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Yes, but I wouldn’t describe it that way. Students should understand that a decimal is a fraction, so dividing decimals is dividing fractions. For example, to calculate 2.16 ÷ 0.3, they would know that 2.16 = 216/100 and that 0.3 = 3/10, which is also 30/100. If they were asked to calculate 2.16 ÷ 3 they would use this understanding to rewrite this as (216/100) ÷ (30/100) = 216 ÷ 30. I think this is what you mean by extending the zeros … and of course I wouldn’t expect them to got through this reasoning every time, once they understood it.
Well, it says “Summarize categorical data for two categories in two-way frequency tables.” So I would interpret that as including the possibility that they are given the data in some other form and have to put it in a two-way table. Of course, it doesn’t have to be that way every time.
It’s fine to mention asymptotes. The end behavior of an exponential function is that it approaches an asymptote on one side or the other of the $y$-axis. The two terms are not always synonymous, but they are in this case.
I can’t speak for Jason, but I can give you my thoughts on this problem. There are two concepts at play in this problem: one is the understanding of subtraction as a missing addend problem. That is, understanding 427 – 316 as the number you add to 315 to get 427. The number line is a good model for visualizing this. The other concept is using the base 10 system in subtraction. That is, understanding 427 as 4 hundreds 2 tens and 7 ones, 316 as 3 hundreds, 1 ten, and 6 ones, and subtracting hundreds from hundreds, tens from tens, and ones from ones. I agree with Jason that the number line is not a good model for this understanding. The precise relationship between 100s, 10s and 1s is not so easy to see on the number line, because you can’t really accurately depict a one on a scale which also includes hundreds. The ones and the tens can get confused; indeed, that seems to be exactly the problem Jack was having (although it’s a little hard to figure out which marks are Jack’s and which marks are the student doing the problem). Also, the method presented might be misconstrued as suggesting you have to go in order: first subtract the hundreds, then the tens, then the ones, which really misses the point. I don’t think the problem is completely bad, but I do think it’s a little off key. I’d be inclined to use a number line for something like 23-8, where you can easily see the intermediate numbers 10 and 20. For a 3 digit subtraction I’d want to use base ten blocks, or just the verbal decomposition described above.
I’m a little confused by this question, because the discussion was about pyramids, not cones or spheres. The middle school surface area standards are in Grades 6 and 7, as listed at the top of this thread. You are correct that surface area of cones and spheres is not in the standards.
There are two differences that I can see: (a) the one you noticed, that A.REI.1 explicitly asks students to explain the reasoning and (b) A.REI.1 is not limited to linear equations.
As for (a), “justifying the steps” sometimes ends up being a recitation of rules, e.g. “I added a 5 to both sides.” A.REI.1 is asking for more than that. It is asking for students to see each step as an if-then statement: if x – 5 = 3 then (x-5) + 5 = 3 + 5, therefore x = 8.
This is the most extreme example of acceleration I have heard of. Calculus for all is crazy. As a university professor I see the damage done by this sort of thing all the time: kids coming in with a fragile grasp of algebra or calculus because they have been raced through it, and being placed into remedial courses. This is a recipe for failure in college for many kids. There are some students who are ready for acceleration and they should have that opportunity. The rest are being done a disservice by this sort of thing. It’s an abdication of educational responsibility.
There used to be an excuse for it: the middle school curriculum was often sparse and repetitive and acceleration was the only way out of it. But with the Common Core students have plenty to do in middle school, and if they take it at the right pace they will be truly ready for college.
Yes, a lot of states have put thought into this. Here is a document from the Massachusetts Department of Education outlining various acceleration options: http://www.doe.mass.edu/candi/commoncore/MakingDecisions.pdf.
The Grade 8 standard explicitly limits to systems of two equations in two variables, whereas the high school standard only suggests a focus on such systems, but allows for bigger systems as well. There is also a general uptick in fluency and complexity expectations from Grade 8 to high school. In general, there is some overlap in the topics studied in Grade 8 and high school, with greater depth expected in high school.
Yes, I think abieniek has it right. Although, I confess, I’m not completely clear on what is meant by “the percent proportion.” In the Common Core a percent is a rate per 100, so anything you would do with a rate you can do with percents. This includes, in Grade 6, understanding that 5% of 200 is 10 because $\frac{5}{100} \times 200 = 10$, and, in Grade 7, being able to express the statement “the sales tax is 5% of the price” using an equation such as $T = 0.05P$.
Yes, this sounds basically right. Although I wouldn’t call the spinner a model, but rather a representation of the model. The model itself is just the sample space {1, 2} with the assigned probabilities P(1) = 2/5 and P(2) = 3/5.
No, symbolic logic is not in the standards (and I don’t think it was in many state standards previously).
Here is the standard for convenience of other readers:
HSF-LE.A.3. Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function.⋆
We certainly want students to know this is always true. However, the mathematical proof of this fact uses calculus, so it is beyond the scope of the standards. I don’t know what informal ways of seeing it you have in mind, but they may well fall under the rubric of “observe.”
