math

# Probability

First, sorry some of my pictures are going to be upside down.  I can’t figure out how to flip them

So…Probability.  Likely…unlikely…certain…impossible…fair…unfair…after a few days of working on creating some probability games I felt good about everyone’s understanding of these terms.

It was time to get out my mystery box!

(The following is 100% true!)

On Monday morning I said to my class, “Last summer, there was a lot of cleaning going on around here.  In that big room, between rooms 6 and 7, all the teachers were putting stuff they didn’t want anymore in case someone else might want it before it was thrown out.  I went in there one day and found this:

(I held up the box, a la Vanna White.  I turned it this way and that.) I saw that it said ‘probability’ on it, so I picked it up.  Wanna see what’s inside?”  They did.  I gave it a shake, and showed them the contents.

Someone called out that there was one jar missing.  Most of the class started counting the jars – some by 1s, some by 2s.  I covered up the row that had one missing, hoping people would count by 5s, and 2 children did.  Eventually we agreed that there were 23 jars.  I showed them some samples of the jars, but we didn’t look too closely.

“Do you know what wasn’t in the box?” I asked. “Instructions.  I am sure there were instructions in here at one time,  but they are lost.  I have no idea what we are supposed to do with this box.  Do you think we can figure it out?”  They did.

I sent them back to their tables, and gave a few random jars to each group.  I put an iPad in the middle of each table and turned on the Show Me app to help them capture their thinking as I made my way around to each group. I captured some interesting thinking!  And some inappropriate conversations that had nothing to do with math.  Most of them thought the jars were just for counting.  They thought there had to be some reason some jars had triangles in the bottom and some had circles.   (I think you can see that in these pictures.)

One group started to see some fractions.  One jar has 2 yellow and 2 blue, and they said that half of the beads were yellow or blue. They became my jumping off point for the next half of the lesson. I gathered everyone, we talked about what we say, and then I had the fraction “noticer” explain his thinking last.

We talked a bit about the connection between saying “2 out of 5 are blue” and 2/5 are blue. “I want you to go back and see if you can build on what C is saying about fractions.” They all did.

We talked about this on the board, while I recorded our thinking.  I want to provide photographic evidence, but I label their answers with their names (they love it!  And it allows me to simply take a photo and then I can use that as some specific evidence of their understanding in my notes, and on their reports.)  All of that was still on the board when lunch began.

I have this really grade “Grade 6 helper” at lunch who likes to play games on the board with the class while they eat.  I was supervising between 2 rooms and when I returned to my room, I saw that she had written this on the board:

The next day I started with that picture.  “O saw our ideas from math yesterday, and she gave you this puzzle to think about.  And I notice that she wrote ‘probability’ which is one of the words on our box.  What’s the connection?” We talked about how the word experiment fit into all of it too.

We talked a bit; they went back to desks.  I handed out the jars so that each group had one example of each kind of jar.  …fast forward….this was the end result of our work that day:

One of the interesting things that came out was the need to have the jars labeled so that the recorded answers would make sense.  Honestly, this is a huge step for the class.  They are young and don’t always get that people can’t read their minds.  I am serious about that!  When I read their work and it is disorganized, they are always so annoyed that I am asking them to clarify.  BUT this exercise showed me that they are understanding the need for clarity.

Two groups labeled the jars “A, B, C, D”, one labeled them “1, 2, 3, 4” and one used “997, 998, 999 and 1000”.

Overall, I thought this was a really good bit of good luck.  At one point, I heard one boys ay to his group, “I don’t get it.  Why sell a box like that with no instructions?  It doesn’t make sense.  You have to give people the instructions!”

I want to talk a bit about the process of this lesson.  It took us about 4 days to get everything accomplished that I wanted to accomplish.  Now, I could have covered this material in 1 day, and then conducted some experiments with coin tossing, or random draws of tiles, or “rock, paper, scissors”.  These are all things I have done in the past, and things I still might do this week. At the end of the first day, I wondered if it was all going to work out.  I assessed the lesson and decided that everyone was just a little bit frustrated with not being able to definitively answer the question “What is this box for?” and yet they were still interested in figuring it out. I felt I didn’t really have much to show for the day.  Sometimes other teachers come in my room.  I was glad nobody had that day. However, by the last day, I felt that there was some real discovery going on and everyone was doing some actual inquiry and learning.

Roughly half of my class this year will be roughly half of my class next year. I won’t be able to use this activity with them next year in the same way.

I didn’t think of it until just now, but have not Googled this item. I found out that DIME stands for “Development of Ideas in Mathematical Education”.  From a Google Books entry, I discovered they come from England, and that they are pretty much impossible to find now.  I should have had some recording sheets besides the instructions.  I also found an online version here.  Apparently, we are to shake the jar, and watch how they settle in the triangle. Or here, you win if two blue balls touch. This will likely be our Monday math.

Update: we did try playing some of the games, and it was fun for the students. They still can’t figure out why the box didn’t come with instructions. 😉