1. New Material: The Four Operations 1. algorithms standard (carrying, borrowing, division alg.)‏ non-standard (European, lattice...)‏ 2. properties (ugh)‏ associative, commutative, distributive... using these to understand algorithms 3. models pictorial, number-line, partition, repeated subtraction...

2. New Material: The Four Operations algorithms, properties, models for now – only on whole numbers later: –integers (negative #'s)‏ –rational numbers (fractions)‏ –real numbers (only a little)‏

3. Carrying and borrowing -out of context Tues 6/12: back to Alphabitia –Exploration 3.1 part 1: 3a, 4a, 5a,part 2: 4a, 6 focus on: –understanding relationships –why/how procedures work not: computational efficiency –do these in Alphabitian # system –don't translate back and forth

4. Children's and Alternative Algorithms Addition: –(2 min warm-up) a couple parts of 3.2 #4 –Expl. 3.3- learn the algorithm –->make up and solve new problems see if algorithm works for larger numbers think about why the algorithm works –-> what's going on?

5. Children's and Alternative Algorithms Subtraction: –(2 min warm-up) a couple parts of 3.4 #4 –Expl. 3.5- learn the algorithm –->make up and solve new problems see if algorithm works for larger numbers think about why the algorithm works –-> what's going on?

6. Algebraic Properties (things you know) Addition: a, b are real numbers –commutativity: a + b = b + a –associativity: (a + b) + c = a + ( b + c) –identity: a + 0 = 0 + a = a »0 is the “additive identity” –additive inverse: for any a, there is a number, -a, so that a + -a = 0 –closure: a + b is a real number

7. Algebraic Properties (things you know) Multiplication: a, b are real numbers –commutativity: a·b = b·a –associativity: (a· b) ·c = a · ( b·c) –distributive prop.: a·(b+c) = a·b + a·c –identity: a· 1= 1· a = a »1 is the “multiplicative identity” –multiplicative inverse: for any a, there is a number, 1/a, so that a·1/a = 1 –zero property: 0·a = 0·a = 0 –closure: a · b is a real number

8. why bother? various algorithms and computational tricks are not magic. –we can figure out / explore why they work it ALL comes down to place value and algebraic properties ★ another (still) useful tool: expanded form »use when you want to work with the digits in each place value Ex: 18 + 93 = (1·10 + 8 ) + ( 9·10 + 3 )

9. European algortihm (Friday 6/15) 1 - why does this algorithm work? 2 - Some Fun: why is a number whose digits sum to a multiple of 3, divisible by 3? Not magic. Also not inaccessible

10. two basic ideas ( about the 'why does it work?' question) many possible steps and orders of steps –start with a big picture 623 -158 = [6*100 +(2 +10)*10 + (3+10)] - [(1+1)*100+(5+1)*10 +8 do what you want to do –break things up, add/subtract, regroup –just say the property that allows you to do this.

11. multiplication (Mon 6/18) Expl. 3.6 pt 2 – warm-up, finding patterns Expl. 3.10 – the standard algorithm –often called the area model Expl. 3.11 – alternative algorithms

12. change of pace (Tues 6/19) handout: The Locker Problem, (4.2) –start table on 4.1, work on 4.2 –return to and complete 4.1 as needed when finished learn: –scaffolding algorithm Expl. 3.17 –lattice alg. for multiplication Expl 3.11 –area model

13. number theory which of these do you think are “hard”? Suppose we want x 3 ÷ p to have a remainder of 2 for a natural number x and a prime number p. For example, 2 3 ÷ 3 has a remainder of 2. Write the general rule for finding x (if it exists) for a given p. Every even number (greater than 2) can be written as the sum of two prime numbers. (True/False and proof?) If an integer n is greater than 2, then the equation a^n + b^n = c^n has no solutions in non-zero integers a, b, and c. (True/False and proof?)

14. number theory Why is a number whose digits sum to a multiple of 3, divisible by 3? If p is a prime number, then p n has how many prime factors? handout- some conjectures about numbers with 2, 3, 4, 5, 6, 7, and odd factors.

15. Thursday 6/21 continue working of factorization handout maybe a presentation by a group or two see website for full list of algorithms and references to 'why it works'-type questions.

16. Using the partition model to understand long division 447 / 3 –--> partition model, draw three sets –--> start filling up each set with largest available place values... how many longs in each set? etc.

17. Prime Factorization Factorizations of 135: –9*15, 27*5, 3*45, 3*3*3*5, etc. –prime factorization: unique up to exponents written in terms of increasing prime factors 45 = 3 3 * 5 (or 3*3*3*5 if you prefer) 6 = 2*3, 15 = 3*5, 14 = 2*7 – Each is of the form: p*q. Understand factors of one, understand them all Treat them the same, study p*q

18. so many primes how many primes do you think there are? why? how do you know?

19. so many primes write the first couple prime numbers in order. multiply them. then add one. do you think the result is prime?

20. so many primes If there are not infinitely many primes, there is a finite number of them. Let's give this number a name: ___ Multiply all ___ of these primes and add one. Is the new number prime? If so, what does that mean?

21. so many primes More formally: –If the number of primes is not infinite, it is finite. Assume there are n primes: The whole list: p 1, p 2, p 3,... p n-1, p n. –But p 1 p 2 p 3... p n-1 p n + 1 is prime. –This contradicts our assumption. So there cannot be n primes. something a mathematician thinks is beautiful (we're weird).

22. red tape Exam 2 handout, same as before: –due Wed. 5 pm, my office RLM 10.110 –no collaboration. –your resources: course materials, me, your brain. Anonymous survey about final. –fill out all possible hours you are available on Fri 7/6, and Sat 7/7. Probably no class on Fri.