The best innovation since the Cartesian coordinate system

I wrote previously about the analogy of an algebra equation being like a pan balance, and recommended a website to use for practicing the concept. But there’s no practical way to do this with three-dimensional objects, because the weight of x would have to change for each equation.

However, there is a way to learn algebra with physical objects — you just have to remember the basic rules without any help from gravity. It’s still much easier than the letters and numbers that have plagued generations of teenagers.

Okay, so what is this system? A set of algebra manipulatives includes little squares to represent ones, two different sizes of narrow rectangles to represent x and y, and larger pieces to represent x2, y2, and xy. The most common way of representing negatives is that the back sides of the pieces are a different color.

What’s so great about them? First, you can use them to solve algebra equations. Rather than a bunch of numbers and letters on paper, you have objects to rearrange. 3x + 6 becomes 3 x pieces and 6 one pieces. These help a student learn algebra just as counters help a first-grader learn addition, because they can do concrete actions to each side of an equation, and see what 3x + 6 = 5x – 4 actually looks like.

Second, and even more exciting to me, is that these manipulatives provide a visual representation of multiplying and factoring polynomials. Rather than trying to remember all the apparently arbitrary rules, a student can see that x times 3 forms a 3x rectangle. The trial-and-error of factoring becomes a simpler matter of rearranging the pieces to make a rectangle, adding pairs of opposites as needed.

It’s algebra without anxiety! Using manipulatives makes the mechanics less error-prone, and reinforces the logical concepts automatically. Over time, students gradually transition to doing algebra exclusively on paper, when they’re ready. These simple plastic pieces are a visual aid as innovative as the Cartesian coordinate system, and I wish more teachers would use them in the classroom.

Now, where can you get some? The best designs are three dimensional: Lab Gear and Algeblocks. Unfortunately both are pricey, and the 3D Lab Gear parts are unavailable as far as I know. The set I use for tutoring is called Algebra Models and is comparatively inexpensive. I recommend the “Cooperative Group Set” for one student — the individual set doesn’t have enough pieces for most algebra problems, unless you use a book designed for the manipulatives. It would be nice to have the “Small Group Set” for even more pieces.

Another option, which may not be much of a cost savings over the Algebra Models, is to cut the pieces out of cardstock. There is also a virtual representation of the manipulatives, but that format is far from ideal.

In the future, I’ll write in more detail about how to use these. Which topic in algebra would you most like me to explain?


  1. [...] the most complete representation of algebra rules I’ve ever seen on a computer. It rivals the Algebra Models I wrote about before. It makes algebra so easy a five-year-old can learn it. And kids love the game [...]