Batch 2 - Class 91 - Confirmation Bias, Combining Numbers (Commutative, Associative operations)

• Build a refined model for the Owls and Voles simulation on Excel
• (Alice - 78) The forgetful White Kinight - The White Knight told Alice about a trial where there were three defendants, and

  All three defendants knew who was guilty, and made statements to that effectThe first defendant accused either the second or the third defendant, but the White Knight couldn't remember which oneThe second defendant claimed to be the guilty one!The third defendant either accused himself, or he accused the second defendant, but the White Knight couldn't remember which oneThe guilty defendant lied. Either one or both told the truth, and the White Knight couldn't remember which one

• Answer: We are given that the guilty one lied. If B were guilty, he would have told the truth when he accused himself; therefore, B cannot be guilty. If A were guilty, then all three of them would have lied (because A would have accused either B or C, both of whom were innocent; B would have accused himself, who was innocent; and C would have accused either C, who was innocent, or A, who is innocent). But we are given that not all of them lied, so A can't be guilty either. So C was guilty.
• Squarable Numbers - which ones are not and can you prove it?

• 2-4-6 Puzzle: Teacher challenges the class to guess a rule that he knows. Students can offer triples of numbers and he'll tell them whether they fit the rule or not. To start, put forth a triple 2,4,6 and tell the kids that it fits the rule. Divide the class into groups, and when any group is sure of what the rule is they can call it out. The first group to call out wins. If a group calls out a wrong rule, they are eliminated. Record the triplets students offer for test. Keep each group's tests private. 

  Analyze the sequences tested by students later. See what their hypothesis are and how they are testing itAsk students what they think is going on in their testsDescribe confirmation bias - do we want to hear a "yes" or a "no"Ask for other examples where we may display confirmation bias (eg: medical diagnosis, listening to a TV channel that confirms your beliefs)How do we formulate and test for alternate hypotheses - what are the different hypotheses we could have come up with in a 2-4-6 sequence?
• Look at the source image - what do you think it is?

  Look at the target image - what is it?Now look at the source image - what do you think it is?What else could the source image be? Knowing the target image can make it harder to think about alternate views of the source image.

  

     

  

    

  
• Magician's Trick: Here we will guess which hand is the penny is

  Teacher's Notes: Ask a student to hide a penny in one of her hands and guess which one it is in. Repeat 10 times. Comment that the kid is doing a great job of hiding it, and not letting it show.Now ask the student if they can guess. Have a penny in both hands so they always get it rightAfter 3 tries, which the student will get all right, play into their confirmation bias by reinforcing - "Do you play rock, scissor, paper" - to their "yes", reinforce that the games are similar, and hence the must be really good at itLets try couple of more times. "So are you trying to imagine what I am doing, or are you looking at my facial expressions, or my hands?" - reinforce their bias to be rightFinally let it that you had a dime in both hands all alongAsk the students what they think was going on?Why didn't they ask the other hand to be showed? Would they have asked for it if the first hand was empty?

• All numbers 1..100, both inclusive, are written on a blackboard. At every stage, you are allowed to combine two numbers, and instead of that number write x+y+xy (Teacher's note: Dont introduce algebra as yet, tell kids they replace those two numbers with sum of the two numbers and their product)

  What are the possible values that remain at the end?What are the ways to get a small total or a large total?How do we navigate this problem? It seems too big... Can we try with a smaller problem? What is the smallest problem?What if you combined two numbers but in different order 3 and 5, or 5 and 3. Why? Note that this is not always true - for example in division.For small sequences, kids may come up with the conclusion that order doesn't matter. Try with other sequences like 3, 5, 8Is there any way to show that the answer is not dependent on order? Or can you find a counter-example?Try with a sequence of 4 numbers (18 ways to order them) Lets now get to algebra, and try to combine three numbers a, b and c algebraically. Lets try for four numbers a,b,c and dAdvanced kids should start to realize that this operation is commutative and associative. Ensure they know what that meansTabulate results for sequences upto 6 numbers. Can kids find the relationship between these final values?Can you find a general formula ((n+1)!-1), or for any set of numbers (1+a)(1+b)... -1So now if you were to do it for 100 numbers, what would the answer be? (101!-1)

• (Geoffrey - 64) As soon as the news of bank robbery came in, a police car was sent to High Street bridge. The robbers had made their escape in a car, and the bridge was the only point of escape north of town. The police car took strategic position at point A on the map. Three patrols were assigned to patrol A-B, A-D and A-C, each going back and forth the blocks 200mx100m in size, at uniform speed of 10km/hr. All of them started from point A at 4am. On the first occasion thereafter when the three met at the car together, a radio message advised them that the robbers had escaped from south side of town. What time did the message arrive?

• Answer: Each patrol must have run 72 lengths of blocks, i.e. 14400 metres when they first get together. At 10km/hr, this means about 1:26 hrs, so time must be 5:26 am