Problem Solving and Problem Solving and Exponents: Challenging Exponents: Challenging

the Normsthe Norms

Teodora CoxTeodora CoxSUNY FredoniaSUNY Fredonia

AMTNYS AMTNYS November 14, 2009November 14, 2009

OverviewOverviewProblem Solving

◦Polya◦Schoenfeld

Exponents◦Properties◦Engaging Problems

Problem SolvingProblem Solving

What is …a problem?

an exercise?

an enigma?

George Polya (1887-1985)George Polya (1887-1985)

Problem Solving Phases:

I. Understand the Problem

II. Devise a Plan

III. Carry out the Plan

IV. Look Back

II. Devise a PlanII. Devise a PlanDraw a PictureUse a FormulaSolve a similar or simpler problemMake a tableLook for a patternWork BackwardsRestate the problemGuess and Check

Alan Schoenfeld Alan Schoenfeld

Framework for Analyzing Problem Solving Framework for Analyzing Problem Solving BehaviorBehavior

•Cognitive Resources

•Heuristics

•Control

•Belief Systems

Mathematical Problem Solving (1985)

Properties of ExponentsProperties of Exponents

For all positive integers m and n: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 ProblemLockers ProblemThere 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 Crossing the River ProblemProblemA 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…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 PenniesDoubling PenniesMake 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 PondLilies in the Pond

What is the biggest number What is the biggest number using three digits?using three digits?9999+9+999 x 99 x 99

Number length: 369,693,100 decimal digits

~ 10^9 digits

Four 4’s problemFour 4’s problem

http://www.wheels.org/math/44s.html

Using four 4's and any operations, try to write expressions that have the numbers from 0 to 100 as the answer.

Four 4’s problemFour 4’s problem0 = 44 − 44 = 4 − 4 + 4 − 4 = 4 + 4 - 4 - 41 = 44 ÷ 44 = 4 ÷ 4 + 4 − 4 = (44 − 44)!2 = 4 ÷ 4 + 4 ÷ 43 = (4 + 4 + 4) ÷ 44 = 4 ×(4 − 4) + 45 = (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 GameGuess My Number Game

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

In closing,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 Problem Solving and Exponents: Challenging Exponents: Challenging

the Normsthe Norms

Any Questions? Any Questions? Comments?Comments?

Teodora Cox, SUNY FredoniaTeodora Cox, SUNY FredoniaTeodora.Cox@Fredonia.eduTeodora.Cox@Fredonia.edu