Read this: Shut Up About Harvard.

Media outlets like the New York Times, Boston Globe, the Atlantic, etc. do us a disservice with all their reports on admissions.

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Read this: Shut Up About Harvard.

Media outlets like the New York Times, Boston Globe, the Atlantic, etc. do us a disservice with all their reports on admissions.

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ExteriorAngle is a worksheet that can start the class. Before students see it, they should understand how linear pairs and any triangle’s 3 interior angles sum to 180 degrees.

It goes quickly.

I have a student who resists constructions. Student had to construct a 60 degree angle on a test, and instead of making Euclid Proposition 1, the student drew a right isosceles triangle and split the hypotenuse in 3 with a ruler’s markings.

And when I checked with GSP, that makes a 63.43 degree angle.

It’s good to go through some of these misconceptions from a hardcore doubting Thomas.

When Algebra 1 students find the vertex of a parabola, they use x=-b/(2a).

When Geometry students find the mirror line of the reflection (x,y)–>(b-x,y) they get x=b/2.

So I thought, what if there’s a dilation-reflection composition (x,y)–>(b+ax, y) ? then the mirror line will exist at x=-b/(2a), right?

Not sure. Just noting this to come back and check when my grading is done.

In teaching geometry, lately I’ve become a fan of compositions of transformations–thinking of a Transformation like a reflection about the x-axis as a function that takes one point and outputs its reflection point. Those rules for reflection are easy, especially about the lines y=0, x=0, and y=x.

about y=0 … (x,y)–>(x,-y)

about x=0 … (x,y)–>(-x,y)

about y=x … (x,y)–>(y,x) The inverse function reflection!

And a few years ago, I realized that a reflection about x=0, then reflected about y=0 gave a 180 degree rotation, with the rule (x,y) –>(-x,-y) because 2 flips equals a rotation. And the angle of rotation will be twice the angle between the two axes of reflection (twice 90 is 180).

But I always struggled to understand the 90-degree rotations until now.

Rotate 90 degrees clockwise (x,y)–>(y,-x)

90 equals 45 times 2. So there’s a 45 degree angle between y=x and y=0. So I put (x,y) through the first rule, and it becomes (y,x). Then I put (y,x) through the second rule for reflecting about y=0, and get (y,-x) because reflecting about y=axis means make the second coordinate the opposite sign.

Not sure how useful this is to teaching regular level geometry classes, but it helps me know it, and it might help more kids in honors level geometry classes.

John Benson at Angles of Reflection has a post–a very important one–that’s been sitting at the top of the blog for a long time. Every so often I re-read that and think about it. It offers a telling vignette of how students are taught a rich technique for addition (“near doubles”) but do not use the technique. The students still revert to standard procedures of addition to “get the answer.” Then they tack on the new technique, which means they aren’t really learning about the structure of addition.

This explains something that has troubled me for some time. I frequently hear from parents that their child is not challenged by the curriculum that is being offered in their school. When I have looked at the curriculum I find that it is often very rich and full of many challenges. Perhaps the reason for this disconnect is that when the student looks at the material, the only thing the student is thinking about is how can I get this done and get a right answer, while I am looking at how many different things a student can learn from the variety of approaches taken.

I think the key to cutting off answer-getting in students is to notice that what they want is a good grade for just having the answer, so the teacher strategy is to give a middling grade for just having the answer, and a better grade for employing new strategies.