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 10 Describing and Changing Calculations Next: [Chapter 11. Why Shorthand.] Previous: [Postscript  What is a Variable.] [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 10
Describing & 

The computation of the area of the rectangle can be rewritten with shorthand notation as follows. To introduce shorthand notation, we say the area A of a rectangle is given by its width W times its length L. Here, we use A as shorthand for the area of a rectangle, L as shorthand for its length and W as shorthand for its width. The formula (recipe) for calculating the area A of a rectangle can be written more briefly as


Shorthand notation provides a code for the description of calculations. Formula decoding is required. The shorthand formula A = L ·W is more compact (takes less room) than the wordonly description. This formula is meaningless for us when the role of the letters in this shorthand description is not explained. To understand and to use the shorthand description or formula, you need information. You need to know or find what numbers or quantities the symbols mean or represent. In the above formula, L stood for the length of a rectangle. This has to be said to you or you have to ask. To anyone without this information, the formula remains mysterious.
Talking about and describing computations almost gives us the power to do them. In the area calculation, the area A is obtained from the recipe A = L×W provided the length L and W are given or can be found. Without this information, we can describe or understand a calculation but not use it. The above rectangle example reminds us of the following:
 We can talk about quantities or numbers without doing any arithmetic. We can speak about numbers and quantities even if we have not measured them or do not know their values exactly.
 We can describe calculations without performing them. This description can be done with words alone (throw out the letters) or with mathematical shorthand notation, as convenient.
1.2 Triangles
In words, the area of a triangle is given by one half the length of a base of the triangle multiplied by the height of the triangle. This formula can be justified but at this moment we will not worry about why it holds. We may also write more briefly


We have used single letters in this shorthand description of the calculation. Any mark or squiggle or symbol you can draw and name can serve as shorthand for some number or quantity.
Perhaps, we should use A_{triangle} or another symbol, since we have already used the letter A in the previous rectangle example. Alternatively, we adopt the following rule: while you are reading this triangle example, we use the letter A here as shorthand for the area of the triangle only. More will be said on using and reusing (recycling) shorthand symbols (for example, letters) and the roles they take. Think of them as actors which can perform many parts. They may take only one role in any scene, except for stories and scenes involving identical twins or cases of mistaken identities.
1.3 Circles
The symbol for the Greek letter called Pi is p. In words, the area of a circle is given by the number p times the square of the circle's radius. The square of a number or quantity refers to the number or quantity times itself.
^{10} Geometrically, the numerical value of 5squared is the number of unit squares in a square whose sides are of length 5 units. Similarly, the value of 5cubed is the number of unit cubes in the cube with edges of length 5 units. The ancients thought of numbers in geometric terms involving lengths, areas and volumes, and not in terms of decimal notation.
The square^{10} of 5 for instance is 5^{2} = 5 ×5 = 25. We can also more briefly write

To rewrite or encode this formula in shorthand form, we will first describe the code. Let A be shorthand for the area of a circle.
^{1}Here we must forget the previous meanings and roles of the letter A as the area of a rectangle or the area of a triangle.
Let r be our shorthand for the radius of the same circle.
Then the previous wordonly formula for the area of a circle is written A = p·r ·r or as
A = pr^{2}
In the latter expression, the multiplication signs have been left out (omitted) and r^{2} is shorthand for r·r = the radius r multiplied by itself. The shorthand form of the formula, namely A = pr^{2}, takes up less space than the wordonly form: the area of a circle is given by the number p times the square of the radius of the circle. Here one must ask which is the easiest to understand, the above shorthand or the justgiven wordonly form?2 Changing Calculations
The compact description of formulas using shorthand notation is useful for changing the way calculations are done. Note that when two calculations give the same result, one can be done or written instead of the other. This is the replacement principle. The rules of algebra (more precisely rules which say when two different calculations give the same result) tell us when one calculation can be replaced by another. These rules, to be seen later, are also stated or described with shorthand notation.
2.1 First Box Volume Formula
The volume of a box is given by the height times the width times the length of the box in question. More precisely,

To begin our next line of reasoning, we will group the multiplication as follows.

In shorthand notation, the volume V of a box is given by

2.2 Second Formula
The base of the box is a rectangle with area A = L·W. This gives
V = H ·A 
The symbol A and the product L ·W both represent the area of a rectangle. Here A gives the result of computing the product L ·W. The product tells us the value of A. So in describing the volume calculation, we can replace the symbol A by the product W·L, or viceversa, as convenient.
2.3 Back to the First Formula
Our second and alternate formula for the box volume is V = H·A where A represents the base area. Suppose you met someone who accepted this alternate formula but who doubted our original formula for the volume. What can we do to convince him or her that our original formula says how to compute the volume as well? The following words may help.
To convince the person, we first recall and try to use the base area formula A = L ·W. Let's hope this is accepted. Now if some one gives us the width and length of the base, we can calculate from the rectangle area formula A = (L ·W) and then compute V using the equality V = H·A . This suggests that the original calculation V = H·(L·W) for the volume of the box because the single symbol A and the computation L·W both represent and both can be viewed as shorthand for the same quantity, namely the area of the base. So the symbol A and expression W ·L can each replace the other, whether or not the values of A, L and W are known or not.
In closing, this suggests, we can go back and forth between these two ways of computing the volume of the box. We can use whatever is the most convenient  requires the least amount of work.
3 To Find A Rectangle's Dimensions
The rectangle area formula is easy to compute if you are given the width W and the length L. But can we use the area formula A = L·W to find the width W when the area A and length L are given? The rectangle area formula says
A = L·W 
If you multiply W by a nonzero number called it, and then divide by the same number called it you get back your original quantity W. This is a rule or property of arithmetic with whole numbers and fractions, etc. A description of these properties will be given later. So multiplying and then dividing W by the number L gives the same result as doing nothing to W. This suggests the expression [( W·L)/(L)] when calculated, gives you the width W. Now we can write W = [( W·L)/(L)]. The equality sign is used to signal that the expression on either side of it gives the same result. But the expression W ·L whenever computed is the same as the area A. So we replace the computation of W ·L by A with the understanding that A = W ·L always represents this product W ·L.
Now W can be obtained by calculating [(( W ·L))/(L)]. The latter gives the same result as [(A)/(L)]. So we have a new width formula W = [(A)/(L)] for computing W whenever L and A are given. This formula is correct if A = W ·L for every rectangle you meet. Similarly, the length formula L = [(A)/(W)] can be obtained by interchanging the roles of the actors L and W above.
4 Formulas as Potential Calculations
We have discussed or described two recipes or formulas for calculating areas and volumes without doing any arithmetic. Given the heights, lengths and widths involved, we could compute the areas and volumes. That is easy to do by hand. It is also easy or easier to use a calculator to do the arithmetic for us. Think in terms of potential calculations: formulas describe calculations that could be done (or avoided) as needed. We can postpone calculations, unless we need to do them. Note that when you see a formula for the first time, you may need to practice using it.
5 Further Readings
The following books (and others) cover ideas not included above.
 Mathematics Made Simple by A. Sperling and M. Stuart, Doubleday 1981 edition, ISBN 0385174810.
 Algebra, the Easy Way by D. Downing, 1989, Barron's Educational Series, Inc, 250 Wireless Boulevard, Hauppauge, New York 11788. ISBN 0812041941.
 How to Solve Algebra Word Problems by W. A. Nardi, Simon & Shuster Inc, Gulf+Western Building, One Gulf + Western Plaza, New York, NY 10023. ISBN 06680065745.
6 Two Notions of What is a Variable
6.1 With and Without Symbols
Numbers and quantities which may change or vary are said to be variables. This first notion of a variable does not involve or require the presence of shorthand notation (symbols) to represent the number or quantity in question.
But there is a second notion of a variable employed in mathematics. A symbol or letter which represents a number or quantity is also be called a variable if the number or quantity concerned may change or vary, that is if the number or quantity represented is a variable according to the first notion. While a symbol or letter may be called a variable, not all variables are given or represented letters or symbols. We can talk about numbers and quantities without employing a written symbol for each one.
Remark. A change may be required in mathematics texts and dictionaries to recognize both notions and not just the second.
www.whyslopes.com >> Volume 2 Three Skills For Algebra >> Chapter 10 Describing and Changing Calculations Next: [Chapter 11. Why Shorthand.] Previous: [Postscript  What is a Variable.] [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?
Pattern Based Reason
Online 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 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:
2001  Math Forum News Letter 14,
2002  NSDL Scout Report for Mathematics, Engineering, Technology  Volume 1, Number 8
2005  The NSDL Scout Report for Mathematics Engineering and Technology  Volume 4, Number 4
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 trigformulas for dot and
crossproducts.
LinesSlopes [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.