Real World Usage Examples and Historical Origin of Beginning Algebra (HS Algebra I and II) I have a high schooler who I need to get energized about math. She excels in other sciences, but does not in math. The issue, I learned after some discussion, is that she doesn't find math interesting enough. She sees it as simply processes that you memorize and apply without truly understanding what the end result is about. Let try to better illustrate this point, in chemistry or biology you do an experiment or observe something in nature and then go back to understand what makes these things work. She doesn't see this type of relationship in math. I have looked at essays about 'Why one should learn Math' but they are not much help as they tend to only show how math is used in careers. The best explanations are usually around statistics or focus around advanced level math but she is looking for something to get excited about. What I am hoping to find is a resource to show things like Seven Bridges of Königsberg ( a little more advanced than HS Algebra)and how they apply to specific topics that she is studying.
Here are my specific questions:


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*What are examples to show her things like the Fibonacci number system and what it is and how it appears in nature and history?

*Is there basic information (this is a HS school student) on how someone like Escher used mathematical formulas in some of his pictures or how others used algebra to solve real world problems?

*I would teach her what the original problem was that the mathematician was trying to solve when a particular topic was created and developed are there any good sources for this?


Thanks for all your help.
A NOTE TO MODERATORS: 
I apologize if the question(s) is not specific enough or slides a little off topic for Mathematics. But based on the FAQ I would argue it falls under Math History. But I understand if it has to be closed, I only ask that a better place to ask this be given if possible. Thanks
 A: You can find quite a few good ideas in the better books intended for so-called liberal arts math courses at the undergraduate level. I can think of three that have some good, appropriate material:


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*The Heart of Mathematics, Edward B. Burger & Michael Starbird  

*For All Practical Purposes, COMAP  

*Excursions in Modern Mathematics, Peter Tannenbaum


(Warning: These were chosen very carefully. There are also some perfectly awful books billed as being for liberal arts math courses.)
You might also get some ideas from Discrete Mathematics through Applications, Nancy Crisler & Gary Froelich, a discrete math text written specifically for high school students; it has significant overlap of topics with the COMAP and Tannenbaum books, but it also has some topics more characteristic of first courses in discrete math at the undergraduate level.
A: You could try Plus magazine, which "opens a door to the world of maths, with all its beauty and applications, by providing articles from the top mathematicians and science writers on topics as diverse as art, medicine, cosmology and sport".
A: The three things that make mathematics interesting are WHAT IT GETS AT (i.e., its OBJECTS and their PROPERTIES), the QUESTIONS that can be raised ABOUT what it gets at, and the resulting DISCOVERIES of the PROPERTIES of its objects (answers - called "theorems" after being proven), and NOT its applications or history or reasoning or abstraction or proof. Mathematics is driven by wondering.
For example once we realize that every quadratic has two roots, we can wonder if there are simple CONDITIONS on the coefficients for the roots to be equal, or unequal, or equal except for different signs, or same or opposite signed, or one to be a multiple of the other, or integers, or natural numbers, or non-negative integers, or fractions, or non-negative fractions, or one fraction and one natural, or non-negative squares, or imaginary, or complex, or unit fractions, or prime, or anything else we can dream up. This activity gets us comfortable with conditions in general which are important throughout mathematics, distinguishing between necessary and sufficient conditions, questioning - which is vital throughout mathematics, deriving or obtaining indications of answers, and with the idea that mathematics is the study of the genesis and nature of mathematical OBJECTS and NOT merely memorization and mechanics and the pride of knowing or "reasoning" or cognition or philosophy or tricks and gimmicks.
One last point. When an effect is discovered, such as the Collatz conjecture (http://en.wikipedia.org/wiki/Collatz_conjecture) or the reverse-and-subtract algorithm (that goes to zero for all starting numbers below 1023 and most others), it is great fun to dig for special cases and uniformities, relationships, anything systematic. For example, do the terminating sequences of the reverse-and-subtract algorithm have all possible lengths? One simple uniformity of that algorithm is that the lengths of the sequences for all single digits and palindromes is 1. Questions  of properties and characterizations apply to ALL objects in mathematics. For example, there are four different approaches to the condition for continuity of functions involving: limits, algebra involving epsilon-delta, topology, and simply listing all possible discontinuities.
A: Modular arithmetic and the ensuing "divisibility tests" are pretty easy to ramp up to, and they give students a little boost that they can do something that would otherwise be tedious quickly with a mathematical insight.
If that is too easy, then you could go on to basic material on linear codes over a binary field. I think students at high school level and above can appreciate error correction :)
