Visual Algebra

I will be teaching two instances of my Visual Algebra workshop this summer: grades 7-11, June 27-30, in Oakland; and grades 6-9, July 25-27, in Saint Louis. (More info). This is probably the topic I have presented the most often in the last 25 years.

To many students, not to mention teachers, parents, and administrators, algebra is exclusively about manipulating symbols on paper. This approach is deeply embedded in the culture, but alas, it is only effective with a very small number of students. It is in part responsible for the minuscule number of adults who feel they learned something lasting in algebra class. (I wrote about the limitations of the traditional approach recently, with a specific focus on equation solving.)

Symbol sense is important, but memorized symbol manipulation skills do not in and of themselves develop symbol sense. I have had much better luck with a visual approach. In that approach, symbol manipulation is is not the starting point. It is a stop along the way.

I agree with the Common Core State Standards writers that “real world” modeling is one key to bringing algebra to life. Back in the mid-90’s, there were few curriculum options with a lot of modeling (one of them was my Algebra: Themes, Tools, Concepts book, now free on my site.) Luckily, it is no longer difficult to find materials along these lines. Graphing, manipulatives, and function diagrams provide a helpful complement to the modeling approach. They are what I call visual algebra. I will use this post to provide links to visual algebra materials I created, and to highlight a few new items.


Visit my page on electronic graphing, which links to quite a few concept-oriented activities, including the ever-popular “Make These Designs”.

What’s new? I made a Desmos version of my STAIRS tool, which I had originally created for the Texas Instruments calculators*. (I got some help with JavaScript, and with the Desmos API. Thanks, Brandon! Thanks, Eli!) It makes for a great activity or two on slope and slope triangles. Try it out. I did not make a worksheet for it. Instead, at the bottom of the page, I suggest ways to use the tool. You, the teacher, can pose specific questions to get students started, and/or ask students to set goals for themselves. I would love to hear how this goes in the classroom.


About 26 years ago, I developed a comprehensive hands-on environment for algebra, called the Lab Gear. It was a big success until its publisher was gobbled up by McGraw-Hill. But the Lab Gear is back, from Didax, with a much improved distribution of blocks, and updated Common-Core-compatible books. Going far beyond the one-shot use of algebra tiles, the Lab Gear offers a representation that works for many, many topics in algebra. Read more about it here.

What’s new? I now have 11 short, animated slide shows demonstrating various Lab Gear techniques. I will make one more, and then index them to the books. The slides are mostly intended to help teachers learn how to use the Lab Gear. (As I mentioned in a recent post, it only makes sense to use the slides in the classroom after or along with student use of actual blocks.)

Function Diagrams

This is a parallel axes representation, useful to teach a wide range of concepts, from the basics of operations in middle school to the chain rule in calculus. Because it is initially unfamiliar, it provides a nice change of pace, and an equalizing context in the classroom. Check it out here.

What’s new? Nothing, actually, but I’m hoping someone will create Desmos versions of the function diagram GeoGebra applets. If you do, let me know!

If visual algebra sounds good to you, and you can get the needed funding, sign up for the summer workshop!


* Actually, before graphing calculators even existed, I had created a Grapher in the Logo language, and STAIR was one of the several add-ons I created for it. That’s how old I am.

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