Square Pyramid: Underlying Anatomy

From Tim Brezinski

Snail's Trail Quilt Square

I created this GeoGebra applet based on a quilt square pattern to use in a precalculus class as a visual introduction to the sum of an infinite geometric series. A nice accompaniment is this applet from Irina Boyadzhiev. I also created a Desmos activity with a focus on asking questions which incorporates these two applets.

Rotational and Reflectional Symmetry

I used GeoGebra to create this applet for exploring the rotational and reflectional symmetry of a variety of images.

Euclid: The Game

Complete with a high score list, Euclid: The Game, created by Kasper Peulen using GeoGebra, presents you with a series of compass and straightedge challenges which get progressively more difficult as you move through the levels. You earn a gold medal for completing a level in the minimum number of moves. See if you can get all the way from constructing an equilateral triangle to inscribing a pentagon in a circle!

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.

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.

Indiana Puzzle Quilt

This applet was inspired by a quilt square my mother-in-law made. It provides a nice way of visualizing the  the sum of an infinite geometric series.
Check out the pattern in some real quilts, too!

Update 1/5/2017: See the Snail's Trail Quilt Square

Constructing the Area of a Circle

A beautifully designed applet that walks you through finding the area of a circle by dividing it up into sectors and rearranging them to form a shape which becomes a rectangle as the number of sectors goes to infinity. I recommend using it in conjunction with Steven Strogatz's fabulous article, "Take it to the Limit". (If you like this, you might also like this applet which approaches the same problem in a slightly different way.)

A First Course in Algebra

Dr. Carol J. V. Fisher has created a thorough online Algebra 1 course complete with interactive exercises for every topic. You can also choose to have a worksheet of practice problems and answers generated. I haven't explored the site that thoroughly yet, but there's a link to a beautifully formatted pdf text for each topic and the interactive web exercises are of good quality. I love that there's an option on the trinomial factoring problems to show the graph of the associated quadratic function!


There are also lots of interactive exercises for Geometry and Algebra 2.

Solving triangles

Practice solving 30-60-90 triangles or right triangles, or a mix of both at thatquiz.org, a site which allows you to customize practice on basic elementary, middle, and secondary school math topics. Random numbers are automatically generated for a series of problems based on the difficulty level and problem type(s) you choose.

Turning a Circle into a Rectangle

After reading Steven Strogatz's column Take It to the Limit, I developed this animation (using GeoGebra) of Strogatz's diagram which shows why the formula for the area of circle is what it is, and also provides an approximation for π.

Pythagorean Proof

Jim Morey created this elegant interactive geometric proof of the Pythagorean Theorem.

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.

Interactive Algebra Practice

This site offers an easy to use interface for practicing a fairly wide variety of math topics. I especially like the "graphing" and "formulas" in the algebra section. In these problems, if you choose a difficulty level of 5 or higher you can choose linear graphs, quadratic graphs or a mix. When you have finished the set number of problems, you have the opportunity to redo the problems you got wrong.

Euclid's Elements

This site contains the entire contents of Euclid's Elements and with each proposition is a diagram you can manipulate in order to help you understand the proposition. Very cool.

What's my angle?

This is a nice little site to use for demonstrating how to use a protractor and for allowing students to get feedback on whether they are using the protractor correctly.

GeoGebra

I have only briefly previewed this free software created by Markus Hohenwarter for exploring algebra, geometry, precalculus, and calculus, but I'm excited about the possibilities I see there! The geometric construction tools are reminiscent of those in Geometer's Sketchpad. The algebraic tools seem more extensive than in GSP. Associated with the software is the GeoGebraWiki site, where educators can post resources related to GeoGebra.

The National Library of Virtual Manipulatives

This site from Utah State University has a nice collection of accurately named virtual manipulatives that can be used in K-12 math education. These are organized both by mathematical topic and by potentially appropriate grade level. I would say that the target audience for most is probably grades 3-8, but there are some which can be used by younger or older students. Each manipulative has clear instructions and notes for the teacher or parent. Many also have associated activities which can be accessed at the click of a button.