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Thank you for setting out the three examples, Cathy. I’m sure it will be helpful to other readers of this blog too.
Did you mean to say “horizontal line points right”? Either way, I’m not sure about using the terminology “points right” and “points left”. A number line has arrows at both ends, one pointing in the positive direction and the other pointing in the negative direction. The term “oriented from left to right” was meant simply to indicated that numbers on the right are greater than numbers on the left. This should be a standard convention with number lines, and it should also be a convention that a number line is horizontal; students don’t need to say this every time. Once students start working with coordinates, they have a vertical axis as well as a horizontal axis. I would resist calling these number lines, although obviously they are descended from number lines; rather I would call them axes. And, of course, there is a convention about vertical axes as well, that numbers above are larger than numbers below.
The interpretation in Appendix A is an extension of the standard, certainly not an unreasonable one, but it would be hard to argue that the standards require the theorems you link to, since they don’t list them in the things to be included, and the standard, although it mentions chords, does not mention segments on chords. I think this is a difference between what a well-rounded implementation of the standards might include, and what might appear on an assessment. The standards avoid comprehensive lists, while providing an overall structure in which extensions can find a natural home.
The list is intended to suggest various possibilities, not a comprehensive list of requirements. Different situations will call for different tools, and different pedagogies and curricula will have their own preferences. Technology, for example, is neither required nor forbidden.
And, to speak to question of the authors’ intent, the standards set expectations but do not dictate how those expectations are met or assessed. The debate about assessment is the domain of practitioners and researchers.
It’s true that the number of topics is not much reduced in high school from what is traditional from what is traditional, although there was some reduction. But I would argue that a topic count does not quite capture the focus of the high school standards, because the emphasis on reasoning and seeing structure has the potential to unify what were previously seen by students as many different unconnected techniques. Whether this happens or not depends on textbooks and teachers, of course. But I would hope, for example, that rather than seeing the many different forms of an equation of a straight line as things to be memorized separately, students would see the unifying idea of slope as enabling them to come up with the form they need in a given situation.
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.
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?
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”.
Cathy, I’ve tried to fix the formatting. I’ve noticed that if you are editing with the HTML tab active you need to use the buttons at the top to enter the tags rather than typing them in. The latter causes the tags to be displayed verbatim.
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?
The only domain in the Geometry category that obviously requires algebra is the one on analytic geometry, Expressing Geometry Properties with Equations. You could teach the rest before a high school algebra course.
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.
