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The standards themselves are not being revised, but CCSSO has adopted a way of referring to them that includes uppercase roman letters for the clusters, e.g. 2.MD.A.1. You can read about this at the main standards website, corestandards.org.
I agree! Past the limit you mentioned it becomes rather pointless. I think giving some guidelines in the Progressions would help both under- and over-enthusiastic teachers of algorithms.
Multi-digit for all these standards means “enough digits to reveal the algorithm in all its generality, but not so many as to constitute pointless torture of children.” Exactly where that point is depends a lot on the classroom context, and is a matter of opinion anyway. I would guess that 3 x 3 and 2 x 4 are enough for 5.NBT.5.
Well, if I had to pick just one, I would pick the inclusive one. But this problem is not going to be solved in this forum, I think. Maybe as textbooks start to align more to the common core they will also come up with some common agreements on this sort of thing.
Alexei is correct. Note the emphasis is on rewriting rather than simplifying, however. Indeed, it’s not at all clear which of $\sqrt{18}$ and $3\sqrt{2}$ is simpler, and each might be useful in different contexts.
Sorry, you’ll have to be more specific here. What statement from Appendix A are you referring to? What relationships among segments on secants and tangents?
I’ll leave this for others, but I cannot resist pointing out that the logo of the Institute for Mathematics & Education (ime.math.arizona.edu) has a visual proof of the theorem, as does the Illustrative Mathematics project website (http://illustrativemathematics.org).
Thanks for helping, Cathy. I guess my initial question was how to write the formulas and which ones are expected. It then moved on to wondering whether formulas written with letters should be used at all in Grades 4 and 5. The reference to “specific numerical instances” gives some guidance as to how to discuss how terms can be rearranged but I’m mainly concerned about whether I should write P = 2 ×(l + w) (and other formulas) on the board or not.
Thanks again Bill, I’d missed that bit in the Progressions (p. 22 for others who are interested). Without reading that part of the Progressions I’m sure many teachers will mistakenly read “apply the formula” in the Standards as exactly what you advise against. I’m a little puzzled how it relates to these statements on page 21, “… remembering the basic formula can help to prevent the usual error…” and later on that page “…discussion of formulas such as P = 2l + 2w…”.
I’m guessing that the formulas have value in Grade 4 and 5 as a “summary” of the thinking that students must use – you can discuss them but students aren’t expected to use them in the sense that they substitute numbers into a formula. Am I on the right track?
I think worrying about which formula to use is the wrong focus here. To quote from the progression:
“Apply the formula” does not mean write down a memorized formula and put in known values because at Grade 4 students do not evaluate expressions (they begin this type of work in Grade 6). In Grade 4, working with perimeter and area of rectangles is still grounded in specific visualizations and numbers.
We don’t want students in Grade 4 to think of any of these formulas as a way you have to pass through in order to find the perimeter (in fact, I’m not sure we ever want that). Rather, formulas are a way of describing the calculation you do in order to find the perimeter, and different versions of the formula correspond to different ways of doing that calculation. But the basic understanding of perimeter should not be mediated through a formula; perimeter should be understood as the distance around the sides, which you can find without knowing any formulas.
Phew! Thanks for the clarification on the multiplication symbol, Bill. So should students be using P = 2 ×(l + w) by the end of Grade 4 or can they use one of the other formulas I mentioned?
It depends on the wording of the question. Does it mention functions explicitly at all? If it’s just about equations of lines, then the F-BF.1 and F-LE.2 classifications would not be appropriate. But if the equation is intended to be viewed as defining one variable as a function of the other, then they might be. One reason for having A and F as separate categories is to reinforce the distinction between equations in two variables on the one hand, and functions on the other, a distinction that often seems to get lost.
It depends on the wording of the question. Does it mention functions explicitly at all? If it’s just about equations of lines, then the F-BF.1 and F-LE.2 classifications would not be appropriate. But if the equation is intended to be viewed as defining one variable as a function of the other, then they might be. One reason for having A and F as separate categories is to reinforce the distinction between equations in two variables on the one hand, and functions on the other, a distinction that often seems to get lost.
