Pattern Rule Detectives

Overview

Students investigate a growing pattern based on rows of kūmara plants in a school māra. They identify the variable, describe how the pattern changes, write and evaluate a simple algebraic rule, and connect tables, rules and graphs. The lesson builds on students’ prior work with number patterns, ordered pairs and coordinate grids.

Learning intentions

• We are learning to identify and use a variable.
• We are learning to write and evaluate a simple algebraic rule.
• We are learning to represent a linear relationship using a table, rule and graph.
• We are learning to explain how we know our rule works.

Success criteria

• I can explain what the variable represents in a pattern.
• I can complete a table and write a rule such as (y=2n+1).
• I can substitute a value into a rule correctly.
• I can plot ordered pairs and describe the relationship shown.

Curriculum links

• Pāngarau Taumata 4 — variables in algebra.
• Pāngarau Taumata 4 — show linear relationships through rules and graphs.
• Pāngarau Taumata 6 — relate graphs, tables and equations to linear relationships, providing appropriate challenge for students ready to extend.
• Students practise explaining reasoning, working collaboratively and making connections to a local māra context.

Key vocabulary

Variable, input, output, rule, term, table, linear relationship, ordered pair, coordinate, graph, constant difference, substitute, evaluate.

Lesson structure (50 minutes)

1. 0–5 min · Hook: mystery pattern. Open with the pattern hook and mystery question and display a growing arrangement of plant symbols: 1 plant in row 1, 3 in row 2, 5 in row 3, and 7 in row 4. Students silently predict the next two terms, then use mini-whiteboards to show their answers and explain the change they notice.

2. 5–15 min · Explicit teaching and worked example. Use the worked-example slides to connect the pattern to a table: position (n): 1, 2, 3, 4; number of plants (p): 1, 3, 5, 7. Model that (n) is the input variable, (p) is the output, and the rule is (p=2n-1). Substitute (n=5): (p=2(5)-1=9). Students repeat the reasoning with (p=2n+1) for a second example, checking the first three values. Emphasise that the rule must work for every term, not only the next one.

3. 15–25 min · Pair investigation. Distribute the growing-pattern investigation sheet. In pairs, students complete tables for three growing patterns, identify the constant difference, write a rule using (n), and calculate a later term such as term 10 or term 20. Partners take turns as “recorder” and “checker”, using the prompt: “I know this rule works because…”. Circulate and ask, “What does your variable represent?” and “How could you test your rule?”

4. 25–34 min · Move and match. Place the cards from the Sequence Rule Cards around the room. In groups of four or five, students rotate to two suitable cards, identify the pattern type, and explain whether the relationship is linear. Groups record one example and stand beside the card they think best matches a constant-difference pattern. Invite two groups to justify their choices; correct the misconception that every pattern has a linear rule.

5. 34–44 min · Table, rule and graph challenge. Return to the table-to-graph activity slides. Groups choose the pattern (p=3n+2) or (p=4n-1), complete a table for (n=0) to (5), plot the ordered pairs on the worksheet grid, and write a sentence describing the graph. Pause twice for mini-whiteboard checks: “What is the output when (n=4)?” and “Which coordinate is plotted first?” Students compare graphs with another group and explain how the constant difference appears in the table and graph.

6. 44–50 min · Plenary and exit ticket. Use the discussion and exit-ticket slides to revisit the learning intentions. Students complete the final worksheet question independently: “A pattern has outputs 4, 7, 10, 13. Write a rule using (n), find term 10, and explain what the 3 means.” Invite one student to explain a solution and one student to offer a different checking method.

Resources

• the complete pattern investigation slide deck
• the growing-pattern investigation sheet
• the Sequence Rule Cards
• Mini-whiteboards, pens and erasers
• Pencils, rulers and coloured pens
• Large coordinate-grid display or whiteboard
• Optional counters or small plant-shaped counters for modelling

Assessment

• Listen for accurate use of variable, input, output and rule during pair explanations; note students who confuse the term number with the output.
• Check tables, substitutions and plotted ordered pairs during the investigation. Ask students to verify rules using more than one term.
• Exit ticket answers: outputs increase by 3, so the rule is (p=3n+1); term 10 is (31). The 3 represents the constant increase between consecutive terms. Accept an equivalent explanation supported by substitution.

Differentiation

• Support: provide a partially completed table, a multiplication chart, colour-code input and output, and offer the sentence frames “The variable represents…” and “My rule works because…”.
• Support: allow students to build the first four pattern terms with counters before recording numbers; pair students strategically and read instructions aloud.
• EAL learners: pre-teach vocabulary with gestures and examples, keep (n) and output colour-coded, and permit labelled diagrams or oral explanations before written responses.
• Extension: students create a local māra pattern with a rule of the form (y=an+b), provide four terms and a graph, then challenge another group to identify the rule and justify it.