Algebra Manipulatives

There are four main commercial versions of algebra manipulatives. In order of their appearance on the market, they are Algebra Tiles (Cuisenaire), the Lab Gear (Creative Publications, now Didax), Algeblocks (Southwestern Publishing), and Algebra Models (Classroom Products). All four provide a worthwhile model of the distributive law at a basic level.

However, note that students much prefer blocks to tiles, and moreover, only blocks (the Lab Gear and Algeblocks) allow work in three dimensions. Other than that, the main differences are in the representation of minus, and in the physical handling of the pieces. I return to these issues below, but first here is a quick overview of the four.

My creation! The Lab Gear includes blocks for 1, 5, 25, `x`, `5x`, `x^2`, `x^3`, `y`, `5y`, `y^2`, `y^3`, `xy`, `x^2y`, and `xy^2` — as well as a corner piece used to separate a rectangle or box from its dimensions.

Read about it in:

• Lab Gear Q and A
• How one middle school teacher uses it: The Great Connector
• Early Mathematics
• A New Algebra

And/or see the applet representing the factoring of a trinomial.

For a full presentation of the Lab Gear, see:

• The Algebra Lab Gear books (from Didax)
• My Lab Gear video PD series.
• Algebra Lab: High School
• The Algebra: Themes, Tools, Concepts textbook

Go to the The Lab Gear home page for all the links and for purchasing info.

The Algebra Tiles are inexpensive and widespread. They make it possible to do most of the activities that are needed to introduce and explain the distributive law and factoring.

• The Algebra Tiles do not provide a corner piece. Since the corner piece is a very important support for beginners, I recommend you add corner pieces to your algebra tiles, either by purchasing them separately, or by just drawing them on paper.
• The absence of 3-D blocks is a limitation, but much can be done in two dimensions.
• Likewise, the fact that there is only one variable reduces the range of the model, but it is not fatal.
• The lack of 5, 25, `5x`, and `5y` blocks does not interfere with the key concepts, but it makes it inconvenient to represent expressions involving larger numbers.

The representation of minus is the most controversial part of manipulative models of polynomial algebra. The four commercially available products each handle it in a different way. These ways have consequences in four areas:

• the misconception that -x is negative, and x is positive
• the integrity of the area model of multiplication
• the representation of expressions such as `5x-(x-1)`
• the complexity of the model

The Algebra Tiles model of minus is based on color: one color corresponds to positive numbers, and another to negative numbers. This is then generalized to variables: blue `x`'s represent `x`, and black `x`'s represent `-x`.

• Advantage: the model is simple
• Disadvantages:
  • the model reinforces the misconception that `-x` is negative, and `x` is positive;
  • the area model of multiplication is not geometrically sound when minus is involved;
  • only the simplest expressions involving minus can be represented.

The Algeblocks model of minus is based on position: blocks in the shaded area of the various Algeblocks mats are taken to be preceded by a minus.

• Advantage: the model is simple
• Disadvantages:
  • the area model of multiplication is not geometrically sound when minus is involved;
  • only the simplest expressions involving minus can be represented.

The Lab Gear model involves two different representations, both based on position: blocks in the minus area of the workmat, and blocks sitting on top of other blocks ("upstairs") are taken to be preceded by a minus.

• Advantages:
  • the upstairs representation allows for a geometrically sound representation of minus in multiplication;
  • the two representations can be combined to represent somewhat more involved expressions such as `5x-(x-1)`.
• Disadvantage: it is initially harder to learn than the other models.

The Algebra Models model is based on color. However, at least in the CPM implementation, this is combined with a workmat-like minus area, which makes it possible to do much work with reading and simplifying expressions, equation-solving, and the like -- very much like what can be done with the Lab Gear. Moreover, CPM avoids using minus in the area model of multiplication, and thus the geometric integrity of the model is not undermined. (See next section.)

With the Algebra Tiles, to multiply `(y+3)(y-2)`, you do the same thing that you would do with `(y+3)(y+2)`, except that you use the "minus" colored tiles for -2, and for -`2y` and -6 in the product.

Similarly for `(y-3)(y-2)`, you do the same thing that you would do with `(y+3)(y+2)`, except that you use the "minus" colored blocks for -2, and for -2`y` and -3`y` in the product.

In this model, the rectangles representing `(y+3)(y+2)`, `(y+3)(y-2)`, `(y-3)(y+2)`, and `(y-3)(y-2)`, are all congruent, which is geometrically incorrect, since for example `y+3` clearly should be longer than `y-3`.

The Algebra Models have exactly the same problem.

With Algeblocks, factor blocks are placed on the right or above the center of the quadrant mat if they are preceded by a plus, and to the left or below, if they are preceded by a minus. The resulting product subrectangles are considered to be preceded by a plus for the 1st and 3rd quadrant, or a minus for the 2nd and 4th quadrant.

In this model also, the area model loses its geometric integrity when minus is used:

• The factors are sometimes embedded in the product rectangle, distorting its dimensions.
• Once again, the rectangles representing `(y+3)(y+2)`, `(y+3)(y-2)`, `(y-3)(y+2)`, and `(y-3)(y-2)`, are essentially all congruent, which is geometrically incorrect.

With the Lab Gear, the upstairs representation of minus accurately represents `y-2` as 2 units shorter than y. It is then possible to multiply in the corner piece, and use upstairs blocks in the product, to obtain rectangles with geometrically correct dimensions. However how to do this is not easy to learn in a case such as `(y-3)(y-2)`, and many Lab Gear users stop short of teaching this to their students. This is not a big problem, because the Lab Gear materials encourage a transition from blocks to symbols with the help of the "multiplication table" format for polynomial multiplication ("the box"). That format is visually related to the basic Lab Gear multiplication format, it works fine with minus, and it guarantees that the student does not remain dependent on the blocks.

In the Lab Gear model, variables are blue, and constants are yellow. (I chose yellow to match my then-publisher's Base Ten blocks.) This makes it possible to ask some questions like "Make a rectangle using an `x^2`, six `x`es, and as many yellow blocks as you want. Is a square possible?"

Some teachers have suggested that `x` and `y` should be different colors, for example blue and red. I guess that is conceivable, but then `xy` would have to be purple? Doesn't this imply `x^2y` and `xy^2` would have to be different shades of purple? It may be counterproductive, as students could identify the `xy` block by its color, rather than by its dimensions, which is the more relevant geometric consideration. In any case, it seems unnecessarily complicated.

I encourage you to compare the curriculum materials provided with the four models. In my opinion, the Algebra Tiles manual is confusing and essentially unusable. There is much of value in the Algeblocks binder, which seems in large part inspired by the original Lab Gear books. Moreover, their ingenious (if mathematically flawed) approach to minus does make the Algeblocks easier to use.

But for mathematical correctness, depth, and range, check out the Lab Gear materials. Specifically, my Algebra Lab Gear: Basic Algebra and Algebra Lab Gear: Algebra 1 (Where to get them.) They are the result of many years of work with algebra manipulatives, with both teachers and students, and include all the basic lessons on the distributive law, a rich introduction to equation solving, plus three important features:

• explicit attention to the transition from blocks to symbols
• connections to the graphing of linear and quadratic functions
• a remarkably accessible approach to completing the square and the quadratic formula

For extended work on integer arithmetic, plus a general introduction to Lab Gear at a more basic level, see The Algebra Lab: Basic Algebra. For an approach to more advanced topics, such as solving systems of equations, quadratic connections, and work with algebraic fractions, see The Algebra Lab: Algebra 1. (Where to get these books.) For many lessons using algebra manipulatives, including some seminal ones, and some available absolutely nowhere else (e.g. a model of polynomial long division), see The Algebra Lab: High School, a whole book in PDFs available for free download on this site.

Many of the best ideas I have come up with in this domain have been incorporated (unfortunately without attribution) in CPM's Algebra Connections, a rather wonderful Algebra 1 textbook. This includes everything from the engaging perimeter problems, to using "make a rectangle" puzzles prior to the introduction of the distributive law, to "which is greater?" as a lead-in to equation solving, to the step-by-step introduction to completing the square. Hats off to the authors for their ability to recognize a good activity when they see its effectiveness with students, and many thanks to them for bringing these approaches to a much larger population than I could.

To be honest, I have not seen a lot of pedagogical discussion of algebra manipulatives. Here are some of the topics and questions that do come up.

About using algebra manipulatives

About blocks vs. tiles