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YOU are better than YOU think. Show yourself how: |
-/[]\- Logic
Mastery Do not leave here without it - Logic mastery will develops critical thinking, improve reading and writing, and give a firmer base for work and studies at many levels. Good luck. |
-/[]\- After logic, (a) continue reading Three Skills for Algebra, chapters 8 to 14 and do so alongside site area on solving linear2007 Equations ; or (b) see this calculus starter lesson and Volume 3, Why Slopes & More Math, chapters 2 to 6; For online automated help in senior high school maths & calculus, visit quickmath.com For Automatic Calculus and Algebra Help with derivatives, integrals, graphs, linear equations, matrix algebra, visit calc101.com With overlap, each site quickmath & calc101offers a different range of services, some free, some not, all based on webmathematica. Good luck. |
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Problems Solving Method
Jigsaw Puzzles & Mathematics
Problem solving is like putting together a jigsaw puzzle. In the case of textbook problems, all the pieces are present. Most just need to be fitted together following the clues. In the case of real world problems, there may be missing pieces or extra pieces, and no guarantee that the solution can be done.
In solving a jigsaw puzzle, you fit the pieces together, one at a time, one after another, in some order. The solution follows an opportunistic path with pieces tried here and there, until done or not. In solving a mathematics problem, the pieces to fit together are collectively given by all the rules, methods & tricks you have met in previous lessons and courses. A solution follows an opportunistic path with pieces tried here and there, one at a time and one after another, until all is done or not. Some pieces may be left-over. Each example or solution and each proof or chain of reason met in mathematics may include a piece of information, a trick, that you may recycle in further solutions. Once you know the jigsaw approach or method for solving mathematics problems, rest is routine and opportunistic. There is nothing more to problem solving than watching for and collecting ideas or methods for opportunistic use. Can you do that?
First Hint: Master Logic
Problem solving is like putting together a jigsaw puzzle. In the case of textbook problems, all the pieces are present and just need to be fitted together following the clues, and an possible a picture showing the desired result. In the case of real world problems, there may be missing pieces or extra pieces, and no guarantee that the solution can be done.
Novice problem solvers should examine the following chapters in Volume 2, Three Skills for Algebra.
- Two Logic Puzzles
- Chains of Reason
- Longer Chains of Reason
- Islands and Division of Knowledge
- Painless Theorem Proving.
These appetizers and lessons show how rules and patterns may fit together to arrive at conclusions or solve SOME problems. Problem Solving requires precision reading and writing. Logic Mastery helps with that as well.
Second Hint: Master Fractions
Many applied mathematics problems involving chopping and combining lengths, areas and volumes. So you need to know how to take a proper or improper fraction of a length, area or volume. You need to understand that one length may be 2.5 times or 2½ times or (5/2) times another. Any if you do calculation, you need to do it with care or at least do it with the knowledge that an error in one step makes all that follows wrong. The ability to figure well and precisely, so that you answer is correct, shows or suggests the ability to follow methods, one step at a time and one step after another in any subject, and in problem solving as well.
Algebra Word Problems
one or more variables, that is the question.
If your interest is in solving algebra word problems at the high school level, I would recommend learning how to solve linear equations in one to several unknowns efficiently. The starting point for that could be the site area Solving Linear Equations with Stick Diagrams and chapters 8 to 15 in Three Skills for Algebra.
High school students who can solve linear equations in one unknown are often given word problems where extra variables have to be eliminated to formulate a single equation in one unknown quantity to solve. The trick here is to draw or extract a single equation from the given information. But in most such words problems, it is easier to extract or draw from the given information several linear equations in several unknowns to solve. Each sentence in the word problem gives an equation in one or more unknowns or quantities. Now the algebraic way of writing and thinking can be used to eliminate variables and to solve for the one or more quantities of interest in an effortless fashion.
The algebraic solution of linear equations involves the elimination of variables to obtain say one equation in one unknown. This elimination process may be better done and recorded with algebraic notation. Going directly to one equation in one unknown to solve a problem requires more work to be done with words.
To learn more and to go further, see Solving Linear Equations with Stick Diagrams and chapters 8 to 15 in Three Skills for Algebra.PS. Being good in algebra and beyond requires an efficient command of fractions, what they represent and how to work with them.
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In Volume 2, Three Skills for Algebra, Chapters 8 to 14 and postscript What is a Variable point to a greater & clear use of words in algebra. Chapter 14 introduces a 4th skill for algebra, an elaboration of the third: - The direct and indirect use of formulas, numerically and algebraically, is unifying theme that should be mentioned aloud, with words, in each and every use of formula.