Growing patterns

Classroom resources

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Students will be able to

• 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.

Timing

Predict and Run 8 min · Investigate 15 min · Modify 12 min · Practise and Make 10 min

Before class

Do the Run and Modify boards yourself first. The board draws what each figure adds in orange: that is the whole idea of the lesson.

The rule's two numbers have meanings students can see: the number in front of n is how many each figure adds, and the number added on is what every figure has before that (the first stick, the centre dot, the starting group). The rest of the course builds on this: in y = mx + c, m is the amount added each step and c is the starting amount.

The common mistake is scaling: Figure 10 = 10 × Figure 1. It fails because the parts are shared. Ben's working plants it after the method.

Running the lesson

1. Predict: take a show of hands. 40 (10 × 4) is the common wrong answer. Do not correct it: the board settles it.
2. Run: before moving to Figure 10, ask: what does each figure add, and why only 3? Point at the shared side.
3. Investigate: model the worked example live. Build the table, circle the +3s, then say 'n lots of 3, plus the 1 on the left'.
4. Modify: before the backwards task, ask: is 46 the figure number or the number of sticks?
5. Practise: ask students to check each rule on Figure 1 and Figure 2 before answering.
6. Make: collect different designs that give 52. Is there more than one?

When students get stuck

Answers to the checks

1. Figure 10: 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?
   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.
2. grow the pattern: Go to Figure 10. How many sticks? Was your prediction right?
   
3. Finding the rule: The triangle pattern: Figure 1 has 3 sticks, Figure 2 has 5, Figure 3 has 7. Fill in the table.
   9, 11
   Each new triangle adds 2 sticks: 7, 9, 11.
4. Find the mistake: Ben works out Figure 10 of the squares pattern. Click the line where his reasoning goes wrong.
   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.
5. Your turn: Write a rule for the number of sticks $s$ in figure $n$ of the triangle pattern (3, 5, 7, 9, …).
   2n + 1
   s = 2n + 1: each triangle adds 2 sticks, plus the 1 stick at the start. Check Figure 3: 2 × 3 + 1 = 7.
6. Triangles: How many sticks are in Figure 20 of the triangle pattern?
   41
   s = 2n + 1 = 2 × 20 + 1 = 41.
7. A cross of dots: Each figure of the cross pattern adds a dot to each arm. How many dots are in Figure 15?
   61
   4 new dots each figure, plus the centre: s = 4n + 1 = 4 × 15 + 1 = 61.
8. A starting group: Each figure adds a column of 3 dots to a starting group of 2. Write a rule for the dots $s$ in figure $n$.
   3n + 2
   s = 3n + 2: 3 dots added each figure, and 2 in the starting group. Check Figure 1: 3 + 2 = 5.
9. Backwards: Working backwards: which figure of the squares pattern has 46 sticks?
   15
   3n + 1 = 46, so 3n = 45 and n = 15. Check: 3 × 15 + 1 = 46.
10. 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 $n$.
    4n + 1
    It adds 4 each time, and Figure 1 is 4 + 1: s = 4n + 1.
11. A far figure: A pattern's rule is $s = 5n + 3$. How many are in Figure 12?
    63
    5 × 12 + 3 = 60 + 3 = 63.
12. Backwards again: Using $s = 2n + 1$ (the triangles), which figure has 25 sticks?
    12
    2n + 1 = 25, so 2n = 24 and n = 12.
13. Review: Review: solve $3n + 1 = 31$.
    10
    3n = 30, so n = 10. That is the squares pattern: Figure 10 has 31 sticks.
14. design a pattern: Set $a$ (dots in each new column) and $b$ (the starting group) so that Figure 10 has 52 dots. Show Figure 10.
    
    Your pattern has 10a + b = 52 dots in Figure 10.

Solutions

One possible solution for each task. Other correct programs are fine.

new patterns › Triangles

Open task: any program that does what the step asks.

new patterns › A cross of dots

Open task: any program that does what the step asks.

new patterns › A starting group

Open task: any program that does what the step asks.

new patterns › Backwards

Open task: any program that does what the step asks.

design a pattern

Open task: any program that does what the step asks.

Stretch and support

Stretch: A pattern of tables in a row: one table seats 4, each extra table seats 2 more people. Write the rule, then find how many tables seat 40.

Support: Give squared paper or matchsticks to build Figures 1 to 4. Fill a table before writing any rule.

Coach prompt (optional)

For a student who has done the lesson and is still stuck. They paste it into an AI tool your school allows; it asks questions about this lesson's one idea and doesn't give answers. Students see it on the course recap page only through a link ending in ?coach=1: recap with coach prompts.

You are a patient maths coach for a Year 7 student in Australia. Help me understand ONE idea only, from the lesson "Growing patterns".

The idea: Generating tables of values from patterns: I can Students identify the rule in a visually growing pattern and use it to generate a table of values.

What understanding looks like:
- Correctly generates a table of values from a growing pattern.
- Identifies the rule that generates the values.
- Can find the value for a term not shown in the original pattern.

Mistakes students often make here:
- A student multiplies Figure 1 (Ben's mistake).
- A student writes s = 3n and is one short.
- A student puts the count in as n when working backwards.

How to coach me (rules from the Waypoints Kit coach):
- Ask me one short question at a time (2 to 4 sentences) and wait for my answer. Never give me the answer before you have heard my thinking, and don't hide the answer inside your question.
- Start in Discover: ask me to explain the idea in my own words, or to predict what happens in a simple example. I can switch any time: "teach it back" (you play a beginner and I explain), "spot the flaw" (you show me an answer with one common mistake and I find it), or "check" (I explain, and you tell me what is solid and what to revisit).
- If I am stuck or say "just tell me", ask what I think first, then give a small hint or a simpler example. Explain only after I have had a real go.
- Stay in this idea: don't teach later ideas or add extra content. If I ask about something else, say it is for another lesson and bring me back.
- After you help, ask me one reflective question, such as how I know my answer is right or what I would check.
- Praise my effort and method, not my ability.
- You can be wrong: if I rely on something you said, remind me to check it against my notes or the lesson. If you don't know, say so.
- Use simple words.
- Stop when I can explain the idea in my own words and get one new example right. Then tell me to go back to the lesson "Growing patterns".

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.

What each step is for

• Figure 10: Committing to Figure 10 before seeing it exposes the most common shortcut: multiplying Figure 1 instead of looking at what each figure adds.
• grow the pattern: Seeing each new square in orange shows why the pattern adds 3, not 4: the new square shares a side.
• from pattern to rule: The amount added each time is the number in front of n; what every figure has before that is the number added on. The rule gives any figure.
• new patterns: One change at a time: a different amount added, a pattern of dots, a different starting group, and working backwards from a count to the figure.
• patterns and rules: Finding a rule from a table, using a rule for a far figure and working backwards, mixed; the review is solving an equation from the Equations unit.
• design a pattern: Designing a pattern to hit a target count reverses the lesson: from the rule's numbers to the picture.

Where it leads

Next: tables from rules (any rule, not only patterns), then plotting the table as points, which lie on a straight line. The two numbers of a pattern rule become m and c later in the course.

Where it leads in class: any repeated build (fencing, tiling, seating), and spreadsheet formulas that fill down.