Mathematics Concept & Skill Development Lecture Series: Webvideo consolidation of site lessons and lesson ideas in preparation. Price to be determined. Bright Students: Top universities want you. While many have high fees: many will lower them, many will provide funds, many have more scholarships than students. Postage is cheap. Apply and ask how much help is available. Caution: some programs are rewarding. Others lead nowhere. After acceptance, it may be easy or not to switch. 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. Early High School Arithmetic
Deciml Place Value  funny ways to read multidigit decimals forwards and
backwards in groups of 3 or 6. 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? Early High School GeometryMaps + Plans Use  Measurement use maps, plans and diagrams drawn to scale.  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 humanmade 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 
www.whyslopes.com >> Volume 2 Three Skills For Algebra >> Chapter 4 Longer Chains of Reason Next: [Chapter 5 IslandsandDivisionsofKnowledge.] Previous: [Chapter 3 Chains of Reason.] [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] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42] Chapter 4, Longer Chains of Reason
This chapter explains one version of inductive reason: the recursive or repetitive approach to putting oneway implication rules together, one after another. This chapter ends with a description of the principle of mathematical induction – another method for obtaining conclusions used only in mathematical arguments or computations. There is more to mathematics than just doing arithmetic. Recall that rules, which say that when a first situation occurs so should a second, are called implication rules. Implication rules can be linked together, one after another. A ladderbased story illustrates the underlying idea. It is called induction. This story leads to the notion called mathematical induction, a method of reason or logic used in mathematics after arithmetic to get conclusions (or climb ladders). The method is described first with words, a simple story, and then with some shorthand notation. Romeo and JulietImagine a hero, Romeo, riding a horse towards a tall building (a castle). There is a ladder up the side of the building leading to the room where Juliet lives. The bottom step of the ladder is two meters or more (several feet or more) away from the ground. The ladder is not broken. It is in good condition. A person getting to each step of the ladder can climb to the next. Question: Can an ablebodied individual, Romeo, reach Juliet via the ladder? The answer is yesprovided Romeo can get to the first or bottommost step of the ladder. It is nootherwise. The main logicrelated ideas in this brief story are as follows.
This situation implies we (or Romeo) can reach each step of the ladder. Note that the long ladder may have a finite number of steps, for example 183. Then we (or Romeo) can with enough time and patience, reach the last one, or any step in between. On the other hand, we can imagine a ladder could have an infinite number of steps. For each step we take, a next is possible. For instance, the whole numbers we use for counting do not stop. Each whole number is followed by another — just add 1. Now suppose or imagine we have a sequence of steps, a ladder, which goes on and on without stopping. Then with enough time and patience, we can reach anyone you mention. An example is met in counting. We can begin counting with the number 1, then 2, then 3 and so on. When we begin to count, we may have only a finite number of objects to count. With a long enough life, and enough patience, the count will end. But if we count minutes there will always be one more to count. This minute count will never end. More precisely, each of us counters may end, but the counting of minutes in principle can continue. That is, this minute count can reach any large number you specify in advance with or without you. In principle all minutes after the beginning of the count will be met and counted. To rephrase the above, on a ladder (or road) with finitely or infinitely many steps, the first step needs to be reachable. And from each step, the next step needs to be reachable. When this occurs, any whole number of steps along the road or ladder in question is reachable. CAUTION. The conclusion that all steps can be climbed or reached does not follow from the principle of mathematical induction if the ladder is broken, or if the first step is not reachable
Check for these nasty situations when you want to use this principle to get a conclusion. Reading GuideThe principle of mathematical induction stated below describes the above ladder idea in the algebraic shorthand notation favored in mathematics. The last part of this chapter will not make sense to you if you are not familiar with this shorthand notation. If this is the case, you may skip this description of mathematical induction. Mathematical InductionThe principle of mathematical induction stated below describes the Romeo and Juliet ladder idea in the algebraic shorthand notation favored in mathematics. The last part of this chapter will not make sense to you if you are not familiar with this shorthand notation. If this is the case, you may skip this description of mathematical induction.
Suppose or imagine for each whole number n, there is a situation A_{n}. This gives a step on the ladder. Now the next whole number after a whole number n is given by adding 1, that is n+1. So the next step after A_{n}. is written as A_{n+1}. The principle of mathematical induction says the following: If
then all the situations A_{n}. (where n is a whole number) occur. The word occurs can be replaced by the expression can be reached.The principle of mathematical induction is quite simple. It requires the following: (1) there is a ladder; (2) on the ladder, from each step we can reach the next; and (3) the first step is reachable. When these three requirements are met, the principle of mathematical induction says: all the steps can be climbed or reached. That is all there is to this inductive principle. Question. What can be said about the reachability of A_{n}where n > 4, if we find a ladder for which requirements (1) and (2) are met, and we somehow know A_{4}. is reachable? Hint: Imagine a ladder where the first three steps are broken, but the fourth is somehow climbable. Is the ladder climbable? Selby A, Volume 1A, Pattern Based Reason, 1996. www.whyslopes.com >> Volume 2 Three Skills For Algebra >> Chapter 4 Longer Chains of Reason Next: [Chapter 5 IslandsandDivisionsofKnowledge.] Previous: [Chapter 3 Chains of Reason.] [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] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42] 
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 headtotail
addition of arrows in the plane. Click and drag complex numbers A and B to change their locations.
Pattern Based ReasonOnline Volume 1A, Pattern Based Reason, describes origins, benefits and limits of rule and patternbased 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 storytelling view of learning: [ A ] [ B ] [ C ] [ D ] to suggest how we share theory and practice in many fields of knowledge. Site Reviews1996  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; patternbased 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
crossproducts, 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 howtos 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. 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. 