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Problems Solving Method
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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Professor
Whyslopes
Site value lies in
the difference between its ideas and yours.
Visit the site
entrance whyslopes.com before you
go.
Bon Appetit |
Two common gaps to fill or avoid
- The Old Algebra Gap: Algebra
appears with too few words of explanation in high school and college
mathematics. Chapters
8 to 12 in online Volume 2 put more words into the
explanation and comprehension of algebra. Chapter
14 in Volume 2 with its explicit discussion of the direct and
indirect use a formulas identifies a unifying theme for mathematics
and logic - all rules and patterns will be used forward and backwards.
Chapters
2 to 6 and 12 to 18 in Volume 3 may further ease or avoid the very
challenging use of algebra in the high level mathematics: calculus.
Calculus requires earlier high school mathematics at full strength: (i)
This logically complete but long lesson on complex
numbers shows how to simplify the senior high school
exposition of circular trig functions upto to formulas in the plane
for vectors dot and cross-products. The lesson provides the
route that would have been taken in course design if the key element
of the lesson, a December 2009 invention, had been available in
the 1950s. For further algebra skill development. See the site
coverage of fraction
with units, proportionality,
ratios and rates,
polynomials, quadratics
functions
and straight
line slopes and equations.
- The Arithmetic Gap: An exact and efficient
mastery of arithmetic with decimals and fractions is best (required)
for the high level study of mathematics alone and in science,
technology and business. Pages here on arithmetic
with decimals and integers, on fractions
and solving
linear equations with fractional
operations on stick diagrams may help fill the gap. That
exact and efficient command should be obtained in the last years of
primary school and the first years of secondary school.
Skill mastery in mathematics has to be observable to
be credible. You need to learn to do and record mathematical steps
on paper, one at a time, one after another. Mathematics mastery has
to be seen (on paper) to believed.
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Odds & Ends
Group I
1. Hints for Exams
2A. Exact Arithmetic
2B. Fractions Briefly
3. What is a Variable?
4.. Square Roots
5. Straight Lines
6. Problem Solving Methods
8. Complex No. Applet
7. Trig and Complex No.
9. History of No.s
10. ln(x) and exp(x)
13. Rename the > Sign
14. Problems: Quadratics
15. Problems: Algebra Test
16. Problems: Linear Eqns I
17. Problems: Linear Eqns II
18. Problem Solving Hints
20. Independent Variables
21. Why Logic
22. Why Math
23. The 15 Times Table
24. The 20 Times Table
25. Algebra Formulas
26. On Learning Maths
27. Biology - Growth & Decay
28. Navigation +Time
29 Quibble-What is Algebra
30. Logic in Maths
31. Real Number Operations
Learn More
Group II
Constant Retirement Rate
Road Safety
3 Strikes Law in California.
Math HOW-TOs
9 Steps in Maths
Two Gaps
[ Back ] [ Up ] [ Next ]
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For
Senior
High School & Calculus Students
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Words to clearly
introduce algebra and variables
have been missing in course design. For people who cannot do
algebra,
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the missing words may
explain or ease their difficulties. Volume 2 ,Three
Skills for Algebra, in Chapters
8 to 14 & 18 etc, puts words before symbols to
providing the missing words in a way that enrich the
comprehension of all. Those words form the middle part of a algebra
(and logic) lessons aimed at helping or improving all
of high school mathematics and also calculus course
design & delivery.
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For Avid Readers in School & Out -
Online Books
1. Elements of
Reason. 1996
1A. Pattern
Based Reason 1995
1B. Math
Curriculum Notes 1996
2. Three
Skills for Algebra 1995
3.Why
Slopes & More.Math
1995
Tour their forewords.
Calculus Prep or Help: See Volumes 2 & 3,
and this bigger
Calculus
Guide. If your
calculus questions is not answered here, submit
it. Over time, that may complete the site development of
calculus.
For Parents: Speaking
Skills, Reading
& Writing,
Preparing for Science, ends,
values and methods for work and study, parent- friendly maths
skill development booklets for ages 4-14.
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Mostly
For High
School
Intro to Solving
Linear Equations
- a different paths for junior and even senior high
school students. Question for Tutors: When do
you use and when you skip the stick diagram method
here?
Fraction
Skills, thought-based development, Ages 10 to 14 may need a
tutor. Students who have to understand in order
to do may like the development in all or part.
For Senior
High School Mathematics & Calculus
5
wordy Logic
Chapters
4 curious Algebra
Chapters
Words before & besides symbols. A Key Algebra
forward & backwards Chapter
First Calculus
Preview (1st intro)
Four Calculus
Chapters
(2nd intro)
Intro to Complex
Numbers (long)
Intro to Mathematical
Induction (romantic & wordy at first)
Tutors & Instructors:
These lessons introduce skills differently Would you
recommend them?
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More Topics
1. Decimal
Arithmetic Reference!
2. Integers
- Intro to Signed No.s
3. Fractions
- fully explained.
4. Fractions
with Units
5. Number
Theory,
6. Solving
Linear Equations
7 Formulas
for- & backwards -
8. Proportionality,
Back- & For-wards.
9. Logic
Chapters:
10. Euclidean-Geometry
11. Slopes
& Equations of Straight Lines. (Take
I. See take II below)
12. Why
Study Slopes.
13. Maps,
Plans, Similarity & Trig,
(Take II included here)
14. Quadratics:
Starter lessons
15. Polynomials:
Starter lessons
16 Why
Factor Polynomials:
17 Functions
- Forwards & Backwards.
18. Exponents,
Radicals & logs.
19. Complex
Numbers before trig (new advance/ starter lesson)
20. DC
Electric
Circuits Etc
21. Real
Analysis
22. The
Olde Complex No, Trig
& Vector Section.
23. More
Calculus Stuff
- written after Volumes 2 and 3.
Level I Material: New Stuff
Time and Date Matters
Level I Arithmetic.
Money Matters
Measurement Matters
Matters of Chance (Risk Control)
Logic
Chapters
(leave what's not clear in Level I to Level II)
Using/Making Maps and Plans.
(A variant of
Maps,
Plans, Similarity & Trig, to
appear here).
For Instructors
-
Education
Essays
(opinions,
possibilities, references)
- Free
Advice and Directions for teaching primary & high school maths
will be given in online meeting place with voice &
whiteboard.
- Math & Logic How-TOs
1. Arithmetic
2. Algebra
3. More Algebra
4. Beginner Geometry
5. More Geometry
6. Calculus
7. Show Work or Logic
These may be too dense for students. Offering ideas to change
education makes this site different. Nothing
ventured, nothing gained. Site material is
mathematically correct, and where not, please report
errors. The two level program POMME in the site
entrance implies multiple paths for instruction. Supporting
those paths in turn implies a clear destination for
site development and perhaps a new name.
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