ALGEBRA 186 



to stand for words Tin- mathematical ex- 

 pression is coming into use in the elementary 

 schools through the earnest endeavor of a 

 few of the best teachers of arithmetic. 

 One of the weakest points in the course of 

 s is the lack of a recognized 

 language of arithmetic such as is common 

 to all mathematics beyond arithmetic. This 

 - one of the big obstacles to the -stu- 

 dent beginning algebra. 



The high school teacher must take cogni- 

 zance of this. The translating into mathe- 

 I language of the relations existing in a 

 problem is new to the beginning student in 

 algebra, and the teacher must guide him slowly 

 and clearly through his own clumsy product 

 into concise, accurate and refined method and 

 language of the science. The steps are as 

 folio 



First, he must learn to find the mathematical 

 relations in the problem he reads; second, he 

 by means of mathematical symbols, set 

 forth those relations in the form of an equa- 

 tion; third, he must learn to use this equation 

 as a machine which he must manipulate prop- 

 orly to solve his problem. This is a new view 

 to him and so vital that if he fails to compre- 

 hend it he must stumble through stubble fields 

 in his alirebra career, while if he gains com- 

 mand of it he will fly as in a finely-constructed 

 machine. The early days or weeks in algebra 

 determine his control of the new method of 

 thinking, and so these early days are vital. 



The simple problems in arithmetic given 

 above make good work for the beginning high 

 school student. Below are many further sug- 

 gestions as to how the student passes on from 

 arithmetic to algebra. 



1. He indicates the perimeter of a room 

 which is 17 ft. by 12 ft., thus: 17+12+17+12, 

 or (2X17) + (2X12) or 2X (17+12). 



2. He indicates the perimeter of a room 20 

 ft. long whose width he does not know, thus: 



W+width+ZO+width 



20+w?+20+w> 



(20+u>) + (20+w>) 



2X(20+w>) 



The suggestion is given by the teacher that 

 he may drop the sign X and he writes 

 2(20+10). Tell him that mathematicians have 

 agreed to drop multiplication signs in such 

 cases as this, and between letters and between 

 a digit and a letter, but expect to repeat it 

 many times, for he has years of background 

 to the contrary. The new form should come 

 gradually, not be imposed suddenly. 



ALGEBRA 



The tea* 1. Show me the area of the 



floor of the first room," and the student writes, 

 "17X12." . 



''Show me the area of floor and ceiling." 

 The student writes, "(17X12) + (17X12), or 

 2(17X12)." 



"Show the area of the second floor." 

 "20Xu> or 2Qw" 



"Area of floor and ceiling." "(20Xu>) + 

 (20Xw) or2X(20Xw) or2(20Xu>) or2(20to)." 



"I paid 45c for melons this morning. I paid 

 c cents apiece. How many did I buy?" 

 "45^-c." 



Teacher tells him the -r- is dropped and the 

 fraction form is used from now on to show 



division. The student writes 



c ' 



"A dealer sold 1,200 tons of coal for which 

 he received d dollars. For what did he sell it 



d 



per ton?" The student writes 



1200' 



"It is m miles from the coal mines to the 

 city of Peoria. I traveled the distance in 16 



hours. At what rate did I travel? Ans. ^. 



16 



"I sold 7000 bushels of corn at n cents per 

 bushel and spent $320. What had I left?" 

 U 7000n32000." 



"I worked a number of years at a salary of 

 $140 per month and my expenses were $117 

 per month. What did I save?" 



(140-117) X12Xn 

 (140 117)X12n 

 (140117)1271 



Perhaps by this time many of the class will 

 write the last form immediately, but do not 

 fear to go back and forth from the algebraic 

 to the arithmetical form. It does much to 

 clarify and give real and lasting meaning to 

 the new form. 



Below are suggestions for making situations 

 that would give rise to certain mathematical 

 expressions : 



n+|=6500 







900 (600+7i) 



The teacher directs: "Give a situation that 

 would be expressed by each of the above." 

 Students will give widely different problems: 



1. A man collected a certain sum of money 

 Aug. 1 and % as much on Aug. 2, and col- 

 lected both days $6500. 



2. A man spent % of his month's salary and 

 had $64 left. 



