Rules Into Expressions

Overview

In lesson 21 of 26, students connect familiar linear tile patterns and input-output tables to algebraic expressions. They focus on the variable as a position or input, then write, test, and use a rule to find a later term.

Learning intentions

Students will:

• Connect a visual linear pattern, table, written rule, and algebraic expression.
• Understand that a variable can represent the position number or input.
• Write expressions such as (2n+1) to describe linear patterns.
• Test a rule using a known term and use it to predict a distant term.
• Explain their reasoning using mathematical language, models, and notation.

Success criteria

• I can describe how a pattern changes from term to term.
• I can write an expression using a variable for the term position or input.
• I can substitute a known value to check whether my rule works.
• I can use my rule to find a later term and explain my strategy.

Curriculum links

• Patterns and Relations — oral and written patterns and their corresponding relations.
• Patterns and Relations — constructing and interpreting tables of values and graphs from a relation.
• Variables and Equations — evaluating an expression when the variable has a given value.
• Mathematical processes — communication, connections, mental mathematics and estimation, problem solving, reasoning, and visualization.

Lesson structure (50 minutes)

1. 0–5 min · Hook and retrieval. Display a growing tile pattern with 3, 5, and 7 tiles using the pattern hook slide. Ask, “How many tiles would term 10 have? How could we find term 100 without drawing it?” Students silently notice, sketch, and share one observation; the teacher records different strategies without confirming a rule immediately.

2. 5–12 min · Concrete connection. Revisit the pattern with tiles or a teacher-drawn visual anchor. Open the visual pattern and notice-and-wonder slides and model a table with columns for position, number of tiles, and change. Students build or sketch the first three terms, identify the constant increase, and describe the pattern orally using the sentence frame: “The number of tiles is ___ more than the previous term.”

3. 12–22 min · Explicit teaching. Use the expression modelling slides and the guided notes and pattern-rule worksheet to model how a pattern can be decomposed into “groups of the position number” and a constant part. For example, if term (n) has two tiles per position plus one extra tile, write (2n+1). Emphasise that (n) represents the input or term position, not a fixed number. Students complete guided notes, match each part of the expression to the visual, and substitute (n=1,2,3) to verify the first terms.

4. 22–28 min · Supported example and video connection. Show a short, pre-screened age-appropriate clip about linear pattern rules and algebraic expressions through the video viewing and pause prompts; do not rely on students searching independently. Pause to ask, “What does the variable represent?” and “Where can we see the constant?” Students record one connection between the clip and the tile pattern, then solve one example together using the rule scaffold: “Rule: ___ × (n) + ; Check: when (n=), ___.”

5. 28–42 min · Independent practice and conferencing. Students complete selected questions on the independent practice and evidence checkpoint, then practise an appropriate set of pattern-rule questions in Mathletics. The teacher conferences individually, prioritising students on modified programmes and the student with a math disability. Each student must write a rule, test it with one known term, and use it to find a later term; early finishers find a distant term such as term fifty and justify why their expression is more efficient than drawing.

6. 42–50 min · Share, assess, and exit. Display the discussion and exit prompt slides. Students compare two rules for the same pattern, deciding whether they are equivalent by testing inputs. On the worksheet or a small response slip, each student completes: “A pattern has terms 4, 7, 10, 13. Write a rule using (n), check it for term 2, and find term 20.” Students rate confidence from 1–3 and identify one support that helped them.

Resources

• the complete pattern-rules slide deck
• the guided notes and pattern-rule worksheet
• Mathology patterns-to-algebra resource
• Interlocking cubes, tiles, or counters
• Whiteboard and markers
• Student devices and Mathletics access
• Pre-screened linear-pattern video clip
• Rule scaffold and input-output table displayed or printed

Assessment

• Knowledge and Understanding: Observe whether students identify the variable as position/input, connect each expression part to the visual, and write an accurate rule. Record progress using descriptors such as beginning, developing, or demonstrated.
• Mental Math: Listen for efficient use of constant differences, multiplication facts, and substitution to estimate or calculate later terms. Note whether students can evaluate simple expressions with reduced prompting.
• Problem Solving: Use the evidence checkpoint to document whether each student writes a rule, verifies it with a known term, and applies it to a later term while explaining the reasoning.
• Collect the final response and confidence rating for the PowerSchool evidence record. A demonstrated response includes a correct expression, a valid substitution check, and a correct distant term.

Differentiation

• Provide colour-coded visual anchors: one colour for the variable term, one for the constant, and arrows showing the constant difference. Keep the scaffold visible throughout independent work.
• For students needing support, use smaller inputs, partially completed tables, concrete tiles, multiplication charts, and sentence frames. Read directions aloud, offer dyslexia-friendly sans-serif text, generous spacing, uncluttered pages, and allow oral explanation or scribing.
• For modified programmes, assess recognition of the pattern, completion of a table, and a supported rule such as “add ___ each time” before moving to (an+b). Reduce the number of questions while retaining the evidence checkpoint.
• Seat distractible students close to instruction, give them a defined role such as pattern builder or checker, use brief private prompts, and provide a visible “first–next–check” task sequence.
• Extend ready students by asking them to create a linear pattern with a chosen distant term, write two equivalent rules where possible, or explain why a non-linear pattern cannot be represented by a constant-rate expression.