Multiplying mixed numbers – Grade 5

[Bill McCallum]

Does multiplication of fractions in Grade 5 include multiplying a mixed number by another mixed number? Or common fractions by mixed numbers?

The NF Progressions and 5.NF.4 seem to only have common fractions by common fractions. Although 5.NF.6 states that multiplication with mixed numbers should be done it is unclear whether it includes the possibilities I mentioned. I know, of course, it is not forbidden if it is not mentioned, but what was the intent?

[Bill McCallum]

Duane, sorry, this is going to be a long answer, and in the end it will be the same sort of “figure it out for yourself” answer that I’ve been giving lately. But I will try to throw some light on the matter.

You are using the terms “mixed number” and “common fraction” in ways they are not used in the standards. Indeed, the first term only occurs twice and the second not at all. This is not unusual. One often hears people talk about fractions and mixed numbers as if they were different types of numbers, with their own ways of being added and multiplied.

By contrast, the standards use the word “fraction” to refer to a particular sort of number (one that you get by dividing the interval from 0 to 1 into b equal parts and putting a of them together), not to a particular sort of expression. That number can be expresssed in different ways. It can be written in the form numerator/denominator (“fraction” in conventional terminology) or in the form whole number + fraction less than 1 (“mixed number”).

From that point of view, a flippant answer to your question is: if students are multiplying fractions they are multiplying mixed numbers. However, there is more to it than that. Although any requirement about fractions potentially includes fractions expressed in mixed number form, one has to think about how the complexity of numerical expressions grows across grade levels. A fraction expressed as a mixed number is really being expressed as a sum (with the plus sign missing), so a product of two mixed numbers is really a product of sums. You can deal with this a couple of different ways: compute the sums first then multiply (i.e. “convert the mixed number to a fraction first” in conventional terminology) or multiply out the sum using the distributive law. Exactly where you expect which level of complexity in expressions and their manipulation is not a matter for the NF progression per se, but rather part of the general question about the progression in numerical expressions. The standards leave some room for different decisions by curriculum writers there.

So, to the specifics of your question, 5.NF.4 is not really about the form of the number, but about the number itself. Students should be able to multiply fractions, and that includes multiplying mixed numbers, if only by the first method above. But, although the standards do not impose a limit here, considerations of expression complexity do. These are largely up to curriculum developers. The grand culminating standard 7.EE.3 expects fluency with number in all form, but how you get there could vary from curriculum to curriculum. I would certainly expect students to see some products involving mixed numbers in Grade 5, but not fluency with the most complicated ones (e.g. mixed number by mixed number).

[Bill McCallum]

A further thought, which should perhaps have been my first answer: mixed numbers are basically not a very important subject in the standards. Students should be able to interpret expressions in mixed number form, and should be able to deal with them, but they should not be a separate subject, with long lists of drill exercises associated with them. They will arise naturally in word problems, as in 5.NF.6. That is enough. There shouldn’t be a separate module called “multiplying mixed numbers”. A student who avoids them entirely by always converting them to a/b form is doing just fine.

[Bill McCallum]

Thanks Bill. I think the final sentence of your first reply sums it up: some exposure would be good but fluency can wait for later.

In essence, and I know we’ve discussed this in other posts, the wording of the Standards regarding fractions is difficult to interpret. Sometimes “fraction” is used inclusive of common fractions, mixed numbers, and decimal fractions – sometimes not. In a recent post you’ve also stated that whole numbers are fractions (http://commoncoretools.me/forums/topic/bar-graphsline-plots/#post-1581). Overall, this elastic approach to the topic makes it a real struggle to identify what is required for any given standard.

I understand that the aim is to shake teachers out of their beliefs about different forms of fractions being different numbers, but at some point teachers, curriculum developers, and test-makers need to make marks on a page. They need to know what forms are expected for a given task at a given grade, even if they understand conceptually that all forms are equivalent. The “complexity of numerical expressions” you mentioned is exactly what needs to be known – leaving it open for potential conflict between curriculum developers and test-makers is not ideal.

The terminology of “common fraction”, “improper fractions”, or “mixed numbers” may be reduced or absent from the Standards to avoid creating misconceptions but these terms are very clear, concise, and convenient labels. Although the Standards cannot be altered to increase clarity, perhaps the NF Progressions could be – explaining what such terms are, how they relate to each other, and how they apply to each relevant standard would be a great help for others not reading this blog.

[lhwalker]

I’m thinking that the complexity question would depend on the students we teach. I like the idea that  “A student who avoids them entirely by always converting them to a/b form is doing just fine.”  That tells me I would need to make sure every student mastered that much, but I would also want to raise the bar for those who are capable, including higher complexity in our discussions and assigning challenge problems.

[Bill McCallum]

I agree, and also want to make clear that students should eventually be able to interpret and work with fractions in mixed number form.

[lhwalker]

Awhile back, we had a hierarchy of curriculum topics: review/mastery/introductory.  I’m thinking that once we get the progressions firmly nailed down, a hierarchial list might help clarify questions like these in such a way that “eventually” is set up to happen.

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