September 29th, 2017, 05:54 PM  #11  
Newbie Joined: Sep 2017 From: Iowa Posts: 9 Thanks: 0  Quote:
 
September 29th, 2017, 06:00 PM  #12  
Senior Member Joined: Aug 2012 Posts: 2,134 Thanks: 621  Quote:
So the expression $x^2$ can be thought of as a process or recipe. Put in one number, get out another. If the rule is $x^2$ then this machine outputs the square of its input. Whatever the rule is, that's what the machine does. Another feature of the function machine idea is that the machine is a black box. We do not know or care how the machine works, whether there are gears inside perhaps elves with calculators. All we can ever know about our machine is what it outputs for a given input. The only aspect of a function machine that we can observe are its inputs and outputs. The most important feature of a function machine is that it always produces the same output for a given input. So if you put in 3 and it outputs 47 today, it will do exactly the same thing tomorrow and the day after. That property defines a function. Last edited by Maschke; September 29th, 2017 at 06:03 PM.  
September 29th, 2017, 08:04 PM  #13 
Senior Member Joined: May 2016 From: USA Posts: 1,250 Thanks: 516 
We use letters in algebra as convenient symbols to represent numbers that are either not yet known or else are not yet specified. Any letter will do, minuscule or majuscule, Greek or Roman. Students have trouble with this: they want to know what number x is, but the whole point is that we don't know yet. When the student finally begins to get that x is just an abbreviation for "a particular but as yet unspecified number", we introduce function notation where letters are used as abbreviations for formulas (more precisely, rules). Using letters in two different ways causes further confusion among students. In terms of teaching elementary algebra, it would have been better to use majuscule letters for functions and minuscule letters for numbers. When we see a formula like $d = r * t$, it allows us to capture a potentially infinite number of numeric examples in a single abbreviated formula. $f(x) = x^2$ simply means square x once you know what number x represents. Frequently, using function notation is simply a convenient abbreviation. I write $f(x) = \dfrac{\sin(x)}{x}$ and can thereafter just say $f(x)$ instead of writing down that fraction. Last edited by skipjack; September 30th, 2017 at 12:45 AM. 
September 30th, 2017, 03:16 PM  #14 
Newbie Joined: Sep 2017 From: Iowa Posts: 9 Thanks: 0  
September 30th, 2017, 04:06 PM  #15 
Math Team Joined: Oct 2011 From: Ottawa Ontario, Canada Posts: 13,791 Thanks: 970 
Can be looked at as a li'l puzzle. Teacher preparing questions writes down: 3 + 5 = 8 Then erases the 5 and replaces it with an x: 3 + x = 8 Writes above on blackboard: Ok class, what number does x replace? 
October 1st, 2017, 12:30 PM  #16  
Newbie Joined: Sep 2017 From: Iowa Posts: 9 Thanks: 0  Quote:
One instructor early on asked if I ever did sudoku puzzles? I said that I do not. She said that they work the mind in a similar way.  
October 1st, 2017, 12:33 PM  #17 
Senior Member Joined: May 2016 From: USA Posts: 1,250 Thanks: 516  What denis means by his example is that algebra is merely following up the consequences of that little puzzle.

October 1st, 2017, 12:38 PM  #18 
Math Team Joined: Oct 2011 From: Ottawa Ontario, Canada Posts: 13,791 Thanks: 970  
October 2nd, 2017, 12:01 PM  #19  
Senior Member Joined: May 2008 Posts: 298 Thanks: 81  Beer soaked ramblings follow. Quote:
Worked Examples At an introductory level, you will not become proficient at using mathematics by simply 'reading' a mathematics book, cover to cover. A better approach is to read just one section at a time, then put pen to paper and follow the methods which are demonstrated in the Worked Examples, line by line. An index of Worked Examples is provided at the end of the book for easy reference. Graphs Can you imagine someone who has never seen a car before, attempting to understand what the controls, gears, steering, etc. look like from a verbal description? Understanding would be enhanced enormously by the provision of some welllabelled sketches and diagrams. Visualising mathematical functions/equations is not easy, especially for beginners. When the function is graphed, much of the vagueness and abstractness is removed. In fact, many of the properties of the function are revealed. In Worked Examples, whenever appropriate, graphs are plotted over an interval by calculating a table of points, then drawing the graph. Unfortunately, in many economic applications, this process is often a very timeconsuming exercise and errors are all too easily made in calculations. In this text, the use of ...etc.  
October 4th, 2017, 03:31 AM  #20  
Senior Member Joined: May 2008 Posts: 298 Thanks: 81 
Beer soaked ramblings follow. Quote:
I was very impressed. A few years later, I saw this news about a young blind girl who took on mathematics in college at a regular university. That was inspiring. As inspiring when Paul Schaefer commented on how inspiring it was to hear about a news where this 80 something woman was peeping at her neighbor for her erotic needs. Then I tried solving rubik's cube using just touch markers as if I was blind. I realized just how blessed those who can see. I say you sir are blessed with sight and mobility. Your perceived obstacles are only obstacles until you move them out of the way. From another author: To recognize a conic section, you often need to pay close attention to its graph. Graphs powerfully enhance our understanding of algebra and trigonometry. How ever, it is not possible for people who are blindor sometimes, visually impaired to see a graph. Creating informative materials for the blind and visually impaired is a challenge for instructors and mathematicians. Many people who are visually impaired "see" a graph by touching a threedimensional representation of that graph, perhaps while it is described verbally. Is it possible to identify conic sections in nonvisual ways? The answer is yes, and the methods for doing so are related to the coefficients in their equations. As we present these methods, think about how you learn them. How would your approach to studying mathematics change if we removed all graphs and replaced them with verbal descriptions? Last edited by jonah; October 4th, 2017 at 03:36 AM.  

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