ALGEBRA 



187 



ALGEBRA 



3. A farmer had 900 bushels of grain and 

 sold 600 bushels at one market and n bushels 

 at another. How many bushels had he left? 



The teacher and children should express for 

 each other algebraically situations they have 

 in mind, and each try to fit situations to the 

 other's expressions. Then each one should tell 

 what situation he had in mind when he set 

 <lo\vn the expression. 



When the student has learned to translate 

 a problem into an equation, he has accom- 

 plish* <1 the first big step in algebra. The sec- 

 ond step is a study of the meaning, use and 

 control of the equation, the algebraic machine. 

 The equation as a machine is a new thought 

 to the student. Indeed,' in arithmetic he has 

 violated with impunity the law of balance in 

 the expression of his problems, and he may do 

 so as long as he keeps awake to the relations 

 in his problem regardless of the form; but in 

 algebra he places himself at the mercy of the 

 equation; having once made it, he surrenders 

 to it to take him where it will. This is all new 

 to the beginning algebra student; indeed, this 

 subject is a veritable fairy land; the magic 

 wand is no greater wonder than this same 

 equation which takes up his burden, releases 

 his mental energy and carries him through to 

 nd if he but manipulates each crank and 

 button skilfully. 



The solution of an equation depends upon 



(1) changing the equation to desirable form 

 to find the value of the unknown quantity, and 



(2) keeping the balance through all changes 

 made in form. 



The Negative Quantity. Another element 

 in algebra new to the student beginning algebra 

 is the negative quantity; the idea that number 

 extends on both sides of zero above and below 

 is indeed new to him. The idea of the negative 

 quantity can be illustrated in various simple 

 ways: 



1. A force acting in opposition to a force which 



ing a desired effect. 



2. A debt. 



8. Money spent when one desires to save. 



4. Being carried west when one desires to 

 travel east. 



6. Game "tug of war." The force exerted by 

 each side Is negative considered from point of 



of opposing side. 

 6. Friction as opposed to moving force. 



combination or so-called addition of 

 positive and negative quantities presents only 

 a simple proh!> 



1 A croup of boys are playing at moving n 

 ! wagon some distance north. Three boyi 

 pull north, one with a force of 12 pound* 



other 16 pounds, another 20 pounds; another 

 pulls south 18 pounds, another pulls south 14 

 pounds. Combine the forces and the result 

 shows +12+ + 16+ +20+- 18+- 14= + 16. The 

 wagon moves with a force of 16 pounds in the 

 desired direction. Many algebras do not use 

 the small sign, and in these the problem would 

 appear 12+164-201814=16. 



2. Four brothers go into partnership to as- 

 sume the debts of all and use the money of 

 all. A has $7,000 and owes $9,000. B has 

 $20,000 and owes nothing. C has no money 

 and is in debt $2,000. D has $30,000 and his 

 debts are $4,000. What is the result of the 

 combination? Using small signs it appears: 

 +7000 + -9000 + +20000 + -2000 + +30000 + 

 -4000, or: 



+ 7000 



9000 

 + 20000 



2000 

 + 30000 



4000 



A negative quantity will cancel or destroy an 

 equal positive quantity and vice versa. 



3. The thermometer was at 12 above zero 

 at 10 A.M. It rose 17, and then fell 9. Indi- 

 cate the changes and result, or combine the 

 forces. + 12+ + 17+~9= + 20. 



