Substitution and Formulae

Overview

In this 60-minute lesson, students apply substitution to expressions and formulae, including positive and negative values, while maintaining order of operations and appropriate units. This is lesson 14 of 19 in the Year 9 Maths 2026 Plan and builds on students’ understanding of algebraic notation, operations and measurement formulae.

Learning intentions

• WALT substitute positive and negative values into algebraic expressions and formulae.
• WALT use brackets and order of operations to evaluate accurately.
• WALT interpret formula results in practical contexts and include suitable units.
• WALT use a spreadsheet to investigate how changing an input changes an output.

Success criteria

• I can replace each variable with its value, using brackets when needed.
• I can calculate an expression or formula accurately and explain my order of operations.
• I can give an answer with appropriate units and decide whether it should be exact or rounded.
• I can use a spreadsheet to test inputs and describe patterns in the results.

Curriculum links

• Mathematics and Statistics — Mathsteasers: higher-order thinking questions that challenge advanced learners and deepen understanding.
• Mathematics and Statistics — Mathsteasers / Alignment: applying relevant mathematical ideas in connected and increasingly challenging contexts.
• Mathematics and Statistics — Mathsteasers / Additional resources for advanced learners: reasoning, generalising and explaining mathematical relationships.
• Mathematical processes: students represent, calculate, communicate and reason with algebraic and measurement relationships.

Lesson structure (60 minutes)

1. 0–5 min · Hook and retrieval. Open with the hook and learning intentions slides showing the expressions (3 + 2^2) and ((3 + 2)^2), then ask, “Why are these not the same?” Students complete the two calculations on mini-whiteboards and briefly explain the order of operations they used.

2. 5–13 min · Model substitution. Use the substitution modelling slides to model a substitution ladder: copy the expression, replace every variable, bracket negative values, then calculate using order of operations. Demonstrate (3x+4) when (x=-2), and (2a^2-b) when (a=-3) and (b=5). Students copy one worked example and complete two teacher-led whiteboard checks. Question: “What might go wrong if the negative value is not bracketed?”

3. 13–23 min · Formula matching. Display the formula matching instructions and provide prepared formula cards and context cards for perimeter, area, speed and cost. In pairs, students match each formula to a suitable context and identify the meaning and unit of each variable. They then choose one match and explain why the formula is appropriate. Circulate, checking that students distinguish distance, time, speed, length, area and cost units.

4. 23–40 min · Guided practice and substitution ladder. Distribute the substitution and formulae practice worksheet. Students work through the substitution ladder: simple numerical expressions, positive and negative values, then contextual formulae involving perimeter, area, speed and cost. Pause at 30 minutes for a whole-class whiteboard check of one negative-value question and at 36 minutes to discuss exact versus rounded answers. Students self-mark the completed section using the displayed answers, correcting errors in a different colour and writing the cause of one error.

5. 40–52 min · Spreadsheet investigation. Open the spreadsheet investigation slides and demonstrate a table with input values, a formula and an output. Students use a spreadsheet in pairs to change one input at a time for a selected perimeter, area, speed or cost formula. They record at least three input-output pairs, describe what changes, and identify whether the result should be displayed exactly or rounded. Students compare their findings with another pair and explain one pattern using mathematical language.

6. 52–60 min · Plenary and exit check. Return to the plenary and exit-ticket slides. Students complete three final questions: substitute (x=-4) into (2x+7); evaluate a supplied measurement formula and include units; and explain why brackets are useful when substituting a negative number. Students hold up answers for a final check, then hand in the worksheet or photograph their exit responses.

Resources

• the substitution and formulae lesson deck
• the substitution and formulae practice worksheet
• Prepared formula cards and context cards
• Mini-whiteboards and pens
• Calculators
• Spreadsheet software or a browser-based spreadsheet
• Devices, one per pair
• Projector or interactive display
• Contextual problem examples involving perimeter, area, speed and cost

Assessment

• Use mini-whiteboard checks to identify errors with brackets, negative values and order of operations before students work independently.
• Question pairs during matching and spreadsheet work: “What does this variable represent?”, “What unit belongs in the answer?” and “Which input did you change?”
• Review the self-marked substitution ladder and final responses to identify students needing further practice with negative substitution, formula interpretation or rounding.

Differentiation

• Support students with a substitution frame: copy the formula, write each value beside its variable, substitute using brackets, calculate, then check the units. Begin with simple numerical values and allow calculators.
• Provide dyslexia-friendly copies of the worksheet: clear sans-serif font, generous spacing, short numbered instructions, highlighted variables and worked examples. Read questions aloud and allow students to explain answers orally.
• Pair students strategically for spreadsheet work and provide a partially completed table for students who need reduced working-memory demands. Pre-teach terms such as variable, substitute, exact, rounded and unit.
• Extend advanced learners with rearranged or multi-variable formulae, negative and decimal inputs, and constraints such as “find an input that gives an area between 40 and 50 square centimetres”. Ask them to justify a general pattern rather than only report calculations.