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G1A
Place Value Visualized
Numerals:
We use just ten numerals—0 to 9—to write all our numbers, no
matter how large. When we get beyond 9, we just start
combining those ten numerals to make larger numbers. A
simple example is the number 14. The last numeral in 14, the
4, means 4 units. The preceding numeral, the 1, means 1 group
of ten. In the number 24 (which we read as “twenty four”), the
last numeral again means 4 units, and the preceding numeral,
2, means 2 groups of ten each. It is precisely because that 2 is
the next to the last numeral in the number 24 that we know it
has to mean a quantity of groups of ten. This system of writing
numbers based on place value can continue to larger and larger
numbers. For example, in the number 324 (which we read as
“three hundred twenty four”), the last numeral again means 4
units, the preceding numeral again means 2 groups of ten each,
and the numeral before that means 3 groups of a hundred
each.
Some people find this easier to understand by thinking in terms
of coins. In the number 24, the 4 means 4 pennies, and the 2
means 2 dimes, since each dime is worth 10 pennies.
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The important thing is to understand that when we write a
number with more than one numeral in it, the meaning of each
numeral depends on its placement in the number. A numeral
on the far right means that number of units. The numeral just
before it means that specific number of tens. And the numeral
just before it means that specific number of hundreds. So the
number 324 means 3 hundreds, plus 2 tens, plus 4 units. That
is 300 plus 20 plus 4. As another example, 957 means
900+50+7.
The numeral 0 is essential to this number system. The number
3 means simply 3 units. But in the number 30, the 3 means 3
groups of tens, and the 0 means no units are added to those 3
tens. The presence of that 0 after the 3 causes a big change in
the meaning of the digit 3. By the same logic the number 300
means 3 hundreds plus no tens plus no units. And in the
number 507 we have 5 hundreds plus no tens plus 7 units. The
0 inside the number 507 is just as important as the 0 at the end
of 30 or the two 0’s at the end of 300. If we were to leave the 0
out of 507 it would be just 57. 57 is a different number, much
smaller than507.
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Abstraction:
It is a fairly simple task for children to learn to count, that is, to
recite number names in sequence. If we teach this carefully,
they will also learn that each number name is associated with a
specific quantity.
But to understand our numbering system properly the
children must learn how these ten simple numerals, 0 to 9, are
recycled into multiple meanings, and how these meanings are
determined by place value.
We teach our children that the numeral 1 represents the
quantity one. To make this real for them we teach them that
the numeral 1 is used to define one item. We want them to see
that 1 can mean one marble, or one apple, or one anything.
This is meant to demonstrate to them that the concept one is
an abstraction, not limited to any one thing.
This abstract idea of “one” becomes more subtle when we
teach them that it can also mean one group of things, such as
one family with several members, or the collection of red
marbles in the bag.
We then apply this abstract notion of the meaning of a number
to explain the use of place value in our system of numerical
notation, and in the system of number names we use in
counting.
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The point is simple. (At least it is for us!) In the number 1 the
numeral 1 means one item. But in the number 12 the numeral
1 means one group of ten items. And in the number 146 the
numeral 1 means a group of a hundred items.
It is essential that our children understand how the meaning of
a number is determined by this combination of two details: the
intrinsic value of the numeral, and the position of the numeral
in the number. A numeral at the right hand end of a number
stands for that quantity of units. A numeral just one space to
the left stands for that quantity of tens. A numeral one more
space to the left stands for that quantity of hundreds.
To facilitate the children’s comprehension of place value, we
will sometimes provide this column header:
Tens of
Thousands
Thousands
Hundreds
Tens
Units
Ultimately we want the children to realize that all things
mathematical have abstract traits which can be manipulated in
many situations. This is why mathematics are so useful for
working with and understanding reality.
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1+1=
0+0=
0+1=
1+0=
0+5=
5+0=
0+10=
10+0=
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10+4=
14=10+4
16=10+
12=
+2
18=
+
24=20+4
26=20+
23=
6
+3
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27=
+
57=50+7
32=30+
86=
+6
41=
+
60=60+0
80=
+0
50=
+
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324=300+20+4
417=
+10+7
659=600+
+9
192=100+90+
234=200+
+
521=
+20+
476=
+
+6
735=
+
+
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570=500+70+0
507=500+0+7
57=50+7
In a one digit number, the
numeral defines a quantity of
units.
7=7
1=1
3=3
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In a two digit number, the last
numeral defines a quantity of
units, and the preceding
numeral defines a quantity of
tens.
34=30+4
72=70+2
In a three digit number, the last
numeral defines a quantity of
units, the preceding numeral
defines a quantity of tens,
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and the numeral before that
defines a quantity of hundreds.
629=600+20+9
Every time an additional digit is
added at the beginning of a
number, it defines an even
larger quantity.
2535=2000+500+30+5
3916=3000+900+10+6
5281=5000+200+80+1
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6527=6000+500+20+7
The number 6527 is read as six
thousand five hundred twenty
seven. So, in a four digit
number, the first digit is
thousands, the second is
hundreds, the third is tens, and
the last is units.
This system can keep growing
without limit. A five digit
number starts with tens of
thousands, followed by
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thousands, hundreds, tens, and
units. 46527 is read as forty-six
thousand five hundred twenty
seven.
A six digit number starts with
hundreds of thousands,
followed by tens of thousands,
and so on. 346527 is read as
three hundred forty-six
thousand five hundred twenty
seven.
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Written out horizontally
6 5 2 7
can be read vertically as
6 0 0 0
+
5 0 0
+
2 0
+
7
Adding 6000+500+20+7 is what
the number 6527 means.
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Two teaching items precede the unit.
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