Instructor Note: Show practically. For few different number of coins. Show why with one or two coins (distance that the centre of outside coin travels)
(MC Chap 0, Prob 15) Jar full of beads
Instructor notes: Show practically
(Moscow - 23) Patterns of ball being passed around
(Contributed by Rhea) A girl goes to a shop, pays a Rs 10 note and buys a Rs 5 pencil. The shopkeeper doesn't have change, and gets change from the trader next door. When the girl's gone, the trader comes back and informs the shopkeeper that the Rs 10 note he gave is fake. The shopkeeper gives the trader another Rs 10 note, and has to throw away the fake note. How much loss has the shopkeeper had to incur?
Rs 10
Instructor Notes: Surprisingly, all kids said 5 or 15. Useful to have them step back, and make them see that ultimately only Rs 10 have been lost. Also worked to show the whole transaction with real coins.
(Contributed by Rhea) Christmas song - One gets 1 gift the first day, 3 gifts the second (1+2), 6 gifts the third (1+2+3) and so on for 12 days. What is the total number of gifts?
364
Instructor Notes: Add all; or 1x12 + 2x11 + 3x10 and so on.
Revisit Sequences
Arithmetic - Idli eating guy from Class 4
Sum of first n elements, sum of 1+2+3+4...
Exponential - king giving rice away, yeast, doodling binary trees from class 3
Binary - rabbit trying to get to pizza, ball dropped from class 3
Math in Music (INSTRUCTOR NOTE: WORKED WELL FOR STUDENTS WHO HAD SOME EXPOSURE TO MUSIC, NOT FOR OTHERS)
Show wave formation in string - use a guitar. Show distance computation between successive Cs, and explain in terms of closed waves
Show wave formation in water flutes - use water flutes. Show distance between successive Cs, and explain in terms of semi-open waves
Draw up the theory (high level), that frequency of successive Cs are double and the 12 notes are equally spaced as a geometric sequence
Ask kids if they understand chords, and why some notes sound well together (both different octaves, as well as fifth/harmonic)
String vibrations correspond to chords - following table shows frequency of different notes/octaves, and successive frequencies in a vibrating string (C note). Observe the correspondence to notes highlighted in red
To be incorporated: MC Diaries Y1, Chap 7 - Discussion of the Day, Porblem 2 (b), Problem 4, Problem 5
To be incorporated: MC Diaries Y1, Chap 8 - Even odd, Pictorial Pg 66 - This chapter is about odd and even, and relatively basic - might be good for 4/5th graders
To be incorporated: MC Diaries Y1, Chap 9 - Problem Set Problem 4* - Nelli faces two doors, one which will lead her to 20 random number, and another will lead her to 21 random numbers. Once she chooses the door, she will have to cross one number so that the remaining numbers add up even. Which door should she choose? (21)
To be incorporated: MC Diaries Y1, Chap 9 - Problem Set Problem 7* - 6 guards start out at six corners of a hexagonal tower. Every 15 minutes, two random guards get bored and move to an adjacent corner. Prove that all guards can not land up at the same corner (Color the corners alternately white and black - number of guards at white towers is always odd, and same with black)
(MC Chap 1, Prob 8) Can a 5x5 square checkerboard be covered by 1x2 dominos?
Answer: No; Follow up - mutilated 6x6 chessboard with opposite corners removed
Instructor Notes: Ensure kids see the number of colored and white squares and why it can't be done
(MC Chap 1, Prob 1) Eleven gears are placed in a plane, connected to each other in a circle. Can all gears rotate simultaneously?
Asnwer: No; Clockwise and anticlockwise gears have to alternate
(MC Chap 1, Prob 6) Mahika and her friends are standing in a circle, such that both neighbors of each child are of the same gender. If there are 5 boys in the circle, how many girls are there?
Answer: 5; alternate
Instructor Notes: Kids tend to get the alternate pattern, but may not be able to prove that thats the only option. Work with them to show what happens if there are two boys or two girls next to each other.
Homework problems:
(Khan Academy) 10 prisoners are given a chance of survival by the jailer. The jailer tells them that the next morning, he will have them stand in a queue (so that the last person can see the nine people in front, the second last people can see eight people in front of him, and so on), and then place either a red hat or a purple hat on each person's head. The person hence will be able to see the color of hat for people in front of him (but not his own hat). The jailer will then start from the last person, and ask each person "What is the color of hat on your head". The person can answer "red" or "purple". If the person's answer is correct, he will be released, else he will be killed. The jailer will then ask the same question to the second last person, and so on. The prisoners have the night to figure out the strategy so that maximum number of prisoners can survive (on a guaranteed basis). Figure out a strategy to save the maximum number of prisoners.
Answer: "Last guy calls the odd number of hats in front". Every person there on keeps a count of odd or even parity.
Instructor Notes: Take example to explain, let kids do it, try for smaller number to illustrate induction, parity thinking
References:
Mathematical Circles (Russian Experience), by Dmitri Fomin, Sergey Genkin, Ilia Itenberg
The Moscow Puzzles, by Boris A. Kordemsky
A First Mensa Puzzle Book, by Philip J Carter, Ken Russell