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I guess I would just say “five plus four over nine.” Well, that could still be interpreted as a mixed number. People sometimes say something like “quantity five plus four,” with a pause afterwards, to indicate they are referring to a single quantity. So you could say “quantity five plus four [slight pause] over nine.” Probably I would just avoid having to say this verbally at all.
The standards don’t actually mention the set model for fractions, so exactly when to introduce it is partly up to the judgement of the curriuclum writer. The reason for not introducing it in Grade 3 is that it can cause confusion as to what the whole is. If eat 6 out of 12 bananas, that’s 1/2 of the bananas. But notice that in order to interpret this I need to regard the 12 bananas as a whole. That seems a little alien, as opposed to area of length models, where I can clearly see the rectangle as a whole, or the length from 0 to 1 on the number line.
The set model is really more related to multiplication of a whole number by a fraction: $\frac12 \times 12 = 6$. That doesn’t happen until Grade 5, so that’s why you will see some people say it should go there.
But I think you could also start working with in Grade 4, as a preparation for multiplication of whole numbers by fractions. It depends on exactly how you introduce it.
There are different types of number line. In Kindergarten students might place numbers on a line, but it is often more like a “number row” than a number line … that is, the precise placement is not attended to. In Grade 2 Common Core, the number line is really meant to have a measurement aspect to it. You have a 0 and 1, then you mark of the line in lengths equal to the length from 0 to 1 in order to get 2, 3, etc. At least, you start doing that in Grade 2 … and in Grade 3, the measurement aspect becomes salient when you start talking about fractions. I suspect that all the different sources you are looking at are really talking about different incarnations of the number line.
In my view “fluent” means “fast and accurate” and I don’t see a meaningful distinction between this and “automatic.” But, as you point out, people do make quite suble distinctions between all these words. I think a lot of these distinctions are more meaningful for assessment and curriculum then they are for standards themselves. The standards require fluency; implementers of the standards will be making more fine-grained decisions about how to teach and how to measure it.
Thanks for pointing this out … I’m going to pass it on the blueprint authors for comment. And yes, I try to include the update date, but sometimes I forget!
I’m a little confused by this comment, because 1.NBT.4 does explicitly call for “adding a two-digit number and a multiple of 10,” which seems to be what you are asking for? Can you clarify your concern?
Hi Lane, could you explain a bit more? Maybe give an example of what people are concerned that students might do.
Duane,
Sorry for the long delay in replying to this, but it made me realize I needed to get that revised version of NBT finished. It is now posted. Could you take a look and see if it helps with this confusion? Happy also to answer more questions, now that it is done.Well, remember that course placement is not included in the standards; I guess you are probably looking at Appendix A, correct? I could imagine introducing exponential functions with base e quite early as a standard function in science and finance, and delaying the explanation until later.
Agree with abieniek, and I would only add that this is an explicit requirement in Grade 3.
3.NF.3c: Express whole numbers as fractions, and recognize fractions that are equivalent to whole numbers. Examples: Express 3 in the form 3 = 3/1; recognize that 6/1 = 6; locate 4/4 and 1 at the same point of a number line diagram.
It could be either, I think. It would be a natural part of a classroom where students are talking about their reasoning and asking questions about others’ reasoning. Certainly no need to write an essay!
Yes, it seems we should document this in the progressions document, thanks. It’s interesting that both examples are online tools for skip counting. It’s probably easier to program if you don’t insist on multiples.
On a related note to algorithms, reading through the NBT Progressions, page 3 notes a distinction between “general methods” and “special strategies”. General methods are defined as applicable to all numbers (in base-ten) but not necessarily efficient. They may be efficient but it is not always the case. Special strategies are defined as applicable only to certain cases or applicably to more cases only with “considerable modification”.
The example given on page 3 for a special strategy is 398 + 17, which is rewritten as (398 + 2) + 15. A general strategy example is given as combining like base-ten units, i.e. 300 + (90 + 10) + (8 + 7). Another example of a special strategy is given on page 7 (margin) where you start with one number then count on tens then ones individually, e.g. 46 + 37 –> 46, 56, 66, 76, 77, 78… and so on.
The special strategy given on page 7 doesn’t seem all that difficult to extend to three-digit numbers (i.e. count hundreds, tens, then ones) and beyond, or by adding instead of counting (as noted on page 7). Time-consuming, yes, but not requiring considerable modification. It’s not all that different from counting by ones which was defined on page 3 as a general method. Given its close similarity to counting by ones, and its applicability to all cases, what makes this strategy “special”?
Also, a distinction is made between algorithms and strategies (p.3), with strategies being broken into special and general as discussed above. The top example in the margin of page 7 shows the standard addition algorithm but it is labeled as a general method, i.e. a “strategy”. So I’m confused – is the diagram showing an algorithm or a strategy?
Not sure what to say here. I agree with some of what Steve says, disagree with some other other things he says, and think he misses some important structural elements of the standards. Everybody could write standards that they personally think are better than the Common Core. (Including me.) But that’s not the point of having common standards. Different people with different opinions settled on an agreement in 2010 about expectations for what students should know at the end of each grade level or course. Until we prove we can implement an agreement (not at all clear yet) I’m not interested in relitigating old arguments.
