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Well, I would need to know exactly which methods you are talking about here. I don’t know which method CME uses. The argument I remember from Archimedes involves balancing the areas of different slices … this is arguably making use of Cavaleieri’s principle, whose essence is slicing. Happy to answer in more detail if you can be more explicit.
The diffence is certainly small, but small differences in effective interest rates can add up over time: see http://www.investopedia.com/articles/basics/04/102904.asp.
Kristin has pretty much said it all. Here are a couple more thoughts. The vertical line test and tables of ordered pairs are tools in service of the understanding expressed in your second quotation above. As long as they remain tools, that’s fine (although personally I think the vertical line test is excessive codification of a simple visual observation). The first quotation you give refers to “time normally spent on exercises” on the vertical line test or tables of coordinate pairs. The key words here are “time” and “exercises.” Once a tool becomes the subject of a set of exercises devoted specifically to it, it becomes a topic in its own right, disconnected from the understanding it originally served (as Kristin says in her last paragraph).
Yes, I agree it is an important standard. One way not to implement it would be to get too bogged down in formality and terminology (like insisting that students keep referring to the properties of equality by name, for example). I would have students get in the habit of talking through their solutions:
“If $x$ is a number such that $x^2 – 3x – 4 = 0$
then $(x-4)(x+1) = 0$ because $x^2 – 3x – 4 = (x-4)(x+1)$ no matter what $x$ is.
for all $x$ (by the distributive law). This means that either $x-4=0$ or $x+1=0$, so $x =4$ or $x=-1$. ”At first I would want students simply to understand that solving an equation is a flow of if-the statements; then I would start asking why each step was correct (distributive property, zero-factor property). And then I would raise the question of the converse: you’ve shown me that if $x$ is a solution to the equation it has to be 4 or $-1$, but does that tell me that 4 and $-1$ have to be solutions? How do I know they are solutions?
Maybe one of these days I will write a blog post on this.
This is a case where coherence trumped conciseness in writing the standards. The skill and the piece of knowledge named by these two standards are the same, but the context is different. In A-REI.4b the context is solving equations, and you want students to know that sometimes solving equations leads to complex numbers. In CN.7 the context is the complex numbers as a system, and you want students to know that they can sometimes provide solutions to equations that didn’t have solutions before. These are two different understandings, two different ways of viewing the same piece of knowledge, and approaching that piece of knowledge from both sides seemed worthwhile.
Sorry, I don’t know how I missed this one, thanks for bringing it to the front again. Yes, the equation for an ellipse or a hyperbola is a quadratic equation, so is included in this standard.
In the standards, “fluently” means “fast and accurate.” So assessing fluency is always going to involve some observation of how long it takes a student to do a calculation. As abienek points out, that observation could be made by the teacher listening to a student solve a problem; or, it could also be made by a timed test. In talking to teachers I’ve found that timed tests are a very controversial topic: some people love them, others hate them. I don’t have a strong opinion either way. I do think that if you use timed tests you should try to make the fun and competitive, not scary and stress-inducing. That’s possible in athletics, so it should be possible in math. I also think they are probably not necessary in a classroom where students are explaining their solutions a lot; you can probably tell pretty well from that who is fluent and who is not.
I’m sure Bill will add his own opinion again if necessary, but there was some talk of fluency and “knowing from memory” back in this discussion: http://commoncoretools.me/2012/04/02/general-questions-about-the-standards/#comment-1428
That’s a curricular decision, not a requirement of the standards. Personally I’m with you … I think the set notation is too much baggage at this level.
Good eye! It has been fixed in my files, but apparently I didn’t upload the latest version. Will try to get to that.
I think Grade 6 is early for dimensional analysis. That comes in in the high school standard:
N-Q.1. Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays.
I agree the problem is weak, and I agree with Alexei’s analysis of 7.NS.A.1 and 7.EE.B.3. I would only add that grouping standards into cluster headings, which carry important meaning on their own, has the effect that you sometimes see an overlap in the meaning between two standards. So, the cluster 7.NS.A is about operations with rational numbers, and that includes problem solving, whereas 7.EE.B is about problem solving, and that includes working with rational numbers.
Lane, I think this is a useful way of helping students remember the rules, and that’s especially needed for remedial students. There’s a bit of sweeping under the rug going on here, because if you spell it out, you are saying that
(opposite of 2) times (opposite of 3)
is the same as
opposite of (2 times (opposite of 3)).
We can actually prove this using the distributive property, because that property tells me that
(opposite of 2) times (opposite of 3) plus 2 times (opposite of 3)
is the same as
((opposite of 2) + 2) times (opposite of 3)
which is just zero times (opposite of 3), namely zero.
But if I add two numbers and get zero, they must be opposites!
Of course, I’m not suggesting that you have to go over all this with your remedial students!
