Properties of division – 3.OA.5

[Bill McCallum]

Bill, I’ve asked a related question before ( http://commoncoretools.me/2012/04/02/general-questions-about-the-standards/#comment-2296 ) but the answer ventured more into subtraction than division: how are students to apply the properties of operations to divide? Standard 3.OA.5 gives examples for multiplication (commutative, associative, and distributive) but nothing is mentioned for division. What is expected? Is it just that students use the fact that division is the inverse of multiplication? If so, how is this different from 3.OA.6?

[Bill McCallum]

Duane, the answer here parallels the one there. Division is defined in terms of multiplication; $a \div b$ is by definition the number that gives $a$ when multiplied by $b$. So every division problem can be viewed as a multiplication problem (and vice versa). If I calculate $300\div 25$ by saying “I know $100 = 4 \times 25$ and $300 = 3 \times 100$, so $300 \div 25$ must be $3 \times 4$,” I am implicitly using the associative property $3 \times (4 \times 25) = (3 \times 4) \times 25$. Once again I must emphasize that it is not intended that students necessarily be able to name the properties. A student who can explain the reasoning above or illustrate it using drawings or equations meets the standard.

[virgama]

Hi Bill,

What do you suggest districts adopt for division basic fact fluency?  In other words, should there be an expectation for timed division facts being mastered before leaving third grade as there is for multiplication?  Likewise, what do you suggest for subtraction in second grade?

Thanks,
Matt

[Bill McCallum]

The Grade 3 multiplication standards relevant to this question are

3.OA.7. …. By the end of Grade 3, know from memory all products of two one-digit numbers.

and

3.OA.6. Understand division as an unknown-factor problem. For example, find 32 ÷ 8 by finding the number that makes 32 when multiplied by 8.

So division goes hand in hand with multiplication, so knowing the single digit multiplication facts from memory also means being able to call on that memory to perform the corresponding divisions.

How all this relates to timed tests I don’t know. Note that the standards don’t explicitly call for timed tests. It seems to me that there are other ways of assessing whether students know their single digit multiplication facts from memory. And the division facts require an extra step: to calculate 32 ÷ 8 you might need to try out a couple of multiplication facts, which would take longer. That is, the standard do not expect students to know division facts from memory independently of the multiplication facts, but rather to know them as a bundle linked to the multiplication facts.

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