# Circle on the Road: March 9, 2013

**Math Teachers’ Circles**: A1: Parity Party., B1: The DBD Oil Spill.

**Kindergarten to 3rd Grade Students:** A2: Function Machine Garden., B2: Coloring, Handshakes, Maps, and More: Exploring Combinatorics.

**4th to 6th Grade Students**: A3: This is Math? This is Math!, B3: Verbal Arithmetic.

**7th to 9th Grade Students**: A4: The Cube Coloring Problem., B4: Pascal’s Triangle in Sidewalk Chalk., A6: The Cube Problem., B6: Cryptography: Making and Breaking Codes.

**10th to 12th Grade Students**: A5: The Spider and the Fly., B5: Operation Cookie Jar.

# Math Teachers’ Circles

### A1: Parity Party

An integer can be even or odd. This statement isn’t very profound, but if a quantity can vary while retaining its evenness or oddness then we might find this observation pretty helpful. In fact, this ‘keeping its parity’ phenomenon possesses a certain – albeit modest – power. What’s more, it’s a stepping stone to a much greater power – that of invariants. In this session we’ll investigate parity and other invariants occurring in and helping to solve a few games and problems.

### B1: The DBD Oil Spill: Engineering Pathways to Math Circle

Ten days ago, a big underwater oil well from DBD Petroleum Company ruptured near the coast of Paradise Island. DBD sent a robot to take pictures of the damaged pipes, and using these images they reported to the public an estimate of 8000 barrels per day being discharged into the ocean. The government of Paradise Island has contracted your team to evaluate the DBD estimate of oil discharge. Using what you’ve learned in your mathematics classes, how will your team approach this problem? Come explore a solution with us!

# Kindergarten to 3rd Grade Students

### A2: Function Machine Garden

In this session, the leader will tell a dramatic story about a Creature who steals a child’s valuable treasure, then issues a mathematical challenge that the child must conquer to win the treasure back. The students will join the fictional child in an obstacle course of function machines in the Creature’s garden; each solution is a clue. In a function-machine problem, students use given input and output numbers to deduce a corresponding rule. For young children (ages 5-7), the challenge is to deduce the rules of multiple function machines. For children ages 8-10, the challenge is to deduce rules and their inverses. For middle-school students (11-13) the challenge is to deduce rules, combine them to form a compound function, and then create its inverse.

### B2: Coloring, Handshakes, Maps, and More: Exploring Combinatorics

In this session, we will investigate combinatorics combinations (counting ways of selecting several things out of a larger group, where order does not matter) using different examples.

# 4th to 6th Grade Students

### A3: This is Math? This is Math!

Come and engage in interactive games, puzzles, and other activities which are accessible to fourth to sixth graders, but will provide food for thought for years afterwards. No prior experience necessary.

### B3: Verbal Arithmetic

In verbal arithmetic, a normal arithmetic calculation has had all the digits replaced by letters. How can we restore the original numbers? During the seminar, we’ll work our way through a bunch of verbal arithmetic problems, while learning helpful techniques for solving them.

Verbal arithmetic problems are both educational and entertaining. There is a lot of detective work involved in solving them. At the same time, working on these puzzle-like problems gives students a good chance to deepen their understanding of numbers and arithmetic operations.

# 7th to 9th Grade Students

### A4: The Cube Coloring Problem

Cubie is a 10 × 10 × 10 cube who falls into a bucket of paint during his birthday party. In this workshop we will help Cubie find out how many of his unit cubes have paint on one or more sides, and how many are not painted at all. We will then turn the question around and ask how to paint the unit cubes so that they can be assembled into a larger cube, with paint showing on all the outer sides. The question gets interesting when we use more than one color, and want to assemble the larger cube whose outside is all one of these colors.

### A6: The Cube Problem

A net of a polyhedron is a planar representation of a three-dimensional shape. In this session, we will learn how to translate between a net and a cube so that we can solve problems where values are assigned to faces and vertices in a certain way.

### B4: Pascal’s Triangle in Sidewalk Chalk

We will be exploring patterns and probabilities in Pascal’s triangle while creating giant fractals in sidewalk chalk. Math can happen anywhere!

### B6: Cryptography – Making and Breaking Codes

We will all learn some simple ways to make secret codes and some simple ways to break secret codes. We will practice our coding and decoding skills, learn how computers translate letters into numbers and more.

# 10th to 12th Grade Students

### A5: The Spider and the Fly

The spider and the fly are sitting in the coordinate plane. The spider sits at (1,1) and the fly sits at (2013, 12). The spider is no ordinary spider. It can move two different ways. From a point (a, b), the spider can move to (a + b, b) or (a, a + b). The fly is terrified and sits perfectly still. Will the spider ever catch the fly? We’ll see how Euclid can help the spider, discover a cool tiling pattern for a bathroom and then send the spider hunting on a donut. Audience: 8th – 12th grade, teachers are welcome.

### B5: Operation Cookie Jar

The question is not ‘Who took the cookies from the cookie jars?’, but whether he could have emptied every one of the fifteen cookie jars following all the cookie jar rules. Yes, there are rules about all those jars and cookies. You can’t just use any old grabs to gobble them up. Come to the Cookie Station, and we will tell you the rules so that you can figure out a strategy (or two) to get all the cookies! And there are prizes for doing just that: COOKIES, of course.