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Page 8 of the NF Progressions document provides an example of multiplying a whole number by a mixed number in Grade 4. Is this an accurate example as the Standards only use the term “fractions” (4.NF.4c) not “fractions and mixed numbers” as is used elsewhere (5.NF.6).
Similarly, page 9 provides an example of converting a mixed number to a decimal fraction, and converting 2.70 to 2.7. Is this beyond what students should be doing? Should conversions be kept to amounts less than 1?
(Sorry about the double-posting: I also put this in the blog post discussing the Fractions progressions just before I realized I could start a thread here in the forum.)
Duane, thanks again for these careful questions. The standards regard mixed numbers and decimals as fractions; or, more precisely, a mixed number such as 3 1/2 is a sum of fractions, namely 3/1 + 1/2, and a decimal such as 0.3 is just a different way of writing the fraction 3/10, not a different sort of number from a fraction. So the Grade 4 example is an example of multiplying a fraction by a whole number.
The same philosophy pervades the second example you mention, which illustrates both 4.NF.5 and 4.NF.6:
4.NF.5 Express a fraction with denominator 10 as an equivalent fraction with denominator 100, and use this technique to add two fractions with respective denominators 10 and 100.
4.NF.6. Use decimal notation for fractions with denominators 10 or 100.
Aha! Thanks for the clarification, Bill. I have to admit that it’s a little difficult to always know when “fraction” is being used inclusively in the Standards to mean “common fractions, mixed numbers, and decimal fractions”, and when it is being used to just mean “common fractions”. When “fraction” is set against “mixed number” as it is in places such as 5.NF.3, 5.NF.6, and 4.NF.3c it is clear what is meant – 4.NF.4c is not as clear in comparison, but the examples in the Progressions (and your explanation) have helped.
I agree it can be confusing, and that’s partly because we are trying to accommodate old usage at the same time as promoting new usage. In the Common Core, a fraction is a certain type of number on the number line. The name of the number doesn’t change what it is, so in that sense mixed numbers and finite decimals are fractions (also some infinite decimals of course, but that’s a story for later). The terms “mixed number” and “decimal” really refer to certain ways of expressing fractions. So really every time we we talk about them we should say something like “fraction expressed in mixed number form” or “fraction expressed in decimal form.” But that would get old very quickly, so we use the shorter terms.
And, the terms “proper fraction” and “improper fraction” are deprecated entirely in the Common Core.
There’s a discussion of this point here.
“Kim says:
Hello. I tried to find the standard associated with simplifying fractions in the lowest terms. However, I couldn’t. Can you explain when teachers teach this skill to which grade students?
Bill McCallum says:
April 13, 2012 at 10:51 am
Kim, the Standards do not require simplifying fractions into lowest terms, since it is not a mathematically important topic. To quote the Fractions Progression , “It is possible to over-emphasise the importance of reducing fractions …. There is no mathematical reason why fractions must be written in reduced form, although it may be convenient to do so in simple cases.”
Indeed, there are situations where simplifying fractions gets in the way of understanding. For example, insisting that the answer to 1/10 + 3/10 be written as 2/5 gets in the way of the most important understanding that we want students to come away from this problem with, namely that this addition works the same way as whole number addition, with the unit 1 being replaced by the unit 1/10.”
My question relates to the above question/answer quoted from the old stream…
While we are not “requiring” students to simplify fractions to have what is considered a correct answer… is it reasonable to expect them to “see” it both ways?
To be more specific, when considering assessment that could be multiple choice, would we excpect a student to be able to choose an asnwer written as a mixed number for a question that requires them to add something like 1/8 + 5/8 + 3/8?
Yes, students should be able both to “recognize and generate equivalent fractions” (4.NF.1). The multiple choice question you propose is a nice idea.
I wonder why we then still have the footnote, “Grade 4 expectations in this domain are limited to fractions with denominators 2, 3, 4, 5, 6, 8, 10, 12, and 100.” We can’t ask students to compare 7/4 and 8/5, for example. Or, is it ok if students come up with fractions with denominators other than these in the process of their solutions? In other words, the restrictions apply to problems we give to students…
The limitations are intended to provide some guidance to assessment, and generally to build some restraint into the system. It is certainly possible to design problems around the standards in this grade level that are limited exclusively to such fractions. But a limitation on what is expected is not the same as a limitation on what is allowed, as you point out.
Dr. McCallum,
I’m sorry to post this here, but I’m not sure where else to ask this question. I noticed yesterday when directing my math coaches to your blog that my question on this strand and your reply was not visible to them. I could only see it if I logged in. Do you know if there is something I can change to make my comments visible to the public?
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