Introduction to Quadratic Functions

Active learning works for quadratic functions because students need to move between symbolic and visual representations to grasp non-constant change. Handling physical graphs and real-world examples helps them anchor abstract ideas like vertex and symmetry before formalizing them algebraically.

• Grade 10
• Mathematics
• 3 activities
• 30 min–60 min

Learning Objectives

1. Identify the standard form of a quadratic function and classify functions as linear or quadratic based on their equations.
2. Compare the graphical representations of linear and quadratic functions, recognizing the parabolic shape of quadratic graphs.
3. Explain the relationship between the algebraic form of a quadratic function and the geometric properties of its parabolic graph.
4. Analyze the effect of the coefficient 'a' on the width and direction of opening of a parabola.
5. Differentiate between linear and quadratic functions by examining their equations and graphical characteristics.

Gallery Walk

⏱ 60 min·Small Groups

Gallery Walk: Parabola Scavenger Hunt

Students take photos of parabolic shapes around the school or in the community. They overlay a coordinate grid on their photos and identify the vertex, axis of symmetry, and intercepts of the 'real' parabola.

Prepare & details

Differentiate between linear and quadratic functions based on their equations and graphs.

Facilitation Tip: During the Gallery Walk, position yourself near the most visually complex parabolas to model precise vocabulary as students describe what they see.

Setup: Wall space or tables arranged around room perimeter

Materials: Large paper/poster boards, Markers, Sticky notes for feedback

• Understand
• Apply
• Analyze
• Create
• Relationship Skills
• Social Awareness

Think-Pair-Share

⏱ 30 min·Pairs

Think-Pair-Share: The 'a' Value Challenge

Pairs are given several quadratic equations with different 'a' values. They must predict how the 'a' value will change the shape and direction of the parabola before using graphing software to check their predictions.

Prepare & details

Explain why the graph of a quadratic function is always a parabola.

Facilitation Tip: For the 'a' Value Challenge, circulate with a small whiteboard to sketch competing student ideas about the effect of 'a' and ask guiding questions.

Setup: Standard classroom seating; students turn to a neighbor

Materials: Discussion prompt (projected or printed), Optional: recording sheet for pairs

• Understand
• Apply
• Analyze
• Self-Awareness
• Relationship Skills

Inquiry Circle

⏱ 50 min·Small Groups

Inquiry Circle: Projectile Motion Simulation

Groups use a digital simulation to launch projectiles at different angles and speeds. They collect data on the path of the object and work together to identify the vertex (maximum height) and x intercepts (landing points).

Prepare & details

Analyze the significance of the 'a' coefficient in the standard form of a quadratic function.

Facilitation Tip: Set clear time markers during the Projectile Motion Simulation so students focus on collecting data rather than perfecting movements.

Setup: Groups at tables with access to source materials

Materials: Source material collection, Inquiry cycle worksheet, Question generation protocol, Findings presentation template

• Analyze
• Evaluate
• Create
• Self-Management
• Self-Awareness

Teaching This Topic

Teachers should begin with concrete examples before moving to abstract rules, using both hands-on graphing and digital tools. Avoid rushing to the vertex formula; instead, build understanding through repeated exposure to how 'a', 'h', and 'k' shift the parabola's position and shape. Research shows that students solidify their grasp of quadratic functions when they repeatedly connect equations to real-world contexts like projectiles or satellite dishes.

What to Expect

Successful learning looks like students confidently identifying key features of parabolas and explaining how coefficients shape the graph. They should connect the algebraic form to graphical behavior and use precise vocabulary when describing transformations and intercepts.

Watch Out for These Misconceptions

Common Misconception

During the Gallery Walk: Parabola Scavenger Hunt, watch for students labeling any point on the y-axis as the vertex.

What to Teach Instead

Have students place a sticky note directly on the vertex of each parabola they sketch and write its coordinates, reinforcing that the vertex is the turning point rather than an intercept.

Common Misconception

During the Think-Pair-Share: The 'a' Value Challenge, watch for students believing a larger 'a' value makes the parabola wider.

What to Teach Instead

Provide a set of printed graphs with 'a' values ranging from 0.1 to 5 and ask students to sort them from narrowest to widest, then justify their ordering using y-values at x=1.

Assessment Ideas

Quick Check

After the Gallery Walk: Parabola Scavenger Hunt, provide a list of six functions (four quadratic, two linear) and ask students to circle the quadratics and rewrite them in standard form if needed.

Exit Ticket

During the Think-Pair-Share: The 'a' Value Challenge, collect index cards with students' sketches of an upward-opening parabola labeled with the vertex and axis of symmetry, along with a sentence explaining how the 'a' coefficient would change if the parabola became steeper.

Discussion Prompt

After the Projectile Motion Simulation, pose the question: 'How does the shape of the path change when you adjust the initial speed or angle?' Have students discuss how the quadratic equation models these changes in real time.

Extensions & Scaffolding

• Challenge: Ask students to derive the standard form from a vertex form equation without graphing, then verify their work by plotting key points.
• Scaffolding: Provide a partially filled table of values for students to complete when graphing parabolas with fractional coefficients.
• Deeper exploration: Have students research and present on how quadratic functions model phenomena in engineering or physics, focusing on the role of the vertex and axis of symmetry.

Key Vocabulary

Quadratic Function	A function that can be written in the form f(x) = ax^2 + bx + c, where a, b, and c are constants and a is not equal to zero. Its graph is a parabola.
Standard Form of a Quadratic Function	The form f(x) = ax^2 + bx + c, which clearly shows the coefficients a, b, and c.
Parabola	The U-shaped curve that is the graph of a quadratic function. It is symmetrical and opens either upwards or downwards.
Vertex	The highest or lowest point on a parabola, representing the maximum or minimum value of the function.
Axis of Symmetry	A vertical line that passes through the vertex of a parabola, dividing the parabola into two mirror-image halves.