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It’s important to remember that we are talking about a system of simultaneous equations at each step, not just about one equation. Your goal is to show that at each step the new system of equations has the same solutions as the old system. Specifically, in this standard, you start with two equations $A=B$ and $C=D$. You want to show that this system of equation is equivalent to the system $A+eC = B+eD$ and $C=D$ (this is the sum of the first equation and $e$ times the second equation). Well, if $A = B$ and $C= D$ then $A + eC = B + eD$ and $C=D$, so any solution of the first system is a solution of the second. And if $A+eC = B + eD$ and $C = D$, then $eC = eD$, so subtracting $eC$ from the left of the first equation and $eD$ from the right, we get $A = B$. And we still have $C =D$. So any solution of the second system is a solution of the first. So the two systems have the same solution and are equivalent.
Your debate has nicely captured the way the meaning of rule evolves over the grades. In Grade 4 it means what your friend says: a prescribed sequence of arithmetic operations. The rule could be either recursive (start with 1 and keep adding 3s) or explicit (think of a number, double it and add 1). This leads to the Grade 8 notion of using a rule to define a function. At some point the concept of a rule becomes merged into the concept of a function. Your example illustrates this.
The standards themselves do not specify which definition to use, but the progression does, as you point out. It probably makes sense for everybody to agree on this, and the progression is as close to an official interpretation as one can get, I suppose, since the progressions were written by members of the original Work Team.
I hope the 7–12 geometry progression will be out by the end of the summer. (That’s a hope, not a promise.) It will necessarily be shorter than Wu’s document; the progressions are not intended to spell out every point, but to provide some exegesis of the standards.
I don’t completely understand the question, but I’ll try to say a few things that might help.
First, the phrase “linear relationship” does not occur in the standards, but the phrase “proportional relationship” does. The concept of a proportional relationship is a precursor the concept of a function. One important difference is that when you define a function you designate one of the variables as the input variable and the other as the output variable. Also, as students start to study functions, they start to think of them as objects in their own right. Later in high school they use a letter to stand for a function, and they perform various operations on functions. In Grade 8 the focus is on simply understanding a function as something that takes inputs and yields outputs. Yes, the domain is important, but truth be told the same is true with proportional relationships. So when you say “a table that is a Linear Relationship” you have to be careful. In some cases the variables may only take on whole number of values (e.g. the number of baseball cards), and then it would no more be appropriate in this case to ignore that restriction than it would be in talking about the domain of a function.
Also, it’s important to be clear about the distinction between a function and an equation that defines the function. The equation $y = 2x + 3$ can be viewed as defining a linear function, with $x$ specified as the input variable, $y$ specified as the output variable, and the equation understood as giving the value of $y$ in terms of $x$. But it can also be viewed as an equation in two variables whose solutions form a straight line; an algebraic description of a geometric object. The it is not correct to say that the equation is a function. Rather one should say that the equation defines a function, but also has other uses.
I took a quick look at these. Arizona and Kentucky both attach practice standards to individual content standards, whereas Colorado attaches them at a higher level, so it’s hard to compare Colorado with the other two. It’s not surprising to me that people come up with different ways of attaching practice standards to content standards, because the practice standards really live in curricular implementation of the standards. For any given standard, you can imagine it being touched in at numerous different moments in the curriculum, and in different ways, with different styles of teaching. All of these differences would bring different practice standards to mind. I would expect more agreement if people started tagging practice standards to particular moments in a particular curriculum.
I get these concerns from users occasionally, but I usually can’t diagnose them because the files work for me. I may have forgotten to print to pdf before I uploaded this one … that usually cleans out a lot of the problems. I’ll try to get to that soon.
The July 2013 update to the Algebra progressions document seems to be a corrupted PDF file. Macintosh users opening this in Acrobat find huge swaths of text and images missing on pages 4 and 10-14. I found it could be viewed using Preview program and then re-saved. Can you get a better version of this document made and then make it available on the Arizona site?
Xyzzy
I think the notation should be delayed until the point that the students see the need for it. You can say “the translation that takes A to B” or “the rotation clockwise about O through 30 degrees” for quite a long time. So, for example, I would avoid notation in Grade 8. As for high school, I think it’s up to curriculum writers. I can imagine writing a curriculum which avoids it almost entirely (well, I can imagine trying, anyway). I can also imagine careful deployment of notation being very useful.
If you do introduce notation, I would be in favor of not having it tied to coordinates. For example, for translations, I would write something like $T_{A,B}$ for the translation that takes $A$ to $B$, rather than finding the coordinates of $A$ and $B$ and writing $T(2,5)$. For rotations, I would write something like $R_{\angle AOB}$ for the rotation about $O$ through the angle $\angle AOB$, rather than figuring out the coordinates of $O$ and the angle measure of $\angle AOB$ and writing $R_{(2,1),30^\circ}$. There are a couple of reasons for this. First, coordinates are unnecessary and might not be present. Second, knowing the position of $A$ and $B$ or the measure of $\angle AOB$ is unnecessary, and might be a distraction from the proof. I would want students to get used to the idea that the points, lines, and angles from which they construct the transformations should be points, lines, and angles already existing in the situation, not ones they have to come up with numbers for.
It’s about both, and serves to connect fractions with area. Fundamentally 3.G.2 is about partitioning, which is necessary both for an understanding of fractions and for an understanding of area. The passage in the progression doesn’t limit the ways in which you can use area to represent fractions to only arrays of unit squares. In fact, that passage should also refer to 3.MD.6, Measure areas by counting unit squares (square cm, square m, square in, square ft, and improvised units).
In the Common Core rigid motions are used to define the idea of congruence, and to prove theorems about congruence, such as the criteria for congruence of triangles. This is different from the common use of transformations to study frieze patterns and tilings of the plane. Glide reflections figure prominently in the latter, but not so much in the former. Of course, this doesn’t forbid curriculum writers from naming and using glide reflections. But that is a a side trail from the main goal in the Common Core.
Formally, this is the existence of a multiplicative identity and the distributive law,
$$
y + y + y = 1\cdot y + 1 \cdot y + 1 \cdot y = (1 + 1 + 1) y = 3y.
$$
But in this example I think it would be fine if students also saw this informally as 3 $y$s. And it’s important to remember the footnote on page 23: students need not remember the formal names for the properties. The main point is that they should use them. Thinking of seeing $y$ as $1 \cdot y$ is useful in many manipulations, for example $xy + y = (x+1)y$.We are focused mainly on getting the remaining progressions out in draft form (hopefully by the end of the summer). The editing for final production will take place in the fall. I can’t say exactly when it will be finished.
It might be worth looking at these materials. (I haven’t had a close look myself.)
Once students start using letters to stand for numbers in a systematic way, anything they can do with numbers they can also do with letters standing for numbers. The exponent rules are important in all sorts of situations, for example working working with exponential functions (A-SSE.3c) and with polynomials and rational functions (A-APR). The sort of problem you mention here strikes me as more a sort of algebraic calisthenics—not directly required by the standards, but possibly useful in generating fluency with algebraic manipulations. I would use them sparingly, however; it is possible to go overboard with this sort of thing. And it’s not obvious to me that a student who has done plenty of these would be able to notice that, for example $e^{kt} = (e^k)^t$, which strikes me as much more important.
