Increasing and Decreasing Patterns
Warm-up
Show a staircase pattern with linking cubes: 1, 3, 5, 7. What is the rule? (Add 2 each step.) What comes next? What is the 10th term? Now reverse: 20, 17, 14, 11. Rule? (Subtract 3.) What comes next? What will eventually happen if we keep subtracting?
Explore
Pattern investigation: groups receive a set of linking cubes and must build an increasing pattern using at least 5 terms, record it in a table, describe the rule in words, and predict the 8th term. Then create a decreasing pattern from the same starting point. Compare: when does each sequence reach zero?
Formalize
Doubling pattern: 1, 2, 4, 8, 16, 32. Rule: multiply by 2 (double). How does this differ from the staircase (additive) pattern? Use a table to compare growth rates. Doubling is far faster. This is the intuitive idea behind exponential growth, accessible in Grade 3 through concrete patterns.
Increasing and Decreasing Patterns
Connect to multiplication: the pattern 4, 8, 12, 16 follows the rule multiply by 4 (from the term number) OR add 4 each step. Both descriptions are correct. The multiply-by-4 description uses a table: term 1 = 4x1, term 2 = 4x2, term 3 = 4x3. This is the times-4 table in pattern form.
Practice
Students create two patterns each (one additive, one multiplicative), record in tables, describe rules, and predict term 10. Exit ticket: write the rule for 80, 72, 64, 56 and find the next term.
Exit ticket
Students create two patterns each (one additive, one multiplicative), record in tables, describe rules, and predict term 10. Exit ticket: write the rule for 80, 72, 64, 56 and find the next term.
The setup: 1 table seats 4 chairs; tables in a row share sides — 2 tables seat 6; 3 tables seat 8.
Step 1: Build it with square tiles (tables) and counters (chairs) so the geometry is real: end chairs plus 2 per table.
Step 2: Record a T-chart: tables 1, 2, 3, 4 → chairs 4, 6, 8, 10. Read DOWN the chart: chairs grow by +2 each new table (that's the recursive rule).
Step 3: Hunt the direct rule by reading ACROSS: chairs = double the tables, plus 2 (the two end chairs). Test it on every row: 2×1+2=4 ✓, 2×3+2=8 ✓.
Step 4: Cash in the direct rule: "How many chairs for 10 tables?" 2×10+2 = 22 — no drawing, no marching through 9 intermediate rows.
The two-rule vocabulary to install: the GROWING rule (+2 each time) explains the pattern; the DIRECT rule (double plus 2) predicts any term instantly. Both matter; the direct one scales.
The data: a 30 cm candle burns down 4 cm every hour. Heights: 30, 26, 22, 18, …
Step 1: Confirm the rule with differences: each hour −4. A DECREASING pattern — same machinery as growing patterns, negative direction.
Step 2: Extend: hour 4 → 14 cm, hour 5 → 10 cm, hour 6 → 6 cm, hour 7 → 2 cm.
Step 3: The endgame question that makes it interesting: "When does the candle run out?" After hour 7 it stands 2 cm; it can't survive another full −4 hour. Sometime during hour 8 it dies. The pattern doesn't continue forever — the CONTEXT stops it.
Step 4: Contrast with a pure number pattern 30, 26, 22, …, which happily marches into 2, −2, −6 forever. Real-world patterns wear seatbelts; abstract ones don't. Asking "does the story allow the pattern to continue?" is a modelling question — Grade 3 style.
Exit: make your own decreasing pattern and state where (or whether) it must stop.
Step 1: Present three displays side by side: (a) skip-count 3, 6, 9, 12; (b) triangle towers made of 3, 6, 9 blocks; (c) a T-chart of tricycles → wheels: 1→3, 2→6, 3→9.
Step 2: Ask what's the SAME underneath all three. Students should dig out: everything is "groups of 3" — the counting by threes, the blocks per tower, the wheels per trike.
Step 3: Write the shared skeleton once: 1 group → 3, 2 groups → 6, n groups → n × 3. One rule, three costumes.
Step 4: The transfer test: "invent a NEW story that wears the same skeleton." (Tripods and legs; shamrocks and leaves; triangles and sides.) Inventing an isomorphic story is the deepest evidence of understanding a structure.
Why this matters beyond Grade 3: recognizing "same math, different surface" is THE core mathematical habit. Patterns are its training ground.