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Introduction to Problem Solving
Polya wrote, in 1944, in the introduction to his famous book How to Solve It:
Studying the methods of solving problems we perceive another face of mathematics. Yes, mathematics has two faces; it is the rigorous science of Euclid but it is also something else. Mathematics presented in the Euclidean ways appears as a systematic, deductive science; but mathematics in the making appears as an experimental, inductive science. Both aspects are as old as the science of mathematics itself. But the second aspect is new in one respect; mathematics "in statu nascendi," in the process of being invented, has never before been presented in quite this manner to the student or to the teacher ..., or to the general public.G. Polya
How to Solve It, p. vii
(see bibliography)It is exactly this study, the study of methods of solving problems, that is the focus of this course and what do we need to know/model to help our students develop their own problem solving skills.
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1) Complete the exercises listed below.
NOTE: Your solutions must contain the method (or thinking) that you used to solve each problem.2) Choose one where you felt you had to think differently from the way you usually solve problems. Write a brief paragraph discussing:
a) the method you used to solve itb) in light of either the Polya scheme or the Problem Solving Strategies, the way your method fits in (or does not fit in) with these ideas.
Exercises
1. Which weighs more - a ton of feathers or a ton of coal?
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2. If there are 12 one-cent stamps in a dozen, how many two-cents stamps are there in a dozen?
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3. Scott sets his alarm clock for 10:00 and goes to bed at 9:00. How many hours of sleep does Scott get?
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4. Elephant is to small as ________________ is to _____________________ .
a) large: little
b) hippopotamus: mouse
c) turtle: slow
d) lion: timid
5. Cross out the letter after the letter in the word pardon which is in the same position in the word as it is in the alphabet?
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6. In a different language LIRO CAS means "red tomato." DUM CAS DAN means "big red barn, and XER DAN means "big house." What is the word for barn in this language?
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7. John, Alex and Suzanne live on the same road. John lives 10 miles from Alex. Suzanne lives 2 miles from Alex. How far does John live from Suzanne?
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8. Ten full crates of walnuts weigh 410 lb., while an empty crate weighs 10 lb. Ho much do the walnuts weigh alone?
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9. There are 3 separate, equal-sized boxes, and inside each box there are two separate small boxes, and inside each of the small boxes there are 3 even smaller boxes. How many boxes are there altogether?
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10. Marta is planning a 5000 mile car trip. If she plans to rotate the tires on her car ( 4 and 1 spare) so that each tire gets the same amount of wear, how many miles is each tire driven?
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11. After hearing a lecture on the dangers of smoking, Hugo promised to cut back 2 cigarettes a day, smoking 2 less on Monday than on Sunday, etc. He kept his promise for a week. During the whole week (7 days), he smoked 63 cigarettes. How many did he smoke on the day he heard the lecture?
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12. In how many ways can you add 6 even positive numbers (greater than 0) to have a total of 20. Changes in order are not counted as new solutions!!
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13. How many triangles are in the picture of the right?
14. Assume the earth to be a perfect sphere with circumference approximately 25,000 miles. Tie a rope tightly around the equator. Take a second rope that is 6 feet longer than your rope around the equator. Place this rope in a concentric circle around the equator. How much room is there - enough room for an ant? a mouse? a cat? or yourself? to slip under the second rope?
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15. Play the Fruit Game (http://rook.2020tech.com/cgi-bin/nim), which is an Internet version of the classic problem solving game of NIM. If you have never played this game, watch yourself carefully as you develop a strategy for this game.
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16. in the text, pp. 8 - 10, Ex. 4, 5, 6, 7, 8, 9, 15, 21, 22, 24, 26, 27, 28, 29, 30, 34
100 Problems for Primary People
http://westgresham.gresham.k12.or.us/jcooke/psindex.htmBrain Teasers Contest
http://www.eduplace.com/math/brain/K-12 Math Puzzles from the Math Forum
http://forum.swarthmore.edu/k12/k12puzzles/Mathematical Puzzles
http://www2.eng.cam.ac.uk/~tpl/maths/puzzles.htmlPBS Teacher Resource for Problem Solving in Middle School Math
http://www.pbs.org/teachersource/math/middle_reasoning.shtm
Syllabus | Home | Assignments | Bibliography
