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): dx• Matrix (Linear Algebra): MT

• Vector:

Variables have many possible definitions, referents, and symbols.

f

v

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:

3x + 5 = 11

Arithmetic Solution:

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

• What happens to value of as ?x

1x

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

+ 0 1 2 • 0 1 2

0 0 1 2 0 0 0 0

1 1 2 0 1 0 1 2

2 2 0 1 2 0 2 1

a + b = c mod 3 a • b = d mod 3

M3={0,1,2} Cayley tables for operations

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

1.010

1

1.09

1

125.08

1

142857.07

1

61.06

1

2.05

1

25.04

1

3.03

1

5.02

1

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

s w

l

Solution

• where l = w + s

w

s

s

l

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.

(1+5)/2

V

M