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Good morning,
5NBT6 states, “Find whole number quotients of whole numbers with up to four-digit dividends and two-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division; illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.”
Two questions:
1) To have students illustrate and explain the calculation using area models/rectangular arrays for 4 digit dividends would be very tricky because they would have to divide up the spaces into very small units. Can you please provide an example how this would be shown using an area model/rectangular array?
2) You indicate in the standards using “properties of operations” in several places, yet the only two properties referenced in the Progressions documents are “Distributive,” “Commutative,” and “Associative.” Should these be taught in isolation prior to asking them to use them? I know that they are introduced in 3rd and 4th grade, meaning the students should have an understanding of them, but should a whole lesson be taught as a reminder, or should we just reference during a problem, “I used the _____ Property to complete this problem.”
1) There are some great examples of multiplying with larger numbers beginning on page 23 of this document:
Click to access NCSMJournal_ST_Algorithms_Fuson_Beckmann.pdf
In my opinion, we need to continue to give examples of area models for smaller numbers with each block shown to maintain that connection.
Not surprisingly, some of the examples that lhwalker mentions bear a strong resemblance to examples in in the NBT Progression.
kimbergunn, it sounds as you might be thinking that an area model must be carved up into units. When students begin using area models, it seems that initially they should maintain the connection between understanding the connection between units of area and units of numbers, but that certainly becomes unwieldy to show explicitly by drawing unit squares when numbers get large (and we hope the connection has been built in the context of smaller numbers).
The area model on p. 15 of the NBT Progression does not show individual units of area. (It shows a 3-digit dividend and 1-digit divisor, I hope it’s obvious how an area model might be drawn for a 4-digit dividend and 2-digit divisor.) Also, the standard allows an equation as an illustration.
The discussion of introducing the commutative property for addition here (http://lipingma.net/math/One-place-number-addition-and-subtraction-Ma-Draft-2011.pdf) might be helpful in thinking about how to introduce it for multiplication.
Lane and Cathy have pretty well covered things, but just a couple of extra points. First, notice the footnote on page 15 that says students need not use the formal terms for the properties of operations. They should understand that you can add numbers in any order and use this fact, but they don’t have to know the name for it. Second, when you read a standard like 5.NBT.6, it is important not to interpret it is requiring every method listed for every division problem (that’s why the “and/or” is there).
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