Fall 2017 quarter  10/8/2017  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. Handouts: Test  Previous homework  Contest problems 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. Handouts: Handout  Solutions Handouts: Handout: Split the Difference  Solutions: Split the Difference 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. Handouts: Handout We'll be talking about the classic Liars and Truthtellers puzzles, how to solve them, and how to classify them. 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. Handouts: Hats and Doors  10/15/2017  Problems on tilings and the relation of grid colorings to solving them. Handout and homework for 10/22/17 attached. Handouts: Lesson handout.  Homework for 10/22/17. 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. Handouts: Handout  Solutoins Students will be working through a worksheet on Roman Numerals Handouts: Handout: Roman Numerals  Solutions: Roman Numerals
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.
Handouts: Handout An alternate method of multiplication based in binary numbers and the basic ideas of the distributive property. Handouts: Handout  10/22/2017  More tilings and colorings, factorization of polynomials. Handouts: Lesson handout.  Homework. 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. Handouts: Handout  Solutions Students learned the concept of "cuts" versus "pieces". This is demonstrated through cutting logs and cutting bagels. Handouts: Handout: Cutting Logs  Solutions: Cutting Logs 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. Handouts: Handout Another alternate method of multiplication. Compare to previous week's handout. Handouts: Handout  10/29/2017  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. Handouts: Solution to L3.2.  Solution to H3.2.  Solution to H3.3.  Lesson handout.  Homework. 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. Handouts: Handout  Solutions
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.
Handouts: Solutions: Happy Halloween 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. Handouts: Handout Real world random numbers don't look quite as random as some people think. Benford's law is an explanation of some surprising regularity. An assortment of problems ranging from probability to rates to logic puzzles. Handouts: Handout  11/5/2017  Continuing the topic from last week, remainders and divisibility. Handouts: Lesson handout  Homework  Solution to L4.1  Solution to L4.3  Alternative solution so L4.3  Solution to L4.4  Solution to L4.5 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 nonuniform 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. Handouts: Handout  Solutions 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. Handouts: Function Machine Handout  Function Machine Solutions 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. Handouts: Handout An amazing magic trick, in which a magician can untie any tangle just by being told a single rational number. We'll figure out how it works! A look at how we can organize multiple relates sets of objects/people/things. Handouts: Handout  11/12/2017  We introduce the concept of the greatest common divisor and prove some basic statements about it. Handouts: Lesson Handout  Homework  L5.2 Solution  L5.3 Solution  L5.4 Solution  L5.5 Solution 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. Handouts: Handout  Solutions We will continue with the Function Machine handout Handouts: Function Machine Part 2 Handout  Function Machine Part 2 Solutions 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 BanachTarski 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. Handouts: Handout What if we took derivatives of sequences instead of functions? It turns out, we can still learn a lot! An introduction to permutations and combinations. The class started on page 6. Handouts: Handout  11/19/2017  We formulate the Euclidean algorithm and use it to prove some important numbertheoretic lemmas. Handouts: Lesson Handout  Homework  L6.3 Solution  L6.4 Solution  H6.2 Solution  H6.3 Solution 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. Handouts: Handout  Solutions Students will use different strategies and their knowledge of area to solve problems involving a chessboard. Handouts: Chessboard Handout  Chessboard Solutions 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.
A second handout on permutations with some fun games involving switches based on diagonalizability. Handouts: Handout  Solutions  12/3/2017  We further use the Euclidean algorithm to investigate some linear diophantine equations. Handouts: Lesson handout  Homework  L7.2 Solution  L7.3 Solution  L7.4 Solution  H7.2 Solution 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! Handouts: Handout  Solutions We will continue with the game portion of Fun and Games with a Chess Board. 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. Handouts: Handout 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. Handouts: Handout  12/10/2017  Endofquarter game. 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. Handouts: Combinatorial Game Rules  Holiday Math Students will work on problem sets individually. The problem sets will comprise of problems based on everything we worked on this quarter.
Quick, fun, teambased math competition with prizes!
Handouts: Questions  Winter 2018 quarter  1/14/2018  Introduction to graph theory and geometry. Handouts: Lesson handout  Homework We rigorously defined the negation to a logical statement, and used that definition to explore contrapositives and proofs by contradiction. Handouts: Handout  Solutions Students will be working with symmetry and reflections as well as building on topics from last quarter. Handouts: Math With Sticks Handout  Math With Sticks Solutions 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. Handouts: Handout We'll look at geometries with finitely many points, see what's geometric about them, and also find applications to the real world! We will begin our exploration of game theory by starting with the example of subtraction games of varying subtraction sets. Handouts: Handout  1/21/2018  More graphs and geometry Handouts: Lesson Handout  Homework  H9.1 Solution  H9.2 Solution  H9.3 Solution 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. Handouts: Handout  Solutions Students will conduct an experiment with dice to introduce them to probability. Handouts: Probability Handout  Probability Solutions 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. Handouts: Handout The study of projective and affine finite geometries, and their applications to nongeometric problems Handouts: A second look at games, this time with more piles and complications. Handouts: Handout  1/28/2018  Continuation of graphs and geometry Handouts: Lesson Handout  Homework  L2.5 Solution  H2.2 Solution  H2.3 Solution 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. Handouts: Handout  Solutions 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. Handouts: Math Kangaroo Practice 2010  Leap Frog Handout  Math Kangaroo Practice Solutions 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. Handouts: Handout We looked at the strange algebraic objects known as Quandles, and their applictions to classifying knots. Handouts: Week 3 Handout A look at scaling various shapes through an interesting set of objects called "gnomons". Handouts: Handout  2/4/2018  Eulerian graphs and patallelograms. Handouts: Lesson Handout  L3.2 Solution  L3.6 Solution  H3.2 Solution 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. Handouts: Handout  Solutions 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 Handouts: Knights and Liars Handout  Knights and Liars Solutions 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. Handouts: Handout We reintroduced arithmetic functions, now in the context of number theoretic problems, culminating in classifying even perfect numbers. Handouts: Week 4 Handout An introduction to scaling areas and how equally scaling dimensions affects the final area. Handouts: Handout  2/11/2018  Eulerian paths and parallelograms. Handouts: Lesson Handout  Homework  L4.2 Solution  L4.4 Solution 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. Handouts: Handout  Solutions Students will learn how to solve problems when given the final answer but not the starting number. Note, number 4 should say 4 candies. Handouts: Backwards Reasoning Handout 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. Handouts: Handout We continue studying arithmetic functions, proving results on mobius inversion and using them to approximate the size of arithmetic functions. Handouts: 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. Handouts: Handout  2/25/2018  First lesson on game theory and continuation of geometry. Handouts: Lesson Handout  Homework We practiced effective test taking strategies with real Math Kangaroo problems such as drawing pictures, process of elimination, and skipping questions. Handouts: Handout  Solutions 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. Handouts: How to Fight a Dragon Handout 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. Handouts: Bonus Handout The sum of the reciprocals of the primes, plus the prime number theorem (to within a constant factor). Handouts: Handouts: Handout  3/4/2018  Winning/losing positions and circles. Handouts: Lesson Handout  Homework  L6.2 Solution  L6.5 Solution 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. Handouts: Handout  Solutions Students will relate math to rhythm using music notes. Handout made by Assistant Instructor: Ashin! Handouts: Counting Beats Handout 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. Handouts: Handout What are all the ways we can hang a painting with a string using 1 nail? What about 2 nails? n nails? Handouts: Handout  3/11/2018  Stealing strategies and relative positions of 2 circles. Handouts: Lesson Handout  Homework  H6.1 Solution  H7.1 Solution We finished exploring the ideas from last week's packet, and will move on to higher level concepts next quarter. Students will do review questions that cover the entire quarter. Some will be done in class on the board. Handouts: EE: Review Questions Handout  EE: Review Questions Solutions 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. Handouts: Handout A look at the harder questions of previous exams and strategies for approaching questions you cannot initially solve. Handouts: Solutions  3/18/2018  Stealing strategies continuation, along with central angles and arcs on a circle. Handouts: Lesson handout Traditional math dominoes competition for glory and fame! Handouts: Handout  Solutions Students will do an Indivudal Problem Set that covers topics from the entire quarter. 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.  Spring 2018 quarter  4/8/2018  Today we start learning a new powerful proof technique  induction. Handouts: Lesson Handout  Homework Even though we were a little late, we learned some amazing properties of Pi (and Phi!). Handouts: Handout  Solutions Students will work on logic negations. Handouts: Mr. No Part 1 Nim, an example of a takeaway 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. An investigation into a famous gameshow problem involving changing probabilities. Handouts: Handout  4/15/2018  We proceed by applying induction to problems in combinatorics. Handouts: Lesson Handout  Homework  L1.2b Solution  L1.3 Solution We solved the famous puzzle known as Instant Insanity using an innovative and thought provoking technique that utilizes many skills developed in past quarters. Handouts: Handout & Solutions Students will continue with the worksheet from last week and we will be introducing Venn Diagrams. Handouts: Mr. No Part 2
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. Handouts: Handout We'll be learning about Infinity and Set Theory with guest instructor Dr. Omer BenNeria Our first meeting looking at distanceswith 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. Handouts: Handout  4/22/2018  We proceed by applying induction to problems in algebra. Also: more inscribed angles. Handouts: Lesson Handout  Homework  L2.1 Solution  L2.6 Solution We finished up our solution of Instant Insanity and continued with graph theory theorems and applications. Students will learn how to find the mass of mystery items when comparing them to other items on the "scale" Handouts: Challenge Problems  Balance Scale Handout  Balance Scale Solutions
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.
Handouts: Handout We'll be continuing our exploration of rational points by exploring congruent numbers: the areas of rationalsided right triangles. Our second look at noneuclidean geometry and how shapes change under our new definition of distance. Handouts: Handout  4/29/2018  This week, we look at applications of induction to arithmetic. Also: more inscribed angles Handouts: Lesson Handout  Homework  L3.3 Solution  H3.3 Solution 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. Handouts: Handout  Solutions  Challenge Solutions Students will use their knowledge of Balanced Scales to begin to learn about binary numbers Handouts: Binary Notation Part 1 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. Handouts: Handout We investigate the uses of an abacus to improve the speed and accuracy of simple calculations as well as the basic principles of how to use one. Handouts: Handout  5/6/2018  We introduce bipartite graphs, as well as continue with cyclic quadrilaterals. Handouts: Lesson Handout  Homework  L4.3 Solution  H4.2 Solution We used some techniques from modular arithmetic to play tricks on every student! Handouts: Handout  Solutions We will be continuing working with binary numbers Handouts: Binary Numbers Part 2 Handout
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.
Handouts: Handout Handouts: Handout  5/13/2018  We continue with bipartite graphs and geometry. Handouts: Lesson Handout  L5.2 Solution  L5.4 Solution We used physical intution and simple pictures to begin exploring opposing torques on a bar culminating in the Law of Levers. Handouts: Handout  Solutions We will be continuing working with binary numbers Handouts: Binary Number Part 3 Handout 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. Handouts: Handout  5/20/2018  We use the framework of bipartite graphs to investigate the Stable Marriage Problem. Handouts: Lesson Handout  Homework  L6.2 Solution  L6.4 Solution We explored the foundations of probability theory through blind teachers throwing darts. Handouts: Handout  Solution We will be continuing working with binary numbers 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 nonalgebraic (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. Handouts: Handout We'll study several mathematical impediments to crossing a river. Our final look at the relationship between distances, rates, and time with some additional challenge problems. Handouts: Handout  5/27/2018  No meeting today! Holiday!  6/3/2018  We will get to explore some very strange dice hands on. Students will work with word problems that involve logic This weekend we will be finishing up our discussion of the algebraic and transcendental numbers. Last time we defined what algebraic numbers are, and talked about some of their properties. This time we are going to use their definition and properties to prove, among other things, that algebraic numbers are actually extremely rare and yet we don't know many numbers which are not algebraic. This apparent paradox and more will be discussed on Sunday. Handouts: Handout We'll learn what the Continuum Hypothesis says, and use it to prove a somewhat counterintuitive result about small sets in the plane known as "clouds." Handouts: Handout  Supplement  Solutions  6/10/2018  We introduce Discrete Finite automata and go through some basic examples. Handouts: Lesson handout We will be displaying review questions on a powerpoint and moving on to the next problem as a group so that no one gets stuck on a problem. As usual, we'll end our quarter with Math Relays!  Summer 2018 quarter  6/24/2018  Handouts: Handout 
