Boolean Bridges

The Activities

  1. Topic: Division: Book: One Hungry Cat by Joanne Rocklin.
  2. Topic: Logic: I made several pictures of paths and rivers, where you needed bridges in one or more spots in order to be able to make it to the castle.  Each picture corresponded to a Boolean logic formula.  The kids both had to write down a formula given a picture, and draw a picture given a formula.  All the pictures can be found here, as well as the PowerPoint if you want to make your own.

    A v B

    A ^ (B v C)

    A v B v C v D

    (A v B) ^ (C v D)

  3. Topics: Addition, Subtraction, Multiplication:  I did several “math magic” problems where each kid starts with a different number between 1 and 9, I give them instructions, and then at the end they get the same number.  The three “tricks” were “add 6, subtract 8, add 2”, “add 2, add 2, add 2, subtract 6”, and “multiply by 2, add 3, multiply by 5, subtract 15, cross off the trailing 0”.
  4. Topic: Geometry:  Build some number of squares using as few fences (Keva blocks) as possible.  Later, build some number of triangles.

How Did It Go?

One Hungry Cat

Kid 1 said she had heard the book before, but she didn’t remember what happened.  The kids thought it was pretty funny when the cat kept eating all the food.  Kid 2 pointed out that when there were 8 cookies left, the cat could have eaten only 2, but ate all 8 instead.

 Castle Logic

 I started with one bridge, and then showed them the pictures for “A and B” and “A or B”.  I gave them ^ and v as the “and” and “or” symbols; Kid 1 mentioned that & could stand for “and”.  Then I showed them pictures for A v B v C and A ^ B ^ C, and had them write down the formula.  They didn’t have trouble with this, except that Kid 2 left out the v on the first one, and Kids 2 and 3 both wrote v instead of ^ for the second one.  Then we did A ^ (B v C).  We ended up getting two correct answers: Kids 4 and 5 wrote A ^ B v A ^ C, while the rest wrote A ^ B v C.  This is nice, because it’s the distributive low for Boolean formulas.  Of course, no one used parentheses, but before discussing that I showed them the next one, which was (A ^ B) v C.  Again, a bunch of good answers, including both the distributed and undistributed versions.  As it happened, no one wrote down the same thing both times (they randomly varied the order, probably accidentally although it’s possible Kid 6 did it on purpose to signify which came first).  So, I wrote down A ^ B v C for both of them, and explained how the parentheses tell you the precedence.  They may or may not have understood.  Kid 6 finished these quickly and started drawing a picture of a bridge with a funny troll, and then showed it to other kids.

After this, I gave them A v B v C v D and asked them to draw a picture. Four of the kids drew the right thing; Kid 1’s was the nicest, while two of them didn’t initially connect the roads at the bottom (each road went off the paper at a different place).  Kid 6 finished early and started adding to the troll again, so I gave Kid 6 another problem, (A v B) ^ (C v D).  Kid 6 didn’t want to (and didn’t) work on it, but Kid 4 drew a correct picture for it!  I thought Kid 4 had been working on A v B v C v D, and I’m still not absolutely sure she didn’t just do that one incorrectly, but it’s quite a coincidence that she got a correct diagram for the 2nd one; and when I asked, she said that they had been working on the 2nd one.  Kid 2 had the most trouble throughout the activity, and for the final one drew A ^ B ^ C ^ D instead of A v B v C v D.

 Math Magic

The first one was “add 6, subtract 8, add 2”.  Subtracting 8 is still not that easy for many of them.  They were suitably amazed when they got both the original number, and didn’t know how I did it.  I explained how it all added up to 0.  I also showed the chain formula, and explained how I could group together the terms so I got 0.  Not sure exactly how much they understood; Kid 1 was confused because she had started with 2, had gotten down to 0 after subtracting 8, and was getting confused between having 0 as a partial result and the fact that the operations they performed summed to 0.  Next, I did “add 2, add 2, add 2, subtract 6”.  This time, Kid 2 explained how 2 + 2 + 2 = 6, so it canceled.  Finally, I did a hard one: “multiply by 2, add 3, multiply by 5, subtract 15, cross off the trailing 0”.  Multiply by 5 was, not too surprisingly, pretty hard for them.  Initially, they tried to add their number 5 times together, but since most of the numbers were above 10, they had problems.  When I suggested counting by 5’s, that worked better.  Kid 3 is really fast at counting by 5’s, although when I helped her do 15 * 5, I’m not sure she understood what she was doing.  What was most surprising was how difficult it was for them to subtract 15 from 75/85/95.  I’m not sure a single kid got it right on the first try — usually they would say that 75 – 15 = 65.  Similar to some times in the past, Kid 2 was more stubborn about wanting to do the multiplication her way, so it took her longer than anyone else — eventually she took my suggestion to count by 5’s and then did the counting correctly on her own.  I asked the kids if I should explain how it worked but they said it should stay magic for now.  Kid 4 wrote down the numbers from 5 – 225 counting by 5’s while I was helping Kid 2.

Fences 

Interestingly, the kids immediately, without thinking about it, started using shared fences when building multiple squares (so to build 2 squares, you only need 7 fences).  When we got to 3 squares, some kids had done 3 in a row, but they quickly changed to a 2 x 2 grid once we got to 4.  The first time that we got a less efficient answer was 9 squares, where Kids 1 and 2 had an irregular shape that used 26 blocks instead of 24 (it had a part that was only one square thick with thicker parts on either side).  I showed them how you could move squares in order to save some fences.  Next, I asked them to build 1, 2, … triangles.  Kid 3 quickly went to a “gridded” hexagon for 6 triangles.  We were almost out of time and they were having fun, so I said they should cover the whole table with triangles, and they did.  They didn’t work from a single place, so the interfaces between the sections weren’t great, but they did a pretty good job anyway.  Kid 3 mentioned at some point they should all work from one place, but it didn’t happen.  At the end, the kids were happy to get a little wild and knock over all the blocks and not-that-carefully slide them into the box.

Table full of triangles!

Advertisements

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s