Clint's group will look at a number of puzzles and problems, many of them geometric, which can helpfully be approached using the simple tool of grid paper.Handouts: Counting Problems (Mike) | Grid Luck (Clint)
Mike's group will solve problems involving counting various types of objects.
This week we looked at different ways to encode and decode messages using various methods, including monoalphabetic substitution ciphers, Caeser/shift ciphers, and using frequency analysis.
The Jr. Circle meets in Math Sciences 3915A. Handouts: Fun with Ciphers Handout
Walk The DogHandouts: Walk The Dog WS
We will use complex numbers to solve a variety of geometric problems. A short introduction to complex numbers will be given.Handouts: COMPLEX NUMBERS AND PLANE GEOMETRY Work Sheet
In Mike's group, we will finish the counting problems from last week, and take a look at solutions of quadratic and cubic polynomials. Handouts: Egyptian Fractions (Clint) | Monster Equations (Mike)
This week we explored more ways to encode and decode: using frequency analysis, the Pigpen cipher, the Rail fence cipher, the Cardan grille. We also looked at how some numbers, like bank account or credit card numbers, can also have additional information hidden within their digits.Handouts: More Fun with Ciphers
This week we will be looking at what it means for shapes to be similar, as well as exploring how we can add on to a shape to create another that is similar to the original. The Jr. Circle meets in Math Sciences 3915A. Handouts: Similar Shapes and Gnomons Handout
In Mike's group we will continue our study of roots of polynomials. Clint's group will round out our study of Egyptian fractions.Handouts: Egyptian Fractions II | Monster Equations II Handouts: Introductory Problems | Remainder Problems | Modular Arithmetic Problems | Modular Arithmetic Problems, 2 | Divisibility Tests | Hard Problems | Comments about the Hard Problems
In Mike's group, we will conclude our study of cubic equations, and take a look at complex numbers. Handouts: (same as last week's)
This week we will be exploring the ideas of "triangular" and "square" numbers.
Handouts: Triangular and Square Number Handout
We have learned in the last two weeks that Egyptians worked only with fractions of a very special form. Are there any other mathematical objects or operations that were very different for them as well? We will learn about the Egyptian multiplication and another method, the Russian peasant multiplication, that is related to it. We will see that both rely on binary representation of numbers. Handouts: Spivak's Problems
In mathematics, a knot is a closed piece of string in three-space. Two knots are equivalent if they can be deformed into each other. We will explore several ways of showing that certain knots are not equivalent, using invariants.
Suppose you are throwing a volleyball to your partner during the game. What is the way to throw it so that it flies as far as possible? What is the ball's trajectory? In the first part of the meeting, we will solve several problems on this topic.
After the break, June Wang and Sandra Daley from Raytheon will lead the activity on Straw Rockets, where you will build simple models of rockets, launch them and study their properties depending on design.
Please bring scissors to the meeting!
This week we will review and expand on the triangular and square numbers from last week, looking at other sequences of numbers which have specific patterns.Handouts: Patterned Sequences of Numbers Handout
We will look at ways to represent "stringy" structures with algebra, and try to find ways to classify these objects mathematically.Handouts: Knots, Braids and String Things
We will look at ways to represent "stringy" structures with algebra, and try to find ways to classify these objects mathematically.Handouts: Knots and Braids, part II
This week we will bring back some topics covered in the first few weeks, as well as introduce some new challenges, all centered around today's theme: Halloween!Handouts: Halloween Math Fun Handout
We will further explore the world of "greedy algorithms" (not Egyptian fractions!).Handouts: Greedy Is Good
This week we will finish our discussion of cubic polynomials and study complex numbers.Handouts: A Mathematical Duel
This week we will explore a special sequence of numbers known as the Fibonacci Numbers.Handouts: Fibonacci Handout
Last week we looked at some more greedy algorithms for certain kinds of problems. This week we'll examine some algorithms that are NOT greedy, but do things more cleverly instead.Handouts: When Greedy Fails
In tropical arithmetic, the "sum" of two numbers is their
minimum and the "product" of two numbers is their usual sum. The
algebraic and geometric implications of these definitions are in many
ways similar to those of standard arithmetic, but with a number of
surprising twists. We will explore the graphs and factorizations of
tropical polynomial functions and the intersections of tropical curves.
Handouts: Tropical Mathematics | Tropical Mathematics: Solutions | Tropical Mathematics: Graphs
This week in Mike's group, we will do some practice with problems from the AMC 8.
A mathematical graph consists of dots, called vertices, together with line segments between them, called edges. The study of graphs and their properties comprises a field of mathematics known as graph theory. While graph theory is interesting in its own right, it has many applications outside of mathematics which we will explore.
Clint's group will examine a number of issues related to fair proportional division of discrete sets--for instance, how to most fairly assign each U.S. state a number of seats in the House of Representatives proportional to its population.Handouts: Divvying Up Seats Problems | Divvying Up Seats Solutions
This week we will be taking a look at some practice problems from the Math Kangaroo competition.Handouts: Math Kangaroo Practice Problems Handout
We will learn about how to add, subtract, multiply, divide, and draw complex numbers!Handouts: Complex Numbers!
Mike's group will continue discussion of a geometric way of looking at complex numbers. Clint's group will look at some of the mathematical issues surrounding voting and social choice.Handouts: Voting Problems | Voting Solutions
We will solve a variety of problems from past AMC 10 and AMC 12
This week we will build on the problems we solved last week, and move on to some more challenging math problems.Handouts: More Problem Solving Handout
For the last meeting of the quarter, Mike's and Clint's group will be doing Math Relays where teams compete to answer a series of challenging math problems and puzzles quickly, with prizes for the top teams!Handouts: Clint's group relay problems | Clint's group relay answers
This week marks our last meeting of the year! We will be working on problems that span all topics we have covered this quarter.Handouts: Quarter Recap Handout
We all take for granted the fact that there are infinitely many numbers. But infinite sets can behave very unintuitively. Furthermore, not all infinities are the same! We will take a look at a few surprising facts about infinite sets, and introduce some tools that mathematicians use to make sense of the infinite.
Clint's group will take a look at some tricky problems involving the familiar concepts of perimeter and circumference.Handouts: AllTheWayAround
We will solve a variety of problemsHandouts: Problem Solving
This week Mike's group will explore applications of the triangle inequality to problems in plane geometry. Handouts: Triangle Inequality
Clint's group will continue to look at problems involving geometric concepts of perimeter and circumference, with special attention to applications of the Pythagorean theorem, and to understanding the meaning of "pi." Bring your handouts from last week to complete!Handouts: OneMoreTimeAround
We will solve a Math Kangaroo contest from one of the previous years.
Note: the class will be in MS 6627, 2-3:15 p.m. (both groups!) The change in time and place is for this time only.
We will solve a variety of AMC 10/12 problems.
This week Mike's group will continue to look at problems exploring applications of the triangle inequality. Please bring the worksheet from last week.
Clint's and Liz's groups will move from measuring the outsides of shapes, via perimeter and circumference, to the insides, that is, areas.Handouts: How Much Paint--Area
We will be finishing up the Problem Solving worksheet from 2 weeks ago, which has a variety of problems. Please try to have your students look at it and possibly do up to question 10. Handouts: Problem Solving
If you want to put 11 pigeons into 10 pigeonholes, you find you have to put at least two pigeons in the same hole. This seemingly obvious statement is an example of the Pigeonhole Principle, and is a surprisingly powerful idea. We will discuss some generalizations and reformulations of this famous principles, and apply it to problems ranging from easy exercises to problems from the Putnam exam, the most difficult mathematics competition in the country.
This week in Mike's group we will move from the triangle inequality to some additional topics in plane geometry. Handouts: Geometry
In Clint and Liz's groups, we will continue to discuss problems related to the measurement of area.Handouts: FullAreaHandout
We will be having the children build 3D solids based on 2D projections. As well as havig the students build a 3D solid and sketch the 2D projections.
Please have your child bring 20 blocks or 20 (2x2) legos. Handouts: Projections
Graphs have a straight forward definition; they consist of a col- lection of points, some of which may be connected by straight lines or arcs. Despite their simple description, graphs appear in a variety of applications, particularly network design, competition problems and other branches of math- ematics. Topics we’ll cover in the session include the idea of Euler circuits and paths and incidence matrices. We’ll also review how these ideas can be used to solve several classical brainteasers, including the Konigsberg Bridge Problem and The Knight’s Tour.
This week, Mike's group will investigate rigid transformations of the plane and some of their properties. Handouts: Rigid Transformations
This week, Mike's group will continue to study motions of the plane, and investigate their use in problem solving. Handouts: Projections & Levels I | Projections & Levels II | Rigid Transformations
Clint's and Liz's groups will investigate different ways to describe structures built of small cubes.
Graphs have a straight forward definition; they consist of a col- lection of points, some of which may be connected by straight lines or arcs. Despite their simple description, graphs appear in a variety of applications, particularly network design, competition problems and other branches of math- ematics. Topics we’ll cover in the session include the idea of Euler circuits and paths and incidence matrices. We’ll also review how these ideas can be used to solve several classical brainteasers, including the Konigsberg Bridge Problem and The Knight’s Tour
After working with projections, we will now have the children use what they know about projections and transfer them to levels. As well as taking levels and transferring them to projections. This week will involve a lot of visualization without the use of blocks.
When working on this worksheet, please try to not use building blocks. Handouts: Projections to Levels
Clint's and Liz's groups will continue examining systems of representations for 3-dimensional block structures. Students should bring blocks/cubes/2x2 Legos, and bases for them if possible.
Generating functions encode information about sequences. We will use them to solve recurrence relations (think Fibonacci numbers), prove combinatorial identities and solve enumeration problems (how to count without counting).
This week Mike's group will revisit complex numbers to investigate their bearing on transformations of the plane.Handouts: Complex Numbers and Transformations
We will be continuing with the cubes, but now we will apply them to something more closely related to graphing. We will be "naming" each cube in a 3x3x3 solid and using these names to describe them. Handouts: Number Names
Generating functions encode information about sequences. We will use them to solve recurrence relations (think Fibonacci numbers), prove combinatorial identities and solve enumeration problems (how to count without counting).
Clint's and Liz's groups will look at sequences of numbers and the relationships they have with certain shapes.Handouts: Number Sequences
This week we will be going over rates. Starting off with individual rates and then displaying how working with one or more persons, one can get the job done quicker. Handouts: Teamwork
Mike's group will continue our discussion of complex numbers as they relate to plane geometry.Handouts: Complex Numbers and Analytic Geometry
Clint's and Liz's group will look some more at triangle numbers, other figurate numbers, and examine the number sequences they produce and some interesting patterns therein.
This week Mike's group will investigate patterns of wallpaper using our knowledge of rigid transformations and symmetry.
UPDATE: There was no handout for this week's session. To see all the pictures that we looked at, and find out about their origins (many are quite interesting!), visit the Wikipedia page on Wallpaper Groups:
You can read about the notation we used for symmetry types, and the classification of symmetry groups in the plane.
This Sunday we will be going over questions similar to those on the Math Kangaroo tests to prepare for the upcoming tests. We will help the children work through each problem and learn new methods in approaching how to solve them.Handouts: Kangaroo Problem Solving
This week Mike's group will investigate when a figure can be tiled with dominoes, triominoes, and other tiling patterns.
Clint's and Liz's groups will have their third and final session on figurate numbers and number sequence patterns.
What is linearity? We'll discuss a number of views of linearity and how they interact. The goal will be to see the big picture of what linearity is and how it applies, from computer graphics to physics.
We will continue with Math Kangaroo Practice from last week. There will also be more problems added for further practice. Handouts: Math Kangaroo Practice 2
We will work on old Math Kangaroo problems in preparation for this year's Math Kangaroo contest (Thurs., March 17).
We will first study many curious properties of the golden ratio, and then find this ratio in many geometric objects. In the process we will become quite familiar with the dodecahedron and the icosahedron.
For our las meeting we will be rounding out the quarter with problem solving. Focusing more on questions we discussed and worked on previously in the quarter. Including problems similar to the Math Kangaroo Test.Handouts: final problem solving
Join us for our last meeting of the Spring quarter for a team problem solving competition!
In our first meeting of the Spring quarter, we'll look at several problems involving sums and averages.Handouts: Sums + Averages
We will solve a variety of problems involving induction.
For our first meeting of Spring Quarter, we will be going into plotting points on a plane. We will use city structures to model how to plot "addresses in the city" or points on the plane.
Please have your child bring a ruler for this class as we will be drawing lines. Handouts: City of Descartes-Part I
This week in Mike's group we will take another look at symmetry patterns of wallpaper.
We will discuss the basic properties of the Cantor set.
We'll follow up our work on sums and averages by trying our hand at some simple, and not-so-simple, percentages problems.Handouts: Percentages
This week in Mike's group we will look at how to fold up a wallpaper pattern into a special geometric object. This will help us understand why there are only 17 wallpaper patterns.
We will be continuing with points in the plane. Focusing more on reflections and easing our way into equations of lines. Handouts: City of Descartes - Part II
Suppose you have a finite set of marks on a ruler, where the distance measured by any pair of marks is an integer. Furthermore, if different pairs of marks give different measurements, then this ruler is call a Golomb Ruler. This simple mathematical object has many interesting properties and real life applications, which we will explore in this talk. In addition, we will also consider some of its variants.
We will now be looking at distances in the plane. Relating this to cities as well as routes from one point to another. Handouts: City of Descartes Part III
Clint's group will continue working on percentage problems; Liz's group will race to complete a number of problems on races.Handouts: Races (Liz's group only)
Mike's group will discuss drawing maps on surfaces, and see what happens when we fold them.
Cryptography is the art and science of writing messages in code, and reading coded messages without the key (the latter is often called "cryptanalysis"). Mathematics enters into cryptography and cryptanalysis in at least two ways: it helps in designing encryption methods that are safe, usable, and fast - or helps to attack such methods. On the other hand, mathematics helps understanding structural properties of language which can be used to glean some information about even the most safely encrypted messages.Handouts: Cryptography and Cryptanalysis I: Fun and Games
Liz's group will check out some racing problems involving time, speed, and distance, while Clint's group wraps up work on percentages.
Alyssa's group will be working with permutations, or ordering, of people. We will take a close look at how the ordering is done, focusing on the shifts, or swaps, between people's places. Handouts: Permutations
Mike's group will investigate what happens when the rules of algebra are kept, but we don't say what it is that they are being performed on.
Cryptography is the art and science of writing messages in code, and reading coded messages without the key (the latter is often called "cryptanalysis"). Mathematics enters into cryptography and cryptanalysis in at least two ways: it helps in designing encryption methods that are safe, usable, and fast - or helps to attack such methods. On the other hand, mathematics helps understanding structural properties of language which can be used to glean some information about even the most safely encrypted messages.Handouts: Cryptography and Cryptanalysis I: Fun and Games Handouts: Racing (Clint) | Measuring (Liz)
Alyssa's group will be continuing with permutations. We will be focusing on composition of permutations, inverses, and counting and extending our knowledge to applied problems.
If you did not attend last week, please look over the worksheet from the previous session to ensure your child is not behind. Handouts: Permutations 2
Mike's group will continue to discuss multiplication of symmetries and abstract multiplication.
We will solve a variety of compass-ruler construction problems in plane geometry.
Clint's group will continue work on problems involving races and the concepts of speed, distance, and time. Liz's group will try their hands at some liar/truth-teller problems.
We will take a look at rotations and reflections of an equilateral triangle that move it into itself. We will use permutations to describe these transformations and to study their properties. Handouts: Transformations Pt 1 | Transformation Pt 2
Mike's group will investigate counting things by using symmetry, including pearl necklaces for one's mother.Handouts: Counting with Symmetry
This Sunday we will be taking a close look at areas. We will also derive the relationship between the area of a rectangle and the area of a triangle. Using these relationships to find areas of awkward shapes.
Please have your child bring a ruler for this Math Circle. Handouts: Fun with Areas
This week Mike's group will investigate some examples of groups, and discuss an example related to the counting problems from last week. Mike's group will be led by Daniel Nghiem this week.Handouts: Group Potpourri
This week both groups will have an introduction to binary numbers and their uses--even if you've seen them before, you'll find something new here!
We'll expand our exploration of binary into other number bases, and get a sense for how to do arithmetic in these other bases.
This week Mike's group will find out what day "pi Day" should be in different cases. We will also discuss the Cantor set and some of its amazing properties.Handouts: Base Conversion and the Cantor Set
We will be introducing the idea of sets today to the children. Not only what a set is, but it's properties including components, subsets, unions, intersections and so forth.
Please note: the first session we only went up to page 8 for this first session. Handouts: Introduction to Sets
The concept of probability will be introduced and further elucidated by way of studying examples and solving problems. Specifically, we will discuss the notion of likelihood, the operations that are allowed (and not) for probabilistic reasoning, independence and departures from it, connection to geometry, etc.Handouts: Probability Handout
Clint's group will move from binary numbers to numbers in all sorts of bases; Liz's group will try to understand the relationship between guessing games and what "information" really means.
Mike's group will investigate why 5 is not necessarily a prime number.
We will continue with sets this Sunday. Focusing more on it's applications to problem solving.
Often data in the real world can only be collected for certain values, but what if you want to approximate values that were not given? In this meeting we will be exploring methods for constructing polynomials that pass through each data points, and can be used to approximate, or interpolate, the data.
This week we will celebrate the end of the year with math relays--a sequence of fun, fast-paced math puzzles with prizes for the top-scoring teams!
For our last session, we will do a variety of problems related to the topics we have covered throughout the quarter.
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.