Transcript 1. Adding Using Tiles

Taking the Fear
out of Math
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#3
Addition
Using Tiles
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Key Point
Using tiles, we can represent the number
3 by writing…
Technically speaking, 3 tiles is a quantity,
not a number, in which the adjective is 3
and the noun is tiles.
At the same time, 3 tiles
is a phrase with adjective 3 and with the
noun depending on what the tiles
represent.
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Key Point
For example:
If we are talking about people,
represents 3 people.
If we are talking about apples,
represents 3 apples.
If we are talking about pounds,
represents 3 pounds.
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Key Point
However, since students tend to view
numbers in the form of quantities,
we feel that there is no problem
in letting them assume that 3 and
mean the same thing.
Thus, for example, if we are using tiles,
we will represent the sum 3 + 2 as…
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Our Point of View
The activity below presupposes
that by the time students enter
kindergarten they are able to count
relatively small numbers of objects.
For example, shown a picture such as…
…they will count, “one, two”, and thus
know that there are two tiles.
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Our Point of View
And if they are shown a picture such as…
…they will count, “one, two… three, four,
five”, and thus know that there are
five tiles.
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Our Point of View
Moreover, they might be able to visualize
that…
and
…both consist of 5 tiles.
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Our Point of View
However, they might not be as comfortable
at the beginning with the digits 2, 3, and 5,
nor will many of them be comfortable with
the plus sign and the equal sign.
In other words, they might not have
internalized the fact that…
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Our Point of View
2+3=5
…is just a more concise
(and more abstract) way of writing…
+
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If we use tiles as our noun, the rules of
arithmetic become so self-evident that it
would seem unnecessary to even bother
naming them.
For example, look at the set of tiles below…
It probably seems self-evident to you
that the number of tiles in the set does
not depend on the order in which they
are counted.
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This seemingly unimportant observation
is so important that we call it our
fundamental principle of counting.
Our Fundamental Principle of Counting
The number of objects in a set does not
depend on the order in which the objects
are counted or in the form in which they
are arranged. For example, in each of the
six arrangements shown below, there are
3 tiles.1
1
Note that the color of the tiles is irrelevant.
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Based on this apparently simple
principle, many number facts appear
to be almost self evident.
For example, when a beginning student
looks at the equality 7 + 2 = 4 + 5, the
result is not immediately obvious.
However, in terms of tiles and
Our Fundamental Principle of Counting,
7 + 2 can be represented by…
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The number of tiles does not change if
we shift three of the red tiles
so they are next to the yellow tiles
to obtain the grouping…
…which now represents the sum 4 + 5.
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If we move the two sets of tiles close
together, we see that there are 9 tiles in all
and will be represented by…
1 2 3 4 5 6 7 8 9
1 2
1 2 3 4 5 6 7 81 92 3 4 5
…and it is now visually clear that
7 + 2 = 4 + 5 = 9..
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Moreover, by using other rearrangements
we can show such results as…
1
9
1 2 3 4 5 6 7 8
8+1=9
1
1 3
2
2 4
3 5
4 6
5 7
6 8
7 8
9
1+8=9
1 2 3
1 5
4
2 6
3 7
4 8
5 9
6
3+6=9
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After doing a few problems of this type,
even the most inexperienced students
should be able to see that if we are given a
set of tiles, the total number of tiles
remains the same no matter how the tiles
are rearranged.
This observation will be a big help to them
later when they might be expected to do
mental arithmetic and are dealing with
much greater numbers.
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For example, suppose they want to
compute the sum…
777 + 197
Mentally, it is quicker to add 200
to a number than it is to add 197.
We can take 3 tiles from the set that has
777 tiles (thus leaving that set with 774 tiles)
and add them to the set that has 197 tiles
(thus leaving that set with 200 tiles).
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More visually…
777 tiles
+ 197 tiles
974 tiles
–3
+3
774 tiles
+ 200 tiles
974 tiles
And because the total number of tiles in
the two sets has not changed we may
conclude that 777 + 197 = 774 + 200,
and mentally it is easy to see that
774 + 200 = 974. Hence, 777 + 197 = 974.
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5+3
3+5
In any event, this concludes our
discussion of addition using tiles and
Our Fundamental Principle of Counting.
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