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I think absolute value equations should be treated pretty lightly, and that is reflected in the blueprints. That said, a more detailed elaboration of the blueprints will probably say something about them.
You are correct that it does not mean simulation. I think that sentence might be clearer if it just said “choose a sampling method” although that’s probably not technically exactly the same thing from a statistician’s point of view. If you do a random sample, you are choosing a probability model where every unit in the population has the same probability (that is, a uniform probability model). If you do a stratified random sample, then you could think of that as assigning equal probability to each group and then uniform probability within each group. I think that in the great majority of cases it is going to be a simple random sample for a curriculum aligned to the standards. Of course, an AP Stats course will go more deeply into things.
A fraction does not have to be less on than one, that’s for sure! As for improper fractions, there is no prohibition on writing things like 2 1/2. Indeed, it would be hard to avoid. But the standards do not use the term “improper fraction” because it promotes the misconception that a “proper” fraction must be less than 1. The notation 2 1/2 is just a shorthand for the sum 2 + 1/2, and should be read that way. Then rewriting it in the form 5/2 is accomplished the same way as for any sum of fractions (with the understanding that 2 = 2/1).
The standards also avoid talking about converting between proper and improper fractions, because the word “convert” suggests you are actually changing the number. The number stays the same, there are just different ways of writing it, depending on your purpose. Students should be able to deal with fractions written in any form, but there is no need to insist they write them in one particular way.
I’m not sure you can avoid the term “improper fraction” entirely. I’d be interested to try though.
I’m scratching my head about Bill’s response to kirkkimb’s query as it relates to 6.NS.2 and 3. Those standards require fluency with the standard algorithm for multi-digit decimals. Bill’s response is more of a “reasoning via common fractions” approach instead of an extension of existing skills with the standard algorithm developed in Grade 5.
My interpretation of “extending the zeros” with the standard algorithm is shown in this link for dividing 121 by 8:
https://dj1hlxw0wr920.cloudfront.net/userfiles/wyzfiles/8d3dfd8f-9610-413b-9952-8db75f1b71a4.gif
(hopefully this link will persist for a while – I have no other way to show it)
In other words, continuously extending the decimal places of 121 to thousandths (121.000) to assist with recording a decimal remainder. Is this part of what is expected for fluency with the standard algorithm for multi-digit decimals?
What about an expression like 6x + 7y – 2x? I am looking at a book that uses the commutative property as justification for rewriting this as 6x – 2x + 7y and then the distributive property to go from there to (6 – 2)x + 7y and then 4x + 7y. I don’t think the commutative property should be used in this case because the expression involves subtraction, and students do not yet know that subtracting 2x is the same as adding –2x. Do you agree? If so, how should students reason about simplifying this expression?
Thanks Bill, I finally had a decent chance to read through the new draft. I think the explanations are much clearer than before, especially the discussion surrounding “efficient, accurate, and generalizable methods”. There are a few things I’m interested in clarifying though.
One is that on p.14 there is mention of students adding and subtracting through 1,000,000 using the standard algorithm in Grade 4. I recall reading somewhere else on this blog that the standard algorithms need only be extended as far as necessary to demonstrate that they are generalizable. In light of that comment, is this reach to 1,000,000 viewed as what is necessary to accomplish the light-bulb moment or is it simply a border to stop teachers going any further?
Another is there seems to be a mismatch between an explanation on p.9 and a method in the margin. At the start of the fourth paragraph a description is given of the first (presumably top) method shown in the margin. The statement is given that “The first method can be seen as related to oral counting-on… in which an addend is decomposed…[and] successively added to the other addend.” Further down in that paragraph this is shown as essentially: 278 + 100 = 378 –> 378 + 40 = 418 –> 418 + 7 = 425. However, this is not what the method in the margin shows. Instead the method shown is simply an expanded form of the standard algorithm and relies on splitting both addends into hundreds, tens, and ones. Am I interpreting the paragraph text and margin method correctly? (On a related note, something that may help generally in all final versions of the Progressions documents is labeling any figures “Figure 1”, “Figure 2”, and so on.)
A final query is about a term on p.7, 3rd paragraph. What is a “5-group”?
You can certiainly have problems about comparing mixed numbers. Mixed numbers are really sums of fractions, i.e. 3 2/5 is really a shorthand way of writing 3 + 2/5. So, depending on how the student does the comparison, it might call on their knowledge of 5.NF.3 as well as 5.NF.2.
Remember that the standards are goals to be achieved by the end of the year, so you don’t have to think of them as curricular units, nor does the order in which the standards are written necessarily relate to the order of topics in the curriculum.
abienek is right that the discriminant is not required explicitly, although A.REI.4 would be the place to put it. Of course, it’s there implicitly in that it’s the thing under the square root sign in the quadratic formula. No harm in pointing out that that thing has a name, I suppose, but I agree it doesn’t seem necessary and could need to excessive rulifying.
Sorry it is taken me so long to reply to this. First, there is no official definition of π in the standards, so no, you don’t have to take Wu’s approach. I quite like it myself but I have talked to others who disagree. Mathematically, you can do it either way; the miracle is that the two numbers (area of unit circle and constant of proportionality between circumference and diameter) turn out to be the same.
As to your other questions, I think Wu’s limit argument is a bit too much for Grade 7. I would use the argument that rearranges the triangles into an approximate rectangle with length equal to half the diameter and height equal to the radius if I were going to give any argument at all. From this you can get that the area is 1/2 the product of the circumference and the diameter, A = 1/2 Cr. As you point out, the standards do not technically require that you justify the individual formulas C = πd (or C = 2πr) and A = πr^2. But you are almost there at this point. If you have defined π as a constant of proportionality, you may have given an informal justification of why that constant of proportionality exists. Doing so would amount to a justification of the formula for the circumference. And once you have that you can get the area formula by substituting the circumference formula into A = 1/2 Cr.
I don’t see an SBAC definition, but my guess is they are likely to agree with PARCC. It is the more common sense definition from a mathematical point of view.
I don’t see an SBAC definition, but my guess is they are likely to agree with PARCC. It is the more common sense definition from a mathematical point of view.
I don’t think the standards answer this question directly. Assessment of the standards is a different issue from statement of the standards. In this case, the main point is to understand what happens when you multiply or divide by a power of ten. If you can think of an assessment question that gets at this which involves asking the student to give the power of ten, then great! My instinct is that such questions would be more difficult than ones where the power is given, but I would have to see some examples to be sure.
Interesting question. Even when you are interpreting a single data set you probably have some reference data set in mind. If you look at a data set and say to yourself “that data set is very variable (as indicated by its MAD or IQR)” you are implicitly comparing it to a standard data set in your mind that has a smaller “normal” variability. So the point of this standard is to make such comparisons explicit. But that doesn’t mean you would never ask a student to comment on a single data set.
