Real-World Function Uses

Overview

In this lesson (lesson 8 of 9) you will apply different function types to model and solve real-life problems, focusing on finance-style scenarios that use compound growth, as well as physics- and biology-style relationships. Students consolidate skills from earlier lessons by interpreting models, selecting an appropriate function form, and using it to make decisions.

Learning intentions

Students will:

• model real-life situations using appropriate functions (linear, exponential/compound, and step/periodic where relevant)
• identify key features of a function from a context (growth/decay, rate, domain restrictions, and meaning of parameters)
• solve problems by translating between context and algebra/graph/table representations
• check reasonableness of answers using units, scale, and comparison to the scenario

Success criteria

Students can:

• choose a suitable function type for a scenario and justify the choice using evidence from the context
• set up a function rule and use it to calculate an outcome (value, time, or parameter)
• interpret results in context (what the number means for the situation)
• verify their solution with a quick check (substitute back, compare trend, or test limiting cases)

Curriculum links

• General Mathematics/Applied problem solving: solving practical problems involving compound interest/investments and interpreting totals and interest (where finance contexts require compound growth)
• Essential Mathematics: using technology to model real-world relationships and interpreting predictions from an equation (linking function models to predictions)
• Mathematics Methods: recognising and using piece-wise functions as combinations of sub-functions with restricted domains (for real contexts with rules that change by condition)

Lesson structure (60 minutes)

1. 0–5 min · Warm-up prompt. Teacher displays a short “real-life” scenario card: “A phone plan charges a base fee plus a usage amount, then applies a different rate after a threshold.” Students write: which parts look constant, which change, and whether the rule changes by condition.
2. 5–15 min · Mini-lesson: selecting the function. Teacher explains a quick decision framework:

• If change is proportional to current value → exponential/compound-style function
• If change is steady over time → linear function
• If the rule changes at a threshold → piece-wise function with restricted domains Students complete a worked example matching three scenario sketches to function types, using one-sentence justifications.

3. 15–28 min · Finance modelling task (compound growth). Teacher provides a compound interest/investment context (no spreadsheet needed, but calculator algebra allowed). Example: “You invest $1,500 at 5% p.a., compounded quarterly for 2 years. Find the final value and total interest earned.” Teacher guides students to:

• identify principal, interest rate per year, number of compounding periods, and compounding period rate
• write the compound amount formula and compute values Students complete the problem, then state total interest earned and interpret it in words.

4. 28–40 min · Physics-style function (relationship and interpretation). Teacher gives a physics-like scenario requiring linear or exponential modelling (teacher chooses one appropriate to class needs). Example options:

• Linear: “Distance-time data is approximately straight-line: v ≈ constant. Predict distance after a given time.”
• Exponential/decay: “A substance reduces by a fixed percentage each hour. Predict remaining amount after a given time.” Students select the function type, form a rule from given information (gradient/rate or percentage per step), and calculate one prediction. They record assumptions (e.g., constant rate, constant percentage).

5. 40–52 min · Biology-style function (piece-wise or constrained growth). Teacher introduces a biology context with conditions, such as:

• “Population growth follows one rule up to a carrying-capacity point, then a different rule applies,” or
• “Survival probability is constant above a threshold, but decreases below it,” or
• “A treatment effect applies only after a start time; before that, the effect is zero.” Teacher explicitly connects this to piece-wise functions as restricted domains. Students create a piece-wise function rule, state the domain for each part, and answer a single question (e.g., value at a boundary, value at a later time, or comparison of two times).

6. 52–57 min · Consolidation check. Teacher asks three rapid questions on the board:

• “What feature of the context tells you the function changes?”
• “How do you know the parameter has the correct meaning/units?”
• “What would make your answer unreasonable?” Students answer in short notes; teacher circulates and corrects misconceptions.

7. 57–60 min · Exit ticket. Students complete one concise item: “Choose one of the three scenarios. Write the function rule you used and one sentence interpreting the result in context.”

Resources

• Scenario cards (finance, physics, biology with thresholds/conditions)
• Calculator and writing paper
• Graph paper or a blank table-template for values
• Printed function rules sheet (compound growth rule, generic linear rule, piece-wise definition reminders)
• Device with graphing capability if available (optional; use for checking trends rather than replacing calculation)

Assessment

• Formative: teacher observation during scenario matching and rule-writing (checking correct function choice and justification)
• Formative: review of calculations for compound growth (correct identification of number of compounding periods and rate per period)
• Exit ticket: function rule + context interpretation (assesses both modelling and meaning-making)

Differentiation

• Support:
• Provide sentence starters: “This looks exponential because…”, “The function changes at…”, “The domain for this rule is…”
• Provide a partially completed function template for each task (students fill missing pieces)
• Allow use of a function-rule summary sheet during the finance task
• Challenge/extension within the task:
• Ask students to vary one parameter (e.g., double time or change compounding frequency conceptually) and state the expected effect on the outcome
• Add an extra check question: “Which part of your model controls growth fastest?”
• EAL/SEN considerations:
• Emphasise vocabulary: “rate per period”, “compounded”, “threshold”, “domain”
• Use a visual timeline for time-based rules and a “before/after threshold” diagram for piece-wise models