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Take two intersecting lines. Rotate them through 180 degrees around the point of intersection. Each line is taken to itself, and the point of intersection stays put, so each one of the pair of vertical angles is taken onto the other one. Therefore they are congruent (because there is a rigid motion which takes the one to the other).
Oh, one more point: although simplification is de-emphasized, transforming between equivalent forms is not. That is, students should be able to recognize and produce equivalent fractions (4.NF.1); but they should be taught that simplification is a mathematical imperative.
Alexei is correct here. Mixed numbers, fractions, reduced fractions are all different ways of writing the same number. The focus in the standards is on the number itself, not how you write it. Of course, sometimes it might be convenient to write the number in a particular way. But that goes both ways: you might want to write 7/14 as 1/2, but you might equally want to write 2/5 as 4/10 (to see the connection with decimal notation). Choosing a convenient form for the purpose at hand is an important skill, as is the fundamental understanding of equivalence of forms.
And yes, I think both the PARCC and Smarter Balanced assessments will reflect this philosophy. Indeed, if they marked an equivalent form wrong they would themselves be wrong, since a number is a number is a number; if you get the answer right, you should get the points.
Well, if you introduce the ≤ and ≥ symbols then you should also introduce the open and closed dot notation for graphing. The standards don’t call for either of these things, however; it’s really a curricular decision whether to introduce them. Certainly the focus in middle school is on equations, not inequalities.
It certainly falls under the standard, although it is an advanced example. I wouldn’t call it an expectation.
I understand the desire for a compact notation here, but I don’t really have a great idea, and I would worry that introducing one would lead us into the same issues as 2R1. I think it is best to stick to something like a verbal statement “3 goes into 7 two times with a reminder of 1” and then express this mathematically using the equation 7 = 2 x 3 + 1.
I think the point of studying these special triangles is that it is an exercise in reasoning with the Pythagorean theorem in the context of certain special triangles (equilateral, right isosceles). It’s a bit of a miracle that you can express the trigonometric ratios of these angles exactly using square roots, and I wouldn’t want that miracle to get lost in drill, with our without technology.
Are you talking about using matrices, or equations in two variables, to describe the coordinates? I think that is beyond Grade 8. The intent of the standard is that they should be able to give the coordinates of the transformed figure.
I imagine you’ve written this test already by now! But Smarter Balanced is coming out with some new test specs soon that should have some examples.
I agree with abieniek here. Basically this standard is a prelude to understanding relationships between two quantities. If you generate two different patterns, you can then compare them and see how they vary together. Without actually seeing the NC example, I can imagine on person catching 2 fish per day, the other catching 1 fish per day, and then you would notice that the numbers for the first person are twice the number for the second person. Later, in Grade 6, students will make tables of ratios expressing the same fact, and in Grade 7 they will record this using the equation y = 2x. At any rate, in Grade 5, they should be recording the ordered pairs made by pairing the number of fish caught by person 1 versus the number of fish caught by person 2; not the two lines graphing each person’s catch as a function of the number of days (which would come much later).
Are you talking about the Common Core or some other standards? I suspect the latter because the Common Core doesn’t use the phrase “perform well.” I wasn’t sure what you meant by “pedagogical rationale” or “aesthetic reasons.” If you give me an example of what you are talking about I can try to help.
