In this class, we will study the mod n arithmetic, that of a circle divided into n equal parts. For n = 12, 24, and 60, this is nothing else but the arithmetic of the face of a clock.
The students will be given some problems to solve on the material covered in the Spring quarter. The papers will be checked on the fly. The highest-scoring students will receive citations and awards. Please bring a compass, ruler, and sharp pencils to the class. Protractors are not needed this time.
The test and the answer key are now posted here. Please see the attached files.
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?
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.
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.
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).
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!
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.
We will be spending one more week on combinatorics: permutations and combination-style problems, and will be discovering some important identities of combinations.
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!
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.
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.
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.
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.
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.
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.
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.
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
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.
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.
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.
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!
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.
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.
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.
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).
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.
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!
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.)
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!)
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.
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).
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.
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.
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.
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.
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.
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).
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.
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.
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 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.
