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First, I would say the isometry standards in Grade 8 are 8.G.1–3. 8.G.4 is about similarity transformations.
On the question about G-CO.6, I think the words “geometric descriptions of rigid motions” are important. In Grade 8 students might have a fairly intuitive notion of rigid motions; in high school they work with definitions of rigid motions in geometric terms. For example, in Grade 8 you might say that a reflection about a line takes every point to the point on the other side at the same distance from the line. In high school you might say that reflection about the line $\ell$ takes a point P to itself, if P is on $\ell$, and otherwise takes to a different point P’ such that the line through P and P’ is perpendicular to l, intersecting it at O, and PO is congruent to OP’. Predicting effects using this description is partly simply a matter of grasping which rigid motion is being described.
I also agree that predicting effects could include naming the coordinates. It could also include other observations: for example, reflecting about the side of a triangle produces a triangle that shares a side with the original.
So, a quick example would be: don’t quote the population of the United States down to the ones digit. Do you remember that count down to when the population passed 300 million? It was meaningless. We can’t possibly know the population that accurately, given (a) the possibility of error in the census and (b) the fact that people are dying and being born all the time. A quick google search suggests that births and deaths are in the thousands per day. If you google “population of the United States” you get 313.9 million, accurate to the nearest 100,000.
Alexei is right, I think it makes sense to replace “unit cubes” with “rectangular prisms with unit fraction side lengths” here. One of these days I will publish my glitch file!
Although I would point out that you could use unit cubes. For example, you can pack a $\frac13$ by $\frac15$ by $\frac17$ rectangular prism with unit cubes with side length $\frac1{3\times5\times7}$, forming a $5 \times 7$ by $3 \times 7$ by $3 \times 5$ array. But in the end you still have to find the volume of the cube, and the natural way to do that is by seeing how many of them fit into a cube with side length 1. Since that’s also the way you would find the volume of a rectangular prism with unit fraction side lengths, I think it makes more sense to do the latter directly.
Probably way more answer than you wanted!
There are two related standards:
6.EE.6. Use variables to represent numbers and write expressions when solving a real-world or mathematical problem; understand that a variable can represent an unknown number, or, depending on the purpose at hand, any number in a specified set.
and
6.EE.7. Solve real-world and mathematical problems by writing and solving equations of the form $x+p=q$ and $px=q$ for cases in which $p$, $q$ and $x$ are all nonnegative rational numbers.
Both of these occur the margin in the passage you mention in the Progression. The first is about variables and expressions, the second is about equations. Since expressions are used in writing equations, the reference to a “set” in 6.EE.6 could refer to a solution set in the context of 6.EE.7, but it does not have to. The Progression illustrates this with the example of the expression $0.44n$ to represent the price in dollars of $n$ stamps: here $n$ comes from the set of whole numbers. And note that 6.EE.6 does not refer to a “set of solutions” but rather simply to a “set.”
As to your last question, it is certainly true that in many instances where one uses a variable to stand for a single unknown, it will be in the context of writing an equation to find that unknown. That is, the use of variables described in 6.EE.6 to represent an unknown number will arise in the practice described in 6.EE.7 of writing equations to solve problems. There is indeed a close connection between the two standards. Still, it seems worth distinguishing the idea of choosing a letter to represent an unknown number as an important idea in its own right.
That’s very observant to notice the “or” versus “and.” I wouldn’t attach much significance to it, however. I think it’s just an editing inconsistency. The point of the phrase is simply to make clear that there is no demand to contextualize every problem, nor is there a demand that students must work on purely mathematical problems all the time. I suppose we could just have said “problems” rather than “real-world and mathematical problems,” but people tend to see what they want to see, so we wanted to make clear that any sort of problem was permissible.
The standards don’t lay out an entire trigonometry curriculum, and this is a case of something in such a curriculum which is not explicitly called for. G-SRT.6 is a natural place to attach this piece of knowledge, but it is not required by that standard.
Physicists and mathematicians have different approaches to this. You are right that the tradition in physics is to include units in the equations themselves. In mathematics we tend to put the units in the definition of variables. The important thing is to put the units somewhere. So, if you write $d = 65 t -12$, you’d better have said beforehand that $d$ is distance in miles and $t$ is time in hours (and not “let $d$ be distance and $t$ be time”). Then you can deduce from the equation that the units of 65 are miles/hour. There are advantages to the physicists’ approach, but the mathematicians’ approach also had its virtues. For example, when the units are not present in the equation itself, it is easier to see the structure of the equation.
I don’t have a lot to add here, but one comment I would like to make is that CCSS necessitates a rethinking of acceleration policies. Acceleration in middle school was often a response to the repetitiveness of the middle school curriculum. But CCSS in middle school is not repetitive; it is a dense and rich diet of important mathematics. So students who previously hungered for acceleration might now be quite satisfied with a solid implementation of CCSS.
I’m wondering if you are looking at your state’s augmentation of the standards, rather than the standards themselves. The standard in question, on page 71, is
F.LE.4. For exponential models, express as a logarithm the solution to $ab^{ct} = d$ where $a$, $c$, and $d$ are numbers and the base $b$ is $2$, $10$, or $e$; evaluate the logarithm using technology.
There is nothing about laws of logarithms, and furthermore the bases are limited.
Still, I would say that in general there is point introducing laws if you don’t apply them.
This implicit distinction also occurs earlier than Grade 7. As soon as students start working with the number line, they are beginning to move from a discrete counting model of the whole numbers to a continuous measurement model, which allows for the introduction of fractions. You make an interesting comment that this is not mentioned explicitly in the standards. One possible reason is that although the distinction is important background knowledge for teachers, it is not necessarily useful to introduce this terminology with students. For students, the related concepts are counting (discrete) and linear measurement (continuous). They should work with both of these, but it might not be worth while to introduce terminology, since it is difficult to see how the terminology would add to the intuitive conceptions, and the formalization of these conceptions is way beyond K–12.
First, this standard is not necessarily about coordinate algebra at all. It is more about using the basic similarity and congruence criteria to solve problems with more complex figures. A simple example might be showing that the diagonals of a parallelogram bisect each other. Using this diagram, randomly chosen from the internet
one might first observe that $\triangle AOE$ is similar to $\triangle COB$, using the AAA criterion, and then use the congruence of opposite sides (previously proven) to conclude that the two triangles are congruent, and hence that $AO = OC$ and $BO = OD$. Of course, one can condense this argument with a direct appeal to the ASA criterion for congruence, but I quite like breaking it apart this way in this case, since the similarity is what first arises from the basic properties of transversals, and then the congruence depends on a previously established result.
Here is another very nice example, more complicated. It is a good example of looking for and making use of structure, because if you draw auxiliary lines parallel to the sides through E and F you see all sorts of similar triangles and can chain the ratios between them to solve the problem (I won’t spoil it for you by giving the solution).
I realize this doesn’t really give you the guidance you are asking for, but perhaps it will set some ideas in motion for the geometry course.
I don’t think I understand the question. You can solve a set of linear equations in many variables by the elimination method. Or, you can represent the system by a single matrix equation and solve by row operations on the matrix. The latter is merely a notational representation of the former; solving by elementary row operations and solving by the elimination method are the same thing at bottom. The use of matrix notation to represent systems of equations is not required in the standards.
It doesn’t; Grade 6 is the first explicit mention of surface area. Notice this is in 6.G.4, not 6.G.1. Students use nets to find surface area.
