Optimization – Reflections

First, a new resource has been added to the resource page.  An index of all free-response questions from 1971 – 2018 listed by major topics. These were researched by by Kalpana Kanwar a teacher at Wisconsin Heights High School. Thank you Kalpana! They include precalculus topics that were tested on the exams before 1998. These…

The Marble and the Vase

A fairly common max/min problem asks the student to find the point on the parabola that is closest to the point .  The solution is not too difficult. The distance, L(x), between A and the point  on the parabola  is given by And the minimum distance can be found when This occurs when . The local…

Apt Apps – 1

I am a very big iPad user. I’m on my third iPad and use it all day. Some days I run the battery from 100% down to 10% without even watching movies or playing music. I have lots of apps a few of which I have found to be very useful in doing and teaching…

Lin McMullin’s Theorem

Mathematics more often tends to delight when it exhibits an unanticipated result rather than conforming to … expectations. In addition, the pleasure derived from mathematics is related in many cases to the surprise felt upon the perception of totally unexpected relationships and unities. – Mario Livio,  The Golden Ratio I have a theorem named after…

Experimenting with CAS – Chain Rule

Discovering things in mathematics can be facilitated by using a computer algebra system (CAS) available on many handheld calculators and computer apps. A CAS can provide good data with which to draw conclusions. You can do “experiments” by producing the results with a CAS and looking for patterns. As an example let’s look at how you…