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# Growing patterns
Linear relations · Rules, tables and graphs
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## Today
**We are learning to** in a growing pattern, look at what each new figure adds. That gives a rule, and the rule gives any figure without drawing it.
- I can find what each figure of a growing pattern adds, and write a rule like s = 3n + 1.
- I can use a rule to work out a figure I have not drawn.
- I can work backwards from a count to the figure number.
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## Why it matters
> Tiles along a path, seats around joined tables, fence posts and rails: each one grows by the same amount every time. A rule tells you how many you need for any length.
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💬 Discussion
## Predict
**A row of squares is made of sticks. Figure 1 (one square) has 4 sticks, Figure 2 has 7 and Figure 3 has 10. How many sticks does Figure 10 have?**
- 13
- 30
- 31
- 40
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✍️ Activity
## Try it
{{embed: https://studio.learningwaypoints.com/tutorials/_playground-board/index.html?id=deck%2Fmaths%2Flinear-relations%2Fgrowing-patterns%2Fstep-2&code=widget%3A+pattern%0Ashape%3A+squares%0An%3A+1%2C+10%2C+1%0Acount%3A+on%0A&fresh=1}}
[Open this step in the tutorial](https://studio.learningwaypoints.com/tutorials/maths/linear-relations/growing-patterns/?full=step-2#step-2)
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✅ Check
## What happened?
* 31. Each new square adds 3 sticks (it shares a side with the square before), so Figure 10 has 4 + 9 × 3 = 31. Another way: 3 × 10 + 1 = 31.
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💬 Discussion
## Finding the rule
Here is how to turn the squares pattern into a rule.
$$n: 1, 2, 3, 4 \;\to\; s: 4, 7, 10, 13$$
1) $+3 \text{ each time}$ — Each new square needs 3 more sticks: a top, a bottom and a side.
2) $3n$ — So figure $n$ has $n$ lots of 3 sticks...
3) $3n + 1$ — ...plus the first stick on the left, which every figure has.
4) $s = 3n + 1$ — The rule: sticks = 3 × figure number + 1.
5) $n = 10: \; s = 3(10) + 1 = 31$ — Use the rule for a figure you have not drawn.
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✅ Check
## Finding the rule
**The triangle pattern: Figure 1 has 3 sticks, Figure 2 has 5, Figure 3 has 7. Fill in the table.**
| figure n | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|
| sticks s | 3 | 5 | 7 | ? | ? |
* Answer: 9, 11
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✅ Check
## Find the mistake
**Ben works out Figure 10 of the squares pattern. Click the line where his reasoning goes wrong.**
```
1 Figure 1 has 4 sticks.
2 Figure 10 has 10 squares, so it is 10 lots of Figure 1.
3 So Figure 10 has 4 × 10 = 40 sticks.
```
* Answer: line 2
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✅ Check
## Your turn
**Write a rule for the number of sticks $s$ in figure $n$ of the triangle pattern (3, 5, 7, 9, …).**
* Answer: 2n + 1
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✍️ Activity
## New patterns
- **Triangles:** Each task loads a new pattern. Use the board to see what each figure adds.
- **A cross of dots:**
- **A starting group:**
- **Backwards:**
[Open this step](https://studio.learningwaypoints.com/tutorials/maths/linear-relations/growing-patterns/?full=step-4#step-4)
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✅ Check
## Table to rule
**A pattern has 5, 9, 13, 17 dots in Figures 1 to 4. Write a rule for the dots $s$ in figure $n$.**
* Answer: 4n + 1
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✅ Check
## A far figure
**A pattern's rule is $s = 5n + 3$. How many are in Figure 12?**
* Answer: 63
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✅ Check
## Backwards again
**Using $s = 2n + 1$ (the triangles), which figure has 25 sticks?**
* Answer: 12
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✅ Check
## Review
**Review: solve $3n + 1 = 31$.**
* Answer: 10
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✍️ Activity
## Make: design a pattern
Design your own growing pattern of dots: $a$ dots in each new column, and $b$ dots to start with. Make Figure 10 have exactly 52 dots. Is there more than one way?
Stretch: write the rule for your pattern, and say how many dots Figure 100 has.
[Open this step](https://studio.learningwaypoints.com/tutorials/maths/linear-relations/growing-patterns/?full=step-6#step-6)
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✅ Check
## Exit ticket
**A pattern has 6, 11, 16 tiles in Figures 1 to 3. Write the rule and find Figure 20.**
* s = 5n + 1; Figure 20 has 101 tiles.