Division with fractions – Grade 5

[Duane]

Page 12 of the NF Progressions seems to be suggesting that students start to move towards the use of “invert and multiply” for division involving fractions in Grade 5. This is what I’m interpreting by the example given of 5 ÷ 3 being the same as (1/3) x 5. It’s not suggested that students mechanically apply this as a rule, but is linked to the relationship between division and fractions.

In the Standards (5.NF.7a, b) it seems that the suggestion is more towards pictorial models and the relationship between multiplication and division: (1/3) ÷ 5 = (1/15) is the same as 5 x (1/15) = (1/3). I’m assuming that this link is used as an explanation of the result but I see that it could also be used as a method in that (1/3) ÷ 5 = ___ is the same as 5 x ___ = (1/3); an extension of the idea of thinking 6 x ___ = 18 to figure out 18 ÷ 6 = ___.

Are both the fraction-division relationship and division-multiplication relationships to be focused on equally or is there a preference?

[Bill McCallum]

The equation $5 \div 3 = \frac13 \times 5$ is certainly an instance of invert-and-multiply. But that rule does not arise until Grade 6, and in Grade 5 the equation arises in a slightly different way, as you suggest. In Grade 4 students were multiplying fractions by whole numbers, because can be explained in terms of the repeated addition interpretation of multiplication by a whole number. But that does work for multiplying a fraction by a fraction, which students start doing in Grade 5, so in Grade 5 students have to get a meaning for expressions like $\frac13 \times 5$ that did not have a meaning before. They know from 5.NF.3, discussed on page 11, that $5 \div 3 = \frac53$. That is, $latex \frac53$ is the quantity that when you multiply it by 3 you get 5. On the other hand, it makes sense to give $\frac13 \times 5$ the same meaning, since it fits with the idea of taking one third of something. So $5 \div 3$ and $\frac13 \times 5$ are the same thing.

The situation is a little different for $\frac13 \div 5$, because that can be related directly to a multiplication they already know, namely $5 \times \frac{1}{15}$, as you explain.

So, in answer to your question: the ultimate goal is to understand division as related to multiplication. But you have to have the concept of multiplication defined first. To give a meaning to $\frac13 \times 5$, we think of it as “one third of 5”, or “one part of a division of 5 into 3 equal parts”, and then we use the reasoning behind 5.NF.3 to see that this is the same as $\frac53$.

I think all this could be said more clearly in the progression, thanks for pointing it out.

• This reply was modified 11 years, 9 months ago by Bill McCallum.
• This reply was modified 11 years, 9 months ago by Bill McCallum.
• This reply was modified 11 years, 9 months ago by Bill McCallum.
• This reply was modified 11 years, 9 months ago by Bill McCallum.
• This reply was modified 11 years, 9 months ago by Bill McCallum.
• This reply was modified 11 years, 9 months ago by Bill McCallum.
• This reply was modified 11 years, 9 months ago by Bill McCallum.
• This reply was modified 11 years, 9 months ago by Bill McCallum.
• This reply was modified 11 years, 9 months ago by Bill McCallum.
• This reply was modified 11 years, 9 months ago by Bill McCallum.
• This reply was modified 11 years, 9 months ago by Bill McCallum.
• This reply was modified 11 years, 9 months ago by Bill McCallum.

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