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Introduction
Technology has transformed the way students interact with mathematical ideas, making concepts more accessible, visual, and engaging. Desmos, a free online graphing calculator that transforms how students interact with math. With its intuitive design and powerful visualization tools, Desmos takes abstract algebra, and geometry concepts and puts them into motion.
Whether you’re teaching linear functions, exploring parabolas, or diving into trigonometry, Desmos gives learners the ability to manipulate, test, and animate math in real time. It’s not just a tool; it’s a creative playground for critical thinking and exploration. In this post, we’ll explore how Desmos bridges the gap between conceptual understanding and visual intuition, and why it’s become a must-have in modern secondary math classrooms.
Standards Implications
Standards Implications
Recently, we worked through several graphing investigations as part of an investigation mathematics education lab and experienced how powerfully it supports mathematical practices and state learning goals, especially those in the Alabama 2019 Course of Study for Algebra I with Probability. Below are some examples of functions throughout the investigations that were later explored.
One of the most compelling alignments is with Standard A1.23:
Identify the effect on the graph of replacing 𝑓(𝑥) by 𝑓(𝑥)+𝑘, 𝑘*𝑓(𝑥), 𝑓(𝑘*𝑥), and 𝑓(𝑥+𝑘) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and explain the effects on the graph, using technology as appropriate
Each Desmos investigation, whether focusing on linear growth, exponential decay, or projectile motion, encouraged us to manipulate parameters dynamically and observe effects in real time. For instance:
In Investigation 1: Getting Started with Desmos, we explored slope and intercept through interactive sliders in y = mx + b [f(x) +k], visually discovering what it means for a line to be steep, flat, increasing, or decreasing.
Investigation 2: Exponential Functions with Desmos led us into the possibilities of exponential functions and decay. Using Desmos, we were able to model half-life scenarios and developed fluency with the form f(x)=ab^x [k*f(x)].
Investigation 3: Quadratic Functions with Desmos dove into quadratic behavior, simulating real-world physics as we analyzed height-time graphs for a falling or thrown object. The open-ended nature of sliders for initial velocity and height created space for genuine inquiry and debate.
An enriching moment came during Investigation 3.2: Throw Down. It was briefly discussed whether a ball thrown downward from the 10th floor would take less time to hit the ground than one thrown upward. By using Desmos, it didn’t just confirm the correct answer, but it helped grapple with misconceptions like negative time or maximum height. These often things discussed in physics that trip people up. If I were to take the class again, instead of just numerically soling the situation, I would have much appreciated if this application was present for the lecture of the topic. Through this lab investigation discussion, questions like:
“How fast would you have to throw a ball to keep it in the air for 10 seconds?”
“What does the symmetry of the parabola mean in real life?”
...appeared and left myself and others incentivized to find out why. It became more than just doing math but instead using math to explore physical realities. Additionally, through each investigation, we were able to demonstrate the Standard for Mathematical Practice (SMPs) through the following:
SMP 1: Make sense of problems and persevere in solving them
We weren't handed answers but through building and testing models, we were able to refine our solutions based on visual feedback.
SMP 2: Reason abstractly and quantitatively
By visually coordinating the assignment, we were able to move fluently between equations, tables, and graphs, interpreting each representation.
SMP 5: Use appropriate tools strategically
Desmos became a thinking tool, not just a graphing utility. It lets us pose what-if questions and get immediate visual evidence.
SMP 7: Look for and make use of structure
By observing patterns in sliders or vertex movement, we internalize the structure of function families and saw the different dynamics of vertex form, standard form, and factored form.
Conclusion
Conclusion
Desmos shifted my own thinking in two major ways:
Exploration comes first. Instead of explaining concepts up front, one should build and play. Patterns will then emerge organically.
Desmos bridges the gap between “seeing” and “knowing.” Multiple representations coexist. One can move seamlessly among algebraic, graphical, and tabular forms.
Desmos is more than a calculator, it’s a toolkit for discovery. When paired with thoughtfully designed investigations and meaningful teacher prompts, it turns standards into experiences and practices into habits of mind. As we continue preparing our students for the demands of a data-rich, tech-centered world, tools like Desmos ensure that mathematics remains not only relevant, but exciting, empowering, and personal.
References
Alabama State Department of Education. (2021). Alabama Course of Study: Mathematics (2019 revision, updated June 2021). Montgomery, AL: Author. Retrieved from https://www.alabamaachieves.org
Desmos Studio PBC. (n.d.). Desmos Graphing Calculator. https://www.desmos.com/calculator
Lab Investigations:
Martin, W. (2025). Lab: Graphing Applications – Investigation 1. Auburn University.
Martin, W. (2025). Lab: Graphing Applications – Investigation 2. Auburn University.
Martin, W. (2025). Lab: Graphing Applications – Investigation 3. Auburn University.
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