Algebra: 9 Talking about Numbers or Quantities (Variables versus Constants)

Chapter 9
Talking about Numbers or Quantities

3 Mathematical Usage of Words

The above examples show how everyday words are used to describe numbers and quantities. Our next task is to say further or more precisely how the words variable, constant, known and unknown are used to both describe and refer to numbers and quantities.

See the postscript: What is a Variable

A. Variables Versus Constants

To say that a number or quantity is variable means that the number or quantity may vary or change. To say that a number or quantity is constant means that its value remains unchanged. For example:

The letter p » 3.14159... stands for or denotes a constant - a value or number which will never change.
The time of day is always changing. So time is varying. It is an example of a number or quantity which is always increasing and therefore variable. When you ask what time it is, you will get an approximate answer.

To complicate matters further, numbers and quantities may change in one period and not in another. The height of house increases slowly as it built, remains constant while it used, and decreases rapidly if it torn down. So this height may be variable in some situations and constant in others. In everyday life and in mathematics, when a number or quantity is called a constant, we expect its value not to change in the situation at hand. Similarly, when a number or quantity is called a variable, we should expect or suspect that its value may change.

More examples: Your height is a variable or it was a variable while you were growing. The speed of a car or a bicycle is an example of a variable (a variable number or quantity that is). The speed of a car can be almost constant. The zero speed of a stationary car or a parked car is constant - in one reference system at least. Note that a number or quantity can be variable in one situation, and constant in another. We can further talk about a previously constant or a previously variable number or quantity.

In summary, the terms constant and variable can be used to talk about and describe numbers and quantities. A constant is a number or quantity whose value is expected not to change - whose value should not change. A variable is a number or quantity whose value does or might change. The use of these terms is flexible and context dependent. What is constant in one situation may be varying or changing in another.⁵

⁵In some algebra texts and in some dictionaries, the term variable means or refers to the letters that appear in formulas. That use of the term variable departs from the use and meaning given above. In my view, the mathematical usage of everyday words should be in the first instance linked and extracted from their ordinary usage. Where the mathematical usage has departed from the everyday usage, we need to ask if that departure is necessary, and whether or not the departure should be corrected. Documenting reasons or possibly causes for such departures could be material for a thesis in linguistics.

B. Known Versus Unknown

Numbers and quantities can be known or unknown. You may know your own height, age and weight, but I don't know your personal measurements. To you these quantities are known. To me they are unknown. Whether they are known or not depends on the company you keep - that is to whom you speak. When you see the instruction find the unknown, you should ask the question: unknown to whom? Note further in solving an equation, the solution of the equation goes from being unknown to being known.

This is a note mainly for people who know how to solve equations.See the following chapter or chapters to learn how.

There is only one number x solving the equation 2x = 10. Before you solve this equation, its solution, the number x is unknown to you. The solution is x = 5. When or as you solve the equation (or see the solution), the number x becomes known.
When you are only speaking about the solution x of the equation 2x = 10, the solution is given by a constant. The letter x stands for the constant, non-changing number 5.
Now in two different problems in which you solve for x, their solutions x are often given by different numbers (constants). Thus the value of the solution x may change as you go from one problem to another. From this perspective, the solution x can be also called a variable.

For the sake of variety in our speech, numbers and quantities are also called parameters. A parameter is another name for a number or a quantity. When we say a number or quantity is a parameter, we have no immediate expectation that the number or quantity in question will be constant nor that it will be variable. The term parameter gives a vague expectation somewhere between constant and variable. We can talk about numbers and quantities in precise and imprecise ways.

Chapter Sections: [ Up ] [ 9 Numbers & Quantities ] [ 9 Everyday Words ] [ 9 Words Math Usage ] [ 9 Precision or Not ] [ 9 Numbers & Quantities ] [ 9 Changing Units ] [ 9 Further Readings ]

Next Section: 9-Talking about Numbers or Quantities, Approximate Knowledge, Precision or not.

Next Topic: What is a Variable:

A. More Arithmetic a must for algebra etc

D. Logic In Mathematics

G. Algebra with Take Home Value

I. Vectors & Functions

Decimal Lesson - Reference
Counting & Addition   (8 lessons)
Comparison to Subtraction (9 lessons)
Multiplication ( 11 lessons)
Long Division (12 lessons)
Decimals and Primes (8 lessons)
-Primes & Composites
-Primes Factorization
-Greatest Common Divisors & Multiples.
-Prime Factorization Aids (Learn how to find factors quickly)
-Prime Factorization Examples
-Counting & Generating. Factors.
-Divisibility Rules and Remainders for Division by 2, 3, 5, 9 and 11.
Integers (12 lessons) Intro to Signed Numbers
Fractions (< 20 lessons)  Essential Skills & Concepts
Ratios & Fractions (3 lessons): Similarities & Differences
Units in calculations Fractions with Units

B. Basic Algebra

Solving Linear Equations
- in one unknown. Intro with stick diagrams?
the normal way & with good nttn. (the nttn that reappears in Gaussian Elimination. |
-in more unknowns: simultaneous equations essentially one unknown. the let algebra do the work view of word problems.
- still in more unknowns: Gaussian Elimination via substitution, by equality or comparison, by operations on equations.

C. More Algebra

Words before symbols: See if U like the lengthy chapters 8 to 12 in Volume 2, Three Skills for Algebra
What is a Variable. The answer here is a simple prequel to the modern mathematics viewpoint.

First, every rule & pattern U meet in math, logic & science will be used forwards and backwards. Get a head start with this theme by reading Chapter 14 in Three Skills for Algebra. Second, in the study of Proportionality Relations (3 dense lessons here) finding the proportionality constant gives an initial backward use of the proportionality formula.

Talking about words before symbols and the forward and backward use of formulas gives words to make algebra simpler & clearer.

If you can not read or write precisely, you will have difficulty in following instructions. One wordy remedy is given by chapters 2 to 5 in Three Skills for Algebra. Where does Logic or a geometric model for reason Appear in Mathematics? The answer lies in Euclidean-Geometry In North America, Euclidean Geometry disappeared from high school mathematics as it was too hard. The light treatment here is a possible remedy.

E. More Geometry

The Pythagorean Theorem. Chapter 17 from in Three Skills for Algebra uses algebra and geometry   to show why the Pythagorean equation for right triangles holds. Its forward and backward use is common exercise.. At a more theoretical level, the Pythagorean theorem leads the discovery that not all lengths can be fractional multiples of a unit length. That geometrically implies a need for and even existence of irrational numbers.
Analytic Geometry: Common Practices with Maps and Plans drawn to scale give coordinate-dependent base for senior high school development of similarity, trig, vectors and straight lines.
Complex Numbers: This lesson on Complex Numbers draws on Euclidean and Analytic geometry. Sbortcuts simplifiy trig identities, the cosine law; and   trig formulas for 2D dot- and cross-products.

F. Logarithms, Exponentials,
Roots & Powers

Logarithms, exponentials, rational and real powers for secondary students. This complete Operational Viewpoint. (Sufficient for the precalculus forward and backward use of compound growth and decay formulas in biology, physics, chemistry, personal finance, and calculus. To learn more, if you study calculus, see chapter 19 of Volume 3, Why Slopes and More.Math

In Volume 2, Three Skills for Algebra, chapters

identify methods useful in money computations, methods needed for calculus. Your teachers or other writer may present the same ideas with greater clarity and detail - A site to do.

H. Polynomial & Quadratics

Analytic Geometry: - Slopes and Lines - Take 1.   Take 2 appears in site section Maps and Plans.   Two views are better than one. I may combine them later. -In my school days, slopes appeared year after year.   This Why Slopes calculus preview on graphs of functions y = f(x) explains why. Enjoy.
Quadratics and Polynomials: - Operations on Polynomials: Meet a light and ultraquick geometric introduction to multiplication, addition and subtraction of polynomials. Then see how the foregoing combine to permit long division of polynomials. Compare Fractions with Units. Enrichment: A Plus: The Geometric introduction here gives or is almost identical to a justification for column methods in decimal arithmetic.
Geometric Derivation of the Quadratic Formula:   The account here gives a starter lesson for the more algebraically harder geometric-free derivation. If you study physics, chemistry or trigonometry, you will need to know about quadratics, their factorization and the quadratic formula.
Technical Value: The study of polynomials high school mathematics has technical value as part of the senior high school mathematics preparation for calculus. This simple account of Why Factor Polynomials   (Chapters 2 to 6 in Volume 3 .Why.Slopes.&.More.Math.) will give a context for the study of polynomials, their factorization, and sign analysis of functions, all in a way that should improve your algebraic thinking and reasoning skills.

Vectors in the Plane (2 simple lessons)
- Navigation with vectors or arrows
- Sum of Motions
- more lessons to be added later.
Operations on movement or vectors along the line and in the plane have value in mathematics in defining and implying the properties of real and complex numbers before the assumption of those properties as axioms. Vectors and their properties appear in physics, its mathematical description and formulation.
- Functions - Forwards & Backwards. Here is a full technical reference (24 lessons) for use in a calculus or precalculus course as needed. In it, the set viewpoint of functions expression of modern pure mathematics. comes from the set-based codification and
In the mathematics education reforms of the 1960s in North America, primary and secondary school mathematics were expressed in terms of sets. That expression has now retreated from primary and secondary school texts. But it still lingers on, and can be very useful, a source of clarity and precision, in the situations where it should be retained: Counting with the aid of sets and functions; the description of functions; the high school account of probability theory; and in the discussion or illustration of ideas in logic.

J. Pre-Calculus Skill Check

- Arithmetic Skill Check. In the calculus courses I taught 1983-89, too many students had weak skills in arithmetic. I would give and carefully correct these exercises to tell students what they needed to review and master.
- All the skills and concepts in Chapters 1 to 24 or Volume 2, Three Skills for Algebra: Look for those you do not understand and fill the gaps. Do so quickly while balancing this advice with your other duties. Good luck.

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Chapter 9 Talking about Numbers or Quantities

3 Mathematical Usage of Words

A. Variables Versus Constants

B. Known Versus Unknown

C. What is a Parameter?

Chapter 9
Talking about Numbers or Quantities