Substitution Stations

Overview

In this third lesson of Algebra Detectives: Solve, Simplify, Share, students apply their understanding of variables and algebraic expressions by replacing variables with known values. Through a relay, research stations and paired presentations, they practise accurate calculation, explain errors and communicate mathematical reasoning.

Learning intentions

• WALT substitute known values into algebraic expressions and formulae.
• WALT use brackets correctly when substituting positive and negative integers.
• WALT calculate accurately and communicate an answer with an appropriate label.
• WALT explain and evaluate mathematical thinking collaboratively.

Success criteria

• I can replace a variable with its given value.
• I can use brackets to show my substitution clearly.
• I can calculate the result accurately, including with negative integers.
• I can label my answer and explain how I know it is reasonable.

Curriculum links

• Mathematics and Statistics — Mathsteasers: higher-order thinking questions that deepen understanding for advanced learners.
• Mathematics and Statistics — Mathsteasers / Alignment: challenge tasks connected to classroom mathematics content.
• Mathematical reasoning, problem-solving, representation and communication within the refreshed New Zealand Curriculum.
• Mana values: Manaaki through supportive collaboration, Niwha through perseverance and Atamai through thoughtful reasoning.

Lesson structure (45 minutes)

1. 0–5 min · Hook and retrieval. Display the question, “Which is safer: replacing a letter silently or showing every step?” using the opening hook slide; students individually simplify a familiar expression such as (3n+2) when (n=4), then compare answers with a partner. Teacher listens for misconceptions about replacing a variable and explains that today’s focus is substitution, not solving for an unknown.

2. 5–12 min · Model and substitution relay. Use the worked-example slides to model (4a-3) when (a=5), then (2p+q) when (p=-2) and (q=7), emphasising brackets, order of operations and labels. In three groups of three or pairs with one rotating checker, students complete a short relay on the substitution practice worksheet, passing the pencil only after the next person checks the substitution line.

3. 12–27 min · Error-analysis research stations. Set up three stations from the error-analysis station tasks and display the instructions with the station instruction slides; each task includes a worked solution containing an error, such as omitting brackets around (-3), misusing order of operations or giving an unlabeled answer. Students investigate, annotate the error, correct the solution and record a short explanation; each group assigns roles of reader, recorder and reporter, rotating at each station.

4. 27–36 min · Compare and justify. Present the extension prompt on the comparison challenge slide and support students to substitute values into two expressions, such as (2(x+3)) and (2x+6). All students test at least two values and explain whether the expressions give equal results; advanced learners test several positive, zero and negative values and look for a reason rather than relying only on examples.

5. 36–42 min · Pair presentations. Show the presentation checklist on the sharing and feedback slides; pairs choose one corrected station example and prepare a clear explanation using “I replaced…”, “I used brackets because…” and “My answer is labelled…”. Pairs present to another pair or the class, while listeners identify one accurate step and ask one clarifying question.

6. 42–45 min · Exit check and reflection. Display the final prompt using the plenary slide and distribute the function machine cards for a quick individual input-and-output check, selecting a rule and an integer input. Students show the substitution line, calculate the output, label it appropriately and complete the sentence, “The step I must remember next time is…”.

Resources

• the Substitution Stations slide deck
• the substitution and error-analysis worksheet
• the Function Machine Cards
• Mini-whiteboards and pens
• Highlighters or coloured pencils
• Station signs and timer
• Calculators for students who require them

Assessment

• During the relay, check whether students replace every variable and use brackets for negative values; pause for a quick re-model if several students make the same error.
• At stations, assess annotations for accurate calculation, identification of the error and a mathematical explanation, not simply a corrected answer.
• Use the exit check to identify students needing further support with substitution, order of operations or integer calculations before the next lesson.

Differentiation

• Colour-code variables and their known values throughout the model, worksheet and station tasks. Provide the frame: “The expression is ____. I substitute ____ for ____, so it becomes ____. Therefore, ____.”
• Begin with positive integer substitutions, then progress to zero and negative integers. Offer a multiplication grid, number line, calculator or partially completed first step where appropriate.
• Allow students to explain orally to a partner, record a digital explanation or present with a scribe. Read station instructions aloud and provide uncluttered copies for students who need reduced visual load.
• Extend confident learners by comparing two expressions for several values, including negative values, and explaining when their results are equal using algebraic reasoning rather than only a table.

Extension

• Ask students to create two different-looking expressions that always produce the same result, then prove their claim by substituting values and simplifying.
• Invite advanced learners to design an error-analysis card for another pair, including a tempting mistake, the correct solution and a clue rather than the answer.