Wednesday, January 22, 2014

Solid Geometry: Euler's Formula (Polyhedral Formula)


Gratuitous picture of Kal-el.  He was so proud of this work and wanted to make sure he was pictured.  One can't have one's little brother inadvertently getting credit for one's favorite work!  This work is invented by me.  Don't go digging through your albums looking for it and panicking that you may have missed a whole section because it is not in there.  Kal-El has been very interested in the faces, vertices, and edges of three-dimensional shapes due to a math puzzle book he works on for fun outside of school.  I noticed yesterday that he could benefit from a way to check his work when he counts up faces, vertices and edges.  It wouldn't be very "Montessori" to just TELL him the equation, now would it?  


Here is a close up of the labels I made for the top of each column (Name, Etymology, Polyhedra, Non-Polyhedra, Faces, Vertices, Edges, BLANK).  When you set up the work you will want them in this order.

Materials needed:

basket of primary or primary/elementary solids and stands, cards for column labels, name labels, etymology labels, box of operation signs (or cards), box of colored bead bars from the snake game, box of grey/white bead bars from the snake game.

Procedure:

1.  Choose a solid from your basket of solids.
2.  Decide if it is a polyhedron and place it in the appropriate column on its stand if applicable.
3.  Find the card with its name and place it in the "name" column.
4.  Find the card with its etymology, read it aloud, and place it in the "etymology" column.  (Our name and etymology cards are from ETC Montessori.)
5.  Count its faces, if any, and put the corresponding COLORED bead bar in the "faces" column.
6.  Count the vertices, if any, and put the corresponding COLORED bead bar in the "vertices" column.
7.  Count the edges, if any, and put the corresponding GREY/WHITE bead bar or bars in the "edges" column.
8.  If you look across the row bead bars you will see that you have created a small version of the "subtraction snake game."  Find the final value of the snake and record it in the column that has the blank card at the top.  You can add the operation symbols at the top of the work between the labels at this time if you wish.  You could also wait and do it near the end.  They aren't necessary to find the value of the snake because the bead bars themselves indicate the operation.


9.  Kal-El quickly noticed he was recording a difference of "two" very frequently.  When he finished labeling all of the solids he again commented on how many "twos" he had recorded.  I asked him if he noticed anything special about the differences that were NOT two.  He said, "They are NOT polyhedra."  At this time I asked him to completely remove all of the bead bars that belonged to non-polyhedra (for clarity).   



10.  I asked Kal-El to state what he learned about polyhedra and he said, "The edges subtracted from the faces and the vertices always equals 'two'."  At this time I flipped over the "blank" card.  It says, "Euler's Formula  F+V-E=2." The relationship between the written formula and his statement was NOT obvious to him.  In fact he tried reading it:  "Euler's formula 5+5-3=2?"  I re-read it correctly and asked him, "What do you think F stands for?" etc.,  This would be a good alternative time to add the operations symbols at the top.


11.  To wrap things up, Kal-El recorded his statement about his work and Euler's formula in his math binder.

If you are looking for a fun, easy place to read more about these aspects of solid geography try this link over at Math is Fun.  After we were done I looked at it with Kal-El.  He enjoyed seeing a NEW SOLID listed among the other familiar simple solids (Torus).  He was also very interested in the distinguishing characteristic of prisms (I have lesson planned on that for Friday).  Also of interest where the prisms and platonic solids that are not in our basket.  I don't own the platonic solids but we do have the prisms.  I'll add those to the basket when I do the lesson on prisms.  In the meantime, I'm in the market for some platonic solids if anyone has any suggestions.

Montessori Monday

10 comments:

  1. I think Kal El is smarter than me...... I'm sure that you are :)

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  2. What a great lesson! Thanks for sharing!
    I love that first picture, he looks so proud of himself.

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  3. VERY cool!!! That is the quintessential Montessori extension ;)

    I have had my eye on these ones, but have not yet bought them:
    http://woodenpolyhedra.web.fc2.com/woodenpolyhedra5.html

    For building them, I want this set, but I abhor plastic - we use them at other people's houses - time to get our own set:
    http://www.amazon.com/Polydron-Platonic-Solids-Set-50/dp/B00551JZSM

    :)

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    1. Thank you. Oh boy, do I buy both? So tempting.

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  4. This is pretty cool. I took a look at Jessica's link and wow, those are cool!
    I think we are just treading water over here at the moment. :) I am feeling tired, and we aren't motoring along. You guys are doing really solid stuff!

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    1. don't feel bad! We have done that many times, too! A tired teacher makes for a rough homeschool all the way around - it's better to take it slow and refresh and let yourself get excited and ready to go at it strong again! :)

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    2. I think that Fundamental Needs work looked pretty solid and spot-on to me!

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  5. I love this! :) It is so inspiring, and the geometric solids keep getting moved up more and more to the top of my 'wish list' of items for the classroom. You know, geometry is something that is so fun if you can see it - not just easier to understand, but truly fun! :) Thanks for sharing the link - I am going to enjoy going back to it for help in re-learning (so that when I teach my children, they don't know that mommy had to learn it all over again! hahaha)

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