Download Day 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...
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.