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Thinking you mean 8.F.B.4 for the first question, the initial value of a function is its output when the input is 0 so yes, that is usually going to be the y-intercept.
And, thinking you mean F.IF.A.3 for the second question, I think the initial value is still going to be f(0) if f is defined at 0, but notice the term “initial value” is not used in this standard, and the domain might not contain zero. For example, you could have a sequence a_1, a_2, . . . whose domain is the set {1, 2, 3, . . . }. I think in this case you would call a_1 the initial term of the sequence to avoid confusion.
The standards say in which grade a certain skill or concept should be achieved and is fair game for assessment. That doesn’t necessarily mean you don’t start working on that skill or concept until that grade. The standard algorithm for division is a case in point. It is the culmination of a long progression of strategies and algorithms that stretches over many grades. In the Illustrative Mathematics curriculum students start seeing the standard algorithm in grade 5, but are not expected to have it down until grade 6. I think it would make sense for your 5th and 6th grade teachers to get together on this!
There is certainly overlap. But in A.CED.A.2 the relationship between does not necessarily have to be a functional relationship. An assessment item satisfying A.CED.A.2 might involve a
budget constraint, for example. This would not align to F.BF.A.1.a (unless you also asked to express one of the quantities as a function of the other). F.LE.A.2 is more specific than F.BF.A.1.a about what needs to be done specifically with exponential functions, e.g., you should be able to derive one from two input-output pairs. It’s probably true that any assessment item aligning to F.LE.A.2 would also align to F.BF.A.1.a.I think this is a case where you have to distinguish between what is assessed and what is in the curriculum. I don’t think you could have anything other than the exact forms given on an assessment aligned to the standards. On the other hand, I can see having a word problem that leads to one of your forms and then has a discussion about x/2 = 1/2 x or (x + 7)/3 = 1/3(x + 7) would be o.k. Both of these are facts the students know from their work with rational numbers. To my mind, there’s a distinction between what you expect students to do routinely and what can come up in a discussion. Full-blown fluency with linear equations in any form doesn’t happen until Grade 8.
I agree with Cathy that laws of exponents are not an algorithm, and maybe this is the source of some misunderstanding in this discussion. If, instead of having students do
root(24)
= root(4 x 6)
= root (4) x root (6)
= 2 root (6)we insisted they do
24^(1/2)
=(4 x 6)^(1/2)
= 4^(1/2) x 6^(1/2)
= 2 x 6^(1/2)then I agree that would be pointless. If they are to undertake this transformation at all, there is no reason to insist on one notation or the other. And, once students have learned that a^(1/2) = root(a), then it doesn’t matter whether you write root(ab) = root(a) root(b) or (ab)^(1/2) = a^(1/2) b^(1/2); either way you expressing a property that comes under the heading of “laws of exponents,” albeit in a disguised form in the first instance.
The question is, what is the purpose of this sort of problem in the first place? I can imagine it being put to service in a sequence of lessons whose ultimate goal is to get students to understand transformations like sqrt(a^2 – x^2) = a sqrt(1 – x^2/a^2), which you need to derive certain indefinite integrals in calculus. I can also imagine a curriculum writer deciding that calculus is too far off for that purpose to justify such problems in, say, an Algebra I course. Either way, this is a curricular decision, not a standards decision. The naked activity of simplifying numerical square root expressions, taken by itself, was not deemed sufficiently important to merit its own standard. If that activity becomes valuable in a curriculum designed to meet the standards, then so be it. That which is not mentioned is not thereby forbidden; it’s just that you have to have a standards-based argument for including it.
If the goal is solely to get from root(24) to 2 root(6), then I don’t think the standards convey any preference whether it should be done using the square root symbol or using exponential notation. The main point I was making in 2014 is that this goal is not sufficiently important by itself to merit mention as a separate topic within the standards. Simplifying radicals, if it is to be included as an activity in a curriculum aligned to the standards, should have some higher purpose than merely simplifying radicals. One possible such purpose is to support understanding of the properties of exponents.
Mindy, a transformation doesn’t have to be made up of reflections, rotations, translations, and dilations, although those are certainly the main ones under study. For example a vertical stretch is a transformation (that is, stretch every vertical line by, say, a factor of 2). Or a shear transformation, which slides every point along a horizontal line a distance proportional to its height above the x-axis (think of shearing a square into a parallelogram of the same height). Here is an activity in the Illustrative Mathematics middle school curriculum which illustrates some of these transformations. It’s in Grade 7, in the section on scaled copies, but it paves the way for the work on rigid transformations in Grade 8.
My instinct as that versions 2 and 3 are fine but version 1 is outside scope. For version 2, although it’s true that students are not adding beyond 100 in grade 2, they are supposed to know that the 3 digits of a 3 digit number represent amounts of 100s, 10s, and 1s (2.NBT.A.1), and that’s what really comes into play in version 2.
That’s better than “improper.” But I’m wondering why you need a separate term at all. My first instinct would be to use the term “fraction” and point out that some fractions are less than 1 and some are greater than 1 (and, for that matter, some are equal to 1 . . . 5/5 = 1 is an important thing to know).
I don’t think this is exactly what you are looking for, but the SAP Coherence Map might be useful in tracing these progressions.
So anyway, the basic idea here is this: I know that rectangle which is 1/n by 1/m has area 1/nm because I can fit nm of them in a unit square. So then I know that a rectangle with dimensions a/n and b/m has ab of those little rectangles, so its area is ab x 1/nm = ab/nm. In other words, the area of a rectangle with fractional side lengths is the product of the side lengths. Of course, curricula often treat this as completely obvious, which is a shame, because the reasoning is fun.
