Category Archives: Uncategorized

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


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.

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.