Problem Solving and Exponents: Challenging the Norms

Problem Solving and Exponents: Challenging the Norms Teodora Cox SUNY Fredonia AMTNYS November 14, 2009

Overview • Problem Solving • Polya • Schoenfeld • Exponents • Properties • Engaging Problems

Problem Solving What is … a problem? an exercise? an enigma?

George Polya (1887-1985) • Problem Solving Phases: • Understand the Problem • Devise a Plan • Carry out the Plan • Look Back

II. Devise a Plan • Draw a Picture • Use a Formula • Solve a similar or simpler problem • Make a table • Look for a pattern • Work Backwards • Restate the problem • Guess and Check

Alan Schoenfeld • Framework for Analyzing Problem Solving Behavior • Cognitive Resources • Heuristics • Control • Belief Systems • Mathematical Problem Solving (1985)

Properties of Exponents For all positive integers m andn:am •an = am + n (am) n = am • n (a • b)m = am • bm a–n = 1/an a0 = 1, when a is not 0

Lockers Problem There are 1,000 lockers in a school and they have been numbered from 1 through 1,000. The students decide to try an experiment. Students will walk into the school one at a time. The first student will open all of the locker doors. The second student will close all of the locker doors with even numbers. The third student will change all of the locker doors that are multiples of 3 (change means closing lockers that are open, and opening lockers that are closed.) The fourth student will change the position of all locker doors numbered with multiples of four and so on. After 1,000 students have entered the school, which locker doors will be open, and why?

Crossing the River Problem A man wishes to cross the river with his dog, goat, and (large) cabbage, but the small boat he has access to can take only one of his possessions besides himself. To complicate matters, for obvious reasons, the goat cannot be left in the company of the dog or the cabbage, unless the man is also present. Advise the man how he should proceed.

A Grain of Rice… Long ago in India, there lived a raja who believed that he was wise and fair. But every year he kept nearly all of the people’s rice for himself. Then when famine came, the raja refused to share the rice, and the people went hungry. Then a village girl named Rani devises a clever plan. She does a good deed for the raja, and in return, the raja lets her choose her reward. Rani asks for just one grain of rice, doubled every day for thirty days. Through the surprising power of doubling, one grain of rice grows into more than one billion grains of rice — and Rani teaches the raja a lesson about what it truly means to be wise and fair.

Doubling Pennies • Make an offer to do the dishes. You will charge 1cent the first day, and each day you will charge twice as much as the day before. How much will you earn in 2 weeks? $143.35

Lilies in the Pond • A lily doubles its size every day. If in 30 days it covers the pond how long did it take to cover half the pond? • 29 days

What is the biggest number using three digits? 999 9+9+9 99 x 9 9 x 99 • Number length:369,693,100 decimal digits ~ 10^9 digits

Four 4’s problem Using four 4's and any operations, try to write expressions that have the numbers from 0 to 100 as the answer. http://www.wheels.org/math/44s.html

Four 4’s problem • 0 = 44 − 44 = 4 − 4 + 4 − 4 = 4 + 4 - 4 - 4 • 1 = 44 ÷ 44 = 4 ÷ 4 + 4 − 4 = (44 − 44)! • 2 = 4 ÷ 4 + 4 ÷ 4 • 3 = (4 + 4 + 4) ÷ 4 • 4 = 4 ×(4 − 4) + 4 • 5 = (4 × 4 + 4) ÷ 4 How many of the numbers between 0 and 100 can you write using four 4’s and at least one exponent? e.g. 0 = 4^4 − 4^4 = (4 − 4)^44; 1 = (4^4)/(4^4)

Guess My Number Game • I think of a number from 1 to 1,000,000. • Your task is to figure out my secret number. • You can only ask me questions of the form • “Is your number smaller than n?”, where n is a number. • I will then answer yes or no. • After asking me finitely many questions, you have to guess my number. • If you guess it correctly, you win. Otherwise, you lose. Martin Escardo, 2009

Guess My Number Game Related Questions: • Can you always win? Why or why not? • If so, how fast? You can always win after 20 questions, rather than a 1,000,000 questions in the worst case. Martin Escardo, 2009

Number Guessing Game (applet)http://www.cut-the-knot.org/blue/Cards.shtml

In closing, Take advantage of opportunities to encourage students to ‘see’ exponents frequently: For example, 1) Note how special 16 is: and 2) Riddle: I was born on of the 2 x 5 month and I am years old. Now my age is the product of two consecutive primes. 3) Name the power game: http://www.sporcle.com/games/White/1_10_to_the_1_10

Problem Solving and Exponents: Challenging the Norms Any Questions? Comments? Teodora Cox, SUNY Fredonia Teodora.Cox@Fredonia.edu

Problem Solving and Exponents: Challenging the Norms