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ALGEBRA FOR BEGINNERS.
Algebra is too often looked upon as a "bugbear" by pupils just beginning the work. If the first few lessons on the subject can be made to appear as a review of arithmetical processes with the substitution of letters for figures, the rest of the work will be easy.
In fact, the pupil will naturally be pleased at each new phase of the work, for he learns to solve problems that are more difficult when limited to figures. He will like the freedom given to his reasoning power, how if x equals this or that, 2x will equal, etc. Algebra has a fascination about it that arithmetic has not, yet everything depends upon the start.
Proceed very slowly at first. Get the few general truths about the signs, the fundamental operations, the parenthesis, etc., well fixed in the pupil's mind, and then let the development be more rapid.
Any hesitancy on the part of the pupil to understand operations should be overcome by frequent substitutions of figures for letters. In so doing, the process is explained.
THE FIRST LESSONS.
5+5. Here we have used what process? (Pupil) Addition.
Suppose we wish to perform the same process with a and b. How would you write it? (Pupil) a + b.
In algebra, we use letters as well as figures in all the processes. We cannot always perform the operations as we might with figures, so many times we have to simply indicate the process.
Thus, a divided by b is indicated as , or a b.
How would you show that m and n are to be added? x and y b and c?
53. Here we have used what process? How would you show that b is to be taken from a c from x?
Suppose we wish to multiply 4 by 8, how would we show the process? (Pupil) 4 × 8. 84's.
If we show that a is multiplied by b we would write ab.
Show that c is multiplied by d. x by y. Who can give us two forms for showing that a is to be divided by b? (Pupil) a ÷ b, %.
In algebra, we use both forms.
To the Teacher:
Give problems like the following to make the pupil more proficient in using letters instead of figures:
1. Mary has a oranges and Susan has b oranges. How would you express the number of oranges both have ?
2. If apples cost b cents a peck, what will 3 pecks cost? How would you express the cost of the 3 pecks? What will c pecks cost?
3. If 9 books cost c dollars, what will 1 book cost? 4. If a is the cost of b dinners, what is one meal worth? etc.
The first letters of the alphabet a, b, c, d, e, ƒ, etc. are used to denote known numbers in algebra, while the last letters, x, y, z, are used for unknown numbers, or numbers which are found by solving problems.
1. John has 468 sheep and Frank has 1391. How many have both?
We know how many John has, so we may desig nate the number by a. Frank's may be expressed by
b. We do not know, without working the problem, how many both have, but we may call the number x.
Then, x = a + b
x= 468 +1391
x = 1859.
2. Mr. Moore has $40 and Mr. Gray has 3 times as much. How much money have both?
NAMES OF PARTS OF QUANTITIES.
There are four parts of numbers that we have to deal with in algebra, (1) the sign, (2) the literal part, (3) the exponent, and (4) the coefficient.
There are but two signs, the plus (+) and the minus (-). The minus sign is never omitted, so when no sign is expressed, the plus sign is understood.
The Literal Part.
The literal part is the part indicated by one or more letters used as factors.
In 3ab, ab is the literal part.
The exponent has been used in arithmetical problems, denoting powers of numbers. It is a small figure written to the right and above any factor to show the power to be taken.
In 9ax2, 2 is the exponent, showing that x is to be raised to the second power.
The coefficient is the figure on the left of the literal part and indicates how many times the literal part is to be taken.
In 4ab, 4 is the coefficient, and it shows that ab is to be taken 4 times.
If no coefficient is expressed, 1 is understood.
The terms of an expression in algebra are the parts separated by the plus or minus signs.
In 4a+3b9c, the terms are 4a, 3b, and 9c.
The literal part of each term gives the name to the term.
In 9a, the a gives the name to the term.
Supose we add 4a, 5a, and 6a.
4a + 5a + 6a = 15a.
Here the terms are similar.
Suppose we add 5a, 3, and 6c.
Here the terms are dissimilar.
The literal parts show whether or not the terms are similar. If they have the same name, they can be added; if they have different names, the process can only be expressed.
1. Only similar numbers can be added or subtracted to make one term.
2. In the addition or subtraction of dissimilar numbers, the operation can only be expressed.
For the Pupil :
1. Can 4 sheep and 8 oranges be added?
2. Can 4 sheep and 8 sheep be added?
3. Can 4a and 3b be added to form one term? What
is the only thing that can be done?
4. What part of a term gives it its name? 5. 5b+3b= ? 6a+3b= ?
4ab2ab ? 3ab - bc = ?
6. Define term, similar term, dissimilar term, coefficient, exponent, sign, literal part.
An algebraic expression is one or more terms taken together as one quantity.
Thus, a, a +c+d, xy, are expressions.
A number or quantity containing but one term is called a monomial.
5a, 2, 3c, 15x are all monomials.
A number containing two or more terms is called a polynomial.
6a4b3c, 16ab9cd, 3c + 4d + 3ƒ + 3c2 + 9x – 4d, are polynomials.
A polynomial containing two terms is called a binomial; three terms, a trinomial.
3ab5cd is a binomial.
POSITIVE AND NEGATIVE NUMBERS.
The signs plus (+) and minus (-) are used in algebra to express opposite meanings or conditions.
I climb 10 feet. If I express the distance I climb as
+ 12 feet,
I gain $4.
12 feet means a descent of 12 feet.
$4 express a loss of $4.
I walk east 3 miles. If + 3 miles means east from a point, - 3 miles means west of the point.
If +10° means above zero, 10° means below zero.
and express opposite conditions or meanings.
If a number is preceded by a plus sign or no sign at all, we call it a positive number.