Nov 29, 2015 - Math Circle 2.0

Chess board coordinate review

The students first reviewed the coordinates of the chess board grid.  We picked a spot on the board and then identified the coordinates.  We also tried the reverse.

Missing school bus

Travis hid a school bus on the grid.  The students had to search for the school bus using the coordinates.  Think battleship.  Once the students found the first part of the school bus, the rest of the bus was quickly found.

Shapes with the grid

We used chips on the grid to make rectangles and triangles. 

nov 22

Solving some equations.

How A, B, C pay $1 each to the bus driver who has no change, having only $2, $3 and $4 dollar coins.

Cross 8 digits from 13571357135 so that the remaining number is the largest.

Homework:

1. Find, how many candies are in square and in round boxes for each of the cases:
• ▢+▢+◯=13;         ▢ +▢+◯+◯+◯=19;
• ▢ +▢+▢+◯=15;  ▢ +◯+◯+◯=21;
• ▢ +▢+▢+◯=6;    ▢ +◯+◯+◯=18;
2. Write these Mayan numbers in our common (Arabic) notation:
   
   
   
   
   

nov 8

Continue with rooms: find criterion that one can do a circle.

Solving some equations, like x+2y=4; x-y=1.

Bacteria in a glass problem

A, B, C pay $1 each, having only $2, $3 and $4 dollar coins.

Areas under a graph?

Experiment: water; stones

Nov 15, 2015 - Math Circle 2.0

Number line

We added numbers together.  We also plotted every students age on the number line.  We noted that we would need a longer number line to plot the parents' ages.

Shapes from cutting paper

When you draw a certain number of lines on a paper, how many pieces of paper do you have?  

The students determined the answers for 0 to 3.

Homework:  if you draw 4 lines, how many pieces of paper would you make if you cut along the line?  Is there a pattern developing.

Robot chess king

The king is restless. How many different squares can he travel to from square 1b. What about squares 1a and 1c.  It was charted out graphically.

Nov 8, 2015 - Math circle 2.0

Clapping patterns.  We switched it up by having each student write out a clapping patterns for the other students to perform.

Number line.  Numbers decrease to the left.., increase to the right.  We also used our number line to measure objects like pencils and our hands in class.  Discussed what the number line is measuring when the object lands between two whole numbers on the number line.

Robot chess king.  We identified a start point and an end point on the chess board using rank and file.  We chose a start location and and end location.  We asked the students to figure out how many different ways the king can get to the destination if he could only move x number times. Normal chess moves apply....

Clapping, Number line, Counting coins, and Roboths Math Circle Nov 1 2015

Clapping Patterns

Right and left hands clap on table

First identify right and left hands. 

Then write R and L on hands to help students

Then try to clap against table 

R-L-R-L-

4/4 time, "-" = 1/8 rest

One at a time 

Then 

R-L-RRL-

Number Line; 

What is it? And what do the arrows mean?

Write 1 then 2,3,4,5 on line, starting at the third mark

Coins: 

Place the number of coins on each number to Link counting to numerals 

What numbers are bigger and smaller? 1 vs 2, 3 vs 4, Etc

What is smaller than 1? How many pennies is that?

What is smaller than 0?

Chess board: ranks and files with numbers and letters

Homework 

How many pennies is -1?

If the king is on 1d, what spaces could he reach in one move? In two moves?

Patterns, Rhythm, and a visit to the Store Math O 2.0 18 Oct 2015

Patterns

Finish the pattern. Starting with a few shapes, repeat the shapes

Music

Inspired by Steve Reich's Clapping Music

What is a nickel and a penny worth?

How can you buy a 5c banana?

Robots in the wild

Kids are robots. What instructions are required to get from Ellie's to  Dizas house

Patterns, Clapping Patterns, Addition and Subtraction, Coney, Magnets. Math O 2.0 25 Oct 2015

Clapping

this means 'right left left right'

Blue = Clap, White = no clap

(Too confusing)

Blue = Right hand, White = Left Hand

Lego patterns in sets of 4

Addition and subtraction 

Many variations of:
* Addition
  * We have two blue legos and two white. How many total?
* Subtraction:
  * 6 legos, take away 2, how many are left?
  * Start with 6 legos. Look over there! I hid some legos, how many did I hide?

Buying food

Each kid gets 5 pennies and two nickels.

We have many different items. How many pennies do they cost?

A carrot costs 4c for 1/2. How much is a whole?

Experiment 

Which items are magnetic?
What makes an item magnetic? (remains unanswered, may return to this next time)

oct 25

• Proportions:
• Blob ate three times as many mushrooms as Glob, and Clob ate two times as many as Blob. Together they ate 40 mushrooms. How many each ate?
• Pete had $3, and Paul $4. They bought together 14 rabbits. How many each gets?
•  Find three (or more ways) to cut a square - with straight lines - into two equal parts.
• Do they always intersect at the same point?
• What about a triangle? 
• Learning to count auf Deutsch:
• Funf-und-zwanzig - 25
• Sieben-und-dreißig - 37
• Zwei-und-zwanzig - 22
• Ein-und-zwanzig - 21
• Ein-und-vierzig - 41
• Mayan numbers

• Cut and fold...
• Can one wrap a 1x1x1 cube into 3x3 piece of paper?
• Old puzzle:
  
  

Experiments: sliding magnets

Homework:

Alice and Bob drove 500 miles. Alice drove 3 hours, and Bob 7 (with the same speed) - how many miles each of them drove?  

What numbers are Sieben-und-vierzig, Drei-und-dreißig, Vier-und-vierzig, Funf-und-funfzig?

Find at least three ways to cut by a straight line this triangle into two parts of equal area:

oct 18

• Continue with the areas of the lattice polygons. Work on Pick's formula. 

Reminders: 

Pick's formula for the area of lattice polygons - polygons with vertices in the nodes of a lattice - is:  Area=(number of nodes inside the polygon)+  (number of nodes on the boundary)/2 -1

Cavalieri principle asserts that the area of a figure - viewed as as stack of parallel intervals - does not change if one slides these intervals back and forth. 

• Padlock puzzle
• Linguistics
• Work with compasses; measure distances
• Proportions:
• Split 9 apples so that A gets 2 times as much as B; 
• split 8 pears so that A and C gets the same, B gets 2 times as much as A;...

Experiments
Laser- conical sections

Homework

• How to split 21 peaches between Anne, Amy and Aman so that Aman gets twice as many as Amy, but half as much as Anne?
• M. Argan takes three kinds of pills, one of each each day. He left for a week long travel with 7 pills of each kind. Unfortunately, in the box they all got mixed together, and they all look the same. How M. Argan can still make sure he takes one of each pill each day?
• Find the areas of each of these three figures:

Robots, Coins, Patterns, and a Prism. Math Circle 2.0 Oct 10 2015

Robots

Last week we re-programed the robot so that it could backwards.

Now, the robot not only understands North and West, but also South and East.

Furthermore, now instead of giving one instruction at a time, we give the robot all commands. To do so, first we write out the steps, then have the robot read the steps, and then see what the robot does 

Can we get the robot back from the NW to SE?

Coin Algebra

How many pennies in a nickel? How many in two nickels?

What is a dime worth in terms of pennies? In terms of nickels?

Started with a dime, a nickel, and a penny

trade nickel for five pennies. How many pennies?

Trade dime for ten pennies, how many pennies?

How many nickles can we get

Patterns

Lego chains

Drawing zigzags

1. What is the core pattern? How many times can we repeat this pattern given the pile of legos below?

Draw Triangle, circle square, and repeat. 

Experiments 

Split light into colors with a prism

What color is sunlight? Where do the colors of the rainbow come from?

If we split into ROYGBIV, where do brown, grey, and black come from? Why is the table brown, but we only get ROYGBIV from the prism?

And look at the Sun. How has the shadow changed in 3 weeks since solstice?

Homework 

Continue the patterns:

oct 10

• Continue with the areas of the lattice polygons: introducing half-integer areas. Are integers and half-integers all what we can encounter?
• We'll do a few more tree-climbing caterpillars... Where are the leaves hanging?
• Four German phrases and their translations are given (in a wrong order):

Er liest die Nachrichten     He reads a newspaper

Sie liest eine Zeitung       She eats a sandwich

Sie isst ein Butterbrot      He reads the news

Er liest eine Zeitung        She reads a newspaper

   Find which sentence translates which, and translate into German: Peter eats a sandwich.

• Proportions:
• More pizza: split it into four parts, give three to Alice, Bob and Clara, and continue: split the remaining piece into 4 parts etc. How much each is getting?

Experiments

• 2D version of Chasles theorem using optical illusions...

Homework

• Cut and fold from a single piece of paper (no glueing!):

• Four German phrases and their translations are given (in a wrong order):
Peter fährt sein Fahrrad    Lisa sees her brother
Lisa sieht ihr Bruder       Lisa likes her bike
Peter sieht sein Fahrrad    Peter rides his bike
Lisa mag ihr Fahrrad        Peter sees his bike

Find which sentence translates which, and translate into German: 
She sees her brother.




• Find areas of these polygons:
  




• Caterpillar crawls around the tree. Show the graph of his height, and show where he gets some leaves to munch on:

Roads, Robots, Patterns and an eclipse. Math O 2.0 Sept 27 2015

Towns and Roads

Each road is 1 mile long, how long is the path from bob to king?

K ------ B

Gary to King to Bob to Liza to Bob to King? 

K ------ B

|             |

|             |

G           L

How about with another road?

K ------ B

|             |

|             |

G--------L

Robots ... 

Homework review

Make 2 N, 2 W cards, line them up for each path

Can robot get back (recall, robot only understands N, W)?

What would we have to teach the robot to get him home?

Took a while but they figured it out!

South, the opposite of North

East, the opposite of West

Patterns with Legos

Patterns with sets of 2, 3, 4, 6 colors, repeating 

Experiment: 

The hardy watched a lunar eclipse!

sept 27

• Continue with the areas of the lattice polygons: introducing half-integer areas. Are integers and half-integers all what we can encounter?
• We'll do a few more tree-climbing caterpillars...
• Proportions:
• Pete gave $2, Anne gave $3; they got 15 candies. How they should split them?
• $2, $3, $4; 18 marbles
• $1 and $2; 1 pizza; how to split?
• Train homework
• Frog exchanging positions; 3+3. What about 4+4?
• Cut and fold...
  
  

Experiments

• we'll try to align the shadows of two pencils with given points on the table.
• drop the card to a landing plot: how? 
• we'll continue tracing the shadows of the sun (trying to get analemma)

sep 20

• Finding the areas of lattice polygons
• Train: how much travelled in 15 min, 30 min, 45 min, 6 min...
• Tree; worm crawls around it. Draw its height, how it changes in time.
• How many maxima? what they correspond to?
• Minima?
• Stone moving game: define a "stationary" configuration. For what number of stones a stationary configuration exists?
• Frog jumping puzzle

Experiments:

• shadows of two pencils: is the distance getting longer as we move them?
• Thee mirrors; rays reflected right back

Homework

• Find the areas of these polygons:
  



• The train (still!) makes 100 miles per hour. Find how many miles it makes in 6 minutes, in 12 minutes, in 21 minute...
  



• A caterpillar climbs a tree. Here is the plot of her height. Can you draw the tree?



• We take some stones and do the following operation: take all the stones from one of the 5 slots and place them one by one on the other slots, clockwise. The we take the stones from the next slot and repeat...
  We call a configuration stationary if after one steps it repeats itself: for example, 4,3,2,1,0 is a stationary configuration. Or 1,0,0,0,0 is.
  Can you find other stationary configurations?
  

Math Circle 2.0 Sept 20, 2015: Graphs, Robots, Algebra, Cells, and Sun

Graphs, Robots, Algebra, Cells, and Sun 

Graphs

Similar to last week (four points, how many connecting lines), but with objects (round clay dishes are towns, below) and strings (pipe cleaners).

The story is that the king in his Lego castle has asked Bonita the Banjo player from Batesville to come play. Then asks Guy the Guitar player from Galax and .... Then Filament the fiddle player from Finland ... They each live in different places. How many ways can the musicians get to the castle.

• connect two towns. 
• define, find, and count: how many nodes (towns)? and edges (lines)?

• Now with three towns ...

• And with four?

Robots

Robots are going to walk from town to town.

• Create a 3 x 3 grid of points
• Pink pipe cleaner is robot, understands two commands: North and West
• students take turns instructing robot to get from bottom green square at SW to white square at NE

How many ways can robot get to NE corner? Students found NNWW, NWNW, WNWN, WWNN. 

Homework: Are there others? (Hint: yes)

Algebra

count the number of circles on one square Lego.

How many circles on two square Legos?

Experiments

1. Earth's movement: rotating on its axis and around the sun. 

note: we are starting at the fall equinox

Materials:

• cardboard + graph paper
• sticker
• south-facing window
• pencil

1. place sticker on window
2. paper so sun's shadow lands on it
3. mark the shadow location at 11 AM, 11:30 AM, 12 PM
4. Repeat each Sunday morning for one year.

sep 13

• Finish the bridge crossing puzzle: 
• find how long each pair is crossing;
• find how many time a pair is crossing
• optimize
• Find areas of lattice polygons:
• Find fractions of 60 (minutes of an hour); of 100 (miles). 

Experiment: trace the shadow of an object as the light source goes along straight line.

Math Circle 2.0 Sept 13 2015: Symmetry, Graphs, Algebra, (Robots if time) and Cells

Symmetry, Graphs, Algebra, (Robots if time) and Cells 

Symmetry

Which letters are symmetric?
Draw a line through the axis of symmetry for each letter

A B C D E F G H I J K L M N O P Q R S T U V W X Y Z

Which letters have

• vertical
• horizontal
• radial symmetry?

How to draw a line of symmetry for B? F? O?

An "O" has radial symmetry ... infinite lines of symmetry

 

 

 

 

 

 

 

 

 

 

 

Graphs

Start with two locations, the castle where the king lives and Philo, where the guitarist lives.

How many straight paths can Gary the guitarist take to the castle?

Bob the Banjo player lives in another town called Memphis (Memphis, TN, only about 3 km from the castle).

How many lines can you make among Philo, the Castle, and Memphis?

Ichabod the Instrument repairman lives in the center. How many straight paths among all four locations?

How many lines can you make b/w n dots

(Next time)

How many 

* Vertices?

* Edges?

* Triangles?

E=V+T-1 (Eulers Formula)

How many V+T?

Algebra

Gnomes each eat 1 apple. Dwarves each eat 2 cabbages.

A Gnome arrives at your tavern. If you are the inkeeper, how many apples do you need to provide? How many cabbages?

A Dwarf arrives. How many apples will you need total? How many cabbages?

Another Dwarf arrives.

...

Experiment

What is a house made up of (bricks). They all look the same but make something bigger.
What are plants made of (cells)? How can we see them? Where can we find them?

Roots?
  * Lets get some roots.
  * Cant see them. Lets look at a radish
  * Cant see the cells.
Lets try to stain them (vodka to clear, dry, rub permanent marker, vodka to rinse)

Which cells are bigger than others?

 Preview: