Forum Replies Created
-
Posts
-
Solving this equation might be seen as a combination of:
F-TF.3 (+) Use special triangles to determine geometrically the values of sine, cosine, tangent for π/3, π/4 and π/6. . . .
and
F-TF.8 Prove the Pythagorean identity [sin(θ)]^2 + [cos(θ)]^2 = 1 and use it to find sin(θ), cos(θ), or tan(θ) given sin(θ), cos(θ), or tan(θ) and the quadrant of the angle.
6.EE.8 and 7.EE.4b are both about inequalities in one variable. A number line can show solutions for that type of inequality. When students deal with inequalities with two variables, the coordinate plane is needed for graphing solutions.
Alexei Kassymov seems to be talking about what Wikipedia calls “normalized histograms” and what the Guidelines for Assessment and Instruction in Statistics Education Report (GAISE) report calls “relative frequency histograms.”
In the Standards, as in the GAISE report (see p. 35), bar graphs are for categorical data with non-numerical categories, while histograms are for measurement data which have been grouped by intervals along the measurement scale.
Very sorry not to reply to this much earlier!
I think you’re seeing -5 x -2 as -(5 x -2). They’re not identical expressions. To see that they’re equal, you can use the multiplication rules in grade 7 together with the understanding that -(-a) = a from grade 6.
So, in grade 4, students extend and use their understanding of place value in several arenas: multi-digit addition and subtraction; multi-digit multiplication and division (using 3.OA.7, fluency with single-digit multiplication); and decimal notation for fractions (using understanding of fractions from grade 3).
This allows greater depth than would have been possible in grade 3.
Thanks, Karen. What you and CarrieW seem to be getting at is: Do the standards really intend to have a jump from 1,000 in grade 2 to 1,000,000 in grade 4?
I think that what is going on is related to focus (treating fewer topics in more depth). Here are the descriptions of the critical areas in grades 2–4:
In Grade 2, instructional time should focus on four critical areas: (1) extending understanding of base-ten notation; (2) building fluency with addition and subtraction; (3) using standard units of measure; and (4) describing and analyzing shapes.
In Grade 3, instructional time should focus on four critical areas: (1) developing understanding of multiplication and division and strategies for multiplication and division within 100; (2) developing understanding of fractions, especially unit fractions (fractions with numerator 1); (3) developing understanding of the structure of rectangular arrays and of area; and (4) describing and analyzing two-dimensional shapes.
In Grade 4, instructional time should focus on three critical areas: (1) developing understanding and fluency with multi-digit multiplication, and developing understanding of dividing to find quotients involving multi-digit dividends; (2) developing an understanding of fraction equivalence, addition and subtraction of fractions with like denominators, and multiplication of fractions by whole numbers; (3) understanding that geometric figures can be analyzed and classified based on their properties. . . .
The NBT overview in grade 4 says:
Generalize place value understanding for multi-digit whole numbers.
Use place value understanding and properties of operations to perform multi-digit arithmetic.
The NF overview in grade 4 includes:
Understand decimal notation for fractions, and compare decimal fractions.
So, in grade 4, students extend and use their understanding of place value in several arenas: multi-digit addition and subtraction; multi-digit multiplication and division (using 3.OA.7, fluency with single-digit multiplication); and decimal notation for fractions (using understanding of fractions from grade 3).
Whether or not to have a place value review of the kind you describe is a curricular decision. Work with rounding might serve some or all of the same purposes because students need to use closest numbers (thus they need to compare numbers) and the meanings of places when rounding.
I can’t see that any grade 3 standard indicates that students need to work past 1,000 for NBT.
For grade 3, students add and subtract within 1000 and multiply one-digit whole numbers by multiples of 10 in the range 10–90. For that, they need only numbers within 1000.
On page 29 of the CCSS document, the footnote for NBT in grade 4 says, “Grade 4 expectations in this domain are limited to whole numbers less than or equal to 1,000,000.”
The updated NBT Progression (as part of a larger document) is here: http://mathematicalmusings.org/wp-content/uploads/2019/02/Progressions_CC_to_RP_02072019.pdf.
Have you looked at the NBT Progression here: https://www.math.arizona.edu/~ime/progressions/? It was written by CCSS writers.
This is related to use of different forms mentioned in 7.EE.1, 7.EE.2, and 7.EE.3 as well as:
6.EE. 4 Identify when two expressions are equivalent (i.e., when the two expressions name the same number regardless of which value is substituted into them). For example, the expressions y + y + y and 3y are equivalent because they name the same number regardless of which number y stands for.
Understanding equivalences such as (x + 7)/3 equivalent to (x + 7) times 1/3, (x + 7) times 1/3 equivalent to 1/3 times x + 7/3 (which is in the form px + q where p and q are rational), or x/2 equivalent to 1/2 times x relies on properties of operations as well as:
6.NS.1 Interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions, e.g., by using visual fraction models and equations to represent the problem.
The latter builds on
5.NF.3 Interpret a fraction as division of the numerator by the denominator (a/b = a ÷ b).
and
4.NF.4 Apply and extend previous understandings of multiplication to multiply a fraction by a whole number.
a. Understand a fraction a/b as a multiple of 1/b.
In grade 7, px + q and p(x + q) may include rational p and q (as opposed to nonnegative rational p and q in grade 6 expressions) and equivalences may need to rely on:
7.NS.2 Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.
a. Understand that multiplication is extended from fractions to rational numbers by requiring that operations continue to satisfy the properties of operations, particularly the distributive property, leading to products such as (–1)(–1) = 1 and the rules for multiplying signed numbers. Interpret products of rational numbers by describing real-world contexts.
Two comments:
In the Standards, the laws of exponents are not algorithms. From CCSS glossary:
Computation algorithm. A set of predefined steps applicable to a class of problems that gives the correct result in every case when the steps are carried out correctly. See also: computation strategy.
In the Standards, learning times tables is not a mindless task. See the Operations and Algebraic Thinking Progression at https://math.arizona.edu/~ime/progressions/.
It’s nice to see that people are still reading this blog.
I don’t know what is meant by “standard simplifying roots algorithm” but I suspect that it too relies on the laws of exponents (whether or not it uses exponential notation). Before students identify radicals as exponential expressions, they might use laws of exponents restricted to integers and the meaning of the radical sign in working with radicals, as discussed in the grade 8 section of the Expressions and Equations Progression, which is here: https://math.arizona.edu/~ime/progressions/.
You might be interested in the Number System and Number Progression (also here: https://math.arizona.edu/~ime/progressions/).
