Monday, July 28, 2008

Have you used Algebra tiles?

I introduced integer tiles to my Algebra classes early last fall, and then quickly gave it up. Several students balked at using such middle school measures for studying math, and my arguments that being able to represent math statements in many different ways, including with concrete objects, failed to persuade. In a class of insecure freshmen still figuring out their relative positions in the class, and in some cases still stinging from having been placed in Algebra rather than in Geometry after the placement test, using materials perceived as childish just wasn't socially acceptable.

I quietly dropped the project, only including a problem on modeling integer subtraction as an extra credit problem on a unit test some time later. Not one student got it right. Later in the year, when a number of students continued to demonstrate confusion about combining signed integers and combining like terms, I sometimes wished I'd stuck with the manipulatives a little longer.

Now I'm trying to make up my mind about whether - and, if so, how - to use tiles in my Algebra classes in the fall. Apart from the probable social issues to deal with, I'm wondering about the efficacy of Algebra tiles. A point I've picked up in passing while reading this summer (I'm sorry I can't recall where!) is that the same students who are likely to have much trouble with elementary Algebra are also likely to have difficulty picking up how to manipulate Algebra tiles.

There is, after all, no magic involved. The rules for representing addition and subtraction of integers with bi-colored tiles are not self-evident or trivial. Even for me, the representation of subtraction problems by adding the necessary number of "zero pairs" came as a bit of a surprise. And while I then found the very idea to be very cool and exciting, that is more than I can take for granted that my students will, even if they aren't unable or unwilling to master the rules of the game.

So, what are your experiences with Algebra tiles? How to you go about changing the image of tiles as being all too elementary? And assuming that you have gotten your young charges to take the tiles seriously, how much do you feel that the students learn this way that they do not learn just as well by simply reiterating the formalism of signed numbers and like terms?


Alane said...

I used some homemade algebra tiles while teaching 7th graders. (It was really just red & blue tickets - like you might get at a basketball game.) The kids "got" them OK, but it took a lot of reconstructing to be able to call on that later in the year. ("Remember how we did this... remember how we did that....")

I tried something else that I like better, though less concrete. I told them to think of positive numbers as girls and negative numbers as guys. Imagine they're at a dance, and everyone has to pair up. -5 + 3 means five guys and three girls. There will be 2 guys without partners. That worked well. Later in the year, when I wanted to remind them of it, all I had to do was say, "It's like the girls and guys," and they remembered immediately.

Good luck!

jd2718 said...

I like the two-sided chips for addition of integers,

but I prefer to use drawings where the algebra tiles are called for. I offer an area model for multiplying polynomials.

Did you mean the chips, or the algebra tiles (with an x^2 tile, an x tile, unit tiles, etc)?

H. said...

@Jonathan, I mean both or either: For any kind of manipulative, 1) does it work, and 2) how do you persuade kids to take it seriously?

Mr. K said...

I've used algebra tiles as an end of year exercise for 7th graders.

I understand the reluctance to use manipulatives, and I'm not sure where to start with that.

The way I started is to just pass them out, and let them play with it a bit. Notable observations that they should be able to make for themselves:

* There are three different shapes.
* Each shape has a red side.
* Even though there are three shapes, there are only two different lengths represented.
* The long side is not an exact multiple of the short side. (they can see this more readily by matching up several sticks with the small squares).

Once they've figured this out, explain that the short length is one, and the long length is X. Ask them to figure out the areas of the 3 different shapes.

Then give them some worksheets with various tiles drawn for each problem, and ask them to come up with the expressions for each group of tiles. Then reverse it, and have them draw tiles for given expressions.

You can then go to adding expressions, multiplying expression, and factoriing.

Then, you go back and do the whole thing with negatives (i.e. flipping tiles).

Sarah said...


My experience was similar to yours. I tried a lesson with older students, was rejected, and was new enough to teaching and to tiles that I didn't push them at that time.

Later in the year, after talking to Dan Greene about the General Math class, I started using more manipulatives for the lower level math classes.

Most of my freshmen kept rejecting them. But by then, I knew enough of the tiles to be able to bring them back to some older students who were struggling with the material. (Especially signed arithmetic.) The older students were more willing to use them then. I got the "That's all it is?" reaction a lot. By that point in the year, they didn't have time to practice and then be weaned off the manipulatives. But I was glad to let them use them when they wanted to.

I don't think I'll require use of Algebra tiles in my class next year. And my process of modeling them for the entire class rarely worked. (Way too much lecture, not enough doing.) But I'll keep them accessible and be ready to model them to a small group at a time, "Just to give you another way to think about this."

Sarah said...

Also, very much a fan of National Library of Virtual Manipulatives. I especially like how you can change the length of the x-variable, so it really does feel unknown. With actual tiles, I always face the "but x=5" struggle. (From myself, my mother, and my sister. All abstract math-minded people.)

H. said...

@Alane - Using guy-girl pairings to help students remember things did work for teaching what it means for a relation to be a function- students remembered that far better than any other analogy! I might try it for signed integers too, then - it's not the same class as the one doing functions, so little danger of confusion.

@Sarah - I love the NLVM site too, but have mainly used Grapher and Algebra Balance Scales, and never the app for Algebra Tiles until now. That may just solve the problem, though! My new math ed researcher acquaintance mentioned that students she has tutored were less resistant to using manipulatives when they were on a computer - and our laptop cart might provide just what's needed to help students get over their fear of being underestimated. The "Just another way to think about this" line is a keeper, too. Thanks :)

@Mr. K - Thanks for the details. And for your collection of cool signed number activities from earlier entries, too, linked here for the convenience of having them in one place.

H. said...

@Sarah - NLVM has color chips for modeling integer subtraction, too!

Jackie said...

I use manipulatives quite a bit in my high school classes - pattern blocks, dice, spinners, tiles for modeling probability...

I think one of the ways to get kids to use them is having them available for kids to use whenever they feel the desire - in addition to having them for specific activities.

I too use an area model for multiplying polynomials - and for factoring.
The algebra tiles for operations with integers worked for some kids, for some a number line worked, for some ... I think the trick is finding a way to introduce a few ways without confusing those who understood a the first way.

Glenn said...

I used algebra tiles. I had the same issues as you did, but I forced it through and kept hammering on it. When the learners got to the factoring stage, many were glad I kept using them.
I have a real problem with the NLVM algebra tiles though. It is the best source for virtual algebra tiles (I found 5 different sites that offer algebra tiles), however it truly suffers from a lack of -1.
You can not model (x-1)(x+1) for instance. This means that you are severely limited in what you can model.
This site allows negatives, but it generates the equations. You can not match what the examples in the book or homework problems.
This one uses them for solving, but not for factoring. Another example of single purpose java.
This one: is probably the best one I found. It is generic, allowing any equation to be modeled. It has negatives and positives, and it could be used to model addition, subtraction and factoring. It CANNOT be used to model multiplication without a lot of explaining (no side bars to put the two terms like NLVM) has. Oh, and the colors don't show on my projector. My giant windows wash out the red and blue on the black background.

So in the end, we have a bunch of tools that are all single purpose tools that all look slightly different and none can do what an elmo and a cheap set of plastic tiles can do.

I have written the NLVM many time requesting a -1 tile, perhaps of more people do it, they will get the message. It is the best tool, even with that flaw.

H. said...
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H. said...

Thanks, Glenn! I think I'll be using cardboard tiles for the most part (can't hog the laptop cart the whole time) but using computers when introducing the manipulatives. NLVM integer chips first, hoping that this approach to dealing with negative numbers carries over when we do algebra tiles later.

Some of the links appear to get cut off when published here, while they show up complete in Google Reader, so here they are over again: The Strader site worked well for factoring. The HRW site for modeling equations is neat and clean, but unlike the NLVM Scale Balance it does not provide immediate feedback on whether an operation performed by the user preserves the equality of both sides. The x-power site affords no obvious way of rotating the tiles, making it unclear whether it can really be used for factoring - but it was very neat for representing signed algebraic expressions.

Is it a good idea to include formal assessments of students' skills at modeling with manipulatives? Should students be graded on this?

jd2718 said...


I've used the two-sided chips for adding integers, and used them well.

(I ditched them for subtraction because their using requires building an abstract, artificial construct)

When I've used them I've called them "toys" not "manipulatives," and little speech up front, everyone is going to get the toys, and for some of you the toys will make addition clearer. Some others already know this stuff. That's ok, everyone gets to play.

If you get bored with what we are doing you can work your problems with paper and pencil, or you can build with the toys (but if you have a tower and it falls over, you'll lose them).

And then I teach addition. Somewhere I have a worksheet or two that gives a bunch of integer addition questions, shows how the chips work, and requires the kids to draw pictures of their chips for each one.

So the kids who think they are babyish work ahead, drawing what the chips would look like for each example. Others create the example, then copy it. And a few (usually boys) stack their chips as high as they can.

The drawing part, I like that. Especially when kids do this:


= ++

and I make them share it out.

The algebra tiles, I don't like using them. Too much set up/explanation. We can draw equivalent pictures (which I do for multiplying, and later for completing the square)


jgonz413 said...

I have a class of 9th graders who were close to failing their state exam last year and so they have 1 and 1/2 hours of math everyday. They had trouble with distributive property so I brought out Algebra tiles and some hated them, others got it and some were so excited because they finally understood. I used them with my on-level Algebra kids for 1 step equations and I'm not sure how helpful they were. I will continue to use them because I definitely have some kinestic learners - so I think it does help - keep you posted.