ALGEBRA 



185 



ALGEBRA 



Most arithmetics write it 7+?= 15. This is 

 algebra, but the authors fail to use good alge- 

 braic expression. The "?" is not good here. 

 The word number or some symbol to stand for 

 the word number, as n, the first letter of the 

 word, is desirable. 



In multiplication the child says 6X4=24; 

 8X9=72. The teacher says, "I multiplied 7 

 by some number and got 56," and writes as she 

 says it: 



7Xnumber=56, or, 

 7Xn =56, 



tin -n asks "What is the number?" The prob- 

 lem appears: 



7Xn=56 

 n= 8 



This is algebra in thought and expression, ap- 

 pearing early in arithmetic. 



The area of a rectangle is 63 ; the length is 9. 

 What is the width? The mathematics of the 

 problem is this: 63 is the product of 9 and 

 some number, and its best expression is, 

 63=9Xio. Again this is algebra in thought and 

 in expression, although found in the middle 

 grades of the elementary school. The expres- 

 sion soon takes this form: 

 9u>=63 

 w= 7 



I need $18 to buy a coat. I have $12. How 

 much more must I get? Such problems come 

 early to the child. It is arithmetic, and it is 

 also algebra, and when given its best expres- 

 sion appears as 12+n=18; when solved it is: 

 12+n=18 

 n= 6 



In all the above illustrations the child meets 

 the algebraic thought in his arithmetic, and 

 should be taught the best expression for it, 

 such as given above, which is the so-called 

 algebraic expression. Indeed, it is the only 



tnatical expression for such conditions. 

 The solution of each of these problems is 

 arithmetical. It is reached directly by logic; 

 it is not reached by the manipulation of an 

 equation, which latter method belongs to nip - 

 bra. The child says in arithmetic, "If 9u>=63, 

 w equals % of 63, or 7, or he writes: 

 9ti>=63 



w=% of 63, or 7. 



In algebra he says, "9u>=63." Then he 

 divides each member of the equation by 

 and finds w=7. He uses the equation as a 



I line. He has 9u>=63 as a balance; 

 knows that if he divides both sides of the 



balance by 9, the quotients form a balance. 

 Thus it is seen that in the solutions of the 

 problem lies the great distinction between 

 arithmetic and algebra. 



Farther on in the grades the student finds 

 such problems as this: 



An agent working at 13% commission earned 

 $117. What was the amount of his sale? 



The mathematics of the problem is this: 



$117=SaleX.13 

 The solution is: 



.13 



.*. Sale=$900 



This is arithmetic or algebra, as you please, in 

 thought and form, but arithmetic in solution. 



Another illustration, from seventh or eighth 

 grade: 



The area of a circle is 850 (square units, feet, 

 yards, etc.). What is its radius? 

 Area=TR 2 



R2- 850 

 3.1416 



.'. R = 



Here we have the best arithmetical thought 

 and form, and so have we excellent algebraic 

 thought and form. Indeed, again we say this 

 thought and form belong not alone to algebra 

 but to arithmetic and to mathematics in gen- 

 eral. 



This has not been recognized because the 

 old "rule method" has been followed through 

 the centuries in arithmetic the telling how to 

 do the problem, setting forth rules and classify- 

 ing all problems under these rules. This 

 method calls for no expression of the relations 

 in a problem, but rather shows and calls for 

 processes. Under it, the student begins by 

 adding or dividing or finding square root or 

 whatever the rule dictates. Under the method 

 suggested above (but not common enough 

 to-day) the student attacks the conditions 

 th.it create his problem, and translates them 

 into mathematical language, which is the lan- 

 guage we have so long called algebraic. With 

 it as his tools, the student uses the equation 

 'teal sentence and the signs and 

 symbols that serve the purpose of placing be- 

 fore the eye the relations that exist in the 

 problem. He does not hesitate to use a 

 letter to stand for a number any more than 

 he does to use +, , X, =, and so forth 



