Generating Patterns and Identifying Their Features

Rules Tell You What to Do; Features Are What You Discover

In this lesson:

• Generate a number or shape pattern from a starting point and a rule
• Notice features of a pattern that the rule never states
• Explain informally why a feature will keep happening

Learning Objectives for Today's Lesson

By the end of this lesson, you should be able to:

1. Generate a sequence from a start and a rule
2. Extend a shape pattern's repeating unit
3. Identify a feature the rule doesn't state
4. Explain why that feature continues
5. Distinguish the rule from the features

You Already Know How to Do This

Skip-count by 3s starting from 0, out loud, for five seconds.

0, 3, 6, 9, 12, 15, ...

Today's first pattern does exactly that job — it just starts somewhere else.

The Standard's Anchor: Start 1, Add 3

1 → 4 → 7 → 10 → 13 → 16 → 19 → 22 (each arrow: +3)

Your Turn: Start at 2, Add 5

Generate at least 8 terms.

Where did each new number come from — the starting number, or the one you just wrote?

A Different Rule: Multiply, Not Add

Start at 3, rule "Multiply by 2": 3, 6, 12, 24, 48, 96, ...

Compare to "Add 5": which grows faster? Is the gap between terms constant?

Vocabulary: starting number, rule, term, sequence

Mystery Rule: Working Backward From the Sequence

Sequence: 5, 8, 11, 14, 17, ...

What is the starting number? What is the rule?

What Does "Add 3" Never Tell You?

1, 4, 7, 10, 13, 16, 19, 22

Look carefully. The rule only says "add 3." What else is true about these numbers?

Take one to two minutes. Write down everything you notice.

Rule vs. Feature: A Critical Test

The rule: "Add 3" — tells you what to DO.
A feature: "alternates odd, even" — describes what you SEE.

If the rule were "alternate odd and even," what comes after 13? Even — but which even number?

The feature can't say. The rule can: 16.

Why Does the Odd/Even Alternation Never Break?

1 is odd, odd + odd = even, so term 2 (4) must be even.
4 is even, even + odd = odd, so term 3 (7) must be odd.

Second Example: Start at 2, Add 4

2, 6, 10, 14, 18, 22, 26, 30 — every term is even.

Why? Starting number 2 is even. Adding 4 (even) to an even number: even + even = even.

Could you ever get an odd number in this sequence? Why not?

Watch Out: Does a Feature Really Continue?

Claim: "All the terms in 1, 4, 7, 10, 13, 16, 19, 22, ... are less than 100."

Check it — generate more terms. Does the claim survive?

False. The sequence keeps growing forever; it will eventually pass 100 (and every other bound).

Practice: Generate, Notice, and Explain

For each, do all three steps: generate 8 terms, name a feature, explain why it continues.

1. Start at 1, "Add 2"
2. Start at 7, "Add 10"

Shapes Can Form Patterns Too

Circle, square, triangle, circle, square, triangle, circle, square, ...

What shape comes next? How do you know?

Spotting Features in Shape Patterns

Circles are in positions 1, 4, 7, 10, ... — the same numbers as our "Add 3" sequence!

Growing Pattern: Dots in Rows

1 dot, then rows of 3, 5, 7, 9 — each row has 2 more dots than the last.

Running totals: 1, 1+3=4, 4+5=9, 9+7=16, 16+9=25

Every running total is a perfect square!

Practice: Two Shape Pattern Problems

1. Repeating: triangle, triangle, circle, triangle, triangle, circle, ... Identify the core, extend it 3 more shapes, and name a feature.
2. Growing: rows of 2, 4, 6, 8 dots. What is the rule? What do you notice about the totals?

Capstone: The 20th Shape, No Drawing Allowed

Pattern: circle, square, triangle, repeating. What is the 20th shape? Explain — don't draw all 20.

Use the repeating-unit length and reasoning about the remainder.

Four Errors Worth Watching For

1. Wrong term to build from: use the number you JUST wrote, not the start
2. Rule/feature mix-up: the rule says what's next; a feature can't
3. Unproven feature: needs a reason that works for every term
4. Remainder-0 confusion: it's the LAST shape, not the first

Key Takeaways: Rule vs. Feature

A rule tells you what to DO. A feature is something you DISCOVER — and a real mathematician explains why it keeps happening.

✓ Apply the rule to the most recent term, every time

✓ A feature needs a reason that works for every term, not just the ones seen

What's Next: Two Patterns, Paired Together

5.OA.B.3 — Generate two numerical patterns from two rules, pair up matching terms, and graph the pairs on a coordinate plane.

This is the first step toward functions — a rule that turns inputs into outputs.