Los Angeles Math Circle

LAMC Calendar // 2017-2018 Academic Year. Jump to:

For meetings prior to Fall 2017, visit the Circle Archive.

AdvancedBeginnersEarly ElementaryJunior Circle
We begin our exploration of Egyptian Fraction Representation by studying the details of unit fractions. How do they add? What types of fractions can they represent (all of them!)? We close off by analyzing a recursive algorithm which allows us to slowly construct any fraction by smaller and small unit fractions that will (hopefully) terminate at some point.
Handouts: Handout | Solutions
We continued our study of unit fractions, and showed that all fractions can be written as a finite sum of unit fractions, showing that the Egyptian Fraction Representation exists for all fractions. We then wrapped up by proving other interesting properties of EFR.
Handouts: Handout | Solutoins
We did many word problems associated with percentages that required them to be written as fractions and a small degree of algebra to keep track of percent/fractional multipliers. We wrapped up with a brief visit to compound interest, hinting towards continuous compound interest.
Handouts: Handout | Solutions
We explored the strange hexahexaflexagon which has many more sides than a flat object normally does, 6 in total. We studied the patterns of the hexahexaflexagon: differeing orientations, which sides are connected to others, and what sides compose "main circuits" of the hexahexaflexagon.
Handouts: Handout | Solutions
We solved classic problems of how long it takes multiple people to accomplish a single task at different rates. Pipes filling pools, rabbits eating carrots, and non-uniform burning strings. Through this, the students learned that adding together varying rates of time is not straightforward and required quite a bit of algebra, percentages, and fractional multipliers.
Handouts: Handout | Solutions
We began the study of what many computer scientists call the remainder operator. Modulation of numbers over a given base, which leads to the development of equivalence relations where we can find numbers such as 2 and 8 equal to 0.
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We continued our study of modular arithmetic, and rigorously defined the equivalence relation between numbers under a given modulo. We found that this equivalence relation obeys many of the same properties as the traditional equals sign, which leads to a new structure of numbers.
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We will played subtraction games such as Nim, Epmty and Divide, Chomp, and Dynamic Nim. All of these games could be solved using parity, powers, and inductive gamestate reduction, as the students quickly learned so they could beat their instructors!
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We end the quarter by playing Hackenbush, Toads and Frogs, Cram, and Kayles in competition for small prizes. Also, we did some fun Santa's Sleigh dimensional analysis and Dreidel probability.
We rigorously defined the negation to a logical statement, and used that definition to explore contrapositives and proofs by contradiction.
Handouts: Handout | Solutions
We used strangely shaped pizzas and number patterns to predict (and prove) how a pattern will always continue, even at the trillionth term. Then, we formalized this pattern analysis as the logical statements that compose proofs by induction.
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We continued our study into induction by looking at number patterns, and we took a more quantitative (numeric) approach to proving the logical statements that compose a proof by induction.
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We studied geometric number sequences (such as triangular and pentagonal numbers), and then predicted new terms based on the successive differences of previous terms. Afterwards, we formalized successive differences of sequences notation and explored the beginnings of discrete differentiation.
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We delved into logical deduction problems, where we compare multiple logical statements, and determine their values (true or false) based on the value of some combination of them.
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We practiced effective test taking strategies with real Math Kangaroo problems such as drawing pictures, process of elimination, and skipping questions.
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We introduced the concept of graphs as sets of vertices and edges. Then we proved a number of lemmas and theorems associated with vertex degree, bipartiteness, and Eulerian Paths.
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We finished exploring the ideas from last week's packet, and will move on to higher level concepts next quarter.
Traditional math dominoes competition for glory and fame!
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Even though we were a little late, we learned some amazing properties of Pi (and Phi!).
Handouts: Handout | Solutions
We solved the famous puzzle known as Instant Insanity using an innovative and thought provoking technique that utilizes many skills developed in past quarters.
We finished up our solution of Instant Insanity and continued with graph theory theorems and applications.
We investigated the optimal algorithms to make change with different denominations of currency and how those algorithms can carry over to analyzing trees in graph theory, a structure that is ubiquitous in data science.
We used some techniques from modular arithmetic to play tricks on every student!
Handouts: Handout | Solutions
We used physical intution and simple pictures to begin exploring opposing torques on a bar culminating in the Law of Levers.
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We explored the foundations of probability theory through blind teachers throwing darts.
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We will get to explore some very strange dice hands on.