Monday, January 14, 2013

Montessori Short and Long Bead Chains - Skip Counting

Along with the Decanonmial, the Long and Short Chains (and beaded squares and cubes) are my favorite Montessori Math materials.

Most home educators, however, look at these chains in terms of traditional skip counting as shown at Math is Fun. There skip counting is defined as "counting by a number that is not 1". The counting proceeds and ends at random numbers. 2's go to 20 or 100, 3's go to 36 or 90, 4's go to 48 or 120. Or you will see the fill-in-the-blank worksheets like this one from Super Teacher Worksheets.

Montessori skip counting with beads is far more than random counting schemes and indiscrimiate practice of math facts. The Montessori Short Chains are the precise squares of a number and the Montessori Long Chains are the perfect cube of a number.


Take the Short Chain of 3. 
Skip Counting you are saying:   "3,   6,   9".

But if you fold up the chain you get:

3 sets of 3    
which is the same as

3 x 3   (three by three or three times three )
which is the same as

3(three squared) 
~  9 beads  ~ 

Try a larger number - Short Chain of 6.
Skip Counting you are saying:   "6,   12,   18,   24,   30,   36".

But if you fold up the chain you get:

6 sets of 6    
which is the same as

6 x 6 (six by six or six times six)   
which is the same as

6(six squared) 
~  36 beads  ~  

HERE is a video that shows the Montessori presentation with a short chain.


So if you continue on to the Long Chains, you now are working with the cube of a number. The Long Chains make sets of squares and the squares stack up on top of one another into a cube.

 So take 3 again.

If you take 3 sets of 3 sets of 3,
which is 3 x 3 x 3   (three times three times three)

in other words 3 sets of squares of 3,
which is 3 x 32   (three times three squared)

You get  3(three cubed)
~ 27 beads ~


And here is 6 again.

If you take 6 sets of 6 sets of 6,
which is 6 x 6 x 6   (six times six times six )

in other words 6 sets of squares of 6,
which is 6 x 62   (six times six squared)

You get  6(six cubed)
~  216 beads  ~

HERE is a video that show the Montessori presentation with the progression from the Short Chain to the Long Chain.

I think A Montessori Classroom says it best with Every Day Counts - the Bead Chains:
At its simplest, this activity is good for practicing the order of the numbers. That after 29 comes 30, and after 39 comes 40, etc. On the next level, it illustrates skip counting; counting by twos or threes or fives. On another level, an older child will see the multiplication tables. And finally, this material illustrates the squares and cubes of these numbers and their relationships with each other.
I have a confession to make! It took me til I was in my late 30's and working with my children with Montessori materials to know that the square of a number really makes a square and the cube of a number really makes a cube! Now dontcha feel smart?



NOTE:  So keep in mind when you are purchasing or making the Montessori Bead Chains - you also need the Montessori Beaded Squares and Cubes to do the complete presentation.

Free Long and Short Chain Arrows printables at Hope4ME
Montessori Print Shop shows How to make a Montessori Bead Chain Chart
Discovering Homeschool - Worksheets for Short Bead Chains and Bead Squares
Plain Counting Chain Arrows from MontessoriMom

Skip Counting  - Moteaco
Short Chains - Montessori Album
Skip Counting:  Linear Counting - Wikisori
Short Chains, Long Chains - Montessori Commons
Sensorial Experiences with the Chains - Shu-Chen Jenny Yen

Linked to:
Montessori Monday, Live and Learn


  1. This is a brilliant way to teach skip counting, etc.!!! Thank you!!!
    I, too, did not understand "real" squares and cubes until working with our children. :)

  2. ooh-ooh-ooh - don't forget preparation for geometry (point, line, plane, spatial object that becomes another point) --- and oh my! The elementary work that builds upon this material is, well, I can't think of a word for it!

    Just the visual sensorial impression of having this material displayed in the environment is profound!


    1. I can just see you jumping up and down ooh-ooh-ooh LOL! That's what I love about you Jessica --- but now you're reminding me that I really need to get back to that Keys of the Universe training program of yours - there is still so much more I need to learn, esp. about Montessori Geometry!

  3. Love this! Thanks for the refresher.

  4. I LOVE the Montessori bead materials, and this is fabulous, Suzanne! Thanks so much for your great post with so many resources! I featured your post at the Living Montessori Now Facebook page:


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