You shall teach them diligently to your children, and shall talk of them when you sit in your house, and when you walk by the way, and when you lie down, and when you rise. -Deuteronomy 6:7

Tuesday, September 10, 2013

Montessori Long Division (aka Test Tubes or "Racks and Tubes")

Prerequisites for this work include:
- Use of the single-digit division board
- Use of the large bead frame (we didn't use this since we do not have one.)

It is advisable to only introduce this material after the child has a full understanding (from the prerequisites) of what is happening in the division process. 

The "racks and tubes" or "test tubes" are materials which build upon previous Montessori materials.  After working with single-digit division this is a good next step to work with a really large number!

I also presented the stamp game division (I'll post about stamp game soon!) before introducing this material.

Children love to work with large numbers and it's exciting to finish a problem this big!

Presentation of materials:

Above:  Millions is represented by the black bowl, then down the line to the hundred-thousands,
ten-thousands, one-thousands and so on...

1. Write a number (dividend) for the child, such as 3,957,246.

2. Set up materials as above, with the millions positioned on the left.  The child should already know that, in division, we always start with the highest number. 

 3. Explain as needed, that, "there are always ten beads in each tube."  Count them and show how to skip count by 2's as you take them out.


 4.  Now begin setting up the problem with the dividend.  In the problem, the dividend is 3,957,246.  First, the child will take out "3" of the beads that represent the millions and place them in the corresponding dish.

5.  Continue with the remaining numbers - 9 beads for the hundred-thousands dish, 5 beads for the ten-thousands dish... and so on.

6. After all the dishes are filled with beads representing the dividend, look at the rest of the problem.  Ask the child, "What is the divisor?" (Answer:  3!)

7.  Place three of the skittles (see below) in the place holders.

8. Remind the child, "Each skittle gets an equal amount." as you take the appropriate amount of beads out of each tube and put them in the bowls. 

9.  Take the millions bowl with the millions beads in it, and divide them up among the skittles.

10.  Write the answer on the sheet with the problem written on it.  The answer is "1" - each of the skittles get "1" million.
11. Clear the board by putting the beads you've counted from the bowl back into the tubes and moving the millions slightly over to the left.

12.  Bring forward the hundred thousands and divide the quantity in the bowl in the same manner, writing the partial quotient when finished and clearing the board again.

 13.  Bring down the 10's thousands (5) and notice that they cannot be equally divided - there are two left over. (See below)

14.  Take the two that cannot be distributed and put them in the NEXT LOWER CATEGORY BOWL (thousands)

15.  As needed, help the child realize that we can change the 2 ten thousands for 20 1,000s (this should be easy to understand if the child has worked on division with the stamp game!)

Pouring two of the thousands bead tubes (10 beads in a tube) replaces the two
ten-thousands beads that were returned.

16. Distribute those beads into the board, write the partial answer (9) and clear the board and place it to the left.

17. Bring down the hundreds beads into the board and notice that you cannot even give one to each skittle. 

18.  When there is no sharing possible, write zero on the answer sheet and put the beads into the ten bowl. (below)

19.  Exchange and continue as before.

20.  When finished, place commas in the right place and review the problem with the child.

21.  If there happens to be a remainder (not equally shared beads in the 1 category), show the child how to write r=(n) on the paper to the right of the quotient.  (My child knew how to do this already from her work with the stamp game division.

Aim of this exercise is to introduce the child to long division!

So teach us to number our days that we may get a heart of wisdom.
-Psalm 90:12

See This for Another Example


  1. Dear Sheri,
    Thank you for the information you have shared through your blogs. It has immensely helped me when I use Montessori materials to work with my children.

    1. Thank you Geetha! I haven't updated it in such a long time, but I am glad to know that what remains on this blog is helpful. I'm very grateful for those Montessori teachers who helped me along the way.