The Map is Not the Territory

So, the seven-year-old has been having a spate of self-initiated math lately.  First there was her 'map of angles' and then having her dolly write an essay on 'What Infinity Means to Me'.  So, I guess I wasn't too surprised to hear her from the other room giving her dolly another math lesson:

In her most patient, teacherly voice:

"I'm drawing a simple house, Amelia.  Everything in the house is mostly 90 degrees...you don't have to be exactly accurate but it just has to be good...a triangle window and here's the front porch....compare these two houses.  Can you fix this one?  Good job!"

[Her explanation to me when I asked her later what she was doing: "I drew a house without angles and with angles and Amelia had to fix the house without angles up!"]

After Amelia's success, she continued the lesson with this explanation:

"There are even angles in nature -- straight up and down trees, but some are even 80 degrees, slanting.  The old ones are 50 or 40 degrees."

Later, I got a look at her drawings:






































In the larger house I see her thinking through the angles all starting from the bottom left vertex/corner of the house, which is forward movement from her original representations in the Map of Angles post.  Below the big house is the 'house with angles' at the bottom and what I think is the 'house without angles' (all wonky looking) above that (I thought I saw a different drawing with the same ideas but that, apparently, has gotten lost in the shuffle.)

Another quiet moment found her exploring the structure of an isosceles triangle. 

"See, there are eight of these triangles on each edge [above] and fifteen squares on the bottom edge," she told me.  She also called the line she drew from the top vertex to the center of the bottom edge a "diameter" which she knows is how you divide a circle in half.

In addition to all our sidewalk math adventures over the last year, we've learned more about identifying and classifying geometric shapes in the Beast Academy 3A series but it's been a while since we did the polygons chapter.  We got through skip counting which was perfect and, after entering the perimeter chapter decided to take a break.  This drawing really shows me she's thinking very specifically about the length of each edge.





















And, finally, although this may seem more in the 'art' category, I know for sure that drawing three-dimensionally has all kinds of math involved in it, I just don't know what kind, lol!  Six or nine months ago she tried to sketch Platonic solids and really didn't do it very successfully.  I think her eye has come a long way:

Her milk box:

























An Asian ceramic bowl with some paper flowers in it:
























Her electric pencil sharpener:

















I love seeing (and hearing) the world through her eyes.

It all started with a lovely morning walk to the hardware store.  We needed a long tape measure for our 'measure the house' project that's been brewing for a while.

"Look, Mama!  A triangle!  Except...I didn't know triangles could do that.  I knew that two triangles make a square..."





Our eyes were open at that point, and we found other triangles on our journey.  Here are two with a square -- an almost- trapezoid in the wild!  (I say almost because of the curved bottom edge -- yes, there are shapes all around us but some of them are truly geometric and the others are not really geometric, like the 'triangular' yield sign with it's curved corners, and still others are 'natural' shapes which have their own wonderful rules.)

When my daughter was in preschool, one of the moms (who happened to be a science ed specialist) did a project with the whole class. It was brilliant. She taught the kids how to work a catapult (made with a plastic spoon, I think) and, over two years later, my kid still talks about that day.

So, I was not too surprised when the girl said recently: "I want to make a catapult!" A quick Google search turned up instructions for one made out of craft sticks and tape. Now that is my kind of project. Historically I've not been too handy with 3-dimensional construction, but I'm up for anything that requires tape!

We started by making three congruent triangles and followed the directions through the next step.  Suddenly, recognition!

"Hey look at what we've made!" I exclaimed.  "It's a...?"

"Tetrahedron!" the kid laughed.

























Not surprising -- triangles are the strongest shape, and it was fascinating to see how strong they really were, even given the modest construction materials. 

We used a simple origami box we had lying around as the catapult cup, and wondered if a wider arm would fling objects further.


























We love our new catapult and have had many hours of enjoyment flinging unifix cubes around the living room.  Now that is math in action, lol!