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I agree with the calculus teachers! After all, for $f(x) = x^2$, it is true that $f(x) < f(y)$ for any $x
Yes, if graphing is involved. The idea of the modeling star is that it flags standards likely to be involved in modeling problems.
No, these are both high school topics. See
S-CP.A.2. Understand that two events A and B are independent if the probability of A and B occurring together is the product of their probabilities, and use this characterization to determine if they are independent.
and
S-CP.B.9 (+) Use permutations and combinations to compute probabilities of compound events and solve problems.
Note that the latter is a (+) standard, and so beyond the college and career ready threshold. So certainly not necessary in Grade 7!
(By the way, you can answer a lot of these questions using a word search on the pdf of the standards, available at corestandards.org.)
I think all occurrences of the term “place value” in the standards could be replaced by the term “place value notation” without changing the meaning. Place value notation (as I’m sure you know!) is a the system of writing numbers where, for whole numbers,
- Each number is represented as a sequence of digits 0–9.
- The number is the sum of the values of the digits.
Each digit is assigned a value equal to the digit times a power of 10, the power being 1 for the right most digit, then 10, 100, 1000, etc. as you move successively to the left.
For decimals the system is extended by putting a decimal point at the end and adding digits to the right of the decimal point, whose values are the digit times 1/10, 1/100, etc.
The place value system is this system of notation. So, “the value represented by a digit” is not synonymous with “place value” … rather it is determined by place value (notation).
Students learn about whole number remainders in Grade 4:
4.OA.A.3. Solve multistep word problems posed with whole numbers and having whole-number answers using the four operations, including problems in which remainders must be interpreted. Represent these problems using equations with a letter standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding.
In Grade 6 they study division of fractions, which would include 14.6/3. They are not required to know about infinite repeating decimals until Grade 8, so they might express the answer as a fraction rather than a decimal, as in $4 \frac{26}{30}$. In the Common Core finite decimals are treated as a way of writing certain sorts of fractions, namely those that can be written with denominator 10, 100, and so on. There is no explicit requirement that they express this problem as division with remainder, but it seems a natural extension of their Grade 4 work to be able to say that 14.6 = 3 x 4 + 2.6, and to interpret both the quotient and the remainder in a context.
It also seems to me that, although knowledge of infinite repeating decimals is not required until Grade 8, simple examples such as 1/3 = 0.333 … could appear earlier. But it is not necessary, since you can always just use fraction notation.
Well, I don’t know the whole story here, and getting caught in the middle of a battle between a grandparent and a teacher is the last thing I want to do, but maybe you could use MP6 in your response to this teacher. Attending to precision includes attending to precision in the asking of questions. From your description it does seem that your grandchild answered the question correctly as posed, since the question did not specify a particular order in which the product must be written (nor would that have been a good idea). I agree that it is unacceptable that a mathematically correct answer should be marked wrong.
Now, I think I know where the teacher is coming from (you probably do too). The teacher has in mind that the student should be thinking of 10 groups of 2, and 3.OA.5 does suggest this would be written as $10 \times 2$:
3.OA.5 Interpret products of whole numbers, e.g., interpret $5 \times 7$ as the total number of objects in 5 groups of 7 objects each.
Still, your grandchild’s answer is not wrong, because $10 \times 2 = 2 \times 10$.
It seems to me all this could come out in classroom discussions, and this would be the appropriate place to discuss the answer. That is, without saying the answer is wrong, you could ask the student to explain their thinking, and see how they decided to write $2 \times 10$, and that could lead to some good discussion.
I certainly hope your 6th grader is not being exposed to statistics at a college level! Grade 6 statistics in the Common Core is meant to be an introduction to some basic ideas of data and variability. It is actually pretty important for people to have that ability these days, since we are presented with statistics all the time in newspapers, reports about polling, discussions of important issues like climate change and polling bias, and so on. Not to mention the pervasiveness of statistics in more specialized jobs in the scientific, technical, medical, and biological fields.
But really the purpose of this blog is not to have general discussions about the importance of learning mathematics and statistics, but rather to answer specific questions about the standards. So if you can point to specific standards that you have questions about, that you think your child’s curriculum might not be treating correctly, then I’d be happy to answer them.
It’s possible that the curriculum your child is experiencing is just spending too much time on this, so I’d be interested to know if you think that is the case.
I need a more specific question to be able to answer this. The purpose of this blog is to help people understand the standards; what does a particular standard mean, what’s an example that illustrates it, and so on. It looks like you are having a problem with a particular curriculum. Not all curricula that claim to be Common Core aligned really are, and even with curricula that are aligned I don’t think I can take on answering every question about every curriculum. (This is an entirely volunteer effort.) Still, I’d be happy to try to help if you can give me something more specific. I would encourage you to read the Grade 6 statistics standards themselves and see if you think the work your child is doing is related to them. You can read them at corestandards.org or illustrativemathematics.org.
I think there’s a difference between “Identify and describe” and “increase their knowledge of.” Notice that it is a particular sort of trapezoid being described, one with two non-parallel sides of equal length. Students might see such shapes around the classroom, or build them up out of triangular and rectangular tiles, or has some tiles of that shape available, as in the picture on the next page. So in that sense they will become familiar with them. And maybe they will have a name for them, but the standards don’t specify that.
Why do they need integer operations? Students can see that the distance between $(-1,3)$ and $(2,3)$ is 3 without knowing how to subtract $-1$ from $2$. For example, they could plot the points on the plane and count the units to get the distance. This is in fact good preparation for integer subtraction. I think maybe you are thinking about the formula for distance, which would indeed require integer subtraction. But you don’t need to use the formula.
Simplifying radicals is one of those high school topics that has evolved into a cancerous growth on the curriculum, starving other more important topics for resources. What is important is for students to understand and use the laws of exponents. When they get to algebra they should be able to see that $\sqrt{x^2y}$ is the same as $x\sqrt{y}$ (if $x$ and $y$ are positive). Seeing that $\sqrt{45} = 3 \sqrt{5}$ is a sort of rehearsal for this, and as Alexei points out comes quite appropriately N-RN.2. But treating such simplification as an end in itself, accompanied by long lists of problems, is an example of misplaced priorities. That’s why the standards don’t make explicit mention of it.
Yes, the first proof is just wrong, and also misguided. The similarity of circles follows very directly from the definition of similarity in terms of dilations, as explained above. Trying to go via similar triangles strikes me as extremely irrational. And the review of triangle similarity is out of sync with the standards.
