Select Page

1. Home
2. Knowledge Base
3. Mathematics
4. Passage to Abstraction

#### Material

A fame with a support that allows it to stand.
There are 4 wires across, each with 10 beads.
Green beads on the first (top) wire, for units.
Blue beads on the second wire, for tens.
Red beads on the third wire, for hundreds.
Green beads on the fourth wire, for units of thousands.
On the left side of the frame, the categories are marked:
The top number is “1” then “10” then “100” and then “1,000”
The first three numbers are written on a background of white, and the fourth is on a gray background; this is because the 1,000 is the unit of the next hierarchy of numbers.

#### Presentation

##### Introduction
1. Invite the child to the lesson.
2. Introduce the small bead frame.
3. Ask the child to bring the materials to a table.
4. Bring the golden bead introduction tray.
5. Place a unit bead on the table.
6. “What is that? Right, one!”
7. Slide one green bead at the top, to the right.
8. “This is also one!”
9. Slide the bead back to the left.
10. Repeat the process for the ten bar with a blue bead, hundred square with a red bead, and thousand cube with a green bead (bottom row).
11. Make random small numbers with the bead frame with one category at a time. Slide the beads to the right, and see what number they make, such as “Oh, it’s 20!” or “Wow, there is 300!”
12. Ask the child to make random numbers too, and ask him to say aloud what numbers he makes.
13. Return the golden beads to the shelf, but keep the small bead frame on the table and proceed to the next presentation: frame and notation paper.
##### Frame and Notation Paper
1. Bring two papers, pencils, and underlays on a tray to the table.
2. Count the unit beads, moving them to the right, one at a time:
“1, 2, 3, 4, 5, 6, 7, 8, 9, 10.”
3. “Ten unit beads is the same as one 10!”
4. Slide one 10 bead to the right.
5. “These beads can go home.”
6. Slide the unit beads back to the left.
7. Count the 10 beads, moving them to the right, one at a time:
8. “10, 20, 30, 40, 50, 60, 70, 80, 90, 100.”
9. “Ten 10’s is the same as one 100!”
10. Slide one 100 bead to the right.
11. “These beads can go home.”
12. Slide the 10 beads back to the left.
13. Count the 100 beads, moving them to the right, one at a time:
14. “100, 200, 300, 400, 500, 600, 700, 800, 900, 1000!”
15. “Ten 100’s is the same as one 1000!”
16. Slide on 1000 bead to the right.
17. “These beads can go home.”
18. Slide the 100 beads back to the left.
19. Count the 1000 beads, moving them to the right, one at a time:
“1000, 2000, 3000, 4000, 5000, 6000, 7000, 8000, 9000, 10000.”
20. “There are more number than that, but that’s just where the frame stops.”
21. Slide the 1000 beads back to the left.
22. Examine the notation paper.
23. Briefly talk about the sides of the paper.
24. Show the child how to fold the paper, corner to corner, edge to edge.
25. Starting with the unit beads, count one bead at a time, and write the number on the paper before moving onto the next number.
26. Move a unit bead to the right and count “one.”
27. Write 1 on the green line of the paper that is under “Simple units.”
28. Repeat this process for the beads, but stop after writing 9.
29. Count “ten” then move over to the blue line and write a 1 on the blue line. “One ten.”
30. Continue counting, for example, “twenty” then write a 2 on the next row on the blue line and say “two tens.”
31. Continue this process; go to the red line at “one hundred” and write a 1 on the red line, counting up to nine hundreds; go to the green line for “one thousand” and write a 1 on the green line under “Of Thousands Units.”
32. “We have more numbers, but the paper just ends here.”
##### Fill in the Zeros
1. Flip the paper over to the other side.
2. Perform the same process as with Frame and Notation Paper, but this time, once you reach 10, put the 1 on the blue line, and a 0 on the green line. Continue this process for all numbers, placing 0’s accordingly.
##### Form Numbers
1. Fold paper over and crease the edges, as the other two sides have been used.
2. Make a number using the bead frame, then write it down on the lines, such as 4396.
3. Read the problem out loud.
4. Repeat this process several times; ask the child to do it too.
5. Make a number using the bead frame that includes a zero, then write it down on the lines, such as 970.
6. Read the problem out loud.
7. Repeat this process several times; ask the child to do it too.
8. Reverse the process by writing the number first, then making the number on the frame.
1. Invite the child to the lesson.
2. Use the same paper as before.
3. “This is special paper, and we want to use it as much as we can. We will write off where we left off!”
4. Write a number such as 5236, and ask the child to write it on his paper too.
5. Write an addend such as: +1622
6. “I didn’t bring a straight edge, but I bet you can draw a nice line right here without one!”
7. Draw a line underneath the 2nd addend.
8. Ask the child to make the first addend using the bead frame; count to verify it.
9. Make the second addend yourself using the bead frame; leave an inch or so between the 1st and 2nd addend; count to verify it.
10. Slide the 2nd addend units to the right so that they are now together with the 1st addend units.
11. Count the beads on the right.
12. Write the answer on the paper, under the line, on the units wire, such as “8.”
13. Continue this process for the other categories.
15. Repeat this process for a few more problems until the child is secure;
Draw colored designs in one row after each answer to separate one problem from the next.
1. Invite the child to the lesson.
2. Continue using the notation paper from where you left off.
3. Write a number such as 3476, ask the child to write it too.
4. Write an addend such as ÷ 2749. Ask the child to write it too.
5. Draw a line underneath the 2nd addend.
6. Ask the child to draw one too.
8. Take over to show the child how to add the 2nd addend. (This time you will need to carry over to the next category)
9. Begin by counting the unit beads just as you did for static addition.
10. For example, if you already have six beads, and you are going to add the 9 beads from the 2nd addend, you will have to stop at 4 as that makes 10.
11. Count and slide the unit beads to the right until you are out of unit beads, making a 10:
“One, two, three, four” then elongate your four so it sounds like “foooooooooooooooooour” as you slide on ten bead to the right, and slide all the unit beads back to the left.
12. Continue counting and sliding the unit beads to the right from where you left off “Five, six, seven, eight, nine.”
13. Count the unit beads on the right.
14. Write the answer on the paper, under the line, on the units wire, such as “5.”
15. Repeat this process for the other categories.
1. Invite the child to the lesson.
2. Write a number with more than 2 addends such as:
2583
+1649
+2478
3. Ask the child to write the problem too.
4. Repeat the same process as you would with just two addends, but this time you are simply using three or more addends.
##### Static Subtraction
1. Invite the child to the lesson.
2. Write a problem such as:
8579
-5237
3. Ask the child to write it too.
4. Ask the child to make the first number using the bead frame.
5. “Alright, we are going to take away 7 units.”
6. Count and slide 7 units to the left.
7. “How many tens do we need to take away? Right, three! You can do that!”
8. Let the child proceed with taking away from the rest of the categories.
9. Write the answer starting with the units category.
##### Dynamic Subtraction
1. Invite the child to the lesson.
2. Write a problem such as:
8356
-2688
3. Ask the child to make the first number using the bead frame.
4. Begin to take 8 away from 6, counting and sliding one unit bead to the left at a time.
5. Once you reach 6, elongate “siiiiiiiiiiiiiiiiiix” and slide one 10 bead to the left, and all the units back to the right.
6. Continue counting where you left off “seven, eight” sliding the unit beads to the left as you count them.
7. Move on to the next category, repeating this process and exchanging when necessary.
8. Write the answer starting with the units category.
“We have 8 units, 6 tens, 6 hundreds, and 5 thousands. That’s our answer! 5668!”
10. Write another example for the child to do independently while you watch.

#### Exercises

Child works independently.

#### Purpose

To lead the child to add and subtract without using the material.
To apply everything which the child has learned towards abstraction.

None.

#### Age

5 1/2 +

This notation paper makes real clear the purpose of the zero.
Draw designs through entire rows between problems in order to separate them.