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 15. Solving Linear Equations Next: [Chapter 16. Painless Theorem Proving.] Previous: [ Chapter 14. Forward and Backward Use of a Formula.] [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 15. Solving Linear EquationsVolume 2, Three Skills for Algebra Here are some more examples in which we solve equations. Our aim is to become familiar or at ease with handling and manipulating equations. So we look at the algebraic solution of equations containing one or more unknown numbers. 1 One Unknown1.1 First ExampleWhen we let x = 5, we have 2x = 10 and 4x ¹ 15. Suppose now we forgot the value of x which made 2x = 10, could we find the value of x from the equation 2x = 10? The answer is yes. We can solve for the unknown or forgotten value of x as follows:
1.2 Second ExampleProblem: Find the value of x which satisfies the equation 7x+9 = 65. Solution: The aim is to manipulate (or change or massage) the given equation
1.3 Third ExampleProblem: Find the value of x which satisfies 5x+6 = 117. Solution: The aim is to manipulate the given equation
The solution
Here are some more examples in which we solve equations. Our aim is to become familiar or at ease with handling and manipulating equations. So we look at the algebraic solution of equations containing one or more unknown numbers. 2 Algebraic Shorthand SolutionIn a play or movie, the roles are more important than the actors, stars excepted. That is, any role can be played by any actor. But after the cast is selected, each role is usually played by only one actor, and each actor usually plays only one role. Once a play (or scene) is finished, the actors can take roles in another play (or scene). Likewise, in algebra, we have choice in the selection of the shorthand notation in which a problem or its solution is posed. But after the selection, the choice should be fixed at least temporarily. Once the problem and solution have been treated, the shorthand in it can be recycled in another plot. 2.1 Third Example Revisited
The role of x in the third example can be played by any other
letter, for instance y. We will repeat the third example problem
with y in place of x. (This is mathematics ad nauseum.) Problem: Find the value of y which satisfies 5y+6 = 117. (This problem is identical to the previous one, except the shorthand symbol for the forgotten or unknown number is now the letter y instead of the letter x. The solution is identical. It is given or repeated next. Excuse the repetition, but you must see that it is a repetition.) 2.2 An Algebraic PatternEach of the above examples has the form ax+b = c in which the numbers a, b and c are given, and x is initially unknown. In the first example, the roles of a, b and c were played or given by 7, 9 and 65. That gave the equation 7x+9 = 65. In the second example 5x+6 = 117, the number 5 is used in place of a, the number 6 plays the role of b and the number 117 is given by c. General Problem: Find x if ax+b = c. ALGEBRAIC SHORTHAND SOLUTION. We follow the pattern set in the previous examples. First we subtract b from both sides of the equation ax+b = c. This gives
Check: When x = [(cb)/(a)], we see ax+b = a·[(cb)/(a)] = (cb)+b = c as hoped. The recipe
The formula x = [(cb)/(a)] describes and gives a solution to many problems of the form ax+b = c. We can further use this recipe without repeating each time, the reasoning that led to it. EXTRA. The above formula for x can be used to solve the equation axd = c by putting b = d. The equation axd = c can be rewritten as ax+(d) = c since subtraction of d can be replaced by the addition of the number d. Here are some more examples in which we solve equations. Our aim is to become familiar or at ease with handling and manipulating equations. So we look at the algebraic solution of equations containing one or more unknown numbers. 3 Systems with More UnknownsEquations with more than one unknown can be solved if they are manipulated or massaged into a simpler form. For equations with more than two or three unknowns, we can obtain complicated formulas for their solution, but the equations can be solved more efficiently without these formulas. In this case the method of solution is easier to remember than the formula. This contrasts with the expectation that I have (accidentally) built earlier. The shorthand description of calculations that could be done, that is, formulas for obtaining numbers and quantities, are useful tools. But we should not insist on using them all the time. Your own experience or that of others is needed to say when a purely algebraic approach appears best.Problem: Find the unknown or forgotten values of x,y and z when
Check:
4 Simplified ProblemsAdditionMultiplication Method for nontriangular systemsIn the previous problem, we could find the unknowns one at a time. One method for solving equations is to change, massage and manipulate them into a form where we can find the unknowns one at a time. We can do this by adding multiples of equations together.Example: Solve
Solution: Keep the first equation as is, and add it to the second. This gives 7 = 2x+y and
Example: Solve the following system (set) of equations
5 Examples with Three Unknowns
Example: Solve the following equations (a1), (a2) and (a3).
A Solution: First subtract equation (a1) from (a2). This implies
yields equation (b2) below.
Second, add equation (b1) to equation (b3). This gives
The above steps have eliminated z from the last two equations. Third, from equation (c2) subtract two times equation (c3). This implies
Finally, we may change the order of equations. This yields the more suggestive system of equations:
This last step was optional. Now we can do the following.
Exercise: Solve
Note that you can and should check your answer (values for x, y and z) satisfy each equation. If one is not satisfied then there is an error somewhere in your work, the solution or the check. Exercise: Solve
See the difference a "small" change in the problem makes. 6 Useful Arithmetic Rules & Advice6.1 Useful Arithmetic RulesIn the previous examples we have used rules or properties of arithmetic to help us in our algebraic manipulations. A short description of them follows. These rules or properties are all related to the idea that if two expression are supposed to give the same number when computed, then using one of them in place of the other in larger expressions should not change the values given by the larger expressions.
When a, b and c are shorthand symbols or expressions representing real numbers, the following properties are useful in solving equations.
www.whyslopes.com >> Volume 2 Three Skills For Algebra >> Chapter 15. Solving Linear Equations Next: [Chapter 16. Painless Theorem Proving.] Previous: [ Chapter 14. Forward and Backward Use of a Formula.] [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. 