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I think 3.OA.7 is pretty clear on this point:
3.OA.7. Fluently multiply and divide within 100, using strategies such as the relationship between multiplication and division (e.g., knowing that 8 × 5 = 40, one knows 40 ÷ 5 = 8) or properties of operations. By the end of Grade 3, know from memory all products of two one-digit numbers.
It’s true that 3.OA.7 has an additional focus on knowing one digit by one digit products from memory, but “fluently multiply and divide within 100” includes one digit by two digit products less than 100.
Well, I’m sure I’m not telling you anything you don’t already know (or haven’t already heard from me), but it’s nuts not to have a straight CCSS middle school curriculum available. Look in particular at the group of students who takes pre-algebra in Grade 7 and then takes it again in Grade 8 to remediate. From your description of pre-algebra it seems it is CCSS Grades 7 and 8 squashed into one year. So, instead of taking CCSS Grade 7 in Grade 7 and CCSS Grade 8 in Grade 8, these students take both CCSS Grades 7 and 8 in Grade 7 and then take them both again in Grade 8. Sorry to be blunt, but it’s this sort of madness that got us into the mess we are in now.
Massachusetts has some reasonable acceleration plans here. One of them is to take both Algebra I and Geometry in Grade 9. This is a non-acceleration acceleration scheme that recognizes the time it takes for most students to learn mathematics.
If you look at all of 8.EE.8, you will see both procedural and conceptual components, in particular in the stem of the standard, which says “Analyze and solve pairs of simultaneous linear equations.” Students should both learn procedures and understand why they work; different curricular approaches might handle this in different ways, and the standards don’t dictate a particular curricular approach.
To your specific question, consider, for example, the following (abbreviate) description of the method of substitution.
If both equations are true, then the equation you obtain by isolating one of the variables in one of the equations is also true, and therefore so is the equation you get by substituting in the other equation. You can solve this equation for one of the coordinates, and you can use that to find the other coordinate.
Is this a description of a procedure, or is it a passage of reasoning? Really it’s both. Whether you focus on getting students to understand it as a procedure first, or on getting them to understand the reasoning process, is a curricular choice.
You are right, 7÷3 = 2 1/3 is really a Grade 5 understanding, although 7/3 = 2 1/3 is fine for Grade 4. Bear in mind also that writing something like 7÷3 = 2 R 1 is simply wrong … the thing on the right is not a number, and this usage leads to strange things like 7÷3 = 9÷4 (because they are both “equal” to 2 R 1). Correct usage would either use words like “3 goes into 7 twice with a remainder of 1” or, if you want to write a correct equation, you would write 7 = 2 × 3 + 1.
There’s a discussion of this point here.
Bill, did you mean to post that here instead?
There’s a discussion of this point here.
I was thinking of both of these as being linked to 6.EE.7, since you could use the formula to set up an equation of the form $px=q$ for both my example and Eleanore’s example. I thought mine was a bit simpler because in Eleanore’s case you would get your $p$ as the product of two of the side lengths. But both fit with 6.EE.7, given that you already know the formulas from 6.G.2
The Grade 3 multiplication standards relevant to this question are
3.OA.7. …. By the end of Grade 3, know from memory all products of two one-digit numbers.
and
3.OA.6. Understand division as an unknown-factor problem. For example, find 32 ÷ 8 by finding the number that makes 32 when multiplied by 8.
So division goes hand in hand with multiplication, so knowing the single digit multiplication facts from memory also means being able to call on that memory to perform the corresponding divisions.
How all this relates to timed tests I don’t know. Note that the standards don’t explicitly call for timed tests. It seems to me that there are other ways of assessing whether students know their single digit multiplication facts from memory. And the division facts require an extra step: to calculate 32 ÷ 8 you might need to try out a couple of multiplication facts, which would take longer. That is, the standard do not expect students to know division facts from memory independently of the multiplication facts, but rather to know them as a bundle linked to the multiplication facts.
Nancy, there is some discussion of the topic here:
http://commoncoretools.me/forums/topic/mixed-numbers-and-measurement-grade-3/Bill suggested you could skip the mixed numbers and just use whole numbers on the line plot. So students could measure using mixed numbers (as in “four and one-half inches”) but not write them (as in “4 1/2”). It is an awkward standard to work with.
Thanks for the clarification. Your first example illustrates the difficulty in describing a function abstractly; you might also say “f gives the amount of money paid to a taxi as a function of the number of miles traveled.” Also, I would add units and say “the amount of money (in dollars).”
I’m not sure what the suggestion is here: the task says “let $f(t)$ be the number of people, in millions, who own cell phones $t$ years after 1990.” This strikes me as correct: $t$ is a number (an input to the function $f$) and $f(t)$ is a number (the corresponding output). You could also say something like “let $f$ be the function such that $f(t)$ is the number of people ….” But it seems an acceptable abbreviation to say it the way it is in the task. It wouldn’t be correct to say “let $f$ be the number of people …” because $f$ is not a number, it’s a function.
Thanks Jim, that’s exactly correct, and let me take this opportunity to remind everybody that the order of the standards does not dictate the order in which they are taught (see page 5 of the standards). In particular, a thoughtful curriculum will probably intertwine the OA and NBT standards in any given grade. The OA domain is intended to highlight work on number and operations that serves to prepare students for algebra. But this work goes hand in hand with the work on the base ten system and computation with the operations in NBT.
