ISTE and Content Standards.

  • 1.4.a Design Process
    Students know and use a deliberate design process for generating ideas, testing theories, creating innovative artifacts or solving authentic problems.
  • Calculus 
    i. Sketch the graphs of functions and their gradient functions and describe the relationship between these graphs.

    ii. Apply differentiation and anti-differentiation techniques to polynomials

Assessment

Analyzing Real-World Probabilities Using Digital Tools
Task Overview:
In this assessment, students will design an original function that models a real-world scenario (e.g., the growth of a plant, the trajectory of a ball, or a financial model). They will sketch the function(using GeoGebra, or something similar), calculate its derivative, and analyze how the derivative informs the behavior of the model over time. Finally, they will reflect on how using a design process helped them refine their model and how the calculus concepts apply to solving real-world problems.
Questions to be explored:
  1. How does your function behave at various points? (e.g., is it increasing, decreasing, or stationary?)
  2. What do the maxima, minima, and points of inflection tell you about your model?
  3. How does the derivative of your function help you understand real-world changes in your scenario?
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Steps in the task:
  1. Identify several real-world situation you want to model using a polynomial function. Examples can be the movement of vehicles, limited movement of celestial bodies, etc.
  2. Create a function that represents the situation. Write a justification for why your function makes sense, further justify that this can be considered a reasonable model of your real world situation. Creatively find a way to show your function. This can be done in GeoGebra, Desmos or Wolfram. But more creative solutions can be justified(in particular for rocketry and movement of celestial bodies). 
  3. Differentiate your function to find its gradient function. Sketch the gradient function using your same choice of technology. 
  4. Analyze how the derivative helps explain changes in your model (e.g., turning points, rates of change). Critical Points, Inflection points, and concavity should all be part of your presentation, as well as any relevant calculations and their interpretations. 
  5. Apply anti-differentiation to interpret the area under the curve for a specific interval in your model. 
  6. Reflect on how using a deliberate design process helped you refine your model and answer questions about the situation.
  7. Create a visual presentation from your works, making sure to use technology creatively to put create a thoughtful exploration of your work. 

Supporting Activities.

Exploring the Relationship Between Functions and Their Derivatives
 ​Exploring and applying Differentiation and Anti-Differentiation
  • Learning Outcome:
    Students will be able to graph a function and its derivative, and explain the relationship between the shape of the function and the gradient at different points.
  • Activity Description:
    Students will select a simple polynomial functions and use graphing software (such as Desmos or GeoGebra) to visualize both the function and its derivative. They will analyze how the behavior of the function (increasing, decreasing, stationary points) is related to the values of the gradient function (positive, negative, zero). Students will then present their findings in a visual format (e.g., a poster or digital presentation), highlighting key points such as where the function is at a maximum, minimum, or inflection point and how the derivative graph corresponds to those points.
  • Learning Outcome:
    Students will demonstrate the ability to apply differentiation and anti-differentiation techniques to polynomial functions.
  • Activity Description:
    Given several polynomial functions, students will calculate the derivative and integral (anti-derivative) by hand and confirm their results using a symbolic calculator or software. For example, given several functions,  then check their answers using technology. They will also be tasked with interpreting the integral in terms of area under the curve for a specified range. Their work will be submitted as a written explanation, showing both manual calculations and software verification. Further students will be tasked with showing that the anti-derivative and derivative are inverses of one another in their own words.