Math and Fun: Not an Oxymoron in a Flipped Classroom

In middle school and high school, I heartily looked forward to science classes each day.  They were a chance to ‘experiment’ – to play, test, follow curiosity – and yet, somehow, through this, learn.  What a radical idea: the combination of natural interest and discovery and education!  But all sarcasm aside, teachers are testing more ways to make learning fun, memorable, and meaningful.  Labs need not be limited to a science classroom.  Indeed, flipped learning as a strategy for any class attempts to take advantage of technology to give students the best of both worlds: interesting lessons and non-drudgery homework in the form of class ‘explorations’ or other active learning strategies.  There is huge potential in using technology for this purpose, and as teachers, we can continue to learn by trial and error, using our imaginations (and the scope of the internet) to craft new lesson sequences and see what is effective.

Here is my attempt at creating such a lesson for a high school trigonometry or Pre-Calculus class learning about the Unit Circle.

Learning Objective

Students will be able to label the Unit Circle on their own, using intuition.  They will be able to identify the patterns that the trig functions follow.  Furthermore, they will be able to explain how we derive trig values from special triangles on a circle of unit radius.  Finally, when given a major angle measure, they will be able to return the corresponding sine, cosine, and tangent values, or vice versa, given trig values to match with angle measures.

Digital Resources for Flipped Elements

This lesson will work off of Bloom’s Taxonomy.  Students will begin with “homework” to do the lowest level of thinking to memorize the basic values of the unit circle.  They will have at their disposal multiple tech tools to do this, as students learn in a variety of ways.

Visual learners might want to utilize these programs on Desmos and Geogebra, which help students to connect the dots between shapes/spaces and trig values.


Desmos Unit Circle Exploration: I especially like the use of colors on Desmos – never hurts to make math prettier and more appealing!


Geogebra Unit Circle Exploration: Note how there is a slider which helps students visualize the effect of radian measure on trig value.

Students more inclined to numerical or algebraical thinking might want to use a chart (especially an interactive one like this), which shows patterns among trig values to ease memorization.

Clicking “Show” or “Hide” reveals the corresponding trig value in the chart, making this like a matching game.

The objective of this beginning stage is to gain comfort with the idea of radians, triangles inscribed in a circle, and the connection between angles and trig values.  At this point, students should develop a base knowledge of trig values in relation to certain radian measures, but I would not expect them to understand the significance of these values.

Active Learning Strategies – Interactive, Collaborative, Digging Deep

The next part of the blended lesson is the in-class work, which here will consist of active student participation in a “lab” to get to the root of the importance of the unit circle.  Students would work in groups, following a lab procedure that would lead them through activities looking at recognizing and questioning the qualities of the circle.  At sporadic points throughout the lab, I might draw the class together to make important points or to give them hints on how to proceed, but their background knowledge should give them the foundation to proceed more or less independently.  Finally, at the end, we would come together as a class to review our findings and discuss how this new material will benefit us in the coming unit.

How Will the Lesson Flow?

Using a flipped lesson sequence frees up time to focus on deeping students’ learning in class.  By intentionally guiding students’ focus from basic to more advance studies of the circle, my hope is that they will not feel overwhelmed by the unit circle and instead give each aspect of it its due diligence and attention; ultimately, this should pay off in gaining both a general knowledge and a more application-based ability concerning the unit circle.

Why Use the Blended Model for this Lesson?

The goal of this lesson is for students to develop an intuitive understanding of the unit circle in order to be fully prepared to deal with all things trig-related in the following units.  The unit circle is hugely important and often not covered in due depth.  It may seem that math teachers’ insistence on knowing by heart the values on the unit circle is overboard, but really – they are not deceiving you in telling you that knowing this material like the back of your hand will pay you back several fold as you continue in math.  More critical even than memorizing the values of the circle and a much more thorough way of learning the circle, in fact – is being able to derive these values.  From what do they stem?  How are radians related to measures of a circle?  How are trig functions tied to coordinate points?  How do we convert between trig functions, or even undo trig functions with inverses?  These are the deeper questions that students should be able to answer.

Featured image: Bob B. Brown

What’s Your Sine?

1. Learning Objective

Students will be able to find the derivatives of sinusoidal functions.

2.  Digital Resource(s)

Before coming to class, students will watch an online video that I will create of me proving \frac{d}{du}sin(u) = cos(u) and \frac{d}{du}cos(u) = -sin(u).

3. Active Learning Strategies

Once students have an idea where the equations come from, they can use the freed up class time to solve problems where they are asked to find the derivatives of sinusoidal functions.  One active learning strategy that I might employ is Numbered Heads Together.  I will split students into groups of three or four and have them collaboratively work on problems that I will provide.  One randomly selected student from each group will then provide a brief explanation of how their group went about solving the problem.  Students who would have difficulties solving the problems on their own will have the benefit of group members to help them out.

4. Lesson Flow

The video explains to students where the equations \frac{d}{du}sin(u) = cos(u) and \frac{d}{du}cos(u) = -sin(u) come from.  I will begin class by demonstrating to students how we can use these equations to find the derivatives of various sinusoidal functions.  The example problems will require the use of techniques taught in previous lessons such as the power rule, the product rule, and the chain rule.  After the example problems, students will be split into groups for the Numbered Heads Together activity where they will solve some problems in groups.  Finally, students will be given some homework problems to work on individually for the last part of class.

5. Benefit for Students

As a math teacher, I dislike presenting a formula or equation to students and asking that they believe it on blind faith.  I really believe that students benefit from seeing where formulas come from and why they work the way they do.  However, in many classes (especially calculus), there are simply too many equations to offer a formal proof for each one during class time.  By flipping this lesson, students are able to see where the equations \frac{d}{du}sin(u) = cos(u) and \frac{d}{du}cos(u) = -sin(u) come from, and I won’t have to use class time to prove the equations.  Instead, students can use class time to collaboratively work on problems where they will apply the equations.  This is preferred to having students work on problems at home, where they would not have a teacher or classmates to help them out if they get stuck.

Image credit: Creativity103

There can never be too many labs….

1) Learning Objective

Students will be able to demonstrate knowledge of cellular respiration through a tennis ball lab experiment through data analysis and graphing.

2) Digital Resources Used

The first part of the lesson would be the students on their devices at home watching a video that I created through EDpuzzle. The video would be an introduction and full overview of the process of cellular respiration, I would add checkmarks of questions that the students had to answer and they couldn’t skip ahead they have to watch all of the video. I most likely would not grade these questions but see it as a form of informal assessment for me to know where my students are at (it would all depend on where in the unit we are at). This would free up what would be classroom lecture time to insert a lab which I wish I could always do more of!

3) Active Learning Strategies

For me the best active learning strategies are getting students involved in a lab experiments in class. Currently I do not have the abilities to operate a flipped classroom so I can only do a small amount of labs, but in my opinion labs are one of the best hands-on and active way for students to learn content. The lab that I would do is a tennis ball lab that connects to cellular respiration. The students are in groups and they have to squeeze a tennis as hard as they can for 30 seconds take a rest and do it again. The students have to do this five times in a row and keep track of how many times they can fully squeeze the tennis ball. The connection is that over time your muscles fatigue with has to do with cellular respiration, glycogen storage, and anaerobic fermentation.

4) Lesson Flow

The lesson will flow in the direction of starting with the video and questions at home so that when students come into class they have received and reviewed the content, therefore they are ready to begin the lab. I would most likely do a review at the beginning of the class of content that pertained to the lab and then the rest of the class period would be devoted to the students lab time. A block class period would be perfect to also start a discussion and analyze the data however it could be done over a class period and a half or two full periods.

5) Benefit to Students

This format allows students to be exposed to the material outside of class so that by the time they come into class they know what they know and what they do not know. This allows time for more questions and one-on-one time with students in class. The content offloading also allows for more hands-on, minds-on activities in the class, which for me means LABS LABS LABS. Labs are wonderful opportunites for students to demonstrate their knowledge and learning in a more active format. It also helps students that struggle with learning content from a screen or a person another way to learn material.

Image Attribution: Allison Meier