Repeating and Increasing Patterns
Warm-up
Show a circular arrangement of coloured tiles (no clear starting point). What pattern do you see? Where does the core start? This is harder than a linear pattern: students must look for the repeating unit without a left-to-right anchor. Discuss strategies.
Explore
Pattern stations: (1) Positional patterns: a grid where row 1 is one colour, row 2 two colours, row 3 three colours. Describe the rule. What does row 7 look like? (2) Increasing number patterns: 4, 7, 10, ?, ?, ?. Extend and describe the rule. Find the 10th term. (3) Cultural pattern replication: students recreate a simplified armband pattern from a reference image.
Formalize
Compare a repeating and an increasing pattern side by side: ABABABAB vs. 2, 4, 6, 8, 10. Both have a rule. Both can be extended. What is different? (The repeating pattern cycles; the increasing pattern always grows.) What is the same? (Both have a constant rule that generates every term.)
Repeating and Increasing Patterns
Connect increasing patterns to addition: 3, 6, 9, 12. Each step adds 3. This is the same as repeated addition of 3. Preview: skip-counting by 3 IS an increasing pattern. The hundred chart shows this: colour multiples of 3 and describe the visual pattern.
Practice
Students create one repeating and one increasing pattern, record both in three representations (objects, drawing, numbers), and write the rule in words. Exit ticket: what is the 8th term of the pattern 3, 5, 7, 9?
Exit ticket
Students create one repeating and one increasing pattern, record both in three representations (objects, drawing, numbers), and write the rule in words. Exit ticket: what is the 8th term of the pattern 3, 5, 7, 9?
Step 1: Build two cube patterns side by side. Pattern A: red, blue, red, blue, red, blue. Pattern B: 1 tower, then 2, then 3, then 4.
Step 2: Ask what each pattern will "do next." A: another red (the SAME chunk cycles forever). B: a tower of 5 (each step GROWS by a rule).
Step 3: Name the two species: REPEATING patterns cycle a core; INCREASING patterns change by a consistent rule. The test question: "does it loop, or does it grow?"
Step 4: Classify a mixed lineup quickly: clap-stomp-clap-stomp (repeating); 2, 4, 6, 8 (increasing, +2 each time); square-circle-square-circle (repeating); staircase towers (increasing, +1).
Why the distinction matters: predicting a repeating pattern means finding WHERE IN THE CYCLE you land; predicting an increasing one means applying the growth rule. Different tools, so students must diagnose the species first.
Step 1: Build the staircase pattern: step 1 is 1 cube, step 2 is a tower of 2, step 3 a tower of 3, step 4 a tower of 4.
Step 2: Say the rule in words two ways: "each step is one cube taller than the last" (recursive — how it grows) and "the step number tells the tower height" (direct — what any step IS).
Step 3: Predict step 6 both ways. Growing: 5 then 6 — a tower of 6. Direct: step 6 → 6 cubes, no stepping needed.
Step 4: The bonus question with real depth: "how many cubes did we use for ALL SIX steps?" 1+2+3+4+5+6 = 21. Pair the ends to make it slick: 1+6, 2+5, 3+4 — three pairs of 7 → 21.
The direct rule is the star: it answers "step 100?" instantly (100 cubes), which the growing rule can only reach by 99 patient additions.
Step 1: Find the change: 5 → 8 is +3; 8 → 11 is +3; 11 → 14 is +3. The pattern grows by 3.
Step 2: Extend three terms: 17, 20, 23. Each is one more jump of 3.
Step 3: The backwards question — "what came BEFORE 5?" Run the rule in reverse: 5 − 3 = 2. The pattern could have started at 2.
Step 4: The membership question, which is where the thinking lives: "Will 30 ever appear?" March forward: 23, 26, 29, 32 — it skips right over 30. So no. Ask WHY: starting at 5 and adding 3s gives 5, 8, 11, … — check each against 30. (Grade-appropriate answer: "we jumped from 29 to 32, so 30 got skipped.")
Skills stacked in one string of numbers: find the rule, extend, reverse, and PREDICT MEMBERSHIP — the last one being genuine mathematical reasoning.