Expected value and Dice

Saw this post on twitter tonight:

The exercise for the students is a neat one involving biased and unbiased dice.  If you read the Twitter thread a little further you’ll see suggestions about running Chi squared stats, too.  All great stats examples.

In 2003 and 2004 I was involved in an television game show called “Play for a Billion.”  On the show 1,000 contestants tried to guess a 6 digit number from a number that I had selected.   If anyone did successfully guess the number I had selected they would have won one billion dollars.

I did an interview for the show, but it didn’t make it on air.  A first cut of that interview is below.  I thought it might be to use for a stats example because I picked my six digit number using dice.  To determine if it would be ok to use the dice I had to run through a bunch of stats that were pretty similar to what the exercise above is asking the students to do.

This project was one of the most fun math-related projects that I’ve worked on in my career.

What learning math sometimes looks like part 3: Multiplying in binary

We’ve spent the last week or so in our number theory book talking about arithmetic in bases other than 10.  One of my favorite activities to help kids learn about other bases is using Duplo blocks to model arithmetic in binary.  This activity has seemed to help both of the boys get a little bit better understanding of place value.

The section we were covering in the book today was multiplication in other bases.  Unfortunately what I thought was going to be one last short example in subtraction took more time than I expected.  That problem made our discussion of multiplication shorter than it needed to be.    When I got home tonight I thought it would be good to do revisit multiplication so we took out the Duplo set to work through examples of multiplication in binary.

We got off to an interesting start when my son choose the example 101_2 times 100_2.  He recognized that multiplication by 100_2 just added zeros to the original number.  This was an interesting observation since we’d not talked about that specific idea this morning.  I wanted to try out another example to see what would happen when this trick wasn’t there to help.  Turned out that the trick was getting in the way a little:

I think my son was quite surprised to see that his method at the end of the last movie didn’t work.  One of the things about multiplying in other bases is that you lose your number sense a little bit and it isn’t easy to see when you’ve arrived at the wrong answer.  That’s at least part of what makes these exercises in other bases such a nice way to build up the ideas of place value – that’s really the only thing you can focus on in these problems!

We looked at the problem again and tried to figure out where things had gone wrong the last time.  Going through it a bit more slowly helped see that several numbers last time were accidentally combined into one.   Having found our way through this problem, I gave him one last problem to work through.  He seemed to have a little better sense of the multiplication process by the end of the exercise:

This was a really interesting process to watch.  A little trick that he learned was limiting his ability to understand how to multiply.  It was hard for him to see that this trick wasn’t helping, though, since the wrong answers aren’t so easy to see when you are working in different bases.   When we walked through the problem the second time it was a little easier for him to see what went wrong since he knew that *something* had gone wrong.  It was nice to see him work through the last problem completely on his own after all of this work.

Finally, just for completeness, here are two videos where we do addition and subtraction in binary with duplo blocks: