Tables, Graphs, and Rules

Overview

In Lesson 12 of 26, students connect tables, graphs, diagrams, and rules that describe the same linear relationship. Building on prior work with patterns and ordered pairs, students focus on how each representation communicates change and use evidence to justify equivalent matches.

Learning intentions

Students will:

• interpret a linear relationship in a table, graph, diagram, and written rule;
• identify how the relationship changes as the input increases;
• construct and analyse a table and graph from a rule;
• explain and defend a representation match using mathematical evidence.

Success criteria

• I can match three representations of the same relationship.
• I can describe the change using words such as “increase,” “constant,” and “starting value.”
• I can plot points from a table and identify the pattern in the graph.
• I can use values, points, or a rule to explain why a match is correct.

Curriculum links

• Patterns and Relations — oral and written patterns and their corresponding relations.
• Patterns and Relations — constructing tables of values, graphing relations, and analysing graphs to solve problems.
• Algebraic representations — distinguishing and connecting expressions, equations, variables, and values.
• Mathematical processes — communication, connections, problem solving, reasoning, and visualization.

Lesson structure (50 minutes)

1. 0–5 min · Attention hook and retrieval. Teacher displays the opening comparison and retrieval prompt: “How could one change be shown in four different ways?” and briefly revisits input, output, ordered pair, and rule. Students silently predict what a table, graph, diagram, and rule might show, then share one idea with a partner.

2. 5–12 min · Explicit teaching. Teacher models the relationship “Start with 2 and add 3 each time” using a table, ordered pairs, plotted points, a simple arrow diagram, and the rule (y=3x+2); teacher highlights the constant change and starting value on the four-representation modelling slides. Students complete guided notes and use the frames: “The input is ___,” “The output changes by ___,” and “The starting value is ___.”

3. 12–20 min · Visual representation detective. Teacher displays two or three representations at a time and asks students to identify evidence before naming a match. Students use a dyslexia-friendly comparison chart in the guided comparison and matching worksheet to record: what I notice, what changes, and evidence that connects the representations. Use the sentence frames: “I know these match because ___,” and “The table value ___ appears as the point ___.”

4. 20–30 min · Mathology matching activity. Teacher gives the four students a small set of Mathology representation cards and assigns rotating roles: reader, matcher, checker, and explainer. Students match tables, graphs, diagrams, and rules, then pause after each match to verify at least two corresponding values or points. Teacher conferences with each student and records whether the student identifies a match independently, with prompting, or not yet.

5. 30–39 min · Guided practice and Mathletics. Teacher works through two examples on the guided-practice and error-analysis slides, including one example where a graph has the correct starting value but the wrong rate of change. Students complete the first questions on the guided practice and independent questions and then use Mathletics for targeted practice, checking each answer by substituting one input value or comparing a plotted point.

6. 39–47 min · Independent evidence checkpoint. Teacher asks each student to complete the checkpoint on the three-representation checkpoint: match three equivalent representations and defend one match using a value, ordered pair, starting value, or constant change. Students work independently; teacher provides only the agreed scaffold and asks, “What evidence proves your match?”

7. 47–50 min · Share and exit reflection. Teacher displays the closing reflection prompt and invites two students to explain different forms of evidence. Students complete: “A table communicates ___; a graph communicates ___; a rule communicates ___,” and rate confidence from 1–4.

Resources

• the Tables, Graphs, and Rules slide deck
• the guided comparison and matching worksheet
• Mathology representation-matching cards or activity
• Teacher-selected short linear-relationships video from the provided search results
• Whiteboard, markers, and mini-whiteboards
• Grid paper, pencils, rulers, and coloured highlighters
• Student devices and Mathletics access
• Guided notes with large, uncluttered examples

Assessment

• Knowledge and Understanding: Observe whether students identify input, output, ordered pairs, starting value, and constant change. Progress is shown by matching representations and explaining at least one connection accurately.
• Mental Math: During retrieval and matching, check whether students calculate successive outputs and recognize constant differences efficiently. Note independent, prompted, or not-yet performance.
• Problem Solving: Use the evidence checkpoint to assess whether students select relevant information, match three equivalent representations, and justify one match. Record evidence in the PowerSchool descriptors as secure, developing, or requiring further instruction.
• Collect the worksheet and exit reflection to plan the next small-group lesson.

Differentiation

• Provide colour coding across all representations: input in blue, output in green, starting value circled, and constant change boxed. Keep one example visible throughout the lesson.
• For students needing support or with a math disability, reduce the number of cards, provide partially completed tables and graphs, allow a calculator for checking, and use one-to-one prompting. Accept oral explanations or pointing before requiring written responses.
• Use dyslexia-friendly formatting: clear sans-serif font, 14-point minimum text, generous spacing, short lines, uncluttered charts, bold key words, graph paper with strong axes, and optional read-aloud instructions. Do not rely on colour alone; add symbols or labels.
• For distractible students, seat the group close to the teaching display, use brief timed tasks, assign active roles, and provide a visible “notice → match → prove” routine. Students ready for challenge can create a new rule and produce its table, graph, and diagram for a partner to match.