Download Lecture 1

Congruences and equations
Mathematics Unit 8:
A transition to Algebra
Variable/Equality Concept
• What is algebra?
– How would you characterize when a student
moves from studying arithmetic to algebra?
– What are the key underlying concepts of
algebra?
Variable & Equality Concepts
• Consider the following equations - how
does the use of variable and equality differ
in these examples?
1. A = L•W
2. 3x = 21
3. a2 + b2 = c2
4. n • (1/n) = 1
5. y = kx
Variable & Equality Concepts
(continued)
1. A = L•W
– Formula: A, L, and W stand for quantities unknown and variable
2. 3x = 21
– Equation: variable is unknown but has 1 value, solve
3. a2 + b2 = c2
– Identity: variable is argument of function
4. n • (1/n) = 1
– Property: generalizes an arithmetic pattern - identifies instance of
pattern
5. y = kx
– Function: x is argument of function, feel of variability
Conceptions of Variable
• Variable
– a changing number
– a literal number assuming a set value
– a symbol for which one substitutes names for some
objects (name-object distinction)
– symbol for an element of a replacement set
– mathematical symbol, merely a mark on paper, no
concern for what it represents
• Equal: Can you identify three different concepts
of = ?
=
• = as equivalent
• = means a rule
• = means solve
Variable Represents...
•
•
•
•
•
Unknown Number (common student conception)
Points (Geometry): AB = BC
Propositions (Logic): p  q
Function (Analysis):  f dx
Matrix (Linear Algebra): MT

• Vector: v
Variables have many possible definitions,
referents, and symbols.
Conceptions of Algebra: correlates
with uses of variable
• Concept 1: Algebra as Generalized
Arithmetic
– Variable as pattern generalization
– Key instructions - translate and generalize
– Fundamental in Mathematical Modeling
Example
• Generalized Arithmetic
2 • 3 = 3 • 2 or x • y = y • x
• Modeling
T = -0.4Y + 1020
where T = world record time (seconds) in the mile
run and Y = year
Concept 2: Algebra as Procedure
Algorithms for solving class of problems
• Variable as unknown or constant
• Key instructions - simplify and solve
• Didactic Cut - Arithmetic Equation vs.
Algebraic Equation
Example
• Simplify: |x-2| = 5 so
x-2=5
-(x - 2) = 5
Didactic Cut
• Arithmetic - single variable where
operations can be reversed using +,-,•,/.
Algebra Solution:
Arithmetic Solution:
3x + 5 = 11
3x + 5 = 11 (solve in head)
Didactic Cut
• Occurs when two variables introduced
• Since we must “deal” with unknown, we
can’t just reverse operations
• Example: 3x + 5 = 2x – 1
• How about 1.623x + 3.452 = 4.568
Concept 3: Algebra as
Relationships among Quantities
• Variable as argument - represents domain value or
parameter - represents a number on which other
numbers depend
• Key instructions - relate and graph
• Variables vary which is critical distinction from concepts
1 and 2
• Generalization of algebraic patterns, rather than
arithmetic patterns
• Notions of independent and dependent variable exist
• Functions flow immediately
Example
1
• What happens to value of x as
x  ?
Example
• Functions: f(x) = 6x - 8
Example
• y = mx + b
• Students are not clear whether m, x, or b
is the argument.
Concept 4: Algebra as Structures
• Variable as abstract symbol - marks on
paper
• Abstract Algebra view
– fields of real and complex numbers
– rings of polynomials
– properties of integral domains
– groups and study of structure
M3={0,1,2} Cayley tables for operations
+ 0 1 2
0 0 1 2
1 1 2 0
2 2 0 1
a + b = c mod 3
• 0 1 2
0 0 0 0
1 0 1 2
2 0 2 1
a • b = d mod 3
Modular equations
• 2x+1=0 (mod 3)
• x2 = x (mod 3)
• x3 = x (mod 3)
Conundrum
• Want students to have referents (usually
real numbers) for variables
• Want students to operate on variables
without going to level of referent
Two major questions:
• What constitutes enough facility?
• Should concept or facility come first?
Perspectives for viewing Algebra
Algebra, and therefore variable, must be
viewed from multiple perspectives
• Generalized Arithmetic
• Vehicle for solving problems
• Means to describe and analyze relationships
• Key to characterization and understanding
of math structures
Rational Number Representation
• Fraction Form a/b – extension of integers
• Decimal Form 0.ab – numeration system
that extends base ten numeration
• Percent Form ab% - parts of a hundred,
useful in commerce
• Ratio and Proportion Conception –
applications throughout Math
Decimal to Fraction Conversion
• Terminating Decimal
d = 0.23 = 23 / 100
• Repeating Decimal
d = 0.142857
106 • d = 142857.142857
999,999 • d = 142857
d = 142857/999999
d = 1/ 7
• Non-repeating Non-terminating Decimal
d = 0.10100100010001………..
• How can we write this d as a rational number in fraction
form?
Exploration
• How can we determine if the decimal
representation of a fractional number terminates
or repeats without actually converting the
number to decimal form?
• Use your TI-73 to convert the following fractions
to decimal form. Look for patterns answering the
question above.
• 1/2 , 1/3 ,1/4 , 1/5 , 1/6 , 1/7 , 1/8 , 1/9, 1/10
• Test your conjecture using other fractions such
as 2/3, 3/4 , 5/72 .
Answers
1
 0.5
2
1
 0. 3
3
1
 0.25
4
1
 0. 2
5
1
 0.1 6
6
1
 0.142857
7
1
 0.125
8
1
 0. 1
9
1
 0. 1
10
Terminatesden 21
Repeats
den 31
Terminatesden 22
Terminatesden 51
Repeats
den 6 = 2*3
Repeats
den 7
Terminatesden 23
Repeats
den 32
Terminatesden 10 = 2*5
Fraction to Decimal Conversion
Theorem
• Let a / b be a fraction in simplest form.
• If b = 2m • 5n for m, n  Z, then a / b has a
terminating decimal representation.
• If b has a prime factor other than 2 or 5,
then a / b has a repeating decimal
representation.
Verification
Ratio and Proportion
Ratio: An ordered pair of numbers a:b with b  0.
• Compare relative sizes of two quantities
• a:b is equivalent to a/b
• a:b = c:d if and only if ad = bc
Proportion: Statement that two given ratios are
equal.
• Standard Algorithm: Given a,b, and c, find x if
a/b=c/x
Proportional Reasoning
A form of mathematical reasoning that
involves
• Covariation
• Multiple comparisons
• Inference and prediction
• Useful in modeling real world phenomenon
Aspects of Proportional Reasoning
•
•
•
•
•
Mathematical or Quantitative Aspect
Linear relationship y = m•x
Multiplicative in nature
Missing – value problems a:b = c:x, find x
Compare two rate pairs to deduce which is
greater, faster, more expensive, etc.
Psychological or Qualitative Aspect
•
•
•
•
Does this answer make sense?
Should it be larger or smaller?
Comparison not dependent on specific values
Multilevel comparative thinking – ability to store
and process several pieces of information, then
compare according to predetermined criteria
• Piaget’s Formal Operational Level Of Cognitive
Development
• Mental flexibility – use multiple perspectives
Proportional Reasoning –
A Bridge to Algebra
• Tightly interwoven with fractions and multiplication
• Underlying concept for relative increase/decrease,
slope, Euclidean Algorithm, linear equation, constant of
proportionality, intensive quantities and rates, functions
and operations, and measurement
• Proportions serve as bridge between common numerical
experiences and patterns in arithmetic and more
abstract relationships in algebra.
• Multimodal associations – translations between and
within modes of representation, such as table, graph,
symbol, picture and diagrams
Skill vs. Concept Acquisition
• Students need to automatize certain commonly
used mathematical process.
• The most efficient methods are often the least
meaningful and therefore are to be avoided
during the initial phases of instruction.
• Confuse efficiency of a / b = c / x so ax = bc
with meaning of proportional reasoning.
• Algorithm is mechanical process devoid of
meaning in a real – world context.
Unit-Rate Method:
How Much for One?
• Intuitive appeal: Children purchase many
and calculate unit prices.
• Multiple of Unit Rate: Sandy paid 90¢ for
each computer disk. How much did she
pay for a dozen?
• Determine Unit Rate: Sarah bought a
dozen computer disks for $10.80. How
much did each disk cost?
Unit-Rate Method: Solutions
• Sandy paid 90¢ for each computer disk. How much did
she pay for a dozen?
• Sarah bought a dozen computer disks for $10.80. How
much did each disk cost?
Factor of Change Method
• “Times as many” mentality a:b = na:nb.
• Example: If Al paid 3.60 for 4 disks, how
much did 12 disks cost?
• Restricted to rate pairs that are integral
multiples – at least intuitively.
Eudoxus’ Conception of Proportion
(Carraher, 1996)
Greeks of antiquity had not yet invented
fractions, but were able to handle ratios and
proportions through integer operations on
quantities.
• Constructive Approach: A problem was
solved when the solution was demonstrated
geometrically.
• Number was conceptualized in terms of line
segments.
Book V Of Euclid’s Elements
Magnitudes are said to be in the same
ratio, the first to the second and the third
to the fourth, when, if any equimultiples
whatever of the second and fourth, the
former equimultiples alike exceed, are
alike equal to, or alike fall short of, the
later equimultiples respectively taken in
corresponding order.
Eudoxus
In Modern Notation:
• Given the segments A, B, C and D
• A:B = C:D if and only if for any m,n  Z+
• One of the following holds:
– mA > nB and mC > nD
– mA = nB and mC = nD
– mA < nD and mC < nD
Eudoxus’ Definition
• Eudoxus’ Definition places ratio and
proportion squarely in the context of
perceptual judgment.
• Children develop ratio and proportion
concepts by visually comparing lengths.
• Historic significance – Greeks used to
represent incommensurable quantities.
Golden Ratio
• Also called golden mean or divine
proportion is  = (1+5)/2  1.618.
• Myth – Greeks considered  essential to
beauty and symmetry.
• Fact – Nature exhibits  in body
proportions and equiangular or logarithmic
spiral.
Greek Pursuit of Balance
• Sought a length s, called the geometric mean,
that strikes a balance between two line
segments of different lengths l and w.
l  w = s  w or l/s = s/w
Hence l•w = s2, geometrically s is the side of a
square with the same area as a rectangle of
area l•w

l
s

w

Solution
•
l s

s w
where l = w + s
Golden Rectangle Construction
• Construct 1 x 1 square.
• Bisect side of square at M.
• Construct segment extension of side by
striking an arc of length MV.
V
M
(1+5)/2
