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Yes, we’ve been slow. But I have promised our funder the Algebra and Functions progressions in high school will be out by the end of November.
Duane, yes, this is basically correct. Although the italicized must adds a severity which is more a matter of how the standards are interpreted and implemented. In practice students will vary in where they are in relation to the standards, and a wise implementation must take that into account.
I think this is just a wording issue. What you call calculating, the standards call estimating, because strictly speaking you can’t calculate the rate of change from a graph, since there is always some uncertainty in reading values off a graph. Of course if the graph visibly goes through grid points then it’s natural to make the assumption that those represent exact values.
Duane, I think there’s some room for maneuver exactly where this type of problem is introduced in the progression from 4.NF.3.c, which makes clear that mixed numbers are included in fractions, and 5.NF.6. And the particular task you cite might well be better aligned with 5.NF.6.
I hope someone else can provide resources, I don’t know of any off the top of my head, although googling “acceleration versus enrichment in mathematics” brings up a few interesting-looking hits. But one point I would bring up is that the acceleration schedule will have to change for many states that have adopted the Common Core, because the Common Core is already accelerated relative to many previous state standards. So, if you previously started accelerating in Grade 5, you might well push that forward a grade or two under the Common Core.
Of course, my own opinion, expressed at the top of this thread, is that acceleration is greatly overused. It’s the modern equivalent of the insane speed-reading fad that was pervasive in my youth. It encourages all the wrong values: superficiality, thoughtlessness, expectation of a quick answer, lack of perseverance—all flaws that will cause problems when a student reaches college. The student who can survive acceleration with love of and ability to make use of mathematics is very rare indeed. Much better for gifted students to have extension materials that enrich on-grade topics rather than race through them.
But that’s just my opinion.
Leandra, sorry I missed this. Are you still having this problem? I think the entries should be visible to the public, but sometimes there is a problem that people are viewing cached versions of the site. Did you suggest to them that they refresh the page in their browser?
It would be artificial to restrict estimation to two-step word problems. Note also the cluster heading in 3.MD, “Solve problems involving measurement and estimation of intervals of time, liquid volumes, and masses of objects.” Estimation is important, but shouldn’t become a huge focus, with lots of time devoted to formulaic methods for handling it. The emphasis at this level is on understanding the base ten system, which is itself the basis for number sense and good estimation skills.
Well, the footnote is clear enough, it says Grade 3 students should know order of operations, so yes, they should be able to interpret 5+3×10 correctly. The thrust of the comment in the progressions document is that one need not expect fluency with this right away. I agree the exact boundaries are not really spelled out here. Perhaps a good rule of thumb would be to expect students to be able to deal with fairly simple expressions like 5+3×10, but not to be too aggressive about insisting they can deal with more complicated expressions. After all, it’s never wrong to put parentheses in, for either the student or the teacher. Perhaps the progressions document should be more specific here.
Here’s the standard in question
G-CO.1 Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc.
These definitions will eventually be in the geometry progression, when it comes out. For perpendicular line, I would say that two lines are perpendicular if the adjacent angles formed when they meet are congruent. (Or, perhaps you have already defined right angle this way, and then you say they are perpendicular if they meet at right angles.) Your definitions of circle and line segment sound fine to me. There a couple of different approaches to defining what an angle is, depending on the stage of development. One possibility is to define it to be the region between two rays emanating from a point. This is pretty nice conceptually, but people might find it weird for an angle to be a region. You can also define it to be the two rays, but then you have to specify which angle you mean, then one that is less than or equal to 180º or the other one.
Thanks Joann, I like your characterization of the blog, and your question is a good one. On the hand, the concept of mode is not in the standards. So we don’t have exercises where students have to remember the definition or where they have to pick the mode out of artificially constructed data set.
On the other hand, the peak (or peaks) of a data distribution are certainly parts of its overall shape, as you suggest, and one can imagine situations where you might ask students to look at a data distribution and ask them questions about that. For example, you could look at an age distribution and ask which age group has the most people in it. However, you can do this without making a big deal about the vocabulary item; it’s simply a matter of looking at a data distribution and being able to answer questions about the situation it represents.
Here is the standard:
G-GPE.5. Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point).
I’m not sure exactly what you mean by “using generic coordinates” but here’s the way I see this working. Take two parallel lines and cut them with a vertical transversal and a horizontal transversal (let’s assume the lines themselves are not vertical or horizontal). These lines form a “slope triangle” with each line: a right triangle whose ratio of rise over run gives the slope of the line. (It would be easier if I drew a picture, but I’m lazy.) Using the theorems about corresponding angles and AAA similarity, you can show these two triangles are similar, and that means the slopes are the same.
In the examples I have seen of empty number lines, distance still matters. The number line starts out empty, and then a student who is, for example, subtracting 37 from 52 might put down a 37 and a 52, then indicate a jump from 37 to 40, a jump from 40 to 50, and a jump from 50 to 52, and add up all the jumps. They don’t have to get the placement exactly right, but there is still the idea that there is a precise placement. This is not so different from the number line with a 0 and a 1 marked, where a greater or lesser degree of precision will be required, depending on the task at hand. For example, a student might argue that 3/4 > 2/3 by placing each on the number line and recognizing that 3/4 is closer to 1 than 2/3. There are various ways a student could indicate the argument without being exactly right in placing the numbers.
Duane, thanks for pointing this out. For some reason my RSS feed has stopped working, so I hadn’t seen these (and I was wondering why people stopped commenting!). I do have a spam filter, but obviously it’s not working perfectly. I’ll look into it.
The limitations are intended to provide some guidance to assessment, and generally to build some restraint into the system. It is certainly possible to design problems around the standards in this grade level that are limited exclusively to such fractions. But a limitation on what is expected is not the same as a limitation on what is allowed, as you point out.
I don’t read it that way. Between the two bold faced parts of the sentence there is that “such as”. I read this as saying that “a given number of angles or a given number of equal faces” are examples of attributes, not a list of requirements that apply to both the verbs “recognize” and “draw”. So I think curriculum writers get to use their own judgement here.
