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Topic: Algebra Tiles
Replies: 23   Last Post: Feb 12, 2002 6:42 PM

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Posts: 308
Registered: 12/3/04
Re: Algebra Tiles
Posted: Feb 9, 2002 9:50 AM
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Paper caps can be used to represent quadratics two different ways on the
integer domain.
Method 1
Use a rule that places n objects in n cups or to include the coefficient,
a*n cups. For the balance to work, the objects need to be much heavier
than the cups, like lead fishing weights.
Method 2
Place wadded up (n-1)cups inside the n cups.

It should also be obvious that cubics can be demonstrated with n layers of

And n^4 is n stacks of n layers of n cups of n fishing weights.

It is perhaps less obvious how to do exponents. But since the balance
method is really a trial and error method, you can improvise the extensions
to the above as needed. The limitation is as much on the domain and range
as it is on the type of equation. Even for the simple linear equations,
the method is limited by practical consideration to the integers less than
about 10^6 unless you can to larger scales, then you begin to lose accuracy
on the smaller problems......

I think that it also unfortunate that you suggest that
"It would be nice if they all had experience with manipulatives "
Addition, subtraction, multiplication, and division with manipulatives are
all good pre-K activities. If you follow that lead you'll end up with a
lot more people like my son. He's in the fifth grade in private school
doing prealgebra. His teacher and he were discussing the other day that
he'll be doing calclulus in the 9th grade. I think there is actually a
physiological reason why this works. Much of math is learning a second
language. It is well understood that complete fluency in multiple
languages is much easier to attain before age 6. Essentially all you have
to do is let the child have the opportunities to use the other languages
and they just get learned. Ever try teaching algebra to a group of
retirees? Pretty tough row to hoe. So rather than struggling with how to
teach college students very simple stuff, how about some advocacy of doing
it right in the first place?

It's a little like the US Navy Diver Hot Suit program. By defining the
method of keeping scuba divers warm, they blinded most workers to the much
easier solution and gave more than a few people some serious skin burns.
The real problem is that scuba air is dry and that the surface area of the
lungs is wet and 10x the skin area. The heat of evaporation of water is
about 2600 joules per gram and the heat loss in the lungs simply exceeds
what can reasonably conducted through the skin. The solution was......
Well wait a minute, let's see if y'all can figure it out and that doesn't
mean find the answer with

Anyway, I'm left with a sense that too much of the problem definition on
this list is how to band aid situations that never should have come to be
and far too little about how to change the system to do it right in the
first place.

> From: Bonnie Blustein <>
> To:
> Subject: Re: Algebra Tiles
> Date: Thursday, February 07, 2002 20:37
> I have used Algebra Tiles with 9th graders in beginning Algebra, in the
> context
> of the CPM math program.
> We used them first with adding, subtracting, multiplying, dividing

> You encourage
> the students to "build it" (with the tiles), "draw it" (on paper), and
> "write it" (in mathematical
> symbols). Some of them have "learned the rules" in pre-Algebra, and

> doing this.
> However, most of those students also get the rules confused. After doing
> this a day or so,
> you make the physical manipulation optional but continue to have tiles
> available. There are
> always at least a few kids who continue to use them for awhile, but as

> get problems
> involving numbers with larger absolute values (and later non-integers)

> do have to internalize
> the generalized procedure.
> So then when it comes to linear equations, and a "balance" model with

> cups (for the
> variable) and tiles for the integers, it is an extension of what they did
> before. (x is one cup,
> 2x is two cups, etc. The kids generally come up with the idea of an
> upside-down cup being
> -x) You can present it as a question of how many tiles belong in each

> (same number in
> each!) so there are the same number of tiles on each side. There is a
> problem in that most
> kids are not familiar with a balance scale but they know "Libra." Some

> see-saws but
> a lot of playgrounds have taken these out for safety reasons, so you

> assume anything.
> None of this "automatically" conveys concepts -- in my opinion, anyway,

> teacher has to guarantee
> that the students make the leap from the concrete to the abstract -- but

> helps many of them to have
> something concrete to build on.
> Of course the physical model is severely restricted in terms of the

> of equations that can be
> modeled. However, my feeling is that many students have trouble

> the idea of what it means
> to solve an equation in the first place, and that activities like this

> help them at this stage.
> I have tried a little of this with community college students. Some are
> very resistant to using manipulatives,
> but others like them a lot. It would be nice if they all had experience
> with manipulatives in the middle-
> school grades (or at least in HS) but they haven't. I am going to try

> this more with a "slowed-down"
> algebra class this spring semester (2 semesters to cover Beginning

> The IMP secondary program also has some nice problems/activities. For
> example, they introduce a
> "problem of the week" involving a king and 8 bags of gold and someone who

> stealing gold, and
> you need to use a balance scale (as efficiently as possible) to figure

> which bag of gold is lighter than the others. There are
> some followup activities to that. So when they get to equations, there

> some background to the "balance"
> concept.
> Bonnie


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