I am the creator of the Lab Gear, a manipulative environment for learning algebra. It is more expensive than the better-known algebra tiles, in their various versions, but it is much more effective in the classroom. At least that is the opinion of those teachers I know who have used both. As one of them put it, ” I have been having so much fun with my students using Lab Gear this year. The 3D-ness of it totally blows the other (cheaper) algebra tiles that I used last year out of the water!” (You can read a detailed comparison of the various algebra manipulatives here. For a basic introduction, see these videos.)
All algebra manipulatives work well to illustrate the distributive law in polynomial arithmetic. But if you want to go beyond illustrating and involve students at a deeper intellectual level, you should follow the Lab Gear puzzle-like approach, starting with “Make a Rectangle”. (I discussed this in Factoring Trinomials. See also the Lab Gear path to completing the square.)
In this post, I’d like to return to a side trip, one that is pretty much exclusively in the Lab Gear curriculum materials: perimeter problems, such as this one.
What is the perimeter of this shape, which is made up of a 1 and an x2 block? One student may answer 3 + 3x + (x − 1). Another: x−1+ x + x + x + 1+ 1 + 1. The best way to see if the two students agree is to simplify the expressions. And yes, both simplify to 4x + 2, so they do agree. While gratuitous “simplify” problems are not popular among students, when they are debating the results of their perimeter problems, they care about the outcome and vigorously engage in debates about them.
But problems of this type are just the start. They can be followed up by the reverse: can you arrange Lab Gear blocks to make a shape with a given perimeter? Here are teachers exploring a question of this type:
Similar challenges can be asked about the surface area of 3D creations, and the unit can culminate with student-created problems, such as these:
Of course, middle school students being middle school students, they might come up with out-of-control creations, such as this one:
As you can see, combining like terms around this whiteboard is probably beyond what can be accomplished in a single class period!
— Henri
(Thank you, Liz Caffrey, for the photos!)