…I’ll tell you why. It’s so that we can establish and teach a standard level of logical discourse. These kind of proofs help children learn how to create logical arguments and communicate their findings.

Recently there was a debate among the math teachers in my teaching team about a math problem that some teachers were misinterpreting. Some said that the wording was misleading. Communication is important in math, and there is a standardized logical basis to it. (The answer was 2/3, guys!)

I believe that the Common Core Curriculum Standards do not sufficiently address issues of mathematical reasoning, the need for precision in mathematical discourse, and the role of proof throughout the geometry curriculum. While reading the Common Core State Standards for Mathematics, I noticed that the word “logic” is used only once in the entire 93-page document.

The growing appreciation of the important role of exploration with technology, experimentation, and conjecture in mathematical thinking may have obscured the fact that reasoning is the foundation of mathematics. While science verifies through observation, mathematics verifies through logical, deductive reasoning. Students need to be consciously aware of the distinction between exploring topics and providing more rigorous arguments. In particular, they need to use terminology in a precise manner and be able to specify their hypotheses, particularly if these are implicit.

While I fully agree that exploration and application are essential, I see the geometry in Integrated Math 2 as a unique opportunity for an EAB teacher and student to emphasize the way in which pure mathematics works, its foundation on deduction, reasoning, and proof. In fact, I argue that it is the only instance (aside from Computer Programming, if we offered it in a formal course) in which students have the opportunity to use logic to construct a functional system. Logic is a topic in the soon to be abandoned Math Studies course, but the students do not have the opportunity in this course to construct a functional system, instead, they use the concept on a basic knowledge and comprehension level to complete truth tables.

This is my rationale for focusing as intensely as I do on logical reasoning and structured proof.

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The cumulant generating function takes the use of MGF, Mx(t), one step further. It is primarily useful in finding the expected value and variance of a random variable. The cumulant generating function of X is denoted as Kx(t). It is defined as the natural logarithm of the MGF of X. In math notations, the relationship is as follows:

Kx(t) = ln [Mx(t)]

Kx(t) has a few useful properties, but the two I’m about to derive are the most relevant.

Property #1: The first derivative of Kx(t) evaluated at t=0 gives us E(X). Refer to derivation #1 in the attached file.

Property #2: The second derivation of Kx(t) evaluated at t=0 gives us Var(X). Refer to derivation #2 in the attached file.

Kx(t) is most helpful when the MGF is expressed as an exponential function, and when asked to calculate the variance. Refer to example #1 in the attached file where I found the Var(X) using Kx(t) and also Mx(t).

In general, when given the MGF of a random variable, we can use Kx(t) to find E(X) and Var(X) instead of making use of the MGF directly. As can be seen in the attached file, using the MGF to find the Var(X) was a much longer method than just using Kx(t) so yes, it sometimes can be used as a shortcut depending on how the MGF is expressed.

Working on some calculus drills to improve speed and accuracy. This is the first one… plain ol’ derivatives:

$latex\sqrt{x}\$

clearly this is not working… okay, I’m just trying to type a square root of x without going to codecogs for my code block…

how about…

[latex]sqrt{x}[latex]

nope. Okay, do I even *have* Jetpack with MathType installed? IDK.

I’ve got all these great news ideas and problems for/about probability and counting… but need to work out my mathtype so I don’t have to go to a website or create images for my equations. Stand by…

Darth Vader was a survivor, and so this approach, which employs the “survival function” will be referred to as The Darth Vader Approach.

The survival function is what is left over from your probabilities once you have summed up the probabilities up to a given value of your random variable *X = x*. In this way, we can define the survival function as follows:

The Darth Vader Approach utilizes the survival function to find the expected value of a random variable by simply summing from zero to infinity, and it works in the discrete case:

…and in the continuous case:

…and in mixed cases, we simply add the discrete and continuous portions together.

What is the advantage of this approach?

It is a short-cut to the expected value, especially in cases where the cumulative distribution function F(*x*) is known, but the probability density function, *f*(*x*) is not known.

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