Los Angeles Math Circle

LAMC Meetings Archive // Fall 2007 - Spring 2017

For the current schedule, visit the Circle Calendar

Fall 2007 - Spring 2008Fall 2008 - Spring 2009Fall 2009 - Spring 2010Fall 2010 - Spring 2011Fall 2011 - Spring 2012Fall 2012 - Spring 2013Fall 2013 - Spring 2014Fall 2014 - Spring 2015Fall 2015 - Spring 2016Fall 2016 - Spring 2017
A projection for the purpose of this worksheet is a 2D picture that shows us the "shape" of a side of a cube. In this handout, students will be given projections of the top, front, and left side of a solid that fits in the given dimensions of a cube. Can they identify the correct solid from this information? Or, is there more than one possible solution?
Handouts: Handout 10
We will visit the planet Heptadium in a galaxy far, far away. The planet makes one full spin around its axis in 7 heptahours, so the folks inhabiting the planet divide the dials of their clocks into 7 parts. A heptahour is divided further into 49 heptaminutes, a heptaminute has 49 heptaseconds. We will study the Heptadium way of timekeeping and the related mod 7 and mod 49 arithmetics.
Handouts: handout | some answers and hints
In this lesson, we look at casting of nines as an application of modular arithmetic.
Handouts: Casting out Nines, and other types of Math Magic
We will start this year by studying graphs, specifically when two graphs are the same and paths inside the graph.
Handouts: Graphs 1
We will be exploring counting-type problems, building up to factorial, permutation, and combination situations.
Handouts: Combinatorics I
When messages are being sent in every day life, errors could occur and affect the accuracies of the messages. In this session, we will look at the how one can mathematically detect and correct these errors automatically.
Handouts: Smart Code I
First, we will finish discussing the previous handout, from Problem 12 on. Be prepared to present your solutions of Problems 12 - 16 at the board! Next, we will begin learning coordinates on lines and surfaces, from a segment of a straight line to a sphere. Also, come ready for a great naval battle. A good understanding of coordinates will be needed to win!
Handouts: handout
In this week, we explore more properties of Modular Arithmetic, including multiplication tables, inverses, zero divisors, and powers of numbers
Handouts: Circles and Tables and Sleeping
We will be delving a bit deeper into combinatorics, including combination and permutation-type problems, and deriving general formulas for combinations and permutations (n choose k).
Handouts: Combinatorics II
In this session, we will continue to talk about particular examples of error-correcting codes and study their properties using matrices.
Handouts: Smart Code II
Today we will study Planar graphs, including Euler's formula for planar graphs.
Handouts: Graphs 2
This week, we will continue with projections and ciphers. We will be focusing on projections that do indeed have multiple solid representations.
Handouts: Handout 2
Today we will look at some problems in geometry, focusing on inequalities and areas.
Handouts: Geometry 1
We will study the most elementary, and the most important, functions of one variable, linear, affine, absolute value, and quadratic.
Handouts: handout
We will be spending one more week on combinatorics: permutations and combination-style problems, and will be discovering some important identities of combinations.
Handouts: Combinatorics III
In this session, we will explore different kinds of "shapes" one can obtain by "gluing" together polygons.
Handouts: Exercises for Surface Classification
This week we look at division and modular arithmetic, as well as solve some problems with a sleepwalking Bishop.
Handouts: Dividing Zero and the Sleepwalking Bishop
This week we continue with projections but add levels to the mix. Instead of just looking at 2D projections of the front, top, and left sides of a solid, students will not imagine their solid sliced into levels. They will learn how to interchange between the two.
Handouts: Handout 3
We will continue the study of functions and graphs.
Handouts: handout
We will be learning about the pigeon hole principle and how to use it to solve problems. Combinatorics knowledge will be necessary for parts of the handout, so please make sure you've solved as much of the previous weeks' handouts as possible before Sunday. Problems 14 and 15 of Combinatorics III were assigned as homework on 10/15.
Handouts: Pigeonhole Principle
A cipher is an algorithm to encode or decode information. This week we will explore different ciphers used throughout history. The children will learn not only how they work, but how to encode and decode fun messages!
Handouts: Handout 4 (Corrected)
Today we look at take-away games and game strategies.
Handouts: games
This week we will start the study of number bases.
Handouts: Number Bases 1 | Number Bases 1 (Sol)
This week we will attempt to explain what all the possible surfaces are, discussing the notions of orientation and Euler characteristic.
Handouts: Exercises for Surface Classification
At the beginning of the class, we will solve a Halloween cryptarithm. Then we will continue working through the fourth handout starting from Problem 8. If time remains, we will prove the Pythagoras' theorem and use it as a tool for figuring out distances between points.
Handouts: handout
This week, we will continue our study of the pigeonhole principle, delving into more challenging problems than last week. There may be a short quiz at the end of class. HOMEWORK DUE (at the beginning of class!): Problems 1-12 and 16 from last week's handout, Pigeonhole I.
Handouts: Pigeonhole II
Today we will continue our study of number bases, focusing on the binary, ternary, and hexadecimal number systems.
Handouts: Number Bases 2 | Number Bases 2 (Sol)
This week we will investigate solving quadratic equations mod p, and the interesting questions that arise.
Handouts: QR Handout | QR Chart | QR Worksheet
We look at how to form logical negations properly, contraposition, and proof by contradiction.
Handouts: Negations
Today we will be doing a Halloween themed worksheet with a variety of different problems.
Handouts: Handout 5
This week we will continue last week's discussion of quadratic residues, and look at some problem-solving type exercises.
We will have a quiz at the beginning of the class. Once finished, we will learn two different proofs of the Pythagoras Theorem and apply the theorem to figuring out distances between points in the Euclidean plane, see Handout 5 from page 7 on.
Handouts: quiz
After finishing up our study of the pigeonhole principle, we'll begin to dive into divisibility. To make sure we're all on the same page, we'll also be doing a quiz/individual problem solving check-in. HOMEWORK DUE: Pigeonhole II handout, Problems 7-8, 13-16. Please bring a pencil, scratch paper, and all previous handouts in a 3-ring binder to class.
Handouts: Divisibility I
Today we will solve problems involving inequalities.
Handouts: Inequalities 1 | Inequalities 1 (Sol)
Today we will introduce coordinate planes and number lines.
Handouts: Handout 6
An introduction to graph theory, and an introduction to how to catch runaway students.
Handouts: Graph Theory 1
Handouts: Probability I
For the first hour, we will solve a variety of problems, including taking roots of numbers, drawing a graph of a function, constructing a right angle by means of a rope, and figuring out the winners of a fencing competition. In the second part of the class, we will prove the triangle inequality for the Euclidean plane. We will further use it as a tool for finding the shortest paths between points in the Euclidean plane and on a cylinder.
Handouts: handout
We will be delving further into divisibility this week, learning to solve a variety of wonderful problems!
Handouts: Divisibility II
Today we will study arithmetic and geometric means, and the famous inequality relating the two.
Handouts: Inequalities 2 | Inequalities 2 (Sol)
Today we continued with coordinate planes. Children expanded on their ability to identify and plot points by continuing for directions, learning how to reflect across the x and y axis, as well as finding the locations of points based of given information.
Handouts: Handout 7
In this lesson, we look at more properties of graphs, including the handshake lemma
Handouts: Graph Theory 2
This week we will investigate finding probabilities by comparing areas, discussing certain paradoxes that arise.
Handouts: Geometric Probability
We will learn that there are infinitely many geodesic lines ("straight lines") connecting two points in general position on a cylinder. We will use this information to construct triangles on a cylinder. We will also learn that there exist two types of circles on a cylinder. One of them is just like a circle in the Euclidean plane, the other looks more like the number 8.
Handouts: handout
We will be finishing up our study of divisibility this week! HOMEWORK: Problems 1-10 from Divisibility III handout. Quiz solutions will be posted this week! Divisibility III handout: http://www.math.ucla.edu/~radko/circles/lib/data/Handout-439-538.pdf Quiz (check back for solutions): http://www.math.ucla.edu/~radko/circles/lib/data/Handout-440-538.pdf
Handouts: |
Today we will study how to analyse games, and introduce the concept of P and N positions.
Handouts: Games 1 | Games 1 (Sol)
Handouts: page_one
Enjoy the break!
In this talk, we will look at generating functions, lattices paths and random walks.
Handouts: Probability II
We will be solving fun problems where a group of people must cross a river with certain restrictions in place! We'll also review the major things we've learned this quarter and maybe even tie it all together with a great magic trick to impress your friends with.
Handouts: water_crossings
We will continue our study of games, focusing on the famous game of Nim.
Handouts: Games 2 | Games 2 (Sol)
We will come up with a strategy to defeat a dragon with several heads and tails in the Magic Land.
During the first hour, we will try to find a winning strategy for a game they sometimes use for starters in a game theory course. During the second hour, we will learn what the average value is and apply the knowledge to solving various problems.
Handouts: handout
We will be looking at games on graphs this meeting, including Sprouts, Hackenbush, and chase games
Handouts: Games on Graphs
We will have and end-of-quarter problem solving competition. There will be prizes for the top scoring teams.
Returning students typically say this is their favorite Math Circle of the quarter...and we're sure new ones will enjoy it just as much! We'll put you in a team of approximately five people and you will be solving as many problems as you can...as quickly as you can...for real prizes! Have a great winter holiday break!
Today we will end the Fall quarter by having a relay.
The students will be given a test on all the topics covered during the quarter. The duration of the test is an hour and a half. Then, the test will be quickly graded and the highest-scoring students will be given awards.
Handouts: test
We will work on some interesting AMC problems from the past.
Handouts: AMC Problem Session Answers
We will begin this quarter with a few classes on probability theory. The probability of an event is a number in between zero and one, ends included, so the working knowledge of fractions is a necessity. We will give the students a 15-min-long diagnostic quiz at the beginning of the first class to estimate how good they are with fractions. The degree of their success will tell us how to split the class time between learning fractions and using them to compute probabilities. Since the results of the quiz will only be available after the first class, we will study fractions till the end of the first hour. We will begin learning the probability theory during the second half of the class.
Handouts: handout | diagnostic quiz
Welcome back, everyone! Hope you had a wonderful winter break and holiday season. We will begin the quarter by looking at graphs!
Handouts: Graphs I
Today we will review various counting techniques. This is to prepare for learning probability, which will be a focus this quarter.
Handouts: Combinatorics 1
We explore combinatorics through taxicabs, pachinko, and choosing items
Handouts: Combo1
We will work out some of the mathematics behind Einstein's theory of relativity, including the concept of a spacetime diagram, the theory of Doppler shifts, and Einstein's famous equation E=mc^2. Very little prior knowledge of physics will be required (but we will assume some familiarity with high school algebra).
Handouts: Special Relativity
We will finish our discussion from last week, as well as look at some more challenging problems.
Handouts: Combinatorics 1+2 | Combinatorics 1+2 sols
After a small number of participants, we delayed the due date for the first contest. The new due date is Jan. 20. There will be prizes to the top participants.
Handouts: Contest
We will be continuing our study of graph theory this week, with topics such as connectedness and Eulerian graphs.
Handouts: Graphs II
For the first 70 min. of the class, the students will be given a Math Kangaroo test for 5th and 6th graders from one of the years passed. Even if your child doesn't plan to take part in the actual Math Kangaroo competition this year, it's very beneficial to solve the problems, they are non-trivial and fun! After a 5 min. break, we will run a simulation of the Monty Hall Paradox. Please see the following URL for more. http://en.wikipedia.org/wiki/Monty_Hall_problem It is possibly the most famous paradox of elementary probability theory. To make it more striking, we will use ten doors (or rather playing cards) instead of only three from the original version.
This week we look at more identities with Pascal's triangle, and choice with repetition.
Handouts: Combo2
Handouts: Handout 2
This week in the High School group we will investigate graphs colorings, planar graphs, and Eulerian and Hamiltonian circuits.
Handouts: graph_colorings | graph_coloring_map
This week we use taxicabs to explore Catalan numbers
We will be continuing our study of graphs this week! Note that we will be picking up from the previous handout, so please make sure you have completed problems 1-4 from Graphs II and bring the handout to class!
Handouts: Graphs III
We will solve a few word problems on fractions at the beginning of the class. Then we will continue our study of probabilities, learning the properties of complementary, mutually exclusive, and independent random events.
Handouts: handout
Today we will begin our study of probability.
Handouts: Probability 1+2 | Probability 1+2 Sols
Handouts: Handout 3 | empty bar chart
We will study permutations and combinations and then apply them to counting probabilities.
Handouts: handout
Today we will finish our handout from last time, as well as give students some time to work on the monthly contest.
We will be continuing our half-quarter-long study of graph theory, building up to, proving, and applying Euler's Formula this week. HOMEWORK: Graphs III handout (available on previous week's posting), #1-15, 20-21.
Handouts: Graphs IV
This week we will do a practice exam for the Math Kangaroo exam, and go over solutions for a few problems.
Handouts: Practice Exam | Solutions
We will go over the AMC 10/12A from this year.
Today we will study conditional probability.
Handouts: Conditional Probability 1+2 | Conditional Probability 1+2 sols
We will be finishing up our study of graphs this week, with a focus on directed graphs, and a bit of review on what we have done so far this quarter. There will be no meeting the following week (2.17) in honor of Presidents' Day.
Handouts: Graphs V
We will continue learning combinatorial probability. In particular, we will finish studying the handout for the previous class.
Handouts: handout
In this lesson, we look at successive differences and sequences.
Handouts: Differences
This week will be teaching the kids about what probability is and how to figure out some basic problems.
Handouts: Handout 5
Happy President's long weekend. Math circle will resume the following week.
Enjoy your holiday!
We will introduce some probability theory and then we will discuss how these ideas can be used to study networks in real life.
We will begin a study of geometry-type problems by learning about finding angles. These kinds of problems have traditionally been a favorite of contest writers.
Handouts: Finding Angles
Today we will continue our study of conditional probability (see last week for the handout). Also, the monthly contest is due (it is below). We expect everyone to attempt solutions to the contest.
Handouts: Monthly Contest (due Feb. 24)
In this week's lesson, we look at how induction can help solve problems involving the cutting of Pizza.
Handouts: Pizzas and Induction
We will first recall the basic compass-and-ruler constructions we have done in the past. We will further move to proving the existence of an incircle, circumcircle, etc. for any triangle in the Euclidean plane.
Handouts: handout
This week, we will be teaching the kids how to develop good strategy to solving word problems through logic, numbers, and pictures.
Handouts: Handout 6
This week we will discuss the diagonalization of quadratic forms in order to draw quadratic curves in the plane, and their relationship with bilinear forms.
Handouts: Quadratic Forms
We will be working on geometry problems, many of which have been taken directly from past Math Kangaroo contests. This week's lesson will be much less proof-based and will require much less prior knowledge of geometry than last week's.
Handouts: Angles
We will begin with Problem 6 of the previous class handout and proceed to the end of it. The very few students who have worked the handout to the end during the previous class will be given a new one.
Handouts: handout
Today we will use what we have learned about probability to begin talking about random variables.
Handouts: Random Variables 1+2 | Random Variables 1+2 Sols
Here were cover more induction problems related to other topics from the year.
Handouts: Induction II
Continuing word problem strategy.
Handouts: Handout 7
In this lecture, we will take a look at the murky surroundings of Euclid's 5th Postulate, possibly the most controversial scientific statement of all times. The Great Theorem of Fermat, a proverbial symbol of mathematical complexity, stood open for 358 years. Conjectured by Pierre de Fermat in 1637, it was proven by Andrew_Wiles with the help of his former student Richard Taylor in 1995. The Poincare Conjecture, another very hard math problem made famous outside of the scientific community by its conqueror Gregory Perelman's rejection of the $1,000,000 prize money, was proven in less than 100 years. It took humanity over 2,000 years to realize that the 5th Postulate is indeed a postulate and cannot be derived from other axioms of Euclidean geometry. The breakthrough can only be compared to the Copernican Revolution in astronomy that replaced the geocentric model of the universe with the heliocentric one. Both discoveries broke the millennia-long paradigms. The Copernican Revolution brought about Newtonian mechanics. The discovery of non-Euclidean geometry paved the way to the Relativity Theory of Einstein. In the lecture, we will derive from the axioms the formula for the sum of angles of a triangle in the Euclidean plane and on a sphere. We will see that the first formula, traditionally considered to be the more elementary of the two, is in fact much harder to prove. We will also visit a planet that has only one pole, figure out the shape of the Universe, and some more.
We will be looking at more geometry problems in the vein of contests like the Math Kangaroo, this time with a focus on area. (note: the title and date on the handout are incorrect. The date should read 3/10/13 and the title should concern areas, not angles. This was corrected on the version that was passed out in class.)
Handouts: Areas
The students will solve a Math Kangaroo test from one of the years past and a few extra problems collected for us by Samir's mom, Sarita.
Today we will continue our study of random variables, including the ideas of expected value and variance.
Math Kangaroo practice with word problems.
Handouts: Handout 8
Please join us for our last meeting of the Winter Quarter for a fun team problem-solving competition!
We will be finishing off the quarter by working through a full-length Math Kangaroo contest. (Math Relays have been pushed back to next quarter, but we will still do them!)
Today we will study probability questions that involve picking multiple numbers at random.
Handouts: Monthly Contest Solutions
The class will be split into four teams competing against one another in solving the problems given by the instructors. The winning team will get some prizes.
Handouts: problem set
We will be starting off the quarter with Math Relays!
At the end of the previous quarter, we considered the following problem. Prove that medians of any triangle in the Euclidean plane intersect at one point and that the intersection point divides them in the ratio 2:1 counting from the corresponding vertex. We will take time to give three different proofs to the theorem, each coming from a distinct, and very important, branch of mathematics. During this class, we will begin learning the geometry of weights that will take us from the workings of a lever to solving the original problem and beyond.
Handouts: handout
Pick's Theorem relates the area of a lattice polygon to the number of lattice points on its boundary and the number of lattice points in its interior. We prove this theorem from the ground up, by starting with rectangles, building up to general triangles, and then using triangulations. This is the first of two weeks, the second of which will discuss generalizations of this theory to higher dimensions (Ehrhart Theory).
This week we will review some geometry concepts in preparation for the coming weeks.
Today will be an introduction to understanding perimeters which will lead into a discussion about area for next week.
Handouts: Triangle inequality (ignore problem #7)
Handouts: Pick's Theorem and Ehrhart Theory
This time, we will finish the handout on geometry of masses we have started last time. We will use the geometry of masses to get the first proof of the medians theorem and then to generalize it to 3D and 4D. This will take us about an hour. In the second half of the class, we will solve some fun problems not requiring any special knowledge.
Today we will study transformations of the plane, leading to learning about inversions.
Today the kids will apply their knowledge of area for rectangles to different types of problems. They will also learn about the relationship that a rectangle and a parallelogram share with area.
Handouts: Spring Handout 2
We look at Dartboards, and some simple probability problems
Handouts: Probability 1
Handouts: distance
We will finish our first encounter with the geometry of masses by finally proving the medians theorem in 2D. We will further generalize it to 3D and 4D. To take a break from studying theoretical stuff, we will further solve a bunch of problems that require no special knowledge, but are a lot of fun! If time remains, we will switch back to theory and approach the median theorem from a different direction. It will be linear algebra this time.
Handouts: handout
Today we will continue our study of inversions, showing how the concept can be used to solve problems in geometry.
Some basic set theory, and applications to probability
Handouts: Probability 2
This week we will explore a new method to multiple 2 digit numbers called Egyptian Multiplication.
Handouts: Spring Handout 3
This week the High School Group will investigate writing integers as the sum of two squares.
Handouts: distance2
We will look at decimal expansions and continued fraction expansions of rational, algebraic and transcendental numbers.
Handouts: Rational, algebraic, and transcendental numbers
Some students will be called to the board to discuss their solutions to the problems from the second handout. Then we will learn a formal approach to proofs and use it to prove a few facts regarding medians of triangles in the Euclidean plane. Some students, as well as their parents, keep asking us to add more algebra to our classes. To cater to their needs, we will begin learning quadratic equations at the end of the session (if time permits).
Handouts: handout
Today we will start our study of infinite numbers, including examining the famous Hilbert's hotel 'paradox'.
This week will look at where the area of a triangle comes from as well as how that formula relates to the parallelogram. Please bring a straight edge and colored pencils to class.
Handouts: Handout 4
In this section, we apply our knowledge from the previous two weeks to solve somewhat paradoxical probability problems.
Handouts: Probability 3
This week the high school group will investigate a formula for estimating the size of the factorial of an integer.
Handouts: Stirling's Formula
This time we will finish the previous handout. We will prove a few more geometric statements before switching to algebra. Some parents have asked us to teach more algebra, so here it is. We will take a first look at quadratic equations at the end of the class.
Today we will continue our discussion of the different types of infinity by discussing cardinal numbers.
This week we will be studying perpendicular bisectors and circles using our new knowledge of distance.
Handouts: Distance4
We use origami and the ideas of scissor equivalence to describe geometric concepts.
Handouts: Geometry 1
This week we will be learning in more detail about altitudes as well as why a square represents a rectangle with the biggest area for a given perimeter.
Handouts: Handout 5
Handouts: Handout 7
This week our guest speaker will tell us about how to study geometry when the only thing you know is the distances between different points.
Handouts: Geometry when you only know distances
We will take another look at Problem 5 from the first handout, Problem 8 from the second handout, and solve a bunch of new problems.
Handouts: handout | Proof of Theorem 1
Handouts: Geometry 2
We will be having fun with pi this week!
Handouts: pi
Today we'll finish our discussion of infinite numbers.
We conclude our exploration of geometry by looking at symmetries
Handouts: Geometry 3
We will take two classes to review the material we have learned during this academic year. This is the first of the two.
Handouts: handout
We will be finishing up our look at circles and pi this week!
Today we will start a study of trigonometry from a geometric perspective.
We will investigate paths in geometric objects ("spaces") to answer the question: how many (essentially different) ways are there to get from point A to point B in the given object? It turns out that the best answer is not given by a number, but by algebraic structures called groups and groupoids that take into account how a given path can be broken down into pieces - and what happens when we go in circles. Our main examples of spaces will be graphs, and surfaces.
Handouts: Paths in space notes
We will continue reviewing the material studied during this academic year.
Handouts: handout
We will investigate paths in geometric objects ("spaces") to answer the question: how many (essentially different) ways are there to get from point A to point B in the given object? It turns out that the best answer is not given by a number, but by algebraic structures called groups and groupoids that take into account how a given path can be broken down into pieces - and what happens when we go in circles. Our main examples of spaces will be graphs, and surfaces.
We will be finishing up the year with some strategies for math games!
In this handout, we look how the pigeonhole principle can be applied in various mathematical situations.
Handouts: Pigeonhole Principle
Today we will continue studying trig, seeing how it is useful in problem solving.
Like Math Relays...but maybe even more fun!
We will end our year with relays.
We will meet at the lawn across the Math Science Building and take some pictures of the class. From there we will proceed to the classroom and take the year-end test.
Handouts: test | solutions