🧱 See It
Build a pattern out of tiles, one step at a time. Each new step adds the same number of tiles as the step before it.
Step 1
Step 2
Step 3
3 tiles, then 5, then 7 — 2 new tiles get added every step.
✏️ Draw It
The same pattern as a table. Once you know the jump is always +2, you can figure out a step you haven't built yet.
| Step | Number of Tiles |
|---|---|
| 1 | 3 |
| 2 | 5 |
| 3 | 7 |
| 4 | ? |
Step 4 keeps the +2 jump going: 7 + 2 = 9.
Here's what that growth looks like as a simple bar chart — each bar is one step, and the height is the number of tiles:
Bars climbing at a steady rate — that's what an increasing pattern looks like on a graph.
🔢 Write It
Turn the pattern into a rule using step (often written n) for the step number:
tiles = 2 × step + 1
Check it against what you already built:
- Step 1:
2 × 1 + 1 = 3✓ - Step 2:
2 × 2 + 1 = 5✓ - Step 3:
2 × 3 + 1 = 7✓
step is the jump. The number added or subtracted at the end shifts the whole pattern up or down.💡 Worked Examples
Example 1 — Use the rule to jump ahead: How many tiles at step 10, using tiles = 2 × step + 1?
- Replace
stepwith 10:tiles = 2 × 10 + 1 - Multiply first:
tiles = 20 + 1 - Answer:
tiles = 21
Example 2 — Find the rule from a table:
| Step | Value |
|---|---|
| 1 | 5 |
| 2 | 8 |
| 3 | 11 |
Do this in two separate passes — don't try to see the whole rule at once.
- Pass 1 — find the jump: 5 → 8 → 11. Each step adds 3. The jump is 3, so the rule starts with
3 × step. - Pass 2 — check what's left over: at step 1,
3 × 1 = 3, but the real value is 5. That's 2 more, so add 2. - Rule:
value = 3 × step + 2 - Verify: step 2 →
3 × 2 + 2 = 8✓. Step 3 →3 × 3 + 2 = 11✓
🧠 Quick Reference
Step 2: Check step 1's actual value against "1 × jump" to find what needs adding or subtracting.
Do these as two separate passes, not all at once.