The Challenge
How can students use derivatives to help real people make better decisions?
One of the biggest challenges in teaching Calculus is helping students see beyond the formulas. While students may be able to differentiate functions correctly, they often struggle to answer a simple question: "Why does this matter?"
To bridge that gap, my Grade 12 Basic Calculus students became mathematical modelers. Instead of being given a function, they were challenged to investigate a real-world problem, gather data, create a mathematical model, analyze it using derivatives, and make a recommendation supported by evidence.
The project transformed derivatives from an abstract topic into a practical tool for decision-making.
Becoming Mathematical Modelers
Students followed a seven-step mathematical modeling process:
https://www.youtube.com/watch?v=xHtsuOB-TPw- Define the problem.
- Make assumptions.
- Define variables.
- Build a mathematical model.
- Differentiate and solve.
- Analyze the model.
- Report the results.
Students had the freedom to choose authentic situations involving optimization, efficiency, performance, or rates of change.
This ownership immediately increased engagement because students were investigating questions that mattered to them or to people in their community.
A Student Example: Managing Client Capacity in an Accounting Firm
One group partnered with a local accounting services business that wanted to understand how workload affects service quality.
The students asked:
"What is the ideal number of clients that the firm can handle while maintaining high-quality service?"
Using survey responses collected from clients, students organized and analyzed satisfaction data, created a quadratic regression model, and applied derivatives to locate the turning point.
Their findings suggested that service quality peaked when the firm handled approximately seventeen (17) clients per month. Beyond that point, service quality began to decline, indicating potential workload strain.
Rather than simply reporting a mathematical answer, students were able to make a practical recommendation that could help the business maintain quality service while managing growth.
Technology That Supported Learning
Technology was used throughout the investigation, not simply for presentation.
- Freeform
Students used Freeform as a collaborative thinking space to brainstorm problems, organize research, identify assumptions, and plan their investigations.
- Numbers/Google Sheets
Students used Numbers/Google Sheets to organize datasets, create tables, visualize trends, and analyze relationships between variables.
- Google Forms
Students collected authentic data directly from stakeholders through surveys and questionnaires.
- GeoGebra
Students visualized both the original function and its derivative, helping them connect algebraic computations with graphical representations.
- AI Website Builders
Students transformed their final reports into websites that communicated their findings to a broader audience.
- Keynote (Alternative for Presentation)
Students used Keynote to communicate their findings through visual explanations, graphs, and animations, through a public presentation.
- Safari and Apple Intelligence Tools
Students researched authentic datasets, gathered contextual information, and refined their written explanations while maintaining ownership of their thinking.
Technology became a tool for thinking, analyzing, visualizing, and communicating rather than simply a tool for presenting information.
Sharing Findings with an Authentic Audience
The project culminated in a public presentation at the end of the term. Students presented their mathematical models, derivative analyses, recommendations, and websites to their classmates and, whenever possible, to the individuals, organizations, or stakeholders connected to their chosen problem.
This final presentation challenged students to communicate complex mathematical ideas in ways that non-mathematicians could understand. Students were expected to explain not only their calculations, but also the reasoning behind their assumptions, the limitations of their models, and the real-world implications of their recommendations.
Through this process, students learned that mathematics is not simply about arriving at an answer. It is also about communicating evidence, defending conclusions, and helping others make informed decisions.
Making Mathematical Thinking Visible
One of the most valuable aspects of the project was the emphasis on reasoning. Students were expected to explain:
- why they chose their problem
- why their assumptions were reasonable
- why a particular model fit the data
- what the derivative represented in context
- what limitations existed in their analysis
- how their recommendation could be applied in the real world
This moved the focus away from finding answers and toward making sense of evidence.
Student Agency Through Choice
Although all students were studying derivatives, no two projects were exactly alike. Students explored topics connected to business, education, productivity, health, sports, social media, and service organizations.
The mathematical process remained consistent, but the context was driven by student interest. This balance between structure and choice allowed students to experience mathematics as a flexible tool that can be applied across disciplines and situations.
Why This Matters
When students ask, "When will we ever use derivatives in real life?", this project allows them to discover the answer themselves.
Too often also, they encounter optimization problems that feel disconnected from reality.
In this project, students discovered that derivatives can help answer meaningful questions:
- How many clients should a business accept?
- How much should a company spend on advertising?
- How much practice produces the best performance?
- What workload maximizes productivity?
- When does growth begin to slow down?
By investigating authentic questions, collecting data, building models, and communicating recommendations, students experienced Calculus as it is used outside the classroom. They learned that mathematics is not only about solving equations. It is about understanding situations, making informed decisions, and improving the world around them.
Teacher Resource
The attached project framework/task design can be adapted for any optimization or rate-of-change investigation. Teachers may substitute their own contexts, data sources, and digital tools while maintaining the same mathematical modeling process. They may also explore possible integrations/collaborations with other subjects such as Entrepreneurship and ICT subjects.
The result is a learning experience that combines mathematics, inquiry, technology, communication, and authentic problem-solving in a way that makes Calculus meaningful for students.
Attach up to 5 files which will be available for other members to download.