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. modelspictorial, number-line, partition, repeated

subtraction...

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)

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

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?

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?

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

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

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 )

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

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.

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

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

number theory

• which of these do you think are “hard”?• Suppose we want x3 ÷ p to have a remainder of 2 for a

natural number x and a prime number p. For example, 23 ÷ 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?)

number theory

• Why is a number whose digits sum to a multiple of 3, divisible by 3?

• If p is a prime number, then pn has how many prime factors?

• handout- some conjectures about numbers with 2, 3, 4, 5, 6, 7, and odd factors.

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.

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.

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 = 33 * 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

so many primes

• how many primes do you think there are?

• why? how do you know?

so many primes

• write the first couple prime numbers in order.

• multiply them.• then add one.• do you think the result is prime?

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?

so many primes

• More formally:– If the number of primes is not infinite, it is

finite. Assume there are n primes: • The whole list: p1, p2, p3, ... pn-1, pn.

–But p1 p2 p3 ... pn-1 pn + 1 is prime.–This contradicts our assumption. So

there cannot be n primes.• something a mathematician thinks is

beautiful (we're weird).

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.