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Bill, in a previous response to another poster (http://commoncoretools.me/2011/08/12/drafty-draft-of-fractions-progression/#comment-1887), you agreed that there is an easier method for adding mixed numbers than to convert them to improper fractions. This alternative is to add the wholes, add the fractions, then add the totals together. e.g. 2 1/5 + 1 3/5 = (2 + 1) + (1/5 + 3/5). This method can be adapted for subtraction too, but presents some (small) difficulties if regrouping needs to occur. I assume that Grade 4 students should be able to subtract any mixed numbers, such as 3 2/5 − 1 4/5, and not just non-regrouping ones, such as 3 4/5 − 1 2/5.
Is this where converting mixed numbers to improper fractions proves its worth in Grade 4? If not, I’m having trouble seeing why students would need to convert mixed numbers to improper fractions at all in Grade 4.
In that previous post I was referring to a specific calculation, where it did indeed seem that it was easier to use the properties of operations to add the mixed numbers than to convert them to fractions. But I don’t think it is a good idea to have one preferred method all through Grade 4. Even if it were true that it is always computationally more efficient to do it one way, computational efficiency is not the only goal here, or even the main goal. The main goal is developing a solid understanding of fractions as numbers. Part of this is seeing that fractions, mixed numbers, and decimals are not different types of numbers, but are rather different ways of writing the same number. It may be that this is reinforced by seeing that a computation done two different ways gives the same answer.
It’s not so much that a method is more efficient than another, but that one method requires a lot more mental processing than another. Due to the limits of writing fractions in the correct format on this blog, I’ll need to refer to page 7 of the NF Progressions and the example of converting 47/6 to 7 5/6. If we imagine reading the equation from right to left we pretty much have the process that students need to perform to convert mixed numbers to improper fractions. It seems like a lot of extra, error-prone work that has to be done for each addend, compared to the alternative of adding wholes then adding fractions. This is especially the case once you get denominators greater than 10, or wholes greater than 10, as students then need to go beyond their basic multiplication facts to convert.
The equivalence of different formats for fractions could be demonstrated in other ways (e.g. using region models or number lines) if that is the main goal. I’m just not sure that students will get the equivalence message, or understand the point of conversion, after working through the computation. I think there will be relief, but probably not appreciation of equivalence.
In a grade level packed with so many big ideas already it’s hard to see the value of the exercise. I can see the value of seeing equivalence pictorially, but not via computational conversion – does computational conversion lead on to grander things in Grade 5?
Duane, thanks for this comment. Note that the example of converting 47/6 isn’t coming from a sum and that the CCSS say “Grade 4 expectations in this domain [NF] are limited to fractions with denominators 2, 3, 4, 5, 6, 8, 10, 12, and 100.” Are you chaining together two examples from the progression to extrapolate a single method for adding mixed numbers that you think the progression is recommending? The standard about adding mixed numbers doesn’t prescribe a single method:
4NF.3c Add and subtract mixed numbers with like denominators, e.g., by replacing each mixed number with an equivalent fraction, and/or by using properties of operations and the relationship between addition and subtraction.
Another reason to mention converting mixed numbers to fractions (and vice versa) is that it is yet another example of decomposing or composing a unit. For fractions with base 10 or 100, this can be used to as preparation to extend ideas of the base-ten system, e.g., 2 and 1/10 is 21 tenths.
Your second-to-last paragraph reminds me of Stanley Erlwanger’s famous case study of Benny, the pseudonym of a student who was learning about fractions and decimals in the 1970s. It’s posted here, together with an introduction: http://www.uky.edu/~mfi223/EDC670OtherReadings_files/ErlwangersBenny.pdf. One of the things that Benny believed was that if you did something (e.g., converted mixed numbers to improper fractions) pictorially and computationally, it didn’t matter if you got different answers from different methods. That sort of belief is countered by MPS1 which says that students can explain correspondences between different representations of the same thing and between different approaches to computing the same thing.
Cathy, thanks for replying. My assumptions are based on what I read in the Standards and the Progressions. The first example in the Standards is of conversion. The Progressions starting from page 6 with the heading of “Adding and subtracting fractions” only seem to mention conversion in reference to mixed numbers and improper fractions – no other method seems to be mentioned. I add 4/4 and 4/4 and come up with 2/1, what can I say?
Seriously though, if not for Bill’s and now your comments, I would still assume that conversion is the preferred method based on those two documents. I know now that there is not a single method proposed/preferred – perhaps it would be a useful addition (pardon the pun) to state explicitly, with examples, in the Progressions what the different methods are.
Aside from that, the thrust of query is why link conversion to addition and subtraction in the first place? It can be done but is it necessary? Is it useful? In Grade 4? I think seeing improper fractions and mixed numbers in different ways is useful for the reason you mentioned (composing/decomposing units, looking at decimal fractions) but proposing that it be used for addition and subtraction as it is in 4.NF.3c… why? In Grade 4, as you point out, students can use denominators of 12. To be mean, I could give students a question such as 13 7/12 − 6 4/12 or even just 16 7/8 − 12 2/8 and tell them to convert to improper fractions first and they would absolutely hate me! Of course, I wouldn’t actually do this – except, perhaps, to point out where conversion becomes a chore.
My fear is that because conversion for addition/subtraction is mentioned as an option, assessment writers may assume that all students should have been taught it, so all students should be assessed on it – the “and/or” in 4.NF.3c will be irrelevant and simply become an “and”.
Duane, assessment developers also get guidance from http://illustrativemathematics.org/standards/k8. Check out the Peaches task. Two solutions are given (with and without conversion). Neither is labeled “preferred.”
The standard is written to indicate that the methods are examples: “Add and subtract mixed numbers with like denominators, e.g., by replacing each mixed number with an equivalent fraction, and/or by using properties of operations and the relationship between addition and subtraction.”
The “e.g.” is there for a reason, it indicates that the method is not prescribed. In the sentence, one reason to put the conversion method before the other method is that the sentence is hard to read if the order is switched (too many “and”s).
The focus of the standard is “can the student add and subtract mixed numbers”? I think that we would all be unhappy if an assessment developer misinterpreted the sentence but I don’t think it’s easy to misinterpret in the way you fear. I realize that the US has a history of poor quality tests (e.g., http://www.educationsector.org/publications/margins-error-testing-industry-no-child-left-behind-era), but some of the conditions of test development have been changed.
In the progression, line 5 of page 7 says: “Students use this method to add mixed numbers with like denominators.” I think the problem is that the sentence can be interpreted as saying “Students MUST use this method” and there’s no example that shows another method. We can put one in.
Thanks Cathy – that Peaches task ( http://illustrativemathematics.org/illustrations/968 ) is exactly the type of example that I think should be given in the Progressions, although it wouldn’t necessarily have to be as long. It’s clear, well illustrated and also shows an alternative method for conversion, where the whole number is converted to a fraction with a denominator of 1 which is then turned into an equivalent fraction. Wonderful!
For the sake of my own curiosity though, where is conversion with addition/subtraction headed to? Using the examples from the Peaches activity, Method 1 and Method 2 are probably equally cumbersome if any type of regrouping is required – conversion probably doesn’t provide much of an advantage, if any. When I need to explain to students why they are learning how to convert mixed numbers to improper fractions to add or subtract, what can I say? Is there a point further down the mathematical trail where conversion will make the effort worthwhile?
Yes, I originally thought subtraction requiring regrouping could be an instance where conversion proves its worth with operations. But a student with sound understanding of addition and subtraction could count on from 1 3/4 to 2, then to 2 1/4, then add the jumps to get 2/4.
As you said, following the same process as the subtraction algorithm for whole numbers will also give them the answer but this is probably as time-consuming as conversion. That is, neither method gives an advantage. Is there a clear benefit in later years for converting mixed numbers before calculation? I think having an extra trick up your sleeve is okay as a rationale but if there was something startlingly good about it then that’d help me explain the purpose to students.
I’m not sure that the analogue with whole number subtraction and the possible meanings of “conversion” are clear. Let’s suppose we’re computing $2 \frac 13 – \frac23$.
There are at least three options (I’ll put fewer steps than a student might and put parentheses to push the analogue that I’m trying to make):
A. $(2 + \frac13) – \frac23 = \frac73 – \frac23 = \frac53$ (converting the entire mixed number to an improper fraction, i.e. decomposing two 1s as six thirds)
B. $(2 + \frac13) – \frac23 = (1 + 1 + \frac13) – \frac23 = (1 + \frac13) + (1 – \frac23) = (1 + \frac13) + \frac13 = 1 + \frac23$ (using properties of operations)
C. $(2 + \frac13) – \frac23 = (1 + 1 + \frac13) – \frac23 = (1 + \frac43) – \frac23 = 1 + \frac23$ (decomposing one 1 as three thirds)
Versions A and C both involve writing an improper fraction (though one could do a variant of C that didn’t), but C is a closer analogue to decomposing a unit as done in the subtraction algorithm for whole numbers. So, when you get to decimal fractions, you can use an analogue of C to mimic what’s happening in the subtraction algorithm:
$(2 + \frac1{10}) – \frac2{10} = (1 + 1 + \frac1{10}) – \frac2{10} = (1 + \frac{11}{10}) – \frac2{10} = 1 + \frac9{10}.$
The advantage of just decomposing a 1 as in C is that it’s the analogue of what students learned in multi-digit subtraction, so the same explanation of decomposing a unit of the minuend (in this case a 1 of the 2 + 1/3) applies for fractions and later to fractions expressed in decimal notation. (In case you’re familiar with Liping Ma’s book Knowing and Teaching Elementary Mathematics, I’m thinking about the discussion on pp. 8–9.)
Thank you for setting out the three examples, Cathy. I’m sure it will be helpful to other readers of this blog too.
I have a question about grade 3 NF with regard to mixed numbers. Mixed fractions are not mentioned in the grade 3 NF domain. I see 4/1 and 4/4 but not 41/2. However, when we get to MD in grade 3, we find mixed numbers in the line plots. We are aligning our grade 3 (other grades too) curricular resources to the CCSS. We skipped chapters in our book on mixed numbers. Then when we got to line plots with fractional locations, we were confused. Could someone please provide guidance on this? Thank you in advance.
Nancy, there is some discussion of the topic here:
http://commoncoretools.me/forums/topic/mixed-numbers-and-measurement-grade-3/Bill suggested you could skip the mixed numbers and just use whole numbers on the line plot. So students could measure using mixed numbers (as in “four and one-half inches”) but not write them (as in “4 1/2”). It is an awkward standard to work with.
Page 10 of the Fraction Progression document is the following statement (which is also included in the videos on Illustrative Mathematics):
In grade 4, students calculate sums of fractions with different denominators where one denominator is a divisor of the other, so that only one fraction has to be changed.The progression document then gives an example of adding fractions with denominators of 3 and 6 (not 10 and 100).
In the actual CCSS on page 31, the footnote says
Students who can generate equivalent fractions can develop strategies for adding fractions with unlike denominators in general. But addition and subtraction with unlike denominators in general is not a requirement at this grade.I am wondering where in 4th grade teachers see that they are responsible for addition of fractions where one denominator is a divisor of another. I only see 4.NF.3a which could extend to this, but does not say anything about explicit about divisors that are multiples of another. Although I agree with the footnote, the supporting materials are very specific about grade 4 that does not seem to be included in the actual standards. I might just be missing something.
Thanks in advance for your time.
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