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.
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."
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.
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.
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.
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.
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)
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.
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.
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.
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.
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.
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!
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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?
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!
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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!
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.
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!
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.
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.
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.
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.
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.
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.
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.
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.
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?
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.
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!
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.
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.
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!
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!
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.
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.
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.
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.
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.
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.
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!
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.
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!
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.
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?
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.
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.
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.
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.
