Whole number times fraction vs. fraction times whole number (grades 4 and 5)

[bcohen]

Bill,

There is a lot of confusion and debate in NY regarding standard 4.NF.4.  Specifically, there is disagreement regarding whether, for example, 3 x 1/5  and  1/5 x 3 are both included within this standard, or if this standard includes only the former, leaving the latter for 5.NF.4.

Understanding that the two are arithmetically equivalent (commutative property and all), they are conceptually very different.  The former being multiple copies of a fraction (which could be solved by repeated addition) and the latter being a fraction of a set (which could not be conceptualized as repeated addition).  The former supports the fourth grade fraction work on adding (and subtracting) fractions with like denominators… the latter does not.

Further supporting the second interpretation (that 3 x 1/5 is a grade 4 expectation, but 1/5 x 3 is a grade 5 expectation) is an earlier blog post where you stated, “A natural place to consider set models would be Grade 5, where students start multiplying whole numbers by fractions, and so they could interpret in 3/5 x 10 = 6 terms of a set of 10 objects.”  (http://commoncoretools.me/2011/08/12/drafty-draft-of-fractions-progression/#comment-1786)

Finally, all of the examples in the standards, the Progressions document, and Illustrative Mathematics also support this interpretation.  The exhaustive list of examples provided are:

5 x 1/4  (from the standards)
3 x 2/5  (from the standards)
5 x 3/8  (word problem in the standards)
6 x 4/15  (word problem from Illustrative Mathematics)
5 x 1/3  (Progression for NF)
7 x 1/5  (Progression for NF)
11 x 1/3  (Progression for NF)
3 x 2/5  (Progression for NF)
43 x 2 3/4  (Progression for NF)

Is this interpretation of 4.NF.4 (that it includes multiple copies of a fraction, but not finding a fractional part of a number) accurate?

If so, this really needs to be made explicitly clear with test-writers (and curriculum authorship teams) who seem to miss the distinction entirely.

Yours in making the Common Core “common”,

Brian

[Bill McCallum]

Brian, I’m keen on knowing the answer too. My best guess is that only the repeated addition model is expected in Grade 4.

One related thing I’ve been considering is how to interpret 4 x 1/5 when no context is provided. One way is to think of it as a set problem (what is 4 multiplied by 1/5) and the other is as repeated addition (what are 4 groups of 1/5). Do you think that if there is no context that students should be able to still get the correct answer by approaching it more or less abstractly?

[Cathy Kessel]

Tad Watanabe’s comment touches on similar issues:

p. 7 I think it is important for the document to emphasize that “multiplying by a whole number” means, for example, “5 x 1/3,” but not “1/3 x 5.” Too often, teachers don’t distinguish the multiplier and the multiplicand, so they would think both of these are “multiplying by a whole number.” (http://commoncoretools.me/2011/08/12/drafty-draft-of-fractions-progression/#comment-171)

I notice however that Brian’s question concerns test developers and curriculum developers. It’s possible that curriculum developers might want to include 1/3 x 5 in some way, even if it test developers did not include it. 

Duane, when you say “abstractly,” do you mean “without drawing a visual fraction model (tape diagram, number line diagram, or area model)”? This seems a bit hard to regulate. Even if a question testing 4.NF.4a or 4.NF.4b asked for an equation (assuming an equation is abstract and a diagram is not), a student might draw a diagram before writing the equation.

But maybe I’m missing your question entirely. A central idea is that students understand fractions as numbers on the number line (3.NF.2) as well as things that can be used to represent quantities.

[Bill McCallum]

I agree with the consensus here: 3 x 1/5 in Grade 4, 1/5 x 3 in Grade 5. As Brian says, the two are conceptually quite different: the first is what you get by putting 3 segments of length 1/5 together, the second is what you get by dividing a segment of length 3 into 5 equal parts. The fact that these are the same can be shown by some reasoning on the number line; it’s not obvious. As Cathy says, this doesn’t prevent curriculum writers from developing a pathway that starts talking about simple instances of fraction x whole number before Grade 5, but it does constrain assessment developers.

[Author]

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