Forum Replies Created
-
Posts
-
This is an error in the standards (I’ve noted it before in these pages, but, of course, that’s difficult to find!). It should say “squares with unit fraction side lengths.”
Thanks for the reply. A few thoughts in response.
First, I haven’t kept track of all the PARCC changes and I think you are probably more up on them than I am, so I take your point there.On testing I have mixed feelings. I get the frustration people feel about too much testing and I think it would be a healthy move to reduce the amount of testing (and that seems to be happening). I don’t think it is a healthy move to opt out altogether. So there has to be some balance in between (something this country is having trouble achieving in all sorts of areas). What I don’t know is where that balance point lies. Partly it will be determined by political forces, of course, but is their empirical evidence to help decide? You say 4.5 hours is too long for a high school test. As someone who grew up in a country where the high school test was 3 hours long, I’m inclined to agree. But how do we come up with these numbers? Do we just take the average of everybody’s gut feelings? I wish we had a more empirical approach.
Finally, on the question of responsibility: well, I have pretty much devoted my life since the standards were written to helping teachers understand and implement them and advising curriculum writers, assessment writers, and policy makers on what I think is their proper use (spoiler: standards are not curriculum and standard are not assessment frameworks). I can’t control the extent to which my voice is heard. Illustrative Mathematics, the non-profit I went on leave from my university position to found, has just completed writing a complete, freely available, grades 6–8 curriculum, and are hoping to continue on to high school. Stay tuned!
I could imagine a classroom activity where students try to draw three-dimensional shapes and compare their work, but I don’t think it would be reasonable to put that on a summative assessment. (I think assessment in grade 2 should be mostly formative anyway.) Note that in general the standards are not assessment frameworks; they are just statements of the things we want students to know and be able to do at each grade level. The PARCC and Smarter Balanced assessments make judgements about limits on assessment, although they don’t offer guidance here because they start in Grade 3.
I was about to write a long reply to this and then I discovered that George Bergman has already done it for me. Short version: there is no universally accepted convention that would require the parentheses, but they would indeed remove ambiguity. In this case I think the context (well-known formula for adding fractions) removes the ambiguity anyway. See the end of George’s article for a discussion of PEMDAS and its interpretations.
First let me say that having grown up with a fairly traditional education in Euclidean Geometry in Australia I have never heard of “points of concurrency” as a topic. So I agree with Kristie Donavan!
I’m assuming this refers to the various theorems about medians, altitudes, angle bisectors, and side bisectors of triangles all intersecting at a point. The only one of these that is explicitly called out in the standards the one about medians. Constructing inscribed and circumscribed circles suggests also studying the concurrency of angle and side bisectors, although I think there is latitude in curriculum about how far you go with that. I myself would not advocate remembering all the names of the points where various lines intersect, and that is certainly not required by the standards, although of course it is not forbidden either.
Generally speaking the high school standards were designed to allow states some latitude in curriculum.
To sketch the answer to (B) first: given two non-vertical parallel lines, draw a vertical transversal and a horizontal transversal so that they intersect at a point not on either of the lines. The transversals form a right triangle with each of the lines, and the slope of each line is the quotient of the lengths of the vertical and horizontal sides. Using the fact that alternate interior angles are congruent, you can use the AAA criterion to show that these two triangles are similar. That means the corresponding sides are related by the same scale factor, so the quotients of the lengths of the horizontal and vertical sides are the same.
As to (A), I agree there would be a danger of circularity of you defined the notion of parallel lines in terms of slope. So it would be a good idea not to do that! A standard definition is to say that two lines are parallel if they are either identical or do not intersect at all.
The standards set expectations for what kids should know at the end of the course; they are not markers for particular topics along the way. The sort of question you ask here comes up when people try to arrange the standards into a curriculum, but there are many cases where that doesn’t really make sense (this one, for instance). That said, assessment writers have to make decisions about what sorts of questions belong to which standards, and I think your suggestion here is reasonable that 6.EE.1 would be assessed with simpler expressions than 6.EE.2c.
As for your question about bases, I don’t see any reason to restrict the base to whole numbers.
The usage of ratio in that PARCC question is incorrect, I agree. I can’t see the table, but why didn’t they just say the sales tax is a fixed percentage of the purchase (assuming that they give percentages in the table)?
As for the confusion about rate and unit rate, it’s not a problem for multiple choice items (as long as they get it right!). For student produced response items, assessment writers will have to make some decisions. I think it would be reasonable to accept an answer with the units even though only the unit rate was asked for. I don’t think it would be reasonable to accept an answer without units when the rate was asked for.
If you want students to give coordinates of reflected or rotated points then you have to restrict to reflections and rotations where that is possible given what they know, so yes, that limits what you can do. You could have rotations about points other than the origin in multiples of 90°, and you could probably dream up other situations where special placement of the points or symmetry would make it possible, but basically you are right.
I think you mean the Geometric Measurement Progression, right? The full quote is “An angle is the union of two rays, a and b, with the same initial point
P. The rays can be made to coincide by rotating one to the other about P; this rotation determines the size of the angle between a and b.” So you need to specify a direction from one ray to the other in addition to the rays. I think the meaning is clear enough, but a more formal definition would be something like “An angle is defined to be the union of two rays, a and b, with the same initial point P, along with a direction of rotation from one ray to the other.” Would that be better? I worry that it would sacrifice clarity for precision.
