Friday, November 22, 2013

"LEAST number to select to be CERTAIN of getting at LEAST..." -- Those annoying logic problems!

OVERVIEW

You've seen these on SATs, GREs, etc..
Where exactly does this fit into the Common Core?

Should the underlying Pigeonhole Principle and/or specific strategies/methods be taught to middle or high schoolers?

THE PROBLEM
There are thirty solid-colored scarves in 30 identical unmarked closed boxes, one per box. The 30 scarves consist of:
6 red
7 blue
8 yellow
9 green

What is the LEAST number of boxes that need to be opened to be CERTAIN of getting at LEAST

(A) 2 of the same color [5]
(B) 4 of the same color [9]
(C) one of each color [25]
(D) 4 of each color [28]
(E) 2 green and 2 blue [25]

REFLECTIONS

• Oh those nasty but critical keywords which I wrote in uppercase. Initial instruction must focus on developing meaning for these.

• Those who have had experience teaching this may want to share their strategies. Here are some I've used...

* Act it out with coins in a bag or any concrete objects. Let's say the goal is to get at least one of each color and the student or you selects 4 boxes or objects. I might say, "Do you want to stop? This is surely the LEAST number!" Were hoping students will respond with, "But we can't be sure!"
OR I might grab all of the boxes and say, "Ok, now we are CERTAIN!" Hopefully someone will respond...

* Without getting into the Pigeon Hole Principle I have used what I call the WORST POSSIBLE LUCK (CASE) approach. For example, if the goal is to get one of each color the WORST CASE would be getting the same color repeatedly. Don't like that approach? This is subtle and requires patience, time and several examples. Some learners will grasp it way before others but acting it out with objects and making a game out of it really has made a difference for my students.