Pattern Rules Using Words and Numbers
A pattern rule is the mathematical description of a pattern. Once you know the rule, you can generate any term without listing all previous terms. Start at 4, add 7 each time: the 10th term is 4 + 9x7 = 67. This is genuine algebraic reasoning in Grade 3. The rule is a compact description of an infinite sequence. Students who can state a rule in words and use it to predict distant terms have taken the first step toward writing algebraic expressions.
Rule as words, then numbers
Words first: this pattern starts at 3 and adds 5 each time. Numbers next: start = 3, add = 5, so term n = 3 + (n-1)x5. Students do not need the formula: they need to understand the process. Given term 1 = 3, what is term 7? Add 5 six times: 3+5+5+5+5+5+5 = 33. Or more efficiently: 3 + 6x5 = 33. The efficiency comes from understanding the rule.
Term number vs. term value
In the pattern 4, 8, 12, 16: term number 1 has value 4; term number 3 has value 12. Many students confuse these. A table of values makes the distinction explicit and is the most reliable tool. Position (input): 1, 2, 3, 4. Value (output): 4, 8, 12, 16. This input-output structure is the precursor to the function concept and to coordinate graphing.
Rhythm and song patterns
The BC curriculum connects pattern rules to predictability in song rhythm and patterns. A drum pattern with rule beat-rest-beat-rest (AB repeating) has a period of 2. A pattern beat-beat-rest (AAB) has period 3. Extending these: what is the 17th beat in a AAB pattern? 17 divided by 3 = 5 remainder 2: it is the 2nd element of the core = A (beat). This is the same reasoning as any repeating pattern, applied to music.