Henri Picciotto's

Math Education Newsletter

July 2014

In this issue...
The Lab Gear is back
Common Core updates
Responses to my Common Core analysis
Rethinking triangle congruence and similarity
Curricular materials
Soccer angles
Complex numbers (including games!)
Doctor Dimension (functions, rate of change, etc.)
GeoGebra applets
About teaching
... or just scroll down to read it all

The Lab Gear Is Back

Good news! I have a new publisher for the Lab Gear, which will once again be available for purchase after a bit of a hiatus. This is even better than it sounds, because Didax (the publisher) agreed to a new configuration that includes the 3D blocks as part of the basic package. Those had been unavailable for many years. For more information on the Lab Gear's disappearance and its return, see this blog post. If you are considering a purchase, go to the Didax site.


I imagine that many readers of this newsletter have not heard about the Lab Gear, as it has not been advertised for many years. In a nutshell: the Lab Gear is a manipulative environment for algebra, which I developed in the 1990's in response to the nationwide move towards "algebra for all". There had been a history of algebra manipulatives dating back to the 1970's, and my design combined the best of previous models with some of my own ideas. The Lab Gear and the accompanying curriculum have been imitated since then, but in my humble opinion they remain unequaled in mathematical depth and pedagogical savvy. You can read more about it here. For a history and a (rather technical) comparison of algebra manipulative models, click here.

Finally, if you are a Lab Gear user, I would appreciate it if you would reply to this e-mail with a sentence or a paragraph about how it's worked out for your students. I might use it in my blog, or on my Web site, or when advertising the return of the Lab Gear. Thanks in advance!

On to this month's newsletter. (Not that I publish the newsletter monthly! I aim for maybe eight issues a year, depending on how much else is going on in my life.)


Blog Posts

Here are links to posts on my Math Education Blog that you might find interesting.
If you are so moved, you may comment on the posts, and/or subscribe to the blog.

Responses to My Analysis of the Common Core

Many people confuse the Common Core State Standards for Mathematics (CCSSM) with the test mania that dominates public school education. Because politicians' obsession with yearly high-stakes standardized testing is a poisonous attack on both students and teachers, I wholeheartedly support resistance to those tests. However I encourage those who resist to remember that the standards are not the tests, and the tests are not the standards.

In fact, the Standards are a much more complicated topic than the tests. For one thing, many of the goals of the CCSSM are excellent, both in their emphasis on the so-called "mathematical practice standards" and in some of the proposed curricular redirection. A few months ago, I wrote an in-depth analysis of the CCSS for high school math, including both the strengths and weaknesses of the document. I received an overwhelmingly positive response, which I summarized in this blog post.

Alas, it has not been easy to circulate the document, as the conversation is extremely polarized, and my analysis is apparently too complicated to fit on one side or the other. If you can share it with colleagues, please do!

Triangle Congruence and Similarity Under the Common Core

One curricular shift that the CCSSM call for is to make transformational geometry foundational in grades 8-10, and to derive the criteria for triangle congruence and similarity from that. This is a good idea for many reasons, but the authors of the Standards did not offer much guidance on how to actually carry it out.

Never fear! I have written an eight-page document wherein I suggest an approach to this which is founded on (a) the CCSSM-mandated assumptions, and (b) an approach that is logically tight, and based on geometric construction. The document is geared to teachers and curriculum developers, but it will hopefully provide the basis for curricular materials. Read more about this, and get a link to the document, here.

Complex Numbers

Speaking of the Common Core, I was disappointed that complex numbers appear there mostly as an optional topic. To me, they complete a quest that starts in kindergarten, and should be part of every student's education, not because they will "need them", but because every educated adult should know about them. (How many people "need" the quadratic formula, for example?)

Part of the reason for complex numbers' low status is probably that the topic is usually taught poorly. A strongly visual introduction to complex numbers is very accessible, and provides excellent review of basic trigonometry. Some decades ago, I created a complex arithmetic computer game of sorts, which not only reinforces the concepts underlying this topic, but is also quite fun for almost any student. I just recently found a way to make a GeoGebra version of the game. Read more about it here. If you can't wait to play the game, rush to this page.

World Cup Fever

My book Geometry Labs (free download) includes an interesting hands-on activity about the mathematics of soccer, intended for a normal 9th-10th grade geometry class (Lab 1.10.) My Web site includes a full analysis of the problem at a more advanced level. And my friend Scott Steketee, in celebration of the World Cup, created a Geometer's Sketchpad applet to facilitate an exploration on the computer. Read about all this here!

Popular Posts

In the previous issue of this newsletter, I linked to three of the most popular posts ever on my blog. Here are two more, also selected from the top ten.

- Acceleration: The first of a series of four posts on the topic of hyper-acceleration: students being introduced to math subjects at a younger and younger age, often in ways that are educationally counter-productive.
- Enrichment: A discussion of what makes for a rich curricular activity in math class.

New on

Visit my Web site!

Doctor Dimension

Activities featuring a flat scientist, whose life is all about filling two-dimensional containers with two-dimensional liquids. The activities range from Algebra 1 or 2 work on rate of change, to precalculus work on piecewise functions and concavity, to calculus work on continuity, differentiability, points of inflection, and a preview of integration.

Read about all this, plus a comparison with a superficially similar (and wonderful) Desmos activity here, and here. Or go directly to the Doctor's Web page.


A Geometric Construction page, including a whole unit with a teachers' guide, plus a GeoGebra page with tools limited to straightedge and compass construction. Also an applet / file for making triangles given SSS, SAS, etc. and the accompanying worksheet ("Constructing Triangles".) And finally an applet that makes clear why SSA does not guarantee congruence.

Applets Galore

Doctor Dimension and triangle congruence are among the many excuses for new GeoGebra applets on my Web site. For example, I added a bunch of applets to my Geometry of the Parabola page. In order to make it easier to find the applets, I started a Directory, which I hope to gradually grow and organize.

Browsing My Site is vast, and keeps growing. Visitors can find what they're looking for by searching, of course, but I hope that those who are just browsing can get a lot out of the various directories and site maps I offer. In addition to the Applets Directory mentioned above, here are some possible starting points:

- Annotated Map
- Comprehensive Map
- Grade Level Map

- Graphing Calculator
- Learning Tools
- Middle School
- Teaching

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