Linear Modelling

Overview

Students practise building and using linear models to describe a real situation and to make predictions. They interpret slope and intercept in context and justify whether a model is appropriate.

Learning intentions

• Students will be able to create a linear equation from a table or two points.
• Students will interpret slope and intercept as meaningful quantities in a context.
• Students will use a linear model to make and check predictions.
• Students will communicate reasoning clearly when choosing and using a model.

Success criteria

• I can write a linear equation in the form (y = mx + b) using two data points.
• I can explain what the slope and intercept mean for the situation.
• I can predict a value from the model and state the appropriate units.
• I can check whether the prediction makes sense using residuals or a quick comparison to data.

Curriculum links

• Connect linear relations to real-world situations and interpret their key features.
• Use patterns, tables, and graphs to determine equations for linear relationships.
• Apply mathematical reasoning to justify model choices and prediction accuracy.
• Communicate findings using appropriate mathematical language and representations.

Lesson structure (15 minutes)

1. 1 minute — Warm-up prompt Show a quick scenario on the board: “A rideshare charges a base fee plus a cost per kilometre.” Ask: “What would ‘base fee’ and ‘cost per km’ correspond to on a graph?”

2. 3 minutes — Mini direct instruction (slope/intercept) Briefly connect the context to (y = mx + b): base fee as (b) (the starting value) and cost per km as (m) (rate of change). Remind students that slope shows how much (y) changes for a 1-unit increase in (x).

3. 5 minutes — Modelling from two points Provide a small table (or two points) for the scenario, such as distance vs total cost. Model the steps for:

• finding the slope (m) from the change in (y) over change in (x)
• finding (b) by substituting one point into (y = mx + b) Emphasise checking that the equation reproduces the given data.

4. 3 minutes — Guided practice (predict and interpret) Students work in pairs to predict the cost at a new distance (e.g., 8 km). They must include:

• the substitution into the equation
• the numerical answer with units
• a one-sentence interpretation of what that result means in context

5. 2 minutes — Quick check for reasonableness Ask students to compare the prediction to the trend. If possible, include one extra data point and have them say whether the model seems reasonable (for example, “Is it on the same general line?” or “Is the error small?”).

6. 1 minute — Exit ticket Each student writes one equation for their model and one sentence explaining slope or intercept in context (not just the calculation).

Resources

• Projector/board with the scenario prompt and a small data table (2–3 points)
• Student copies of the table and the prediction question
• Graph paper or a simple blank coordinate grid (optional)
• Calculators allowed (paper steps still expected)
• Timers visible to students for the timed steps
• Markers/whiteboard for teacher worked example
• Worked example of slope/intercept from the first two points (teacher copy only)

Assessment

• Formative: teacher circulates during pair work to check whether students correctly identify slope/intercept and set up (y = mx + b).
• Formative: exit ticket checks for correct equation and a context-based explanation of either slope or intercept.
• Observation: listen for use of mathematical language (rate of change, starting value, prediction, units).

Differentiation

• Support: provide a sentence frame for interpretations, e.g., “The slope means that for every 1 unit increase in ___, ___ increases by ___.” Offer a partially completed slope calculation for students who need it.
• Support: allow students to use a calculator but require them to show the substitution and explain the meaning of at least one parameter.
• Extension: ask advanced students to compare two possible linear models (e.g., using different pairs of points) and discuss which fits better and why.
• EAL/SEN: use clear visuals (table-to-graph prompt) and pre-teach key terms as everyday words (rate, starting cost) while keeping the mathematical definitions tied to the model.