Unit Circle Practice

Sam Shah's applet for practicing finding special angles on the unit circle and their sines and cosines

Visualizing radians

Sam Shah shared a GeoGebra applet created by a colleage of his which does a beautiful job of demonstrating what a radian is. He wrote about it here. I made minor modifications, mostly so that the word radian doesn't actually appear. Here's that slightly modified version.

Evaluating Inverse Trig Expressions

I created this applet on GeoGebraTube to help students practice evaluating inverse trig functions of special angles by visualizing both the unit circle and the graph of the inverse trig function.

Variation on John Golden's GeoGebra Ferris Wheel

I love John Golden's GeoGebra Ferris Wheel, which he's written about here and I've written about here. I also love how GeoGebraTube makes it so easy to start with the great work of someone else and create a variation on it. Here's my variation on John's creation for a user who's slightly more experienced with transformations and with sine and cosine functions. Rather than asking the user to come up with parameters for a given function, it requires the user to come up with the whole function. I also added the option to hide the actual function so that the user could begin by thinking about what it would look like.

We had a great discussion in precalculus class today based on a single randomly generated wheel. We went wherever student questions and answers took us and ended up covering lots of ground. The particularly interesting stuff from my perspective came when those who saw the graph as a shifted sine wave were fighting it out with those who saw it as a shifted cosine wave and those who saw it as a flipped cosine wave. Finally lots of people were seeing lots of ways that they might write an equation for a sinusoidal wave. At the very end of class we generated a new wheel and everyone tried to write a function to go along with it. Class was over and people were pleading, "Please--try my equation!"

Inverse functions graphically

Taylor Russell's inverse function applet provides a very nice visualization of the fact that the graph of an inverse function is obtained by switching the x- and y-coordinates of every point. You input the original function, so it is extremely flexible. As an added bonus, you can also plot the reciprocal function and see that it is not the same as the inverse. I used this applet in combination with Emily Alman's Joe the Math Guy comic for my most successful introduction to inverse functions ever.

Mathmo

Mathmo is a review tool for A-level maths developed by the NRICH project at the University of Cambridge. It is advertised to work in Chrome, Safari, and on mobile devices. There are questions on wide range of topics in a typical American high school curriculum, though the range of question types within a topic is very limited. In some topics (logarithms, for example) there are a few different types of questions, but in most there is a single question type where just the specifics (numbers, functions, etc.) vary. You can ask for random questions from the wide range of syllabus topics or can choose your own specific topics to build up a set of questions. You work the problems on paper (or in your head) and then push the check answer button to compare your answer with the given one. If you want several questions on the same topic, you can add the topic multiple times to your question list or can click the new button from within a particular question.

I did experience a couple of minor bugs. Sometimes, the first time you look at a question you see the code rather than the mathematical notation. Clicking (or tapping) the question changes the code to notation. The description says that the color of the question changes once you indicate whether you got the question right or wrong. I didn't experience that either on the iPad or in Chrome.

Radian measure practice

This is one of the simpler of a huge number of elegant GeoGebra creations from Daniel Mentrard. (Those available in HTML rather than Java are here.) You're given an angle which is some integer multiple of pi/12 and asked to place it on the appropriate place on a circle. You then push the test button to see if you've placed it correctly. You can choose between "Stage 1" and "Stage 2" problems. Stage 1 has only positive angles. Stage 2 includes negative angles.

Transforming f into g

Note: If you're not seeing both graphics windows on GeoGebraTube, you can download the applet and run it on your desktop.

In this applet (inspired by Steve Phelps' What's My Rule series) you select a parent function, f(x),  from among 10 possibilities and click a button to indicate the maximum number of transformations (up to 4) that you'd like to have performed on f to produce a new function g. You see only one point on g, but you can move the corresponding point on f, to determine the relationship between the two. When you believe you have found the parameters that describe the transformation, you can show the graph of the transformed function that your parameters create and see if the lone point moves along it.

Here's a Java version of the applet.

Ferris Wheel

This is a full-featured and beautifully designed GeoGebra applet from John Golden that allows students to practice fitting parameters to a cosine function which models the height of a ferris wheel car above the ground as a function of time. You can watch the ferris wheel spin as the height curve is generated and it provides an endless source of practice since you can always generate a new ferris wheel.

Daylight Hours Explorer

This Daylight Hours Explorer is from a fabulous collection of astronomy simulations and animations at the University of Nebraska at Lincoln. Exploring hours of daylight over the course of a year at various locations on the earth is a great way for students to think carefully about parameters of trig functions.

Visualizing Radian Measure

Ted Coe's Unwrapping a Circle applet on GeoGebra Tube is a visually powerful way of demonstrating what a radian is. I've also posted a slight modification of the applet optimized for use as a demonstration.

Spinning Out Sine and Cosine

A nice animation from the Wolfram Demonstrations Project showing the relationship between a circle and the graphs of sine and cosine. I like the way the cosine graph is show on its side. I'm going to try using it (after editing it in Mathematica to remove the sine and cosine labels) as an introduction both to the unit circle and the trig graphs. It looks like a good candidate for the technique of showing an animation with no commentary and then asking for student questions, writing them on the board, and asking for answers to/discussion of any of the questions.

Law of Sines and Law of Cosines

This applet can be used to provide convincing evidence for the Laws of Sines and Cosines and, once the laws are established, to provide practice problems. I like to begin by showing just the measure of an angle and its opposite side and turning on the help to see the ratio of the sine of the angle to the length of the opposite side. Repeating this for the other two angle-side pairs and dragging the vertices around to create new triangles suggests that these three ratios are always equal for a given triangle. Similarly, sliding the dot along the line reveals the expressions and calculations associated with the Law of Cosines. To provide practice problems, turn off the help and choose any three of the six possible pieces of information. Drag a vertex (or two or three) to create a new triangle. Check answers by revealing the remaining angle and side measures.

Modeling with sine and cosine

In this applet, you look at the graph of a data set (for water level in Cape May, NJ on October 13, 2009) and try to fit a sine function and a cosine function to the data. You can type in your function to see how well it fits the data. Some instruction on how to determine the parameters is provided.

Writing Equations for Sine and Cosine Functions

This applet I created with GeoGebra has 40 problems, which get progressively more difficult, in which the student must write an equation for a graph which is a stretch and/or translation of a sine/cosine graph. To check whether an equation is correct, the student types in the equation and looks at whether the graphs match up.

Blank Axes for Trig Graphs

Blank axes designed to make it easy to create standard trig graphs for insertion into word processing documents. The full power of Geogebra is behind it. To copy the graph you've made, choose "Graphics View to Clipboard" from the Edit menu.

Introduction to the Unit Circle

These are three versions of an applet that I created using GeoGebra to introduce students to the unit circle. One version presents the idea in terms of distance traveled around the circle. The other two present the idea in terms of the measure of the central angle. There is a radian version and a degree version.

Quizlet

Of the flashcard sites I've explored, this one is the best I've found for allowing the input of mathematical notation. You need to register (free) to create cards, but others can use your cards without registering. There are lots of good options for working with cards you've created. The interface for basic card viewing and flipping works very smoothly and quickly. Want to try it? Experiment with my cards on sines and cosines and evaluating inverse trig expressions.