Algebra Tiles

[PeterPan]

http://media.glnsrv.com/pdf/products/sample_pages/sample_JMT070.pdf

 

The new Horizons pre-algebra uses algebra tiles. The link above has a sample where they show how they do them. And doesn't MUS do something similar? Thing is, I don't get WHY one needs to fiddle with it like that. Is that somehow easier than just using the distributive property and doing the multiplication? I don't know, I'm just missing it. Anyone have any experience with this to comment?

[8filltheheart]

I don't use the tiles b/c my kids just don't need visuals to understand what is happening. But, I can give you a simple explanation based on what MUS uses them for.

 

If you have 6*9 and were using tiles, the 6 rows 9 times would make a nice little rectangle When dealing with equations like x^2+ 7x+10 which factors into (x+2)(x+5), again you will end up with a nice little rectangle b/c the 2 10 rods across and the 5 10 rods up require 10 little 1 rods to fill in the missing space. (I have no earthly idea how to represent that visually here! )

 

However, if you have x^2 +7x+3, it will not form a nice little rectangle b/c it does not factor out neatly.

 

Does that make any sense? Personally, I think it makes it more complicated to understand than if kids simply understand mathematically what is happening.

[Dana]

I know that I didn't see why I couldn't add x and x^2 when I was starting algebra. They both had x. Algebra tiles would have helped me see the difference.

 

For some students, using the tiles as a start to solving linear equations would help. You can have a physical representation for the symbolic algebra.

 

I don't know that I'd use them much for factoring.

 

If a student needs them for the transition from concrete to abstract, it's wonderful that they're an additional tool. If a student doesn't need them, then that's great.

 

I know they'd have helped me with understanding like terms and with solving linear equations.

[regentrude]

Never used any "visuals" for algebra with either of my kids.

[siloam]

Hands of Equations also uses visual/hands on to represent the equation, the the word problem book is quite challenging.

 

Looking at these visuals I don't get them right off, but that doesn't mean they aren't valuable. My oldest can do simple algebra in her head, but give her a problem she can't do it her mind and she has a hard time translating what she is thinking to paper. The manipulatives let her play with what is going on in her mind, bridging the gap real and conceptual.

 

Heather

[PeterPan]

Oh I like HOE and think it's good for anyone. It's just these algebra tiles that seem to add a layer of work without improving the conceptual understanding.

[Kuovonne]

Oh I like HOE and think it's good for anyone. It's just these algebra tiles that seem to add a layer of work without improving the conceptual understanding.

 

HOE only works with linear equations. Algebra tiles allows you to work with quadratic equations (x squared). Because x is a bar and x^2 is a square, the student can easily see that they are different quantities yet related.

 

Algebra tiles also help demonstrate the distributive property when factoring and multiplying. When I learned algebra, I learned (x+1)(x+2) = x^2 + 3x + 2 and could factor/multipy both sides; but I didn't see *why* until I built it with algebra blocks as an adult and saw the rectangle with the algebra blocks.

 

Also algebra tiles (vs. MUS blocks used in the same way) are unique that "x" rods are not co-measurable with the "one" blocks.

 

If your kid understands algebra without algebra tiles, that's great. You don't need them. But they can help make some algebra concepts tangible.

[5LittleMonkeys]

We haven't gotten to Algrebra yet but I know my dd11 will need something like this. Numbers on a piece of paper mean nothing to her and formulas are hard for her to remember. If she can get it in her hands and manipulate it and actually see it then she can understand. Before I moved her back in level we were working on fractions with MUS. I started out just trying to teach her the same way as her older sister...without the manipulatives. She couldn't get it because she couldn't visualize it in her head. Once I gave her the manipulatives she got it and had some visual reference to fall back on.

[PeterPan]

Interesting explanation Kuovonne! Well we'll just have to see what happens with dd. Thanks!

[Kuovonne]

Okay, I took a look at the .pdf file linked. It looks like in that example, a the problems are very simple: x(x+1); x(x-1); and x(x+2). I agree that at that stage the manipulatives probably aren't necessary if the student understands the distributive property. I guess that the purpose of the lesson is to get students familiar with the manipulatives and negative numbers. Algebra tiles are much more useful when both factors are x plus (or minus) a whole number.

[PeterPan]

Whoa, now you're sparking my curiousity!!! I'm trying to envision how this would work. And yes, THAT is why I posted that link, because it seemed cumbersome. But this is really nifty. Hmmm. See I had this fleeting idea that I could have dd do the Horizons Pre-Algebra along with the MM pre-algebra pages and hopefully between the two nail things in her mind. I've done some of the Dolciani pre-algebra with her, and I'm afraid she may get lost in the nuances and miss the point. For instance we were doing absolute values, and while she totally go them (and enjoyed the C level problems), there just seemed to be this sort of disconnect. I remember having that myself with Dolciani growing up. I think part of the trouble was not really buying into their whole number line way of viewing math. If you don't like that, then the whole thing was shot.

 

Well that's neither here nor there. I was just trying to determine if the Horizons pre-algebra is good or whether they've launched into a popular pony trick in order to LOOK appealing to homeschoolers. I'm open to other ways of viewing the math for dd, since I don't know if Dolciani will click or not. Nuts, there's no need for Dolciani. It made me vomit all the way through school; no reason it has to her. But she's too good at thinking to do less than a good program. I like the MM sheets we've done so far for the thought process, so I thought those plus the Horizons (for a more basal program) might be just the ticket. The MM sheets aren't meant to stand on their own.

 

I'm just talking out loud. I clearly don't know what I'm doing. :)

[Capt_Uhura]

OHElizabeth, which MM sheets are you using and why do you say they aren't meant to stand on their own?

 

Capt Uhura

[PeterPan]

The MM site says her pre-algebra and algebra 1 worksheets do not include instruction and are meant to supplement. That's what I was going on.

[PeterPan]

Also algebra tiles (vs. MUS blocks used in the same way) are unique that "x" rods are not co-measurable with the "one" blocks.

 

 

Can you explain this a little more? I thought I understood, and after watching the MUS demonstration, I realized I don't. :)

[8filltheheart]

Can you explain this a little more? I thought I understood, and after watching the MUS demonstration, I realized I don't. :)

 

Essentially the only difference is that there are no demarcations on the rods. Since the rods are meant to represent variables, they have no set value. You can use cuisenaire rods or mus's rods but if you use a cuisenaire 100 block and 7 10 rods and 10 one rods for the example I gave in my OP, only the 1 rods actually represent their actual value. The rest are meant to represent x.

 

(It is sort of like using the colored cuisenaire rods without the little notches in them)

[Capt_Uhura]

THanks OHElizabeth for the clarification!

[Kuovonne]

Also algebra tiles (vs. MUS blocks used in the same way) are unique that "x" rods are not co-measurable with the "one" blocks.

 

Essentially the only difference is that there are no demarcations on the rods.

 

It's actually slightly more complicated than that. Not only do algebra tiles have no demarcations, their length is not a multiple of the "one" block.

 

With the MUS blocks with the algebra inserts, it is sometimes physically possible to build an incorrect rectangle for factoring.

For example, x^2 + 7x + 10 could be build as two different rectangles using MUS blocks (one correct, and the other incorrect) but with algebra tiles, only the correct rectangle can be build.

 

The tray on the left contains the correct factoring with both algebra blocks (top) and base 10 rods (bottom / same as MUS blocks). The tray on the right contains the incorrect factoring. Notice that you can make an incorrect rectangle with the base 10 rods, but not the algebra blocks.

 

If you want to learn more about algebra blocks, I found this web page informative:

http://www.mathedpage.org/manipulatives/alg-manip.html

[PeterPan]

Ok Kuovonne, I've spent the last 1/2 hour looking at your pics and the MUS demonstration here https://www.mathusee.com/about-us/demonstration-video/ That link you gave was helpful. I think I actually read around his site a bit ages ago. In any case, it now makes sense to me that they're discussing area, which of course seems pretty useful and like a reasonable way of pondering things. However I'm still not sure I get your pictures. How many green singles do you have in those pics? I couldn't tell. Are there 10 in both? Are you saying you added a single to the iteration on the right? I just missed where the error is. It seems by definition (according to that mathed site) you're to make a square. I'm not sure where the rectangle comes into it.

 

If the length of the x or x2 units are not in similar units to the singles, then how would you get consistent results for the corner single quantities???

 

And are your algebra tiles a particular brand? Just wondering. That mathed site mentioned quite a few brands.

[Kuovonne]

However I'm still not sure I get your pictures. How many green singles do you have in those pics? I couldn't tell. Are there 10 in both? Are you saying you added a single to the iteration on the right? I just missed where the error is. It seems by definition (according to that mathed site) you're to make a square. I'm not sure where the rectangle comes into it.

 

All four representations in my picture have the same number of blocks: one "x^2", seven "x", and ten "ones". For algebra blocks, the orange square represents x^2. The orange rods represent x. (Notice that the x^2 has a length and width of x.) The green blocks are "one." Notice that the "x" rod has a length of "x" but a width of "one." For the base10blocks/Crods, the hundred flat is "x^2", the orange(10)crod is "x" and the little marshmellow blocks are "one."

 

In the picture on the left, x^2 + 7x + 10 is factored correctly (x+2)(x+5)with both algebra blocks and base10blocks/Crods. Notice that each rectangle has a height of (x+2) and a width of (x+5). Remember the dimensions of the x rod is "x" by "1". When you multiply (x+2)(x+5) using FOIL before simplifyig, you get x^2 + 5x + 2x + 10. You can see these four groupings within the rectangle. The x^2 is the x^2 block in the top left. The 5x is five x rods in the top right. The 2x is the two x rods on the lower left. The 10 is the ten "one" blocks in the lower right.

 

In the picture on the right, the exact same blocks are arranged into rectangles. Notice that with the base10blocks/Crods it is possible to form a rectangle with those blocks, implying an incorrect factoring of (x)(x+8).

However, it is *not* possible to form this incorrect rectangle with the algebra blocks, because the algebra blocks are not co-measurable. In this picture, the algebra blocks do not form a tidy (incorrect) rectangle, those "one" blocks stick out too far or not enough.

 

If the length of the x or x2 units are not in similar units to the singles, then how would you get consistent results for the corner single quantities???

 

The length and width of the x^2 block is "x". The length of the "x" block is "x," but its width is "1". That is how you can fill in the corner with "one" blocks.

 

 

And are your algebra tiles a particular brand? Just wondering. That mathed site mentioned quite a few brands.

 

My algebra tiles are the AlgeBlock brand. I happen to like them because

1) They are on the same scale as Crods and base10 blocks (which I also own) and can be used with them.

2) They provide an "x" as well as a "y". (Technically I used the "y" blocks in my example, but they work the same way.)

3) They work in three dimensions. There are x^3, y^3, x^2y, and xy^2 blocks. The rods and flats all have a depth of "1" which is mathematically correct.

 

Of the different algebra tiles mentioned, these blocks are my favorite and the only ones I own. Unfortunately I think they are too pricy for homeschool purchase. I got lucky and found some used. On the other hand, it is very easy to print your own algebra tiles if you want tiles but not the expense.

Edited February 7, 2011 by Kuovonne