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In this Year 10 mathematical modeling activity, students will explore the theoretical aspects of quadratic equations by modeling the shape of water streams from a hose as they vary the angle of the nozzle. This activity focuses on developing mathematical models to describe the behavior of water streams. Use TI Npire CAS to define the quadratic function with given parameters, then vary the angle of projection with a slider.

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1. Introduction: Begin by discussing the concept of quadratic equations, specifically the standard form, y = ax^2 + bx + c. Explain that quadratic functions often model parabolic shapes, and in this activity, they will explore how changes in the angle of a water hose nozzle can be represented mathematically.
2. Model Development: Have students work in pairs or small groups. Provide them with theoretical data that describes the behavior of a water stream from a hose at different angles. For each angle, they should have data points that include the maximum height reached by the water and the horizontal distance it travels before hitting the ground.
3. Equation Development: Instruct students to use the provided data to develop a quadratic equation that models the shape of the water stream as it varies with the angle of the nozzle. Encourage them to fit the equation to the data points.
4. Graphing: Have students create graphs of their quadratic equations using graphing software or calculators. Discuss how the graphs change as the angle of the nozzle changes.
5. Analysis and Discussion: Lead a class discussion about the results. Ask students to compare their equations and graphs, focusing on how different coefficients (a, b, and c) affect the shape of the parabola. Encourage them to make connections between the mathematical model and real-world situations involving water streams.
6. Conclusion: Summarize the main findings of the activity, highlighting the mathematical modeling process and its relevance to understanding how changes in parameters (angle in this case) can be represented mathematically.
7. Extensions (optional): For advanced students, you can introduce more complex scenarios, such as the impact of air resistance or varying water pressure on the shape of the water stream. This can lead to discussions on more advanced mathematical modeling techniques.