Variables In Action

Overview

In this 60-minute Year 9 lesson, students develop algebraic language and use variables to represent unknown or changing quantities. They connect input-output patterns to algebraic notation, then substitute values into simple expressions. The lesson builds on prior work with number patterns, operations and order of operations.

Learning intentions

• WALT describe the roles of variables, constants, coefficients, terms and expressions.
• WALT represent a linear pattern using algebraic notation.
• WALT substitute values into simple algebraic expressions accurately.
• WALT explain the meaning of an expression in context.

Success criteria

• I can identify the variable, coefficient, constant and terms in an expression.
• I can write an expression for an input-output rule or linear pattern.
• I can substitute a value for a variable and calculate the result.
• I can show clear working and check whether my answer is sensible.

Curriculum links

• Te Mātaiaho Mathematics and Statistics, Phase 4 Algebra: using variables and algebraic notation to represent linear patterns.
• Te Mātaiaho Mathematics and Statistics, Phase 4 Algebra: substituting values into expressions and interpreting results.
• Mathematical and statistical thinking: noticing structure, making generalisations, explaining reasoning and communicating mathematically.
• Mathsteasers: using challenging pattern questions to deepen understanding and encourage higher-order thinking.

Prior knowledge

Students should be familiar with the four operations, order of operations, positive and negative integers, equality, and recognising or continuing simple number patterns. Quickly check that students understand multiplication written as repeated addition, such as (3n=n+n+n).

Lesson structure (60 minutes)

1. 0–8 min · Hook: input-output machines. Open with the input-output machine hook and display the machine rule “multiply by 3, then add 2”; students complete a table for inputs 1, 2, 5 and 10, then discuss how to describe the rule for any input. Ask: “What could we use instead of writing every possible input?” Teacher listens for “a letter” or “a variable” and introduces the idea that a variable can represent an unknown or changing quantity.

2. 8–20 min · Explicit modelling: algebra language. Use the vocabulary and modelling slides to model the rule as (3n+2), where (n) is the input. Define variable, coefficient, constant, term and expression, using colour coding: in (4x+7), (x) is the variable, 4 is the coefficient, 7 is the constant, and (4x) and 7 are the terms. Teacher models substitution: if (x=5), then (4x+7=4(5)+7=27); students annotate the same example and explain why brackets are used.

3. 20–30 min · Guided practice: think, pair, share. Display the guided examples in the guided practice slides. Work through (2a+6) when (a=4), and (5p-3) when (p=2), pausing before each step. Students answer mini-whiteboard questions: identify the variable and coefficient in (7m+1); identify the terms in (3y-8); calculate (2b+5) when (b=6). Address errors immediately by asking, “What does the letter stand for?” and “Which operation must happen first?”

4. 30–40 min · Pair activity: pattern detectives. In pairs, students use the pattern and substitution worksheet to complete input-output tables and match each table to an expression. They explain their rule to another pair using the sentence frame: “The rule is ___ because each output is found by ___.” Circulate and question: “How can you test your rule with a second input?” and “Would your expression still work for zero?”

5. 40–53 min · Independent practice: differentiated pathways. Students complete the appropriate section of the differentiated practice worksheet. Support students complete identifying parts of expressions and substitution with positive whole numbers. Core students write rules for linear patterns and substitute positive and negative values. Extension students compare two expressions, such as (3n+4) and (4n+1), and determine when they give the same output, explaining their reasoning. Teacher conferences with a small support group and uses concrete input-output examples before returning to symbols.

6. 53–60 min · Plenary and exit ticket. Revisit the summary and misconception-check slides in the plenary and exit-ticket slides. Students complete the exit ticket independently: a) In (6x+9), name the variable, coefficient, constant and terms. b) Evaluate (3n-4) when (n=7). c) Write an expression for “five more than twice (t)”. Answers: a) variable (x), coefficient 6, constant 9, terms (6x) and 9; b) (17); c) (2t+5). Collect responses to plan the next lesson.

Resources

• the complete algebra introduction and practice deck
• the pattern and substitution worksheet
• Mini-whiteboards, pens and erasers
• Board or interactive display
• Coloured pens for annotating expressions
• Calculators for selected students who need checking support

Assessment

• Use mini-whiteboard responses to check vocabulary, substitution and order of operations during guided practice.
• Question pairs while circulating: “What does the variable represent?”, “How do you know your rule works?” and “Where might an error have occurred?”
• Sort exit tickets into secure, developing and beginning understanding. Look particularly for incorrect coefficient identification, omitted multiplication and substitution errors.

Differentiation

• Support: provide a vocabulary bank, colour-coded examples, partially completed tables and the sentence frame “When ___ equals ___, I replace ___ with ___.” Read instructions aloud and allow students to rehearse answers verbally.
• EAL learners: pair each term with a consistent visual or gesture, explicitly teach “unknown”, “changing”, “multiply” and “evaluate”, and provide worked examples with minimal written language.
• SEN and processing support: use enlarged print, uncluttered worksheet sections, extra processing time and one question at a time. Permit calculators after the substitution method has been shown.
• Extension: invite students to create an input-output machine with two possible expressions, then prove whether the expressions always, sometimes or never produce the same output.