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Very late to the party here, but I agree with Cathy and Lane. There is no explicit restriction to the number of equations, therefore there is no requirement to write a single 2-step numerical expression. On the other, some students will be ready for this and should be encouraged.
My guess is that these publishers have not read the glossary to the standards. The Grade 3 standards call for fluency with multiplication within 100 (3.OA.7), knowing single digit multiplication facts from memory (3.OA.7), and multiplying single digit numbers by 2-digit multiples of 10 (3.NBT.3). Multiplication within 100 is defined in the glossary to mean “Multiplication or division of two whole numbers with whole number answers, and with product or dividend in the range 0–100.”
So that excludes all two-digit by two-digit multiplications except 10 x 10. And it excludes the more difficult single-digit by two-digit multiplications.
Hang on, which of the things that Solon did are you referring to? Inscribe the laws on large wooden slabs, or leave the country? According to Herodotus, Solon “left his native country for ten years and sailed away saying that he desired to visit various lands, in order that he might not be compelled to repeal any of the laws which he had proposed” [emphasis added]. I’ve always thought 10 years was about the right length of time for a revision cycle.
As for community ownership, I think that’s beginning to happen, in the messy way that such things do happen in this country. Our discussion on this blog is part of that process.
Lane, there isn’t anything specifically about these as you point out. They could come in as an advanced example of
A-CED.1. Create equations and inequalities in one variable and use them to solve problems.
But the standards treat inequalities with a light touch, leaving the heavy work with them to advanced courses.
I guess I would choose
F-BF.5 (+) Understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents.
Note that this is a (+) standard so might be beyond the coursework that some students take.
Joanna, to address your last question on the importance of converting repeating decimals to fractions, I would say that the important thing here is not so much actually doing the converting as understanding why it can always be done. Repeated reasoning with the conversion can lead to this understanding. So, to achieve the understanding, you need to do a few conversions, just to see how it works. But it is the seeing how it works that is important point.
Hi Andy, sorry for the long delay in replying … this blog got away from me for a while. Your points (i)—(v) are basically right for me. As for the difference between speaking ex cathedra or as private opinion, I have tried as hard as possible on this blog to express my views of the standards as someone reading them along with everybody else. Of course, I have insights into the intentions of the authors, having been one of them, and I am happy to share those insights. But I think if the standards are going to work then we have to treat them as a document owned by the community.
On (ii), my guess is that different curricula will take different approaches. I see the partial products algorithm as a natural precursor to the standard algorithm, where you compress some of the partial products by noticing you can sum them as you go, for each digit in the multiplier.
No, notice that it mentions only numerical expressions, so that would not include expressions containing variables.
I agree this standard is a little opaque. But I’m also having trouble understanding what your question is. Here’s the standard:
Generate two numerical patterns using two given rules. Identify apparent relation- ships between corresponding terms. Form ordered pairs consisting of corresponding terms from the two patterns, and graph the ordered pairs on a coordinate plane.
The standard then goes on to give an example:
For example, given the rule “Add 3” and the starting number 0, and given the rule “Add 6” and the starting number 0, generate terms in the resulting sequences, and observe that the terms in one sequence are twice the corresponding terms in the other sequence. Explain informally why this is so.
So, students would generate 0, 3, 6, 9, etc., and then generate 0, 6, 12, 18, etc., and then they would notice that all the numbers in the second pattern are twice those in the first pattern. Notice that this is not something explicitly given to them … it is a consequence of the fact that 6 is twice 3. Later, in studying the proportional relationship $y = 2x$, students might make tables of $x$ and $y$ values where they notice the same thing: adding 3 (or any other number) to a value in the $x$ column results in adding 6 (or twice that number) to the value in the $y$ column. The process of forming ordered pairs and graphing them is preparation for making tables and graphs of relationships between varying quantities.
I think your example of looking at $x^6-y^6$ in two different ways is an excellent example of seeing structure in expressions, and it is no more complicated than the identity mentioned in the “for example” part of A-APR.4. It’s certainly very reasonable for classroom discussion. As for assessments, I don’t know what limits the assessment consortia will set on types of identities. I hope we don’t end up with some long list of identities students have to memorize. In some strange way that can work against seeing structure, because the list becomes the object instead of the expression. But you are quite right that it does not make sense to limit to only the identities explicitly list. That would be a strange way to interpret A-APR.4, for example, which only lists one identity as an example. It would be odd if that one identity made the list but difference of cubes did not.
There is no prohibition on these in the standards, but they are not required. So the answer to your question really depends on entire curricular design. High school courses can include material not in the standards, but the material in the standards should come first. As a practical matter, I suspect for most students there is enough in the standards to take up a full year without including extra material, but an advanced course or a course for STEM-intending students might want to go further.
