Modelling with Quadratic FunctionsActivities & Teaching Strategies
Quadratic functions come alive when students connect them to motion and optimization problems before abstract symbol manipulation. Active learning lets students test assumptions, gather data, and revise models in real time, which builds both conceptual understanding and procedural fluency.
Learning Objectives
- 1Design a quadratic function to model the trajectory of a projectile, identifying coefficients related to initial velocity and gravity.
- 2Analyze the vertex of a quadratic model to determine the maximum height or optimal value in a given scenario.
- 3Evaluate the limitations of a quadratic model when applied to real-world projectile motion, considering factors like air resistance.
- 4Calculate the parameters of a quadratic function that best fits a set of data points representing a parabolic relationship.
- 5Critique the appropriateness of using a quadratic model for different types of real-world optimization problems.
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Pairs Activity: Projectile Trajectory Launch
Pairs select launch angles for small balls or paper aircraft, measure height at timed intervals using rulers and stopwatches. Plot points on graph paper, sketch the quadratic curve, and locate the vertex for maximum height. Compare predictions with actual outcomes and adjust for initial velocity.
Prepare & details
Design a quadratic model for a projectile motion problem, identifying key parameters.
Facilitation Tip: During the Projectile Trajectory Launch, ensure each pair uses the same initial height to standardize comparisons when they present their graphs.
Setup: Flexible workspace with access to materials and technology
Materials: Project brief with driving question, Planning template and timeline, Rubric with milestones, Presentation materials
Small Groups: Optimization Fencing Challenge
Groups receive a fixed length of string as fencing to enclose maximum rectangular area. Test dimension pairs, calculate areas, tabulate data, and graph the quadratic relationship. Identify the vertex algebraically and geometrically to verify the optimal shape.
Prepare & details
Analyze the significance of the vertex of a quadratic model in an optimization context.
Facilitation Tip: For the Optimization Fencing Challenge, provide rulers and grid paper so groups can visualize rectangles accurately when scaling dimensions.
Setup: Flexible workspace with access to materials and technology
Materials: Project brief with driving question, Planning template and timeline, Rubric with milestones, Presentation materials
Whole Class: Model Data Collection Relay
Class divides into relay teams to collect trajectory data from shared launches. Each team measures one aspect (height, time, angle), compiles class dataset, fits a group quadratic model. Discuss deviations from ideal parabola due to real-world factors.
Prepare & details
Evaluate the limitations of a quadratic model when applied to real-world phenomena.
Facilitation Tip: In the Model Data Collection Relay, assign roles clearly so every student measures, records, or plots to keep the relay moving efficiently.
Setup: Flexible workspace with access to materials and technology
Materials: Project brief with driving question, Planning template and timeline, Rubric with milestones, Presentation materials
Individual Task: Vertex Application Worksheet
Students receive scenario cards with optimization problems, write quadratics, complete the square for vertex form, and interpret results. Extend by sketching graphs and predicting outcomes for varied parameters.
Prepare & details
Design a quadratic model for a projectile motion problem, identifying key parameters.
Setup: Flexible workspace with access to materials and technology
Materials: Project brief with driving question, Planning template and timeline, Rubric with milestones, Presentation materials
Teaching This Topic
Teachers know students grasp quadratics best when they start with concrete measurements and move toward symbolic generalization. Avoid launching straight into equation forms; instead, have students sketch predicted paths, estimate vertices from graphs, and only then formalize the quadratic expression. Research shows that students who build models from real data develop stronger number sense and are better at validating their own results.
What to Expect
By the end of these activities, students will confidently derive quadratic models from data, interpret parameters physically, and critique model limitations. You will see students using graphs to predict outcomes, adjusting equations based on evidence, and explaining why the vertex matters in context.
These activities are a starting point. A full mission is the experience.
- Complete facilitation script with teacher dialogue
- Printable student materials, ready for class
- Differentiation strategies for every learner
Watch Out for These Misconceptions
Common MisconceptionDuring Projectile Trajectory Launch, watch for students assuming all maximum ranges occur at 45 degrees.
What to Teach Instead
Circulate during launches and ask groups to compare their maximum ranges for angles 30°, 45°, and 60°; have them plot these points on a shared class graph to see the trend together.
Common MisconceptionDuring Optimization Fencing Challenge, watch for students assuming the vertex always represents a maximum value.
What to Teach Instead
Ask groups to test both revenue and cost scenarios using the same quadratic form; have them present how the leading coefficient changes the vertex’s meaning in context.
Common MisconceptionDuring Model Data Collection Relay, watch for students believing real-world data fits quadratics perfectly.
What to Teach Instead
After the relay, display the raw scatter plot alongside the quadratic model; ask students to identify deviations and discuss possible causes such as measurement error or external forces.
Assessment Ideas
After Projectile Trajectory Launch, give students a new launch height scenario and ask them to write the quadratic equation, calculate the vertex, and explain what it means in context.
During Optimization Fencing Challenge, collect each group’s sketch of the optimal rectangle, the quadratic equation they derived, and a written sentence explaining why the vertex location matters for maximizing area.
During Model Data Collection Relay, pause after collecting data points and ask students to predict the vertex of their quadratic model before graphing; listen for reasoning that connects the data’s trend to the equation’s parameters.
Extensions & Scaffolding
- Challenge: Ask early finishers to adjust their projectile model to include air resistance by subtracting 0.1t^3 from height(t) and observe how the vertex shifts.
- Scaffolding: For struggling students, provide pre-printed grids with key points plotted so they focus on connecting the equation to the graph.
- Deeper: Introduce a third activity where students compare quadratic models to linear and exponential fits for the same dataset, discussing which model best explains the trend.
Key Vocabulary
| Quadratic Function | A function of the form f(x) = ax^2 + bx + c, where a, b, and c are constants and a is not zero, whose graph is a parabola. |
| Vertex | The highest or lowest point on a parabola, representing the maximum or minimum value of the quadratic function. |
| Projectile Motion | The motion of an object thrown or projected into the air, subject only to the acceleration of gravity (in idealized models). |
| Optimization | The process of finding the maximum or minimum value of a function, often used to find the best possible outcome in a given situation. |
| Parabolic Trajectory | The curved path followed by a projectile, which in the absence of air resistance is a parabola. |
Suggested Methodologies
Planning templates for Mathematics
5E Model
The 5E Model structures lessons through five phases (Engage, Explore, Explain, Elaborate, and Evaluate), guiding students from curiosity to deep understanding through inquiry-based learning.
Unit PlannerMath Unit
Plan a multi-week math unit with conceptual coherence: from building number sense and procedural fluency to applying skills in context and developing mathematical reasoning across a connected sequence of lessons.
RubricMath Rubric
Build a math rubric that assesses problem-solving, mathematical reasoning, and communication alongside procedural accuracy, giving students feedback on how they think, not just whether they got the right answer.
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