Patterns to Graphs

Overview

In lesson 6 of the 10-lesson unit Algebra for Kapa Haka, students investigate visual patterns based on hypothetical rehearsal formations and resource growth. They move from noticing change to describing arithmetic sequences, writing rules, organising values in tables, plotting coordinates and evaluating how well a model represents a situation.

Learning intentions

• WALT identify and describe patterns in visual and numerical sequences.
• WALT find the common difference and write a term-to-term or position-to-term rule.
• WALT organise a relationship in a table and represent it on a graph.
• WALT interpret a model and discuss its limitations.

Success criteria

• I can identify the common difference.
• I can write a term-to-term or position-to-term rule.
• I can complete a table and plot accurate points using labelled axes.
• I can interpret a graph and explain when a model may not be realistic.

Curriculum links

• Pāngarau — Tau me te Taurangi: use patterns, relationships, sequences, rules and algebraic representations.
• Pāngarau — Tau me te Taurangi: connect representations in words, tables, diagrams and graphs to reason about change.
• Pāngarau — Tauanga: organise, represent and interpret data, while considering the limits of a representation.
• Marau ā-Kura: use locally appropriate contexts and te reo ā-iwi where suitable; the formations in this lesson are abstract mathematical arrangements only.

Lesson structure (60 minutes)

1. 0–8 min · Pattern noticing. Open with the pattern-noticing hook and display three or four visual patterns involving hypothetical rehearsal formations or growing resources. Students silently record “what changes?” and “what stays the same?”, then share observations with a partner. Avoid presenting the patterns as actual iwi or hapū choreography.

2. 8–18 min · Model tables, sequences and axes. Use the worked sequence and graph examples to model how to list terms, identify the common difference, complete a table and write a term-to-term rule. Demonstrate a position-to-term rule for a simple linear pattern, then model plotting ordered pairs with the position on the horizontal axis and the number of people/resources on the vertical axis. Ask: “What does one step to the right mean?” and “What does the gradient or rate of change represent here?”

3. 18–35 min · Build a pattern. Place students in groups of four, with roles of builder, recorder, checker and reporter. Give each group counters or linking cubes and one scenario card, such as a hypothetical formation adding the same number of spaces each round or resources increasing by a fixed amount. Students build the first four or five stages, photograph or sketch them, and describe the growth orally and in writing. Circulate and question: “How do you know the difference is constant?” and “Could your rule predict stage 10?”

4. 35–48 min · Convert to tables and graphs. Distribute the patterns, tables and graphs worksheet. Groups record stage number and total, complete a table, identify the common difference, write a term-to-term or position-to-term rule, and plot the points. Students label both axes, include units or a clear context, and write two sentences interpreting the graph. Offer a spreadsheet option for groups ready to compare representations. Hold brief graph conferences with groups that need checking.

5. 48–55 min · Gallery interpretation. Display group graphs around the room. Students rotate in pairs, leaving one comment or question on two graphs: “I notice…”, “I predict…”, or “I wonder whether…”. Discuss why points may be discrete rather than joined, and when a straight-line model might become unrealistic—for example, limited people, space, time or resources.

6. 55–60 min · Exit ticket and share-back. Students complete the final section of the patterns, tables and graphs worksheet: identify the common difference in (4, 7, 10, 13), write a rule, plot or describe the next coordinate, and state one limitation of a model. Invite two students to share different valid rules or interpretations. Collect completed tables, graphs and written responses as assessment evidence.

Resources

• the complete pattern-to-graph slide deck
• the patterns, tables and graphs worksheet
• Counters and linking cubes
• Scenario cards for hypothetical formations and resource growth
• Graph paper, rulers and pencils
• Coloured pens or pencils
• Optional spreadsheet tool
• Visual timer and role cards

Assessment

• Listen for accurate identification of what changes and check students’ explanations of the common difference during group construction.
• Check tables, rules, axis labels, coordinates and graph points during circulation and graph conferences.
• Use the exit ticket to assess sequence reasoning, rule-writing, representation and understanding of model limitations. Record spoken explanations where written output does not show understanding.

Differentiation

• For dyslexic learners, provide the uncluttered worksheet and graph paper, pre-labelled axes, large accessible sans-serif font, colour coding for stages and totals, and visual instructions read aloud or provided as audio. Accept oral pattern descriptions before written rules.
• For students with ADHD, use manipulatives, a visible timer, movement during the gallery rotation and specific group roles. Break the task into short checkpoints: build, record, rule, graph and explain.
• Support students who need more structure with a partially completed table, a word bank and sentence starters: “The common difference is…”, “The rule adds…”, and “The graph shows…”. Pair students strategically and check one stage at a time.
• Extend confident learners by comparing a linear pattern with a non-linear growth scenario. Ask them to explain which model is more appropriate, whether points should be joined, and what practical constraints could make the model fail.

Extension

• Create a second pattern with changing differences, then compare its table and graph with the linear pattern.
• Explain how the model could be improved using a realistic constraint, such as a maximum group size or finite supply of resources.
• Connect the representation to science data, digital technologies or visual arts by explaining how the same relationship could be communicated in another format.