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Appetizers and Lessons for Mathematics and Reason
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20 pages in French: Algèbre  
 Définition d'une variable
  
La raison basée sur les  règles et modelés

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geometric development

Coordinates, relative or absolute?

Coordinates may be given relative to a choice of unit length and direction (a unit vector) along the coordinate axes of a map. Or, equivalently, coordinates may be given relative to a choice of unit length and a choice of positive direction for the coordinate axes. In both cases, these relative coordinates are ordered pairs of signed numbers.

Coordinates may also be given absolutely relative to a choice of unit length and choice of positive direction for each coordinate axis. For example, a point in a planar map may be determined by absolute coordinates

[+5 cm, -6 cm]

where here the unit of length is the centimeter cm. Implicit here (a first example) is the multiplication of the unit length by a signed number. Implicit here (another first example) is a multiplication of the unit and unit vectors along the coordinate axes by signed numbers. 

Addition of vectors (displacements) in the plane and more specifically collinear vectors in a line, their multiplication by signed numbers (coordinates) and their representation as signed numbers multiplies of a unit vector implies is or consistent with definition of the addition and multiplication of the signed number multipliers alone, apart from their role in representing collinear vectors as multiples of a given unit vector.  

Advanced Theory

  • Unsigned Reals Numbers - use of unsigned decimals as coordinates.
  • Signed Coordinates - Introduction of real numbers by prefixing signs to hitherto unsigned numbers.
  • Plane Vectors - Navigation - use of arrows or vectors in describing piecewise linear paths in the plane; Head-to-tail addition; Associativity of in place head-to-tail addition.
  • Horizontal Vectors & Adding Vector Multiples of unit vectors]. Addition of horizontal, more generally collinear, vectors that represent displacements, AND properties of this addition - commutativity included.
  •  Adding Signed Numbers. The addition of signed numbers A and B is defined so the addition of multiples A and B of a  vector  equals the multiple A+B the vector.

  •  Multiplying Signed Numbers. The product or multiplication of signed numbers is defined so the multiplication by signed number A of a signed number multiple B  of a vector is equals the multiple AB of that vector.

  • Distributive Law for Reals. The sum of collinear vectors given by multiples A and B of a nonzero k should not change if k = c m where m is another vector. Two methods of expressing the sum as multiple of c lead to the distributive property  (A+B)C = AC + BC for signed real numbers.

  • [Real Numbers Axioms] The foregoing considerations imply a superset of the real number axioms assumed in modern mathematics curricula (or derived in a context free manner in pure mathematics.)

  • Modular or Remainder Arithmetic for real numbers- Here is real number generalization of modular or remainder arithmetic for whole numbers. 

Signed Real Numbers
operational development

Real or Signed Numbers may be introduced as Coordinates with signs 
in 2D and then 1D
(Using maps & coordinates to introduce rectangular coordinates)

Unsigned numbers and absolute or relative lengths may be used in ordered pairs to locate points on rectangular map, with say the origin at a bottom-left corner. If that small map is extended or placed on a larger rectangular map in which the origin of the small map is not at the bottom, left corner of the larger one, the small map ordered pair, coordinate system may be extended through the use of negative numbers written in the manner -5 with negative signs in the super-prescript before a whole number 5. Here positive numbers, for example 6, may be employed in place of and used like unsigned numbers.

Issue: The use of signs + and - in the super-prescript position before whole numbers and decimals appeared in my mathematics education, but was not used in practice with unsigned fractions (a/b) to generate signed fractions. With the latter, signs were employed in prefix position and not in prefix, superscript position). The superscript placement of signs, positive and negative, prefixes appears to be optional. It stems from or before the Modern Mathematics curricula of the mid-1950s. 

Arithmetic with Signed (Real) Numbers 

See how to develop an operational command of arithmetic signed numbers (integers, rational, decimals and in general). This may be met or illustrated first with integers, then rational numbers and decimals. Here decimal arithmetic is done exactly or approximately

 

Additive Inverse - Negative of a Number A:

For A = sign(A) length (A) is nonzero, the negative of A is -A = co-sign(A) length(A) = the additive inverse of A. If A is 0, the negative of A is 0 and additive inverse of A is zero. (Saying how to calculate A defines it.)

 

Addition of Signed Numbers:

The sum of two signed (a.k.a real) numbers A and B is given as follows

  • If A and B have the same sign then

    A+B = (common sign)( Magnitude(A) + Magnitude(B))
    = (common sign)(sum of the addend's magnitudes)

    Here the magnitudes are unsigned real numbers given by decimal or fractions etc.
  • If A and B have opposite signs and are equal in magnitude (length) then A and B
    are additive inverses with B = -A and -A = B, and

    A+B = 0
  • If A and B have opposite signs and unequal in magnitude (length) then

    A+ B = (sign of Biggest)( Biggest - Smallest)
    = (sign of longest) (Longest - Shortest)

If sign(A) is + or +1 then co-sign(A) is - or -1. And if sign(A) is - or -1 then co-sign(A) is + or +1.

Subtraction

The rule B - A = B + (-A) allows all subtractions of a signed number A to be expressed (rewritten) as additions involving the negative inverse of A.

Product of Signs:

(+)(+) = +
(+)(-) = -
(-)(+) = -
(-)(-) = +

Multiplication of Signed Numbers:

Next if A and B are signed numbers, their product

AB = (sign A)(sign B) [(length of A)] [(Length of B)]

                     = [(sign A)(sign B)] [(magnitude of A)(magnitude of B)]

Call this the multiply the signs, multiply the lengths for multiplication pf pairs of signed numbers. Take the product AB to be zero if A or B is zero.

 

Remark:  If we define the multiplication of arrows V by signed numbers A  as follows

A times V  = { the vector of length  (length A)(Length V)  with the same direction as V if A is positive and the opposite direction if A is negative. 

The the law of signs for multiplication is consistent with the associativity of this "scalar" or signed real number multiplication of arrows:   

A times (B V) = (AB) times V

whenever A and B are signed numbers and V is an arrows.   This property illuminates the law of signs and provides a geometric motivation for it.  The definition of products of signed numbers could also be presented after the definition of  products of signed numbers and arrows was defined.  There-in lies a longer route, but one that might appeal to some as more natural. 

Leading Questions:   How many times does the first arrow go into the second collinear arrow?

Division of Signed Numbers:

Example First Arrow Second Arrow No of Times
 A   = = =>     = = = = = =>   2 or +2
 B  <= = =  = = = = = =>   -2
 C   = = >  <= = = = = =  -3
 D   <= =    < = = = = = =   3 or +3
 

Leading Question:  How many times does a arrow of length q divide or go into a arrow of length p when (i) they are collinear with the same direction; and (ii) they are collinear with the opposition directions.

Answer for (i)  is  p/q 

Answer for (ii) is - (p/q).

Now each signed number may be identified with an collinear arrow in the positive or negative direction of a coordinate axis.

Example
Revisited
First Arrow
or Number
Second Arrow
or Number
No of Times
 A   = = => (+3)     = = = = = =>  (+6)    2 or +2
 B  <= = =  (-3)  = = = = = =>  (+6)  -2
 C   = = >    (+ 2)  <= = = = = =  (-6)   -3
 D   <= =      (-2)  < = = = = = = (-6)   3 or +3

In general, if A and B are signed numbers, their quotient

A/B = (sign A)(sign B) [(length of A)/(Length of B)]

                         = [(sign A)(sign B)] [(magnitude of A)/(magnitude of B)]

Check that in revisited examples A to B above. That suggest the following rule or convention for the division of signs.

(+)/(+) = (+)(+) = +
(+)/(-) = (+)(-) = -
(-)/(+) = (-)(+) = -
(-)/(-) = (-)(-) = +

 

Multiplicative Inverse (Reciprocal):

If A is nonzero, then the multiplicative inverse (a.k.a reciprocal) of A is

A-1 =

1
A

  =   

sign(a)

.

       1        
length(A)

Division

The rule B/A = B (1/A) allows division involving a signed number A to be expressed (rewritten) as products involving the multiplicative inverse of A.

Comparisons of Signed Numbers:  

Greater in Magnitude Comparison: 

The magnitude (or length) of the signed Numbers  -10, +5, -1, 0, +3 can be compared. We see that -10 has the largest magnitude, namely 10, while 0 has  the small magnitude and that is 0. Here -10 is greater in magnitude than say +5 while 5 is greater than + 3 in magnitude.

Less Than Comparison and the LESS THAN sign < 

Examples:

  • Observe  15 = 10 + 5 or 10 = 15 -5. Here 10 is 5 less than 15. We say 10 is 5 LESS THAN 10, 
    and write 10 < 15 (by 5)

  • Observe  2 = -4 + 6 or -4 = 2 -6.   Here -4 is 6 LESS than -2, and we write -4 < 2  (by 6)

  • Observe -8 = -15 + 7, or -15 = -8  - 7.  So -15 is 7 less than -8, and we write  -15 < -8 (by 7)

The by N part  in parentheses gives the difference. The part  is optional. 

 Definition (Algebraic Form): a first signed number A is less than a second signed number B and we write  A < B by when   A = B - C for some positive number C

More Than Comparison and the MORE THAN sign > 

Examples

  • Observe  15 = 10 + 5. Here 15 is 5 more than 10. We say 15 is 5 more than 10, 
    and write 15 > 10

  • Observe  2 = -4 + 6.  Here 2 is 6 more than -4, and we write 2 > -4 

  • Observe -8 = -15 + 7. So -8 is 7 more than -15, and write -8 > -15

Definition (Algebraic Form):In general a first number A is more than a second number B and we write  A > B when the first number A is given by the second number B  plus a positive number C.  That is,  when A = B + C exceeds B by a positive number C.

Remark (Name Change Suggestion): Instead of calling the sign >,  the greater than sign, teachers and students should call it the more than sign. That may help because primary and junior high school students learn to compare unsigned number by magnitude and not by the more positive idea. The name change is consistent with calling the sign <, the less than sign. See below.  (The webpage Reference: Rename the Greater Than Sign written earlier suggests calling > the more positive sign instead of greater than sign.  However the phrases (i)  -10 is +4 more positive than -14 and (ii) -10 is greater than -14 are  as appealing to my ear as the phrase  -10 is +4 more than -14. 

To Do: Add or link to a lesson explaining how to use the more than or more positive than concept to manipulate inequalities - to obtain properties of inequalities - how they are preserved or reversed under addition of terms and multiplication by signed numbers.

 

 

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

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  <|  (o)   (o)   |> 
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||
   / \_ 

For Senior High School  & Calculus Students

  <| (o)   (o)   |> 
 \     | |      / 
\___ _/

||
 -/[]\- 
||
   / \_ 

Words  to clearly introduce algebra and variables have been missing in course design. For people who cannot do algebra, 
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. 

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 Scienceends, values and methods for work and study,  parent- friendly maths skill development booklets for ages 4-14.

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? 

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  
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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Road Safety Message   Walk on a side walk. If that is not possible, try  not to  walk on a road with your back to the traffic.
Try to see what  trucks, cars, buses or bicycles are coming, so that you may step out of their way.  Put safety first. .

Support for Technical Mathematics from Number Theory to Calculus Prep

A. More Arithmetic a must for algebra etc D. Logic In Mathematics G. Algebra with Take Home Value I. Vectors & Functions
Decimal Lesson - Reference  
Counting & Addition
   (8 lessons)
Comparison to Subtraction
  (9 lessons)
Multiplication
( 11 lessons)
Long Division  (12 lessons)
Decimals and Primes (8 lessons)
-Primes & Composites 
-Primes Factorization
-Greatest Common Divisors & Multiples.
 
-Prime Factorization Aids 
(Learn how to find factors quickly)
-Prime Factorization Examples
 
-Counting & Generating. Factors

-Divisibility Rules and Remainders for Division by 2, 3, 5, 9 and 11.
Integers (12 lessons) Intro to Signed Numbers
Fractions (< 20 lessons)  Essential Skills & Concepts 
Ratios & Fractions (3 lessons):  Similarities & Differences
  
Units in calculations
Fractions  with Units
B.  Basic Algebra
Solving Linear Equations  
- in one unknown. Intro  with stick diagrams?
the normal way
 & with good nttn.
(the nttn that reappears in Gaussian Elimination. |
-in more unknowns: simultaneous equations essentially one unknown. the let algebra do the work view of  word problems.
  - still in more unknowns:  Gaussian Elimination via substitution, by equality or comparison, by operations on equations
C. More Algebra
Words before symbols: See if U like the lengthy chapters 8 to 12 in Volume 2, Three Skills for Algebra  
What is a Variable.  The answer here  is a simple prequel to the modern mathematics viewpoint.
First, every rule & pattern U meet in math, logic & science will be used forwards and backwards.  Get a head start with this theme by reading  Chapter 14 in Three Skills for AlgebraSecond, in the study of Proportionality Relations (3 dense lessons here) finding the proportionality constant gives an initial  backward  use of the proportionality formula.
 Talking about words before symbols and the forward and backward use of formulas gives words to make algebra simpler & clearer.  
If you can not read or write precisely, you will have difficulty in following instructions.  One wordy remedy  is given by chapters 2 to 5  in Three Skills for AlgebraWhere does Logic or a geometric model for reason Appear in Mathematics? The answer lies in  Euclidean-Geometry    In North America, Euclidean Geometry disappeared from high school mathematics as it was too hard. The light treatment here is a possible remedy.
E.  More Geometry
The Pythagorean Theorem. Chapter 17 from  in Three Skills for Algebra uses algebra and geometry   to show why the  Pythagorean equation  for right triangles holds. Its forward and backward use  is common exercise..  At a more theoretical level, the Pythagorean theorem leads the discovery that not all lengths can be  fractional multiples of a unit length. That geometrically implies a  need for and even existence of irrational numbers.
Analytic Geometry:
Common Practices with  Maps and Plans drawn to scale  give coordinate-dependent base  for senior high school development of similarity, trig, vectors and straight lines.   
Complex Numbers: This lesson on
Complex Numbers  draws on Euclidean and Analytic geometry. Sbortcuts simplifiy  trig identities, the cosine law; and   trig formulas for 2D dot- and cross-products. 

F. Logarithms, Exponentials,
Roots & Powers

Logarithms, exponentials, rational and real powers for secondary students. This  complete Operational Viewpoint. (Sufficient for the precalculus forward and backward use of compound growth and decay formulas in biology, physics, chemistry,  personal finance, and calculus. To learn more, if you study calculus,  see chapter 19 of Volume 3, Why Slopes and More.Math

In Volume 2, Three Skills for Algebra, chapters
  1. Geometric Sums Etc,
  2. Notation For Sums,
  3. Personal Money Maths and
  4. Some Finite Mathematics
identify methods useful in money computations, methods needed for calculus. Your teachers or other writer may present the same ideas with greater clarity and detail - A site to do.

H. Polynomial & Quadratics

Analytic Geometry:   -  Slopes and Lines - Take 1.   Take 2 appears in site section Maps and Plans.   Two views are better than one.  I may combine them later.  -In my school days, slopes appeared year after year.   This Why  Slopes calculus preview on graphs of functions y = f(x) explains why.  Enjoy.
Quadratics and Polynomials: Operations on Polynomials:
Meet a light and ultraquick geometric introduction to  multiplication, addition and subtraction of polynomials. Then see how the foregoing combine to permit long division of polynomials.    Compare Fractions  with Units. Enrichment: A Plus:  The Geometric introduction here gives or is almost identical to a justification for column methods in decimal arithmetic. 
Geometric Derivation of the Quadratic Formula  The account here gives a starter lesson for the more algebraically harder geometric-free derivation. If you study physics, chemistry or trigonometry, you will need to know about quadratics, their factorization and the quadratic formula.
Technical Value: The study of polynomials  high school mathematics has technical value as part of the senior high school mathematics preparation for calculus.  This simple account of Why Factor Polynomials   (Chapters 2 to 6 in Volume 3 .Why.Slopes.&.More.Math.) will give a context for the study of polynomials,  their factorization, and sign analysis of functions, all in a way that should improve your algebraic thinking and reasoning skills. 
Vectors in the Plane (2 simple lessons)
- Navigation with vectors or arrows
- Sum of Motions
- more lessons to be added later.
Operations on movement or vectors along the line and in the plane have value in mathematics in defining and implying the properties of real and complex numbers before the assumption of those properties as axioms.  Vectors and their properties appear in physics, its mathematical description and formulation. 
Functions - Forwards & Backwards.  Here is a full technical reference (24 lessons) for use in a calculus or precalculus course as needed. In it, the set viewpoint of functions expression of modern pure mathematics.  comes from the set-based codification and
In the mathematics education reforms of the 1960s in North America, primary and secondary school mathematics were expressed in terms of sets. That expression has now retreated from primary and secondary school texts. But it still lingers on, and can be very useful, a source of clarity and precision, in the situations where it should be retained: Counting with the aid of sets and functions; the description of functions; the high school account of probability theory; and in the discussion or illustration of ideas in logic. 

J. Pre-Calculus Skill Check

Arithmetic Skill Check.  In the calculus courses I taught 1983-89, too many students had weak skills in arithmetic. I would give and carefully correct these exercises to tell students what they needed to review and master.  
-  All the skills and concepts in 
Chapters 1 to 24 or Volume 2, Three Skills for Algebra: Look for those you do not understand and fill the gaps. Do so quickly while balancing this advice with  your other duties.  Good luck.

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