Introduction to Functions

Active learning works for quadratic functions because students need to see how changes in coefficients transform the graph. By manipulating equations and observing results, students build intuition about the parabola that static notes cannot provide. Movement and collaboration turn abstract symbols into concrete, visual understanding.

• Secondary 3
• Mathematics
• 3 activities
• 20 min–40 min

Learning Objectives

1. Explain the definition of a function as a rule that assigns exactly one output to each input.
2. Identify the domain and range of a function given a specific set of ordered pairs or a graph.
3. Calculate the output value of a function for a given input value using function notation.
4. Differentiate between a relation that is a function and one that is not, using the vertical line test.
5. Construct a real-world scenario that can be represented by a function, defining its input and output variables.

Ready-to-Use Activities

Inquiry Circle

⏱ 40 min·Pairs

Inquiry Circle: The Parabola Transformer

Using graphing software like Desmos, students work in pairs to explore how changing 'a', 'b', and 'c' shifts the parabola. They must complete a 'mission' to create a parabola that passes through specific points or has a specific turning point.

Prepare & details

Explain the concept of a function as a special type of relation.

Facilitation Tip: During The Parabola Transformer, circulate to ensure groups change only one coefficient at a time so students isolate each variable's effect.

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

Gallery Walk

⏱ 30 min·Small Groups

Gallery Walk: Match the Graph to the Equation

Place several parabolas on the walls and give groups a set of equations. Students must use their knowledge of intercepts and turning points to match them, leaving a written justification for each match on the wall.

Prepare & details

Differentiate between dependent and independent variables in a functional relationship.

Facilitation Tip: For Match the Graph to the Equation, provide graph paper and colored pencils to help students track their matching process visually.

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

⏱ 20 min·Pairs

Think-Pair-Share: Symmetry Secrets

Show a graph with two x-intercepts. Ask students to find the x-coordinate of the turning point without the equation. After pairing, the class discusses how the axis of symmetry is always exactly halfway between the roots.

Prepare & details

Construct examples of real-world relationships that can be modeled as functions.

Facilitation Tip: During Symmetry Secrets, listen for pairs to reference the axis of symmetry formula x = -b/(2a) before sharing their observations with the class.

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

Teaching This Topic

Experienced teachers begin with concrete examples before abstract formulas, using graphing calculators or digital tools to show immediate transformations. Avoid starting with the general form y = ax^2 + bx + c; instead, isolate y = ax^2 to emphasize the role of 'a' first. Research shows that students grasp symmetry better when they fold paper parabolas along their axis before calculating it algebraically. Always connect the visual shape to the algebraic expression to prevent students from treating them as separate topics.

What to Expect

Successful learning looks like students confidently identifying parabola features from equations and matching them to graphs without hesitation. They should explain how coefficients affect shape and position using precise mathematical language. Missteps in plotting or interpreting become obvious as groups compare their work during activities.

Watch Out for These Misconceptions

Common Misconception

During The Parabola Transformer, watch for students who assume a larger 'a' value makes the parabola wider.

What to Teach Instead

Direct students to plot y = x^2 and y = 5x^2 on the same grid, then compare the steepness of the curves. Ask them to plot points at x = 1 and x = 2 to see how quickly the values grow with a larger coefficient.

Common Misconception

During Match the Graph to the Equation, watch for students who confuse the y-intercept with the turning point.

What to Teach Instead

Have students highlight the y-intercept on each graph using a colored pencil and label it with 'c'. Then ask them to locate the turning point and label it separately, comparing its position to the y-axis across different graphs.

Assessment Ideas

Quick Check

After The Parabola Transformer, present students with two equations: y = 2x^2 and y = -0.5x^2. Ask them to sketch each parabola on the same grid and explain how the sign and size of 'a' affect the graph's direction and width.

Exit Ticket

After Match the Graph to the Equation, give each student a card with a graph of a quadratic function. Ask them to write the equation in the form y = ax^2 + bx + c and identify the y-intercept and turning point coordinates before leaving class.

Discussion Prompt

During Symmetry Secrets, pose the prompt: 'If a parabola has a turning point at (3, -4), what is the equation of its axis of symmetry? Explain how you know.' Call on pairs to share their reasoning before moving to the next example.

Extensions & Scaffolding

• Challenge students who finish early to create a parabola with a specific turning point and y-intercept, then trade with a partner to match equations to their graphs.
• For struggling students, provide a table of values for y = x^2 and y = 2x^2 side-by-side to highlight how 'a' affects the steepness before they attempt transformations.
• Deeper exploration: Have students research real-world applications like projectile motion and present how the quadratic function models the path, focusing on the meaning of the turning point and intercepts in context.

Key Vocabulary

Function	A relation where each input value (from the domain) is associated with exactly one output value (in the range).
Domain	The set of all possible input values for which a function is defined.
Range	The set of all possible output values that a function can produce.
Function Notation	A way to represent a function using symbols, such as f(x), to denote the output of a function 'f' when the input is 'x'.
Independent Variable	The input variable in a function, typically represented by 'x', whose value does not depend on other variables in the function.
Dependent Variable	The output variable in a function, typically represented by 'y' or f(x), whose value depends on the independent variable.