Adventures in subitizing

The cool thing about doing Quick Images with 5th graders is that you can explain the purpose of it to them.  They understand when you explain the concept of children's mathematical development moving from counting single dots at a time, toward more sophisticated ideas like clumping dots together, or recognizing 5 dots immediately based on their knowledge of that pattern from using dice.  They also enjoy saying the word "subitize" as much as I do.

The down side of QI's and 10 year-olds is that the patterns I had to use were not at all difficult for them.  They enjoyed explaining their thinking, hearing other ideas of how to "see" the solution, and getting to play a fun game, but I found that on day one they weren't really stretching their thinking.

So on day two I modified one of the cards at the end of a string of 3.  We were working on patterns of 4 (four dot clusters).  At this point the kids have all put on their cool jackets, popped the collars, and feel like they can breeze through the quick images - having fun, but not working too hard.  I chuckled inside when I heard the collective "hey!", "what?", "was that..?", and other shrieks of doubt when I snuck this doozy in on them:

Really nothing earth-shattering, and most of them got the answer just as quick as with the others once they refocused, but I could tell they had to do some more mental gymnastics to find this one out.  I only had time to get the solution to this one from a couple of students, but it was fascinating nonetheless.  Rather than the typical counting up, adding on, or simple multiplication, this problem required multiple processes.  Here is how they did it: 

"I recognized the shapes as groups of 4, and saw there were blocks of 8, so I started to just multiply those blocks of 8 times the 4 times across.  But then I noticed some were missing, so I looked at it as 3 groups of 4 on each side (as circled above) and got 3 x 4 = 12, and there were 2 of those, so I had 24.  Then I added the 2 little groups of 3 each, so 6 more.  So I got 30."

"I noticed there were two dots missing right away, so I knew that.  I saw it as groups of 4, and it was 4 groups across and 2 groups high, so there were 8 groups of 4.  That means I had 32, and I know it was 2 less than that from the 2 that were missing, so it was really 30."

When this latter solution was re-voiced, we settled on an equation that looked like this: 2 x 4= 8 (groups), x 4 (dots/group) = 32, - 2 (the dots know to be missing) = 30.

 These kids have some pretty sophisticated strategies for solving these puzzles.  They have asked me when we are going to do some more.  I do have some plans to try some even more complicated dot puzzles like these.  I have and idea to do some with half-dots since we are working on adding fractions, and when we start our intro to geometry we are going to do some where they are given 10 little interlocking cubes on their desk and shown a quick image of a 3-D shape that they must replicate with their pieces.

These kids already have the ability to remember and manipulate the images they see in their minds - that is why some of these patterns are easier for them.  I am extremely eager to see how they might handle the more complex spatial manipulation that comes with the geometric shapes, and how they will group different parts of the shapes together - subitizing with geometry, and actually using the manipulatives in front of them to bring the shapes to life.