Freshman Mathematics. Hour 1

This lesson is part of the Classical Liberal Arts Academy’s High School Diploma Program. This program provides students with 120 hours of instruction in high school English, Mathematics, Science, History and Foreign Language, fulfilling modern high school diploma requirements in the most efficient way possible.


In Freshman Mathematics, a student must receive 120 hours of instruction in a high school level mathematics course. In Freshman Mathematics, we will complete 120 hours of instruction in Algebra to satisfy this requirement. In this first lesson, we will study the definitions necessary for the study of Algebra and serve the first hour of instruction.

  1. Study the lesson Instruction below.
  2. Study the lesson content for mastery.
  3. Complete the lesson assessment.



In this first hour, we will begin the study of the definitions of terms necessary for the study of Algebra. We will discuss the content of articles 1-11 in Ray’s Primary Elements of Algebra.

Article 1. In Algebra, quantities are represented by letters of the alphabet.

2. Quantity is any thing that is capable of increase or decrease; as, numbers, lines, space, time, etc.

3. Quantity is called magnitude, when considered in an undivided form; as, a quantity of water.

4. Quantity is called multitude, when made up of individual and distinct parts; as, three cents, a quantity composed of three single cents.

5. One of the single parts of which a quantity of multitude is composed, is called the unit of measure; thus, 1 cent is the unit of measure of the quantity 3 cents. The value or measure of any quantity is the number of times it contains its unit of measure.

6. In quantities of magnitude1, where there is no natural unit, it is necessary to fix upon an artificial unit as a standard of measure; then, to find the value of the quantity, we ascertain how many times it contains its unit of measure. Thus, To measure the length of a line, take a certain assumed distance called a foot, and, applying it a certain number of’ times, say 5, it is found that the line is 5 feet long; in this case, 1 foot is the unit of measure.

7. The Numerical Value of a quantity is the number that shows how many times it contains its unit of measure. Thus, the numerical value of a line 5 feet long, is 5. The same quantity may have different numerical values, according to the unit of measure assumed.

8. A Unit is a single thing of an order or kind.

9. Number is an expression denoting a unit, or a collection of units. Numbers are either abstract or concrete.

10. An Abstract Number denotes how many times a unit is to be taken. A Concrete Number denotes the units that are taken. Thus, 4 is an abstract number, denoting merely the number of units taken; while 4 feet is a concrete number, denoting what unit is taken, as well as the number taken. Or, a concrete number is the product of the unit of measure by the corresponding abstract number. Thus, $6 equal $1 multiplied by 6, or $1 taken 6 times.

11. In algebraic computations, letters are considered the representatives of numbers.


As you study this lesson, learn to answer all of the questions below. Submit the answers in the Study Center for grading.

  1. How are quantities represented in Algebra?
  2. What is quantity?
  3. When is a quantity called a magnitude?
  4. When is a quantity called a multitude?
  5. What is the unit of measure?
  6. How does one find the value of a quantity when there is no natural unit?
  7. Define numerical value.
  8. What is a unit?
  9. What is a number?
  10. What does an abstract number denote?
  11. What does a concrete number denote?
  12. What do the letters used in Algebra represent?

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  1. Note that we are learning of multitudes in article 5 and magnitudes in Article 6.