Tag: introduction to squaring and cubing

Numerical Decanomial with Paper

Age: 7.75

Date: June 2, 2015

Presentation: Numerical Decanomial with Paper Rectangles and Squares

After the Decanomial layout we did last week (or was it the week before?  The days blur…), Thumper is doing the numerical layout this week.

You can watch a video of how it’s done on youtube:

You’re basically doing the decanomial layout using paper.  Now, when I was making my Cultivating Dharma album, I got confused by the writeup because it was not very clear.  I had to cross reference with the video and other write-ups to come up with my current version.  In the video, you will see that the papers are all the same size.  But after doing some research I thought using graph paper and having sizes that are equivalent to their actual multiplication size (1×1, …10×10) is better.

You have 10 envelopes, on the outside labeled Decanomial 1, 2, 3, etc and one labeled Squares 1-10.  On the inside, you have two sets of numbers for each decanomial:

  • Decanomial 1: 2, 3, 4, 5, 6, 7, 8, 9, 10
  • Decanomial 2: 6, 8, 10, 12, 14, 16, 18, 20
  • Decanomial 3: 12, 15, 18, 21, 24, 27, 30
  • etc.

Basically it has the numbers for that decanomial, assuming you haven’t used it already in the previous one.  For example, decanomial 1 is 1×1, 1×2, 1×3, etc.  “Decanomial 2” is 2×1, 2×2, 2×3, 2×4, etc.  But since 2×1=2 was already in decanomial 1, and 2×2=4 is in the “Squares 1-10” envelope, you don’t need to include these.

In the write up, you have the child lay out diagonally the squares 1-10 first, then you build Decanomial 1, 2, 3, etc.  You can talk about the multiplication table, to pick up the tickets in random and place them etc.

What We Did

Given how old Thumper is, and my own laziness, I really did not follow the presentation.  I basically showed her my write up and said, we’re making this table, which is a paper representation of the bead layout you did last week.  I showed her how you would mark off squares and rectangles on the graph paper and cut them out; reminded he she needed to write the value of the rectangle on the paper; that she does NOT have to mark and cut in order.  She could very well do 1×1, 1×2, 2×1, 1×3, 3×1, etc.  I know she knows half of her multiplication table so there is no need for order for us as part of learning process.  I also told her she would glue these rectangles on our Ikea roll.

IMG_6195
Work in progress

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十項式排列 Decanomial Layout

Age: 4.75 & 7.75

Presentations:

  • Tables Layout
  • Decanomial Layout: Finding Squares
  • Adjusted Decanomial: Commutative Law
  • Adjusted Decanomial: Tower of Jewels
  • Stacking the cubes

Even though I’ve shown Thumper presentations from squaring & cubing, and squares and cubes, we actually never finished some of the introductory exercises in squaring and cubing.  After reading a blog post about it from What Did We Do All Day, I finally found time to do it with the children on Sunday. What Did We Do All Day has a really great post about the exercises and also the variations within the different albums.  I don’t have the NAMC or Montessori R&D albums, though I do have another really detailed AMS album I used as reference when creating my math album this semester.  Ultimately, since my goal was just to show the children these exercises without accompanying written work, I went with  my album.  Partly because I feel I don’t have a strong grasp of how to implement followup work, but also partly because I just wanted Thumper to have the physical experience without all the written work, which often discourages her from working. IMG_6144

These presentations for me are just exercises that are fun and arouses the children’s interest in the squaring and cubing material, all the while giving them a sensorial experience of what it means to square and to cube.  I took all the related exercises together and we just flowed with it.  The whole thing took about 2-4 hours, with lots of breaks in between.  In the classroom, you would do this with at least two children.  I think 2 is a good number.  Anymore and the layouts get messed up way too easily.

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Squares and Cubes….and lots of beads in general

Age: 4.5 and 7.5

Presentations: Making Geometric Shapes, Association of squaring chains with bead stairs (primary), Concept and notation of squares, Notation of squares layout, Finding the totals of squares, Power scales, Chain of 1000 (just work)

Because I’m in the squaring and cubing section part of the album right now in class, I’ve been giving the kids some presentations in this area recently. Last week, I presented the introduction section of squaring and cubing, which are extensions of the short chain work: Making Geometric Shapes.  This is where we bring out the short bead chains, fold them into different polygons, and introduce the nomenclature.

IMG_5387 You can see typical Thumper reaction here at the face of new materials; she wants to play with it and make a star with the 9 chain instead of a nonagon. IMG_5530 The other thing we did that day was Circumscribing One Figure Around Another.  For this presentation, you show them how to circumscribe one shape inside another and introduce those two vocabularies (circumscribed and inscribed).   Astroboy worked on this again today (Thursday).  IMG_5527 Because he did this after my attempt at giving him the Bead Cabinet presentations that were at the end of my primary album (whereas this activity is in my ELE album), he put all those bead bars around the circumscribed shapes.  And of course he was no where super interested in my presentation.  Though he was game and allowed me to finish my presentation and even helped me count.  (He does love anything that has to do with pretending to be a train and lining things up.) My sister asked me what the point of this work is, circumscribing figures.  I think it’s just a sensorial activity.  Letting kids play with their geometric shapes.  The kids also figured out by playing with it that as the shapes got larger by one unit on each side, the difference in area between shapes got larger.

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