[greyhound2]

Yes, that should have been “multiplying n (4 in this case) by the numerator (2)” since multiplying “4 by 2″ means 2 groups of 4 (2 x 4).

So, if I’m understanding correctly, if we want to talk about 2 rows of 4 in this case, instead of 4 columns of 2, we would talk about “multiplying the same number, n, by the numerator and denominator of a fraction . . .” instead of “multiplying the numerator and denominator of a fraction by the same number, n . . . .”

I see how this goes back to my original question of how to say “4 x 2.” If we say “4 by 2” (in an equal groups situation), it means 2 groups of 4 but, unless we specifically interpret it as 4 by 2, 4 x 2 would generally, at least in the U.S., be interpreted as 4 groups of 2 as explained on page 24 of the OA Progression.

Then there’s the fact pointed out in the OA Progression that, in many other countries, 4 x 2 would mean 2 fours. According to the Grade 3 section of the OA Progression, “it is useful to discuss the different interpretations and allow students to use whichever is used in their home. This is a kind of linguistic commutativity that precedes the reasoning . . . arising from rotating an array.”

I’ll remember that 4 x 2 can mean different things in an equal groups situation depending on whether it is interpreted as: a) 4 by 2, b) four twos as in the U.S., or c) two fours as in other countries.

Thank you.

[greyhound2]

Thank you very much for your reply.

I was thinking that it could be advantageous to focus on the rows (2×4 or 2 groups/rows of 4 smaller squares) rather than the columns (4×2 or 4 groups/columns of 2 smaller squares) because the rows correspond to the original thirds before they were partitioned into smaller pieces. I suppose one could still focus on the rows whether one says 2 groups/rows of 4 smaller pieces (2xn) or 4 groups/columns of 2 smaller pieces (4×2), but there seems to be a better connection between saying “2 groups/rows of 4” and seeing the two-thirds as two 1/3 unit fractions.

Focusing on the rows rather than the columns also seems (to me) to align better with this sentence from page 5: “They see that the numerical process of multiplying [each] the numerator and denominator of a fraction by the same number, n, corresponds physically to partitioning each unit fraction piece into n smaller equal pieces.”

I was also thinking that focusing on the rows vs. the columns might make it easier for students to understand why multiplying the numerator, in this case 2, by 4 (2 groups of 4 or 2×4) and the denominator, 3, by 4 (3 groups of 4 or 3×4) results in an equivalent fraction, although I understand that the students, using visual models, are to first develop their own methods/rules/algorithms for generating equivalent fractions, so a student may see 4 groups/columns of 2 (4×2) as readily or even more easily than they see 2 groups/rows of 4. As you said, “Either way is fine, and it’s probably useful to go through both ways.”

Thank you again.

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