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Mathematics and Logic - Skill and Concept Development

Questions: Will these ends and values motivate? Will smaller & more steps in site lessons and lesson ideas build skills and confidence?
Should we emphasize how ideas & methods depend on earlier ones? Does concept & skill mastery need to be checked to be believed? What is a Variable?
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.

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.

For students of reason in society, science and technology: Pattern Based Reason describes origins, benefits and limits of rule- and pattern-based thought and actions. Not all is certain. We may strive for objectivity, but not reach it. Postscripts offer a story-telling view of learning: [ A ] [ B ] [ C ] [ D ] to suggest how we share theories and practices.

Site's Best Lessons

For Logic

These online chapters may amuse while leading 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. Site Theme: Different entry points may be easier or harder for knowledge mastery.

For Arithmetic

Deciml Place Value - funny ways to read multidigit decimals forwards and backwards in groups of 3 or 6, US-CDN, UK-German and Metric SI style.

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.

For 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.

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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 &
Changing Calculations

Volume 2, Three Skills for Algebra

Talking about Two More Skills

Besides identifying numbers and quantities or talking about them, we can also describe and change calculations which include them or which say how to compute them. Simple calculations can be described with words alone or with shorthand notation. More complicated computations need the shorthand notation. Manipulating, changing or massaging (term learnt from a fellow instructor in 1984) of calculations, is best done with shorthand notation. Any description of a calculation with or without this shorthand notation gives what is called a formula. There is more to mathematics than just doing arithmetic precisely.

Examples of Formulas and Shorthand Notation

1.1  Rectangles

Recall the area of a rectangle is given by its length times its width. We can write this as
Area of a rectangle = its length ×its width
This is a longhand description of the computation of the area of a rectangle. If you give me the values of the length and the width, I can compute the area.

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

A = L×W
or
A = L ·W
The shorthand notation takes less space to write than the word-only description. Read the symbols × and · as times or multiply. The symbols × and · are both shorthand codes for times or multiplication. The dot symbol · is preferably to the times symbol × when the latter could be confused with the letter x. Confusion can occur because the letter x which many people write is too similar to the multiplication symbol ×.

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 word-only 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:

  1. 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.
  2. 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.
We will describe a few more calculations before starting to change them.

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

(Area of triangle) = 1
2 		[ (base length) ·(height of the triangle)]
We may write still more briefly that the area of a triangle is given by
A = 1
2 		[B ·H]
This involves some shorthand notation: the letters A, B and H. When you read or decode this shorthand notation, remember B stands for the length of a base of the triangle. Also remember H stands for the height of the triangle above this base. Lastly, remember A stands for the area of the triangle.

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 Atriangle 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 5-squared is the number of unit squares in a square whose sides are of length 5 units. Similarly, the value of 5-cubed 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 square10 of 5 for instance is 52 = 5 ×5 = 25. We can also more briefly write

Area of a circle = p·radius ·radius
Here we are using a letter, the Greek letter p to stand for and be shorthand for a constant, invariable, unchanging number. The number p is approximated by 3.14159

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.

1Here 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 word-only formula for the area of a circle is written A = p·r ·r or as

A = pr2

In the latter expression, the multiplication signs have been left out (omitted) and r2 is shorthand for r·r = the radius r multiplied by itself. The shorthand form of the formula, namely A = pr2, takes up less space than the word-only 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 just-given word-only 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,

volume = height ·length · width
The order in which the multiplication is performed does not affect the result. That is a property of or rule for arithmetic.

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

volume = height ·( length · width).
Note or remember that calculations within a pair of parentheses ( ) are done before those outside the pair.

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

V = H ·(L ·W)
The product L·W inside the parentheses is done first.

 

2.2  Second Formula

The base of the box is a rectangle with area A = L·W. This gives

V = H ·A
where the letter A is our shorthand for the result of the product L ·W. But expression L ·W equals the area of the base. Therefore, an alternate formula for the volume is V = H·A where A stands for the result of L ·W or the area of the base. The alternate formula can be used if the dimensions L and W are given or measured. The alternate formula can also be used if the base area A is given, but the values of W and L are unknown (or forgotten). But whether unknown, known or forgotten, their product L·W must equal the area 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 vice-versa, 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
From this formula we can directly compute A provided the other two quantities L and W are given or known.

If you multiply W by a non-zero 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.

  1. Mathematics Made Simple by A. Sperling and M. Stuart, Doubleday 1981 edition, ISBN 0-385-17481-0.
  2. Algebra, the Easy Way by D. Downing, 1989, Barron's Educational Series, Inc, 250 Wireless Boulevard, Hauppauge, New York 11788. ISBN 0-8120-4194-1.
  3. 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 0-6680-06574-5.

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]

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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?

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. ...
For Geometry

Maps + Plans Use - Measurement use maps, plans and diagrams drawn to scale.

Euclidean Geometry - See how chains of reason appears in and besides geometric constructions.

Coordinates - Use them not only for locating points in the plane or space.

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 assumes no knowledge and some knowledge of the tangent function in trigonometry.

What is Similarity - another view of using maps, plans and diagrams drawn to scale in the plane and space. May buildings in space are similar by design.

For Calculus

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 - this 1995-96 lesson introduces calculus skills and concepts. It may also may be given to introduce further function maxima and minima both inside and at the ends of closed intervals.

Check Arith. Skills - too many calculus and precalculus students do not have strong arithmetic and computation skills. The exercises here check them while numerically hinting at equivalent computation rules.

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