Homework: What is the quickest way to measure 9 minutes with a 4 minute hourglass and a 7 minute hourglass?
Answer: put the egg in. You can run the 7 and 4 minute together. Flip 4 when it runs out. Flip 7 when it ends. When the 4 ends, 7 would have flown for a minute. Flip 7 to get an extra minute.
AttendanceLiza, Damini, Siddhant, Zorawar, Arjun
Class puzzles
When we write 124 in decimal system, what does it mean? Units, Tens and Hundreds place
Why tens and hundreds and not twenties and hundreds? Place values are powers of 10
What if we wrote 124 in base 20? What number would it imply?
If we wanted to write 300 in base 20, how would we write it? (Don't answer the question yet)
What if we were writing in base 6? What digits would we be allowed to use, and what would be the place values?
Lets get to binary counting - base 2. What digits can we use and what are the place values?
Now give them cards with 1,2 , 4, 8, 16 dots
Exercise 1: Ask kids to show or hide the dots randomly. Ask other kids to see what number it means. Does that tally with computation using place value of digits, 1 being show and 0 being hide
Exercise 2: Ask the kids display some numbers that the class suggests. The kids have to collaborate to show the cards that add up to those many dots. Ask the class to now write that number as 1s and 0s.
Observe the pattern: Person at rightmost position always toggles the number. Any other person toggles their number when the number to its right goes to 0
How much can you count using 5 binary digits? 32 values (0 to 31)
How much can you count using your ten fingers on two hands? Ask some kids to try by writing the numbers on their hands.
What are things you see in daily life that are binary? Look at things in the room - switch, kids (standing or sitting), books (open or close), lights (on or off).
How can computers store binary values? Charge, magnetic field...
How can we send a binary number to someone?
Let kids think of ideas - remember, you only have 0s or 1st to send
Introduce idea of modulator - something that converts binary digits to sounds - say low pitch and high pitch
Ask kids to hold any number, and make those sounds
Ask rest of the kids to "demodulate" i.e. figure out which number it was
What would happen in base 3? base 6? How would you represent 231 in base 6?
Lets come back to the original problem of writing 300 in base 20. How should we write it? What digit values do we need? Any idea how we can get that many digits?
Puzzle: If you have a balance and 4 weights, how many different weights can you weigh
Kids may first come up with 1, 2, 4, 8 weights to get to 15 measurements
Notice that this is equivalent to having 0 or 1 against each weight - just like binary counting
Homework Problem:
In the problem with 4 weights, what if you can put the weights on either of the balances?
What system of counting does this look like - how many values can each weight take?
-1, 0 or 1 depending on whether you are keeping it on left side, skipping it, or keeping it on right
3 values per weight -hence ternary system
What are the weights we should use in ternary system - same as place values of ternary system (1,3,9,27)
So how many measurements can we support - 1+3+9+27=40