Graphs Tell Stories

Overview

This is lesson 17 of 19 in the Year 9 Maths 2026 Plan. Students represent contextual relationships using tables, ordered pairs, graphs and rules, then interpret rate, gradient and intercept. The lesson builds on prior work with coordinates, algebraic rules and proportional relationships.

Learning intentions

• WALT represent relationships using tables, graphs and rules.
• WALT distinguish between proportional and non-proportional relationships.
• WALT interpret gradient or rate and intercept in context.
• WALT communicate what a graph shows, including its limitations.

Success criteria

• I can plot ordered pairs accurately and label axes with suitable scales and units.
• I can identify patterns and connect a table, graph and rule.
• I can explain what a rate or intercept means in a real situation.
• I can critique a graph and describe one limitation or possible misleading feature.

Curriculum links

• Mathematics and Statistics — Mathsteasers: higher-order thinking questions that challenge advanced learners and deepen understanding.
• Mathsteasers / Alignment: challenge tasks connect with mathematical topics and existing textbook learning.
• Additional resources for advanced learners: reasoning, explanation and comparison tasks provide appropriate challenge.
• Mathematical communication and problem solving are developed through representations, critique, justification and contextual interpretation.

Lesson structure (60 minutes)

1. 0–7 min · Hook and retrieval. Display a distance–time graph and ask, “Which journey is faster, and how can you tell?” Open with the graph-story hook and retrieval questions. Students discuss with a partner, then use mini-whiteboards to plot two ordered pairs and identify the horizontal and vertical variables. Address common errors such as reversing coordinates or omitting units.

2. 7–17 min · Model the connections. Use the slides to model a taxi fare relationship: a fixed starting charge of $4 plus $2 per kilometre. Build a table, plot selected ordered pairs, describe the rate, identify the intercept and write the rule (C=2d+4). Students annotate their own copy of the relationships and graphs worksheet and explain what each number means in context. Emphasise that a graph is a model of a situation and may only be valid within a particular domain.

3. 17–32 min · Represent a context. Place students in pairs and distribute the worksheet. Each pair chooses one dataset: taxi fares, household water use or distance travelled over time. Students complete the table, select a sensible scale, label axes and units, plot points by hand, and describe the pattern. They then decide whether the relationship is proportional or non-proportional, supporting their decision with evidence from the table, graph and rule. Teacher conferences with pairs, checking scale, plotting accuracy and interpretation.

4. 32–42 min · Digital comparison. Students enter the same data into a spreadsheet or graphing tool and compare the digital graph with their hand-drawn version. Open the digital graphing instructions and comparison prompts. Students identify any differences caused by scale, joined points, missing values or an inappropriate graph type. Ask: “Would it be reasonable to extend this graph indefinitely?” Students record one limitation and one improvement.

5. 42–53 min · Graph critique and challenge. Display two versions of a graph showing the same data, one with a misleading vertical scale. Students complete the critique questions on the worksheet, explaining how the visual impression changes. Advanced learners investigate a piecewise context, such as a delivery charge with different rates after a threshold, or compare two relationships and determine when one becomes cheaper or faster. Invite students to justify conclusions rather than relying on visual appearance alone.

6. 53–60 min · Share and exit check. Invite two pairs to explain how they interpreted rate and intercept in different contexts. Students complete the final written interpretation on the worksheet: “The rate is ___, which means ___. The intercept is ___, which means ___. The graph is limited because ___.” Finish with the plenary questions and exit response. Collect responses to identify students needing support in the next lesson.

Resources

• the relationships, graphs and rates slide deck
• the relationships and graphs worksheet
• Graph paper
• Rulers and pencils
• Mini-whiteboards and pens
• Calculators
• Spreadsheet or graphing software
• Contextual datasets for taxi fares, water use and distance–time
• Projector or interactive display

Assessment

• Use mini-whiteboards to check ordered pairs, axis conventions and recognition of proportional relationships.
• During conferencing, ask students to explain the meaning of rate and intercept without accepting context-free answers.
• Assess the graph critique and final written interpretation for accuracy, use of units, connection to context and recognition of limitations.

Differentiation

• Provide guided axes, partially completed tables, coordinate templates and a worked example on the worksheet for students who need additional structure.
• Offer table-to-graph matching, a reduced dataset and partner talk before writing. Read instructions aloud, use clear sans-serif text, generous spacing and uncluttered graph layouts for dyslexic learners.
• Allow students to use a spreadsheet or graphing tool after first attempting key points by hand. Provide sentence starters: “The rate shows…”, “The intercept represents…”, and “This graph cannot show…”.
• Extend advanced learners with piecewise relationships, competing rates or a critique of deliberately misleading graph scales. Require algebraic and graphical justification.

Extension

• Compare two taxi companies with different starting charges and rates. Determine the distance at which their costs are equal and explain the result graphically.
• Create a context where the relationship changes rate. Represent it with a table, graph and piecewise rule, then explain the domain of each section.