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As a math coach, I am trying to advise my teachers on the expectations of the standards. I understand the most important thing in adding and subtracting mixed numbers is understanding why you do what you do. My 5th grade teachers asked if they are supposed to teach regrouping when adding and subtracting fractions. My thoughts were to advise them to teach it as one strategy, and focus on students understanding of when to use that strategy (when it is efficient and when it is not), rather than teaching regrouping as the way to do it all the time. I would recommend that they also teach students that converting to improper fractions can also be efficient,depending on the problem. Am I on the right track? Any other insights would be appreciated. Thanks!
If I understand the question correctly, you are saying that sometimes students should add fractions expressed as mixed numbers by grouping the whole number parts together and adding them, and grouping the fractional parts together and adding them, and then putting the two results together. And, other times they might just want to expressed the fractions in purely fractional form and add the numbers together in that form. And, they should have some judgement about when to do which.
If that’s what you are saying, then I wholeheartedly agree!
This is the question I was just looking for an answer to online! So I still need some clarification. If you have changed to a ‘purely fractional form’ does that mean in older terminology that you have changed it to an ‘improper fraction’? Also, let’s say students had to do that step and then they subtract to find the solution… what do they do next? Should they change back to a mixed number… should they simplify and if so first or last… I basically feel that kids needs to know how to make sense of fractions and interchange them as needed especially since with TESTING… many questions will have multiple choice answers. What if the PARCC assessment has a LCD answer or a simplified answer? I hope I’m making sense! I just found your blog and it’s super helpful!
My understanding is that simplifying is really deemphasized in the CC (for example, the quote below for the progressions document). PARCC seems to automatically detect equivalent forms of numbers in the answers. Also, one of their released items uses 5/15 as an outcome that students need to evaluate for value (not representation) reasonableness. Appears that they are not going to rely on simplification as a source of errors to catch.
“It is possible to over-emphasize the importance of simplifying fractions
in this way. There is no mathematical reason why fractions
must be written in simplified form, although it may be convenient to
do so in some cases.”Alexei is correct here. Mixed numbers, fractions, reduced fractions are all different ways of writing the same number. The focus in the standards is on the number itself, not how you write it. Of course, sometimes it might be convenient to write the number in a particular way. But that goes both ways: you might want to write 7/14 as 1/2, but you might equally want to write 2/5 as 4/10 (to see the connection with decimal notation). Choosing a convenient form for the purpose at hand is an important skill, as is the fundamental understanding of equivalence of forms.
And yes, I think both the PARCC and Smarter Balanced assessments will reflect this philosophy. Indeed, if they marked an equivalent form wrong they would themselves be wrong, since a number is a number is a number; if you get the answer right, you should get the points.
Oh, one more point: although simplification is de-emphasized, transforming between equivalent forms is not. That is, students should be able to recognize and produce equivalent fractions (4.NF.1); but they should be taught that simplification is a mathematical imperative.
…but they should not be taught that simplification is a mathematical imperative.
Forgive me if this seems like a silly question, but the language of 5.NF.2 includes mixed numbers, right? I was discussing this with a colleague the other day and noticed that 5.NF.2 just says “fractions” but it follows 5.NF.1 (and is the application of the skills in 5.NF.1), so it would make sense to think that 5.NF.2 includes word problems with mixed numbers. The example in the G3-5 Fraction progression doc (page 11) does not have mixed numbers, so I am not sure what to do.
Any insight would be much appreciated.
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