## Think Curriculum!

**Teachers: **
Looking for the true and joyous way to think about and teach a curriculum topic?

**Students: **
Looking for the true and joyous way to understand a curriculum topic?

These essays and short video presentations remove the intellectual clutter and reveal the true simplicity and elegance of thought behind various tidbits of the standard curriculum. Let's return the joy to what we teach and what we learn!

**Algebra**

- FOIL or FILO or IFLO or OLIF
- On Logarithms
- Graph Quadractics with Absolute Ease (VIDEO!)
- Fitting Quadratics to Data (VIDEO!)
- Graphing Factored Polynomials (VIDEO!)
- Deriving the Quadratic Formula (VIDEO!)
- Completing the Square (VIDEO!)
- The Natural Way to the Equation of a Line (VIDEO!)
- Synthetic Division: How to understand it by NOT doing it! (VIDEO!)
- Why All Quadratic Graphs are U-Shaped (VIDEO!)
- The Binomial Theorem (VIDEO!)
- Splitting the Middle Term (VIDEO!)
- Square Roots and Addition (VIDEO!)
- Fitting Exponentials to Data (VIDEO!)
- The Discriminant of the Quadratic – YOU DON’T NEED IT! (VIDEO!)
- Which Mean do you Mean? (VIDEO!)
- Parentheses Numbers (VIDEO!)
- Why Thrown Objects follow Quadratic Paths (VIDEO!)
- Why do you Flip the Inequality Sign when Multiplying by a Negative? (VIDEO!)
- Nutting your way through a standard Quadratics Exercise: Example 1 (Video!)
- Nutting your way through a standard Quadratics Exercise: Example 2 (Video!)
- Buying Gloop Puzzle (VIDEO!)
- The Vinculum (VIDEO!)
- Polynomials and the Reverse Galley Method (Essay!)
- Experimental PD on Logarithms
- FACTORING: Some Sample Essays
- Synthetic Division

**Arithmetic**

- Does 0.999… equal 1 or does it not? (VIDEO!)
- Why is negative times negative positive? (VIDEO!)
- Dividing by Zero (VIDEO!)
- What is zero to the zeroth power? (VIDEO!)
- Explaining Tricky Exponents (VIDEO!)
- The Euclidean Algorithm (VIDEO!)
- Why is dividing by a fraction the same as multiplying by its reciprocal? (VIDEO!)
- The Truth About Rationalizing the Denominator (VIDEO!)
- A Cute Finger (and toe!) Multiplication Trick (VIDEO!)
- The Danger of Square Roots (VIDEO!)
- i is not the square root of negative one! (VIDEO!)
- Divisibility Rule for 7 (VIDEO!)
- Divisibility Rules for 3 and 9 (VIDEO!)
- Primes, Twin Primes, Triplets of Primes: How Many? (VIDEO!)
- Why is Negative times Positive Negative? (VIDEO!)
- A Formula for Pascal’s Triangle (VIDEO!)
- Square Roots and Addition (VIDEO!)
- Are Factor Trees Unique? (VIDEO!)
- Euclidean Algorithm and Jug Filling (VIDEO!)
- Fundamental Theorem of Arithmetic: Proof! (VIDEO!)
- Dividing by Nine: Something Cute (VIDEO!)
- Dividing by 97 – and other such numbers (VIDEO!)
- Integer Triangles – and more! (Essay!)
- Benford’s Law (VIDEO! and ESSAY!)
- Anomalous Cancellation in Fractions
- Mersenne Primes (VIDEO!)
- Perfect Numbers (VIDEO!)
- Egyptian Fractions and Fibonacci’s Greedy Algorithm (VIDEO!)
- Pigeon Hole Principle (VIDEO!)
- Sums of Cubes (VIDEO!)
- Partition Numbers: An Accessible Overview (VIDEO!)
- SURPRISING FIBONACCI APPEARANCES!
- The Sum 1+2+…+N: A Visual Approach (VIDEO!)
- There are more Triangular numbers than Square numbers! (VIDEO!)
- Divisibility by 11 – some more! (VIDEO!)
- Average Age Puzzle (VIDEO!)
- Which Mean do you Mean? (VIDEO!)
- A Few Thoughts about Troublesome Zero (VIDEO!)
- Line Multiplication (VIDEO!)
- Multiplication without Multiplying! (VIDEO!)
- Sums of Squares Formula (VIDEO!)
- Divisibility Rules Galore! (VIDEOS)
- Divisibility Rules for 1,2,4,5,6,8,10,12,14,… (VIDEO!)
- The PIGEON-HOLE principle (ESSAY!)
- Fractions are Hard!

**Calculus Concepts**

- Fun with the Intermediate Value Theorem
- What is i raised to the i-th power? (VIDEO!)
- Deriving Euler’s Formula: e^ix = cos(x) + i sin(x) (VIDEO!)
- What is e? (VIDEO!)
- e and Compound Interest (VIDEO!)
- The Basel Problem / Zeta (2) and Wallis’s Product (VIDEO!)
- The Geometric Series Formula (VIDEO!)
- Benford’s Law (VIDEO! and ESSAY!)
- Fractions Take Up No Space (Rationals have Measure Zero): VIDEO!
- An Euler Gem: Distinct Partitions and Odd Partitions (VIDEO!)
- There are more Triangular numbers than Square numbers! (VIDEO!)
- A Volume/Surface Area Query in Calculus (VIDEO!)

**Geometry**

- On 30-60-90 and 45-90-45 Triangles (VIDEO!)
- The Two Pancakes Theorem (VIDEO!)
- Wrapping a Rope around the Earth Puzzle (VIDEO!)
- Pi for a Square! (VIDEO!)
- On Pythagoras’ Theorem (VIDEO!)
- Viviani’s Theorem (VIDEO!)
- Finding the Center of a Circle (VIDEO!)
- When Perimeter equals Area (VIDEO!)
- Integer Triangles – and more! (Essay!)
- Pythagoras in 3-D: TWO WAYS! (VIDEO!)
- Heron’s Formula for the Area of a Triangle (VIDEO!)
- Everting a Torus! (VIDEO!)
- What is Fractional Dimension and What is a Fractal? (VIDEO!)
- A Leap-Frog Puzzle using Vectors (VIDEO!)
- Angles in a 5-pointed Star (VIDEO!)
- Equidistance and Circles for Triangles (VIDEO!)
- Curry’s Paradox and the Notion of Area (VIDEO!)
- Pi equals 2 and SQRT(2) equals 2: Two paradoxes! (VIDEO!)
- Geometry Formulas Explained (VIDEO!)
- Halfway between a Cylinder and a Cone (VIDEO!)
- Geometry Puzzle: The Size of an Arc (VIDEO!)
- Wire-Cutting Problem (VIDEO!)
- The Brownie Puzzle (VIDEO!)
- Radian Measure (VIDEO!)
- The Medians of a Triangle – via Vectors! (VIDEO!)
- Mobius Bands / Mobius Strips (VIDEO!)
- Proving the Reflection Property of an Ellipse (VIDEO!)
- Pythagoras Galore! (Videos and Chapter Materials)
- An UnIntuitive Pythagorean Surprise (VIDEO!)
- Tangent Theorems for Circles via Pythagoras (VIDEO!)
- Shortest Paths via Pythagoras (VIDEO!)
- The Equation of a Circle (VIDEO!)
- Do you Believe the Area Formula for a Triangle? (VIDEO!)
- Symmetry: Smart Phones, Friezes, and Fractals
- Is the Composition of Two Rotations another Rotation?

**Miscellaneous**

- How to Think Like a School Math Genius
- Squangular Numbers and Theon’s Ladder (ESSAY!)
- Fibonacci Formula (VIDEO!)
- Converting Units (VIDEO!)
- Benford’s Law (VIDEO! and ESSAY!)
- Perfect Numbers (VIDEO!)
- What is Fractional Dimension and What is a Fractal? (VIDEO!)
- 25 People in a Square Grid Puzzle (VIDEO!)
- Fractions Take Up No Space (Rationals have Measure Zero): VIDEO!
- A Leap-Frog Puzzle using Vectors (VIDEO!)
- A Gory Puzzler (VIDEO!)
- Assessment Thoughts (ESSAY!)
- A Centroid Paradox? (VIDEO!)
- The Medians of a Triangle – via Vectors! (VIDEO!)
- Mobius Bands / Mobius Strips (VIDEO!)
- MATHEMATICAL LETTERS
- How to think like a School Math Genius: FIVE PRINCIPLES (Video!)
- Nim and the Powers of Two (VIDEO!)
- The PIGEON-HOLE principle (ESSAY!)
- Integer Triangles
- Impossible Braids!
- WITHOUT WORDS and MORE WITHOUT WORDS solutions
- The call to “GO BACK TO BASICS”
- Helpful Home Hints from a Math Guy

**Miscellanious**

- Intersection Math Puzzle (VIDEO and ESSAY!)
- Path Walking Puzzle (VIDEO!)
- Dots on a Circle Puzzle (VIDEO!)
- A Harmonic Series Paradox (VIDEO!)
- Everting a Torus! (VIDEO!)
- The Medians of a Triangle – via Vectors! (VIDEO!)
- MAA AMC Curriculum Inspirations
- The National Math Salute (VIDEO!)
- Clocks: Have you ever noticed … ?? (VIDEO!)
- Hyperbolic Sine and Cosine (VIDEO!)
- A lead-in to ten-adic numbers

**Numbers**

- The Euclidean Algorithm (VIDEO!)
- Partition Theory
- Primes, Twin Primes, Triplets of Primes: How Many? (VIDEO!)
- Why is Negative times Positive Negative? (VIDEO!)
- Divisibility by 11 (VIDEO!)
- Graphing Complex Numbers (VIDEO!)
- Are Factor Trees Unique? (VIDEO!)
- Euclidean Algorithm and Jug Filling (VIDEO!)
- Fundamental Theorem of Arithmetic: Proof! (VIDEO!)
- Anomalous Cancellation in Fractions
- Average Triangular, Square and Pentagonal Numbers (ESSAY!)
- An Euler Gem: Distinct Partitions and Odd Partitions (VIDEO!)
- The Sum 1+2+…+N: A Visual Approach (VIDEO!)
- Locker Puzzle (VIDEO!)
- Divisibility Rules Galore! (VIDEOS)

**Permutations and Combinations**

- Why is 0! equal to 1? (VIDEO!)
- The Binomial Theorem (VIDEO!)
- A Formula for Pascal’s Triangle (VIDEO!)
- Permutations and Combinations (ESSAY!)
- Multi-Choosing (VIDEO!)
- Hats and Derangements (VIDEO!)
- Derangements: Everything about them and More! (ESSAY)
- Parentheses Numbers (VIDEO!)

**Pi**

**Pre-Calculus**

- Guide to Absolute Values
- Logarithms: Brief History and Brief Math (VIDEO!)
- LOGARITHMS: More Mathematics about them (VIDEO!)
- Graphing Complex Numbers (VIDEO!)
- Change of Base Rule for Logarithms (VIDEO!)
- Basic Transformations of Graphs: Translations (VIDEO!)
- What is a Vector? (VIDEO!)
- The Binomial Theorem (VIDEO!)
- The Geometric Series Formula (VIDEO!)
- Why the Graph of an Inverse Function is a diagonal Reflection (VIDEO!)
- What is a Function? (VIDEO!)
- Fitting Exponentials to Data (VIDEO!)
- Fractions Take Up No Space (Rationals have Measure Zero): VIDEO!
- A Brief Introduction to PARAMETRIC EQUATIONS (VIDEO!)
- Hyperbolic Sine and Cosine (VIDEO!)
- Matrices and Vectors
- Wrapping-Paper Slicing

**Probability**

- The MONTY HALL Problem (VIDEO!)
- Bertrand’s Paradox (VIDEO!)
- Twinkle Twinkle and Math (VIDEO!)
- American Pennies are Biased? (VIDEO!)
- The Boy-Boy Paradox (VIDEO!)
- Path Walking and Probability Theory (VIDEO!)
- Sicherman Dice (VIDEO!)

**Pythagorasâ€™s Theorem**

- Generating Pythagorean Triples (VIDEO!)
- An UnIntuitive Pythagorean Surprise (VIDEO!)
- Tangent Theorems for Circles via Pythagoras (VIDEO!)
- Shortest Paths via Pythagoras (VIDEO!)

**Quadratics**

- An extraordinarily quick way to sketch quadratics
- Graphing complex solutions to quadratics
- Deriving the Quadratic Formula (VIDEO!)
- Completing the Square (VIDEO!)
- Why All Quadratic Graphs are U-Shaped (VIDEO!)
- The Discriminant of the Quadratic – YOU DON’T NEED IT! (VIDEO!)
- Wire-Cutting Problem (VIDEO!)
- Why Thrown Objects follow Quadratic Paths (VIDEO!)

**Statistics**

- Standard Deviation Formula Explained! (VIDEO!)
- “Which Serum?” Question (VIDEO!)
- Line of Best Fit: Least Squares Method (VIDEO!)

**Trigonometry**