august 30 - circle 2.0

For the lesson of August 30, 2015.....

1. Symmetry

The 2.0 group reviewed the symmetry lesson of last week.  We also worked together to complete a pattern on a grid using the symmetry worksheet below.


2. Number line

After symmetry, we introduced the concept of the number line.  We identified numbers on the number line.  We then used the number line to determine which number was greater. At the end of the topic, we tried simple addition.

3. Money

The 2.0 group sorted a bag full of pennies, nickels and dimes into the three types of coins.  We then learned that one nickel is equal to five pennies.  For homework - how many pennies are needed to have the same value as 5 nickels?

august 30

• Work out the cross problem: tile the plane; find an integer lattice; cut the cross. 
• What is the area of a square: 1, 4, 9,... But the cross's area is 5; how we can make a square of the area 5?
• Try the same trick with other pentaminos? P, T,  
• Take white and black stones, same number of each. Place them on a circle; find a place so that when you start from there and add 1 for each black stone, and subtract 1 for each white stone, you never go below 0...
• Try to do the trains (using the Boston Metro visualization widget)
• Slice of bread with two cuts can be split into 3 pieces. What about a bagel? A pretzel?

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?
• Here is a solution (there are many!) of the cross puzzle.
  
  
  
  Can you cut and reassemble each of these two pentaminos, to get squares?

OK, train again: if it makes 100 miles per hour, how many miles it makes in 15 minutes? 30 minutes? 45 minutes? One hour and 15 minutes?

august 23

Warm-up:

• 100 heavy guys want to cross a wide river. Two kids happened to be around with a boat which can carry no more that two kids or one adult. Can one get all the big guys across?
• Alice, Bob and Clara ate apples. Alice and Bob together ate 5 apples (we call it A+B=5). Further, A+C=3, B+C=6. How many apples ate each of them?

Same if A+B=3, B+C=2, C+A=2.

Graphs of functions: 

• Train moving from A to B...

Homework:

• Partition the cross with straight cuts into several parts from which make a square:

• Solve:
• A+B=5, B+C=5, C+A=4;
• A+B=5, B+C=5, C+A=6;
• what about A+B=5, B+C=5, C+A=5?
• A train goes with speed 100mph. 
• Plot where it will be in 1h, 2h, 3h, 4h. 
• What about 30min? 1h30min? Draw as many points on the graph as you can...
• What if the train makes a 1 hour long stop after two hours? Show where it will be after 1, 2, 3, 4 hours...

    

may 24

Can one tile the whole plane with 2 monominos, 2 dominos, 2 triminos etc (all - linear).

Recall Nim's; move to Nims with 2 piles; Nims with 2 ends...

Intermission: CCXLVIII-CXXII=?

Split payment fairly...

Homework

may 17

• which of these tiles can tile the whole plane?
  


• Analyze Nim₄, Nim₅. What about Nim_1,3 (you can take 1 stone or 3 stones, but not 2...)?


• Can you partition with one straight cut these figures so that each part has the same area of red and white?
  
  
  
• Split payment fairly...

Homework

• Can one tile the plane using these shapes?
  
  


• Create the table of winning and losing in the Nim_1,4 game (one can take 1 or 4 stones). Who wins if you start with 99 stones? 100? 101?


• Cutting the shapes! - the problem above is still unsolved!

may 10

• Nim (or Nim₂, where one is allowed to take away 1 or 2 stones, other Nims are possible) - the complete solution is easy: if one faces the pile with the number of stones there divisible by three, one loses given the opponent plays wisely, and wins otherwise. We established that using the recursion, an awfully useful trick.
• This led to the question, how you determine divisibility by three? The sum of digits trick helps...
• Draw projections of some shapes
• Split into 3 parts led to the question: can one tile the plane with these pentaminos?

Homework:

• One can tile (cover without overlaps or holes) the whole plane with the r pentaminos, for example as shown below:
  
  Now, can you tile the whole plane with T-pentaminos? Or with y's? or z's?
  

• Is 198407392 divisible by 3? What about 38002978643001?
• Who wins the game of Nim₃ starting with the pile of 101 stones?

may 3

• We finish the three different coins puzzle (how to order them with fewest weighings on a scale)
• We'll work on the Nim game: what is the best possible strategy? Who wins if one starts with 100 stones?
• Vectors! we will add more than one vector; and see what kind of configurations sum up to zero...
• We play more of the "pick a symbol" game...
• Time permitting, we'll talk about splitting the payment for rice bowl...

Homework!

• We established the pattern of whether the person wins or loses facing a pile in the Nim game - one wins if the pile has 1 or 2 or 4 or 5 stones; loses with 3 or 6... 

  1	2	3	4	5	6	7	...	...	99	100
  W	W	L	W	W	L	W	...	...	?	?

  Can you say what happens when you have a pile of 99 stones? of 100?
• Roman numerals strike again: Find CIX+LXXI-XL=?
• Split into three equal shapes: