Instructions to the Square Root Board

The square root board shows in a simple way the essence of squaring a number and determining its square root.

For both operations, children should already have multiplication experience.

Taking square roots becomes very easy with the board, because one can see that is it the opposite of squaring. In order to square a number, it must be multiplied with itself: 2×2=4
One can see that this operation always looks like a square on the board; thus it is called “squaring”.

To find out the root of 4, we count the beads on one side. There are 2 beads, which means the root of 4 is 2:

This works easily with all numbers from 1 to 9. Instead of beads, we can also use pebbles, blocks or buttons:

With numbers greater than 9, we need beads in various colors. Maria Montessori developed the following simple system:
Green = 1
Blue = 10
Red =100
Light green = 1000
Light blue = 10,000
Light red = 100,000

We see the board just like an x-y graph system. Each bead on the x-horizontal line is multiplied with each bead on the y-vertical line. The result is placed in the right color at the point where the two lines meet. Here you can see what 12×12 looks like:

And these photos show the process step by step: 12 = 2 green beads for the ones and 1 blue bead for the ten:

Then we place the same beads on the vertical line:

Then we can start multiplying: 1×1 =1 :

10×1=10 and 1×10= 10 :

10×10=100 and that’s the complete square:

To find out the result, we add the value of all beads: 4×1 plus 4×10 plus 1×100 equals 144. Here:

Even with large numbers, this technique remains simple. One can observe a specific variety of color patterns which increase with regularity. For example, here is 122 squared:

After all beads are placed, we can start adding: (1 x 10,000) + (4 x 1000) + (8 x 100) + (8 x 10) + (4 x 1) = 14,884

The root of 14,884 is one side of the square, 122:

After we have learned to use the board to SEE the way squares grow, we can now learn the opposite process – finding the root of a number. First, we lay the square pattern using the appropriate amount of beads and then count one side.

The square root board helps to understand what the concept of square root actually means. It takes away the fear from this process usually only taught on paper.

I remember that I could not understand square roots when I went to school and I am happy to have found the square root board!
Try it!

3 thoughts on “Instructions to the Square Root Board

    • HI, Thanks for your comment, as I look through this post that I wrote a long time ago, I discovered I made a mistake, that’s probably where you got confused. 4 x 1 is of course 4 – not 1!!!
      So, I will correct this sentence in the blog “After all beads are placed, we can start adding: (1 x 10,000) + (4 x 1000) + (8 x 100) + (8 x 10) + (4 x 1) = 14,881” TO “After all beads are placed, we can start adding: (1 x 10,000) + (4 x 1000) + (8 x 100) + (8 x 10) + (4 x 1) = 14,884” Please also know this is my OLD BLOG that I’m not usually updating anymore, please go to my NEW BLOG page

  1. Thank you for sharing – I haven’t had a chance to practice this since my training, so this is an excellent post for jogging my memory!

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