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I’ll check with the main author, but my guess would be that the progression was meant to cover the standards which (as I suspect that you know) begin at K, not PK. Or is your question meant to be about why are there no PK standards?
PK territory is covered in the National Research Council report Mathematics Learning in Early Childhood: Paths Toward Excellence and Equity. This can be downloaded free of charge here: http://www.nap.edu/catalog.php?record_id=12519. In addition to discussion of research on early childhood mathematics, it’s got recommendations about the early childhood workforce and its professional development, and is accompanied by a podcast.
Re “look like”: I checked Illustrative Mathematics and don’t see a task addressing this standard.
Re “mean”: here’s a relevant piece from the Algebra Progression:
Students work with solving systems of equations starts the same way as work with solving equations in one variable; with an understanding of the reasoning behind the various techniques.(A-REI.5) An important step is realizing that a solution to a system of equations must be a solution of all the equations in the system simultaneously. Then the process of adding one equation to another is understood as “if the two sides of one equation are equal, and the two sides of another equation are equal, then the sum of the left sides of the two equations is equal to the sum of the right sides.” Since this reasoning applies equally to subtraction, the process of adding one equation to another is reversible, and therefore leads to an equivalent system of equations.
I think that part of the issue may be use of “cluster” as a verb. jrhiglenn, as you point out, the GAISE document does not contain the word “clustering.” It does, however, use “cluster” as a verb in two of the instances you noted: “Random selection tends to produce some sample means that underestimate the population mean and some that overestimate the population mean, such that the sample means cluster somewhat evenly around the population mean value (i.e., random selection tends to be unbiased)” and “Names clustered by length”.
The 6-8 Progression example also uses “cluster” as a verb: “Which measure will tend to be closer to where the data on prices of a new pair of jeans actually cluster?” Another example illustrates the “clustering” mentioned in 8.SP.1: “the points are closely clustered about the line” (p. 11, shown under a line in a scatterplot).
It sounds as if you’ve been looking for a term from statistics rather than statistics education. An analogue: the CCSS uses terms from mathematics education that are not used in mathematics (e.g., “counting on,” “add within 20”).
“Clustering” is used in the American Statistical Association’s GAISE report (Guidelines for Assessment and Instruction in Statistics Education), which is listed in the CCSS document under “works consulted” and can be downloaded here: http://www.amstat.org/education/publications.cfm. It’s also used in the 6–8 Statistics and Probability Progression for the CCSS which is here: http://ime.math.arizona.edu/progressions/.
Some comments on the issue that I think Jim has moved on to: differences between what’s in 1.OA.3 and 1.OA.6.
1.OA.3 comes under the cluster heading “Understand and apply properties of operations and the relationship between addition and subtraction.”
1.OA.6 comes under the cluster heading “Add and subtract within 20.” As noted in 1.OA.6, a student might do this by counting on rather than using strategies that involve properties of operations.
Within the examples of strategies shown for 1.OA.6 are uses of the properties of operations as described in 1.OA.3.
Making ten (e.g., 8 + 6 = 8 + 2 + 4 = 10 + 4 = 14): Going from 8 + 2 + 4 to 10 + 4 is a use of the associative property. It seems to me that making a ten in calculating sums within 20 always involves use of the associative property unless one of the numbers is already 10. It seems to me that “use the associative property to add” is more generic than make-a-ten. Also, make-a-ten is a strategy that a student might identify, but “use the associative property to add” might not be so readily identified.
Creating equivalent but easier or known sums (e.g., adding 6 + 7 by creating the known equivalent 6 + 6 + 1 = 12 + 1 = 13): Going from 6 + 6 + 1 to 12 + 1 is also a use of the associative property.
[Edited to remove duplicate and correct one equation, Bill McCallum, 3/21]
Here’s one example from the OA progression (p. 15):
For example, a student can change 8 + 6 to the easier 10 + 4 by decomposing 6 as 2 + 4 and composing the 2 with the 8 to make 10: 8 + 6 = 8 + 2 + 4 = 10 + 4 = 14.
This method can also be used to subtract by finding an unknown addend:
14 – 8 = box, so 8 + box = 14,
so 14 = 8 + 2 + 4 = 8 + 6,
that is 14 – 8 = 6.
Related: A strategy for word problems (so connected with 1.OA.1) that draws on the commutative property:
Students re-represent Add To/Start Unknown box + 6 = 14 situations as 6 + box = 14 by using the commutative property (formally or informally).
This is from the Appendix of the Progression under Level 3: Convert to an Easier Equivalent Problem. Start Unknown problems are described in Table 2 of the OA Progression. Students work with such problems in Grade 1 but need not master them until Grade 2.
I’m writing “box” for the symbol that looks like a little square.
I wasn’t meaning to suggest that students work only with the number line (as opposed to graphs in the plane). But if they do, a teacher might use an unmarked length (e.g., a ruler with its marks not showing) or one of those large compasses for classroom use to illustrate the location of the two places on the number line that are 3 units from the origin.
Note the geometry standards for grade 5 in the cluster “Graph points on the coordinate plane to solve real-world and mathematical problems.” These appear in the CCSS document after the grade 5 OA standards, but that is not meant to indicate that they necessarily appear later in instruction.
Giving some examples in the order in which the associated standards appear (which is not necessarily the order in which associated abilities might be learned).
In A-REI, under the cluster heading: Represent and solve equations and inequalities graphically
11. Explain why the x-coordinates of the points where the graphs of
the equations y = f(x) and y = g(x) intersect are the solutions of the
equation f(x) = g(x); find the solutions approximately, e.g., using
technology to graph the functions, make tables of values, or find
successive approximations. Include cases where f(x) and/or g(x)
are linear, polynomial, rational, absolute value, exponential, and
logarithmic functions. (emphasis added)This might be done for f(x) = |2x + 3| and g(x) = 10, finding an approximate solution to f(x) = g(x), that is |2x + 3| = 10, via technology as in the standard, although I’d hope that students could do it by hand.
Some other situations that might involve solving |2x + 3| = 10 below.
In F-IF:
Analyze functions using different representations
7. Graph functions expressed symbolically and show key features of
the graph, by hand in simple cases and using technology for more
complicated cases.
b. Graph square root, cube root, and piecewise-defined functions,
including step functions and absolute value functions. (emphasis added)
A key feature of f(x) = |2x + 3| – 10 is where it intersects the x-axis. To find this by hand (assuming it’s a simple case), solve 0 = |2x + 3| – 10.
Or, understand its graph as a translation of f(x) = |2x + 3|. Students learn about translations in Grade 8, in particular that they preserve length, and this is continued for graphs of functions in F-BF:
Build new functions from existing functions
3. Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x),
f(kx), and f(x + k) for specific values of k (both positive and negative);
find the value of k given the graphs. Experiment with cases and
illustrate an explanation of the effects on the graph using technology.
Include recognizing even and odd functions from their graphs and
algebraic expressions for them.
For example, take f(x) = |x|.
Graph f(2x), f(2x + 3), f(2x + 3) – 10, noting x-intercepts of each.
If you search the CCSS document for the term “array,” you will find one example that involves different numbers of rows and columns: “The apples in the grocery window are in 3 rows and 6 columns. How many apples are in there?” (There are others in the Operations and Algebraic Thinking Progression.) The phrasing used is “3 rows and 6 columns” rather than “3 by 6.” That avoids the need to know whether “3 by 6” means “3 rows and 6 columns” or “6 rows and 3 columns.”
As discussed in the OA Progression (p. 24), seeing one array as a rotation of another (e.g., an array with 3 rows and 6 columns is a rotation of an array with 6 columns and 3 rows) can be used to illustrate the commutative property. So, at some point, students see that e.g., an array with 3 rows and 6 columns has the same number of objects as one with 6 rows and 3 columns.
Yes, that sentence could be less terse.
I think it helps if you read it in the context of the paragraphs that precede it. In particular, for two-dimensional shapes students “develop competencies that include . . . creating and maintaining a shape as a unit, and combining shapes to create composite shapes that are conceptualized as independent entities (MP2).” Students do similar things for three-dimensional shapes but more slowly. “Arch” is one such shape.
If you haven’t already done so, I suggest checking out the acceleration thread. That also discusses middle grades: http://commoncoretools.me/forums/topic/acceleration/page/2/#post-1750
The problem is that any finite sequence can be continued in a variety of ways and by itself doesn’t determine a unique pattern. There are illustrations of this in the short section on “the problem with patterns” in the Functions Progression. It can be downloaded here: http://ime.math.arizona.edu/progressions/
No, sorry. It will be posted on the Progressions page (http://ime.math.arizona.edu/progressions/) when it is and announced on this blog.
