--- title: "Growing patterns" subtitle: "Linear relations · Rules, tables and graphs · Lesson 1" --- # Growing patterns **Name:** {{field-24}} **Date:** {{field-15}} **Class:** {{field-15}} > **Key question:** 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. :::teacher-note **In one sentence:** Students find the rule for a visually growing pattern from what each figure adds, use it for a figure they have not drawn, and work backwards from a count (AC9M7A05, node 1). **Students should be able to say:** 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. **Width:** the unmarked parts are the floor every student does (predict, run, investigate and make). Typical adds the modify practice; wider adds the real-world contexts. They add contexts, not difficulty. Students work in the online tutorial alongside this sheet. ::: ## Predict | Figure | 1 | 2 | 3 | |---|---|---|---| | Sticks | 4 | 7 | 10 | How many sticks does each new figure add? Where do they go? :::keep-together **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 My prediction, and why: {{lines-3}} ::: :::teacher-note **What happens:** 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. ::: ## Run Try it on the board in the tutorial. Was your prediction right? What was different? {{lines-2}} ## From pattern to rule :::keep-together ### Finding the rule Here is how to turn the squares pattern into a rule. 1. $n: 1, 2, 3, 4 \;\to\; s: 4, 7, 10, 13$ — Count the sticks in the first few figures and put them in a table. 2. $+3 \text{ each time}$ — Each new square needs 3 more sticks: a top, a bottom and a side. 3. $3n$ — So figure $n$ has $n$ lots of 3 sticks... 4. $3n + 1$ — ...plus the first stick on the left, which every figure has. 5. $s = 3n + 1$ — The rule: sticks = 3 × figure number + 1. 6. $n = 10: \; s = 3(10) + 1 = 31$ — Use the rule for a figure you have not drawn. **The triangle pattern: Figure 1 has 3 sticks, Figure 2 has 5, Figure 3 has 7. Fill in the table.** {{rows 1}} | figure n | 1 | 2 | 3 | 4 | 5 | |---|---|---|---|---|---| | sticks s | 3 | 5 | 7 | | | ::: :::teacher-note **Answer:** 9, 11 Each new triangle adds 2 sticks: 7, 9, 11. ::: :::keep-together ### 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. ``` Line: {{field-15}} Why is it wrong? {{lines-2}} ::: :::teacher-note **Answer:** line 2 Line 2: the squares share sides, so Figure 10 is not 10 separate copies of Figure 1. Each new square adds only 3 sticks: 3 × 10 + 1 = 31. ::: :::keep-together ### Your turn **Write a rule for the number of sticks $s$ in figure $n$ of the triangle pattern (3, 5, 7, 9, …).** Answer: {{field-15}} ::: :::teacher-note **Answer:** 2n + 1 s = 2n + 1: each triangle adds 2 sticks, plus the 1 stick at the start. Check Figure 3: 2 × 3 + 1 = 7. ::: :::typical ## 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.** Work in the tutorial. Plan first: what will you change, and how will you test it? {{box-4}} ::: ## Patterns and rules :::keep-together ### 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: {{field-15}} ::: :::teacher-note **Answer:** 4n + 1 It adds 4 each time, and Figure 1 is 4 + 1: s = 4n + 1. ::: :::keep-together ### A far figure **A pattern's rule is $s = 5n + 3$. How many are in Figure 12?** Answer: {{field-15}} ::: :::teacher-note **Answer:** 63 5 × 12 + 3 = 60 + 3 = 63. ::: :::keep-together ### Backwards again **Using $s = 2n + 1$ (the triangles), which figure has 25 sticks?** Answer: {{field-15}} ::: :::teacher-note **Answer:** 12 2n + 1 = 25, so 2n = 24 and n = 12. ::: :::keep-together ### Review **Review: solve $3n + 1 = 31$.** Answer: {{field-15}} ::: :::teacher-note **Answer:** 10 3n = 30, so n = 10. That is the squares pattern: Figure 10 has 31 sticks. ::: ## 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. Work in the tutorial. Plan first: what will you change, and how will you test it? {{box-4}} ## Exit ticket :::keep-together **A pattern has 6, 11, 16 tiles in Figures 1 to 3. Write the rule and find Figure 20.** {{lines-3}} ::: :::teacher-note **Answer:** s = 5n + 1; Figure 20 has 101 tiles. :::