Extent of 5.NBT.6 and 5.NBT.7

[Bill McCallum]

I’m wondering about the expected limits for division in Grade 5. Fluency in the standard algorithm is not expected until Grade 6 (6.NS.2) though I’m assuming that, based on Bill’s comments about other algorithms, some foundational work with the standard algorithm before that time would be useful.

Bearing that in mind, and looking at 5.NBT.6 and 5.NBT.7 together, what are the expectations for using the standard algorithm with questions that involve decimal fractions in some way? I’m looking for specific guidance on each type given by these examples:

a) 29 / 4 = 7.25

b) 1.20 / 6 = 0.20

c) 10 / 0.2 = 50

d) 3.6 / 0.9 = 4

I know that what is not mentioned is not thereby forbidden, but most of the discussion on page 18 of the NBT Progressions seems to focus on examples B, C, and D and using reasoning to get the quotients, rather than following an algorithm. An example like A above does not seem to be mentioned anywhere, for any method.

[Bill McCallum]

Examples A, B, and D doesn’t seem to fall under either 5.NBT.6 or 5.NBT.7. The former requires whole number quotients of whole numbers, and the latter is limited to the fraction divisions under 5.NF.7, unit fractions by whole numbers and whole numbers by unit fractions. But I agree that any of them could arise at by reasoning about the corresponding multiplication problem.

[Bill McCallum]

Examples A, B, and D doesn’t seem to fall under either 5.NBT.6 or 5.NBT.7. The former requires whole number quotients of whole numbers, and the latter is limited to the fraction divisions under 5.NF.7, unit fractions by whole numbers and whole numbers by unit fractions. But I agree that any of them could arise at by reasoning about the corresponding multiplication problem.

[Bill McCallum]

Thanks Bill. Just to clarify – is example C okay? It’s similar to an example on p.18 of the NBT Progressions (7 / 0.2) because it has 0.2 as the divisor. But would 0.2 really be considered a unit fraction (i.e. 1/5) and, if so, how does 0.125 (i.e. 1/8) fare?

[Bill McCallum]

I was thinking C was o.k. because 0.2 is a unit fraction, but of course you are right that this requires a conversion. That conversion is fairly simple, and well within the purview of Grade 5; your example of 0.125 is somewhat trickier.

[jegtitle]

Regarding 5.NBT.7, please clarify …and divide decimals to hundredths. Does this mean divisors, dividends and quotients to hundredths? I’ve used drawings and models, but using 5.NBT.1’s understanding of place value seems to help my students.

[Bill McCallum]

Yes, I would say that this limits the division problems to ones where divisors, dividends and quotients all be decimals that with no non-zero digits beyond the 100ths place.

[Anonymous]

Hi Bill,
I’m still confused about 5.NBT.7 based on the posts. Your last post says that “dividends, divisors, and quotients” can all be decimals limited to the hundredths place. Should it be dividend or divisor (e.g., 4 / 0.25 or 0.25/4) as opposed to dividend and divisor (0.16/0.4)?

Also, does 5.NF.7
“Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions” constrain 5.NBT.7 at all?

[Anonymous]

Bill’s reply in this thread (from 3/25/13) states that 5.NBT.7 is limited to the fraction divisions under 5.NF.7: unit fractions by whole numbers and whole numbers by unit fractions. So, it seems like the example of 0.16/0.4 would not fall under 5.NBT.7, but the other examples (4/0.25 and 0.25/4) would.

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