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On page 5 of the Fractions Progression, the caption for the top picture explains that “the whole on the right is divided into 4 x 3 small rectangles of equal area, and the shaded area comprises 4 x 2 of these, and so it represents 4 x 2/4 x 3.
Is there a way of reading “4 x 3” and “4 x 2“ aloud (e.g., “4 groups of 2,” “4 times as many as 2,” etc.) that would be more in keeping with a conceptual understanding? (I understand that students are not the audience for the caption, but I’m trying to imagine how students might verbalize their thought processes and actions and asking myself if there are particular phrases or word orders that would be more consistent than others with a student’s emerging conceptual understanding.)
You previously wrote (here) that “the concept of fraction equivalence . . . is developed more fully in Grade 4, where students reason directly with visual fraction models to see that taking, say, 3 times as many copies of a unit fraction one-third the size gives you the same number.” In the case of the example on page 5, it would be “4 times as many copies of a unit fraction one-fourth the size gives you the same number.” I understand that, but I’m having a hard time matching that phrase up conceptually with 4 x 2/4 x 3. I see the picture on the top right as 2 groups of 4/3 groups of 4 (2 x 4/3 x 4).
As students are deepening their understanding of equivalent fractions, is there a difference between seeing the two thirds as partitioned into 4 groups of 2 vs. partitioned into 2 groups of 4?
Thank you, Mr. McCallum, for all the work you’re doing and for reading my questions.
This a very interesting question. To summarize, you are saying the to you the diagram says
$$
\frac23 = \frac{2 \times 4}{3 \times 4}
$$
rather than
$$
\frac23 = \frac{4 \times 2}{4 \times 3}.
$$
First, your way of seeing it is fine and ends up with the correct understanding. The difference between your way of seeing the diagram and the way expressed in the progression is the difference between the two ways of viewing a $3 \times 4$ array: as 3 rows of 4, or 4 columns of 3. You are looking at the rows: there are two green rows of 4 squares each in an array consisting of 3 rows of 4 squares each, so the fraction is $(2\times4)/(3\times4)$. The other way to look at this is that there 4 columns of 2 green squares each in an array consisting of 4 columns of 3 squares each, so the fraction is $(4 \times 2)/(4 \times 3)$.Either way is fine, and it’s probably useful to go through both ways.
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.
I was a bit confused by your last paragraph, because I think of the phrase “multiplying 2 by 4” as meaning $4 \times 2$, not $2 \times 4$. But I take your overall point.
I was a bit confused by your last paragraph, because I think of the phrase “multiplying 2 by 4” as meaning $4 \times 2$, not $2 \times 4$. But I take your overall point.
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.
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