Observe and Ask

This simple Desmos sketch is designed to elicit observations and questions that will lead naturally to an introduction of the ideas of domain and range and an exploration of power functions. I show it to a class without any explanation and as I move the a-slider I ask for questions (which I write on the board, but don't answer). If students want the k-slider moved or if they want to see a particular value of a or k, I oblige. Eventually I ask students to begin trying to answer some of the questions and see where that takes us.

Compass and Straightedge Construction Challenges

Ancient Greek Geometry from Nico Disseldorp's Science vs. Magic is a simple but elegant applet which gives compass and straightedge constructions the feel of a game. You start with two points and are given a list of things to construct, along with the number of steps in which you ought to be able to complete each construction. Click on two points and you create a line segment between them. Click one point and drag and you get a circle centered at that point. You score when you complete each construction, and score more when you do it within the number of steps prescribed. The first task is Proposition 1 of Book 1 of Euclid's Elements. The applet works equally well on a tablet or a standard computer.

Intuitive Notion of the Limit - One-Sided Limits

This applet, which is part of Marc Renault's GeoGebra Calculus Applets collection, provides an intuitive introduction to the idea of a limit of a function as x approaches c from the right, left, or both.

The red, green, and blue lines along the x-axis, y-axis and the function which shrink as you decrease δ help with the visualization of motion along the function toward the point where x = c and the corresponding motion toward the limiting values on the axes. Choose from the good library of example functions that are included or enter your own. (I like to start with my own rational function which looks linear but actually has an unseen hole to emphasize from the very beginning the difference between the value of a function and its limit at a point.)

When I used this with a class for the first time, my students asked to see the examples with "Exploding" in the title and they reacted audibly to the motion of the green lines on the y-axis as the function approached its asymptote. My experience has been that when introduced to the idea of limits on a graph there are always a few students who have a lot of difficulty visualizing the various ways in which certain things approach approach other things, and starting with this applet seemed to really help.

Puzzle

Can you figure out how the p and q sliders control the lines? To check your answer, try typing in equations of the lines (in terms of p and q) in the blank rows underneath the sliders and see if the lines that your equations produce match the lines that are already there. (When you've found something that works--but not before, that's cheating!--you can scroll down to rows below the blank ones to see if your equations are the same as the ones that were actually used to produce the lines.)

Operations on functions graphically

A.B. Cron has created a series of GeoGebra applets that demonstrate operations on functions graphically. You can enter any two functions (f and g) and then, from their graphs, determine points that will be on the graph of, for example, h = f + g. After plotting a number of points, you can check the box to show the graph of h to check your work. The adding functions applet has links to the applets for subtracting, multiplying, dividing, and composition.

(Links updated 7/30/2016)

Finding logs

This is a clean, simple applet by Michael Borcherds that provides practice finding logs. It keeps track of how many you got right on the first try and how much time you've spent. To restart the count, refresh the page.

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.