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 >> Work and Study Tips >> S Adding words to algebra Next: [V Reasons and Motivations for Logic and Mathematics.] Previous: [R Why Learn Mathematics Skills.] [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22][23] [24] Adding Words to AlgebraWhile some calculations may be described with words  think of the perimeter calculation example, many calculations can be represented or given by an algebraic expression or formula that is best seen and grasped in a silences in a glance. Formulas like the compound interest or growth formula and like the quadratic formulas fall in this category of being seen and grasped in silence. The use and development of formulas for lengths, areas, volumes, speed, distance, time, simple interest, compound growth and even or odd numbers, introduces students to the shorthand role of letters and symbols in mathematics. A. Naming and Identifying FormulasOutside of algebra, a picture is worth a thousand words. Inside, formulas to awkward to read aloud term by term may also be worth a thousand words. However, both pictures and formulas may be named or identified with words. Outside of algebra, the phrase Mona Lisa identifies a painting of Leonardo da Vinci. Inside of algebra, formulas may be identified by name
Inside algebra, descriptive or identifying phrases also identify formulas.
In these phrases, the words formulas, expression and calculation may be used interchangeable. In dealing with fractions, we may speak of general and/or efficient methods or formulas for adding, subtracting, multiplying and dividing. At the higher level in the mathematical subject of calculus, the phrases product rules, quotient rule, chain rule, integration by parts identify operations with words.
Words to name or identify formulas and operations is a simple way to
expand the role of words in mathematics. B. Talking about Numbers, Amounts and QuantitiesFollowing the appearance of algebra and logic in and apart from mathematics education, the oral element introduce above may expand. The first skill for algebra which I introduced in fall 1983 emphasized that we can talk about numbers, amounts and quantities without doing any arithmetic and even apart from or parallel to the use of letters and symbols to denote them. So we may talk about and describe numbers, amounts and quantities as being known or not, measureable or not, private or not, confidential or not, forgotten or not, constant or not, and varying or not. Some of these descriptive terms are not usually part of mathematics, but their presence suggests how we may understand and explain the concepts of a numbers, amount or quantity being known or not, constant or not, or variable or not. I have written an essay What is a Variable that informally introduces and expands the concept before algebra begins. While pure mathematics and logic introduce technical definitions of what is a variable, those definitions are too complex for people just learning algebra. In the first instance, when a letter that denotes a number, amount or quantity that is unknown, constant or variable then the letter too will be called an unknown, constant or variable, respectively. And in the use of formulas, letters often denote numbers or measure whose value is to be given or found. What I am advocating here is a simpler use of language, a use closer to everyday use. While numbers, amounts and quantities may be described or talked about apart from algebra or the use of letters to denote them or their values, doing so while denoting them by letters or symbols expands the role of words in algebra and so in mathematics. C. Using Formulas etc Forwards and Backwards.A theme that transcend algebra. A Fourth Skill For Algebra. Every formula met in mathematics, accounting, science, technology etc may be used directly and indirectly, that is forwards and backwards. The simple message that the forward and backward use of formulas (direct and indirect use) is part of high school mathematics and beyond names a required skill and allows us to recognize, identify and thus emphasize the most frequent pattern in high school mathematics and beyond. This message needs to be given explicitly and early in secondary mathematics. Otherwise the underlying skill become part of the hidden, or silent and unspoken, agenda in mathematics courses. Teachers: Consider combining the www.purplemath.com a two page lesson on solving literal equtions with the message above and the examples and exercises indicated below. The page banner above was Forward and Backward use of equations but it now reflects the purplemath lesson, Solving Literal Equations. First Site ExampleDirect and Indirect Use of the Rectangle Area Computation Formula Volume 2, Chapter 10, in discussing Direct use of A =WL assumes W and L are given. Indirect use assumes A and one of W and L is given, and leads to the calculation or formulas W = A/L or L = A/W. The explanation of those formulas is a step towards algebraic reasoning  the direct and indirect or forward and backward use of formulas. More Examples: Formulas for perimeters and areas of squares, circles, triangles, rectangles etc can be used forwards and backwards. Finding the value of a proportionality constant k say in an equation y = k x represents an indirect or backwards use of an equation, a prerequisite to further forward and backward use of the equation y = kx. The calculation of parameters a and b in y = ax + b (or y = mx +b) represents another backward use of a formula or equation. Quebec students in secondary III have met the forward and backward use of the Pythogorean equation c^{2}=a^{2}+b^{2} where c is the length of the hypotenuse and the two numbers a and b are the lengths of the other two sides (legs) of a right triangle. To Do: : Post some details and exercises here to further illustrate and emphasize the forward and backward use of common formulas. Chapter 10 before the forward and backward use of a formula goes further in showing how to describe a the calculation of a box V = H(WL) and show how to employ substitution (a new concept for students) to go between this formula and V = HA where A = WL. Details are given in the chapter. The details may be easier to grasp if numerical examples are added to this exposition. Seeing how a box volume formula V = hA and V = h (WL) can be transformed into each other illustrates and may introduce the notion of equivalent expressions. The law applied here is A = WL is a geometric law rather than an algebraic law (like the distributive law). None, the idea that an expression represents a number or quantity and that there may be more than one ways to compute the number or quantity is key to the notion of equivalence. Students thus see how substitution in formulas leads to new formulas, how arithmetic patterns may be used to use formulas directly and indirectly, and how algebraic solutions may be more general or powerful than arithmetic solutions. Algebraic Exercises:
The exercises will be easier after reading the first sections of Chapter 15 and Chapter 14 in Volume 2, Three Skills for Algebra. Remark. In arithmetic to calculus and beyond in mathematics and
the physical sciences, and in the use of implication rules in or from
logic, all rules and formulas will be used forwards and backwards, some
time bothways in the same problem, especially those involving
proportionality constants. In the latter, finding one represents
a backward use of a formula, after which the formula
may used forward or backwards.
use.

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. 