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Division with Racks and Tubes

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  5. Division with Racks and Tubes

Material

Seven racks, each containing ten test-tubes.

Three racks are painted white for the simple family:
One with green beads for units.
One with blue beads for tens.
One with red beads for hundreds.

Three racks are painted grey for the thousands family:
One with green beads for units of thousands.
One with blue beads for tens of thousands.
One with red beads for hundreds of thousands.

One rack painted black for the million family, with green beads for units of millions.

Seven containers or bowls:
3 painted white outside (one green, one blue, and one red on the inside).
3 painted grey outside (one green, one blue, and one red on the inside).
1 painted black outside (green inside).

Three division boards: (designed like Unit Division Board)
One with a green strip for the unit. (only board used in primary)
One with a blue strip for the tens. (used later in elementary)
One with a red strip for the hundreds. (used later in elementary)

Three sets of skittles: 9 each of green, blue and red.
Paper and pencil.

Presentation

  1. Invite the child to the lesson.
  2. Ask the child to bring the material to a table.
  3. Bring the large bead frame.
  4. Introduce the materials; compare the test tube racks colors to the large bead frame colors.
  5. Point out how they all correlate; the tube of green beads in the white rack is units, the tube of blue beads in the white rack is tens, the tube of red beads in the white racks is hundreds; the tube of green beads in the grey rack is thousands, etc.
  6. Return the large bead frame to its shelf.
  7. Write a problem on your paper, and ask the child to write it as well, such as: 573,296 ÷ 4 =
  8. Set the bowls in front of their corresponding racks at the top center of the table.
  9. Start with the units and count and place that number of beads in the bowl.
  10. Repeat this for each category.
  11. One category at a time, starting with the units, move the bead rack and bowl to the right of the table, and verify that each category is correct.
  12. Place the division board in the middle of the table.
  13. Place the skittles on the table.
  14. “We are going to share out four times!”
  15. Starting with the largest category, which is the category closest to you, bring the rack and bowl to the top center of the table.
  16. Share out that category to the skittles on the board.
  17. After sharing out the category, write the answer.
  18. Clear the board and place the shared beads back into their tubes.
  19. Move the next rack and bowl next to the previous rack and bowl.
  20. Places the remainder bead(s) that need to be exchanged into the bowl of the new category which is now adjacent.
  21. Count 10 beads in the test tube of the new category for each bead of the previous category that was placed in the bowl.
  22. Pour these 10 beads into the bowl.
  23. Now that the previous categories’ beads have been exchanged, place the previous categories’ beads back into their tubes.
  24. Move the previous category to the left side of the table..
  25. Center the new category in the top middle of the table.
  26. Repeat the process: share out, write the answer, clear the board, place the rack and bowl of the next category adjacent to the current one, and make exchanges as necessary with the remainder.
  27. After the exchange, move the previous rack and bowl to the left of the table and center the new rack and bowl at the top middle of the table.
  28. Continue repeating the process until you arrive at the answer.
A Zero in the Answer
  1. One another day “I’ve got a new problem I want to show you.”
  2. Write a problem that will result in the answer having a zero, such as:
    9216 ÷ 3 =
  3. Repeat the same process as before, but emphasize that when you can’t share out anything, that category is 0 and that a 0 must be written as the answer for that category, just as any other number would be.
Moving Toward Abstraction
  1. Ask the child to set up the division with racks and tubes material.
  2. Bring two papers, pencils, and underlays on a tray.
  3. Ask the child to watch, and to write what you write.
  4. Draw a vertical line in the 2nd box of 2nd row.
  5. “And I’m going to draw a line up here, like a flat roof on a house.”
  6. Draw a horizontal line from the top of the vertical line, about 8 boxes long; you have now made a standard division line.
  7. Write a number and ask the child to write it too, such as 89764.
  8. “We’re going to share out this number, so it’s inside the house.”
  9. “Outside, we’re going to put the number we share out to. We have 4 skittles here, so let’s write a 4 outside.”
  10. Ask the child to place the beads in their bowls as before.
  11. Ask the child to place the skittles on the board.
  12. Ask the child to share the beads out to the skittles.
  13. “Alright, 8 shared out 3 times is what? Right, 2!”
  14. Write a 2 in the first box above the line.
  15. Point to the 4.
  16. “4 taken 2 times is what? Yes, 8.”
  17. Under the first box, write: – 8
  18. Draw a line under both the -8 and the box adjacent to it on the right.
  19. “8 take away 8 is what? Right, 0!”
  20. Write a 0 under the line.
  21. Wait for the child to clear the board and watch as he moves onto the next category.
  22. “9 shared out 4 times is what? Right, 2.”
  23. Write a 2 above the line in the 2nd box.
  24. “But we have a remainder, I’ll show you what to do.”
  25. “Let’s take this number down.”
  26. Write the 9 next to the 0.
  27. “4 times 2 is what? Right, 8.”
  28. Write -8 under the 9; draw a line under both the -8 and the box adjacent to it on the right.
  29. “9 take away 8 is what? Yes, it’s 1.”
  30. Write a 1 underneath the -8.
  31. “Look, that’s what’s in the bowl! One!”
  32. “Let’s exchange it!”
  33. Wait for the child to make an exchange.
  34. “There sure are a lot of beads in there now!”
  35. Wait for the child to share them out.
  36. “How many beads did everyone get? Yes, 4.”
  37. Write a 4 above the line in the third box.
  38. To the right of the one, bring down the 7.
  39. “4 taken 4 times is what? Right, 16!”
  40. Write -16 below the 17.
  41. Draw a line under both the -16 and the box adjacent to it on the right.
  42. “What is 17 take away 16? Yes, one!”
  43. Write a 1 underneath the 6 in -16.
  44. “Look, that’s what is in the bowl, one!”
  45. Wait for the child to exchange.
  46. Wait for the child to share out.
  47. “How many did everyone get? Yes, 4.”
  48. Write a 4 above the line in the fourth box.
  49. To the right of the 1, bring down the 6.
  50. “4 taken 4 times is what? Yes, 16.”
  51. Write -16 below the 16.
  52. Draw a line under the -16 and the box adjacent to it on the right.
  53. “16 take away 16 is what? Right, it’s nothing! Just a zero.”
  54. Write a zero in each box under the -16.
  55. “There is nothing to exchange. What does that leave us with? Yes, just a 4 to share out.”
  56. Wait for the child to share out.
  57. “4 shared out 4 times is what? Yes, 1!”
  58. Write a 1 above the line in the fifth box.
  59. To the right of the 0, bring down the 4.
  60. “4 taken 1 time is what? Yes, just 4.”
  61. Write a -4 under the 4.
  62. Draw a line under both the -4 and the box adjacent to it on the right.
  63. “4 take away 4 is what? Yes, o!”
  64. Write a 0 under the -4.
  65. “Look, there is no remainder. We are all done!”
  66. “Let’s read the problem. 89764 shared 4 times is 22441.”

Exercises

Child works independently.

Purpose

To do division with large numbers.
To lead the child towards abstract division.

Age

6

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