1. of “Redistribution” (i.e., “Borrowing and Carrying”)
Teaching You How to Use Color Tiles (in this case poker chips) To Easily Teach Children the Otherwise Difficult Concept
of “Redistribution” (i.e., “Borrowing and Carrying”)
in Adding and Subtracting
This works for one child or large groups of children at their tables or desks, as long as you can see what they are each doing, because you can “see” what they are all thinking as they work with the chips. It doesn’t take long to watch each child to be sure he or she is doing it right. You need roughly one standard box of poker chips for every two children to share. This works for any age child who can count by ones and by tens to one hundred. The full rationale and explanation of this method can be found at “The Concept and Teaching of Place-Value.”
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2. We are just going to say to start with that one blue chip is worth ten white ones. And what that means will be… =
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3. You will see later why that is important and useful.
… whenever you have ten white ones, you can trade them for a blue one, and whenever you have a blue one, you can trade it for ten white ones.
You will see later why that is important and useful.
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4. = And we will also say one red chip is worth ten blue ones.
That means whenever you have a red chip, you can trade it for ten blue ones, and whenever you have ten blue ones, you can trade them for a red one.
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5. Now knowing this, we can use chips to add and subtract.
And that means a red one can be swapped for 100 white ones, because if we swapped a red one for ten blue ones, and then swapped each of those blue ones for ten white ones, you would have 100 white ones for all ten blue ones, or for that one red one.
Now knowing this, we can use chips to add and subtract.
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6. But first you have to be able to combine different color poker chips to show numbers. For example five white ones is white ones is 8.
But you can show 12 by having either 12 white ones or by having one blue one (which is 10) and two white ones.
37 would be
214 would be
452 would be
Etc. If you are a parent working with a child or children, make sure they can represent any number from 1 to 999 with the poker chips or color tiles, whichever you are using.
Then we can begin simple addition.
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7. Here are 2 and 5. How many are there all together?1
Here are 2 and 5. How many are there all together? 3 2 1 5 4
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8. Here are 2 and 5. How many are there all together?
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9. 2 1 5 4 3 7 6
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10. 2 1
So = 7 5 4 3 7 6
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11. Subtraction would be the opposite
Subtraction would be the opposite. To subtract four from seven, you would take away four chips (shown here crossed out), which leaves three chips, showing that 7 – 4 = 3.
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12. Here are 7 plus 5 more. How many are there all together?3 4 1 2 7 6 5
Here are 7 plus 5 more. How many are there all together? 3 2 1 5 4
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14. 4 2 1 3 7 6 5 10 9 8 12 11
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15. 4 2 1 3 7 6 5
So = 12 10 9 8 12 11
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16. That gives us a lot of white chips on our table, and we don’t need that many because we can swap out ten of them for a blue one. See…
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17. twelve
eleven
TEN =
Which is still 12.
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18. Subtraction works in the opposite way
Subtraction works in the opposite way. If you want to take 9 from 12, and you only have a blue one and two white ones, you cannot take 9 white ones away until you change the blue one into 10 white ones.
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19. twelve
eleven
TEN =
To subtract 9 from 12, you would trade in the opposite way – change the blue ten into 10 white chips first, so you then would have 12 white chips.
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20. 4 2 1 3 7 6 5
Then remove nine white ones, which will leave three. So 12-9 = 3. 10 9 8 12 11
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21. Now let’s add 13 to 14, using poker chips.
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22. How many are there all together?
twelve
eleven 10
fourteen
thirteen
Here are 14 and 13.
How many are there all together?
eleven
twelve 10
thirteen
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23.  Previous
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24.  Previous
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25. 10 20
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26. 2 1 10 4 3 5 6 20 7
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27. 2 1 10 4 3 5 6 20 7
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28. 2 1 10 4 3
= 27 5 6 20 7
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29. 28 2 1 10 5 3 4 20 8 6 7 =
and 35 10 2 1 20 3 5
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30. 10 20 30 40 50
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31. 2 1 10 5 4 3 20 8 6 7
50 and ? 30 10 9 40 11 12 13
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32. 2 1 10 5 4 3 20 8 6 7
50 and 13 30 10 9 40 11 12 13
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33. But we don’t need all those chips, particularly all those white ones
But we don’t need all those chips, particularly all those white ones. We can swap out ten of the white ones for one more blue one. That will give us…
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34. Six blue ones (ten, twenty, thirty, forty, fifty, sixty) and three white ones = 63
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35. This is another way you might have counted them too.52 51 10 55 54 53 20 58 56 57
This is another way you might have counted them too. 30 60 59 40 61 62 63
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36. If you are a parent working with a child, give the child some other addition problems where they can convert 10 white ones to a blue one.
For example: ; ; ; ; ; etc.
And give them some subtraction problems where they have to convert a blue chip to 10 white ones in order to complete the subtraction.
For example: 35 – 19; 86 – 39; 51 – 47; 91 – 45; etc.
Just practice until it is second nature.
If you are the student working on your own, do additions and subtractions yourself and check your answer with a calculator.
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37. After doing a bunch of these with the children and walking around the room looking at each child’s work as the child does it, to be sure they are understanding -- which they will, possibly with some assistance from you -- it is now time for the big “test” and the wonderful “aha” moment for them.
They can do this on their own just by giving them the question, but I will walk you through what they will be thinking – as you can “watch” them think it through by the looks on their faces and what they do with the chips.
So do NOT tell them how to do it. Let them work on it on their own and figure it out by themselves so that they will gain the understanding.
It doesn’t do any good for them to know what you understand; that is not the same thing as understanding what you know. And it is not the same thing as understanding even what they know. This whole presentation is about teaching place value for understanding.
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38. Here is 100.
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39. How much is 100 – 37?
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40. How much is 100 – 37?
If you are doing this by yourself, stop with this slide until you figure out the answer by yourself first.
If you are a parent doing this with children, let them figure this out by themselves. Do NOT help or tell them how to do it. They will figure it out on their own. Let them.
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41. Hey! I said no peeking till you have figured it out yourself.
After you do that, you can go ahead and check your answer.
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42. So what can you do to take 37 away from it?
You can’t take much away from this, since there is only one chip, right?
So what can you do to take 37 away from it?
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43. You can trade this red chip for ten blue ones.
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44. We still can’t take 37 away from this, but….
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45. We can trade one of the blue ones for ten white ones.
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47. Now we can take away 37: by taking away three blues and seven whites.
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48. Which will leave six blues (ten, twenty, thirty, forty, fifty, sixty) and three white ones = 63.
So 100 – 37 is 63.
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49. Now the point is to show how color value and place value are basically the same thing, because each column is just like a different color:
The “ones” column tells how many white chips there are; the “tens” column tells how many blue chips there are, and the “hundreds” column tells how many red chips there are.
We are going to add 286 to 341.
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50. Consider each number column to tell you how many chips you have of the colors shown.
This column tells how many red chips you have. It is the hundred’s column.
This column tells how many blue chips you have. It is the ten’s column.
This column tells how many white chips you have. It is the one’s column.
So the number 341 would be 3 4 1
286 would be 2 8 6
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51. Consider each number column to tell you how many chips you have of the colors shown.
This column tells how many red chips you have. It is the hundred’s column.
This column tells how many blue chips you have. It is the ten’s column.
This column tells how many white chips you have. It is the one’s column.
So the number 341 would be 3 4 1
286 would be 2 8 6
And if you add them together, you get 5 12 7
Which looks like five hundred twelvety seven, which, of course is not a number.
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52. But now we can trade ten of those blue ones for a red one.
That will leave us two blue ones, which is 20, and it will give us six red ones, which is And we will still have the seven white ones.
So the columns will look like …
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53. Consider each number column to tell you how many chips you have of the colors shown.
This column tells how many red chips you have. It is the hundred’s column.
This column tells how many blue chips you have. It is the ten’s column.
This column tells how many white chips you have. It is the one’s column.
So the number 341 would be 3 4 1
286 would be 2 8 6
And if you add them together, you get 5 12 7
Swap ten blues for a red and you get
5 + 1
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54. So that means 341 plus 286 equals 627, and doing it on paper in columns is the same as doing it with different color poker chips:
341
+286
627
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55. Then practice this with your children, adding different two- and three- place numbers. Then do the same thing with subtracting two and three place numbers. And once they get good at both, let them do it without the colors. When they can do that, they will be able to know how to use place-value and redistribute numbers (“borrowing and carrying”) in addition and subtraction.
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56. Two Caveats
Teaching concepts is often difficult because concrete specifics that can be shown, told about, or held are not themselves the concepts, but embody the concepts. I will be using the visual aids of Power Point slides and my oral narrative to go through concrete steps with you as you work tactilely with the poker chips yourselves, but all these concrete examples are only tools to help guide your thinking – and it is your own thinking that will help you learn. The same is true for how this method guides children’s thinking into “seeing” the abstract ideas from the specific cases used. The method I will demonstrate also will let you see whether the children have developed the abstract idea being taught.
Notice, this does not teach the concept of “place-value” itself and how it is derived, but teaches how to use it in adding and subtracting. The concept of place-value itself requires a different kind of explanation and guided thinking. For a demonstration of that concept and how to teach it, see
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