Discrete Mathematics is Essential Mathematics

in the 21st Century:Rationale and Examples

for High School

NCTM 2009

Eric W. Hart

Time Magazine - Natl Standards“For example, I learned a lot of calculus, which hasn't proved that useful in my career. But I do remember being confronted at a Time Inc. meeting on digital strategy with the simple question of how many direct two-way links there were in a fully connected network of 50 nodes. It was a long time before any of us could figure out even how to begin figuring it out. Tomorrow's careers are likely to require more knowledge of networks, probabilities, statistics and risk analysis.”– Walter Isaacson, Time Magazine, 4/15/09

Discrete Mathematics

• What

• Why

• How/Where

What is Discrete Mathematics?

Three major topics in PSSM

• Combinatorics systematic listing and counting

• Iteration and recursion sequential step-by-step change

• Vertex-edge graphs modeling paths and networks

Solve problems like …

• How many different computer passwords can be created with five letters and two numbers? (Combinatorics)

• How fast does money or a population grow? (Recursion and Iteration)

• What is an optimal route for plowing snow from neighborhood streets?(Vertex-Edge Graphs)

Other DM Problems & Topics

• When buying music on the Internet, how can you efficiently find the song, securely pay for it online, and efficiently download it?(Mathematics of Information Processing)

• When electing a new mayor, how can you most accurately reflect voters’ opinions and avoid costly runoffs?(Mathematics of Democratic Decision Making)

• Organize information in rectangular arrays of rows and columns (Matrices)

Key Themes of DM

• Discrete Mathematical Modeling

• Optimization

• Algorithmic Problem Solving

Why Discrete Mathematics?

• Useful — Widely used in business, industry, and government

• Contemporary — particularly relevant in today’s digital information age

• Educationally Powerful — problem solving, modeling, reasoning, representing, engaging, and provides students success doing mathematics

Discrete Mathematics for All

“Discrete mathematics should be an integral part of the school

mathematics curriculum.” (NCTM, 2000, p. 13)

How/Where?

• Integrate DM into other strands and courses– Vertex-Edge Graphs geometry– Recursion algebra and geometry– Math and voting Social Studies– DM topics richer Consumer or General

Mathematics courses

• Separate DM course– 4th year course alternative to precalculus– DM and Stat course

How/Where?

• National Recommendations– NCTM– Achieve

• State Standards– IA– NJ– MN– AZ– other

NCTM Navigations Books on Discrete Mathematics

Navigating through Discrete Mathematics in PreK-5

and Navigating through Discrete

Mathematics in 6-12

Navigating through Discrete Mathematics in PreK-12

• Focus: how to implement PSSM’s recommendation that “discrete mathematics should be an integral part of the school mathematics curriculum”

• 6-12 Book• PreK-5 Book• Detailed grade-specific recommendations• Classroom activities• Mathematics background

PreK-5 Book

Introduction1. Counting PreK-22. Counting 3-53. Vertex-Edge Graphs PreK-24. Vertex-Edge Graphs 3-55. Iteration and Recursion PreK-26. Iteration and Recursion 3-5

6-12 Book

Introduction1. Counting 6-82. Counting 9-123. Vertex-Edge Graphs 6-84. Vertex-Edge Graphs 9-125. Iteration and Recursion 6-86. Iteration and Recursion 9-12

Recursion

Recursive View of Functions

• Linear

• Exponential

• Polynomial

Recursive View of Functions 1

Describe patterns shown in the table.

A B

0 5

1 7

2 9

3 11

Look down the column of B’s

NEXT = NOW + 2

Bn+1 = Bn + 2

A B

0 5

1 7

2 9

3 11

Look across from A to BB = 5 + 2A

A B

0 5

1 7

2 9

3 11

Recursive View ofLinear Functions

• Explicit form: B = 5 + 2A• Recursive form:

NEXT = NOW + 2, start at 5 Bn+1 = Bn + 2, B0 = 5

• Slope seen concretely in recursive form• Rate of Change seen concretely• Note also: arithmetic sequence

Recursive View of Functions 2

Describe patterns shown in the table.

A B

0 5

1 10

2 20

3 40

Look down the column of B’s

NEXT = NOW * 2

Bn+1 = Bn * 2

A B

0 5

1 10

2 20

3 40

Look across from A to B

B = 5 * 2A

A B

0 5

1 10

2 20

3 40

Recursive View ofExponential Functions

• Explicit form: B = 5 * 2A

• Recursive form: NEXT = NOW * 2, start at 5 Bn+1 = Bn * 2, B0 = 5

• Potent comparison to linear – add constant vs. multiply by constant at each step

• Note also: geometric sequence

Common ApplicationsLinear and Exponential

Recursive point of view gives powerful information and insights

Distance-Rate-Time Compound Interest

Represent each situation with a table, graph, and equations (recursive and explicit)

Distance traveled from DT Chicago:1 hr to get out of town (30 mi), then cruise at 70 mph

Money saved: $1000 initial deposit, 6% interest compounded annually

Tables and Equations

Distance traveled from DT Chicago:1 hr to get out of town (30 mi), then cruise at 70 mph.

NEXT = NOW + 70start at 30

d = 30 + 70(t – 1)

Money saved: $1000 initial deposit, 6% interest compounded annually.

NEXT = NOW(1.06)start at 1000

Time (t) Dist (d)

1 30

2 100

3 170

4 240

A=1000×1.06t

Time (t) Amt (A)

0 1000

1 1060

2 1123.60

3 1191.02

Distance traveled from DT Chicago:1 hr to get out of town (30 mi), then cruise at 70 mph.

Recursive Form Gives Potent

NEXT = NOW + 70

• add constant at each step

• constant difference between steps, 70 mph constant, that’s what it means to have:

constant rate of change

linear function

arithmetic sequence

repeated addition: multiplication

d = 30 + 70(t – 1)

Money saved: $1000 initial deposit, 6% interest compounded annually.

Information and Insights

NEXT = NOW(1.06)

• multiply by constant each step

• not constant difference (but constant ratio, and I wonder about quadratics), which means:

not constant rate of chg

exponential function

geometric sequence

repeated mult: exponentiationA=1000×1.06t

Dn+1 =Dn + 70 an+1 =1.06an

Distance traveled from DT Chicago:1 hr to get out of town (30 mi), then cruise at 70 mph.

NEXT = NOW + 70

d = 30 + 70(t – 1)

Graph: line (constant rate,slope)

y-int: meaningful?For this situation, the graph starts at (1, 30)

Money saved: $1000 initial deposit, 6% interest compounded annually.

NEXT = NOW(1.06)

Graph: definitely not a line, even though it looks linear, doesn’t go up same amount for each unit over (mult, not add, a constant).

y-int: initial dep, (0, 1000)

A=1000×1.06t

Common ApplicationsLinear and ExponentialRecursive point of view gives powerful

information and insights

Distance-Rate-Time Compound Interest

Distance traveled from DT Chicago:1 hr to get out of town (30 mi), then cruise at 70 mph.

Money saved: $1000 initial deposit, 6% interest compounded annually.

Recursive View of Functions 3

Describe patterns shown in the table….

Time n Inst speed

at time nAvg spd during

each secDist fallen

during each sec

Total dist fallen after

n secs

0 sec 0 ft/sec 0 ft/sec 0 ft 0 ft

1 32 16 16 16

2 64 48 48 64

3

4

n

Skydiver falls at 32 ft/sec/sec. Ignore all other factors.

Time n

Inst speed at

time n

Avg spd during each

sec

Dist fallen during

each sec

Total dist fallen after n secs

0 sec 0 ft/sec 0 ft/sec 0 ft 0 ft

1 32 16 16 16

2 64 48 48 64

3 96 80 80 144

4 128 112 112 256

n Add 32 each sec:

32n

Start with 16, add 32 each sec:

16+32(n-1)=32n-16

32n-16 T(n)=T(n-1)+D(n)

T(n)=T(n-1)+32n-16

T(n)=(4n)2

T(n)=16n2

Quadratic Function: T(n) = an2 + bn + c

T(n) 1st difference 2nd difference

0 16 32

16 48 32

64 80 32

144 112

256

c a + b 2a

a +b + c 3a + b 2a

4a + 2b + c 5a + b 2a

9a + 2b + c 7a + b

16a + 2b+ c

Recursive View of Functions

• Linear

• Exponential

• Polynomial

Set Theory and Logic

Example: Internet Searching

Library of Congress Online Catalog

• Help page on “Boolean Operators”http://catalog.loc.gov/help/boolean.htm

• AND, OR, NOT

• Venn diagrams

Modular Arithmetic

Examples:

• Check digits for ID Numbers– UPC– ISBN– Credit Cards

• Cryptography– Secure online transactions– Encrypted email or file storage

12-Digit UPC

Check Digit Algorithm:1. 3 (sum of even position digits)

+ (sum of odd position digits) RIGHT-TO-LEFT

2. Equivalent to 0 mod 10?Yes? OK. No? Error.

Note: Use all 12 digits in the algorithm. Last digit is is the check digit. Check digit is chosen to make the result = 0 mod 10.

Try it: 0-49387-10018-7

Cryptography

“the science of transmitting and managing information in the presence of an adversary” (Koblitz, AMS Notices, Sept, 2007)

Mathematics:modular arithmetic, number theory, abstract algebra

Cryptography on the Internet

Cryptography on the Internet

RSA Public-Key Crypto

• Everyone knows the public encryption key

• Only the designated recipient knows the secret decryption key

• Works and is secure because it’s easy to multiply two numbers, but hard to factor a number

RSA method

• Multiply 2 large primes, p and q, to get a large composite number, n.

• Compute r = (p – 1)(q – 1)• Find a number e (for encrypt) that has a

multiplicative inverse d (for decrypt) under multiplication mod r.

• Encrypt: raise to power e, reduce mod n• Decrypt: raise to power d, reduce mod n• Public encryption key – e and n

Private decryption key – d

RSA method works because:

• Special case of Euler’s Theorem:

(Me)d = M mod n

• To break the code, an adversary would need to factor the very large number n, and factoring is hard.(Factoring n would give p and q, which would give r, which would give d, since e is public, and thus the message could be decrypted.)

Modular ArithmeticExamples:

• Check digits for ID Numbers– UPC– ISBN– Credit Cards

• Cryptography– Secure online transactions– Encrypted email or file storage

Vertex-Edge Graphs

• Diagrams consisting of points (vertices) and segments or arcs (edges) joining some of the points.

• Connections matter, not size and shape• Model and solve problems related to

– Networks– Paths– Relationships among a finite # of objects

(like conflict or prerequisite relationships)

Critical Path – PERT

Discrete Mathematics is Essential Mathematics

for All Students• Counting

– How many?

• Recursion– Sequential change

• Vertex-Edge Graphs– Networks, paths, relationships

• Set Theory, Logic, Modular Arithmetic– Math of information processing & the Internet

• Math and Voting– Preferential voting