Wonderful Course for NYC/Long Island Teachers

In my last post, I mentioned an amazing l course I took a few summers ago. Each class a different experienced teacher shared their best ideas and lessons. I learned so much, and wished that this had been part of my teacher training.

I took the 9 – 12 version of the course, but they are offering a version geared to grades 6 – 9 starting on October 3. The course is FREE and meets for 15 classes and 3 computer labs over the course of the year at SUNY Old Westbury. I recommend it highly!

Follow the link for all the details.

The Institute of Leadership Training for Teaching Mathematics and Technology

or download the brochure & application.

Brochure                              Application

A Hodgepodge of Ideas?

When I started teaching Precalculus, I thought it was no wonder the students were confused. It felt as if we jumped from topic to topic – complex numbers, trigonometry, logarithms, polar coordinates, etc. – and spent only a few days on some of them. It was just last year that I was finally able to really help my students see the connections.

For most of the course, we are studying the functions. My students will need to know them all pretty well for calculus. And the functions share many characteristics: they have a domain and a range and a graph and can be shifted right or left, up or down, or reflected in the x– or the y-axis (although a reflection in the y-axis may not return another function.) The notation for these shifts and reflections is exactly the same, no matter what the function:

y – k = f(x – h) translates h units horizontally and k units vertically

y = – f(x) reflects the function in the y-axis and yf(- x) reflects the function in the x-axis

y = cf(x) stretches or shrinks the function vertically and y = f(cx) stretches or shrinks the function horizontally

Students need so see how these are used with a number of different functions before they get they idea.

\begin{array}{l}\begin{matrix} y=f\left( x \right) & \quad \ y=4f\left( x \right) & \quad y=f\left( 4x \right) \\ \end{matrix}\\\begin{matrix} y={{x}^{3}} & y=4{{x}^{2}} & y={{\left( 4x \right)}^{2}} \\ y=\sqrt{x} & y=4\sqrt{x} & y=\sqrt{4x} \\ y=3x+2 & y=4(3x+2) & y=3\left( 4x \right)+2 \\ \end{matrix}\end{array}

Those functions whose reflections in the line x is also a function have an inverse and the inverse works the same for all these functions: linear, square root, cubic, exponential, etc. When you reflect in this line, the y‘s become the x‘s and the x‘s become the y‘s. As a result:

  • We have an algorithm for finding the equation of an inverse that calls for swapping the x and the y and solving for the new y.
  • The dimensions of the domain of the parent function become the dimensions of the range of the inverse function. The dimensions of the range of the parent function become the dimensions of the domain of the inverse function.

The y‘s become the x‘s and the x‘s become the y‘s. The range becomes the domain and the domain becomes the range. The second bullet is a direct consequence of the first. Inverse functions are so useful that we truncate the trigonometric functions in order to create them, and we actually have a totally artificial, man-made inverse of the exponential function, namely the logarithmic function.

Some functions have a y-intercept, an x-intercept, vertical or horizontal asymptotes, or periods and amplitudes. Some are algebraic and some are transcendental. Each is needed to model some relationship in the real world: periodic functions for sound waves or business cycles, exponential (or logistic) functions for population growth, etc.

In A Tour of the Calculus, David Berlinski presents and discusses the Mandela of the Functions. I share the image from this book with my students. Here is a link to a web site where you can see how the functions come full circle. Page down once to see the diagram.

The Mandala of the Functions

All of these are based on our Cartesian coordinate system, but there are functions whose equation are much simpler if expressed in terms of distance from the origin (pole) and direction from the positive x-axis (initial ray). Polar coordinates and the trigonometric form of a complex number have a lot in common, namely the Pythagorean Theorem. The work we have done on building the unit circle from the 45-45-90 and the 30-60-90 triangles sets the stage. This year, for the first time, my students knew the special right triangles by heart and so were easily to solve both of these types of problems.

As I wrote this I wondered why it was that I finally made these connections myself. Was it “just” another year of experience and reading books & blogs? I think I owe much credit to a summer course I took at a local college. For two weeks, a different very experienced teacher came in each day and shared their best lessons and ideas. It was an amazing experience and I still refer back to those notes frequently. Thank you, Art!

Making Sense of Quadratics

As a student, I never understood why we set quadratics equations equal to zero and factored to solve them. This really irked me. An article a few years ago in Mathematics Teacher inspired this set of Math Labs. I have different groups of students work on each of these and present the results, then provide each student with a photocopy of the completed set. Each group also puts their graph on a poster-sized sheet of graph paper, and these stay up on my side wall throughout the year. We frequently come back to them, to remind them of the concepts such as double roots that these illustrate.

Labs 1 through 4 are parabolas, Lab 5 extends the lesson to show that the product of three linear functions is a cubic.

Playing Games in Calculus

Last year I taught calculus for the first time. I quickly found that my students loved anything I could turn into a game. I found this activity on Maria Anderson’s Teaching College Math blog. I adapted it for use with my smart board. Given an expression, students had to move it under the correct approach:

Only Product Rule       Only Quotient Rule         Only Chain Rule

Product Rule plus (what?)       Quotient Rule plus (what?)   Chain Rule plus (what?)

Here is the pdf that Maria provided

and here is a link to the the smart notebook that I turned it into.

Derivative Game

Thanks to Maria for this and all the other ideas she has provided. She has consolidated her blogs, including teaching College Math, into this one:  Maria’s consolidated blog Click the Mathematics button on her page if you want to see more.

 

Like a Butterfly’s Wings in China

A single book that I read as a fifth grader has had quite an impact on my life. Carry On, Mr. Bowditch is based on the life of Nathaniel Bowditch, born in Salem in 1773. Nat loved math (he taught himself algebra and calculus) and showed me how it could be important. He was interested in celestial navigation, and honed these skills during five voyages that took him around the world. Nat famously steered his ship home to Salem during a blinding snowstorm one Christmas, confident in his calculations.

As he used the published navigational tables, Nat found thousands of errors and realized that any of them could cause a shipwreck. He recomputed all of the tables and added clear instructions. Having decided to “Put in the book nothing I can’t teach the crew”, Nat worked patiently with the unschooled crew members. By the end of the voyage, each could calculate lunar observations and plot the ship’s position.

Nat has been my inspiration, both in my studies of mathematics and in my efforts to teach it.

Bowditch’s American Practical Navigator, published in 1802, was just updated in 2002 and is still in use today. Yes, Amazon has it. Here are pix of the copy my husband found for me many years ago.

ImageImage

Do you know what a haversine is?

First Baby Step –

So I’m one of the almost 200 teachers who signed up for the new blogger initiative. Thanks to the initiative for making this happen, and to Julie for her easy-to-follow instructions!

I teach math at a private high school in New York, and am still a relatively new teacher. Other teacher blogs have helped me sort out the contradictory advice on what a “good teacher” does. I’m looking forward to joining in on the conversation!