Search Loci: Convergence:
The bottom line for mathematicians is that the architecture has to be right. In all the mathematics that I did, the essential point was to find the right architecture. It's like building a bridge. Once the main lines of the structure are right, then the details miraculously fit. The problem is the overall design.
"Freeman Dyson: Mathematician, Physicist, and Writer". Interview with Donald J. Albers, The College Mathematics Journal, vol 25, no. 1, January 1994.
Using Historical Problems in the Middle School
Problems from 19th Century American Textbooks
The second group of problems is from old American textbooks.
1. If a barrel of ale will last a family of 6 persons for 2 months, how many persons would drink 9 barrels in a year?
First state this: 1 barrel per 6 people per two months. If you divide 1 by 6 and by 2 you get .0833333...Then you state that nine barrels per X people per 12 months, or 9 divided by 12 equals .75. Then divide .75 by .0833333. By doing this you get 9 persons, which is the answer.
2. If by paying a teacher $3.25 per week a certain school would be kept 11 1/2 weeks, how long might it be kept if she be paid only $2.875 per week? (Question: When would a teacher have been earning somewhere near $3 per week?)
First I multiplied $3.25 which she was paid by 11 1/2 weeks, which gave me $37.75. Then I divided that by $2.875 to see the number of weeks, which was 13.
3. A teacher receives 77 pieces of money for his month’s salary. The number of half dimes (nickels) received is 85 5/7% of the number of dimes; the number of dimes equals 87 1/2% of the number of quarters; the number of quarters equals 40% of the number of halves; and the number of halves equals 55 5/9 % of the number of dollars; what was his salary?
First I converted the percents to fractions.
Nickels = 6/7 dimes
Dimes = 7/8 quarters
Quarters = 2/5 halves
Halves = 5/9 dollars
Total 77 pieces.
I first guessed 19 dollars and did try and check. My answer was way off because I needed more pieces of money. I tried higher numbers of dollars and finally got the correct answer. I guessed 36 dollars and multiplied that by 5/9. I got 20 which was the number of halves. I multiplied 20 by 2/5 and got 8 which was the number of quarters. I multiplied 8 by 7/8 and got 7 which was the number of dimes. Then I multiplied 7 by 6/7 and got 6 which was the number of nickels. I added all of the pieces of money together and got 77. The salary was $49.00. (Note how much higher this was than in the previous problem, where the teacher made somewhere near $12 for a month. Does this have to do with the fact that here the teacher is male, while in the previous problem the teacher is female?)
4. A man bequeathed $900 to three friends; the first must have a certain portion, the second must have twice as much as the first, the third $28 more than the first. How much did each person receive?
X + 2X + (X + 28 ) = 900
4X + 28 = 900
4X = 872
X = $218
Thus: First gets $218
Second gets 2 x 218 or $436
Third gets $218 + $28 = $246
5. If 1 pig is worth $3 and 5 pigs are worth 2 sheep and 5 sheep are worth 2 cows and 10 cows are worth 3 horses, what is the value of 12 horses?
First state the problem simply: 1 pig = $3, 5 pigs = 2 sheep, 5 sheep = 2 cows and 10 cows = 3 horses. 12 horses ?
I then calculated: 1 x 3 = 3; 3 x 5 = 15; 15/2 = 7.5; 7.5 x 5 = 37.5; 37.5/2 = 18.75; 18.75 x 10 = 187.5; 187.5/3 = 62.5; 62.5 x 12 = $750, the answer.