Infinite Sequences in Calculus

So, when Calculus moves beyond just derivatives and integrals, one way to ease into infinite series is to first look at infinite sequences. In the future, I want to do this with spreadsheets, but for now I use graphing calculators. Here’s an exploration that uses TI83’s or 84’s to get at two key ideas: how to tell if a geometric sequence will converge, and how to tell if an alternating sequence will converge.  With those two tools, students can do a lot to make sense of the infinite series that make up Calculus C. It’s your choice how rigorous you wish to make this (do you require them to work out the limits analytically or just express it in common English?) and here is the pdf: SeqConv2019

College, College, College

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.

The Remote Exterior Angle of a Triangle

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.

Angles Ain’t That Simple

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.

Connecting Geometry To Algebra 1

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.

“Two Flips Equals A Rotation”

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.

Do Charter Schools Outperform Public Schools In New York City?

Source: Do Charter Schools Outperform Public Schools In New York City?

Cutting Off The Path To Answer-Getting

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.

Schools are the Hearts of Communities

Although Anthony Cody gives us much to think about in his questions and suggestions about Bernie Sanders’ run for President as a Democrat, I want to address one point that speaks to my experience:

School closures are devastating to communities, and have been focused in African American and Latino neighborhoods. This results in community decay and spurs gentrification. Research shows federal efforts to “turn around” high poverty schools have not succeeded. This policy should be halted, and schools in these communities should be supported with wraparound social services to directly address poverty, not destroyed.

I grew up in a rural, poor community and went to small public schools with few resources. Looking back, it is clear to me that the school was the heart of the community, and that is true for schools in the city as well as the country. From the sports teams, to the Pancake Suppers, to the local votes on millage increases to pay for repairs and new buildings, to the Quiz Bowl teams that travel to neighboring schools, schools provide identity, purpose, and a sense of the future to a community. To lay off half the staff or shut down a school is for America to apply a scorched earth policy against its own citizens…this might make sense if the Wehrmacht were rolling in.

But there ain’t no Wehrmacht today. There’s just elites tearing apart American communities… and it is even more shameful they are focusing this on communities of historically oppressed minorities.

(0.5)!

So, it is AP Calculus prep time, my least favorite time of the year.

This year I am lucky to have one student, just one, who will NOT take the AP exam. So, I asked him, “Do you think it would be a boring waste of time to do the old AP exam your classmates are doing?”

“Yes,” he answered. So, I pulled down the old Calculus textbook by Paul Foerster–whose explorations are essential to my class–and thumbed through it for a good problem that would take him some time to do…

The factorial function and the Gamma function!

He set to it, doing some improper integrals, and some integration by parts and then he used his calculator to integrate the factorial function to get 0.5! = 0.886.

The book said, find a simple expression involving pi that equals 0.5! and after a little time, he found.

Moreoever, he learned that (-1)! and (-2)! were infinite but (-0.5)! and (-1.5)! were finite.

And I learned these facts, too!

Back in college, I saw the gamma function in my textbooks but spent zero minutes using it, much less investigating its definition. So having a student who was not constrained by the AP test prep (and I’m only spending a week and a half on prep) gave me the opportunity to broaden and deepen my knowledge of mathematics… as well as this one student, and after the exam is over and done with, I’ll use this experience to push the rest of the students to learn (0.5!). It will fit in with the “Integrals and Statistics” unit I’ve composed for the month & a half after May 6.

This is what studying math brings you: new discoveries, new connections, a sense of wonder at how the world of numbers makes sense.

Thank you, Paul Foerster, for your wonderful books!