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Mathematics and Logic - Skill and Concept Development

with lessons and lesson ideas at many levels. If one site element is not to your liking, try another. Each one is different.

30 pages en Francais || Parents - Help Your Child or Teen Learn
Online Volumes: 1 Elements of Reason || 2 Three Skills For Algebra || 3 Why Slopes Light Calculus Preview or Intro plus Hard Calculus Proofs, decimal-based.
More Lessons &Lesson Ideas: Arithmetic & No. Theory || Time & Date Matters || Algebra Starter Lessons || Geometry - maps, plans, diagrams, complex numbers, trig., & vectors || More Algebra || More Calculus || DC Electric Circuits || 1995-2011 Site Title: Appetizers and Lessons for Mathematics and Reason

Mathematics Concept & Skill Development Lecture Series: Webvideo consolidation of site lessons and lesson ideas in preparation. Price to be determined.

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Are you a careful reader, writer and thinker? Five logic chapters lead to greater precision and comprehension in reading and writing at home, in school, at work and in mathematics.
- 1 versus 2-way implication rules - A different starting point - Writing or introducting the 1-way implication rule IF B THEN A as A IF B may emphasize the difference between it or the latter, and the 2-way implication A IF and ONLY IF B.
- Deductive Chains of Reason - See which implications can and cannot be used together to arrive at more implications or conclusions,
- Mathematical Induction - a light romantic view that becomes serious.
- Responsibility Arguments - his, hers or no one's
- Islands and Divisions of Knowledge - a model for many arts and disciplines including mathematics course design: Different entry points may make learning and teaching easier. Are you ready for them?

Early High School Arithmetic

Deciml Place Value - funny ways to read multidigit decimals forwards and backwards in groups of 3 or 6.
- Decimals for Tutors - lean how to explain or justify operations. Long division of polynomials is easier for student who master long division with decimals.
- Primes Factors - Efficient fraction skills and later studies of polynomials depend on this.
- Fractions + Ratios - See how raising terms to obtain equivalent fractions leads to methods for addition, comparison, subtraction, multiplication and division of fractions.
- Arithmetic with units - Skills of value in daily life and in the further study of rates, proportionality constants and computations in science & technology.

Early High School Algebra

What is a Variable? - this entertaining oral & geometric view may be before and besides more formal definitions - is the view mathematically correct?
- Formula Evaluation - Seeing and showing how to do and record steps or intermediate results of multistep methods allows the steps or results to be seen and checked as done or later; and will improve both marks and skill. The format here allows the domino effects of care and the domino effects of mistakes to be seen. It also emphasizes a proper use of the equal sign.
- Solve Linear Eqns with & then without fractional operations on line segments - meet an visual introduction and learn how to present do and record steps in a way that demonstrate skill; learn how to check answers, set the stage for solving word problems by by learning how to solve systems of equations in essentially one unknown, set the stage for solving triangular and general systems of equations algebraically.
- Function notation for Computation Rules - another way of looking at formulas. Does a computation rule, and any rule equivalent to it, define a function?
- Axioms [some] as equivalent Computation Rule view - another way for understanding and explaining axioms.
- Using Formulas Backwards - Most rules, formulas and relations may be used forwards and backwards. Talking about it should lead everyone to expect a backward use alone or plural, after mastery of forward use. Proportionality relations may be use backward first to find a proportionality constant before being used forwards and backwards to solve a problem.

Early High School Geometry

Maps + Plans Use - Measurement use maps, plans and diagrams drawn to scale.
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- Coordinates - Use them not only for locating points but also for rotating and translating in the plane.
- What is Similarity - another view of using maps, plans and diagrams drawn to scale in the plane and space. Many human-made objects are similar by design.
- 7 Complex Numbers Appetizer. What is or where is the square root of -1. With rectangular and polar coordinates, see how to add, multiply and reflect points or arrows in the plane. The visual or geometric approach here known in various forms since the 1840s, demystifies the square root of -1 and the associated concept of "imaginary" numbers. Here complex number multiplication illustrates rotation and dilation operations in the plane.
- Geometric Notions with Ruler & Compass Constructions :
1 Initial Concepts & Terms
2 Angle, Vertex & Side Correspondence in Triangles
3 Triangle Isometry/Congruence
4 Side Side Side Method
5 Side Angle Side Method
6 Angle Bisection
7 Angle Side Angle Method
8 Isoceles Triangles
9 Line Segment Bisection
10 From point to line, Drop Perpendicular
11 How Side Side Side Fails
12 How Side Angle Side Fails
13 How Angle Side Angle Fails

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www.whyslopes.com >> Volume 3 Why Slopes - A Calculus Intro Etc >> Chapter 21 Arrow Addition Next: [Chapter 22 Complex Numbers.] Previous: [Chapter 20 Vectors and Complex Numbers.]   [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] [28] [29]

Chapter 21. Arrow Addition

The terms arrow and vector will be used interchangeably.

Map Addition Method

In navigation, drawing an arrow on a map from a point A to a point B represents a linear
displacement or movement between them, that is the tail point A and the head point B.

FOOTNOTE: Physically, the arrow from A to C should lie on the path a taut line or string would follow between the two points.

To show a second displacement from the head point B, put the tail of one second arrow at B. The result of these two movements, is a nonlinear movement from a tail point A of the first arrow to the head point, say C, of the second arrow.

The straight arrow joining the tail point A of the first arrow to the head point C of the second is a third arrow called the sum of the first two. It represents a linear movement from the points A to the point C. The foregoing describes the head to tail map addition method for adding two arrows together when the head of one is at the tail of another.

Two of the solid arrows and dotted lines parallel to them in the above diagram form the sides and diagonal of a parallelogram. The three solid arrows form a triangle. Another triangle is formed by the dotted lines and the diagonal arrow of the parallelogram. The rotation of these triangles and the parallelogram will have a deep consequence in the following overlapping discussion of complex, sines and cosines.

Reading Guide

Too much may be said in this chapter. If you get lost in the details, read this chapter lightly or go on to the next chapter. One aim of this chapter was to make fuss about some technical details - gaps in the author's comprehension perhaps.

Parallelogram Addition Method

When two arrows have the same tail points, they determine a parallelogram as well. They can be added by moving without a change in direction, one the arrows to the head of the other. This gives the parallelogram method for adding or summing two arrows with the same tail points/origins.


Observe the presence of a two triangles and a parallelogram.

The dotted lines indicate positions of two of the solid arrows, after a movement to the head of the other without a change of direction. We will describe each displacement as a parallel movement of one along the other. The arrows before and after movement altogether form the sides of a parallelogram. The addition of one arrow to a second is represented by the parallel movement of the tail point of one to the head of the other. The formation of the parallelogram implies that which is added to which is immaterial, the result will be the same. Either way, the solid arrow along the diagonal of the parallelogram gives the (linear) arrow sum of the other two.

Arrow Components


The above diagram shows how the arrow from A to B can be regarded as the parallelogram sum of a horizontal arrow and a vertical arrow. The horizontal and vertical arrows are respectively called the horizontal and vertical components of the initial arrow from A to B. These components depend on the choice of directions for the so-called horizontal and vertical axes. The initial arrow is the map and parallelogram sum of the two component arrows. In the representation of arrows, an arrow can be viewed as the map addition of its vertical component to horizontal component arrows. Here the tail of the vertical component is moved to the head of the horizontal component. The arrow could be also be viewed as the map addition of the horizontal component to the vertical one. Which map addition is shown on a diagram is immaterial.

Component Addition Method

The following diagram shows the map addition of two solid arrows, namely the tail to head addition of the arrow from B to C to the arrow from A to B gives the same result as parallelogram addition of [i] the sum of the vertical components to [ii] the sum of the horizontal components.

The following diagram show the parallelogram addition of two arrows, gives the same result as parallelogram addition of [i] the sum of the vertical components to [ii] the sum of the horizontal components.

This implies the component method for computing the components of the map or parallelogram sum of two arrows. Compute the horizontal and vertical components of the sum by adding the horizontal and vertical components, respectively, of the summands. The sum itself is then give by the map or parallelogram sum of its components.

Coordinates of Points and Arrows

Recall the rectangular and polar coordinates of points in a plane. These coordinates can be measured \rm with ruler and protractor. Measurement of these coordinates in a few examples is enough to convince us of their existence.

In the coordinate plane, each point is represented by a pair of rectangular, alias Cartesian coordinates $(a,b).$ Each point $(a,b)$ in the plane determines a vector with tail is at the origin $(0,0)$ and head at the point $(a,b).$ This vector represents the linear displacement from the origin to the point.

Polar coordinates $[r,\theta]$ define for each point a distance or length $r$ to the origin $0=(0,0)$, and an angle $\theta.$ The latter is a measure of the angle between the positive half of the horizontal axis, the line segment joining the origin to the point. Points in the plane can now be located using rectilinear displacements from the origin, or using counterclockwise angular displacements from the horizontal axis and a distance from the origin. Note the Pythagorean theorem implies \[r=\sqrt{|a|^2+|b|^2}\]

For convenience, we will also represent the point $(a,b)$ in the Cartesian coordinate plane by its polar coordinates $[r,\theta].$ That is, we write $[r,\theta]=(a,b)$ when both determine the same point. Here square brackets are reserved to indicate the polar coordinates of a point while round brackets indicate rectangular or Cartesian coordinates of the same point.

The use of square and round brackets is an alternative to the use of subscripts or the use function notation $h(r,\theta)$ and $g(a,b)$ to respectively represent the points determined by polar and rectangular coordinates.

A point or arrow with tail at the origin be specified by giving rectangular or polar coordinates, or both.

Note the origin $(0,0)=[0,\theta]$ is a special case for polar coordinates. It is located by $r=0.$ The value of the angle $\theta$ has no effect and can be selected arbitrary. The arrow of zero length has no defined direction. Adding it to any arrow yields that arrow.

Coordinates and Components


Each point $(a,b)$ regarded as an arrow or vector, is the sum of two components: the horizontal component is determine by the point $(a,0)$ and the vertical component is determined by the point $(0,b).$

Coordinate Addition Method

The arithmetic sum of the rectangular coordinates $(a_1,b_1)$ and $(a_2,b_2)$ is given by their arithmetic sum $(a_1+a_2,b_1+b_2).$ So we write \[(a_1,b_1)+(a_2,b_2) = (a_1+a_2,b_1+b_2) \] For example, $(2,3)+(5,7)=(2+5,3+7)=(7,10).$

Arrows equal in length to the components of two arrows being added are indicated in the following diagram. The previous conclusions drawn illustrated by this diagram.

The rectangular coordinates $(a_1,b_1)$ and $(a_2,b_2)$ respectively identify two arrows from the origin $0=(0,0)$ of the plane. We will compute the sum of the horizontal and vertical components of these two arrows. The horizontal components are $(a_1,0)$ and $(a_2,0).$ Their sum is $(a_1+a_2,0).$ This gives the horizontal component of the parallelogram sum of the two arrows specified by the rectangular coordinates $(a_1,b_1)$ and $(a_2,b_2).$

Similarly, the vertical components of the two arrows specified by $(a_1,b_1)$ and $(a_2,b_2)$ are $(0,b_1)$ and $(0,b_2).$ Their sum is $(0,b_1+b_2,).$ This gives the vertical component of the parallelogram sum of the two arrows determined by the rectangular coordinates $(a_1,b_1)$ and $(a_2,b_2).$ Finally, the parallelogram sum of the two arrows specified by $(a_1,b_1)$ and $(a_2,b_2)$ is the sum of its components $(a_1+a_2,0)$ and $(0,b_1+b_2,).$ The latter sum yields the arrow associated with the rectangular coordinates $(a_1+a_2,b_1+b_2).$ Thus the component method for addition of arrows agrees with the arithmetic method

\[(a_1,b_1)+(a_2,b_2)=(a_1+a_2,b_1+b_2)\]

Thus the addition of arrows can be represented and done in terms of coordinates - The axes need not be orthogonal.

Real Multiples of Vectors

Multiplication By Positive Numbers or $-1$

Observe the addition of a point $(a,b)$ to itself yields $(a+a,b+b) =(2a,2b).$ The product of $(a,b)=[r,\theta]$ with a positive number $c$ is taken to be the point \[c\cdot (a,b) =(ca,cb)=[rc,\theta]\]

When $c=n$ is a whole number, multiplication by $c=n$ corresponds to adding the point $(a,b)$ to itself $n$ times. This can be shown using the principle of mathematical induction.

More generally, when $c=\frac nm$ is a rational number, multiplication by $c$ correspond to adding the point $(\frac1m a,\frac 1m b)$ to itself $n$ times. Here adding the point $(\frac 1m a,\frac 1mb)$ to itself $m$ times yields the original point $(a,b).$ Finally, multiplication by -1 is assumed to reverse the direction of an arrow and the linear displacement that the arrow represents. This reversal does not change the arrows length. Two reversals presumably yield the original direction. See the next diagram.

Note that integer multiples of the an arrow preserve the slope or direction or angle of the arrow. We further assume

$(-a,-b)=[r,\theta+180^\circ]$ if $(a,b)=[r,\theta].$

In general, the product of a point $(a,b)=[r,\theta]$ in the plane with a real number $c>0$ is $(ca,cb)=[cr,\theta]$ if $c>0$ and $(ca,cb)=(|c|r,\theta+180^\circ) = (-|c|a,-|c|b)$ if $c<0.$


www.whyslopes.com >> Volume 3 Why Slopes - A Calculus Intro Etc >> Chapter 21 Arrow Addition Next: [Chapter 22 Complex Numbers.] Previous: [Chapter 20 Vectors and Complex Numbers.]   [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] [28] [29]

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Road Safety Messages for All: When walking on a road, when is it safer to be on the side allowing one to see oncoming traffic?

Play with this [unsigned] Complex Number Java Applet to visually do complex number arithmetic with polar and Cartesian coordinates and with the head-to-tail addition of arrows in the plane. Click and drag complex numbers A and B to change their locations.

Pattern Based Reason

Online Volume 1A, Pattern Based Reason, describes origins, benefits and limits of rule- and pattern-based reason and decisions in society, science, technology, engineering and mathematics. Not all is certain. We may strive for objectivity, but not reach it. Online postscripts offer a story-telling view of learning: [ A ] [ B ] [ C ] [ D ] to suggest how we share theory and practice in many fields of knowledge.

Site Reviews

1996 - Magellan, the McKinley Internet Directory:

Mathphobics, this site may ease your fears of the subject, perhaps even help you enjoy it. The tone of the little lessons and "appetizers" on math and logic is unintimidating, sometimes funny and very clear. There are a number of different angles offered, and you do not need to follow any linear lesson plan. Just pick and peck. The site also offers some reflections on teaching, so that teachers can not only use the site as part of their lesson, but also learn from it.

2000 - Waterboro Public Library, home schooling section:

CRITICAL THINKING AND LOGIC ... Articles and sections on topics such as how (and why) to learn mathematics in school; pattern-based reason; finding a number; solving linear equations; painless theorem proving; algebra and beyond; and complex numbers, trigonometry, and vectors. Also section on helping your child learn ... . Lots more!

2001 - Math Forum News Letter 14,

... new sections on Complex Numbers and the Distributive Law for Complex Numbers offer a short way to reach and explain: trigonometry, the Pythagorean theorem,trig formulas for dot- and cross-products, the cosine law,a converse to the Pythagorean Theorem

2002 - NSDL Scout Report for Mathematics, Engineering, Technology -- Volume 1, Number 8

Math resources for both students and teachers are given on this site, spanning the general topics of arithmetic, logic, algebra, calculus, complex numbers, and Euclidean geometry. Lessons and how-tos with clear descriptions of many important concepts provide a good foundation for high school and college level mathematics. There are sample problems that can help students prepare for exams, or teachers can make their own assignments based on the problems. Everything presented on the site is not only educational, but interesting as well. There is certainly plenty of material; however, it is somewhat poorly organized. This does not take away from the quality of the information, though.

2005 - The NSDL Scout Report for Mathematics Engineering and Technology -- Volume 4, Number 4

... section Solving Linear Equations ... offers lesson ideas for teaching linear equations in high school or college. The approach uses stick diagrams to solve linear equations because they "provide a concrete or visual context for many of the rules or patterns for solving equations, a context that may develop equation solving skills and confidence." The idea is to build up student confidence in problem solving before presenting any formal algebraic statement of the rule and patterns for solving equations. ...

Senior High School Geometry

- Euclidean Geometry - See how chains of reason appears in and besides geometric constructions.
- Complex Numbers - Learn how rectangular and polar coordinates may be used for adding, multiplying and reflecting points in the plane, in a manner known since the 1840s for representing and demystifying "imaginary" numbers, and in a manner that provides a quicker, mathematically correct, path for defining "circular" trigonometric functions for all angles, not just acute ones, and easily obtaining their properties. Students of vectors in the plane may appreciate the complex number development of trig-formulas for dot- and cross-products.
Lines-Slopes [I] - Take I & take II respectively assume no knowledge and some knowledge of the tangent function in trigonometry.

Calculus Starter Lessons

Why study slopes - this fall 1983 calculus appetizer shone in many classes at the start of calculus. It could also be given after the intro of slopes to introduce function maxima and minima at the ends of closed intervals.
- Why Factor Polynomials - Online Chapter 2 to 7 offer a light introduction function maxima and minima while indicating why we calculate derivatives or slopes to linear and nonlinear curves y =f(x)
- Arithmetic Exercises with hints of algebra. - Answers are given. If there are many differences between your answers and those online, hire a tutor, one has done very well in a full year of calculus to correct your work. You may be worse than you think.


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