Another Way to Solve Quadratic Equations

Here is a recent video on how to solve quadratic equations.  It depends on the fact that the two roots lie symmetrically about the x-coordinate of the vertex.  https://www.popularmechanics.com/science/math/a30152083/solve-quadratic-equations/  It eliminates the guessing or systematic trials by using in simpler quadratic equation and is quite elegant.

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Do We Really Need Fraction Bars?

Do we really need fraction bar article

Do We Really Need Fraction Bars

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A Year of Nights of Wondering VIII: Stuck Edition

My students have taken their seats and are beginning their daily quiz.  One says, “I’m stuck, Mr. Hatton.”  I murmur something about doing the best she can.  She repeats, “No, I’m stuck.”  I remind her  about the online quiz makeup problems. Again, “I am really stuck.”  I look over.  She shows me that she can’t get her backpack off.  She is  ‘stuck’ in the webbing.  She had tied the chest strap with a slip knot and couldn’t get it undone.  The nylon webbing knot was tightly compressed.  After a minute or two, I managed to get the end of a key though a loop and prise the knot loose.  This student’s math teacher had helped her get ‘unstuck.’

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Creativity Conference II – What Inspired Me

I was inspired  by the smart, sensitive and, yes, creative presenters at our second Creativity Conference here at Southern Oregon University.  Last year I listed what I had learned.  This year I will list what stirred me to action.

  1. Art, its disciplines and the mini-skills necessary for its practice, can inform learning in any field even with difficult (college students) or sad (refugee children) people.  I learned from the  Jessica Hunter-Larsen and Dez Stone Menedez from Colorado College that the discipline of seeing necessary to art can be transferred to other areas. Physics was an example.  I will be using this idea when I introduce graphs of functions and also the unit circle in precalculus.  Another thing I learned was that techniques used in teaching art can help break the barriers typical college students have to risk taking and owning their own learning.
  2. The classroom is for thinking – the higher level the better.  This I learned from Dr. Weiping Hu.  I have always said that I liked test days because I knew the students were thinking about math.  Dr. Hu’s presentation has inspired me to have more active, higher level thinking exercises in my classes.
  3.   The practice and joy of doing art improves learning and much more.  I was inspired by the Chula Vista elementary school district that committed fifteen million dollars to arts education to enrich the experiences of its students any of whom are traumatized by the fear of and the actuality of deportation of their parents, relatives and friends.  This was presented by Ivonne Chand O’Neal and Harun Tadik.
  4.  Through the tangle of creativity measures, three simplifications stood out.  Dr. Dean Simonton’s idea the creativity equals novelty times utility times surprise.  Michelle Neumayer’s prerequisites of safety, connection and learning.  And Dr. Jonathan Feinstein’s focus on creative guiding principles.  The first I will use as we develop ways to measure creativity.  The second I will use as a lens into my and others’ teaching practices.  And the third I will use as I engage with creative endeavors and products.
  5. There was so much more.  My notebook is full of light bulbs – ideas to pursue and “TD,s” – things to do – look up references, change class exercises, and more.

Thanks to everyone involved for such stimulating ideas and such grace.

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Precalculus Essay Test

As something to do while waiting for new advisees at our first summer registration event, I attempted to turn a typical precalculus exam into an essay test – no calculations, just words.  Actually also diagrams.  You can see it at the end of this post.

What I found out:

  • Students would have to read too much.
  • The exercise helped me clarify what I really wanted them to know and is therefore recommended.
  • Sometimes just doing the problem is sufficient evidence of understanding.
  • I am dependent on the unit circle model for explaining and understanding trig functions.  Many of the problems were really to see if the students really knew it.
  • The questions were somewhat labored since I was converting a standard test rather than starting from zero.
  • The questions demanded precise language – never a bad discipline.
  • Students have to be primed and practiced for such an exam.
  • The test will be hard to grade.  I would need explicit rubrics for each problem – something I do naturally on the standard tests.

Precalc Test

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Changing the Limits of Integration

It is possible to change the limits of a definite integral to any limits you want by using a linear u-substitution.  To go from the integral from a to b to the integral from a’ to b’, use a u-substitution of u = mx + c, with m = (b’ – a’)/(b – a) and c = (a’b – b’a)/(b-a) with du = mdx.  This technique is used in Gaussian Quadrature for instance.

This is simply a change of scale and a translation.  I was interested in what it looked like as the shape got deformed, always with the same area, to the targeted limits.  Here we ‘move’ a parabola from [3,6] limits to [-1,1] limits:  Moving Parabola to Adjust Integration Limits Here is the final frame:

Parabola now shifted to [-1,1]

Slightly more interesting was to watch this low three peaked object change to a narrow three peaked shape.  The scaling does not change the number of periods displayed because although the period changes, the plotting interval changes accordingly.

Three Peak Beginning Shape

Three Peak Ending Shape

You can watch the video here. That’s all.  I just used this exercise to learn how to create animations using Python.

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I Love Math – And It Makes Me Glad

I love the game of Go, I love golf, I love my wife, I love teaching, and I love math – and this makes me glad.

Eight years ago, I ended that sentence with the word, “sad.”  Eight years of minor and not so minor health problems. Eight years of increasingly ill golf buddies and relatives. Eight years of friends and relatives passed.  I have the sense that I am intently clinging to those activities and those people who give me the most pleasure. Now I marvel at those few new ideas that enter my brain.  I am paying closer attention to my friends, my students and my wife and sometimes reel in amazement at their creativity, their humor and their courage. I am still looking forward and striving to improve, though now there is a sense of impending limitation and fear of not finishing my small projects. So, I’m glad and grateful that I can still play and think and enjoy other’s company.  As you can see, I am no longer measuring or judging my performance. I am intentionally and self-consciously paying attention. I love living on the Earth in an Earth-designed body with an Earth-responsive brain and it makes me glad.

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