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Is anyone interested in starting a "Dolciani" social group?


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For example, the FOIL method of multiplying binomials mentally is usually presented, as such, and rote memorized.

 

If instead, it is introduced in the context of finding the area of rectangles of side lengths (ax+b) and (cx+d), and then generalized to finding the area of any rectangle; this will undoubtedly provide a more meaningful experience than just memorizing patterns for writing squares of binomials as trinomials.

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For example, the FOIL method of multiplying binomials mentally is usually presented, as such, and rote memorized.

 

If instead, it is introduced in the context of finding the area of rectangles of side lengths (ax+b) and (cx+d), and then generalized to finding the area of any rectangle; this will undoubtedly provide a more meaningful experience than just memorizing patterns for writing squares of binomials as trinomials.

 

Dolciani is certainly more geometric/visual in its presentation but I think your claim is unfair to AoPS Algebra.

 

They have a sidecar on geometric factoring on p115 in Chapter 4(Factoring) and revisit that in a geometric sidebar in Chapter 10(Getting Started with Quadratics) which is immediately after the introduction of the FOIL method and the next part of the text says 

 

Using these gimmicks to understand how to multiply binomials is fine, but don't rely on them to memorize how to multiply binomials. For example, how is FOIL going to get you through expanding the product of (x^2 - x +4) (x^2 - 2x +3)? If you understand how to multiply binomials, extending that understanding to more complicated expressions is easy.

 

Dolciani certainly has a more extensive geometric approach in my 1965 Structure Method Algebra. Chapter 7 is great.The visual proofs in the problem sets for section 7-5 are real gems. Problem 3 showing that (a+b)^2 - (a-b)^2 = 4ab and the others remind me of the visual proofs of the Pythagorean theorem.

 

Dolciani is more visual but I think AoPS does a fine job developing understanding well beyond the algorithmic.

 

Also, I think both groups miss the boat by just not introducing grid multiplication of numbers first and then expanding that to algebra.

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Exactly. Same thing FOIL but easily extended to multiplying trinomials with a 3x3 grid or a trinomial times a binomial with a 3x2 grid. Just like in arithmetic multiplication, I think beginners in algebra have two related problems 1) Rememberring the algorithm 2) A simple working memory issue of tracking where they are in the process and not missing pairs. The grid helps with 1 and completely removes the working memory issue. Once the process is internalized you drop the grid and go on your way.

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Hey, I'll join! Just finished Pre-Algebra last year, and we've started Algebra this year - I'm a former English teacher, highly skeptical of my mathematical abilities, but determined to self-educate as long as this is working for all of us. Predicting I can get at least through Alg / Geometry / Alg II.

 

 

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  • 3 years later...

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