The examples that you’ve provided work, of course, but why those “movements” of terms are mathematically valid cannot be explained without first having an understanding of the balanced approach.

I have no qualms about students using shortcuts, so long at they can explain why they work. If they don’t get that, then they miss why we do math. It’s not rote memory and procedural efficiency that’s valued (at least for me). It’s about having a deep understanding about how numbers relate and what we can do with those relationships to solve meaningful problems. To skip a conceptual understanding of math for the sake of doing things faster, to me, does students a great disservice.

]]>1. It can perform many solving steps in just one step, without having to perform double-writing of terms on both sides.

Exp: Solve 5x – p + q – r + 7 = 2x – s + t + 8 – 9 (10 terms before move)

5x – 2x = -s + t + p – q + r + 8 – 9 – 7.(10 terms after move)

3x = (-s + t + p – q + r – 8)

2. It can easily check the overall transition of terms by observing the simple Rule: “No missing terms, no new terms added, after every move”.

3. It allows smart moves by switching around numerator to denominator, and vice versa.

Example: Solve (m – 3)/(n + 4) = (p + 2)(x – 1).

Move (x – 1) to left side and other terms to right side.

(x – 1) = [(p + 2)(n + 4)]/(m – 3).

4. It will greatly help, when students later have to deal with more complicated equations and inequalities in higher math studies. They don’t have to perform double-writing of terms in every solving step, that wastes time and that is usually main causes for errors/mistakes

5. It always keeps the equation balanced (see proofs in many articles post on Google, Yahoo). It doesn’t divide solving linear equations in many steps that complicates the solving process. ]]>

I don’t know if students respond to the checkpoint/finish line analogy, but I did want them to feel a sense of progression. I’ll ask them!

I’ve prepared two or three versions of quizzes and tests so I can give them back to students. My class has particular groups of students, and some don’t mingle, so I can strategically give them out according to social groups. I hope that’s been working :s

I’ll be sure to post more reflections in the near future!

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