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I think I need a more precise question here. But here are some musings. If I were teaching functions I would certainly give examples where there is not a well-defined output, to emphasize the importance of that aspect of functions. For example, give a table of days and average temperatures and then ask if the temperature is a function of the day (yes) or if the day is a function of the temperature (no, because there is more than one day with a given temperature). That’s different from defining the concept of a relation and then giving students a whole bunch of relations and asking them to sort them into functions and non-functions. That can lead you into territory where the concept of “one output for each input” gets lost in a blizzard of vertical line tests (which I think most students never connect with inputs and outputs).
The are two aspects of the same thing, but the first is about performing a procedure, and the second is about understanding. So, an assessment of the second one might expect a student to come up with an appropriate multiplication equation give a division problem, whereas an assessment of the first one might expect a student to solve a multiplication equation without necessarily relating it to division.
I guess the word “dividing” could cause confusion with the operation of division. If you talk about dividing a circle into quarters, and then later talk about dividing by four, you might end up confusing division by 4 with division by 1/4. Maybe. I don’t think it’s a big deal.
I depends on the questions … if it’s about whether or not a certain standard is being interpreted correctly, for example, then that’s fine. If it’s about the intentions of EngageNY authors, then I don’t think I can answer.
The high school standards do not specify any arrangement of the mathematics into courses; that is up to states and districts. I’m not an expert on PARCC, but I believe they will have end of course tests for Algebra I, Geometry, and Algebra II. These should be relatively independent of each other, but there could be places where they assume that an Algebra II student has had Geometry. For example, the Geometry domain Expressing Geometric Properties with Equations (G-GPE) is in Algebra II in the PARCC framework. You can check out the frameworks here.
The biggest question I have about your proposal is not the order of those courses, however, but the preparation of the students entering Algebra I in Grade 7. The K–8 standards were designed to gives students a solid preparation for algebra. How do you handle acceleration in Grades K–6 for these students?
Illustrative Mathematics has some illustrations of 3.OA.D.9.
As for your example, I guess the question is whether it counts as an arithmetic pattern, and how properties of operations are used make the conclusions. I’m not really seeing that right now, but maybe it works.
I was a bit hasty last time. I shouldn’t have suggested that inequalities were not included in A-CED.1 itself … just that you shouldn’t infer inequalities from an explicit statement listing types of equations to be included.
Here’s the full text of A-CED.1
A-CED.1. Create equations and inequalities in one variable and use them to solve problems.
Include equations arising from linear and quadratic functions, and simple rational and exponential functions.So yes, inequalities are included in this standard. However, for Algebra I, PARCC focuses only on equations, as you noted. For Algebra II, I don’t read their statement as excluding inequalities. It says tasks should have a real-world context, which is appropriate because this standard has a modeling star. And it lists things to be included, which happen all to be equations (consistent with the original standard). But that does not exclude the things not listed. So no, I don’t think PARCC states are disregarding that part of the standard. However, there is a pretty clear signal here that inequalities do not play as big a role as equations.
I think the standard allows either approach. But yes, you are right, the idea is not to treat radicals as a separate thing, but to connect them with rational exponents and the rules of exponents.
Your second interpretation is correct. You might want students to observe that the two points are the same distance from the axis but on opposite sides, or that you could get one from the other by flipping across the axis, or something like that. You might even use the word “reflection” when likening the relationship between the points to reflection in a mirror. But students are not expected to understand reflections as transformations, or use the word, in Grade 6.
Well, this is not quite the same thing, but Illustrative Mathematics is developing course plans and unit blueprints which will be implemented on the website as a navigational tool somewhat similar to the NDSL diagrams.
Any problem where the sum is less than 100 is included here. The point of the “including” was to make sure that these simpler types of problems were not neglected for their power to reveal the role of the base 10 system in two-digit addition.
Yes, I would say that this limits the division problems to ones where divisors, dividends and quotients all be decimals that with no non-zero digits beyond the 100ths place.
I agree with Cathy here. Also, I would add to your definition that a unit is something replicable; you can measure out quantities using copies of the unit. For example, you can measure the length of a table in matchsticks.
