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:

Sept 6 2015 Math Circle 2.0

Homework Review

What is the value of 5 nickels in pennies

Symmetry

Radial Symmetry? Which animals have radial vs bilateral symmetry?

Robots in Urbana

Students can give a lego guy the following instructions: North, West, South, East.
How many ways to get from this corner to that corner?

ref: Math Circle 1.0 Oct 10 2013

Algebra

Gnomes and Elves eat Vegetables. Gnomes eat one corn cob and one tomato. Elves eat two tomatoes. 

sept 6

• We'll start with two problems from homework:
• A family is crossing a dark narrow bridge; at most two of them can cross at a time, and the family has just one flashlight. Mom can run across in 1 minute, dad in 2 minutes; son in 5 minutes and their pet sea lion in 6 minutes. Can they all cross the bridge for 13 minutes?
• Cutting and reassembling
• Place a saucer on 5 tall wooden blocks. How many one can remove to keep the saucer stable?
• Graph the train movement (homework continued).
• Remind about remainders. Examples. How many apples can be divided equally between 2 and 3 people without remainder? 
• Experiments: Mirrors.

Homework

• What about that family on the bridge? still not done!
• Cut and reassemble in a square the Z pentamino from the past homework (august 30).
• One can draw various squares with corners on the nodes of the graph paper (two examples are shown below. Draw more squares (of different sizes, of course) - the more, the better!