Batch 1 - Class 04 - Equations, Ratios, Sense of numbers
Pre-class problems:
Get your own "Fibonacci in nature" sample
Put "+" signs between digits 9 8 7 6 5 4 3 2 1 to get 99. You are not allowed to change the sequence of digits. You may choose to not put any sign between some digits, for example you could use 98+7... kind of formation
There are balls of 10 different colors. All balls are of 10gm each, except one color which are all either 9gm or 11gm (we do not know for sure). There is a weighing scale where we can put any number of balls and read the weight. In one weighing, figure out which balls are odd weight, and whether they are lighter of heavier than others.
Put "+" signs between digits 9 8 7 6 5 4 3 2 1 to get 99. You are not allowed to change the sequence of digits. You may choose to not put any sign between some digits, for example you could use 98+7... kind of formation
Answer: 9+8+7+65+4+3+2+1 9+8+7+6+5+43+21
There are balls of 10 different colors. All balls are of 10gm each, except one color which are all either 9gm or 11gm (we do not know for sure). There is a weighing scale where we can put any number of balls and read the weight. In one weighing, figure out which balls are odd weight, and whether they are lighter of heavier than others.
Answer: Take 1 ball of 1st color, 2 balls of 2nd color, and so on. Weigh the 55 balls. The difference of weight from 550gm is the pile from which those balls were taken. The weight being more or less than 550 denotes heavier or lighter balls resp.
Instructor Notes: Kids typically will start at choosing 1 ball of each color. Then try to get to to different number of balls.
Equations: Instructor Notes: Fundamentals of equations - notation x, y, z. Basic method of solving by eliminating one variable at a time. n linear equations in n variables (show inductively with 1 variable, two variables and so on). Emphasize on formulation of problem in mathematical terms. MULTIVARIATE EQUATIONS SEEMS ADVANCED FOR KIDS OF THIS AGE, SO SKIP ALGEBRA, AND SKIP FIRST PROBLEM. DO REST WITHOUT USE OF ALGEBRA
(Shakuntala - 1 adapted) Meena goes shopping with Rs 16 in her handbag in one rupee notes and 20 paise coins. When she returned she had as many one rupee notes as originally she had coins, and vice versa. She came back with a half of what she started with. How much of each currency did she exactly start with?
Answer: 15 rupee notes and 5 coins
(Shakuntala - 15) A man ate 100 idlis in 5 days, 6 more each day than the previous. How many did he eat the fourth day
26
Instructor Note: Illustrate with beads of two colors, one for first day, and another for each subsequent day. Total the rest (60), and then show that 5xfirst_day must equal 8. Alternately, get kids started with a "guess", say 10, and then move them up or down.
(Shakuntala - 68) 8 spiders and beetles in a box have 54 legs. How many spiders
Answer: 3
Instructor Note: Get kids started with a guess, say 4:4. and then try to reduce excess legs. Or note that 9 beetles is 54 legs, and 4 beetles is same as 3 spiders.
There are some berries. Eldest daughter takes a third, next takes one third of remaining, last takes one third of remaining. Mom distributes 8 remaining and then everyone has equal. What is total number
Answer: 27
Instructor Note: Illustrate the answer with beads. For younger kids, go with hit and trial. Go bottom up with 8 being 2/3 before last step, and so on
Fun exercise - (Moscow - 23) - 12 girls are standing and playing with a ball passing to their right. After a while they get bored, and decide to pass by skipping one, and so on. Draw a pattern that describes this.
Homework: Draw patterns with 13 girls and different skip levels.
(Contributed by Ishartek) - One can combine 4 stubs of cigarette to roll a new cigarette. If there are 16 stubs, how many cigarettes can be rolled
Answer: 5 - 4 initially, and then one from the resulting 4 stubs
Ratios: Instructor notes: Fractions - divide 2 cakes amongst 5 people notion of 2/5 versus 2/10
(FirstMensa - 129) A coin is fixed, and another coin is rolled around it. How many times will the moving coin revolve
Answer: 2
(FirstMensa - 24) Homework: There are nine coins forming a circle in chain, and another 10th coin is rotating around them touching the coins. How many rotations will the external coin make in one full round (Answer: 5). What if its not a circle but still the external coin is able to touch all coins (Answer: 5). What if you are rotating inside (Answer: 1)
Instructor Notes: Try practically with coins/ gears
(MC Chap 0, Prob 15) Homework: There is a jar full of red beads and jar full of yellow beads. A cup of beads is moved from 1st jar to second, and then a cup is moved from 2nd back to first. Which is greater now: the percentage of red beads in second jar, or the percentage of yellow beads in first jar?
Answer: Same
Instructor Notes: Show with balls of different colors in jars
Sense of Numbers : Instructor Notes: Go both with scale to items, and items to scale. Try a game to do one full contest round in typical buckets of scale.
Distance/ Length
Time
Speed
< mm - bacteria, human hair, proton, neutron, cells
< millisecond - Speed of computer
mm/year - rate of sea level rise (3.3 mm/year), moon receding from earth (38mm/year), human hair (120mm/year)
few mm - mosquito
few millisecond - Cycle time of neuron impulse, blink of eye, human response to visual stimuli
few seconds - time for second hand to move :-) heart beat, breath
km/year - Speed of Jakobshavn Isbræ, one of the fastest glaciers, in 2003 (20km/year), Typical speed of Thiovulum majus, the fastest-swimming bacterium (30km/year), World record speed of the fastest snail (86km/year)
few m - human being, football field, Eiffel tower
few minutes - episode of TV serial, length of a song
km/hour - speed of walking man (4-6 km/hr), swimming world record for 50m (8.5km/hr), world record marathon (20.4 km/hr), bicycle (20-25km/hr), Usain bolt's 100m record (37.5 km/hr), speed of car (40-100km/hr)
few km - mount everest, distance to your home, depth of oceans
few hours - (3600 sec) - Exams, soccer game
100km/hr - The top speed of the world's fastest roller coaster, Formula Rossa. (240km/hr), formula one car record (370km/hr) speed of airplane (600-1000km/hr)
few thousands km - earth, moon,
few days - time for nails to grow, vacations, earth's rotation