Polynomials and VEX Drive Motor Control

VEX Robots can be more competitive when they have addressed several drive motor control challenges:

  1. Stopping a motor completely when the joystick is released. Joysticks often do not output a value of  “zero” when released, which can cause motors to continue turning slowly instead of stopping.
  2. Starting to move gradually, not suddenly, after being stopped. When a robot is carrying game objects more than 12 inches or so above the playing field, a sudden start can cause the robot to tip over.
  3. Having motor speeds be less sensitive to small joystick movements at slow speeds. Divers seeking to position the robot precisely during competition need “finer” control over slow motor speeds than fast motor speeds.

These challenges can be solved using one or more “if” statements in the code controlling the robot, however using a single polynomial function can often solve all of these challenges in one step. A graph can help illustrate the challenges and their solution:

Continue reading Polynomials and VEX Drive Motor Control

Solving Math Problems At The Board

Many students perceive their teachers to know more than they really do (Math teachers in particular). If a Math student who just observed a teacher solve a problem at the board is asked “What do you think was going through the teacher’s mind as they solved that problem?”, I suspect the average answer would be something very specific, like “This is a second degree polynomial in standard form, so solving this problem will require precisely six steps, the first of which is…”

In reality, the jumble of thoughts in a math teacher’s mind probably go something more like: “What looks easiest to simplify first? Oh… I see two places, no make that three, that I could start… but which should I choose? Does this look like it will take a lot of room to solve? If so, I had better organize my work a bit more…  OK – now that I have simplified things a bit, what options do I see from here?”

Teachers and mathematicians do not “see” the entire series of steps needed to solve a problem before they start work on it. Instead, they usually seek to take whatever step looks like it will simplify the problem the most, and have no clue (yet) what will follow that. Once that first step has been completed, the appearance of the result influences what is tried as a second step. It is an iterative process, one step at a time, with a re-evaluation of the situation being done after each step. There is much uncertainty in the process, even for a math teacher.

Continue reading Solving Math Problems At The Board

Procedural vs Intuitive Approaches

Life is full of alternatives. Would like fries or coleslaw with your meal? Should you put on your right or your left shoe first? Should you attempt to solve a math problem using algebraic procedures, or your intuitive sense of the situation?

Life is also full of false choices: there are many occasions when you do not have to make a choice unless you wish to. You could have fries with a side order of coleslaw. If you wear loafers, you could slide your feet into both shoes at the same time. And many math problems can be solved quite successfully using a combination of intuitive reasoning and algebraic procedures.

Continue reading Procedural vs Intuitive Approaches

Word Problems… !#$%@;*!!

Word problems can be… frustrating.  Most of their reputation arises from their use of words to describe a quantitative problem.  And if the problem’s author did not choose their words very carefully, you’ve got Trouble (with a capital T). So why are so many word problems assigned? Because they are more similar to the quantitative problems you might encounter in life than many of the practice problems in your textbook: you have to supply some insight and organization in order to arrive at a solution.

Just about every level of mathematics, not to mention chemistry and physics, seems to send periodic spasms of word problems your way.  The SAT makes liberal use of them as well.  Since you cannot avoid them, you had might as well learn how to solve them once and for all.

Continue reading Word Problems… !#$%@;*!!

Improve Your Math Grade: Use Scrap Paper

Never solve math homework problems on the piece of paper you intend to hand in (unless it is a problem that is very, very easy for you).

I used to do my work on the same piece of paper I intended to hand in, and when I encountered a problem I was uncertain about… I froze. I did not dare write anything incorrect on the page I was going to hand in. If I did, I might have to recopy all my work in order to end up with a neat-looking page to hand in. So, I would just skip the problem, telling myself I would ask the teacher about it in class next time. The result was that I greatly slowed my learning how to do that type of problem.

I now know that if I figure out how to do a problem all by myself, I will remember how to solve it for years. But, if I ask someone else how to do a problem I was struggling with, I will probably forget their solution within minutes.  So, if my goal is to do well on a final exam a few months from now, I am better off trying to figure the problem out by myself.

Continue reading Improve Your Math Grade: Use Scrap Paper