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Logic and Mathematics Skill & Concept Development with How-TOs Fran鏰is: 26 pages
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Logic 5 Chapters Arithmetic 10 Steps Algebra 12 Starter Steps & 5 Advanced Steps
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Ages 15+: Why study slopes Polynomials Quadratics Why factor polynomials Logarithms Functions
What is similarity Euclidean geometry leanly Coordinates + complex no.s Vectors DC Electric Circuits
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What is a variable 5. Fraction Operations by Raising Terms Solving Linear Equations: Take I Take II

Online Volumes: 1 - Elements of Reason, 2 - 3 Skills For Algebra, 3 - Why Slopes and
More Math, 1A - Pattern Based Reason, 1B - Skill Development Principles + Troubles

Welcome: Site content may develop critical thinking, improve reading and writing, and build mathematics skills. See online chapters on on logic and pattern based reason.

Teachers: This December 2011, 5-phase framework offers a context for mathematics & logic instruction. Phases 1 to 3 focus on skills with actual or potential value for adult & daily life. College-oriented phases 5 & 4 focus on calculus & preparation for it. Phases 1 to 4 may also serve trades & professions not dependent on calculus.

Site Review: Math resources ... span ... arithmetic, logic, algebra, calculus, complex numbers, and Euclidean geometry. Lessons and how-tos .... provide a good foundation for high school and college ... mathematics. Read more.

Home < More Algebra < 4 Functions << 1 Geometric Introduction of Function Notation

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Geometric Examples of Functions

Dependent and Independent Variables

A function is a computation or assignment rule that describes how one number depends on others. Computation and assignment rules may be given in several ways using formulas, graphs, sets of ordered pairs, tables, arrow diagrams and words, alone and in combination. In the case of redundancy, that is, in the case where different rules may be used, they have to agree.

Area of Rectangles

The area A of a rectangle is given by its width W times its length L. So we write


A = L 路W
This formula shows how to compute area A from the two dimensions L and W of the rectangle. So the value of A depends on or can be computed from the values of L and W. To indicate this dependence of A on the values of L and W we may write

A = A(L,W) - physicist notation.

or

A = F(L, W) - mathematician notation.

where the value of F(L, W) is given by the product L 路W. That is,

F(L, W) = L 路W

We say that A depends on the values of L and W or that A is function F of L and W.

The values of A, L and W vary or change as we consider different rectangles. So the letters A, L and W are placeholders for three variables, namely the area, the length and the width of a rectangle. We will call the letters A, L and W above variables as they are placeholders for numbers or quantities that vary or may vary between rectangles or examples of rectangles.

We may read F(L,W) aloud as F at the ordered pair or point (L,W). But reading aloud is now get awkward. Where it does, reading in silence or a glance may be preferred. See what is practical - what is not too awkward to read aloud.

Numerical Example A: When W = 4 and L =10, we have

A = F(L, W) = L 路W = 4 路10 = 40

The replacement of (L,W) by (4, 10) respectively in the formula for F, that is, in

F(L, W) = L 路W

gives

F(4, 10) = 4 路10 = 40.

Alternatively we may write

F(L, W)| (L,W) = (4,10) = L 路W |(L,W) = (4,10)= 4 路10 = 40.

  • Here we borrow a trick from college mathematics (calculus) we read

    F(L, W)| (L,W) = (4,10)

    as the expression F(L,W) evaluated at the point where (L,W) = (4, 10).
  • We likewise read L 路W| (L,W) = (4,10) as the expression L 路W evaluated at the point where (L,W) = (4, 10).

Numerical Example B: When W = 5 and L = 7, we have

A = F(L, W) = L 路W = 5 路7 = 35

The replacement of (L,W) by (5, 7), a substitution, in the formula for F, that is, in

F(L, W) = L 路W

gives

F(5, 7) = 5 路7 = 35

So F at the point (5,7) has the value 35 or more briefly, F at (5,7) equals 35. Reading aloud or showing how is not always un-awkward.

Remarks:

  • Writing A = F(L, W) says the variable A, its value, depends on those of the two variables L and W; but a second formula
  • F(L, W) = L 路W is needed to describes or specifies how.

Instead of writing two lines (1) and (2) separately, we may indicate both in a single line

A = F(L, W) = L 路W

where three parallel lines = indicate equals by definition.

What is a definition: A definition says what something is. A definition is an answer to the question what is.

Summary: the Area of a rectangle A = F(L,W) where L and W denote the dimensions of the rectangle, say L for length and W for Width, and F(L,W) = L*W. Here (L,W) = (4,5) and F(L,W) = L*W gives F(4,5) = 4*5 = 20. That is evaluation of A = F(L,W) when (L,W) = (4,5) gives A = 4*5 = 20. Here the area A of a rectangle is determined by the values of the ordered pair (L,W). All we have done here is introduce function notation and the function concept into the formula A = L*W. The area A is a function of two variables L and W or the point (L,W) in the L-W plane or first quadrant. (blah, blah, blah, ...) Here A = F(L,W) is a real valued function of two real variables or quantities (L,W).

Perimeter of Rectangles

The perimeter P of the rectangle

depends on the values of W and L as well. So following the physicists, we may write P = G(W,L) to indicate this dependence - that P is also a function of the two numbers W and L. Here we say function to indicate the existence of that dependence. The dependence or function G is given by the formula

G(W,L) = 2W + 2L

and by the formula

G(W,L) = 2(W + L)

Describing a single dependence or function in two different ways is acceptable when and only when the different ways all give the same results.

Area of Triangles

In words, the area A of a triangle is given by one half the length B of a base of the triangle multiplied by the height H of the triangle.

We may write still more briefly that the area of a triangle is given by
A = 1
2 		[B 路H]

To indicate this dependence, we write

A = T(B,H)

where the dependence or function named T is given by

T(B,H) = 1
2 		[B 路H]

If the value of a number or quantity y depends on and is unquely determined by the values of numbers a, b, c and so on  mathematicians will write y = f(a,b,c, ...) to indicate this dependence where the three dots (ellipsis) stand for the other numbers on which y depends. Another letter except y, a, b and c may be used in place of f in the foregoing function notation y = f(a,b,c, ...) for this dependence.

Teachers & Tutors: Site pages offer better or best practices for providing skills - simpler than expected & comprehensive but for exercises. For your charges, your duty is to study them alone or in groups and develop skill building exercises & activities to share. Start now. The effort here is the best I can do. Others are welcome to refine or exceed it. Please do.

Secondary Mathematics for Ages 11+, A Practical Approach for home-tutoring or -schooling, or for schools & colleges with local curriculum control. Study how to include site content - its skill development how-TOs and innovations into present or future lesson plans - some reading required.

Road Safety Messages and Questions: When and why should you face traffic when walking along a road or cycle path? Is it a good idea to hang limbs outside of cars etc? What gives more protection in a crash: a car, motorbike or bicycle? See too, the BBC-Belgium story Texting and Driving - texting & the impossible test - the article links to a gruesome utube video on the subject

The Logic of Injustice: How Texas sent an innocent man to his death - The wrong Carlos. Some judgments are irreversible. Procescution: Where and when prosectors play to win rather than for justice, guilt beyond a reasonable doubt goes unrespected due to prosecutors who putting winning first, those innocence before the law may be convicted. Some procescutors offices in continuing to accuse after a pardon due to reasonable doubt or innocent being shown, may sucessfully oppose compensaton for false convictions by asserting a pardon individual is still under suspicion. Then the pardoned individual or the latter's estate is not compensation for years or decade of improper or false imprisonment, or for execution. Site chapters on Logic
and some in Pattern Based Reason may slowly lead to greater precision in reading, applying and writing laws.

May 2012, Composition Starting: Pre-School and Primary Mathematics - Quantitative Skills, An Intellectual View, Feedback Welcome:

The 8 Most Popular Site Inlinks

20 Times Table - the most popular site page - popular pages - unexpected.
Fractions & Ratios - with lesson on raising terms to introduce & justify times, division & comparison as well addition & subtraction
Parent Center - See below
Volume 1, Elements of Reason - Intro to all site books.
What is a Variable - best for ages 13+
Written work formats for Arithmetic and Algebra - a skill method and standard!
Complex Numbers Visually - best for ages 13+
Natural Logs, Exponentials, Powers, Roots

Division of Labour: This site offers advice and directions with pointers to resources elsewhere, if known, when they help or lessen the need to write more.

Parent Center: Help your child or teen learn:

Parent-friendly Work Booklets for ages 3+ to 13 Use these or others to check or build skills. Other booklets are available but these booklets allow parents unsure of themselves in mathematics to help their children. The selection acquired in Canada is published in the USA. So it has a US orientation. In retrospect, the selection shows parents what to check with the booklets or by other ways, the choice is theirs. But in retrospect, the selection does not cover integral and fractions liquid weights and measures - ask the publishers to correct that! For ages 9 to 12 say, parents may compensate by showing boys and girls how to use weights or mass, and further measures in food preparation. Beyond that children may be shown how to measure and calculate angles, lengths and areas [proportional amounts too] directly or by using maps and plans drawns to scale. Learning how to gather and measure all the ingredients, pots and pans for a dish or a meal, along with cleaning up sets the stage for like activities or experiments in science courses, and in developing organizational skills, gives boys and girls a head start. Good luck. At the other extreme, more comprehensive than light, if your motto is McCainian: drill, drill, drill then Toronto mathematician and actor John Mighton's jump math organization has jump math workbooks for at least grades 3 to 8 for at-home and in-school use - training sessions for teachers available. Jump math has been expanding to cover older students. Jump Math Samples: plus Fractions for Grades 3-4 & Grades 5-6 [Read] Free Resources grades 1 to 8 [unread - likely to be good]. and

Mathematics Skills For Ages 3 to 14 - technical!

Skills with take home value - A few ideas

Basic skills include time-date-calendar Matters; money matters; map, plan and scale diagram matters;counting, measuring and figuring; decision making with logic and likelyhood; being careful and being aware of the domino effect of mistakes; reading and writing with precision.

Is your child able to add, subtract and multiply amounts of money, work with fractions, work with clocks and calendars, work with maps and plans, and measure length, weight-mass and volume? Schools may promote your son or daughter without providing basic skills in reading, writing and arithmetic.

Arithmetic and Number Theory Skills

Algebra Starter Lessons

1 Working With Sets
2 Formula Forward Use - Evaluation
3 Solving Linear Equations - Skip first step with students able to solve 1 eqn in 1 unknown.
4 Computation Rules and Function Notation
5 Real Numbers
6 More Less Greater Than Inequalities and Comparison
7 Axioms Logic and Equivalent Equations
8 Unifying Theme For Algebra
9 Proportionality Backwards and Forwards
10 Examples of Algebraic Reasoning
A Origins of Counting and Figuring Methods
B Real Numbers Extrinsic Development


Site coverage of formuala evaluation format, of computation rules and axioms, and of the forward and backward use of formulas and proportionality relations lessens the amount of natural talent needed to understand and explain algebra.

Geometry - maps plans trigonometry vectors

1 Maps Plans Measurement
2 Euclidean Geometry - Constructions + extras
3 Cartesian and Polar Coordinates
4 Lines and Slopes Take 1
5 What is Similarity
6 Trigonometry first steps
7 Complex Numbers
8 Unit-Circle Trigonometry
9 Lines and Slopes Take 2 with tangent function
10 Intersecting Straight Lines and Transversals
11 Parallel Straight Lines and Transversals
12 Function Translating and Rescaling
13 Vectors
14 Degrees to Radians and Radians to Degrees
15 Arc or Inverse Trigonometric Function

Pre-Teen and young teen mastery of skills and practices which should be common with map-plans-diagrams drawn to scale, contour interpretation included, has actual or potential take-home value for daily- and adult-life in solving routine problems. Elevating some practices to principles, axioms or postualates, provides a base for analytic and Euclidean geometry, an analytic view of similarity, and an efficient mastery of trigonometry and complex numbers. Right triangle trigonometry provide an analytic alternative to solving geometric problems by drawing diagrams to scale.

More Algebra

Natural-Logarithms Exponentials Powers Roots
Five Polynomial Operations
Quadratics Geometrically
Functions
5 Factored Polynomial Sign Analysis Examples
Rewriting algebraic substitution as function substitutions

The first topic leads to a full high school level theory for the forward and backward mastery of growth and decay models and for definition, range and domains of radicals, roots and powers. The next two topics make quadratics and polynomials easier to learn and teach. Site coverage of functions turns vertical and horizontal line rules into computation methods for evaluating functions.

70 Calculus Starter Lessons

Calculus Lessons Elsewhere:

  1. How to Ace Calculus: Street Wise Guide - Mostly Text.

  2. Flash Video for Calculus Phobics

They cover basic topics in ways likely to complement your notes, your textbooks and site material. When Goldilocks trespassed in the house of the three bears, she found three bowls of porridge, two not to her liking, and one just right. Different bears have different tastes. As invited guest here and elsewhere, if one or more explanations is not to liking, try another. It may be better or just right.

Unsolicited Advice

Learning to do and high marks if it comes to easy is often deceptive - light rather than deep. For that reason, students with learning difficulties determined not to let it get in their way may go deeper and farther than those with none. High marks, if the come easy, may be deceptive - provide a too light and not a deep mastery. That could have been your problem in secondary school, one that leads to comprehension shock or difficulties in calculus and more generally in the first year of college. Bon Appetite.

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Home < More Algebra < 4 Functions << 1 Geometric Introduction of Function Notation

[1] [2][3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27]

Logic-Reason for all
Careful Thinking
Chains of Reason
Mathematical Induction
Responsibility
Bodies-of-Knowledge

Arithmetic - Ages 10+
1. Deciml Place Value - fun
2. Decimals for Tutors
3. Prime Factors - quickly
4. Fractions + Ratios
5. Arith with units - science

Geometry
1 Maps + Plans Use
2 Euclidean Geometry
3 Rct +Polr Coordinates
4 Lines-Slopes [I]
5. What is Similarity 		Algebra Starters - the base
1. Better Work Format
2. Solve Linear Eqns
3. Computation Rules
4. Axioms, Item 3 Viewpnt
5. Formulas Backwards 		More Algebra
Logarithms-ax & m/nth roots
Five Polynomial Operations
Quadratics Geometrically
Functions || Vectors too
Arith. Skill Check+Answers 		Calculus Prep/Preview
What is a Variable
Why study slopes
Why factor polynomials
Complex Numbers
Limits + Continuity

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