CTY- Inductive and Deductive Reasoning Syllabus

Inductive & Deductive Reasoning Syllabus TOPICS/OBJECTIVES DAY 1

WHAT (time) • HOW “Recursive Formulas and Inductive Reasoning”

Morning

• •

Afternoon

Find and write recursive formulas from number sequences Define and give examples of inductive reasoning

Cake Cutting Problem (10 min) • Students work independently at first Introductions, Rules, Honor Code (30 min) • Students introduce themselves: name, hometown, math background, CTY experience, etc. • Discuss class rules/expectations, honor code Pre-assessment (70 min) Discussion of Ind./Ded. Reasoning (20 min) • Solicit definitions/examples from students, but don’t give official definitions yet • State the goal of the course Number Devil Chapter 1 (20 min) • Instructor/TA read and explain concepts ND Follow Up (10 min) • Students workout (by hand) examples of 11 x 11, etc. • Try to predict future products • Students guess and then check (111111111111)2 to see flaws of inductive reasoning Intro to Inductive Reasoning (10 min) • Lecture • Relate to Cake-Cutting problem, recursive formulas Introduction to Recursive Formulas (15 min) • Lecture Recursive Formula Practice (15 min) • Students work individually • After checking with TA/Instructor, students move on to RF Assignment Recursive Formula Assignment (10 min) • Students work individually • Also, students will find a recursive formula for the Cake Problem

TOPICS/OBJECTIVES Late Afternoon

WHAT (time) • HOW Work Time (65 min) • Students continue on RF Assignment to hand in • When finished, they will find a RF for the cake cutting problem • Students create their own recursive formulas • As time permits, students will present their pattern on the board and the class will try to figure out the recursive formula • Instructor will give hints as necessary Daily letter (10 min) • Students write to the instructor/TA about the day • How was your first day in class and in the dorms?

TOPICS/OBJECTIVES DAY 2 Morning

“Explicit Formulas and Figurate Numbers” • • •

Afternoon

Late Afternoon

WHAT (time) • HOW

Students will be able to write and distinguish between recursive and explicit formulas. Students will be able to apply the formula for the sum of consecutive integers. Students will be able to apply recursive and explicit formulas to figurate numbers.

Review/Loose Ends from yesterday (25 min) • Do a review problem from yesterday (this was a student suggestion) • Have students continue to present their formulas from yesterday • Hand back Recursive Formula Assignment and discuss problem spots Evaluating Inductive Arguments (25 min) • Instructor presents several slides of inductive arguments • Students rank each arguments as good, fair, bad • Talk about the number of examples needed for a good argument Traffic Jam (35 min) • Do this activity outside if possible and have students work in 2 groups • If, finished early, define explicit formula and have them find the explicit formula for the Traffic Jam and some problems from yesterday Intro to Explicit Formulas/Figurate Numbers (10 min) • Lecture • Introduce Factorials Intro to Figurate Numbers (20 min) • Lecture • Introduce Gauss’s Formula (add first n numbers) Revisit recursive formulas from yesterday (15 min) • Students work independently to find explicit formulas for Cake Problem, Traffic Jam, Chess Problem, and Square Numbers • If students finish early, have them work with Pentagonal Numbers Number Devil Chapter 2 (20 min) • Instructor/TA read and explain concepts Crisscross Cubes (70 min) • Students work in groups with cube manipulatives • If extra time, students can work on the toothpick staircases Explicit/Figurate Assignment (70 min) • Students work individually and turn in work • If done, work on Toothpick Staircases or other supplementals Daily Letter (10 min)

TOPICS/OBJECTIVES

WHAT (time) • HOW • What do you like best and least about being away from home?

DAY 3 Morning

“Deductive Reasoning” •

• • •

Afternoon

Late Afternoon

Students will understand deductive reasoning and be able to distinguish it from inductive reasoning, given examples. Students will practice solving various deductive problem types. Students will be able to distinguish truth from validity in simple deductive arguments. Students will be able to recognize and determine the validity of syllogisms.

Review/Loose Ends from yesterday (30 min) • Review Problem on explicit formulas • Tie explicit formulas to linear equations • Show algebraic derivations for Crisscross Cubes using Gauss’s formula Thought Question (40 min) • Can dogs use inductive reasoning? • Students write independently and then discuss/present • Discuss Pavlov’s Dogs, Beeline, instinct vs. inductive. reasoning Census Taker Problem #1 (15 min) • Students work in groups to solve • Instructor gives hints as necessary Deductive Reasoning (30 min) • Lecture • Tie to census taker problem Number Devil Chapter 3 (55 min) • Instructor/TA read and explain concepts • Students construct Sieve of Eratosthenes (through 100) • Discuss how far you need to check to see if a number is prime • Discuss conjectures – Goldbach and Twin Prime Syllogism Practice (25 min) • Students work individually without instruction • Students share answers with a partner • Go over as a class using Euler Diagrams and/or symbolic logic Writeup for Census Taker Problem #1 (10 min) • Instructor models a thorough written solution and outlines expectations Census Taker Problems #2-7 (55 min) • Students work in groups to solve • Will formally write-up 2 problems of their choice • Will continue later Daily Letter (10 min)

DAY 4 Morning

“Deductive Reasoning” cont. •

• • •

Afternoon

Late Afternoon

Students will understand deductive reasoning and be able to distinguish it from inductive reasoning, given examples. Students will practice solving various deductive problem types. Students will be able to distinguish truth from validity in simple deductive arguments. Students will be able to recognize and determine the validity of syllogisms.

Review Syllogisms/Prop. Logic (30 min) • Have students do Syllogism Practice #1 • Discuss strategies for dealing with “some” statements Contrapositives (25 min) • Lecture – introduce contrapositive • Guided Practice – have students write contrapositives Inductive vs. Deductive Reasoning (35 min) • Students complete worksheet individually • Discuss/debate answers as a class Continue to work on census taker write-ups (25 min) • Students turn in one (or two) formally written up problem Maze Game (90 min) • Activity for both INDE sections • Students split into 4 teams • They work to together to figure out patterns to navigate through a giant maze painted on the floor • Students work as teams to solve bonus sequences Number Devil Chapter 4 (25 min) • Instructor/TA read and explain concepts Continue to work on census taker write-ups (40 min) • Students turn in one (or two) formally written up problem • Students can work on supplemental puzzles when done • SEND+MORE=MONEY and similar arithmetic puzzles Daily Letter (10 min)

DAY 5 Morning

“Truth, Validity, and Soundness” •

•

Afternoon

Late Afternoon

Students will be able to use premises to reach valid conclusions and analyze the validity of conclusions. Students will continue to develop their ability to determine whether syllogisms are valid or invalid

Pass back census taker assignment (20 min) • Talk about good mathematical writing Syllogism #2 (25 min) Introduction to using premises to reach valid conclusions (35 min) • Again, emphasize distinction between truth and validity • Define premises and give examples from geometry and the English language Chart Height Problem (15 min) • Students actively work together to solve BACO logic problems (25 min) • Students do problem1 individually, then go over as a class • Students work on problems 2 and 3 in pairs Focus on mathematical writing – write up 2 solutions to turn in Number Devil Chapter 5 (15 min) • Instructor/TA read and explain concepts • Demonstrate the a square number is the sum of two triangle numbers Thought Question (35 min) • What axioms do you use in your everyday life? Finish Write-up of BACO logic problems (15 min) Daily Letter (10 min) Return BACO logic problems write-ups (20 min) • Discuss using a counterexample for part C • Meet one-on-one with students to discuss quality of write-up Lady/Tiger Problems (60 min) • Students work in pairs, check answers with TA Matrix Problem Packet, if done with above

* * * * WEEK TWO * * * * Day 6 Morning

“Proofs” • Students will be able to explain simple graphical and algebraic proofs.

Writing Syllogisms (70 min) • Students are given a list of 5 elements and they must draw a Venn Diagram • Then students must write both a valid and invalid syllogism Break (15 min) Intro to Proofs (50 min) • Proof of Distributive Property • Provide proofs of 0.999… = 1 • Proof from Number Devil: diagonal length of the unit square • Students prove the triangle and trapezoid area formulas on their own

Afternoon

Number Devil Chapter 6 (15 min) • Instructor/TA read and explain concepts Continue Proof of Area of Trapezoid (10 min) Proof of the Pythagorean Theorem (30 min) • From the curriculum binder • Instructor demonstrates using diagrams on chalkboard Hands-on Activity (30 min) • Students will prove the area of a circle using construction paper and scissors

Late Afternoon

Graphical Proof Assignment (60 min) • Students work individually or in pairs • If done, students can work on chart logic packet or lady/tiger problems Daily Letter (10 min)

DAY 7 Morning

Afternoon

Late Afternoon

“Limits and Fibonacci Numbers” • The students will be able to apply a general understanding of the concept of limits and use relevant notation. • The students will explore the Fibonacci sequence, golden ratio, and a few applications.

Hand-on Proof Activity (25 min) • Students will use paper folding to show that there are 180 degrees in a triangle More Syllogisms (20 min) • Students complete Syllogism Worksheet #3 • Compare answers with a partner, then discuss as a class • Each student draws a Venn Diagram for a particular problem on the board Proof of odd/even sums (35 min) • Rigorously define even/odd numbers • Instructor guides students through proof of even + even = even • Have students try to prove the other 2 combinations on their own • If done, they can prove the multiplication rules Proof by Contradiction (35 min) • Review Fund. Thm. of Arithmetic • Infinitude of Primes • Define Rational and Irrational Numbers Number Devil Chapter 7 (15 min) • Instructor/TA read and explain concepts Proof by Contradiction cont. (25 min) • Square root of 2 is irrational Fibonacci Numbers and the Golden Ratio (25 min) • Define and have students compute the first 17 • Have students make a ratio sequence and find limit Intro to Limits (20 min) • Give “Definition”, walk through several examples, introduce proper notation, tie to inductive reasoning • Students compute a few infinite limits on their own Continue limits (30 min) • See if students can find the infinite limit trick • Explain why this trick works Worm Problems (40 min) • Students work in groups Daily Letter (10 min)

DAY 8 Morning

“Modular Arithmetic” • • •

Afternoon

Late Afternoon

Students will learn how to add and subtract in modular arithmetic Students will learn how to prove the divisibility rules for 3 and 11 Students will experience some of the patterns in Pascal’s triangle.

Finish discussion of limits (45 min) • Do limits where n approaches a fixed number • Discuss the definition of division and 0/0 • Have half of the class do right-sided limit and half do left-sided limit • See if students notice another explicit formula (n+4) and prove using difference of 2 squares Discussion about calculus (30 min) • Relate the two major problems of calculus to limits Divisibility by 3 (25 min) • Introduce rule, tie to inductive/deductive reasoning • Mini-Lecture on modular arithmetic • Students complete worksheet individually on modular arithmetic • Instructor guides students though the proof of divisibility by 3 Fibonacci Patterns (25 min) • Students find and write out first 30 Fibonacci numbers • They color in multiples of 2, 3, 5 (in different colors) • See if they notice pattern and use inductive reasoning to extend the pattern • Instructor guides them through a proof of their findings using deductive reasoning Number Devil Chapter 8 (20 min) • Instructor/TA read and explain concepts Finish Fibonacci Patterns (45 min) Pascal’s Triangle (20 min) • Notice patterns • Color in even/odd numbers Fibonacci Patterns Assignment (60 min) • If finished, work on Pascal’s Triangle or Bottle Counting Problems Mid-Course Self-Evaluation (20 min) • Students write about the performance in certain areas of the course

DAY 9 Morning

“Symbolic Logic” •

•

Students will be able to construct truth tables using not, and, or, if-then, if and only if, and combinations of these connectives. Students will be able to translate English into symbolic logic and vice versa.

Introduction to Symbolic Logic (75 min) • Students follow along with fill-in notes • Concepts: not, and, or, if-then, if-and-only-if, truth tables Painted Cube Project (35 min) • Major Course Project • Students work individually • Formal Write-up is required

Afternoon

INDE Clue (60 min) • Combine with other section of INDE • Students play a real-life game of Clue using the buildings around campus Continue work on Painted Cube Project (25 min)

Late Afternoon

Number Devil Chapter 9 (25 min) • Instructor/TA read and explain concepts Continue work on Painted Cube Project (40 min) Daily Letter (10 min)

DAY 10 Morning

“Symbolic Logic” cont. •

•


Seating Puzzle (10 min) • Students will use a set of premises to figure out where to sit Continue work on Painted Cube Project (120 min)

Afternoon

Number Devil Chapter 10 (20 min) • Instructor/TA read and explain concepts Finish Painted Cube Project (45 min) Daily Letter (10 min)

Late Afternoon

Fractal Video/Talk (90 min) • Students watch a video on fractals and their applications • Instructor gives lecture on Fractal Dimension

* * * * WEEK THREE * * * * DAY 11 Morning

“Limits” and “Symbolic Logic” cont. •

•

Afternoon

Late Afternoon


Carpool Logic Problems (60 min) • Students work individually evaluating truth statements about the axioms • Students present solutions afterwards • Every student writes his/her model on board Summation Lecture (15 min) • Define Sigma Notation and give a few examples • Define an infinite sum as the limit of partial sums • Use “walking” and “pie” examples to illustrate the concept of an infinite sum Finish Summation Lecture (30 min) Koch’s Snowflake Fractal (20 min) • Students create their own snowflake on grid paper • Relate area and perimeter to limits Number Devil Chapter 11 (20 min) • Instructor/TA read and explain concepts Finish Koch’s Snowflake Fractal (60 min) • Explain Perimeter/Area Paradox • Talk about the perimeter and area of Sierpinski’s Triangle • Talk about Coastlines Coastline of Britain (15 min) • Discuss distance of coastlines and relate to fractals Truth Table Practice (30 min) • Instructor presents example • Students try some individually Remainder Jump Game (25 min) • Students play each other in a modular arithmetic game Daily Letter (10 min)

DAY 12 Morning

Afternoon

Late Afternoon/ Homework

“Paradoxes” •

Students will be able to articulate the conflicting lines of reasoning that emerge from paradoxes and to evaluate those lines of reasoning.

Truth Table Warm-up (25 min) • Students will work individually on a truth table review problem • Truth Tables for inverse, converse, contrapositive • We will go over as a class Translating Symbolic Logic Statements (30 min) • Go over first 2 pages orally • Students work on next 3 pages • Go over packet as a class Analyzing Paradoxes (35 min) • Define paradox, give some simple examples • Instructor walks students through Newcomb’s Paradox • Students answer questions in pairs Continue Paradoxes (40 min) • Finish Newcomb’s Paradox • Present the Barber Paradox Hotel Infinity, Part I (20 min) • Instructor/TA read and explain concepts Present the Crow Paradox (30 min) Triangle Area Paradox on graph paper (30 min) • Students draw shapes on graph paper to try to find the error Finish Discussion of Paradoxes (45 min) • Grue Paradox • Hangman’s Paradox • Russell’s Paradox (students act out) Blocker Game (30 min) Daily Letter (10 min)

DAY 13

“Infinity”

Morning

•

•

•

Afternoon

Late Afternoon

Students will be able to find the recursive and explicit formulas for the Tower of Hanoi Problem Students will be able to distinguish between countable and uncountable infinities. Students will be exposed to some of the seemingly paradoxical results of attempting to calculate with infinity.

Tower of Hanoi Activity (75 min) • Students work individually or in pairs • If finished, they can work on “The Hardest Logic Puzzles of All Time” Introduction to Proof by Mathematical Induction (60 min)

Hotel Infinity, Part II (25 min) • Instructor/TA read and explain concepts Infinity Lecture (70 min) • Define countable and uncountable infinity • Present/solicit examples • Present Cantor’s diagonalization argument Analyzing Arguments (45 min) General Formulas (30 min) If finished, continue to work on “The Hardest Logic Puzzles of All Time” Daily Letter (10 min)

DAY 14 Morning

“Reasoning Outside the Box” •

Students will practice creatively in problemsolving

Student Program Evaluations (30 min) Scavenger Hunt (110 min) • Students divide into teams on a scavenger hunt across campus • They will solve various math problems in order to get the next clue

Afternoon

Post-Test (45 min) Write Thank-You Letter to Parents (30 min) If done with above, students will work on: Truthtellers and Liars • Students work alone or in pairs • Use deductive reasoning to determine who is telling the truth and who is lying

Late Afternoon

Hat Riddle (30 min) • Students act out a riddle in order to figure out the solution Lateral Thinking or Rebus Puzzles (40 min) Daily Letter (10 min)

DAY 15

“Closure”

Morning

•

Bye Bye

Seating Challenge (20 min) • Students work on a formula to figure out the seating chart for the day Number Devil Chapter 12 (25 min) Pass out Reading List (15 min) Paper Folding Riddle (20 min) Break (15 min) Lateral Thinking or Rebus Puzzles (45 min)