Exploring Functions and Relations

Overview

Students build on prior lessons in algebra by distinguishing between functions and relations using tables, mappings, and graphs. They practise using the vertical line test to decide whether a relation is a function, linking to how quadratic graphs and equations are solved later in the unit.

Learning intentions

WALT: Differentiate between functions and relations.

• Students will be able to define a function and a relation using clear mathematical language.
• Students will be able to identify whether a relation is a function using the vertical line test.
• Students will be able to explain their reasoning using examples from tables, mappings, and graphs.
• Students will begin connecting relation/function ideas to features of graphs for future work with quadratics.

Success criteria

• I can define a function as a rule where each input has exactly one output.
• I can define a relation as any set of input-output pairs (inputs may have one or more outputs).
• I can use the vertical line test to decide if a graph represents a function.
• I can justify my decision with evidence from the table, mapping, or graph.

Curriculum links

• Algebra — Students identify and graph functions and solve quadratic-related problems using digital tools as appropriate.
• Algebra — Students connect graphical representations to algebraic reasoning, using features of graphs (turning point, intercepts, and ranges).
• Mathematical modelling foundations — interpreting solutions in terms of context (prepared for later AC9M9A05 work).

Lesson structure (45 minutes)

1. 0–5 min · Warm-up recall Teacher displays three input-output statements (e.g. “each x gives one y” vs “an x can give multiple y”) and asks students to vote thumbs-up/thumbs-down. Students record which ones are likely functions and why in one sentence (use the starter: “This is a function because…”).

2. 5–12 min · Teach: function vs relation Teacher explicitly defines relation and function with a mapping diagram: one input → one output (function) vs one input → multiple outputs (relation). Students complete a mini “Sort & explain” on scrap paper: 4 diagrams/tables into Function or Relation, then write one justification.

3. 12–24 min · Guided practice: vertical line test Teacher shows four hand-drawn graphs (some are functions, some are not) and models the vertical line test step-by-step (slide a finger imagined vertical line; check whether it meets the graph once). Students work in pairs on a guided worksheet:

• Part A: For each graph, students answer “Function or Relation?” and circle the evidence line(s).
• Part B: Check with a given table/mapping and match consistent examples.

4. 24–32 min · Dyslexia-friendly check-in + mini assessment Teacher reads instructions aloud and highlights key words (“input”, “output”, “exactly one”, “vertical line”, “more than once”). Students complete 6 rapid questions (no diagrams overload):

• 2 definitions/application items (short answer)
• 4 vertical line test items (tick function/relation + one-word reason: “one” or “more than one”).

5. 32–40 min · Class share: common misconceptions Teacher selects 2 student answers (anonymous) and prompts discussion: “What went wrong?” and “What evidence would fix it?” Students revise one answer from their check-in using a correction code: “Add detail / Use vertical line / Fix definition”.

6. 40–45 min · Exit ticket Teacher gives one new mapping and one graph. Students submit:

• mapping: Function or Relation (with one-sentence justification)
• graph: Function or Relation (vertical line test verdict only)

Resources

• Guided worksheet (visual sorting, mapping diagrams, simple graphs)
• Vertical line test card (laminated strip with “vertical line” prompt)
• Coloured pencils (evidence marking on graphs)
• Whiteboard slides or projector showing 4 model graphs
• Dyslexia-friendly sentence starters strip (see below)
• Timer for steps and short check-in
• Optional: digital tool for graphing (if available) to confirm one example

Assessment

• Formative during guided practice: teacher circulates, listens for correct “exactly one output” language and vertical line evidence.
• Quick check-in (Part 5) marked with a simple rubric: correct classification + reasoning level (definition vs evidence).
• Exit ticket used to plan next lesson grouping (students ready for deeper function/graph connections).

Differentiation

• Support (visual + language):
• Provide sentence starters: “This is a function because each input has…”, “The vertical line hits the graph…”.
• Use simpler graphs first (single-valued, symmetrical examples), then harder ones (multi-valued).
• Offer a guided worksheet version with fewer items per page and larger spacing.
• Read instructions aloud; allow oral responses recorded by teacher or onto a small response sheet.
• Support (Green School concept: clarity and reduce cognitive load):
• Use consistent colour coding: blue for inputs, red for outputs.
• Keep one vertical line per question; model checking once before asking students to check again.
• Extension (advanced learners):
• In guided practice, add a “two-step justification” question: “Classify as function or relation using both the graph and the table. Are they consistent?”
• Extension activity (as requested, advanced learners):
• Advanced learners create a function table from a realistic scenario: Example prompts (choose one): “Parking cost over time” or “Ticket price with a fixed fee and additional cost per hour.” Students must:
• produce a table (inputs and outputs),
• state the rule in words,
• include one non-example that breaks the function rule (e.g. two outputs for the same input) and explain why it is a relation.

Dyslexia-friendly reading options

• Provide “read-aloud” version of worksheet directions (teacher or audio device).
• Use high-contrast print (dark font on light background), large headings, and short lines.
• Allow audio recording for students to capture their justification sentence.
• Offer two-level vocabulary supports on a small card:
• Function: “each input → one output”
• Relation: “each input → one or more outputs”
• Vertical line test: “line crosses graph once = function” / “crosses more than once = not a function”