Attached are the test from the lesson, the homework that was due on 10/08/17 and the problems from the contest held during the second part of the lesson. Homework for 10/15/17 are problems 1c and 4b from the contest, as well as problem 4 from the previous homework.
We begin our exploration of Egyptian Fraction Representation by studying the details of unit fractions. How do they add? What types of fractions can they represent (all of them!)? We close off by analyzing a recursive algorithm which allows us to slowly construct any fraction by smaller and small unit fractions that will (hopefully) terminate at some point.
Today we will learn about the (surprisingly difficult) task of converting the preferences of many individuals into a group preference. We will look at some methods for arriving at a group preference, and discuss some of the underpinning mathematics.
An exploration into the world of "perfect logicians". It is highly encouraged for students to right out the different cases possible for both the 3 and 5 hat variations.
We continued our study of unit fractions, and showed that all fractions can be written as a finite sum of unit fractions, showing that the Egyptian Fraction Representation exists for all fractions. We then wrapped up by proving other interesting properties of EFR.
Today we will be continuing out study of voting systems. We will also take a look at (and prove) Arrow's impossiblity theorem, which gives us important insights into the nature of voting systems.
We did many word problems associated with percentages that required them to be written as fractions and a small degree of algebra to keep track of percent/fractional multipliers. We wrapped up with a brief visit to compound interest, hinting towards continuous compound interest.
Over the course of your mathematical career I'm sure that various teachers, friends, or tutors have told you that infinity is not a number. Why is that? And if it isn't, does that mean that mathematics has nothing to say about the infinite? In this class, we will answer these two questions, as well as addressing many more.
Today we start a block on Number Theory. First topic: remainders. We have all seen them before, but how can we use them, and why do they even exist? Also: integral points on graphs of linear functions and an interesting system of equations.
We explored the strange hexahexaflexagon which has many more sides than a flat object normally does, 6 in total. We studied the patterns of the hexahexaflexagon: differeing orientations, which sides are connected to others, and what sides compose "main circuits" of the hexahexaflexagon.
Students will do math problems, all which are Halloween themed.
Math problems include, problems about working together, cutting shapes into pieces in a creative way, calculating distance in respect to time, creating combinations, and probability.
This week we will continue our tour of the infinite. Using the tools that we developed last week we will find some new bijections, uncover some more surprising results and prove Cantor's theorem, one of the most important, surprising and pradoxical results in modern mathematics.
We solved classic problems of how long it takes multiple people to accomplish a single task at different rates. Pipes filling pools, rabbits eating carrots, and non-uniform burning strings. Through this, the students learned that adding together varying rates of time is not straightforward and required quite a bit of algebra, percentages, and fractional multipliers.
Students will learn the basics of functions. Students will do problems where they have to identify the pattern of the functions and its inputs and outputs.
In this week's meeting we will use the tools that we developed over the past two weeks to look at paradoxes. Since time immemorial mathematicians and philosophers have pondered questions about the infinite and mused over their peculiar implications. Now that we have the tools necessary to have these conversations in earnest, we can discuss, appreciate and resolve some of these paradoxes.
We began the study of what many computer scientists call the remainder operator. Modulation of numbers over a given base, which leads to the development of equivalence relations where we can find numbers such as 2 and 8 equal to 0.
This week we will be concluding our study of the infinite. In this week, our goal is to get as close as we can to proving the Banach-Tarski paradox. This paradox shows that it is possible, using some very clever cuts, to cut a solid 3d sphere into three distinct parts, and rearrange those parts so that at the end one is left with two spheres identical in every way to the first.
We continued our study of modular arithmetic, and rigorously defined the equivalence relation between numbers under a given modulo. We found that this equivalence relation obeys many of the same properties as the traditional equals sign, which leads to a new structure of numbers.
With our study of the infinite complete, we are going to talk about generating functions. Sequences are one of the common themes across mathematics, and generating functions give us a different powerful method for answering questions about sequences by looking at them in a different light. We will introduce the idea of generating functions, and use them to solve a variety of problems.
We will played subtraction games such as Nim, Epmty and Divide, Chomp, and Dynamic Nim. All of these games could be solved using parity, powers, and inductive gamestate reduction, as the students quickly learned so they could beat their instructors!
This week we'll be finding some surprising connections between three disparate things, an ancient puzzle, counting on other bases, and the famous Sierpinski triangle.
We'll look at how to divide by 2, how to subtract, and if we're really making good progress, how to divide by 3.
No, really. And this may be the hardest week of the quarter.
We end the quarter by playing Hackenbush, Toads and Frogs, Cram, and Kayles in competition for small prizes. Also, we did some fun Santa's Sleigh dimensional analysis and Dreidel probability.
Today we are going to take a look at one of the most enduring 'real world' problems which is greatly aided by mathematical study, cryptography. The practice of concealing, decoding, and hiding messages has been around since the dawn of time, but the desire for efficient and unbreakable codes has accelerated in recent years.
We used strangely shaped pizzas and number patterns to predict (and prove) how a pattern will always continue, even at the trillionth term. Then, we formalized this pattern analysis as the logical statements that compose proofs by induction.
This week we are going to going to continue our study of cryptography and cryptographic schemes. We are going to talk more about symmetric schemes, and in particular we are going to talk about a process that lets two people, who have never spoken before, agree on a common secret shared key.
We continued our study into induction by looking at number patterns, and we took a more quantitative (numeric) approach to proving the logical statements that compose a proof by induction.
Students will play a game with Leap Frogs and do some Math Kangroo practice. Please omit the Leap Frog portion of this week. Handout is posted just for reference.
This week we will be concluding our discussion of cryptographic schemes by talking about one of the most exciting developments in cryptography over the last century, public key cryptography. One scheme of this kind if the celebrated RSA algorithm. Using RSA, two parties who have never met can exchange messages in public with total knowledge that their messages are secret.
We studied geometric number sequences (such as triangular and pentagonal numbers), and then predicted new terms based on the successive differences of previous terms. Afterwards, we formalized successive differences of sequences notation and explored the beginnings of discrete differentiation.
Knights can only tell the truth and nothing but the truth, but Liars can only tell lies. Students will discover who is a Knight and who is a Liar in this worksheet
This week in the LAMC we will be rolling dice and taking chances in our survey of probability. We will cover some of the basic rules for those who have no background, and will build up to understanding Bayes rule. this rule is extremely important rule in probability and statistics, and moreover is a rule which can change the way that you think about live, probably.
We delved into logical deduction problems, where we compare multiple logical statements, and determine their values (true or false) based on the value of some combination of them.
This week we will have a special guest, who will present on information theory. Information theory is the mathematics that gives meaning to randomess, and is responsible for letting people talk over the phone, use YouTube, and many more.
Created by one of Junior Circle's other instructors, Kristi Intara. In this handout we examine how to systematically perform calculations to find the day of the week (Sunday, Monday, Tuesday, etc.) a particular date is, e.g., your birthday.
We practiced effective test taking strategies with real Math Kangaroo problems such as drawing pictures, process of elimination, and skipping questions.
Students will learn how to defeat dragons by "cutting off" their heads and tails. However, there is a catch. When you cut one head off two heads grow back! Other rules will also take place in the worksheet.
This week we will finish up our work in information theory, With our expertise of entropy in tow, we will use it to analyize some more problems in probability. Finally we will learn about optimal codes.
We introduced the concept of graphs as sets of vertices and edges. Then we proved a number of lemmas and theorems associated with vertex degree, bipartiteness, and Eulerian Paths.
This week we will talk about the core of programming and computers, the idea of an algorithm. Algorithms are not only extremely common in real life (they are used every time you use anything electronic) but they are also quite interesting mathematically on their own. Today we will start talking about algorithms, agree on a definitions, and come up with lots and lots of examples of algorithms. Note, zero programming experience is required. We will not be programming in a particular language, but we will be talking about programs.
Today we are going to continue our discussion of algorithms, but instead of trying to try and write algorithms to do specific things, we are going to try and talk about algorithms themselves. Despite having obvious applications to computer science, algorithms were originally studied within mathematical Logic. Today we will talk about trying to write programs to analyze programs, and come up against one of the most famous hard problems, the so called Halting problem.
As we have done at the end of every quarter, this Sunday we'll be having a competition style class. You will be broken up into teams, and you will together compete to see who can answer the most questions with the fewest mistakes.
Nim, an example of a take-away game, is very old indeed. Since time immemorial those who know how to win at Nim have confounded friends, baffled enemies, and won numerous bar bets. Today we will letting you in on the secret, so that you too might be able to exercise that same power, if not entertain your friends for a little while. Although we will motivate the discussion with Nim, we will continue to talk about other mathematical games, and find and prove some surprising results.
Finding rational solutions to algebraic equations goes back to the ancient Greeks, but these days this problem finds applications ranging from pure mathematics to cryptography.
We solved the famous puzzle known as Instant Insanity using an innovative and thought provoking technique that utilizes many skills developed in past quarters.
This weekend we will be returning to the roots of mathematics, and study some problems that were solved a very, very long time ago. We are going to be studying plane, and in particular we will by studying what are called cyclic quadrilaterals, quadrilaterals which can be circumscribed by a circle.
Our first meeting looking at distances--with a twist. Normally we only consider the straight line distance (also known as Euclidean distance) or "as the crow flies" distance, but in this case we are looking at the Manhattan distance, where we are restricted.
This week we'll be studying a bit of game theory, the branch of mathematics that explains why we interact with each other the way that we do, and what we can do about it.
There is something that I would like you to do before class this weekend. Please find a half of an hour or so to play the game at the following URL. It frames the discussion that we'll be having, and is a very well developed, interactive tool. Plus, it's kind of fun! http://ncase.me/trust/
You should be able to play it in browser on any desktop/laptop.
We investigated the optimal algorithms to make change with different denominations of currency and how those algorithms can carry over to analyzing trees in graph theory, a structure that is ubiquitous in data science.
Today we're going to be talking about the field of combinatorics. Combinatorics is known as the math of counting, however the counting itself is usually not the point. The point is the clever arguments that allow the counting to be done at all. Combinatorics is a mainstay of mathematical puzzles and competitions alike, as it is an extremely rich field of math which is still elementary.
This weekend we'll be continuing out discussion of combinatorics. Last week we spent a lot of time talking about some introductory problems and reminding you about the idea of a bijection. This week we'll be using that familiarity and background knowledge to tackled even more interesting problem.
This weekend we will have a special guest Dr. Alex Austin will be giving a guest lecture to both the High School I and II groups. In his talk, Dr. Alex will be talking about fractals.
This week we'll be talking about algebraic numbers. Just like the real numbers are often thought of in terms of the rational and irrational, the real numbers can also be broken up into the algebraic numbers (which includes all of the rationals) and the non-algebraic (i.e. transcendental) numbers. This week we'll be talking about the former category, and learning more about the real number line than you ever wanted to know.
