Variables: Maths Mysteries

Overview

Lesson 1 of 4 in Equation Detectives. Students investigate variables as unknowns and changing quantities using mystery cards, balance models and familiar contexts such as scores, costs and growing patterns. They finish by creating several possible values for a variable and explaining their reasoning.

Learning intentions

• WALT recognise a variable as a symbol representing an unknown or changing quantity.
• WALT describe what a variable represents in a real-life situation.
• WALT write simple expressions using a variable.
• WALT explain mathematical thinking using words, numbers, diagrams and symbols.

Success criteria

• I can identify the variable in a situation.
• I can explain what the variable represents and what values it could have.
• I can write an expression such as (x+5), (3x) or (20-x).
• I can justify more than one possible value for a variable when the information allows it.

Curriculum links

• Algebraic thinking: using symbols, patterns and relationships to represent unknown and changing quantities.
• Mathematical representation: connecting everyday contexts, diagrams, tables and expressions.
• Mathematical reasoning and communication: explaining strategies, comparing ideas and justifying conclusions.
• Number knowledge: applying addition, subtraction and multiplication to simple expressions and contexts.

Lesson structure (60 minutes)

1. 0–5 minutes – Hook: the mystery number

Open with the mystery-number hook and display: “I am thinking of a number. It is more than 5, less than 20, and gives a score of 24 when multiplied by 3 and added to 6.” Ask students what information is known, what is unknown and whether there is only one answer. Avoid naming the unknown immediately; invite several guesses and explanations.

2. 5–13 minutes – Build the idea of a variable

Use the variable introduction slides to introduce a variable as a letter or symbol that can stand for an unknown or a quantity that changes. Model examples: (s) for a score, (c) for cost, and (n) for the number of steps in a pattern. Contrast (x+5), meaning “five more than a number”, with (3x), meaning “three times a number”.

Students briefly discuss with a partner: “What could (p) represent?” Take examples such as points, people or pages. Emphasise that the meaning depends on the context.

3. 13–28 minutes – Mystery-card investigation

In pairs, distribute the variable mystery cards and recording sheet. Students cut or separate the cards if required, then solve and record each mystery. Examples include:

• “A game score is (s+10). What does (s) represent?”
• “A movie ticket costs (c) dollars. Four tickets cost (4c).”
• “A pattern has (n) tiles in one row and 2 extra tiles. Write the expression.”
• “You have $20 and spend (x) dollars. Write an expression for the money left.”

Partners must underline the variable, write what it represents, and give a sensible value where possible. Circulate and question: “Could the variable be zero?” “What values would make sense in this context?” “Is the variable unknown, changing, or both?”

4. 28–40 minutes – Balance model and expression matching

Give each pair an algebra balance mat and use counters, linking cubes or classroom objects as informal representations. Model a balance with one hidden or unknown amount and discuss how both sides must have the same value. Connect the model to statements such as (x+3=8), without requiring formal equation-solving procedures.

On the mat, students represent and discuss simple examples such as “one mystery amount plus 4 is equal to 9”. They then match context cards from the worksheet to expressions. Focus on meaning rather than speed or formal notation.

5. 40–53 minutes – Collaborative “What could the variable be?” challenge

Show the challenge instructions on the collaborative challenge slides. Groups of four receive one prompt from the worksheet, such as: “A class earns (p) points each round and has 30 points after several rounds. What could (p) be?” or “A growing pattern has (n+4) shapes. What could (n) be?”

Groups create at least three possible values, write an expression and prepare a brief explanation. They must identify any limits, such as whole numbers, a maximum score or a realistic cost. Groups share one solution; classmates listen for whether the proposed value fits the context.

6. 53–60 minutes – Plenary and assessment

Return to the reflection and exit prompt. Students respond independently on the final section of the variable mystery cards and recording sheet:

• In the situation “A bus has (b) passengers and 6 more people get on”, what does (b) represent?
• Write an expression for the number of passengers now.
• Give one possible value for (b) and explain why it is sensible.

Invite two students to share different valid values. Preview that the next lesson will investigate how equations help detectives find an unknown.

Resources

• the complete Variables: Maths Mysteries slide deck
• the variable mystery cards and recording sheet
• algebra balance mat
• Counters, linking cubes or small classroom objects
• Scissors, pencils and highlighters
• Whiteboard and markers
• Mini-whiteboards, if available
• Group table signs or roles: reader, recorder, checker and reporter

Assessment

• During pair work, listen for accurate identification of the variable and whether students can describe its contextual meaning.
• Check mystery-card recordings for correct expressions, sensible values and explanations.
• Use the independent plenary response to identify students ready for equations and those needing further work with context-to-symbol translation.

Differentiation

• Support: provide a word bank with “unknown”, “changing”, “number”, “cost”, “score” and “number of”; allow students to use a box or blank before introducing a letter.
• Support: pair students strategically, read cards aloud, use counters and offer sentence frames such as “The variable represents ___ because ___.”
• Extension: ask students to write a new mystery where the variable could have at least three values, then state a condition that would make only one value possible.
• EAL/SEN: use visual examples, colour-code the variable and known numbers, reduce the number of cards, and accept oral explanations or labelled diagrams before written expressions.