Solving Quadratic Equations

Overview

Students explore quadratic equations by connecting graphical and algebraic representations, building conceptual understanding alongside procedural fluency. This lesson aligns with the IB MYP Mathematics framework, emphasizing the key concepts of form, logic, and relationships.

Learning Intentions

• Students will understand the structure of a quadratic equation in standard form.
• Students will be able to solve quadratic equations using factoring and the quadratic formula.
• Students will connect the solutions of a quadratic equation to the x-intercepts of its parabola.
• Students will communicate mathematical reasoning clearly and logically.

Success Criteria

• I can identify the coefficients a, b, and c in a standard-form quadratic equation.
• I can solve a quadratic equation by factoring when possible.
• I can apply the quadratic formula to find exact solutions.
• I can interpret solutions graphically as the x-intercepts of a parabola.

Curriculum Links

• IB MYP Mathematics — number and algebra strand, focusing on patterns, relationships, and representational forms
• Using mathematics to model real-world and abstract situations, connecting algebraic and graphical forms
• Applying logical reasoning and showing work systematically as part of mathematical communication
• Developing conceptual understanding of functions and their behavior as foundational preparation for IB DP Mathematics

Lesson Structure (59 minutes)

0–7 min — Hook and Prior Knowledge Activation Display the equation x² − 5x + 6 = 0 on the board and ask students to write down everything they notice and wonder about it. Take 3–4 quick responses to surface prior knowledge about factoring and parabolas, and note any misconceptions to address during instruction.

7–20 min — Direct Instruction: Standard Form and Factoring Introduce the standard form ax² + bx + c = 0, labeling a, b, and c clearly. Walk through two factoring examples as a class — one straightforward (x² + 5x + 6 = 0) and one requiring more attention to signs (x² − x − 12 = 0). Emphasize the zero-product property as the logical foundation: if two factors multiply to zero, at least one must equal zero.

20–30 min — Paired Practice: Factoring Students work in pairs on four factoring problems, progressing from simple integer solutions to cases with a leading coefficient greater than 1 (e.g., 2x² + 7x + 3 = 0). Circulate actively, checking for sign errors and ensuring students are writing full solution steps. Pause at 28 min for a brief whole-class check on problem 3.

30–42 min — Direct Instruction: The Quadratic Formula Introduce the quadratic formula as a universal tool, especially for equations that do not factor neatly. Model two examples: one that produces rational solutions and one with irrational solutions (leaving answers in simplified radical form). Explicitly connect the discriminant (b² − 4ac) to the number and nature of solutions, and link back to whether the parabola crosses, touches, or misses the x-axis. Use a quick sketch on the board to reinforce the graphical connection.

42–52 min — Independent Practice Students independently solve a set of five problems — a mix of factoring and quadratic formula — on a printed or digital worksheet. Problems range from straightforward to moderately challenging, mirroring the rigor expected in IB assessments. Encourage students to choose their method and justify it briefly in writing.

52–59 min — Closing Discussion and Exit Ticket Bring the class back together. Ask: "When would you choose factoring over the quadratic formula, and why?" Take 2–3 responses to reinforce strategic thinking. Students then complete a 3-question exit ticket individually: identify coefficients, solve one equation, and state what the solutions mean graphically.

Resources

• Whiteboard and markers for teacher modeling
• Printed or digital practice worksheet (factoring and quadratic formula problems)
• Exit ticket slips (3 questions, half-sheet format)
• Graphing tool (Desmos or graphing calculators) for visual connection
• Quadratic formula reference card for students who need it during initial practice
• Class set of scientific calculators

Assessment

• Paired practice responses and circulating observations inform real-time instructional adjustments during the lesson.
• Independent practice worksheet is collected or photographed at the end of class to review procedural accuracy and written reasoning.
• Exit ticket directly measures each student's grasp of all three success criteria and informs the next lesson's starting point.

Differentiation

• Support: Provide a partially completed worked example alongside the practice problems, and allow use of the quadratic formula reference card throughout the lesson. Pair students strategically so stronger peers can model reasoning without simply giving answers.

• Extension: Challenge early finishers to derive the quadratic formula themselves by completing the square on ax² + bx + c = 0, then verify their derivation matches the standard formula. Ask them to explain in writing what a negative discriminant tells us about the graph.

• EAL students: Pre-teach key vocabulary (coefficient, factor, discriminant, root, solution) with a brief visual glossary on their desk. Allow extra processing time during the paired activity, and accept solutions that show correct mathematical steps even if written explanation is minimal.

• Students needing extra challenge within the main lesson: Include one non-routine problem on the independent practice — such as forming a quadratic equation from given roots — to stretch thinking beyond procedural application and align with higher IB MYP achievement descriptors.