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It depends what you mean by “all angles of rotation.” Students can make arguments where the angle of rotation is specified by another angle in the diagram. So, you might prove SAS congruence by first translating so that the vertices with the A coincide, and then rotating so that two sides coincide, and then reasoning that the other vertices must coincide as well. This is an arbitrary rotation, but the argument doesn’t require you to ever specify the angle measure of the rotation: you would simply say “rotate by angle AOB” or something like that. So that is well within the standards.
But, giving a coordinate formula for an arbitrary rotation through a given angle measure is beyond the standards.
I don’t know how the district resolved the issue. I continue to think that for many students acceleration leads to shallow grasp of the mathematics, which ultimately leads to them hitting remediation when they get to college. Acceleration is like adderall: appropriate for some students but way over-used. Parents and schools are beginning to appreciate the problems with over-prescription of adderall; it’s time that started appreciating the problems with over-prescription of acceleration.
The standards don’t regard decimals and fractions as different types of numbers between which conversions must be made, but rather as different notations for writing fractions. Note the language of the cluster heading above 4.NF.5: “Understand decimal notation for fractions, and compare decimal fractions.” Thus, students should see 0.57 as another way of writing the fraction 57/100. I think this means that the answer to all your questions is yes (subject to limits of common sense, of course … you don’t want to get carried away with all the ramifications of understanding decimal notation right away).
I agree with abieniek’s ideas for where you might teach this. I would add that you don’t have to assess everything you teach, and since this is not explicitly in the standards it is not required that it be assessed.
Lane basically has it right here: 3.NBT.3 is about 4 x 80, and 3.OA.5 allows for students reasoning from this to 80 x 4. But the emphasis in Grade 3 would be on 4 x 80.
It’s worth quoting the entire text of 2.G.1 here.
Recognize and draw shapes having specified attributes, such as a given number of angles or a given number of equal faces. Identify triangles, quadrilaterals, pentagons, hexagons, and cubes.
The second sentence specifies that the shapes to be recognized and drawn should include the ones listed, but does not limit to those shapes. The core of this standard is the first part of the first sentence: students should have experience recognizing and drawing shapes with specified attributes. The progression gives examples of this that go beyond the list in the second sentence, but should not be interpreted as a required interpretation of this standard. So, basically, this is really up to states to interpret. I would add that the consortium assessments don’t start until Grade 3, so there is really some flexibility here.
The “know relative sizes” in the standard is really about students knowing, for example, that 1 kg is 1000 times as large as 1 g. That said, they can do the comparisons you ask by expressing 0.1 kg as 100 g, or 1/10 m as 10 cm. Expressing larger units in terms of smaller units is part of the standard as well.
Middle school acceleration made some sense when the middle school curriculum was impoverished, as it often was under previous state standards. It makes less sense when the middle school standards are as rich and demanding as they are under the Common Core. I certainly understand all the forces driving middle school acceleration and I also understand that those forces are not going to go away overnight. But parents and schools do not have to submit to those forces. I didn’t with my own children, and they are all happy and successful.
Yes, PARCC is using the inclusive definition, see here:
https://www.parcconline.org/sites/parcc/files/ES%20Table%20Geometry%20EOY%20for%20PARCC_Final.pdf
In the Common Core decimals are treated simply as a different way of writing fractions with denominator 10, 100, and so on. So Grade 5 students can certainly see the equivalence of 3/5 and 0.6 because they can see the equivalence of 3/5 and 6/10. For 5/8 you are getting into 1000ths, so that would have to wait.
I’m not clear on the distinction between your two interpretations, but the examples you came up with are really great, so I guess the second one is best! I’ll get the Illustrative Mathematics team working on more. Can we use yours?
If I understand the question correctly, you are saying that sometimes students should add fractions expressed as mixed numbers by grouping the whole number parts together and adding them, and grouping the fractional parts together and adding them, and then putting the two results together. And, other times they might just want to expressed the fractions in purely fractional form and add the numbers together in that form. And, they should have some judgement about when to do which.
If that’s what you are saying, then I wholeheartedly agree!
Good question, and I think different curricula might approach this differently. Some might choose to introduce some of these Grade 4 concepts in an informal way in Grade 3. But I would point out that the classification into squares, rhombuses, and rectangles really only requires an ability to recognize angles as square angles or not, and an ability to detect when pairs of sides, or all four sides, are equal in length. This could be the right way to approach this with 3rd graders.
Yes, this is not a classification made by the standards themselves, although I do think PARCC did a pretty good job of interpreting the standards with this classification, and I would add that SBAC classifies the standards the same way (except that they merge the supporting and additional categories).
