Solving Real Models

Overview

In the final lesson of the unit, students complete a modelling investigation involving a non-linear relationship. Working in pairs, they use digital tools to analyse data, select and justify an exponential or quadratic model, solve an equation, test its reasonableness and communicate a conclusion in context.

Learning intentions

Students will:

• analyse bivariate data using a digital tool and identify a non-linear pattern
• formulate an equation that models the relationship
• solve the model for a meaningful unknown value
• evaluate the model and communicate evidence-based conclusions

Success criteria

• I can describe the association in a scatterplot, including its direction, strength and shape.
• I can choose and justify an appropriate non-linear model.
• I can use technology to solve or estimate a value from my model.
• I can explain whether my prediction is reasonable and identify a limitation.

Curriculum links

• Statistics — construct scatterplots and comment on the association between two numerical variables in terms of strength, direction and linearity.
• Algebra — experiment with functions and relations using digital tools, make and test conjectures, and generalise patterns.
• Algebra — use mathematical modelling to choose, apply, interpret, evaluate and modify linear, quadratic or exponential models.
• Algebra — recognise connections between algebraic and graphical representations of exponential relations and solve related equations.

Lesson structure (45 minutes)

1. 0–5 min · Hook and retrieval. Teacher displays a scatterplot and asks, “Can a model make a reliable prediction outside the data range?” using the opening prediction question and scatterplot; students independently identify the pattern, recall association language and share one possible limitation.

2. 5–10 min · Investigation briefing. Teacher introduces the scenario and success criteria through the investigation scenario, workflow and success criteria and distributes the modelling investigation workbook. Students read the data context, identify the explanatory and response variables, and predict whether an exponential or quadratic model may be suitable.

3. 10–25 min · Analyse and model. Teacher supports pairs to enter the data into a spreadsheet or graphing tool, create a scatterplot, and test suitable regression models; students complete the first sections of the modelling investigation workbook, recording axis labels and units, describing direction, strength and shape, comparing model fit, and writing a proposed equation. Teacher checks that students distinguish association from causation and do not choose a model solely because it passes through one point.

4. 25–35 min · Solve and test. Teacher prompts students with model-comparison prompts and prediction questions: “What does your equation predict when the response variable is ___?” and “Is that prediction interpolation or extrapolation?” Students use the digital tool to solve an equation or estimate an input for a stated target, substitute or check the result, and record appropriate rounding and units.

5. 35–41 min · Communicate findings. Teacher models a concise evidence-based conclusion using the conclusion scaffold: claim, evidence, interpretation and limitation. Students complete the final worksheet response, stating the model selected, the prediction, why it is reasonable, and one assumption, limitation or possible confounding factor.

6. 41–45 min · Share and exit check. Teacher invites two pairs to compare different models or predictions and displays the plenary comparison and exit questions. Students complete the worksheet exit response: “My model is useful for ___ because ___; it may be unreliable when ___,” then submit their work.

Resources

• the complete modelling investigation slide deck
• the modelling investigation workbook
• Spreadsheet software or graphing technology with scatterplot and regression features
• Student laptops or tablets, one per pair
• Projector or interactive display
• Teacher-prepared bivariate dataset embedded in the worksheet
• Calculator, if required
• Digital timer

Assessment

• During data entry and model selection, question pairs about variable roles, scatterplot shape, model choice and whether predictions are interpolation or extrapolation.
• Check worksheet evidence: correctly constructed scatterplot, labelled variables and units, justified model, equation, solved value, reasonableness check and limitation.
• Use the final response to assess whether students can interpret a prediction in context rather than reporting an unexplained numerical answer.

Differentiation

• Support students with a partially completed data table, a step-by-step digital-tool guide, a choice of two likely model types and sentence starters: “The association is…”, “I selected this model because…”, and “The prediction is reasonable because…”.
• Pair students strategically and provide teacher conferencing for interpreting regression output, rearranging equations or reading coordinates from a graph.
• Provide calculator and speech-to-text access where appropriate; read the scenario aloud and clarify technical vocabulary without simplifying the mathematical demand.
• Extend confident students by asking them to compare two plausible models, calculate or compare residuals, investigate a different target value, or explain how an additional data point could change the model and conclusion.