[#TMWYK] Logic & Equivalence

Some logic and equivalence #tmwyk* this morning:

8yo (trying to get me to do something for her): Please plus please!
Me: What does that mean?
8yo: It means yes.
Me: So, what does a thank you + thank you mean?
8yo: That means yes, too.
Me: How about a no thank you plus a thank you?
8yo: No.
Me: And a no thank you plus a no thank you?
8yo: Maybe.
Me: How does that work?
8yo: Because there are two no's and two thank you's.
Me: You mean there's an equal number of both?
8yo: Yeah.

A little later, about our cats Max and Lucy:

8yo: Max + Lucy = Trouble
Me: That's for sure.
8yo: Actually, cat + cat = Trouble.
Me: So, Max + Lucy = Cat + Cat = Trouble!
8yo: NO!
Me: Why not? If cat + cat is trouble, and so are Max + Lucy, then don't Max + Lucy = cat + cat? They both equal trouble...

She had reached her limit with my questions by then, but I found the whole series of ideas rather fascinating.  First because this was the first time I've had this particular kind of conversation with her. It seems Christopher Danielson's Talking Math with Your Kids is really having a positive influence on my conversational abilities! Second because the whole idea of equivalence in equation form seems to flummox kids. The equal sign often seems to mean "an answer" and not a relationship. It was fun to test/push her a little on this idea and see how it would play out in conversation.

*#tmywk is the hashtag for Talking Math with Your Kids on Twitter. You can also go to Talking Math with Your Kids.  You'll be glad you did!!

What Do You Make of This? (Build What I Have, Redux)

In my recent post Conversational Math: Part Two I described a game that my daughter and I like to play with Cuisenaire rods called "Build What I Have".  It's really fun.  So fun, in fact, that my daughter asks for it every other day or so.
Here's a funny thing that happened today:

It was my turn to give directions.  To start, take a look at how my design looked when I was finished giving directions.  I should point out that there is only one layer of blocks laid out on the board we were using.  I should also point out that we hid our work from each other.  She is trying to replicate my design using only my spoken words as a guide.  Ironically, no gestures are allowed in this game!

Mine.

Here's how I started: "Take three light green rods and put them together so they are all touching and all perpendicular to the bottom of the board.  It should look like a square that is three units across.  Now add a fourth light green rod on top of the square.  It should be placed perpendicular to the other green rods and parallel to the bottom of the board.  It should look like a rectangle now."

So far so good!

Me: "Take an orange rod and place it on top of the green rectangle."
Kid: "Like a teter totter."
Me: "Yeah, I guess so! Now, take two white blocks and put them on top of each end of the orange rod" (etc.)

Then it was time for the big reveal...tah dah!

Hers.

Geez Louise, what happened?!?  Her orange rod IS on top of the light green rectangle, but it appears the kid built up while I built...up?  I suspect we were both right, but I am at a loss to explain why or how.  I think it is accurate to call the kid's work three dimensional, but so is mine because it is built with polyhedra and not 2D lines on a flat surface.

Hmmm....any ideas?