Active Learning for Calculus with Wolfram|Alpha Notebook Edition

Today’s post is an excerpt of calculus lesson ideas from a blog post written by Jordan Hasler and John McNally and published on the Wolfram Blog. The original post can be viewed here.

As you may know from your own experience (or perhaps from the literature on education), passively receiving information does not lead to new knowledge in the same way that active participation in inquiry leads to new knowledge. Active learning describes instructional methods that engage students in the learning process. Student participation in the classroom typically leads to deeper knowledge, more developed critical thinking skills and increased motivation to continue learning. In this post, you will see example activities demonstrating how Wolfram|Alpha Notebook Edition can support active learning methods in your classroom.

Wolfram|Alpha Notebook Edition combines the natural language processing of Wolfram|Alpha with the flexible format of Wolfram Notebooks. Combine text, graphics, natural language computations, interactive visualizations and more in a single place. Whether you’re an educator or a student, Wolfram|Alpha Notebook Edition makes it easy to take an active role in the learning process.

Sample Activities for a Calculus Course

Exploring Tangent Lines

Tangent lines (and their connection to derivatives) are a fundamental concept in calculus and one that students often have difficulty understanding by staring at a formula. However, with Wolfram|Alpha Notebook Edition, students can examine patterns and then make predictions based on their experiences. By actively forming connections from experience, they gain a greater intuition for the concept.

You can ask your students to define a function, say f(x) = x2:

Now find the tangent line to this function at the point (1,1):

Notice that the output contains a variety of information you will incorporate into lessons at some point during the instructional sequence. Any part of the output can be used for future exploration. Suppose you first want to have students explore the patterns that emerge as they consider tangent lines at different points. The last input can be easily modified to do just that:

With three computations performed, you could ask your students to make a prediction based on these examples. For example, what seems to be the relationship between the point chosen and the slope of the tangent line to this curve? By going back and considering patterns in their previous results, many students will pick up on the fact that the slope of the tangent line to this function has been twice the value of the x coordinate in the last three examples.

Connecting Tangent Lines to Derivatives

Since your students already defined f(x) = x2, they don’t need to do so again in the same notebook:

To introduce the important connection between tangent lines and derivatives, you can ask students to compare their previous results about tangent lines with new calculations about derivatives:

By seeing concrete calculations and matching these patterns for themselves, students will be led to wonder if the patterns hold generally. Luckily, symbolic computations can also be done to help answer their questions:

Include Interactive Demonstrations

With concrete examples now grounding their understanding, you can help students learn why they have seen some sort of connection between tangent lines and derivatives. You can bring interactivity into your students’ math explorations by using Demonstrations from the Wolfram Demonstrations Project:

Demonstrations can be browsed through or brought up using natural language inputs.

This is only a brief teaser! Check out the original post here for tons of ideas for algebra lessons, as well as information on how to find further interactive explanations. It’s worth a read!