ORMC Meetings Archive • Fall 2007–Spring 2024

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 Spring 2018 quarter // Filter groups by: AdvancedBeginnersBreaking Numbers into PartsEarly ElementaryHigh School IHigh School IIJunior Circle 6/3/2018 Beginners [grades 5-6]: Non-transitive Dice We will get to explore some very strange dice hands on.Early Elementary [grades 1-2]: Logic Word Problems Students will work with word problems that involve logicHigh School I [grades 9-10]: Transcendental Numbers 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: HandoutHigh School II [grades 11-12]: The Continuum Hypothesis and Clouds We'll learn what the Continuum Hypothesis says, and use it to prove a somewhat counter-intuitive result about small sets in the plane known as "clouds."Junior Circle [grades 3-4]: Modular Arithmetic and Ciphers Handouts: Handout | Supplement | Solutions 6/10/2018 Advanced [grades 7-8]: Discrete Finite Automata (Kyle Hess) We introduce Discrete Finite automata and go through some basic examples.Handouts: Lesson handoutEarly Elementary [grades 1-2]: Powerpoint Individual Problem Set 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. High School II [grades 11-12]: Relays As usual, we'll end our quarter with Math Relays!Junior Circle [grades 3-4]: Final Meeting of Spring Quarter Summer 2018 quarter // Filter groups by: AdvancedBeginnersEarly ElementaryJunior Circle 6/24/2018 Early Elementary [Grades 1-2 in Fall ?18]: Summer session starts Junior Circle [Grades 3-4 in Fall ?18]: Hats and Doors Handouts: Handout Fall 2018 quarter // Filter groups by: AdvancedBeginnersBreaking Numbers into PartsEarly Elementary IEarly Elementary IIHigh School IHigh School IIJunior CircleOlympiad Training 10/7/2018 Advanced [grades 8-9]: Warm-up problems We warm up for the new academic year by solving problems from some recent mathematical competitions.Handouts: Lesson HandoutBeginners [grades 6-7]: A survey of Olympiad problems For our first week of the year, we'll be working on a handout containing olympiad problems from a long time ago. The problems do not follow any particular theme other than general knowledge of algebra.Handouts: HandoutEarly Elementary I [grades 1-2]: Split the difference (Oleg Gleizer) Handouts: Split the Difference handoutEarly Elementary II [grades 2-3]: First Meeting! Ciphers Its our first meeting! We will be introducing ciphers to the class. Only complete pages 1-4Handouts: Ciphers Part 1High School I [grade 10]: Gaussian Integers I We will introduce Gaussian integers in order to decide which integers can be written as a sum of two squares.Handouts: Gaussian Integers IHigh School II [grades 11-12]: Math on Chess Board (Avet Grigoryan) We will start the school year with an an overview of math problems and puzzles involving a chessboard (with and without chess pieces) and solving techniques including tiling, coloring and invariance principle. Handouts: Problems and Select SolutionsJunior Circle [grades 4-5]: Cryptarithms and Logic Handouts: Handout 10/14/2018 Advanced [grades 8-9]: Quadratic Equations I (Vieta's Theorem) We start our exploration of quadratic equations with Vieta's Theorem. As usual, we also do some geometry.Handouts: Lesson Handout | HomeworkBeginners [grades 6-7]: Fractions and Decimals This week we'll develop a theory for converting fractions to decimals and vice versa. Everyone knows that lots of things can be represented by both a fraction or decimal, but most people are at least a little fuzzy on how the two are related. This week we'll cut through the fuzz and by the end you'll be a pro at fractions and decimals.Handouts: HandoutEarly Elementary I [grades 1-2]: Sharing Problems and Birthday Cakes (Oleg Gleizer) We will first check homework focusing on harder problems. Then we will solve a warm-up problem, get back to sharing problems, and end up cutting birthday cakes of various shapes. Handouts: Sharing Problems handoutEarly Elementary II [grades 2-3]: Ciphers Part 2 We wi be continuing past page 4 on our Ciphers worksheet. The goal for this class is to work on our problem solving skills and to try again when our first assumption does not work. (Pages 5-8) (WE DID NOT DO PROBLEM #8)Handouts: Ciphers Part 2High School I [grade 10]: Gaussian Integers II We will wrap up the discussion of Gaussian integers and prove which integers are the sum of two squares.Handouts: Gaussian Integers IIHigh School II [grades 11-12]: Symmetries and Groups (James Cameron) In this meeting we will explore the symmetries of different objects like squares, rectangles, and coins. We will see how these symmetries interact with each other to form a structure called a group.Handouts: Problems and Multiplication TablesJunior Circle [grades 4-5]: Leap Frog A look at how even with the opportunity to team up against another player or all work together this game is still very tough (maybe too tough??).Handouts: Solutions | Handout 10/21/2018 Advanced [grades 8-9]: Functions and Quadratic Equations II We solve some problems on the properties of general functions, and learn to apply Vieta's theorem from last week. As usual, some geometry.Handouts: Lesson Handout | HomeworkBeginners [grades 6-7]: Place-Value Numerals The students ventured away from our decimal system and explored binary, trinary, and hexadecimal base systems. These form the basis of communication between modern computers, and also allow for encryption of English messages much more simply than the decimal system would, so it is important that our students are familiar with them.Handouts: HandoutEarly Elementary I [grades 1-2]: Roman Numerals I (Oleg Gleizer) We will begin to study Roman numerals. Handouts: Roman Numerals I handoutEarly Elementary II [grades 2-3]: Mayan Numbers Part 1 Today will have have an introduction to Mayan Numbers plus Ken-Ken warm up! We will also be collecting and correcting the Cipher worksheet so make sure your student brings it to class! Homework: Finish the Mayan Numbers worksheet.Handouts: Mayan Numbers Part 1 High School I [grade 10]: Gini Index We introduce the Gini index, an economic metric to measure wealth inequality.Handouts: Gini IndexHigh School II [grades 11-12]: Symmetries and Groups II (James Cameron) Having experimented with the groups of symmetries of the rectangle+string model, we will define groups and explore associated concepts such as subgroups, group actions, isomorphisms, orbits, and stabilizers. Handouts: Problems Junior Circle [grades 4-5]: Sets and Functions Part I Handouts: Handout | Later Pages of Handout 10/28/2018 Advanced [grades 8-9]: Quadratic Equations III (Quadratic Inequalities) We continue exploring quadratic equations by completing the square and learning the formula for the roots. We also start working towards quadratic inequalities, in particular seeing when a quadratic function is positive or negative.Handouts: Lesson Handout | HomeworkBeginners [grades 6-7]: Place value systems continued This weekend we will be continuing our study of place-value systems that are not base 10. Last week was a nice gentle introduction to the topics that consisted of a lot of computation. This week we are going to use some of the intuition that we built up last week to solve some more theoretical questions, and come to some surprising conclusions!Handouts: HandoutEarly Elementary I [grades 1-2]: Roman numerals II (Oleg Gleizer) We will continue studying the first handout on Roman numerals. Once finished, we will switch to the second handout. Handouts: Roman Numerlas II handoutEarly Elementary II [grades 2-3]: Mayan Numbers Part 2 We will continue with our lesson on Mayan Numbers and turn in the Mayan Number Part 1 worksheet. For the Mayan Numbers Part 2 worksheet, we did not do number 7.Handouts: Mayan Numbers Part 2High School I [grade 10]: Quadratic Reciprocity I We introduce Legendre symbols and quadratic reciprocity to study residues modulo primes. Handouts: Quadratic Reciprocity IHigh School II [grades 11-12]: Circle Inversions In this meeting, our goal is to construct a strange new geometry where straight lines are circles and triangles have angles that sum to less than 180. We start with circle inversions and then introduce the Poincaré disc model. Handouts: Handout | SolutionsJunior Circle [grades 4-5]: Sets and Functions Part II Handouts: Handout 11/4/2018 Advanced [grades 8-9]: Quadratic equations IV We take a step back from learning fundamental facts about quadratic equations, and practive what we have already learnt on some problems. Geometry also makes an appearance.Handouts: Lesson Handout | HomeworkBeginners [grades 6-7]: Problem Solving Last week we finished talking about place value systems, and this week we'll be letting out hair down and doing some good, old fashioned, problem solving. There is no specific theme for this week but we will be using some of the things that we developed in the past couple of weeks. Handouts: HandoutEarly Elementary I [grades 1-2]: Roman Numerals II, continued (Oleg Gleizer) We will continue to study the Roman Numerals II handout. Handouts: Quiz 2Early Elementary II [grades 2-3]: Logic Gates Part 1 We wi be learning about the different types of Logic Gates and how to use them. We will also be correcting Mayan Numbers Part 2. Homework: pages 1-17Handouts: Logic Gates FullHigh School I [grade 10]: Quadratic Reciprocity II We explore applications of quadratic reciprocity. Handouts: Quadratic Reciprocity IIHigh School II [grades 11-12]: Hyperbolic Geometry I We will continue our exploration of the Poincaré disc and prove facts about hyperbolic lines and shapes. Handouts: Solutions | Handout Junior Circle [grades 4-5]: Sets and Functions Part III Our final look at sets and functions. In our final meeting on this topic we will discuss bijections betwen infinite sets and show how that can help determine equivalence.Handouts: Handout 11/11/2018 Advanced [grades 8-9]: Quadratic equations V (Parabolas) We introduce parabolas -- graphs of a quadratic equations, and solve some problems regarding them.Handouts: Lesson Handout | HomeworkBeginners [grades 6-7]: Permutations This weekend we are going to start a multi-week study of the topic of permutations. We are going to start our study by defining what a mathematical permutation is, learn how mathematicians notate permutations and prove some elementary results.Handouts: HandoutEarly Elementary II [grades 2-3]: We will have class! This is Veteran's Day weekend (a three day weekend) but we will be having class. Today we will be finishing the Logic Gates worksheet! **IF YOUR CHILD MISSED THIS CLASS YOU ARE EXCUSED AS THE AIR QUALITY AND EVACUATIONS LED TO STUDENT ABSENSES.Handouts: Logic Gates FullHigh School I [grade 10]: Tropical Geometry I Instead of our typical definition of addition and multiplication, tropical arithmetic looks at minimum and addition operations. We will graph and find roots of tropical polynomials.Handouts: Tropical Geometry I and IIHigh School II [grades 11-12]: Hyperbolic Geometry II We will keep working in the Poincare disc and discover phenomena peculiar to hyperbolic geometry such as AAA congruence, Lobachevskii's Theorem, and Schweikart's constant.Handouts: Section 2Junior Circle [grades 4-5]: Parity and Puzzles Today, we will look at how certain problems can be resolved by viewing them in the context of parity.Handouts: Handout 11/18/2018 Advanced [grades 8-9]: Quadratic equations VI We use the theoretical properties of quadratic equations established during the last few weeks to solve some concrete problems. Also: more geometry.Beginners [grades 6-7]: Permutations continued This weekend we will be continuing our study of permutations. Now that we have a basic understanding conceptualization of permutations and have some basic notation down, we are going to apply that notation to better understand the 15 puzzle. By the end of the class we won't have solved the puzzle, but we will be a lot closer. Handouts: HandoutEarly Elementary I [grades 1-2]: Roman numerals IIS (Oleg Gleizer) We will discuss the ambiguity of Roman numerlas, will learn how to convert decimals to Roman numerals, and will learn writing dates as Roman numerals. Handouts: handoutEarly Elementary II [grades 2-3]: Solids and Projections Part 1 Today we will be manipulating shapes in our heads and drawing their projections on our papers. This is Part 1. Homework will be announced in class. High School I [grade 10]: Tropical Geometry II We will continue our study of tropical arithmetic by proving a version of the Fundamental Theorem of Algebra for tropical quadratic polynomials.Handouts: Tropical Geometry I and IIHigh School II [grades 11-12]: Hyperbolic Geometry III We look at more peculiar hyperbolic facts, like the hyperbolic Pythagorean theorem and the angle of parallelism. Handouts: SolutionsJunior Circle [grades 4-5]: Practicing with Percentages We will look at several problems dealing with percentage decreases, increases, and changes in general.Handouts: Handout 11/25/2018 Beginners [grades 6-7]: No Class this Week! Have a happy Thanksgiving!Early Elementary II [grades 2-3]: NO CLASS - THANKSGIVING WEEKEND No class this weekend because everyone is on Thanksgiving Break!High School II [grades 11-12]: No class - Thanksgiving Break 12/2/2018 Advanced [grades 8-9]: Individual Problem Solving We see what we have learned about geometry and quadratic equations with an individual problem-solving session.Handouts: HandoutBeginners [grades 6-7]: Permutations Finale Today we are going to finish up our study of permutations and finally resolve the case of the 15 puzzle. After that is done, we will take a look back at what we have done, and take note of some interesting results that we have proved along the way. Handouts: HandoutEarly Elementary I [grades 1-2]: Cut the log (Oleg Gleizer) We will do some log cutting as a first step in learning topology. Handouts: handoutHigh School I [grade 10]: Golomb Ruler Given a ruler, how many inch markings can you remove and still measure each increment between 1 and 12 inches? Is there some way to construct a 12-inch ruler such that each increment from 1 to 12 can be measured in a unique way?Handouts: Golomb RulerHigh School II [grades 11-12]: Electrical Circuits and Random Walks We will have a short review quiz. Then, Aaron Anderson will talk about how electrical circuits correspond to random walks on the vertices of graphs. Handouts: HandoutJunior Circle [grades 4-5]: Review of Fall Quarter Material We will review several of the most difficult topics from this quarter. 12/9/2018 Advanced [grades 8-9]: Math Dominoes Fun and games!Beginners [grades 6-7]: Permutations post Finale and Math Relay This weekend we are going to have our final class for the quarter. For the first hour we are going to take stock of everything that we have proved so far about the 15 puzzle, and the second half will be a class-wide relay with prizes!Handouts: HandoutEarly Elementary I [grades 1-2]: Cut the bagel. (Oleg Gleizer) We will cut some bagels as a step in learning topology. Handouts: handoutHigh School I [grade 10]: Presentations High School II [grades 11-12]: Electric Circuits and Random Walks II We will continue talking about the correspondence between voltage, resistance, and current in circuits with random walks on graphs. Handouts: Challenge ProblemsJunior Circle [grades 4-5]: Math Dominoes A fun competition during class to determine which team of students can solve questions from this topics this quarter and beyond the quickest and most accurately. Winter 2019 quarter // Filter groups by: AdvancedBeginnersBreaking Numbers into PartsEarly Elementary IEarly Elementary IIHigh School IHigh School IIJunior CircleOlympiad Training 1/13/2019 Advanced [grades 8-9]: Weighings, Logic and Geometric Constructions We start a few new topics for the quarter -- weighings, logic and ruler & compass constructions.Handouts: Lesson Handout | HomeworkBeginners [grades 6-7]: Geometry I This weekend we are going to start our study of geometry starting waaaay back at the start of Greek mathematics. This weekend we'll be (re)learning how to use a compass and straight edge. As such, please remember to being a compass and straightedge with you to class today! I hope that you are all as excited to resume the LAMC as I am. Handouts: HandoutEarly Elementary I [grades 1-2]: Squares on a chessboard (Oleg Gleizer) The utlimate goal of this lesson is to count all the squares on the chessboard, 1 by 1, 2 by 2, 3 by 3, and all the way to 8 by 8. Handouts: handoutEarly Elementary II [grades 2-3]: Quilt Mending! First Class of the quarter and we will be doing a brand new topic!Handouts: Quilt Mending | Quilt Mending SolutionsHigh School I [grade 10]: Continued Fractions I We will introduce continued fractions and learn how to calculate them. We will also investigate the relationship between the irrationality of a number and properties of its continued fraction expansion.Handouts: Continued Fractions IHigh School II [grades 11-12]: Math Relay We will have a competition to solve problems for prizes!Junior Circle [grades 4-5]: Special Lesson by Nitya (Futoshiki Squares) Handouts: Handout 1/20/2019 Advanced [grades 8-9]: Weighings, Logic and Geometric Constructions II We continue with the topics of weighings, logic and straightedge & compass constructions.Handouts: Lesson Handout | HomeworkBeginners [grades 6-7]: Geometry II Today we are going to continue our studies of Geometry, and learn more about what you can do using a compass and ruler, and finally talk about geometry as you have seen it in school. Today will be a nice mix of hands on drawing/calculation with the compass and ruler, as well as a bit of proving using claim / reason charts.Handouts: HandoutEarly Elementary I [grades 1-2]: Games on the chessboard (Oleg Gleizer) We will figure out ways to always win in a bunch of fun chessboard games. Handouts: handoutEarly Elementary II [grades 2-3]: Math Kanagroo Practice Part 1 We will be practicing Math Kanagroo questions. We will focus on reading the questions fully, strategies find the answer, and practicing to show our work. Handouts: Math Kangaroo Practice 1High School I [grade 10]: Continued Fractions II We will continue our study of continued fractions with an imporant application in number theory: Given an irrational number, how efficiently can it be approximated by rational numbers? Continued fraction expansions play an important role in solving this problem.Handouts: Continued Fractions IIHigh School II [grades 11-12]: Algorithms We will start a lesson studying algorithms: what are they, and how do they work?Handouts: HandoutJunior Circle [grades 4-5]: Pseudo Paradoxes A look at a series of proven statements. But something about their conclusions seem a bit odd...Handouts: Handout 1/27/2019 Advanced [grades 8-9]: Weighings, Logic and Geometric Constructions III More straightedge and compass constructions, along with harder weighings and logic problems.Handouts: Lesson Handout | HomeworkBeginners [grades 6-7]: Geometry III Today will be out third and likely final Geometry session of the quarter. During the first week we got some practice using the straightedge and compass, during the second we had a gentle introduction to two column proofs, and for this last week we'll be solving problems using some of what we have learned. Not all of the problems look like they are 'classic' straightedge and compass problems, but we will find that using just those two implements, you can do more than you might think. For this final week please bring your straightedge and compass with you to class. Handouts: HandoutEarly Elementary I [grades 1-2]: One more chessboard game, then the abacus. (Oleg Gleizer) After a warm-up, students will figure out a winnign strategy for a fun chessboard game, called Move a Rook into the Corner. If time remains, students will start learning how to use an ancient computer, called the abacus. This will allow students to better understand the working and advantages of the decimal place-value numeral system currently in use by humanity. Handouts: handoutEarly Elementary II [grades 2-3]: Babushka Squares Students will be doing an activity and worksheet that involve manipulating shapes to fit them all in the smallest box possible. If your child missed this week, problem #2 sadly cannot be completed without the manipulatives we used in class. Homework is the beginning of problem #3. All students have to do is create all many 7 square shapes as they can.Handouts: Babushka SquaresHigh School I [grade 10]: Mathematics of musical scales (Olga Radko In this power-point presentation, we will address the following questions: Why do some musical intervals sound pleasant, while others do not? Why do we have exactly 12 notes in an octave of a piano? Why aren't distances between frets on a flute or a guitar equal to each other? The answers, surprisingly, involve deep mathematical analysis involving continued fractions, the problem of doubling the cube, and rational approximations. High School II [grades 11-12]: Algorithms II We will learn about more aspects of algorithms, such as efficiency and computational complexity. Handouts: Challenge ProblemsJunior Circle [grades 4-5]: Pseudo Paradoxes II We will continue with last week's handout. 2/3/2019 Advanced [grades 8-9]: Invariants and Geometric Constructions We focus more on using invariants in combinatorial problems about processes, and also continue with straightedge & compass constructions.Handouts: Lesson Handout | HomeworkBeginners [grades 6-7]: Vectors Today we will be working with vectors, and connecting them to the previous work that we have been doing on geometric constructions. Vectors are in some ways just like the counting numbers, and in other ways are very geometric, unlike the counting numbers. This dual nature of vectors makes them both interesting and useful. We will start to uncover this duality today!Handouts: Handout | Auxiliary HandoutEarly Elementary I [grades 1-2]: The abacus. (Oleg Gleizer) We will use the abacus to study decimal place-value numerals. Early Elementary II [grades 2-3]: Math Kangaroo Practice 2 We will be continuing to practice Math Kangaroo. HOMEWORK: 3rd and 4th grade practice test.. The MK Practice 2 is not homework, please don't have your student complete this at home.Handouts: MK Practice 2 | Practice TestHigh School I [grade 10]: Limits of Sequences We will introduce the formal defnition of a limit of a sequence and develop basic properties.Handouts: Limits of SequencesHigh School II [grades 11-12]: Cantor Set We're going to look at the famous Cantor set, its construction, and some of its oddities.Handouts: HandoutJunior Circle [grades 4-5]: Parity and Proof Writing We will look at several parity related problems and then continue onto two different important topics related to proofs.Handouts: Handout 2/10/2019 Advanced [grades 8-9]: Invariants and Geometric Constructions II Handouts: Lesson Handout | HomeworkBeginners [grades 6-7]: Math Kangaroo This week we are going to do something completely different from what we have been working on the past month. We are making a hard right turn from geometry and instead we will be focusing on problem solving with an emphasis on solving problems for the upcoming math kangaroo competition. You certainly don't have to aim to take the exam to enjoy this weekend's class, indeed When I was still in school, I enjoyed these contests no so much because I was competitive, but more because the questions themselves were often quite beautiful. You do not need anything special for this week's lesson!Early Elementary I [grades 1-2]: The abacus and extra problems. (Oleg Gleizer) We will attenpt to finish the abacus handout. If time remains, we will solve extra problems from the next handout. Handouts: extra handoutEarly Elementary II [grades 2-3]: Final Math Kangaroo Review We will be going over the practice test from last week and possibly doing some extra practice problems while discussing strategies for taking the Math Kangaroo exam.High School I [grade 10]: Sequences II We will continue our practice with formally proving limits of sequences and we will prove some additional properties of sequence limits.Handouts: Limits of Sequences IIHigh School II [grades 11-12]: Cantor Set II We will continue studying the Cantor set, invesitgating properties such as its cardinality and "dimension." Once we develop some notions of dimension, as a bonus we will also look at other fractal sets and their dimensions.Handouts: Handout | Bonus ProblemsJunior Circle [grades 4-5]: IFF and Parity Handouts: Handout 2/17/2019 Beginners [grades 6-7]: No Class Today On account of the holiday, there will be no Math Circle this weekend. See you next week!Early Elementary II [grades 2-3]: NO CLASS: PRESIDENT'S DAY WEEKEND No class this week because of the holiday/three day weekend 2/24/2019 Advanced [grades 8-9]: Invariants and Geometric Constructions III We continue with invariants, semi-invariants and geometric constructions.Handouts: Lesson Handout | HomeworkBeginners [grades 6-7]: Combinatorics I Today we are going to talk about a subject of math that really counts, 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.Early Elementary I [grades 1-2]: Balance scale. (Oleg Gleizer) Students will first take a quiz on the abacus. Then the class will learn how to use a balance scale for weighing objects and solving math problems. Handouts: handoutEarly Elementary II [grades 2-3]: Finding Fake Coins Today we will find fake coins in different scenarios using a balance scale. Handouts: Fake CoinsHigh School I [grade 10]: Integer-Valued Polynomials We characterize all polynomials that have integer outputs for integer inputs.Handouts: Integer-Valued PolynomialsHigh School II [grades 11-12]: The Gini Index (Dan Hoff) We’ll explore a measure of economic inequality known as the Gini Index. In particular, we’ll learn what it is, how to calculate it, and what some of its strengths and limitations are. Handouts: Handout | SolutionsJunior Circle [grades 4-5]: Math Kangaroo Review Handouts: Handout 3/3/2019 Advanced [grades 8-9]: Invariants and the Intercept Theorem We continue our route to triangle siilarity by learning the intercept theorem.Handouts: Lesson Handout | Homework | ReadingBeginners [grades 6-7]: Combinatorics II This week we will be continuing what we stared last week and talk more about combinatorics. We will be starting with a brief review of what we spoke about last week, before moving onto completely new problems.Handouts: HandoutEarly Elementary I [grades 1-2]: Math Kangaroo preparation. (Oleg Gleizer) We will study in preparation for the Math Kangaroo competition, up-coming on 3/21. Early Elementary II [grades 2-3]: Evens and Odds Today we will be doing a worksheet that allows the students to discover the patterns of even and odd numbers. The warm-up will be Ken-Ken. We will be collecting the Fake Coins worksheet from last week.Handouts: Evens and OddsHigh School I [grade 10]: Mass Point Geometry The center of mass of a system of finitely many point masses is relatively easy to calculate. We will explore certain planar geometric problems that can be easily solved when we assign masses to relevant points.Handouts: Mass Point GeometryHigh School II [grades 11-12]: Weighings, information exchange, and other abstractions (Sherry Gong) In this lesson we will do combinatorial weighing and probability problems, with some related problems about information exchange.Junior Circle [grades 4-5]: Planes, Tetraheda, and Cross Sections Handouts: Handout 3/10/2019 Advanced [grades 8-9]: Similar Triangles Our multi-week journey towards similar triangles culminates with the proof of the two main theorems about them.Handouts: Lesson Handout | HomeworkBeginners [grades 6-7]: Pi day! Today we'll be taking a break from our normally scheduled content to talk about everyone favorite geometric constant, pi! Pi is of course one of the most well known mathematical constants and has been studies from ancient Greece until now. We'll do a couple problems about Pi and compute a whole bunch of things geometrically. Handouts: HandoutEarly Elementary I [grades 1-2]: Balance scale and binary numbers. (Oleg Gleizer) We will first use a balance scale to introduce binary numbers. Then, we will study their properties. Handouts: handoutEarly Elementary II [grades 2-3]: Pie Day! Today we celebrate Pie Day as it is coming up next week! We will be collecting last week's worksheet and correcting a couple of problems for credit.Handouts: EE Pi Day WorksheetHigh School I [grade 10]: Pi Day (almost) Pi originated as the ratio of a circle's circumference to its diameter in the plane. We will see how this differs for circles on a sphere.Handouts: Pi DayHigh School II [grades 11-12]: Pi and Probability To celebrate Pi day, we'll look at some probability questions involving pi. For example: suppose you have equally spaced lines and you drop a toothpick. What is the probability that the tootpick crosses a line? Junior Circle [grades 4-5]: Pi Day Based Problems Handouts: Handout 3/17/2019 Beginners [grades 6-7]: Math Relay For our last meeting of the quarter, we will be having another math relay! As we did last quarter, we'll be splitting people up into teams an seeing which team can get through the most problems.Early Elementary I [grades 1-2]: Balance scale and binary numbers, continued. (Oleg Gleizer) We will continue the study of binary numbers using a balance scale. Early Elementary II [grades 2-3]: Creative Student Worksheet Today we will be completing a worksheet that consists of word problems and math questions created by the Early Elementary II students! This is the last class before a 2 week break!Handouts: Creative Student WorksheetHigh School I [grade 10]: Presentations High School II [grades 11-12]: More Pi Problems We will continue the worksheet on problem related to pi. We will find the probability that two randomly selected integers are coprime and calculate some continued fractions.Handouts: Handout | Quiz RubricJunior Circle [grades 4-5]: Pi Day Part II (Area) Handouts: Handout Spring 2019 quarter // Filter groups by: AdvancedBeginnersBreaking Numbers into PartsEarly Elementary IEarly Elementary IIHigh School IHigh School IIJunior CircleOlympiad Training 3/24/2019 Early Elementary II [grades 2-3]: NO CLASS We will not have class today. Remember to re-register for Spring Quarter! 3/31/2019 Early Elementary II [grades 2-3]: NO CLASS We will not have class today. Remember to re-register for Spring Quarter! 4/7/2019 Advanced [grades 8-9]: Combinations We start a new topic of combinations and Pascal's triangle. Also a problem on triangle similarity.Handouts: Lesson Handout | HomeworkBeginners [grades 6-7]: How big are big numbers really? We've all seem some pretty big numbers in our day. Sure maybe you've seen a 81, a 104 or maybe even (if you are very worldly) 3841. But, jut how big are big numbers really? How big is a number like 52!, the number of ways to arrange the number of cards in a 52 card deck? What about the number of times you would have to flip 200 coins before you got all heads? Can you even say which one is larger? Today we'll answer this question and more, by introducing the logarithm, a function that is extremely useful for making sense of the super massive.Handouts: HandoutEarly Elementary I [grades 1-2]: Decimal and binary numbers. (Oleg Gleizer) We will reviev binary numbers and then take a quiz. Then we will start figuring out what is in common between decimal and binary numbers. Handouts: handoutEarly Elementary II [grades 2-3]: Continuation of Student-Made Worksheet Welcome back! Please bring your worksheets from the last time we had class. We will be continuing to go through the problems at all the students created! Homework is to finish at least 25 problems on the worksheet. Reminder to re-register your student for Spring Quarter!Handouts: Creative Student WorksheetHigh School I [grade 10]: Groups of Symmetries We will discuss some surprising symmetries with a visual aid. This week will lead to an in-depth exposition of abstract group theory.High School II [grades 11-12]: Continued Fractions We will continue studying continued fractions in more detail, looking at special properties and details of convergence.Handouts: HandoutJunior Circle [grades 4-5]: Insect World (I) Handouts: Handout 4/14/2019 Advanced [grades 8-9]: Combinations II (Pascal's Triangle) We continue studying combinations, introducing the binomial formula and Pascal's triangle.Handouts: Lesson Handout | HomeworkBeginners [grades 6-7]: Goats Maybe on the whole you felt like last week was entirely too much. Maybe you thought that the numbers that we spoke about last time were too large, logarithms were too confusing and you are ready to take a mathematical break and return to a more pastoral existence. Good news! This next week we are talking about goats. That's right goats, everyone's favorite ornery, stubborn, ravenous livestock. We will find that making sure that goats have enough to eat is more mathematical then you might have thought. This just goes to show that you can try and leave math, but math will always find you!Handouts: HandoutEarly Elementary I [grades 1-2]: Decimal and binary numbers, continued. (Oleg Gleizer) We will continue comapring binary and decimal numbers. We will see that binary and decimal numbers are place-value while roman numbers are not. Early Elementary II [grades 2-3]: Cubes and Nets Part 1 We will be doing a worksheet that introduces nets of cubes to the studentsHandouts: Cubes and Nets Part 1High School I [grade 10]: Introduction to Groups I We introduce the definition and basic properties of a group in the context of the symmetries of a square index card in the plane.Handouts: Introduction to Groups IHigh School II [grades 11-12]: Continued Fractions (Continued) We will continue studying the properties of continued fractions, in particularly proving convergence and the 'best approximation law.' Junior Circle [grades 4-5]: Insect World (II) 4/21/2019 Advanced [grades 8-9]: Combinations III We continue with the problems from last week on pascal's triangle and combinations.Beginners [grades 6-7]: Knights, Knaves and Knormals. Logic puzzles are a mainstay of recreational mathematics, and today we'll be solving problems involving people that always tell the truth (knights), always lie (knaves), and sometimes tell the truth and other times lie (knormals). Solving these problems can be challenging, but we'll learn how to approach them is a systematic way so that you can always find the answer. Although these problems seem like all fun and games, they actually have some connections to mathematical logic; the most fundamental branch of modern math.Handouts: HandoutEarly Elementary I [grades 1-2]: Odd and even numers (Oleg Gleizer) We will study odd and even numbers in the decimal and binary form. Handouts: handoutEarly Elementary II [grades 2-3]: NO CLASS Class canceled due to a lot of students being on Spring Break vacation or celebrating the holiday.High School I [grade 10]: Introduction to Groups II We continue our discussion of groups with a focus on group morphisms and quotient groups.Handouts: Introduction to Groups IIHigh School II [grades 11-12]: Expected number of hacky sacks and temper tantrums (Paul Zeitz) Watching a math circle for 2nd and 3rd graders leads to nontrivial questions about probability and expected value. We will discover interesting and non-intuitive phenomena about randomness. Handouts: HandoutJunior Circle [grades 4-5]: Insect World (III) Handouts: Handout 4/28/2019 Advanced [grades 8-9]: Combinations IV (Stars & Bars) This week we introduce the binomial formula, and the "stars & bars" counting strategy.Handouts: Lesson Handout | HomeworkBeginners [grades 6-7]: Divisibility `This weekend we'll be talking about divisibility. Just about everyone knows what it means to divide two integers, but this week we be doing almost no dividing. Instead we will be a lot more concerned with the question of when two one integer evenly divides (i.e. has no remainder after division) another. Divisibility might not sound like a terribly deep or nuanced topic, but it is, in fact, more nuanced and developed than you would believe. Further, it is a common entry into talking about abstract algebra, one of the largest branches of modern pure math.` Handouts: HandoutEarly Elementary I [grades 1-2]: Odd and even numbers, continued. (Oleg Gleizer) We will continue to study odd and even numbers in the decimal and binary form. If time remanis, we will do a bit of magic related to binary numbers. Handouts: solutions and teacher's notesEarly Elementary II [grades 2-3]: Cubes and Nets Part 2 We will be continuing the topic of nets of cubes. In this worksheet we will explore different routes across vertcies and edges of the cube. Some questions on this worksheet are repeated from Part 1, so we crossed them out. We will also cut out nets of solids and see if they create cubes.Handouts: Cubes and Nets Part 2 | Nets of CubesHigh School I [grade 10]: Electrical Circuits and Random Walks We discuss random walks on graphs as they relate to basic electrical circuit diagrams.Handouts: Electrical Circuits and Random WalksHigh School II [grades 11-12]: Wallpaper Patterns What kinds of patterns can be used as wallpaper. What are their groups of symmetries, and how can we classify them? How many are there? We will attempt to answer some of these questions and learn how to use Thurston's "orbifold notation" for wallpaper patterns.Handouts: HandoutJunior Circle [grades 4-5]: Insect World (IV) Handouts: Handout 5/5/2019 Advanced [grades 8-9]: Combinatorics V (Stars & Bars) We continue with the more advanced verison of "starts & bars", plus various combinatorics problems.Handouts: Lesson Handout | HomeworkBeginners [grades 6-7]: The Chinese remainder theorem This week we are going to continue our discussion of divisibility. Last week we introduced the general idea and solved some interesting problems, but this week we are going to try and turn our attention to proving the useful Chinese Remainder Theorem. This theorem gives the conditions under which you can find a solution to systems of simultaneous contingencies. Although this might sound very complicated, we'll find that it isn't so bad.Handouts: HandoutEarly Elementary I [grades 1-2]: Odd and even numbers III. (Oleg Gleizer) We will continue studying the Odd and Even Numbers packet. Early Elementary II [grades 2-3]: Venn Diagrams A look at how we can organize multiple sets of objects/people/things using our knowledge of venn diagramsHandouts: Venn DiagramsHigh School I [grade 10]: Electrical Circuits and Random Walks II We define effective resistance and introduce Polya's random walk problem. High School II [grades 11-12]: Wallpaper Patterns II After practicing finding the signatures of many different wallpaper patterns, we will move on to classifying all the distinct wallpaper signatures using the "Signature Cost Theorem." As we will see, there are less than 20!Handouts: Handout | SolutionsJunior Circle [grades 4-5]: Olympiad and Riddles Handouts: Handout 5/12/2019 Advanced [grades 8-9]: Combinatorics VI (Young Tableaux) We introduce a new method to encode partitions of positive integers -- Young tableaux, and use them to prove some beautiful bijections on partitions.Handouts: Lesson Handout | HomeworkBeginners [grades 6-7]: Modular Arithmetic and Ciphers Over the past two weeks we have been studying divisibility and the Chinese remainder theorem, but up until not we have been looking at them as purely mathematical problems with no regard towards any applications. This weekend we'll be talking about one very important of molecularity, cryptography. Cryptography is the study of how we can conceal information in such a way so that it can be perfectly uncovered by the right person, and look like gibberish to everyone else. Handouts: HandoutEarly Elementary I [grades 1-2]: Binary magic. (Oleg Gleizer) We will staudy a magic trick based on binary numbers. Handouts: handoutEarly Elementary II [grades 2-3]: Taxicab Geometry Part I We will learn how to plot points on a grid and learn how to calculate distance between pairs of pointsHandouts: Taxicab Geometry 1High School I [grade 10]: The Cantor Set We introduce a few definitions of the Cantor set and relate it to cardinality and a measure on the real line.Handouts: The Cantor SetHigh School II [grades 11-12]: Integer Valued Polynomials In this lesson we will study polynomials with the property that every integer input gives an integer output. Handouts: HandoutJunior Circle [grades 4-5]: Isoperimetric Puzzle Handouts: Handout 5/19/2019 Advanced [grades 8-9]: Combinatorics VII (Young Tableaux) We continue exploring Young tableaux and the partition function.Handouts: HomeworkBeginners [grades 6-7]: Problem Solving and Take Away Games Over this weekend and the next we'll be covering both take away games and problem solving. One class will do problem solving first and then take away games, but both classes will do both over the next two weeks. For problem solving we will be diverging from our usual program and will be tackling some old math competition problems. These problems won't be emphasizing any particular mathematical principle and instead we'll be trying to discover how to approach these problems, and especially how you can make progress when you don't see the answer. For take away games we will have a friend of the math circle and current UCLA math PhD student Jeremy Brightbill give a special presentation on take away games. These mathematical games are typically played between two people where the goal is to take pieces in such a way that either you or the other person has to take the last piece. Over the course of the lesson we'll be playing a lot of games, so this is one lesson that you definitely don't want to miss! Handouts: Take Away Games Handout | Problem SolvingEarly Elementary I [grades 1-2]: Mobius strip We will study the most famous one-sided two-dimensional surface, the Mobius strip, by comparing it to a two-dimensional cylinder. The class includes quite a bit of cutting and gluing. Since many of students do not yet have the necessary hand-eye coordination, the class is taught in the PARENT-AND-ME format. Handouts: handoutEarly Elementary II [grades 2-3]: Taxicab Geometry Part II We will continue to learn about coordinate points and distance using the Taxicab worksheetHandouts: Taxicab Geometry 2High School I [grade 10]: Fractals How do we assign a dimension to a set? We define Minkowski, packing, and covering dimensions and relate these to the Cantor set.Handouts: FractalsHigh School II [grades 11-12]: Bipartite Graphs (Harris Khan) In this lesson we'll solve the "stable matching problem." Imagine two tennis clubs A and B competing in a tournament. Each player has a preference for which person they want to play from the other team. Can we find a pairing that is stable, i.e. where there is no pairing such that both players prefer to play someone else? Junior Circle [grades 4-5]: Towers of Hanoi Handouts: Handout 5/26/2019 Beginners [grades 6-7]: No Meeting this week In observation of Memorial day we will not be having Math Circle this weekend.Early Elementary II [grades 2-3]: NO CLASS - HOLIDAY WEEKEND We have no class today! Homework from last week is to finish as much of the Taxicab II Worksheet as possible. Please have your students come to class with questions about the worksheet.Junior Circle [grades 4-5]: No Meeting (Memorial Day Weekend) 6/2/2019 Advanced [grades 8-9]: Miscellaneous We finish out the year with problems on various topics.Handouts: Lesson Handout | HomeworkBeginners [grades 6-7]: Take Away Games and Problem Solving Today we'll be doing the complement of what we did tow weeks ago. You will have either a guest lecture or a lession on problem solving, depending on what you did not do last week. Early Elementary I [grades 1-2]: How to fight a dragon. (Oleg Gleizer) Students will learn how to use math to fight dragons. Handouts: handoutEarly Elementary II [grades 2-3]: High School I [grade 10]: Oblivious Transfer & Private Secure Computation (Amit Sahai) Suppose an online bookstore has N books B1, ..., BN, and you want to buy a book, but you don't want the bookstore to know which book you're buying. In other words, you want be able to choose an integer i such that 0 < i < N+1, and you want to figure out a way that you can learn Bi, and yet the bookstore learns nothing about the integer i. This is called an Oblivious Transfer (OT). We will use modular arithmetic to construct OT, and see how to use OT to solve an even more general cryptographic problem called Private Secure Computation.High School II [grades 11-12]: Oblivious Transfer & Private Secure Computation (Amit Sahai) Suppose an online bookstore has N books B1, ..., BN, and you want to buy a book, but you don't want the bookstore to know which book you're buying. In other words, you want be able to choose an integer i such that 0 < i < N+1, and you want to figure out a way that you can learn Bi, and yet the bookstore learns nothing about the integer i. This is called an Oblivious Transfer (OT). We will use modular arithmetic to construct OT, and see how to use OT to solve an even more general cryptographic problem called Private Secure Computation.Junior Circle [grades 4-5]: Triangular Numbers Handouts: Handout