# Guessing Games (Age 7)

## The Activities

1. Topics: Geometry, Perimeter: Book: Chickens on the Move by P. Pollack and M. Belviso.
2. Topic: Binary Search: I printed out some charts with letters A-Z and corresponding numbers between 1 and 99, increasing from A to Z, but with a different pattern of numbers on each chart.  The kids took turns being the “puzzle master” and being the guessers — the goal was to find a particular number (which I specified each time) in as few guesses as possible.
3. Topic: Probability: Same as an activity the Age 5 circle did last week.  I had two dice, one with twenty sides (1-20) and another with ten (0-9).  Putting the twenty-sided first, if you roll both of them you get a number between 10 and 209.  First game was, write down a number, then I roll the dice, if your number is higher, you win.  Next was, if you’re lower you win.  Finally, the interesting game is you pick one number, I roll 5 times, and you win if at least one roll is lower than your number and at least one is higher.  We also played variants where you needed, say, at least 2 higher and at least 6 lower out of 10 rolls.
4. Topic: Probability: Each kid had a bag with 10 colored blocks, split between red and green (different number per kid).  They could pull blocks out one at a time, putting them back after each draw, as many times as they wanted.  When they were done, they would guess how many green and how many red were in their bag.
5. Topics: Multiplication, Commutivity: We took the product 3 * 5 * 7 and evaluated it 3 different ways — (3 * 5) * 7, (3 * 7) * 5, and (5 * 7) * 3).  Then I showed them an informal proof of commutivity, using a 3-dimensional figure made of blocks (changing the order of multiplications corresponds to rotating the figure).

## How Did It Go?

We only had two kids this week.  As usual with small circles, both kids were very attentive.  Several of the activities could be viewed as competitions; and indeed the kids did compete.

#### Chickens on the Move

A light book that introduces the concept of perimeter.

#### Binary Search

The kids really liked being the puzzle master.  We used a whiteboard where we started with each letter on the board, and then erased the letter and replaced it with the number after they guessed.  They picked this up really fast.  They didn’t quite do binary search, they guessed where in the interval the next number would be based on its relationship to the endpoints; but it’s not clear that binary search was the best strategy anyway, since the numbers were somewhat uniformly distributed (I did have some fairly significant skew in some of the charts, but since there were only 99 possible values for the 26 boxes, you couldn’t get in too much trouble by not splitting evenly each time).  They got every number in 3-5 guesses.

#### Higher/Lower

They also were pretty good at this activity.  One kid chose a really low number on the first round, probably because they were confused about the rules; but after that, they stuck with 209 or 208 for higher (they thought for a while they had to choose a number that was possible; after I said it didn’t, they chose 1 for the next round).  When I said the rules for the final game, they said right away that they wanted to be in the middle.  However, they chose ~ 50 for the middle at first.  Later they realized 100 was the middle.  Since they got this so quickly, I added advanced rounds where they needed, say, 2 higher and 6 lower.  We used colored blocks to keep track of each result (you got a green if your number was higher, red if lower).  Both of the kids got the general idea, but one of the kids chose better numbers than the other.  I also played a couple times; as it happened, even though my numbers were only slightly better than theirs, it made the difference between winning on losing (not that likely that this would have happened).

#### Bag of Blocks

One of the kids stopped after 15 pulls, 3 red and 12 green.  They guessed 3 red and 7 green, when the answer was actually 1 red and 9 green.  The other kid stopped after 30 pulls (they chose to make it a multiple of ten), got 21 red and 9 green, and guessed 8 red and 2 green (actual answer was 7 red and 3 green — although one of them got dropped on the ground along the way, so there were only 6 red and 3 green).  So there’s still plenty to learn here — they only have a general sense of ratios; and they don’t have a strong sense that more pulls = better results.

#### Commutivity of Multiplication

They sort of already knew that you can do multiplication in any order, although calculating 3 * 5 * 7 was hard enough that I think they were still a bit surprised it came out right.  The proof by 3D figure seemed to work for at least one of them.