Substitution in Formulae

Overview

In lesson 3 of 10 in Algebra for Kapa Haka, students use substitution to evaluate expressions and formulae. They apply this to fictional rehearsal timings and resource quantities, building accuracy with brackets, order of operations, units and sensible answers.

Learning intentions

• WALT replace a variable with a given value.
• WALT write the substituted expression before calculating.
• WALT use brackets and follow the order of operations.
• WALT check whether an answer is sensible and state its units.

Success criteria

• I can replace a variable with a value.
• I can use brackets correctly when substituting.
• I can follow the order of operations.
• I can check my answer, explain whether it is sensible, and include the correct unit.

Curriculum links

• Pāngarau: Tau me te Taurangi — using algebraic expressions and formulae to represent and solve familiar problems.
• Te Reo Rangatira: whakarongo, kōrero, pānui and tuhituhi through mathematical discussion, explanation and written working.
• Marau ā-Kura: contexts can be localised through the kura’s own aspirations, language and tikanga. Today’s kapa haka contexts are fictional and make no claims about actual practice.
• Cross-curricular links: technology, event planning and numeracy for life.

Lesson structure (60 minutes)

1. 0–8 min · Warm-up: substitution cards. Teacher displays short formulae such as (T=3n+5), gives a value for (n), and models matching a variable card to its value card; use the warm-up substitution prompts and teacher-prepared formula cards. Students work with a partner to substitute mentally or with counters, then show answers on answer boards. Ask: “What does the letter represent?” and “What unit might the answer have?”

2. 8–18 min · Explicit teaching. Teacher introduces the routine: identify each variable, replace it with a bracketed value, write the complete substituted expression, calculate using order of operations, then check the unit and reasonableness. Open the worked-example teaching slides and demonstrate (T=4n+6), (n=5): (T=4(5)+6=26) minutes. Contrast this with an incorrect answer caused by calculating in the wrong order. Students copy one worked example and complete two quick examples, holding up answer boards after each step. Teacher checks that students write the substituted expression before calculating.

3. 18–35 min · Paired circuit. Teacher places numbered stations around the room and distributes the rehearsal timing and resource quantity circuit; each pair uses a timer, counters and a calculator only after showing working. Students rotate through fictional problems, including rehearsal timing and resource quantities, recording the variable, substituted expression, answer and unit at each station. Partner A reads and substitutes; Partner B calculates and checks; swap roles at the next station. Circulate and ask, “Why are brackets needed?”, “Which operation comes first?” and “Is your answer sensible?”

4. 35–47 min · Team problem design. Teacher forms teams of three or four and displays the problem-design instructions. Each team designs one fictional substitution problem using a formula, two or three values, a clear context and a unit. The problem must require brackets or more than one operation. Students write the formula, values, answer and a brief checking note on the worksheet. Remind teams that fictional timings and resources must not be presented as authentic kapa haka practice.

5. 47–55 min · Solve and critique. Teacher exchanges team problems and displays the peer-critique prompts. Each team solves another team’s problem, then checks whether the formula was copied correctly, the substituted expression is clear, operations were ordered correctly, and the unit and answer are sensible. Students give one specific positive comment and one improvement suggestion before returning the problem to its designers. Invite two teams to explain different methods.

6. 55–60 min · Exit ticket. Teacher gives the final prompt on the plenary and exit-ticket slide: “A fictional event uses (R=6g+4) resources, where (g=7). Write the substituted expression, calculate (R), state a suitable unit, and explain whether the answer is sensible.” Students complete the response independently and hand it in as evidence for the algebra skills portfolio.

Resources

• the complete substitution lesson deck
• the rehearsal timing and resource quantity circuit
• Teacher-prepared substitution and formula cards
• Answer boards and pens
• Counters or resource tokens
• Timers
• Calculators
• Station signs and pencils

Assessment

• Use circulated questioning to check variable identification, bracket use and order of operations.
• Use answer boards during modelling and the circuit to identify errors immediately; check units and estimates.
• Assess team-created problems, peer feedback and the exit ticket for the algebra skills portfolio.

Differentiation

• Support dyslexic learners with colour-coded variable/value matching, large symbols, short sentences, bold operation signs, an accessible font, worked-example strips and teacher or audio reading. Permit oral explanation or oral calculation, with a partner recording the mathematics.
• Support learners with ADHD through numbered stations, movement, visible timings, defined partner roles and a brief reset break. Offer a quiet workspace and reduce the number of stations if needed.
• Provide a formula bank, partially completed examples and a unit word bank for students needing additional scaffolding. Pre-teach: substitute, formula, input, output, order of operations, unit and estimate.
• Extend advanced learners by requiring two variables to be substituted and asking them to compare how changing one variable affects the result. They can justify the change using calculation and mathematical language.