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[Corrected] The standards go up through Algebra II, but do not include Calculus. They partially cover Precalculus if you include the plus standards. I didn’t understand your comment about not being supposed to teach these courses. There is nothing in the standards that forbids students from going beyond them if they are ready to do so before the end of high school. The policy for such students is entirely up to the school, school district, or state.
I don’t think the two meanings are inconsistent; $5 \times 7$ is indeed the result of multiplying $5 \times 7$, and the student who says “five times seven” isn’t wrong (maybe a smart aleck, but not wrong). In fact, now I think about it a little more, I’m not sure it’s correct to say that there are two different meanings; rather there are two different ways of writing the product.
At any rate, the language of the standards is not necessarily the language you would use with students. The standard describes a certain understanding you want students to have, an interpretation of $a \times b$ as the the number of things in $a$ equal groups of $b$ things each. You are probably not going to bring about this understanding by asking a class of third graders “what is the intepretation of $5 \times 7$?” You might simply work with various situations in which products arise in this way.
Your second interpretation is correct. This standard is the first standard about multiplication and is about understanding the meaning of multiplication. So yes, it’s about the meaning of expression $5 \times 7$. I agree the word product is used with two meanings; as expression here, and as the result of a calculation elsewhere. Do you think this causes confusion? It seems fairly common.
I guess you could model subtraction with a set of objects without it being a word problem per se. For example, you could ask students to count 5 blocks, take away 2, and then count the ones left over, and describe this as “5 take away 2 leaves 3”. The difference between this and a word problem is that you are not asking the student to read (or hear) about a situation and then represent it with a subtraction problem.
Greatest common factors and least common multiplies are treated with a very light touch in the standards. They are not a major topic, and limited to numbers less than or equal to 100 (6.NS.4). For such numbers, listing the factors or multiplies is probably the most efficient method, and has the added benefit of reinforcing number facts. It also supports the meaning of the terms: you can see directly that you are finding the greatest common factor or the least common multiple. The prime factorization method can be a bit mysterious in this regard. And, as you point out, prime factorization is not a topic in the Common Core, although prime numbers are mentioned in 4.OA.4. So, the standards do indeed remove this topic from the curriculum. Achieving the focus of the standards means giving some things up, and this is one of those things. (Of course personally, as a number theorist, I love the topic!)
Good points, Duane. The measurement standard in question is
3.MD.4 Generate measurement data by measuring lengths using rulers marked with halves and fourths of an inch. Show the data by making a line plot, where the horizontal scale is marked off in appropriate units—whole numbers, halves, or quarters.
It’s a very interesting question whether this is introducing mixed numbers or not. A ruler marked in halves and quarters doesn’t generally label the marks with the numbers they represent, and the bamboo shoot diagram you refer to works equally well with only the whole number marks labelled. So you could avoid mentioning mixed numbers altogether in Grade 3. On the other hand, examples like this are clearly pointing in that direction, and provide a good opportunity to talk about them if, for example, a student asks how to label the marks. But work with mixed numbers is not required until Grade 4.
Here’s the diagram for others’ convenience:
I am working on it now, hope to get it done soon. (But my perennial optimism is sometimes confounded.)
Although the standards do not say so explicitly, the examples do suggest that the constant of proportionality is always positive in Grades 6–7. As you point out, this is confirmed by the progressions document on Ratios and Proportional Relationships says on page 11 that
Proportional relationships are a major type of linear function; they are those linear functions that have a positive rate of change and take 0 to 0.
It makes sense initially to keep the constant positive, since although negative numbers have been introduced in Grade 6, most of the quantities being dealt with in proportional relationships are positive.
Once students start dealing with linear functions in Grade 8 and beyond, they become familiar with the meaning of negative slope and negative rate of change, and they can use those terms to describe functions of the form $f(x) = kx$ with $k <0$. I could go either way on whether you call that function a proportional relationship, and I don’t think it much matters which way you go. It’s probably easiest to allow the term, since that’s what people will do anyway. There’s a useful brief discussion of this question at the Math Forum.
I agree as long as “always” means “almost always in the same class” or something like that; not “for ever”. Students who continue with mathematically intensive studies will sooner or later encounter the same letter used differently or the same quantity with a different letter, for example when they take a high school physics class or a college electrical engineering class. We don’t want them to think that the letters somehow have intrinsic meanings. By the same token, the input variable in a function shouldn’t always be $x$ or $t$.
More from Doug Clements about defining attributes:
… the phrase (perhaps unfortunate) of “defining attribute” is for the adult to understand it is defining, but for the child, it is not expected that they know a formal definition, only that they understand “rectangles need to have all right angles” but that their color doesn’t matter. (Until middle school or later, of course, we don’t expected them to know what’s truly defining, such as needing only to say a parallelogram needs “at least one right angle” to be a rectangle, etc.).
Tad, I’ve asked Doug Clements to take a look at this and hopefully he will get to it soon. On the issue of “defining attributes”, I will say that being able to describe defining attributes is not the same thing as being able to give a formal definition. You might want to look at the discussion here to see if it answers some of your questions.
Feel free to put it in a comment here if you don’t mind it being public, and I will make sure it is included with the feedback we consider for the revision. If you would rather send it privately, you can send it to me at william.mccallum@gmail.com.
There’s a discussion of this point here.
In PARCC the Grade 5 MD cluster “Convert like measurement units within a given measurement system” is listed as a supporting standard, which means it is linked to major clusters such as fractions. I see this as signaling that unit conversion will often be embedded in other problems, rather becoming an extensive separate topic on its own. Assessments will likely establish some limits on how many different systems are expected from your list, since this is not specified in the standards themselves.
