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Becky, have looked at the Number System Progression?
I meant to say “we are not expecting very formal proofs” and have edited my response to reflect that.
A good working definition is that “prove” means provide a logical argument appropriate to the grade level. Exactly what “appropriate” means is up for discussion and adjustment as achievement improves. But the fundamental requirement is that the argument be faithful to the mathematics. One might define a linear function to be a function of the form $f(x) = b + mx$ and then adopt a fairly straightforward algebraic proof: over an interval of length $d$, $f(x)$ grows by the constant amount $f(x+d) – f(x) = b + m(x+d) – (b +mx) = md$.
As for assessment, I would expect it to be difficult, but not impossible, to assess this standard with a summative, machine-graded assessment. The ideal would be in-class observation by knowledgeable teachers.
Here’s the link that I think Lane is referring to:
http://commoncoretools.me/forums/topic/absolute-value-equations/#post-1799
But I noticed that this is mostly about absolute value equations, not inequalities. I would also point out
7.NS.1.c. … Show that the distance between two rational numbers on the number line is the absolute value of their difference, and apply this principle in real-world contexts.
It seems to me that this is the key understanding, and could be applied in many contexts, including inequalities. I’m not sure that breaking this down into discrete activities such as solving absolute value inequalities is helpful.
As for compound inequalities, I see that as a notational device which is within the scope of the standards but which does not rise to the level of an explicit mention (but maybe I am misunderstanding the question here).
Here’s the answer from Dick Scheaffer:
Most of the graphics were done on a software program called Fathom. Fathom was designed for statistics education and probably is the favorite among those who teach AP Statistics (whereas most statisticians on college faculty would find it too limiting).
Well, one way to do it would be to indeed prove the SSS criterion. The triangle congruence criteria are quite fun to do with transformations, and remember that in Grade 8 we are not expecting very formal proofs. But another way to go would be something like the following proof, which essentially pulls in the necessary piece of the SSS proof.
Given a triangle whose three side lengths $a$, $b$, and $c$ satisfy $a^2+b^2= c^2$, construct a right triangle with legs of length $a$ and $b$. Then, by Pythagoras’ theorem, its hypotenuse has length $c$. Now put the two triangles together along their sides of length $c$, flipping one of the triangles if necessary to get a kite shaped figure (because of the corresponding sides of lengths $a$ and $b$). Drawing the other diagonal you can see the kite as two isosceles triangles matched along their bases. The base angles of the isosceles triangles are equal, so the opposite angles of the kite that you just joined are also equal. But one of those is the right angle of your right triangle. Therefore the original triangle also has a right angle, and you have proved the converse.
(Probably should have tried to draw a figure for this.)
You need to know that the base angles of an isosceles triangle are equal. That has a very nice proof using reflection about the angle bisector of the vertex.
Thanks Corey. Maybe I will get around to giving a history on my other blog one day. Basically this mixes up two documents, the end-of-high-school expectations produced in summer 2009, and the standards themselves, produced in 2009-10. If you google “NGA press releases”, then search the press releases on “Common Core”, then read those in chronological order, you will get the bare bones of the story, including the composition of the various teams and the various organizations involved. Maybe I will say that over at Diane Ravitch’s blog.
Here is the comment from the statistician I asked, Roxy Peck:
I think that at the grade 6 level, the idea is that kids are working with census data and that they are not interested in generalizing beyond the group that they have data on. The idea of sampling is introduced in Grade 7 and from there on it is reasonable to have students think about the sampling process and whether or not that process is likely to result in a representative sample whenever they are generalizing from a sample to a larger population.
However, in many situations, like the ones that look at relationships between variables in Grade 8, regression and other model fitting is viewed as descriptive rather than inferential. So I think that these kinds of examples are OK, as long as students are trying to describe the relationship in a given set of data and are not generalizing beyond the data set. It is related to the distinction between descriptive statistics and inferential statistics. My take on the common core standards is that all of the modeling relationships between two numerical variables falls in the descriptive statistics realm. But I think it is appropriate to ask students to think about the sampling issues if they are asked to make predictions based on regression models. The assumption that is being made is that the data are representative of the relationship between the variables–something that would follow if the data are from a random sample but which also might be reasonable even when the data are not from a random sample.
I would add that the example in the progression about animal speeds is linked to the Grade 6 standard about describing data sets (6.SP.5), so falls squarely in the domain of descriptive statistics.
I’m going to ask a statistician to take a look at this question, but here is my take on it. When you talk about a representative sample, you are talking about a situation where there is a more or less homogeneous population with a number of different subtype (e.g. ethnic groups in a population of a country). And you want to make sure your sample has the same proportions of subtypes.
But I don’t think you can say that animals form a homogenous group. For each species you might measure average height and weigh, and you have probably already chosen a representative sample within each species in order to get those averages. And then you want to see if there is any relationship between these variables across species. You could just try to get all the species, but I’m not sure what it would mean to make a representative selection of species. Would you try to make sure the percentage of mammalian species represented the true proportion in the world? Or would you go for some sort of geographic representation? I can see that it’s worth discussing these things, but it seems a too complicated example to introduce 6th graders to the idea of a representative sample (or even high schoolers, for that matter).
The Mean Absolute Deviation is intended as a more intuitive precursor concept to the Standard Deviation. It makes sense to take the mean of all the absolute deviations if you want to get a quantitative measurement for the spread; it turns out for rather subtle reasons that taking the square root of the sum of the squares of the deviations (that’s what the Standard Deviation is) is better, but you can’t really explain the reasons at this level, and the more complicated procedure might obscure the underlying idea.
This captures most of the standard, namely comparing the means of two treatments and deciding whether the difference is significant. The standard talks about comparing parameters, and the mean is just one parameter. But it is probably the one you will most often look at when comparing to treatments. You might, however, see a difference in standard deviations, or one of the quartiles, and want to know if that difference is significant as well. Also, I’m not completely sure what you mean by a resampling technique, but I suppose it is the same as as a simulation, which is the word the standard uses.
Sorry for the delay in replying, somehow this one got lost! I really think this is just a miscommunication. We should be keeping things simple at Grade 5, and there was no intention here to suggest nesting of symbols. It was simply a matter of being agnostic about which grouping symbols to use. Later on students learn a hierarchy of symbols, but here they just learn the idea of bracketing things off (or bracing them, or putting them in parentheses). In practice it will probably always be parentheses, and we should probably just have said that.
The first installment of my answer is now posted here.
I try to keep this blog focused on simply answer questions about what’s in the standards. These are all good questions, but I think I’ll answer them over at my other blog, isupportthecommoncore.net (it will probably take me a couple of days … I’ll post a note here when I have done it).
Yes, a probability model and a probability distribution are the same thing, or at least two ways of looking at the same thing. A probability model assigns a probability to every event in the sample space. One way to conceptualize this (later in high school) is through the idea of a distribution, a function on the sample space for which the area under the graph above a certain event represents the probability of that event. (Sorry, compressed a large part of a course into that sentence!) And so, a uniform probability model and a uniform distribution are the same thing (note, however, that we also use the word distribution to refer to a data distribution, a different but related thing … but that’s another story).
The second definition of compound event (from your textbook) is the one used in CCSS. A compound event is an event in a sample space that has been constructed out of two other sample spaces. For example, you have the sample space {heads, tails} for tossing a coin and the sample space {1, 2, 3, 4, 5, 6} for rolling a die, and you construct the space
{(heads, 1), (heads, 2), (heads, 3), (heads, 4), (heads, 5), (heads, 6),
(tails, 1), (tails, 2), (tails, 3), (tails, 4), (tails, 5), (tails, 6)}out of both spaces, and calculate probability of events like “flip heads and roll an even number”, which is the subset {(heads, 2), (heads, 4), (heads, 6)}.
And yes, you might call this particular compound event a multi-stage event too. Although I can imagine cases where the two parts of the compound event happen at the same time (e.g. lightning strikes the tree and I am standing under it).
