Pattern Rules Using Words and Numbers
Warm-up
I have a pattern rule: start at 6, add 4 each time. What are the first 5 terms? (6, 10, 14, 18, 22.) What is the 10th term? Without listing all 10: add 4 nine times from 6. 6 + 9x4 = 42. Check by listing: does this work?
Explore
Students receive a table with position filled in (1 through 8) and must complete the values using a given rule. Then reverse: given values, find the rule. Then: given the rule and two terms, fill in a gap in the sequence. Each reversal requires deeper rule understanding.
Formalize
Rhythm pattern application: clap a pattern (clap-clap-rest repeating, rule: period 3). What is the 20th action? 20 divided by 3 = 6 remainder 2: 2nd in the core = clap. Check by counting. Discuss: this is the same as any period-3 repeating pattern in mathematics.
Pattern Rules Using Words and Numbers
Connect table of values to the input-output idea: position is the input, value is the output. Rule converts input to output. This is a function before it has a name. Plot the table on a simple grid: position on x-axis, value on y-axis. What shape do you get for an additive rule? (A straight line.)
Practice
Students complete 4 tables of values from given rules and find rules from 4 given tables. Predict the 12th term for each. Exit ticket: write the rule in words for the pattern 2, 5, 8, 11, 14.
Exit ticket
Students complete 4 tables of values from given rules and find rules from 4 given tables. Predict the 12th term for each. Exit ticket: write the rule in words for the pattern 2, 5, 8, 11, 14.
Step 1: Find the change: +4, +4, +4. Consistent, so we have a rule worth stating.
Step 2: State the RECURSIVE rule in careful words: "start at 5, and add 4 each time." Both halves are required — the start AND the change. (Test the necessity: "add 4 each time" alone also describes 2, 6, 10, … — a different pattern. Rules must pin down ONE pattern.)
Step 3: Use it to extend: 21, 25, 29.
Step 4: The precision game: give students sloppy rules and have them break them. "The numbers get bigger" — 5, 100, 101 also fits; too loose. "Add 4" — where do we start? A rule is GOOD when every person following it builds the identical pattern.
Exit skill: write a rule so airtight that a classmate who has never seen your pattern reproduces it exactly from the words alone.
Step 1: Execute the rule forward five terms: 50, 47, 44, 41, 38. Every student should produce this identical list — that's what makes it a rule.
Step 2: Reverse the game: here's a pattern — 80, 72, 64, 56 — write ITS rule. Differences: −8 each time. Rule: "start at 80, subtract 8."
Step 3: The stress test — a rule with two operations: "start at 2, double each time": 2, 4, 8, 16, 32. Compare its growth to +4's steady march: doubling EXPLODES. Let students feel that different rule TYPES have different personalities.
Step 4: Mixed lineup — match five patterns to five rules, with one decoy rule that matches nothing. The decoy forces genuine checking rather than pairing leftovers.
The skill being built: fluency translating both directions between the WORDS (rule) and the NUMBERS (pattern). That two-way street is pre-algebra.
The puzzle: a pattern begins 2, 4, … Ask the class for the next number. Camp A says 6 ("add 2"). Camp B says 8 ("double").
Step 1: Let both camps state their rules fully: "start at 2, add 2" versus "start at 2, double." Check both against the given terms: 2→4 works for each. BOTH rules are legitimate.
Step 2: Extend both: A gives 2, 4, 6, 8, 10; B gives 2, 4, 8, 16, 32. The patterns diverge at term three — dramatically after that.
Step 3: The honest conclusion: two terms cannot decide a pattern. "What comes next?" has no single right answer until MORE terms (or the rule itself) are given. Mathematics with too little information gives multiple truths.
Step 4: Repair the puzzle: reveal term three is 6. Now camp B's rule dies (it predicted 8) and camp A survives. More data killed a hypothesis — which is precisely how science works, in miniature.
Habit installed: before extending any pattern, ask "do I have enough terms to be SURE of the rule?"