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I’m a little puzzled by the example given for 4.NF.4b at the top of page 8 of the NF Progressions doc. It doesn’t seem to elaborate on the idea given in the Standards for students to “Understand a multiple of a/b as a multiple of 1/b”. It shows 3 x 2/5 as repeated addition (i.e. 2/5 + 2/5 + 2/5), with this being equivalent to multiplication of the numerator by 3 (i.e. 3 x (2/5)). By the time you get to the end result of 6/5 it is not obvious from this elaboration that 6/5 is the same as 6 x 1/5, except from whatever prior knowledge a student has of a/b being the same as a x (1/b).
I guess what I’m trying to say is that 4.NF.4b doesn’t seem to serve much purpose and the Progressions example doesn’t make the purpose any clearer. How does understanding this particular fact – that a multiple of a/b is also a multiple of 1/b – help a student multiply fractions generally? To draw an analogy from whole numbers, how does knowing that a multiple of any whole number is also a multiple of 1 help a student multiply generally?
I think that the example might work better this way:
3 x 2/5 = 2/5 + 2/5 + 2/5 using the grouped objects interpretation of multiplication
= 1/5 + 1/5 + 1/5 + 1/5 + 1/5 + 1/5 using 4.NF.4b or (more directly) 4.NF.3d
= 6 x 1/5 using the grouped objects interpretation again
= 6/5 using 4.NF.4b.
Here’s the grouped objects example in Table 1 of the geometric measurement progression: “You need A lengths of string, each B inches long. How much string will you need altogether?”
By the way, there is a typo on p. 8 of the NF progression, which was pointed out earlier: http://commoncoretools.me/2011/08/12/drafty-draft-of-fractions-progression/#comment-988: On page 8, the second paragraph under decimals, it states “Grade 3 students learn to add decimal fractions by converting them to fractions with the same denominator. . . . ” That should be “Grade 4 students. . . . “
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