Square of Pythagoras - First Presentation

Thanks to all who inquired.  No, I'm not dead!  It's been almost a year and a half since my last post, and while my life has been moving quickly onward I've been facing it mostly with glazed eyes and an air of confusion.  Such is life with twin boys.

My sons are now 2 and are healthy, mischievous little fellows.  I've begun to do some early Montessori work with them, which you will hear about in an upcoming post.

Ella is now "officially" grade 1, in that she is 6 and that is the typical grade 1 age.  And there ends all that is "typical" about her.

Evidently she's doing some things on a grade 4 level, and she was teaching our 17 year old babysitter about geography.  Her reading is about average, or perhaps a bit above average for her age, and her handwriting in abominable!  Though much of what she writes looks like the hen-scratchings of a preschooler, what she writes is very well done.  I guess we just didn't do enough of the sandpaper letters or the metal insets.

I have a vast backlog of photos and lessons to share and I hope to do so over time.  For today, here was our afternoon lesson: the first presentation of the square of Pythagoras (also called the deconomial square).

The square of Pythagoras is basically the multiplication times table in visual form.  The smallest square is 1x1 cm and is red in colour, which coordinates with the colour of the bead bars.

The next three shapes are green, like the "2" beads and are 2x1 cm, 2x1 cm, and 2x2 cm square.

The next five shapes are pink, representing "3" and are (2) 3x1 cm, (2) 3x2 cm, and a 3x3 cm square.

In the above photograph, Ella is laying out the 4s.

I find that she really enjoys making the square of Pythagoras as she thinks that the pattern is beautiful.  I think so too.

The completed square (as done by Ella).  As you can see it goes from 1x1 cm to 10x10 cm.  It is meant to coordinate with the pink cubes, the brown/broad stair/rectangular prisms, and ultimately can be used as a guide to laying out the bead bars in the deconomial.  Hopefully I will be showing these extensions in the near future.

As for the Square of Pythagoras material itself.  It is really hard to find to purchase. Ideally it is supposed to be made out of wood, but I found it much easier to make it out of foam sheets using a large paper-cutter.  (An office-sized paper cutter can be invaluable when making materials.)  I've also seen beautiful square of Pythagoras lay-outs made of felt or other fabrics, plastic duo-tang covers (though I don't know how easy or difficult it would be to find the colours you want), and painted cardboard or bristol board (though they can tend to warp).

Initially I planned on making it using inches as a base, rather than centimetres.  Bad idea!  It was HUGE and very unweieldy.  I'm very glad I downsized, though what I'm going to do with all that extra foam is beyond me!

The first presentation of the square of Pythagoras is not meant to teach the multiplication table, but like the binomial and trinomial cubes it is a sensorial introduction to a more complicated mathematical concept.  Right now we call all the green rectangles and square the "twos,"  all the pink the "threes" and all the yellow the "fours."  It will be some time before we begin to name individual rectangles 2x1 or 4x3.

To present it I simply named the material, "This is the square of Pythagoras." and I began making the square, beginning with the red, then the green, then pink, and so on.  It didn't take long before Ella got the idea and just took over.

She was rather frustrated by the fact that her square of Pythagoras wasn't as "perfect" as mine, but with a bit of reassurance that it took practice she became satisfied with her results.  I'm looking forward to her reaction to the extentions.

My Completed Pink Tower

Okay, after sanding and sanding and sanding and sanding, going to Dad's to use his belt sander, then went home and sanded some more. In the midst of all that sanding I was repeated told by Dad, "Don't be so picky!" Can you guess which of the two of us is the perfectionist?

(If you happen to be looking for instructions on how to make the pink tower or broad/brown stair, go to Make Your Own Sensorial Materials, Part One and Part Two.

I gave all the blocks a coat of latex primer, then did multiple layers of "cherry blossom pink" acrylic paint. Finally, I gave them all three coats of acrylic varnish. Here's the results:



You can see more detail when I do the pink tower shell-shape:


From a distance, they don't look too bad. But if you take a closer look, you can tell that they weren't professionally made:

On the 10cm x 10cm x 10cm block, you can see the "dip" in the wood. If I had sanded much more, it would not longer have been 10cm x 10cm x 10cm, so I cut my losses and voila!

Here's another view of the 10cm x 10cm x 10cm block. You can tell here that one of the corners isn't really a corner. That was another fault in the log. Also (but not shown) is a large knot in the wood that, despite substantial wood fill and sanding, still has enough of an indentation to make it noticeable.

Just for comparison, I put the teeny-tiny 1cm x 1cm x 1cm block atop the 10cm x 10cm x 10cm block. Theoretically, it would take 1000 of the former to make the latter, and when you look at it this way, you can really see that.

Ella can do the pink stair with little-to-no difficulty, although sometimes it's hard to tell, as she often has a "friend" (be it little monster, Mayas the rabbit, Lambie, or some imaginary person) who just can't figure it out on his/her own, so mistakes are made which Ella has to correct! It's not always clear to me if she's making the mistakes deliberately in order to get to "correct" her friend, or if it happened by accident and she's just using her imaginary friend/stuffed animal as a scapegoat for her mistakes. In some cases, I can tell one way or the other, but sometimes... Well, let's just say I know her better than anyone, but there's plenty of times when I have no clue what's going through that little mind of hers!