1. Theory of Knowledge is a course taken by all International Baccalaureate Diploma Programme students. This course discusses how the student is able to know something. The student is described as an "actor of knowledge" who attempts to find knowledge, where knowledge, as defined by Plato, is "justified true belief".International Baccalaureate Diploma ProgrammePlatojustified true belief The course teaches that there are four Ways of Knowing: sense perception, emotion,sense perceptionemotion reasonreason, and language.language Also used are the following seven Areas of Knowledge, which are put here in the form of a spectrum, the two ends of which are labeled "objective" and "subjective", from left to right respectively: mathematics, natural sciences,mathematicsnatural sciences social sciencessocial sciences, history, the arts, ethics, and spirituality.historyartsethicsspirituality The course teaches nine reasons for justification of things one claims to know: logic, sensory perception, revelation, faith, memory, consensus, authority, intuition, and self-awareness.logicsensory perceptionrevelationfaithmemoryconsensus authorityintuitionself-awareness The Theory of Knowledge

2. Attributes of Successful Problem Solvers (according to Dr. Naimie) Successful problem solvers are: Curious Careful Capable Confident Committed Compatible Creative Circumspect

3. MATH Science Engineering The Math-Science-Engineering Triad Mathematics - The language of science Science - The process of collecting evidence, recording data, and analyzing information to construct theoretical explanations of how things work. Engineering - The application of science and mathematics to safely realize a desired outcome

4. What areas of mathematics are taught at Bow High School? Arithmetic – Order and Arrangement Algebra – Balance and Equality Geometry – Shape and Form Calculus – Change Probability and Statistics - Likelihood

5. Algebraic.vs. Differential Equations Algebraic Equations Algebraic Equations describe the relationship between two or more variables. Some examples are: Differential Equations Differential Equations describe the relationship between variables and the rates of change of the variables. Some examples that contain a single variable, derivatives with respect to time, and constants are:

6. Where Do Differential Equations Come From? Differential equations, like algebraic equations, describe relationships that scientists, engineers, economists, entrepreneurs, doctors, etc. encounter every day! Whenever you are interested in how something changes … you are considering a situation where a differential equation applies. Motion Heat Up/Cool Down Charging/Discharging of a Capacitor Growth/Decay Population Dynamics Chemical Concentrations Stock Market Fluctuations Many Others!

7. Example #1: Constant Acceleration Suppose an object is released from rest at a height H above the floor and that it falls at constant acceleration (g) until it hits the floor. Under these conditions, the rate of change of velocity with time, which is equal to the acceleration by definition, will be constant (and equal to g). Governing Differential Equation: Intuitive Solution: If the rate of change of velocity with time is constant, then the slope of the graph of velocity.vs. time must be constant (and equal to g). Thus, the graph of velocity with time must be linear! The Solution:

8. Case #2: Exponential Growth of a Species Suppose a species of animals has an infinite supply of food and no predators. Under these conditions the rate of change for the population of the species will be directly related to the size of the population, with a constant of proportionality A. At t=0, the population is equal to P o Governing Differential Equation: Solution: Step 1: Rewrite the differential equation such that all terms containing P are on one side and all terms containing t are on the other side, take anti-derivatives, and isolate P :

9. Step 3: Sketch the solution (note that exact shape of graph depends upon the values of P o and A but the general shape does not): Step 2: Determine the constant ( C ) using the initial value of P that is given in the problem (that is … P o ) :