Mike's group will participate in a fun, team-based problem solving contest. We will meet in room 6221 (our usual room), and proceed outside to the court of sciences, weather permitting. Prizes will be awarded to the top teams. Candy will be awarded to everyone.
Far away, in a distant galaxy, there is hotel called Hotel Infinity. True to its name, the hotel has infinitely many rooms. Currently, all the rooms are taken. Then, a guest arrives. Can you find a space for the new guest? What if 10 guests arrive? Can you combine two such hotels into one?
In this first meeting related to our studies of infinity, we will solve a series of problems related to Hotel Infinity.
We will investigate the continued fraction expansions, an alternative format to decimal notation for representing a number which reveals different types of information about that number's properties.
We will look at representations of numbers as fractions and decimals. Is every fraction a decimal? Is every decimal a fraction? How can you tell? How can you convert between the two? What are the properties of each type of representation, and what relationships exist between equivalent representations of the same number?
In this 6-hours-long (3 meetings) mini-course, we shall compare a few 2-dimensional (2D) surfaces, the Euclidean plane, a cylinder, a Moebius plane (a.k.a. an unbounded Moebius strip), a sphere, and, if time permits, the Lobachevsky plane.
We will be doing some or all of the following:
1. Picture Coding
2. A riddle word problem
3. Roman Numerals exercise
4. "How much does it weigh?" worksheet involving a balance
(A worksheet will be posted later in the week)
We will continue the study of infinity and working with problems related to the special Hotel Infinity. We will be expanding this idea of infinity to the Infinity Rockets, ultimately leading up to finding an algorithm of how to fit infinitely many people into a rocket with infinitely many seats.
We will continue what we began last week, looking at representations of numbers as fractions and decimals. Is every fraction a decimal? Is every decimal a fraction? How can you tell? How can you convert between the two? What are the properties of each type of representation, and what relationships exist between equivalent representations of the same number?
In this 6-hours-long (3 meetings) mini-course, we shall compare a few 2-dimensional (2D) surfaces, the Euclidean plane, a cylinder, a Moebius plane (a.k.a. an unbounded Moebius strip), a sphere, and, if time permits, the Lobachevsky plane.
This week we will be working again with balances but children will need to use reasoning to find a fake coin among a group of real coins with the help of the balance. They need to be able to understand the difference between worst/best case scenario and how the fake coin will not always be in the same place.
We will transition from the idea of infinity into sets. More specifically, ideas including maps between sets, one-to-one, onto and equivalence of sets.
This week we will finish our discussion of continued fractions. We will continue working on the handout from last week, along with a few more problems. Basic calculators are again encouraged for this week's session. If you have time during the week, get in a little practice performing the computations we've discussed so far!
We will continue our series on the relationship between fractions and decimals, focusing on how the period of a repeating decimal works. UPDATE: The homework for students is to complete problems 1-7 of the attached handout before next week, if they have not already done so.
In this 6-hours-long (3 meetings) mini-course, we shall compare a few 2-dimensional (2D) surfaces, the Euclidean plane, a cylinder, a Moebius plane (a.k.a. an unbounded Moebius strip), a sphere, and, if time permits, the Lobachevsky plane.
A regular n-gon is a polygon with n sides of equal length and all angles between
adjacent sides equal. An alternative description is that all corners of a regular n-gon lie
on a fixed circle, and the angle between two adjacent corners, as seen from the center
of the circle, is pi/n .
The theme of this session is to construct different regular n-gons.
Having thoroughly picked apart the relationship between fractions and decimals, we'll turn our attention to a generalization of repeating decimals--the fabled geometric series.
In this talk, we will continue with constructing regular n-gon using straight edge and compass. Then we will explore the possibility of construction using only compass.
Following the high school group, this week Mike's group will investigate the construction with compass and straightedge of regular polygons in the plane, and connections with algebra.
For more information on constructing regular polygons, see:
http://en.wikipedia.org/wiki/Constructible_polygon
We'll wrap up geometric series and start a brand new topic: Parity!
Update 10/31: Try to complete problems 1 and 2 of the Parity1 handout before next week. (It's not necessary to complete the others, as there will be time to work on them during the next session.)
The Tower of Hanoi or Towers of Hanoi , also called the Tower of Brahma or Towers of Brahma, is a mathematical game or puzzle. It consists of three rods, and a number of disks of different sizes which can slide onto any rod. The puzzle starts with the disks in a neat stack in ascending order of size on one rod, the smallest at the top, thus making a conical shape.
The objective of the puzzle is to move the entire stack to another rod, obeying the following rules:
* Only one disk may be moved at a time.
* Each move consists of taking the upper disk from one of the rods and sliding it onto another rod, on top of the other disks that may already be present on that rod.
* No disk may be placed on top of a smaller disk.
We will continue with different ways to multiplying 2 numbers. This week we will look at Russian Peasant Multiplication, which, surprisingly, has no relation to Russia or Peasants. However, this will be another good way to show the students how to write numbers as sum of powers of 2.
Mike's group continue with last week's handout on constructing regular polygons with compass and straightedge.
For more information on constructing regular polygons, see:
http://en.wikipedia.org/wiki/Constructible_polygon
Many amazing constructions can be accomplished using straightedge and compass. However, one cannot trisect arbitrary angle or construct cube root 2. In this talk and the next one, we will look at simple ways to execute the above constructions by folding papers, and consider the possibilities when different folding moves are allow.
The Hanoi Tower is a type of algorithm, which is a method used to give a finite list of detailed instructions. Last week, we allowed the children to play with the puzzles but this week we will teach them how to correctly record their steps and movements that way the puzzle can be solved correctly and efficiently.
We will use both Egyptian and Russian Peasant Multiplication to transition into binary notation. This includes writing numbers in binary, adding them and multiplying them.
This week we will be doing a worksheet called cutting logs, which will involve the kids learning how to make appropriate patterns when given an input (number of cuts) and determining the output (number of pieces);this can also be reversed. They will then be asked to put the pattern in an equation form. For example, if I have x cuts, how many pieces, y, will i have? How can I always know the answer without actually have to draw all the cuts?
This week Mike's group will investigate taxicab geometry further, continuing last week's discussions of the analogues of conic sections, and comparing taxicab geometry to the axioms of Euclidean geometry.
During the past two meetings, we used paper foldings to trisect angles and solve cubic equations in general. This week, we will change our perspective (literally) and look at parabolas in the projective plane. In particular, we will develop enough tools to find the common tangent of two parabolas in homogeneous coordinate, the coordinate system of the projective plane.
Hidden in the jungle near Hanoi, the capital city of Vietnam, there exists a Buddhist monastery where monks keep constantly moving golden disks from one diamond rod to another. There are 64 disks, all of different sizes, and three rods. Only one disk can be moved at a time and no larger disk can be placed on the top of a smaller one. Originally, all the disks were on one rod, say, the left one. At the end, they all must be moved to the right rod. When all the disks are moved, the world will come to an end. (No worries here, it will take the monks a few hundred billion years to complete the task.)
In this session, we shall first play with the puzzle and try to figure out the fastest way to solve it. Next, we shall study some auxiliary material needed to better understand the puzzle. This includes place-value numeral systems, like the decimal system we use for counting, the binary system that formed the bedrock of Egyptian multiplication we have learned in the Fall quarter, and pizza slicing as a way to think about fractions.
In the first session of the new year, we'll start exploring the mathematics of making (and breaking!) secret codes.
UPDATE 01/15/2011: For help with solving simple substitution ciphers like the last one on this handout--for example, a list of letter frequencies in English text (and more)--see the Frequency Analysis entry on Wikipedia: http://en.wikipedia.org/wiki/Frequency_analysis.
We will discuss modular arithmetic, the Chinese Remainder Theorem, Euler's Theorem, Wilson's Theorem, and then work through a variety of olympiad-style number theory problems.
We will continue our study of cryptology with a look at the Vigenere cipher, once believed by many to be unbreakable.
NOTE: For homework, complete sections 1-4. On Section 5, begin by numbering the cipher (the first one, which begins "KVX DOGRUXI OM R PHRH-KVBMRZ ...") in 123123123 fashion, and since there is a single letter R, guess either R=a or R=i for the ciphertext letters under number 1. Use your guess to decode the first three lines, but only the letters under number 1. We'll work on it more next week.
We will discuss modular arithmetic, the Chinese Remainder Theorem, Euler's Theorem, Wilson's Theorem, and then work through a variety of olympiad-style number theory problems.
We shall have a Hanoi Tower contest at the beginning of the class. The first student to assemble the puzzle with six disks will be pronounced the Great Master of Disks and will get a $1 reward. We shall proceed to solve the problems form the second handout we haven't solved yet. We shall next study the decimal place-value numeral system currently in use by humanity.
Having looked at some of the classical ciphers, we'll shift attention to the foundation of modern public-key cryptography (and a really cool area of math in its own right), modular arithmetic. UPDATE: Homework is to completely decrypt one or both of the Vigenere ciphers in the Crypto packet (see previous week for handout). Also if you were not present in class, read the Modular Arithmetic packet up to Problem 1, and try to solve Problem 1. (We will continue work on the rest of it next meeting.)
To make sure that no student is left behind, we shall solve a few more Hanoi Tower puzzle problems. Once finished, we shall begin our study of place-value numerals.
This week Mike's group will continue its study of higher dimensional spaces, and look at some counterintuitive aspects of spheres and cubes in high dimensions.
The talk will illustrate two themes. The first is that topology, a branch of mathematics that was developed during the 20th century, can be viewed in part as a natural continuation of mathematicians' efforts, from the 17th century onward, to clarify the foundations of calculus. The second is that, in contrast to sciences in which a new theory supplants all previous ones, in mathematics new developments generally arise as extensions of previous discoveries. Specifically, we will trace a portion of the history of calculus and topology to see how it served as background to research conducted late in the 20th century.
Cryptarithmetic, also know as cryptarithm, alphametics, or word addition, is a math game of figuring out unknown numbers represented by words. Different letters correspond to different digits. Same letters correspond to same digits. The first digit of a number cannot be zero. Deciphering cryptarithms is a great way to further familiarize ourselves with our numeral system (decimal place-value).
We'll take a fond look at one of the world's oldest algorithms, the Euclidean Algorithm, and explore its relationship to modular arithmetic.
UPDATE: Homework is to complete the Euclidean Algorithm handout.
In this class, we shall study geometric series. If time permits, we shall study the Sierpinski carpet and employ a geometric sequence to figure out its area.
In this class, we will study some remarkable features of the Sierpinski triangle. At the end of the class, we will take a look at other fractals, geometric objects of fractional dimensions (hence the name), more from the point of view of art rather than science.
Groups are important algebraic structures which are used in all areas of mathematics, as well as physics and chemistry. The goal will be to define and give some examples of these objects. The main example will be the braid groups, which is a mathematical formulation of what happens when we braid our hair.
Mike's group will investigate Euclid's algorithm for polynomials, reviewing the standard Euclid algorithm as well as long division for polynomials. We will also continue our investigation of intersection numbers.
A rational triangle is a right triangle whose three sides are all rational numbers. A rational number is called a congruent number if it is the area of some rational triangle. It is an ancient problem to decide if an arbitrary integer is a congruent number. Surprisingly, this innocent-looking problem is still open today. In this talk, we will go through the history of congruent numbers and look at its connections to important objects in modern number theory.
Today will be a review of all that we have covered this quarter. We will be keeping the exams to review the responses of each child to help better understand what to do next quarter, but I encourage you go over the questions with you children!
In this class, we will see how states of the Hanoi Tower puzzle can be represented by graphs very similar to the ones formed by the vertices and sides of the Sierpinski triangle approximations. See the handout for more.
To start the quarter off, we will start work with binary notation. We will introduce this concept first by going back to the previous lessons of balancing of scales.
In this class, we will learn how geometry has begun, proceed to Euclid's "Elements" and discuss the notions of an axiom, a point, line, and straight line. At the end, we will do some Greek-style geometry using a rope to perform multiplication and even raising a number to a power!
The distance between two points is the length of the shortest line connecting them. Normally, such a path is a straight line. But in a city consisting of a square grid of streets shortest paths between two points are no longer straight lines (as every cab driver knows). We will explore the geometry of this unusual distance and play several related games.
We are all familiar with the absolute value function. It measures the distance of a number from 0. We will examine the essential properties of this function, define other absolute values on the rational numbers, and classify all such absolute values. This leads to the idea of completions and the p-adic numbers which we discuss briefly. Finally we introduce Hensel's Lemma for solving polynomial equations over the p-adics and mention the local-global philosophy.
This week we will be continuing to work with binary notation. We will show how there are similarities and differences between binary notation and decimal notation place values. We will also start to work with adding and subtracting binary numbers.
The concept of proof lies at the heart of mathematics, yet school mathematics usually spends very little time discussing them. What is a proof? Why do we need them? How do you write them, or verify someone else's? This week begins a series which will examine these issues.
The "Occupy Wall Street" movement was concerned with the inequality of income in the U. S. Are the rich in the U. S. really getting richer and the poor getting poorer? How does the U. S. compare with other countries in this regard? The Gini Index is an important statistical tool for answering such questions. You will use geometry and some calculus to study its mathematical properties. In particular, we will investigate some mathematical models of income distribution and the corresponding Gini indices.
In this class, we will first finish covering the parts of Handouts 1 and 2 we haven't covered yet. If time permits, we will do some more compass and ruler constructions.
In this session we will explore different ways of sampling from populations. How is it done poorly, and how can we do it well? We'll talk about recent polls that give us some ideas for predicting the upcoming presidential election.
Additionally, we'll talk about sampling distributions -- a sampling distribution tells us what to expect for our sample under certain constraints. Information from the sampling distribution allows us to test a research claim or to come up with specific methodology. We will use examples to illustrate the ideas.
We will look at some very useful tools in the proofmaker's toolkit: proof by contrapositive, and proof by contradiction.
UPDATE 04/29: Homework for next week is to complete pages 1-4 of the handout (Proofs3), if not done already.
Starting with Bayes' rule, we'll figure out the probability of winning the 1960's game show Let's Make a Deal. We'll extend the ideas to some examples where Bayesian thinking helps avoid common misunderstandings.
Finally, we will apply Bayesian ideas to statistical inference. Using prior knowledge, we will assess different estimators with an application to baseball batting averages.
We will take a break from our geometry mini-course, do some wizardry studies, save a prince from an evil king and help a prisoner choose between a cell with gold and a cell with a hungry tiger. Don't forget to bring your magic wands!
At the beginning of Guy Ritchie's cult hit, Lock, Stock, and Two Smoking Barrels, the
main character, Eddy, gets trapped in a (rigged) poker game set up by the villain, "Hatchet"
Harry: while all the players are dealt one card face up, and two face down, Eddy's "down
cards" sit directly above a hidden camera.
After a few rounds of play, Eddy is eventually caught in a hand against Harry. He holds
10 of diamond face up, and 6 of spade 6 of heart face down,
while Harry's hand shows
9 of club ? ?
The ve hundred thousand dollar question (as it happened) Eddy faced was this:
What are the odds that Eddy's hand is the better one?
Come to the math circle this Sunday to nd out... and to learn how to attack similar
gambling-type problems using probability.
In this class, we will learn to measure angles with a protractor. We will also learn a few facts about angles and prove the first theorem of the course, the one claiming that angles of any triangle in the Euclidean plane add up to a straight angle.
Suppose that there are two hotels for numbers. The first hotel has rooms labeled 1 through n and each room can take at most one guest. If m numbers get rooms at the hotel and fill the hotel to capacity, then m must equal n. This seems quite natural, but how do we prove it? The second hotel is very strange. Last night, all the natural numbers were guests at the hotel and the hotel was filled to capacity. Tonight, all the fractions have decided to visit the hotel but the natural numbers won't leave. The hotel manager was in a panic because the hotel appeared to be overbooked. Fortunately, the hotel manager's sister is a mathematician and she told her brother not to panic. After rearranging the guests, she was able to fit in not only all the natural numbers, but all the fractions as well. How did she do it and what does this tell us about the infinite?
Last Sunday, we have worked out the proofs of Propositions 1 and 2 with the first class (3:45 - 4:45) and the proof of Proposition 1 with the second class (5:00 - 6:00). This time, we will finish studying the proofs in the last handout. If time permits, we will begin a new topic, Clock Arithmetic.
We'll review problems from last week's individual problem solving, and then begin investigating some of the mathematics of motion and everyday physics.
UPDATE: We will continue the handout next week. For those who weren't here or didn't do so already, try to complete problems 1--10 (everything up to the statement of the Law of Lever).
Enumerative combinatorics is a field of mathematics that deals with counting the number of objects satisfying some combinatorial description. We will learn to count the number of a number of seemingly unrelated objects such as monotonic lattice paths and partitions of integers, and explore their relationships with each other. If time permits, we will also make some necklaces--or rather, count the number of different ways to do so.
In this class, we will learn that time, like money, is a man-made concept, that our clocks and watches do not show time, but rather model the Earth's rotation around its axis, and that time travel is quite possible if you live next to a pole.
We continue our investigation of mechanical topics, looking more at levers and how the concept of center of mass can help us solve surprising problems. (Update: We won't be discussing pulleys this time after all.)
We will first describe sentential logic, a language used to formalize
certain kinds of assertions. Then, we will state and prove the
compactness theorem of sentential logic, a result of fundamental
importance that in some sense allows us to understand infinite
collections of assertions by their finite subcollections. This will be
used to prove a corollary about graph colorings.
