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Yes, it’s certainly true that they need to be whole numbers. Not necessarily greater than one, but certainly greater than zero.
Yes, 8.EE.7b,
Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms
pretty much covers any form of a linear equation.
Your interpretation is plausible in the following sense. First, you can think of a cylinder as composed of a quadrilateral in the sense that you can make a cylinder by forming a rectangle into a tube. Second, given the radius of a cylinder you can use the formula for the circumference to find the side length of the rectangle that made the cylinder (the other side being the height of the cylinder). That said, I’m not sure it is exactly what we had in mind with these two standards. My instinct is to see where this reasoning naturally arises in a fully developed curriculum based on the standards. That might well be in Grade 7 as you suggest, but it might be later. The main point I think is not to try to find a catalog of every formula for volume and surface area, but rather to take opportunities to calculate these things as they arise naturally, using geometric reasoning, as in the idea above of unfolding the cylinder into a rectangle. In the end, it’s the ability to think this way that will stand students in good stead, rather than “knowing the formula.”
Joanna, very interesting thoughts (and nice to know the connection with Jeff, who I still see regularly at our Harvard consortium meetings). It’s a complicated problem, and I don’t have a neat answer. Like you, I would like to see more students get further ahead in mathematics, and I wouldn’t want the Common Core to be interpreted as a barrier to acceleration. In fact, I believe that in the long run the Common Core is an engine for acceleration. Students who experience a faithful implementation of the Common Core in elementary school are more likely to be ready to take off. But we have to wait for that, and in the meantime we have to eliminate phony courses that give the illusion of acceleration without the reality of.
I don’t think there’s any significant difference. It might have been better to use the word “prove” in both places, but we tend to use the word “derive” for formulas.
No, this is not true. In fact, the standards themselves explicitly contradict this. Page 5 of the standards says:
These Standards do not dictate curriculum or teaching methods. For example, just because topic A appears before topic B in the standards for a given grade, it does not necessarily mean that topic A must be taught before topic B. A teacher might prefer to teach topic B before topic A, or might choose to highlight connections by teaching topic A and topic B at the same time. Or, a teacher might prefer to teach a topic of his or her own choosing that leads, as a byproduct, to students reaching the standards for topics A and B.
And in particular it would not make sense to teach OA and NBT separately, since there are many close connections between those two domains. It would make more sense to intertwine them in the curriculum.
Jim basically had it right in his first post, but Dan and Sherry were correct that the coordinates were unnecessary, and that the key point went by very quickly. To my mind the key sentence in Jim’s proof is “then dilate it until the radii match.”
Why is it even possible to dilate until the radii match? Because by definition all the points on the circle are the same distance (the radius) from the center. So that means that if you dilate the smaller circle from the center, all its points will arrive at the larger circle at the same time.
In more detail: Given two circles, translate the first one so that its center coincides with the center of the second circle. If the first circle has radius r and the second circle has radius R, then perform a dilation on the first one from its center with scale factor k = R/r. Since every point on the first circle is a distance r from the center, every point on the dilated circle will be a distance kr = R from the center, so the dilated circle is identical to the second circle.
This would be easier to explain with visual aids, of course.
As for activities to support this, I can imagine having students play around with a dynamic geometry program, and asking them perform similarity transformations that map circles onto each other. At first maybe using the mouse, but then by giving the precise commands: “perform the translation that takes O to O'” and “dilate around O’ with scale factor 1.2”. To find the scale factor they would have to realize it is the ratio of the radii. Then you could ask what it is about a circle that makes this work (it has a constant radius).
You had me worried for a moment there that we had a major blunder in the standards, until I checked and realized you had misquoted the standard in an important way. It actually says “Develop a uniform probability model by assigning equal probability to all outcomes, and use the model to determine probabilities of events.” Can you see the difference? There’s a difference between outcomes and events; an outcome is an element of the probability space (in this case, a name on a list). In a uniform probably model, every outcome is assigned the same probability. So we assign every name on the list a probability of 1/10.
An event is a subset of the probability space; for example, the subset consisting of all students on the list who are in Grade 7. We can use the uniform probability model to calculate the probability of an event by counting the number of outcomes in the event (the number of students in Grade 7) and multiplying the probability of a single outcome.
As for your question about repeated names, I am imagining a list of full names, like a roll. Of course, two students could still have the same name, in which case presumably the school would have a way of distinguishing them. Still, we should make that clear.
There is no intention to exclude these symbols from the curriculum. There is a discussion of this point here. In general, some confusion has resulted from the avoidance of specifying vocabulary and specific symbol usage in the standards. A lot of those decisions are up to curriculum writers.
Your intuition is correct that the middle school geometry standards are more experiential and the high school ones more formal. In middle school students get a feeling for transformations and their properties by playing around with transparencies or dynamic geometry software. In high school they should be able to make arguments using the precise descriptions of transformations. Sorry that the geometry progression hasn’t come out yet; working on that.
As for the question about volume formulas, I would be inclined to take the Grade 8 standard fairly literally; it’s really just about knowing the formulas. Understanding where they come from and being able to give a formal derivation is significantly more advanced than that, which is why it is left till high school. I can see why some might find this interpretation unpalatable, but some formulas are quite simply beautiful and classical, and it doesn’t do any harm to appreciate them for a while without deep analysis (we do the same with art all the time).
I did participate in a review of the Arizona documents, but didn’t participate in writing them. Not sure I remember the specifics of these comments any more.
First, I would say that what you have heard from your administrator is a local interpretation of the standards rather than something that is in the standards. The standards do not dictate curriculum or pedagogy; they certainly say nothing about whether you should be tied to a text book or not.
Second, the high school standards in the Common Core are not divided into courses. So it will be up to states and districts to decide what is in Algebra I, Geometry, Algebra II, Math Analysis (not quite sure what that is), etc. But I would say that the biggest change in the approach to Algebra in the standards is embodied the domains A-SSE (Seeing Structure in Expressions), A-REI (Reasoning with Equations and Inequalities), F-BF (Building Functions) and F-IF (Interpreting Functions). Although the topics within these domains might seem familiar, the emphasis in seeing structure, reasoning, building, and interpreting is a big shift.
Well, there are the progressions documents! When I was growing up (in the previous century) I remember reading books by W.W. Sawyer which really brought out a lot of connections for me. His “Vision in Elementary Mathematics” deals with middle and high school topics, and is available on Amazon. Another nice book at a higher level is Gelfand and Shen’s Algebra. Maybe others could add their suggestions.
