position - 1 (position 1: 3(1)-1=2, position 2: 3(2)-1=5, etc.). Both are complete, correct descriptions of the same pattern. Having multiple valid descriptions is a strength, not a confusion.
Question 2 Multiple Choice
Which is a complete pattern rule for the sequence 10, 8, 6, 4, 2?
A'The numbers get smaller' — that tells you the direction
B'Even numbers' — all the terms are even
C'Start at 10, subtract 2 each time' — that gives the starting point and the operation
D'The pattern is 10, 8, 6, 4, 2' — just list all the terms
A complete rule must tell you enough to generate every term. 'Start at 10, subtract 2 each time' does this: you get 10, then 8, then 6, and so on. 'The numbers get smaller' is too vague — many patterns get smaller. 'Even numbers' is an observation, not a rule (it does not specify which even numbers or in what order). Listing all terms is a description, not a rule — it does not tell you what comes next.
Question 3 True / False
A pattern rule that says 'the numbers go up' is specific enough to identify a unique pattern.
TTrue
FFalse
Answer: False
Infinitely many patterns have numbers that go up: 1, 2, 3, 4... and 1, 10, 100, 1000... and 2, 5, 8, 11... all go up. A useful rule must specify exactly how the numbers change — 'add 3 each time' or 'double each time' or 'each term is its position number squared.' Vague descriptions like 'going up' fail the test of being able to generate the specific pattern.
Question 4 Short Answer
What is the difference between a term-to-term rule and a position rule, and why might you want both?
Think about your answer, then reveal below.
Model answer: A term-to-term rule tells you how to get from one term to the next (e.g., 'add 5'). A position rule tells you how to find any term from its position number (e.g., 'the term is 5 times its position'). The term-to-term rule is easier to spot — you just look at the gaps. The position rule is more powerful — it lets you jump to the 100th term without listing all the ones before it. You want both because the term-to-term rule is easy to discover, and the position rule is efficient to use.
This distinction — recursive vs. explicit — is fundamental in mathematics. Term-to-term rules are recursive (each term depends on the previous one). Position rules are explicit (each term is computed independently). Much of algebra involves converting between these two forms, and students who understand both at the pattern level have a head start.