November 2020 – Algebrawesome

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.

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.