Algebraic Structures: An Activities Based Course MATH 276 UWM MATH 276 UWM

The Handshake Problem  There are 32 people in this room.  If everyone shakes hands with everyone else in the room, how many handshakes will there be?

Please write the following:  Name  Phone number  address  School  Subjects/Grade level you teach  Something you would like me to know

What is a Number? How do we use numbers?

 Count  Order  Compare  Measure  Summarize  Locate  Identify  Operate with numbers  Collect numbers; put them into sets  Identify/describe patterns  Follow rules

Win-a-Row  Game for 2 people: one positive, one negative  Each player has 8 numbers. 4 x 4 game board  Decide who goes first.  In turn, write one of your numbers on the game board. A number may be used only once.  Add each row and columns. Write the sums. If more sums are +, + wins.  After you have played a few games, write addition patterns you see.

Solve the following: 2(3x + 5) = x

Integral Domain - pg. 146 (Z, +, ·)  The integral domain of integers is the set: Z = {..., -3, -2, -1, 0, 1, 2, 3, …} together with the operations of ordinary addition and multiplication which satisfy properties.

Principle of Well-Ordering Every non-empty subset of N+ contains a smallest element.

The Handshake Problem  There are 32 people in this room.  If everyone shakes hands with everyone else in the room, how many handshakes will there be?

Carl Friedrich Gauss

Proof by Induction The first domino falls. If the k domino falls, the k + 1 domino will fall.

If this number is written in standard place value form, what digit appears in the unit’s place? 2009

 Think of a number.  Add seven.  Multiply by two.  Subtract four.  Divide by two.  Subtract the first number you thought of.

When is the Inductive Method of Proof helpful?  You can start with a conjecture.  You want to prove the conjecture for a set with a smallest element.  Inductive proofs are often used for sequence of partial sum patterns k-1 = 2 k - 1

Inductive proof  Prove the conjecture is true for the smallest element in the set.  Prove: If the conjecture is true for n = k, then it is true for n = k + 1.

What is the smallest number evenly divisible by 1, 2, 3, 4, 5, 6, 7, 8, 9, 10?

Prime Numbers and the Sieve of Eratosthenes

Prime Number  An integer n > 1 is prime if its only positive divisors are 1 and itself.  An integer n > 1 that is not prime is called composite.  A prime factorization of a positive integer n is an expression of the form: n = p 1 · P 2 · p 3 · · · p k

Fundamental Theorem of Arithmetic Every positive integer other than 1 can be factored into prime factors in exactly one way, except possibly for the order of factors.

Divisors and Multiples a|b

True or False?  3|9  12|6  0|5  11|11  If a|b and b|c, then a|c  5|6!  11|6!  If c|a and c|b, then c|(a + b)  If c|a and c|b, then c|(a - b)  8|(8! + 1)  6|(6! - 3!)

 You and your sister go to a carnival that has both a large and a small Ferris wheel.  You begin your ride on the large one at the same time your sister begins to ride the small one. Determine the number of seconds that will pass before you and your sister are both at the bottom again.  A. The large makes one revolution in 60 seconds and the small makes a revolution in 20 seconds.  B. The large makes one revolution in 50 seconds and the small makes a revolution in 30 seconds.  C. The large makes one revolution in 12 seconds and the small makes a revolution in 9 seconds. Ferris Wheel Problem

Factor Patterns  5! = = 120  6! = = 720  10! = = 33,628,800  Note that 5! And 6! both end in one zero, and 10! ends in two zeros. Without computing 50!, determine the number of zeros in which 50! ends. (50! ≠ 5! 10!)

Measuring with Index Cards  Draw as many segments as possible, each with a different length, measuring 1 inch, 2 inches, 3 inches,... up to 10 inches using: a.only a 3 x 5 inch index card b.only a 4 x 6 inch index card.

The Division Algorithm Pg. 154  b = aq + r 0 < r < a  aq < b < a(q + 1)

Euclidean Algorithm  Use the Euclidean algorithm to find gcd (15,70)  Use the Euclidean Algorithm to find gcd (276,588)

Factor Patterns  5! = = 120  6! = = 720  10! = = 33,628,800  Note that 5! And 6! both end in one zero, and 10! ends in two zeros. Without computing 25!, determine the number of zeros in which 25! ends. (25! ≠ 5! 5!)

Summary  Algebraic Structures  Sets of Numbers, Integers, N+ (Natural Numbers)  Properties of Integers,  Real Number System (R, +, )  Definitions  Inductive Proof  Prime Numbers; Composite Numbers  Fundamental Theorem of Arithmetic  a|b and properties  Divisors (factors), Find gcd ( )  Multiples, Find lcm ( )  The product ab = gcd (a,b) lcm (a,b)  Division Algorithm  Euclidean Algorithm

Presentations  Teach Euclidean Algorithm  Prove the square root of 2 is an irrational number.  7.12, pg. 152  7.13, pg. 152 with proof  Option (7.3, 7.4, 7.5 Tower of Hanoi)  Teach a lesson: The Locker Problem, Crossing the River, Cuisenaire Trains  Each a lesson on Abundant, Deficient, Perfect Numbers

Even and Odd Numbers Write a definition for: even number Write a definition for: odd number

Rational and Irrational Numbers  The need for multiplicative inverses and rational numbers  The need for irrational numbers  Prove √ 2 is an irrational number. (Pg. 159)  Real Numbers - Filling the holes on the number line  Complex Numbers - A solution for x = 0; √-1 = i

Proof by Contradiction or Indirect Proof  You are taking a true-false quiz with 5 questions.  From past experience you know: If the first answer is true, the next one is false. The last answer is always the same as the first answer. You are positive the second answer is true. On the assumption that these statements are correct, prove that the last answer is false.

Locker Problem  e6/Locker/index.html e6/Locker/index.html

Mental Math Problems  1) 12 x 15  2) 9 x 15  3) 90 x 14  4) 2 x 18  5) 12 x 9  6) 3 x 36  7) 16 x 14  8) 7 x 25  9) 2 x 5 x 0 x 7  10) 12 x 1 x 11

Using the Graphing Calculator  Rule ( X KEY, Y = )  Table ( TBLSET, TABLE )  Graph ( WINDOW, TRACE, MODE, FORMAT )