Rules for Two-Variable Patterns

Overview

In this second lesson of the three-part unit Patterns, Groups, and Sharing, students use cubes or tiles to build and describe growing spatial patterns. They record the linked variables—step number and number of objects—in a table, then connect the spatial pattern to a number pattern and express its rule.

Learning intentions

• WALT create and continue sequential patterns with two variables.
• WALT record linked values in a table.
• WALT describe a pattern rule using words, numbers or pictures.
• WALT explain how the rule works for a later step.

Success criteria

• I can continue a spatial or number pattern accurately.
• I can record the step number and number of objects in a two-variable table.
• I can describe the rule using words, numbers or pictures.
• I can use my rule to find a later term.

Curriculum links

• Mathematics and Statistics — recognising, continuing and describing sequential patterns.
• Mathematics and Statistics — representing relationships between two linked variables in a table.
• Mathematics and Statistics — using mathematical reasoning to explain and justify a rule.
• Mathsteasers — applying higher-order thinking to challenge understanding and deepen mathematical reasoning.

Lesson structure (30 minutes)

1. 0–3 min · Engage and connect. Open with the pattern hook and learning intention and display three growing arrangements: 1 cube, 3 cubes, 5 cubes. Ask, “What do you notice? What might come next?” Students silently predict, then share with a partner. Briefly connect to the previous lesson’s work with patterns, groups and sharing.

2. 3–8 min · Build the pattern. Use the growing pattern examples to model building the first three steps with cubes or tiles. Give pairs cubes or tiles and ask them to build steps 1–4, saying what changes each time. Students identify the constant increase and use the words step, next and increase when explaining their thinking.

3. 8–15 min · Record and formulate the rule. Display the colour-coded two-variable table and complete it together: Step 1, 2, 3, 4; Objects 1, 3, 5, 7. Use arrows to show “add 2 each time”. Ask, “Which variable changes? How does it change?” and “How many objects would there be at step 5?” Students complete the table, connect the objects to the number pattern 1, 3, 5, 7, and state the rule orally, pictorially or numerically.

4. 15–25 min · Independent follow-up. Distribute the two-variable pattern practice sheet. Students continue several spatial and number patterns, complete two-variable tables, and create one pattern for a classmate to solve. They must include the rule for their own pattern. Encourage students to use cubes or tiles first, then record their thinking. Circulate and ask, “How do you know?” and “Will your rule work for the next step?”

5. 25–28 min · Partner solve and explain. Students swap their created patterns with a partner and solve them without seeing the rule. Partners compare answers and explain whether the rule works. Invite one pair to share a pattern and justification using the partner-check prompts.

6. 28–30 min · Plenary and assessment. Return to the final reflection prompt. Students complete an oral or written exit response: “The rule is ___ because each step ___.” Ask two students to explain how they would find a later term, such as step 8, without building every step.

Resources

• the pattern teaching and discussion deck
• the two-variable pattern practice sheet
• Linking cubes or square tiles
• Mini-whiteboards and pens
• Coloured pencils
• Board or chart paper
• Prepared table showing step number and number of objects

Assessment

• Observe pair building and questioning during the concrete activity: can students identify the constant change and continue the pattern?
• Check tables and explanations for correctly linked step and object values, not just correct answers.
• Use the final response to assess whether students can state a rule and justify how it works for a later term.

Differentiation

• Support learners with concrete repeating or growing patterns, partially completed tables, colour-coded columns and arrows showing the change between terms.
• For learners working at an earlier level, keep addition or subtraction rules within 20 and provide sentence frames: “The pattern starts at ___ and ___ each time.”
• Offer cubes or tiles, enlarged tables and oral recording for students who need reduced writing demands. Pre-teach and display step, next, increase, decrease, rule and term; allow students to explain in their strongest language before sharing in English.
• Extend advanced learners by asking them to create a pattern with a changing amount, such as adding 1, then 2, then 3, or to represent the same pattern in two ways. They must justify how their rule works for a later term and explain why a partner’s rule is or is not valid.