Overview

Grade Level: Algebra 1 (Ages 14–15)
Duration: 90 Minutes
Class Size: 30 Students
Curriculum: International Baccalaureate Middle Years Programme (MYP) Mathematics – Criterion A & B Alignment

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Learning Objectives

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

• Recognize and write quadratic functions in standard form ( y = ax^2 + bx + c ).
• Identify parameters (a), (b), and (c), and explain their influence on the graph shape, axis of symmetry, and intercepts.
• Determine the direction and width of parabolas by analyzing the coefficient (a).
• Use Desmos graphing technology to explore quadratic behaviors dynamically.
• Formulate hypotheses regarding the effects of varying (a) and connect quadratic graphs to real-world contexts like projectile motion.
• Communicate mathematical ideas confidently and reflect collaboratively on learning progress.

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IB Curriculum Alignment

MYP Mathematics Criterion A: Knowing and Understanding

• Demonstrate knowledge and understanding of algebraic forms and graphing functions using appropriate terminology.
• Explore representations and interpret contexts confidently.

MYP Mathematics Criterion B: Investigating Patterns

• Formulate and test hypotheses about how parameters influence quadratic function graphs.
• Use technology (Desmos) as a mathematical tool for exploration and confirmation of patterns.

Approaches to Learning (ATL) Skills

• Communication: Explain mathematical reasoning and share findings.
• Critical Thinking: Analyze parameters’ effects and formulate predictions.
• Use of Technology: Implement Desmos to visually test hypotheses about quadratic graphs.

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Materials Needed

• Whiteboard and markers
• Projector and computer for Desmos demonstration
• Student devices with internet access for Desmos
• Printed worksheet (20 problems total, including vertex & axis of symmetry calculations)
• Timers or clocks
• Student notebooks and pencils

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Lesson Breakdown

Time	Activity	Details & Teacher Language	IB Focus & Notes
5 min	Bell Ringer & Engagement	"You might have noticed some functions with (x^2). These are quadratic functions with unique shapes. Today we’ll uncover their secrets."	Activating prior knowledge, clear communication (Criterion A)
10 min	Introduction to Quadratic Form	Write (y = ax^2 + bx + c). "This is the standard form of a quadratic function. Let's identify these letters' roles."	Clarity in terminology and expression (Criterion A)
15 min	Parameters Deep Dive	Discuss: (a) controls parabola direction and width; (b) controls axis of symmetry; (c) is the y-intercept. Example: (y = 2x^2 + 3x + 1). Q&A: "What are a, b, c here?"	Conceptual understanding, formative questioning (Criterion A)
15 min	Graph Shape Exploration	Sketch parabola examples on the board: (a > 0) opens up, (a < 0) opens down. Explore how increasing (a) narrows the curve.	Linking algebra to geometry, promoting conceptual link (Criterion A)
20 min	Technology Integration: Desmos Demo	Students open Desmos. Teacher inputs: \
(y = x^2), (y = -x^2), (y = 2x^2), (y = -3x^2). \
Ask: "How does (a) alter graph direction/width?"	Use of technology; critical thinking & hypothesis testing (Criterion B)
20 min	Student Worksheet Activity	Students (individually or pairs): \

• Identify parameters \
• Predict parabola directions \
• Graph using Desmos \
• Match graphs to equations \
• Reflect on real-world application (e.g., basketball trajectory) | Application and communication (Criterion A & B) | | 10 min| Class Discussion & Reflect | Facilitate sharing: \ "How does (a) impact parabola shape? Why is this useful?" \ "Where in real life do we see parabolas?" | Metacognitive reflection; consolidating conceptual understanding | | 5 min | Closing & Preview | "Today, one number changed the entire graph. Next class, we’ll deepen our understanding by predicting vertex positions and graph shifts without graphing tools." | Encourages curiosity, inquiry, and IB learner agency |

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Detailed Activities Description

1. Bell Ringer & Engagement (5 min)

• Briefly stimulate students’ curiosity using familiar examples or questions about quadratic expressions.
• Informal discussion to recall prior algebraic concepts and curve shapes.

2. Introduction to Quadratic Standard Form (10 min)

• Display and write the quadratic equation (y = ax^2 + bx + c).
• Define each parameter with clear, accessible language.
• Example breakdown on board with student participation.

3. Parameters Deep Dive (15 min)

• Elaborate on the role of each parameter:
  • (a): Determines "opening" direction (up/down) and width (narrow/wide).
  • (b): Influences the axis of symmetry (\displaystyle x = -\frac{b}{2a}).
  • (c): The graph’s y-intercept.
• Use a sample quadratic, emphasize identifying (a), (b), and (c).
• Encourage students to verbalize parameter meanings, promoting ATL communication.

4. Graph Shape Exploration (15 min)

• Sketch varying parabolas on the board (positive (a), negative (a), larger and smaller (|a|)).
• Compare and contrast their shapes, stressing real-life visualisation connections.

5. Technology Integration: Desmos Graphing Demo (20 min)

• Guide students to open Desmos.
• Enter various quadratic equations, observing real-time graph transformations.
• Ask prompting questions like:
  • "What happens when (a) is negative vs. positive?"
  • "How does doubling (a) affect the curve?"
• Use Desmos to verify predictions, deepening comprehension through visual feedback.

6. Student Worksheet Activity (20 min)

• Students complete the 20-item worksheet in pairs or individually:
  • Identify parameters (a), (b), and (c).
  • Predict parabola directions before graphing.
  • Graph quadratic functions on Desmos.
  • Match graphs with equations.
  • Answer reflection questions about real-life applications.
• Teacher circulates, scaffolding when needed and encouraging critical thinking.

7. Class Discussion & Sharing (10 min)

• Invite several students to explain how parameter (a) affects graph shape.
• Discuss real-life parabolic trajectories (e.g., basketball throw).
• Facilitate connections to practical uses and reinforce mathematical terminology.

8. Closing & Next Steps (5 min)

• Summarize the day's main takeaways.
• Preview the next lesson focusing on the vertex and axis of symmetry formulas, preparing students for vertex form exploration.
• Motivate students’ curiosity and anticipation for future learning.

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Differentiation Strategies

Learner Type	Support Strategy
Diverse Learners	Stepwise scaffolded worksheet sections with guided examples and visual diagrams.
Advanced Learners	Extension challenge: Investigate combined effects of (b) and (c) on vertex and intercepts.
Visual Learners	Emphasize sketches, Desmos visuals, and color-coded graphs during explanations.
Kinesthetic Learners	Use physical graph card manipulations or coordinate-graph plotting activities if available.

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Assessment

Formative (During Lesson)

• Q&A exchanges during parameter discussion to gauge understanding.
• Review of student worksheet answers throughout activity.
• Observation of engagement and application during Desmos activity.

Summative (Next Lesson)

• Quiz on identifying graph direction and shape from standard form without graphing tools.
• Evaluate ability to calculate the axis of symmetry and vertex position from (a), (b), and (c).

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Teacher Reflection Questions

• Did students make meaningful connections between the algebraic parameters and the visual features of parabolas?
• How effectively did the Desmos platform support prediction and investigation of graph changes?
• Were students able to explain graphical behaviors without Desmos by the lesson’s end?
• What adaptations can enhance inquiry and differentiation in upcoming lessons?

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Appendix: Desmos-Ready Student Worksheet (20 Problems with Vertex & AOS Focus)

Name: ___________________________

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Part 1: Identify Parameters (6 problems)

For each quadratic: Write values of (a), (b), and (c). Also calculate the axis of symmetry (AOS) and vertex coordinates.

1. (y = 2x^2 + 5x + 1)
2. (y = -x^2 + 4x - 3)
3. (y = 3x^2 - 2x + 6)
4. (y = -4x^2 + 8x - 7)
5. (y = 0.5x^2 + 3x + 9)
6. (y = -2x^2 - 6x + 2)

Use (x = -\frac{b}{2a}) to find AOS. Then plug back into the equation to find vertex ( (x, y) ).

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Part 2: Predict Parabola Direction & Vertex Nature (6 problems)

Circle whether it opens upward or downward, then predict if vertex is min or max.

Equation	Opens Up or Down?	Vertex is Min or Max?	Explain your reasoning
(y = x^2 - 4x + 3)	Up / Down	Min / Max
(y = -3x^2 + x + 7)	Up / Down	Min / Max
(y = 5x^2 - 10x + 2)	Up / Down	Min / Max
(y = -0.5x^2 + 6x -1)	Up / Down	Min / Max
(y = x^2 + 2x + 1)	Up / Down	Min / Max
(y = -x^2 - 8x + 15)	Up / Down	Min / Max

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Part 3: Graph on Desmos (4 problems)

Graph these functions on Desmos, record the vertex and AOS observed, then answer:

• Which open upward?

• Which open downward?

• How does (a) influence the width?

• (y = 2x^2 - 4x + 3)

• (y = -x^2 + 6x - 5)

• (y = 0.5x^2 + x + 2)

• (y = -3x^2 - 2x + 4)

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Part 4: Matching Graphs to Equations (2 problems)

Match each description with the corresponding equation:

• A. Wide parabola opening upward with vertex at (1, -1)
• B. Narrow parabola opening downward with vertex at (2, 5)

1. (y = 0.5x^2 - x + 2) → _______
2. (y = -4x^2 + 16x - 11) → _______

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Part 5: Real-Life Quadratics Reflection (2 questions)

1. Explain briefly why tossing a ball forms a parabolic trajectory.

2. Give one other real-world context where a quadratic function may appear.

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Teacher's Quick Reference Keys

For Part 1 (Example Problem)

• (y=2x^2 + 5x + 1)
  • (a=2), (b=5), (c=1)
  • AOS: (x = -\frac{5}{2*2} = -\frac{5}{4} = -1.25)
  • Vertex: plug (x=-1.25) into (y), calculate vertex (y)

Similar method for other problems.

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Real-Life Connection Example Answer

• The ball's height changes with time influenced by gravity — the quadratic term models this motion as a parabola rather than a straight line.

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Final Teacher Notes

This plan marries inquiry-led, conceptual understanding with real technology use aligned rigorously with the IB MYP criteria and ATL skills framework. It encourages students to become confident mathematical communicators and critical thinkers in uncovering the role of each parameter in quadratic functions. Through exploration, collaboration, and reflection, this unit will lay a strong foundation for more advanced quadratic concepts.

Feel free to customize materials based on your classroom's needs, and consider integrating Desmos Activity Builder to instantly capture student responses for a next-level interactive experience. You’re poised to impress your students and colleagues with this IB-aligned, tech-forward math exploration! 💛