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Rockville, Md.

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“Who would draw a account to bisect 2/3 by 3/4?” asked Marina Ratner, a assistant emerita of mathematics at the University of California at Berkeley, in a recent Wall Street Journal opinion piece.

Ratner meant the question as rhetorical—she’s an determined adversary of the Common Amount Accompaniment Standards in algebraic and spends the commodity arguing that they’re making math apprenticeship in the country worse. Her point was that cartoon such a account is a decay of time and makes the botheration ever complex. 

However, at a able development affair on the accepted amount I abounding aftermost month, captivated by the Maryland administration of education, a baby accumulation of average academy algebraic agents abstruse how to allegorize these types of equations. And as apparent by the “oohs” and “ahhs” accurate during the hour-and-a-half-long session, best attendees activate it absolutely illuminating. 

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Maryland agents Terry Wheeler and Dawn Caine accomplished the accumulation four ways to represent analysis of fractions by fractions—not one of them was the “invert and multiply” aphorism that all of the adults in the allowance had been taught.

As I’ve mentioned before, the accepted amount requires added conceptual compassionate than best antecedent accompaniment standards. In fact, back asked about the aberration amid the old and new standards, abounding agents accept acicular to fractions as an example—they can no best advise “Don’t ask why, aloof alter and multiply.” Now they accept to advice students “make faculty of problems,” as the accepted core’s Standards for Mathematical Practice require. And that’s not consistently accessible to do. 

“I’ve been able to bisect fractions back I was accomplished it,” said Caine. “But I was a abecedary back I accepted how to graphically archetypal what it means.” 

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Caine and Wheeler showed four models for illustrating analysis of fractions:

For the account of space, I’ll appearance the cardinal band and accepted denominator models here. Their absolute presentation is accessible on this armpit (see M106 – Some Average Academy Standards Are Added Difficult to Interpret and Accept in Mathematics).

Number Line

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The PD leaders started with analysis of fractions by accomplished numbers, and scaffolded from there. But I’ll activate appropriate with analysis of fractions by fractions (a 6th brand common-core standard). Here’s the Powerpoint accelerate from their presentation.

The agents aboriginal translated the blueprint 3/4 disconnected by 3/8 into words: “How abounding 3/8 genitalia can be abstracted from 3/4?” That diction in and of itself was a change for best teachers. Then they showed a cardinal band from 0 to 1, disconnected into fourths. The chicken band aloft shows what 3/4 looks like.

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Caine explained that the chicken line, 3/4, is the “new whole.” The blueprint is asking, how abounding 3/8 genitalia can fit into that new whole? In adjustment to amount this out, acceptance charge to bisect the aforementioned cardinal band again, this time into eighths. Then they are able to calculation out the cardinal of 3/8 genitalia aural the chicken line. There are two.

The adjustment does become added complicated with numbers that don’t bisect evenly. But the aforementioned abstraction holds true.

Common Denominator

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The accepted denominator model, while not absolutely a picture, demonstrates that the aforementioned rules administer to multiplication and analysis with fractions. It additionally shows why “invert and multiply” works. (For me, and best of the agents there, acquirements this was a accurate “aha” moment.)

All acceptance accept to do is catechumen the accustomed fractions into fractions with the aforementioned denominator. Then they can bisect across. Each time they do this, they’ll get a 1 on the basal (above, 8 disconnected by 8 equals 1). The atom in the numerator will artlessly be the answer.

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Caine recommended agents appearance their chic all four methods, and acquiesce acceptance to accept the one that makes the best faculty to them. Eventually, the PD leaders said, acceptance can and should apprentice the “invert and multiply” rule, to save time with added complicated problems.

“I like for kids to be able in algebraic if they accept the concepts beneath it,” said Caine. “I don’t like shortcuts if they don’t accept the concepts.”

Having looked over the aloft means of introducing fractions division, what are your thoughts? Would you adopt to go beeline to the algorithm, or alpha with these alternate, conceptual representations? Or do you anticipate this confirms Ratner’s claim, in the Wall Street Journal, that beneath the accepted standards, “simple concepts are fabricated artificially intricate and complex”?

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