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Topic: Simplifying Algebraic Expressions with Subtracted Expressions
Replies: 67   Last Post: Nov 25, 2013 12:57 PM

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 Pam Posts: 1,507 Registered: 12/6/04
Re: Simplifying Algebraic Expressions with Subtracted Expressions
Posted: Nov 16, 2013 4:33 PM

Posted: Nov 16, 2013 2:33 PM by Bob:

>
> So riddle me this...
>
> The original tart problem was as follows ...
>
> "Mrs. Chen made some tarts. She sold 3/5 of them in
> the morning and 1?4 of the remainder on the
> afternoon. If she sold 200 more tarts in the morning
> than in the afternoon, how many tarts did she make?"
>
> Pam's steps were...
>
> 1. They would draw a horizontal rectangular bar,
> divide it into 5 equal parts.
> 3. divide each of the remaining 2 parts in half
> 4. (as well as the original 3 so as to continue to
> have equal portions)
> 5. By shading 1/4 of the "afternoon tarts" (1/10 of
> the whole bar)
> 6. and shading an equal portion of the "morning
> tarts"
> 7. what is left of the morning tarts is equal to 200
> 8. Since there are 5 parts, each part is equal to 40
>
> Where did steps 3 and 4 come from? and step 6? Think
> like a child that doesn't know how to solve this
> problem. You can't get those steps without already
> having rationalized the solution mathematically.

Consider me riddled. This seems so obvious, I don't know how to respond.

>
> Illustrating a solution is not the same as teaching a
> kid to solve. The kids that manage to illustrate
> their solutions in this manner already have solutions
> and move on past this gimmick. The others suffer
> because they don't get how you know how many blocks
> to make. My suggestion (and I know it works) is LAY
> OFF THE MARKINGS. Use the bar to indicate the whole
> and go as far as showing the parts, then finish it
> off with plain old arithmetic.

How do you suppose you know, Bob, where to mark "approximately 3/5" on a line, with any kind of accuracy?

> Showing the markings
> might help a 3rd or 4th grader get comfortable with
> fractions but there comes the time when you have to
> leave *counting* behind and move on to arithmetic and
> reasoning.
>
> And suppose that the problem is this instead...
>
> "Mrs. Chen made some tarts. She sold 3/5 of them in
> the morning and 1?6 of the remainder on the
> afternoon. If she sold 210 more tarts in the morning
> than in the afternoon, how many tarts did she make?"
>
> Yes, I know you can solve it if you divide the bar
> into enough pieces.

Still riddled. This is as easy to solve with a bar diagram as the original. But your point is well taken: some problems are cumbersome to solve with a bar diagram. And what a fantastic introduction into algebra that makes! "We could solve this using a bar diagram, as you have practiced earlier, but let me show you a more generaly applicable method."

>
> But what if it is whole numbers rather than
> fractions, like coprime whole numbers? The number of
> markings gets rather large.
>
> What if it is decimals? 100 markings? 1000?
>
> What if it involves multiplication? Something simple,
> like 2. She sold twice as many tarts in the morning?
> What then?
>
> What if, god forbid, it involves LETTERS!
>
> Don't laugh, there is a problem with kids trying to
> learn algebra and they can't do it by counting.
>
> Illustrating a solution requires that you know the
> solution. You can only know the solution if you can
> rationalize the solution. I said that the problem was
> a hard 5th grade problem, and it is,

It really is quite routine among the problems solved in 5th grade.

> because those
> problem sums are amongst the most challenging
> problems on the PSLE, which is taken at the end of
> 6th grade. And we are talking Singapore challenging,
> not U.S. challenging. Our six graders would faint and
> suffer PTSD if they had to take the PSLE. Pam said
> "Not really. Not if they know how to shade..." I say
> "No. If they know how to solve the problem they can
> illustrate the solution if they can shade, but
> shading won't enable them to solve the problem."
>
>

If you understood how bar diagrams develop throughout the curriculum, you would, well, understand that you indeed have it backwards.

> professional training?
>
> I use the "professional card" in place of the
> argument that if you actually teach students
> mathematics they will use it in some capacity in
> their professional lives. That we don't see even
> traces of gimmicks like this later on is just to make
> the point that this is not mathematics. But my real
> concern, the thing that hits me when I see a gimmick,
> is why? Why a gimmick? I don't even think the kids
> are looking for gimmicks. Nor the parents. Just
> because they don't know any better doesn't mean they
> want a gimmick.
>
> Where does it stop?
>
> Singapore seems to have an answer, the end of 6th
>

Yes.

Pam

Date Subject Author
11/11/13 MVTutor
11/11/13 Robert Hansen
11/12/13 Bishop, Wayne
11/12/13 MVTutor
11/13/13 Jonathan J. Crabtree
11/13/13 Bishop, Wayne
11/14/13 Joe Niederberger
11/14/13 Robert Hansen
11/14/13 Joe Niederberger
11/14/13 Robert Hansen
11/14/13 Joe Niederberger
11/14/13 Robert Hansen
11/14/13 Louis Talman
11/14/13 Joe Niederberger
11/14/13 Robert Hansen
11/14/13 Robert Hansen
11/16/13 Bishop, Wayne
11/16/13 Robert Hansen
11/14/13 Joe Niederberger
11/14/13 Robert Hansen
11/16/13 Bishop, Wayne
11/14/13 Pam
11/15/13 Robert Hansen
11/15/13 Joe Niederberger
11/15/13 Robert Hansen
11/15/13 Joe Niederberger
11/15/13 Joe Niederberger
11/15/13 Joe Niederberger
11/15/13 Robert Hansen
11/16/13 Bishop, Wayne
11/16/13 Robert Hansen
11/17/13 Bishop, Wayne
11/17/13 Robert Hansen
11/17/13 Bishop, Wayne
11/15/13 Joe Niederberger
11/15/13 Robert Hansen
11/15/13 Pam
11/15/13 Robert Hansen
11/15/13 Pam
11/16/13 Robert Hansen
11/16/13 Robert Hansen
11/16/13 Joe Niederberger
11/16/13 Robert Hansen
11/18/13 Louis Talman
11/21/13 Robert Hansen
11/21/13 Louis Talman
11/16/13 Pam
11/16/13 Robert Hansen
11/16/13 Pam
11/16/13 Robert Hansen
11/18/13 GS Chandy
11/17/13 GS Chandy
11/17/13 Pam
11/17/13 Robert Hansen
11/17/13 Pam
11/17/13 Robert Hansen
11/17/13 Pam
11/18/13 Robert Hansen
11/18/13 Robert Hansen
11/18/13 Pam
11/18/13 Robert Hansen
11/25/13 Bishop, Wayne
11/25/13 Robert Hansen
11/22/13 Joe Niederberger
11/25/13 Bishop, Wayne
11/23/13 GS Chandy
11/24/13 Robert Hansen
11/25/13 Bishop, Wayne