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Circling the Tangent

Student work.

Circling the Tangent is a lesson for students in a high school level calculus course or precalculus class.

TSW learn and write the equations for circles.

TSW solve equations to find values from inputs.

TSW define tangent and normal lines.

TSW use definitions of tangents and normals with the circle equation to create a unique artwork.

Instructions.

Students will wite 5 equations. Each for a circle with a radius of 8 units. The centers are the origin and then in each quadrant such that four circles are tangent to both the x and y axises.

Students then use the equation for the circle centered at (8,8) to identify and plot the cooresponding values for when x=0,1,2,4,6,8,10,12,14,15,16.

Next they label the points for the other ones (students may choose to do the center circle after the first 4 are done.

Once the points are plotted, students use a drawing triangle with the right angle on the plotted points and one side going through the center as a radius (or normal).

The other leg of the triangle will be at right angles to the normal and be a tangent. Draw that line to the edge of the quadrant that circle is is.

After one of these is done for each point on a circle, the student will come back and use a straight edge to extend the tangent lines in the other direction.

Students then do the same for the centered circle.

Upon satisfactory completion, creators then color the drawing as they wish.

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How does one teach the foundations of learning.

This junt is from a few years back when I was really focused on algebra learning. The answer to which, “You learn by doing.” Just like everything else and, in particular, human occupations for which machines can not do. It appears I may have not finished it but I do not honestly know if it does complete.

Student work. I got it just because I let him make these things rather than stress about some mandated math facts.

I watched one of my students explain a concept to a classmate the other day and realized that I had succeeded in doing what I needed to as a teacher for him. There is plenty more math he can and will learn. There is always more to learn. That was not the source of my relaxed understanding.  I could see that he was thinking with a solid mathematical logic and could develop new skills with a new found ease. In short, he had learned how to learn and apply math. This student will be leaving the school soon and is rather busy getting everything together so I am not pushing him as hard as I could in math but I honestly feel that I have achieved my primary mission. Besides, he is more interested in programming and origami now so I have no qualms with letting him study and pursue those skills and the problem solving they entail. He wants to make something, not just solve more equations.

To be fair, this student is naturally a deep thinker and one who I have had to argue almost every logical difference in math with. It can be tiresome but he now approaches the subject with a very solid understanding of its structure and dynamic. Not all students are like this. A part of me always wonders why this is so. Math is a playful and fascinating pursuit that we have turned into thousands of drills and practices. One learns far more figuring out a pattern in the units digit of multiples of 3 than a page of multiplication drills. These patterns that are not so difficult to arrive at are the basis for the laws and proofs that build the subject. Solving the little drills just becomes a matter of mechanics. The solutions make sense for how we get to them and the whole study works out the way it should.

This is a long way around to getting to what many of us already know. Teaching math through drills is not teaching or learning of math. It works in despite of itself. We keep doing it because we keep doing it. Grading is easy and it makes for straightforward but deeply flawed and largely pointless testing.Through all these tedious problems and arrays of rules to follow, we suffocate the joy of math. The subject is so profoundly interesting and creative that young minds still enjoy it through all we do to destroy it.

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Starting on the curvature of the world.

Maths can be a great escape that has pragmatic purposes in our lives. A lot is unknown and many worry greatly about where we are headed but that does not mean we should stop our pursuits of knowledge. The opposite is more true in direction. We really must be challenging ourselves to learn more with rigors of definition.


How does one measure the radius of the Earth? People have known a fantastically accurate value for far longer than may make sense to many of us today.
Since radius and circumference are in a pi-portion to each other, if we know one, we know the other. It turns out, one of the great questions of maths peoples throughout history is to identify the size of the Earth.
Eratosthenes is one of the dope names of all times. Say it one time fast…Anyway, he is attributed to the earliest accurate measure and it was down with sticks, rope, and a lot of walking.

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So you wanna online teach during COVID-19 c/o 2020?

super in progress–like everything here. Here are some notes that may or may not help. Comments are welcome!

Like you, I’m not sure what is ahead but teaching is why we are here and that will continue.

I have spent way too much of my time involved with on-line learning to not feel securely knowledgeable in it. While I am not an authority, I don’t know anyone with more understanding on it than I have.

The first step is knowing you will be wrong a lot. Things will be messy and they will not be under your control. The space on the screen is now the extent of your classroom.

For those who are the most worried, you will have an easier time if you trust the culture you developed earlier in the year. Use those mantras and things that signal to the students that you are there. They hear your voice in the words. We all know that nothing is more academically vital than building literacy skills, by reading and writing with your students, you make those skills the vital supports for the class. If that is all you are able to do and it is only one student you can help, then I pray that the worst is past for you, but it is more than enough.

LMS’s

Learning management systems. These are the one of a hatful of sites we use to do school work. There are differences and I have one my favorites but you need to pick one asap. Google classroom will do everything you need and is in google land so it all works well together.

I can’t recall how many different gradebooks I have used but they are apt to be your best communication system to the most involved parties. They also usually have a bunch of functions you never knew about because who pays attention to that presenter anyway. That’s a hard gig. If you like to keep a disciplined order to your class, this could be all you need with strong communication and documentation.

Super Important

The course almost never looks the same to you and the student and they do not respond to it how you think they will. It’s like those instructions you know to give now because you see the same mistake on repeat. The more linear you make the work, the fewer places to get lost.

If you can do shared time, the key part is not you explaining standards but coming to an understanding on how to set up and navigate the system.

As always, this depends on yourself and your class. Now may be a very good time to consider what drives you to teach. Learning and helping people learn was my initial draw. I decided when I came into education more formally, that I really was called to teach problem solving. My expertise in the matter comes from making every mistake possible as I slip in the mud.

If you have a curriculum that you teach and you have made that what you do, stick with that. Try just filling in all of the boxes and attachments on your gradebook. Communicate with students you can reach so y’all can figure out what parts you add show up where on their end and visa versa. This would be a really excellent exercise to review all the documents, resources, questions, and etc. that you usually use.

Class page or class feed?

In general, your established online classes look like a page and have different parts you can go to but the order is static. These take time to create unless you get a general course which are mostly blah. You need something to do. This does not have to be anywhere near completed at the start. It needs to first task but you need to have a good outline of how it will look.

Others, which will be a great number now, look like a blog feed. They are messy and cumbersome to rework but they do work. You post something. The students respond. Repeat. If you get your class that far, you can do a job that you can be proud of. Just teach.