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I’d have to see the text to be able to comment on it. But as a general comment I would say that there is a progression from proportional relationships to functions. At some point in that progression you introduce the concept of a function, and then you can look back and point out that a proportional relationship can be viewed as a function (in two different ways, depending on which quantity you choose as the input). I don’t know where the non-proportional relationships come in, however, that sounds a bit out of place to me.
Andy, you are talking about my August 23 reply, right? I didn’t mean it to be taken any differently from my previous comments on this. To me the unit cubes or the rectangular prisms with unit fraction sides are mathematical objects, and when I talk about packing them this way or that I am talking about a mathematical activity. So, this could be represented by “manipulatives, drawings, computer animation, verbal descriptions …” as in my previous answer. Sorry for the miscommunication.
Lane, no, there is no standard notation for step functions that I know of in current use in school mathematics.
In answer to your first paragraph: Yes! The standard says “Use rational approximations …” not “Find rational approximations …” so it is a mystery to me how people could misinterpret it. Of course, as you say, this might sometimes involving finding them using simple methods of trial and error, as you suggest, but how anybody could interpret this as requiring a systematic method for finding approximations is beyond me.
In answer to your second paragraph, I’m not sure what you mean by “a clever application of solving by elimination.” 8.NS.1 does in fact say “convert a decimal expansion which repeats eventually into a rational number,” so you have to have some way of doing this. One way is to solve a linear equation: if $x = 0.171717 …$ then I can write the equation $100x-x = 17.171717 … – 0.171717 … = 17$ and solve the equation $100x – x = 17$ to write $x$ as a fraction. You are right that there is some cleverness involved in thinking of multiplying by 100 and subtracting the original number, but this in itself is a nice application of MP7, Look for and use of structure.
Students have been solving equations like this since Grade 7:
7.EE.4 Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities.
Converting repeating decimals to fractions strikes me as a nice “mathematical problem” that falls under this standard.
The dividing line been regular standards and (+) standards shouldn’t be viewed as a demarcation in the curriculum, but rather in assessment. There are a number of instances where for reasons of coherence one would want to include some (+) standards in the curriculum. Notice the statement on p. 57:
All standards without a (+) symbol should be in the common mathematics curriculum for all college and career ready students. Standards with a (+) symbol may also appear in courses intended for all students. [Emphasis added.]
That said, I do think there’s a difference between F-BF.4a and A-CED.4. The procedure is the same, but conceptualizing the procedure as “finding an input to a function which yields a given output” is a step up. Seeing functions as objects in their own right, and algebraic procedures as ways of analyzing those objects, is a sophisticated viewpoint.
This is still causing a bit of confusion. I just uploaded a version of the progression with the new language here. It makes clear that adding and subtracting fractions with one denominator dividing the other is not a requirement in Grade 4 except in the case of denominators 10 and 100. But it also makes clear that this is still a possibility in the case of simple additions such as $\frac13 + \frac16$ where the idea is not to present a general rule but to reason directly with simple equivalences and the definition of addition using visual representations such as fraction strips. So, you couldn’t put this on an assessment in Grade 4 but you could put it in the curriculum.
There is that word “simple” in there. The examples are not intended to be exhaustive, of course, but they do illustrate what is meant by that word. I would say that the examples you gave, while certainly falling within the meaning of the standard, are not required by it. Such theorems should certainly be proved, but proofs using congruence and similarity make more sense to me. The analytic proofs would be quite laborious, wouldn’t they?
I think students will be using some sort of technology (not limited to calculators) for all the operations with data: scatterplots, curve-fitting, plotting residuals. They should be looking at realistic data sets, which are too large for anything else. It makes sense to me to analyze residuals while looking at a different curves of fit, in order to see how the distribution varies. Then you eventually just start looking at the curve of best fit and the correlation coefficient.
I agree with Cathy. What you have sketched is an approach to proving the Pythagorean theorem which is quite beautiful.
Requirement?
Here is a link to the comment: http://commoncoretools.me/forums/topic/algorithms-grades-2-5/#post-939. In it, I said
Some think it is the algorithm exactly as notated by our forebears, some think it includes the expanded algorithm, where you write down all the partial products of the base ten components and then add them up. Ultimately this is a question that has to be settled by discussion, not fiat.
I then went on to state my opinion:
My opinion is that the standard algorithm has two key features; like the expanded algorithm it relies on the distributive law applied to the decomposition of the number into base ten components, but in addition it relies on the fact that the order of computing the partial products allows you to keep track of the addition of the partial products while you are computing them, by storing the higher value digit of each product until the next product is calculated.
I don’t see how I could have made it clearer that I was not stating a requirement, just giving an opinion. And I didn’t say anything about benefit, either. That would take a much longer discussion, since benefit or harm would depend on the context: the students, the classroom culture, the curriculum, the time constraints, and so on.
I agree that students can be fluent with methods such as those described by Beckmann and Fuson. They would satisfy
4.NBT.5 Multiply a whole number of up to four digits by a one-digit whole number, and multiply two two-digit numbers, using strategies based on place value and the properties of operations. …
However, as I’ve said elsewhere, I don’t think that the partial products algorithm is included in what people meant when the used the term “standard algorithm” circa 2009, whether they were for it or against it. So I don’t think 5.NBT.5 includes the partial products algorithm. That said, I don’t think kids should be beaten to death with this; the standards form a pathway along which some students will be ahead, some behind. And some of those behind will need to take shortcuts to catch up. They are not a catechism, they are a shared agreement about what we want students to learn.
Lots of questions here. First, on the rhombuses, rectangles and squares, notice the phrase “and others” earlier in the standard. The intention was not to exclude parallelograms or any other particular type of of quadrilateral. Rather, the intention was to not require parallelograms. Listing every single type might have been taken as a requirement. The main point is to begin to see how different types can be included in larger more general category. The exact list of specific shapes is not important.
The cluster heading “Reason with shapes and their attributes” is consistent through Grades 1–3 … not sure what the question is here.
As for revising the standards, your guess is as good as mine. My preference would be a long revision cycle, say 10 years. Not because the standards don’t need revision, but we need time to work with them to do a thoughtful revision that isn’t just a cacophony of everybody’s favorite modifications.
Sorry, I thought I had already replied to this one. I don’t think students have to see repeating decimals before Grade 7; that is, the standards do not require this.
Well, it’s a bit like saying “Get ready for school, and don’t forget to brush your teeth.” Brushing your teeth is implied by getting read for school, but you want to make sure it isn’t forgotten. G-SRT.11 is a brush-your-teeth standard. Same goes for calling out proof in G-SRT.10 … you want to make sure people see that.
I think Appendix A is a little of base here. There’s a discussion of this from about a year ago here.
