Favorite Ideas
- Numbers are a mental abstraction the human mind uses to conceptualize what occurs in nature. You can add numbers forever - they never end. Through subtraction, negative numbers were created to represent a concept of less than zero. Applications of division led us to develop fractions to capture parts of a number.
- The rectangular shape of collections in an array demonstrates the commutative property of multiplication. A 3x5 rectangle is the same as a 5x3 rectangle. When you "square" a number, the resulting figure is a square. Summing all of the odd numbers creates squares (1+3 is 4, 1+3+5 = 9, and adding 7 makes 16).
- Algebra is how we manipulate equations to discover the unknown. x + 3 = 7 is the same as x = 4. Units matter! Conversion formulas are one algebraic application.
- Geometric proofs often come about by a stroke of inspiration. Archimedes estimated pi by inscribing polygons with more and more sides inside of a circle.
- Calculus involves measuring change. A derivative measures a rate of change, such as velocity. An integral sums a series of changes, which has useful applications for calculating the volume of complex geometric shapes.
- In statistics, a normal distribution occurs when sums of various results form a bell shaped curve, centering around the middle value, which occurs at the highest frequency. The width and frequency of the bell curve describe the deviation.
- Conditional probabilities can create confusion, but looking at things from a different perspective can help. Consider a cancer screening where people have a .8% chance of actually having cancer; if they do, there is a 90% chance it will show up in the screening, and if they don't, there's still a 7% chance the test will be positive. Framed a different way, this means if a person's cancer screening was positive, there is a 9% chance they actually have cancer.
- Infinity is often thought of as a large number, but it can also be small. If you are walking to your front door, and each step, you go half the distance between you and the door, your steps will become infinitely small, and you'll never actually reach the door.