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Andy, I sense a certain frustration on your part here, but thanks anyway for bringing these matters up … that’s what this blog is for.
First, I would say that there’s a difference between the case of 4.MD.2 and the case of 4.NF.5, in that the former was open to more than one interpretation. In my earlier reply I advocated using other standards in Grade 4 to decide the interpretation of 4.MD.2, including the Grade 1 understanding of subtraction. In particular, 4.MD.2 should not be interpreted as creating additional operations not already covered by the other operation standards in Grade 4. And note that I didn’t really need to cite the Grade 1 standard because, as you point out, there is warrant for subtraction of fractions in 4.NF.3. However, none of this interpretation work requires a change in the wording of 4.MD.2; it is a question of whether you interpret a standard made up of lists by multiplying all the lists together or by selecting the combinations that are warranted by the other standards.
The case of 4.NF.5 is different, because it’s pretty clear what the words mean, and it doesn’t say “subtract.” As you point out, other standards do say “add and subtract” and this one doesn’t. I agree with you that it’s unfair to expect curriculum writers to interpolate a word that isn’t there, at the same time as admonishing them to stick to the standards, so I would say you should take this standard as written. Note this does not exclude something like 0.7-0.3, which is already covered by 4.NF.3.
The question remains, why limit to addition in 4.NF.5 and not in 4.NF.3? One possible reason is that 4.NF.5 is the only Grade 4 standard where we are adding fractions with different denominators. Given the extra conceptual demand involved, it is perhaps a good idea to limit to addition.
This is the sort of thing I worry about with acceleration that is either unwarranted by the student’s performance, or unsupported by adequate curriculum and instruction.
I agree there are two uses of the word “cluster” here, one referring univariate data and the other referring to bivariate data. But in both cases the word means the same thing, namely an informal notion of data points being close to each other in a group. And that is certainly the sense intended in the standards, not any formal statistical construct.
The standards don’t require fraction notation at all in Grade 2. The relevant standard here is
2.G.3. Partition circles and rectangles into two, three, or four equal shares, describe the shares using the words halves, thirds, half of, a third of, etc., and describe the whole as two halves, three thirds, four fourths. Recognize that equal shares of identical wholes need not have the same shape.
This is not really about fractions as numbers, although it is clearly a precursor to that concept. Fractions as numbers and fraction notation are introduced in Grade 3.
This language in the progression is misleading, and has been fixed in the most recent draft (not yet published). You are right that fraction addition is limited to equal denominators and denominators 10 and 100 in Grade 4.
Examples A, B, and D doesn’t seem to fall under either 5.NBT.6 or 5.NBT.7. The former requires whole number quotients of whole numbers, and the latter is limited to the fraction divisions under 5.NF.7, unit fractions by whole numbers and whole numbers by unit fractions. But I agree that any of them could arise at by reasoning about the corresponding multiplication problem.
Examples A, B, and D doesn’t seem to fall under either 5.NBT.6 or 5.NBT.7. The former requires whole number quotients of whole numbers, and the latter is limited to the fraction divisions under 5.NF.7, unit fractions by whole numbers and whole numbers by unit fractions. But I agree that any of them could arise at by reasoning about the corresponding multiplication problem.
Your method of solving the 4 x 100 meter relay problem certainly fits with the Grade 4 standards, although I also think it also illustrates that the problem is challenging for Grade 4. And it seems to me one could stage things differently; perhaps developing first the understanding that 10.6, 10.60 and 10 6/10 are all different ways of writing the same number, and then adding the numbers in decimal form more directly. Decimals are regarded in the standards as a different way of writing fractions, not as different sorts of numbers from fractions, so the phrase “a fraction with denominator 10” in 4.NF.5 refers equally to 0.6 or 6/10, and an “equivalent fraction with denominator 100” could be written equally as 0.60 or 60/100. Thus, after having become familiar with the meaning of decimal notation one might write the solution to the word problem as 10.6 + 11.33 + 11.9 + 9.98 = 10.60 + 11.33 + 11.90 + 9.98 = 10 + 11 + 11 + 9 + 0.60 + 0.33 + 0.90 + 0.98 and then add the whole numbers and the fractions with denominator 100 as you have indicated. By Grade 5 one might leave out the fraction notation altogether, as indicated by 5.NBT.7.
As to your question about subtraction, multiplication and division, it is a good idea to put the standard in the context of all the other standards at the grade level. Multiplication of fractions is not completed until Grade 5, nor division until Grade 6, so 4.MD.2 should not be construed as introducing additional content. One might make this a general principle in reading the standards; reading standards in isolation can lead to nonsense. It is neither a requirement of English grammar nor in keeping with the weight and focus of the other standards in Grade 4 to produce 48 distinct standards by multiplying the 3 lists
- “four operations”
- “intervals of time, liquid volumes, masses of objects, and money”
- “including simple fractions or decimals” [taking this to mean whole numbers, fractions and decimals].
Thus, word problems involving multiplication and division of decimals are not implied by 4.MD.2.
Subtraction however does not strike me as completely excluded since students since Grade 1 have been encouraged to “understand subtraction as a missing addend problem” (1.OA.4).
The work with numerical fractions is intended to set the stage for algebraic fractions. Note the general formula for addition in 5.NF.1, and the reference to complex fractions in 7.RP.1.
I think of the functions-based approach as an approach were you introduce the concept of a function early, and use it as a springboard for a lot of the work in algebra. For example, you might think of equations in one variable as questions about when two functions (the functions defined by the expressions on either side of the equation) are equal. And you might have such equations arise in a context where both functions have some meaning related to the context and the equation of when they are equal is meaningful in the context.
It falls under both. Cathy’s point is that because of 1.OA.4, Understand subtraction as an unknown-addend problem, properties of operations do potentially apply to subtraction problems because those subtraction problems might be recast as unknown-addend problems. Although it is true that subtraction does not satisfy the same properties as addition, that does not mean that those properties cannot be used to solve a subtraction problem, as Cathy’s examples illustrate.
The whole could be the interval from 0 to 1 on the number line … this is the main representation the progression is heading towards (see 3.NF.2). The fractions progression suggests introducing set models in Grade 4, not in Grade 3.
Well, you can form the surface of the pyramid this way, but I wasn’t thinking of that as a 3-D object. The solid pyramid cannot be formed by cubes and right prisms. That’s why I thought surface area of a pyramid was fair game, but not volume. On the other hand, I think cones include pyramids in Grade 8 (a pyramid is a cone on a polygonal base).
