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First, sorry for the long delay in replying. I got hijacked by my day job for a while.
I guess when there seems to be confusion we should try to go back to the text of the standards and see what we can get from it. The second cluster under 6.SP is called “Summarize and describe distributions.” It doesn’t use the word “construct”, although one could argue that in order to summarize a distribution you need to construct a summary. But, as I said in the earlier post, you could do this using technology, and it seems to me that this would be a strategic use of tools in Grade 6, falling under the meaning of MP5. So my inclination would be to stick with my original interpretation (surprise!) and say that Grade 6 students could be using technology to produce summary statistics.
As the question of box plots, 6.SP.B.5c says:
Summarize numerical data sets in relation to their context, such as by
c Giving quantitative measures of center (median and/or mean) and variability (interquartile range and/or mean absolute deviation), as well as describing any overall pattern and any striking deviations from the overall pattern with reference to the context in which the data were gathered.
Taking the first choice in each parenthetical, we get median and interquartile range. Box plots are a possible way of representing these, so it would be natural to use them in this context. Although I would certainly not expect Grade 6 students to be skilled in producing them by hand, for the same reasons outlined above.
I haven’t actually checked if this contradicts the progressions document or not, and feel I should move on to answer other overdue questions!
It depends which fractions and decimals. First, let me reiterate that the standards do not regard fractions and decimals as different kinds of numbers, but rather different ways of writing the same number. Thus, rather than talking about converting fractions to decimals, I would talk about writing fractions in decimal notation (and vice versa). And this is indeed the language used in the standards:
4.NF.6. Use decimal notation for fractions with denominators 10 or 100. For example, rewrite 0.62 as 62/100; describe a length as 0.62 meters; locate 0.62 on a number line diagram.
So, in Grade 4, students deal with decimals that have one or two digits after the decimal point.
I was thinking C was o.k. because 0.2 is a unit fraction, but of course you are right that this requires a conversion. That conversion is fairly simple, and well within the purview of Grade 5; your example of 0.125 is somewhat trickier.
The standards don’t suggest circle graphs should be taught anywhere, although of course they also don’t forbid the teaching of circle graphs at some point, as long as it doesn’t interfere with the things they do suggest. I agree they are a part of daily life, but I worry a bit about the statement that “students must understand what they represent.” Yes, eventually, perhaps during their schooling, perhaps after. But this sort of rhetoric is what led us into the mile-wide-inch-deep curriculum. There are many wonderful and important things to learn; not all of them can be learned before the end of high school. Focus is important for depth.
Another point to consider is that not every important quantitative representation has to be taught in mathematics class. Circle graphs could show up in history, social studies, or science.
Thanks Bill. Just to clarify – is example C okay? It’s similar to an example on p.18 of the NBT Progressions (7 / 0.2) because it has 0.2 as the divisor. But would 0.2 really be considered a unit fraction (i.e. 1/5) and, if so, how does 0.125 (i.e. 1/8) fare?
Circle graphs are not explicitly mentioned in the standards; categorical data is represented by bar charts, which are introduced in Grade 2. A reason is that the connection with the number line is tighter for bar graphs, since the value in each category is read from the vertical scale, rather than estimated from the area of a sector.
It seems to me you are doing a good job of thinking this through. Yes, you are right, 7/12 doesn’t come until Grade 7. All your ideas for 30% of 45 sound good to me. But I don’t see why you are trying to avoid 0.30 x 45, that is also covered by the NS standards. You are right also that the NS standards have to be interwoven with the RP standards in the appropriate way.
I’m not sure where you are seeing pre-K, but some states added pre-K standards (using their 15%). The standards themselves don’t have any.
Good question, but the standards don’t answer it. This is really a question for hte curriculum writer. I’d be inclined to agree with your implementation here, but another interpretation would not violate the standard.
The sub-parts of a standard draw attention to particular aspects of the standard or emphasize that a particular case must be treated. In this case, S-ID.6c draws attention to linear functions particularly. It’s a bit like saying to your child, make sure you pick up everything, and don’t forget to look under the bed. The second is technically a consequence of the first, but you say it anyway.
I’m not sure about the second part of your question because I don’t know what you have in mind when you talk about “using” a standard. Certainly it’s possible to have a task which addresses some but not all aspects of a standard. And yes, you could just say it is related to S-ID.6, without necessarily naming which part or parts.
I think the discussion shows that you understand this standard very well! In particular, an equivalent system of equations is not necessarily made up equations that are individually equivalent the equations in the original system. A student proof would look something like the discussion here.
