Understanding Functions Deeply

Date: Tuesday, January 20, 2026

Duration: 90 minutes

Class Size: 27 students

Subject: Mathematics (Algebra 1)

Curriculum Alignment: International Baccalaureate Middle Years Programme (MYP) - Mathematics, Criterion A: Knowing and Understanding

IB Learning Objective:
Students will demonstrate understanding of the concept of a function, recognizing whether relations are functions and representing them in multiple formats.
IB Key Concept: Relationships
MYP Mathematics Objectives Covered:

• Explain patterns and relationships in a range of contexts.
• Use appropriate forms of representation to present information accurately.

CCSS Standard Reference: A.FIF.1 - Understand that a function is a rule that assigns to each input exactly one output.

---

Learning Objectives

By the end of this session, students will be able to:

• Define a function in mathematical terms and verbally describe it.
• Distinguish between functions and non-functions based on input-output relationships.
• Use input-output tables, mapping diagrams, and graphs to represent functions.
• Justify whether a given relation represents a function.

---

Materials Needed

• Whiteboard and markers
• Projector/screen for visual aids
• Printed input-output tables for bell ringer and practice (27 sets)
• Individual student notebooks
• Mapping diagram handouts
• Graph paper or graphing apps (optional)
• Exit ticket slips

---

Lesson Breakdown

1. Engaging Starter: Real-world Connections with Input-Output Machines (15 minutes)

• Objective: Spark curiosity by relating functions to everyday machines (vending machines, soda machines).
• Ask: “How does a vending machine work? How do we know exactly what we get when we press a button?”
• Show images or quick video clips of machines taking input and delivering output.
• Distribute a short bell ringer worksheet of 5 input-output tables/examples. Students identify whether each is a function by analyzing if each input gives exactly one output.
• Whole-class brief discussion of answers to create a shared understanding and activate prior knowledge.

2. Direct Instruction (I Do) — Defining Functions (20 minutes)

• Introduce the formal mathematical definition: “A function is a relation where each input has exactly one output.”
• Use a whiteboard to model examples:
  • Input-output tables showing functions and non-functions.
  • Mapping diagrams emphasizing uniqueness of outputs per input.
  • Simple graphs (vertical line test introduction).
• Show non-example relations that fail the function test.
• Connect with IB philosophy by highlighting the importance of precise language to describe relationships in math and real life.

3. Guided Practice (We Do) — Classifying Relations Together (20 minutes)

• Work through 3-4 diverse sample relations as a class.
• Students use mapping diagrams and input-output tables to decide: Function or Not?
• Prompt students to articulate reasoning using IB MYP mathematical communication skills.
• Teacher circulates, asking probing questions and encouraging mathematical dialogue in pairs or small groups.

4. Independent Practice (You Do) — Classify Relations Individually (20 minutes)

• Distribute 4 new input-output tables, mapping diagrams, and small graph sets for students to classify independently.
• Challenge: For at least one relation, ask students to explain why it either is or isn’t a function in written form.
• Provide differentiated support: struggling students receive a teacher-led small group session focused on mapping diagrams and reasoning.

5. Closing - Exit Ticket (15 minutes)

• Prompt: “Explain why this relation is or is not a function.” (Use a relation shown on the board or provided on a slip.)
• Collect exit tickets to assess individual understanding and identify misconceptions for future lessons.
• Wrap up with a brief reflection linking the day’s learning objective to real-world applications, fostering learner profile attributes like Inquirers and Thinkers.

---

Differentiation & Inclusion

• Use varied representations (tables, diagrams, graphs) to cater to different learning styles.
• Small group focused on visual mapping diagrams for ELL or students needing concrete examples.
• Encourage peer collaboration and discussion to enhance language skills and conceptual understanding.
• Allow extended time for independent practice if needed.

---

Assessment & Feedback

• Formative assessment via bell ringer responses, guided practice participation, and exit tickets.
• Immediate verbal feedback during practice to scaffold understanding.
• Teacher observations to refine small group support needs.

---

Reflection for Teachers

• Did students effectively articulate the difference between functions and non-functions?
• Which representations resonated most with the class?
• What misconceptions emerged, and how will these inform follow-up lessons?

---

By carefully blending authentic real-world examples with structured mathematical reasoning and multiple representations, this lesson embodies the IB’s holistic approach to student-centered conceptual understanding and inquiry development.