## Introduction

Even though they cannot make algebra easy, manipulatives can play an important role in the transition to a new algebra course:

- Manipulatives provide access to symbol manipulation for students who had previously been frozen out of the course because of their weak number sense.
- Manipulatives provide a geometric interpretation of symbol manipulation, thereby enriching all students' understanding, and making a powerful connection to another part of mathematics. This also helps justify the underlying logic of algebraic structure, which is unfortunately often seen by students as a bunch of arbitrary rules.
- Manipulatives support cooperative learning, and help improve discourse in the algebra class by giving students "objects to think with" and talk about. It is in the context of such reflection and conversation that learning happens.

However all algebra manipulatives are not created equal. In this article, I will describe the differences between the existing models, and discuss the mathematical and pedagogical implications of each.

## Four Versions

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.

**The Lab Gear**

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**

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.

**Algeblocks**

Algeblocks do have a variation on the corner piece (the quadrant mat) and they do support two variables and three dimensions. They lack the 5, 25, `5x`, and `5y` blocks, which is inconvenient. You can use them to do almost all the basic activities about the distributive law and factoring.

**Algebra Models**

Algebra Models do include two variables, a 5 block, and a variation on the corner piece (the Work Tray.) They are tiles, and thus they are less expensive, and cannot be used in three dimensions. Their main advantage is that they are incorporated in an excellent Algebra textbook: *Algebra Connections*, by the College Preparatory Mathematics program (CPM).

## Mathematical Choices

### Representation of Minus

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.)

### Multiplication with Minus

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.

### More colors?

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.

## Implications for Teaching

### Curriculum Materials

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.

### Pedagogical Arguments

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**

Some professors argue that teaching a whole other representation on top of the usual introduction to symbol manipulation is just too much: it takes precious time, and patient explanations are preferable. There is an element of truth to this: if manipulatives are taught in the "listen to me, and now practice" style, there is little point to that. Algebra manipulatives should be introduced in a guided discovery / problem solving mode, and be the basis of collaboration, reflection, and discussion. With such an approach, it is time well-spent: the resulting understanding reduces the need for excessive practice and re-teaching.

Some apparently stronger students argue that they don't need manipulatives. In some cases that is true, if the student actually understands the geometric model. However in many cases, students who make this claim have a shallow grasp of the symbolic representation and are intimidated by the visual approach. I have found it effective to stress that they should be able to understand and explain the geometric connections. Once they show that, they indeed do not need the manipulatives.

Also note that the manipulatives suggest interesting problems, such as the perimeter and surface area problems in the Lab Gear books. Those are intrinsically worthwhile, and they provide practice in combining like terms. Students do not resent those problems.

**About blocks vs. tiles**

This is a practical question with pedagogical ramifications. Tiles are physically harder to work with, which will lead to quick manipulatives fatigue. This is particularly salient when comparing Lab Gear with any of the other manipulatives: representing (e.g.) `(x+5)^2` requires four Lab Gear blocks (or eight, if you count the ones outside the corner piece.) This can be set up in a few seconds. The same product requires 36 Algebra Tiles (or 48, counting the outside.) This is so tedious as to rule out examples of this type.

In any case, it turns out that students much prefer blocks, and particularly enjoy "Make a Box" and other 3D problems. Such positive attitudes are helpful!

## History of Algebra Manipulatives

The first use of manipulatives to illustrate algebraic ideas was by math educator Zoltan Dienes, who used base ten blocks. Using the "rod" (10) as `x`, and the "flat" (100) as `x^2`, he showed how to use base ten blocks to illustrate the distributive law. For example, `(x+5)(x+2)=x^2+7x+10` can be seen by making a 12 by 15 rectangle, and seeing that its area is 100+7·10+10. The idea was powerful, and launched the idea of algebra manipulatives, but this model falls short when trying to factor: `x^2+7x+10`, represented by 100+7·10+10 can be arranged into one of two rectangles: 12 by 15, or 10 by 18. The first one is the correct factoring `(x+2)(x+5)`, but the second is not: `x(x+8)` does not equal `x^2+7x+10`.

Mary Laycock improved on Dienes' model, by using multi-base blocks. Instead of just working with base 10 blocks, a trinomial factoring had to work in all bases. In other words, the same layout should work whether `x=3`, 5, 6, or 10. She also introduced the "upstairs" representation of minus, which made it possible to represent a product involving minus like `(x-1)(x+1)` in a geometrically correct way.

Peter Rasmussen used base ten tiles, plus 5-, 25-, and 50-tiles for convenience. More importantly, he created the non-commensurable `x`, which solved the problem of false factorings encountered when using arithmetic blocks for variables. He also laid out multiplications in a tray that was a precursor to the corner piece. His MathTiles were placed along the tray's frame in such a way that the student saw them edge-on, thereby suggesting the one-dimensionality of linear measurements of the dimensions of the two-dimensional rectangles inside the tray. His model of minus combined Laycock's upstairs method with a color scheme. The tiles were only painted on one side, so that if a tile is turned over to its unpainted side, it is considered negative. (Number tiles, `x`-tiles, and `x^2`-tiles each had their own color.)

The Algebra Tiles were based on Rasmussen's ground-breaking model, without the upstairs representation of minus. Unfortunately, they dropped the multiplication tray. Fortunately, they did keep the non-commensurable `x`.

The Lab Gear was based on Rasmussen's and Laycock's models, keeping the non-commensurable `x`, and adding a non-commensurable `y`, and blocks for `5x`, `5y`, and 25, for convenience. (Blocks for 10, 100, and 1000 can be readily added to the Lab Gear by using base 10 blocks, which could be quite useful in younger grades.) The corner piece extends the uses of the tray for multiplication into the third dimension, and 3D multiplication is supported by `x^3`, `x^2y`, `xy^2`, and `y^3` blocks. The minus model is extended beyond the upstairs model by the use of the workmat, with its minus area. The workmat is an environment for working with polynomials involving minus signs, as well as for equation solving and inequalities.

Algeblocks are based on the Lab Gear, including the x and y variables, and variations on the workmat. However, the 5, 25, `5x`, `5y`, are missing. The corner piece is replaced by the problematic quadrant mat, and the upstairs representation of minus is abandoned.

Finally, the Algebra Models combine ingredients from all these sources: tiles rather than blocks, `x` and `y` variables, minus indicated by color, the Work Tray instead of the corner piece, a 5 tile, but no 25, `5x`, or `5y`. And of course, no 3D blocks.