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I mentioned this thread to a mathematics education researcher of my acquaintance. She said that parents might be anxious about getting their children ready for calculus in high school given that acceleration traditionally lead to this track. If that’s the case, one way to reassure parents is to indicate pathways that lead to calculus in high school without acceleration in grades 6–8.
The GAISE report (Guidelines for Assessment and Instruction in Statistics Education, http://www.amstat.org/education/publications.cfm) doesn’t give large amounts of attention to stem-and-leaf plots. It puts them at level A (the first of three developmental levels):
The Framework uses three developmental Levels: A, B, and C. Although these three levels may parallel grade levels, they are based on development in statistical literacy, not age. Thus, a middle-school student who has had no prior experience with statistics will need to begin with Level A concepts and activities before moving to Level B. This holds true for a secondary student as well. If a student hasn’t had Level A and B experiences prior to high school, then it is not appropriate for that student to jump into Level C expectations. The learning is more teacher-driven at Level A, but becomes student-driven at Levels B and C. (p. 13)
I suspect that for young children stem-and-leaf plots might be a bit hard to read because the digits convey so much information and the less detailed dot plots might be more helpful in seeing trends. Also, stem-and-leaf plots don’t seem to lead quite so straightforwardly to later types of graphs.
BTW, the president of the American Statistical Association has a blog post “2013: The International Year of Statistics” here: http://www.huffingtonpost.com/marie-davidian/2013-the-international-ye_b_2670704.html
On behalf of multiple people, you’re welcome! I think it makes everyone happy if you’re finding the documents helpful.
Re p. 7: correct, it’s f(–3). (And note that the task comes from Illustrative Mathematics so you can comment on it at the task page.)
Re p. 11: That definitely was an non-optimal sentence—one needed to read “composition” and “composition of a function and its inverse” as two topics joined by “and”. Here’s the rewrite.
Note: Inverse of a function and composition of a function with its inverse are among the plus standards. The following discussion describes in detail what is required for students to grasp these securely. Because of the subtleties and pitfalls involved, it is strongly recommended that these topics be included only in optional courses.
Re p. 18: Agreed.
You can get some answers by searching the forums for “simplif”. All quotes are from Bill McCallum. I think it’s helpful to see all these together because they address the general theme of simplifying in the same way.
Re simplifying polynomials:
. . . the standards do indeed quite consciously avoid the word “simplify”, the point being that different forms of expressions are useful for different purposes, and there is often no mathematical reason to call one of those forms the simplest. This is in accord with MP7, Look for and make use of structure. Students are expected to be able to make strategic choices about what manipulation they perform for the purpose at hand, rather than respond mechanically to commands like “simplify”. http://commoncoretools.me/forums/topic/simplifying-polynomials/#post-675
Re simplifying radicals:
Note the emphasis is on rewriting rather than simplifying, however. Indeed, it’s not at all clear which of √18 and 3√2 [hoping this pastes correctly–CK] is simpler, and each might be useful in different contexts. http://commoncoretools.me/forums/topic/simplifying-radicals/
Re simplifying fractions:
. . . the Standards do not require simplifying fractions into lowest terms, since it is not a mathematically important topic. To quote the Fractions Progression, “It is possible to over-emphasise the importance of reducing fractions …. There is no mathematical reason why fractions must be written in reduced form, although it may be convenient to do so in simple cases.” http://commoncoretools.me/forums/topic/mixed-numbers-in-grades-4-and-5/#post-954
For p. 15, paragraph 3, see how this edit works for you (I’ve included the preceding paragraph):
To prove that a linear function grows by equal differences over equal intervals,F-LE.1b students draw on the understanding developed in Grade 8 that the ratio of the rise and run for any two distinct points on a line is the same (see the Expressions and Equations Progression). An interval can be seen as determining two points on the line whose $x$-coordinates occur at the boundaries of the intervals. The equal intervals can be seen as the runs for two pairs of points. Because these runs have equal length and the ratio of rise to run is the same for any pair of distinct points, the consequence that the corresponding rises are the same. These rises are the growth of the function over each interval.
In the process of this proof, students note the correspondence between rise and run on a graph and symbolic expressions for differences of inputs and outputs (MP.1). Using such expressions has the advantage that the analogous proof showing that exponential functions grow by equal factors over equal intervals begins in an analogous way with expressions for differences of inputs and outputs.
Nick, in the meantime, you might want to check out the elaboration of these tables in the Operations and Algebraic Thinking Progression at http://ime.math.arizona.edu/progressions/
This progression mentions pp. 32-33 of Mathematics Learning in Early Childhood, which does not give a table, but does give the problem types and some of the example problems.The Ratios and Proportional Relationships Progression (which can be downloaded here: http://ime.math.arizona.edu/progressions/) discusses problems that include finding unknown side length in the context of 7.G.1 on p. 10 and gives an illustration on p. 11.
7.G.1 is about solving problems involving scale drawings of geometric figures, so it includes similar triangles as a special case.
Re “interpreting line plots”: I’m not sure, but I think you might be asking about what to teach and assess in the classroom rather than at the end of the grade. The standard is about what students are expected to know at the end of grade, so interpreting line plots might occur in curriculum before the end of the grade. Similarly, students might start their study of line plots using measurements in whole-number units—the standards don’t dictate how students learn about line plots, just what they are able to do with line plots at the end of the grade.
End-of-year assessments can be designed to distinguish between facility with computation and with interpretation—and the hope is that that will actually happen. (Sample items are here: http://www.parcconline.org/; and here: http://sampleitems.smarterbalanced.org/itempreview/sbac/index.htm, though I don’t see any on 4.MD.4.) Also, there are separate standards that involve computation, e.g., the NF standards.
Tad Watanabe’s comment touches on similar issues:
p. 7 I think it is important for the document to emphasize that “multiplying by a whole number” means, for example, “5 x 1/3,” but not “1/3 x 5.” Too often, teachers don’t distinguish the multiplier and the multiplicand, so they would think both of these are “multiplying by a whole number.” (http://commoncoretools.me/2011/08/12/drafty-draft-of-fractions-progression/#comment-171)
I notice however that Brian’s question concerns test developers and curriculum developers. It’s possible that curriculum developers might want to include 1/3 x 5 in some way, even if it test developers did not include it.
Duane, when you say “abstractly,” do you mean “without drawing a visual fraction model (tape diagram, number line diagram, or area model)”? This seems a bit hard to regulate. Even if a question testing 4.NF.4a or 4.NF.4b asked for an equation (assuming an equation is abstract and a diagram is not), a student might draw a diagram before writing the equation.
But maybe I’m missing your question entirely. A central idea is that students understand fractions as numbers on the number line (3.NF.2) as well as things that can be used to represent quantities.
Beth, I think there’s some overlap but also some difference.
In CCSS, a procedure isn’t necessarily a procedure for arithmetic computation. In PSSM, computational fluency is discussed in the Number and Operations strand and “method” refers to arithmetic computations (p. 144) and means “algorithm,” but I know there’s sometimes confusion about the meaning of “algorithm.”
What CCSS means by “algorithm” is discussed in the NBT progression. There “algorithm,” “method,” and “strategy” are each distinguished. The only mention of “procedural fluency” in CCSS occurs in discussion of Adding It Up. That report focused mainly on arithmetic, where “procedure” is likely always to be a reference to “algorithm.” (There are many references to “procedure” in Adding It Up, which can be read online here: http://www.nap.edu/catalog.php?record_id=9822.)
In CCSS, procedures occur in OA and NBT, but also in other areas:
Students strategically choose and efficiently implement procedures to solve linear equations in one variable, understanding that when they use the properties of equality and the concept of logical equivalence, they maintain the solutions of the original equation. (Grade 8 Expressions & Equations, p. 52)
Use the mean and standard deviation of a data set to fit it to a normal distribution and to estimate population percentages. Recognize that there are data sets for which such a procedure is not appropriate. (high school Statistics & Probability, p. 81)
PSSM also discusses fluency with things other than arithmetic computations, e.g., “develop fluency in operations with real numbers, vectors, and matrices” (Number & Operations, p. 290); “write equivalent forms of equations . . . and solve them with fluency” (Algebra, p. 296).
PSSM “fluency” might not be identical to CCSS “procedural fluency” but might be in some cases, e.g., expanding (a + b)(x + y). There’s some subtlety about the meaning of “procedure” in this context. For example, FOIL seems to me to be an algorithm because it gives a specified (and unnecessary) order for performing the multiplications. (There’s no advantage to the order specified by FOIL, FILO would be just as good. I’m not claiming that either PSSM or CCSS advocates FOIL!) The main issue is a systematic use of the distributive property. A systematic use of the distributive property in expanding binomials might be considered a procedure that’s not an algorithm.
In summary: “computational fluency” and “procedural fluency” may not be the same because they may sometimes refer to fluency with different things, but “fluency” is likely to be the same.
“By allowing students to start with a more elaborated method, such as Area Method 1, and then progress to Method 2 or 3 as they no longer need the support of a drawing, students can use methods that allow them to make sense of the algorithm while also working towards fluency.”
This comment was made in the context of multiplication, but is relevant for thinking about division as well. The standards are benchmarks that students reach at the end of a grade. The Progressions sometimes illustrate things that students might do before reaching those benchmarks.
I’ve put some different ways to illustrate the figure from p. 15 of the NBT Progression with equations here: http://wp.me/aJHdC-6K.
Because there is a remainder, using a rectangle might seem a bit strange but it could be done with the inclusion of a unit square. For this example, using an array seems pretty time-consuming.
This is also an opportunity for identifying correspondences between equations and diagrams (MP1), which I didn’t do. In particular, students might be asked about what the 6 corresponds to.
I was puzzled until I looked at the item which has a slot for the answer as b = _______ . The problem states explicitly that b represents the number of boats needed.
Putting (11 + 13 + 3 × 20) ÷ 10 in the slot doesn’t give an expression that’s equal to b unless you assume that 4/10 of a boat is going to carry 4 people. (However, putting that expression in the slot does give an equation.)
On the next page (p. 5), the answer key states that b is 84/10 and also that b is 8 R 4. However, you interpret those (is 8 R 4 equal to 84/10?), that seems to be different from the solution (which is 9). But, using the transitive property of equality (see p. 90 of the Standards) together with the answer key, yields 84/10 = 8 R 4 = 9.
I think the test developers need some comments.
A possible solution (less computing and no letters):
11 + 13 + 3 × 20 = 10 + 1 + 10 + 3 + 6 x 10 = 8 x 10 + 4, so 9 boats are needed.
There’s a similar example on p. 30 of the OA Progression.
Recycling Bill’s earlier answers to questions of a similar nature (http://commoncoretools.me/2011/12/26/progression-for-statistics-and-probability-grades-6-8/#comment-2135):
“that which is not mentioned in the standards is not thereby forbidden”
“that which is mentioned in the progressions is not thereby required.”
However, a main focus is for students in grade 2 is get fluent with addition within 20 (2.OA.2), so it doesn’t seem as work with arrays that have more than 5 rows or columns is much of a priority in grade 2. It may help to compare grades 2 and 3 of the Operations and Algebraic Thinking Progression which is here: http://ime.math.arizona.edu/progressions/. That has a lot of discussion of the subtleties of arrays in the grade 3 section.
Re figures: Maybe we need to include a few more figures in the geometry progression. There’s a figure of an angle on p. 22 of the geometric measurement progression and a figure of two lines on p. 23.
It will be up to assessment consortia about how to assess understandings about line vs line segment and angle vs two line segments with a common endpoint. The geometric measurement progression touches on the latter issue on p. 23 and may perhaps help to suggest why it’s helpful for students to understand angles as two rays with a common endpoint rather than two line segments with a common endpoint. And if two intersecting lines create angles, then students will need to understand lines as different from line segments.
I think that the example might work better this way:
3 x 2/5 = 2/5 + 2/5 + 2/5 using the grouped objects interpretation of multiplication
= 1/5 + 1/5 + 1/5 + 1/5 + 1/5 + 1/5 using 4.NF.4b or (more directly) 4.NF.3d
= 6 x 1/5 using the grouped objects interpretation again
= 6/5 using 4.NF.4b.
Here’s the grouped objects example in Table 1 of the geometric measurement progression: “You need A lengths of string, each B inches long. How much string will you need altogether?”
By the way, there is a typo on p. 8 of the NF progression, which was pointed out earlier: http://commoncoretools.me/2011/08/12/drafty-draft-of-fractions-progression/#comment-988: On page 8, the second paragraph under decimals, it states “Grade 3 students learn to add decimal fractions by converting them to fractions with the same denominator. . . . ” That should be “Grade 4 students. . . . “
