A. Decimal Addition - Column Methods
Physical Interpretation
Addition Via Counting
To illustrate this, take for instance two bags of marbles. Say one bag
has 13 marbles and another 15. Ask your child to put the two bags
together, and count how many there are. Now take two bags, one with 13
buttons and another with 15 buttons. Ask your child again to count how
many there are. This shows the total number of objects does not depend on
their type or kind. Redo this addition experience with varying numbers.
Addition Table
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1
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2
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3
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4
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5
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6
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7
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8
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9
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10
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1
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2
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3
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4
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5
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6
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7
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8
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9
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10
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11
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2
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3
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4
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5
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6
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7
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8
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9
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10
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11
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12
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3
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4
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5
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6
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7
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8
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9
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10
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11
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12
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13
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4
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5
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6
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7
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8
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9
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10
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11
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12
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13
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14
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5
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6
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7
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8
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9
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10
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11
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12
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13
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14
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15
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6
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7
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8
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9
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10
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11
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12
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13
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14
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15
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16
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7
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8
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9
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10
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11
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12
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13
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14
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15
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16
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17
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8
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9
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10
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11
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12
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13
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14
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15
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16
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17
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18
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9
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10
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11
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12
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13
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14
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15
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16
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17
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18
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19
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10
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11
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12
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13
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14
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15
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16
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17
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18
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19
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20
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The entry for 2 + 3 is 5. That comes from counting. If we have two
items and three more items in non-overlapping groups than by
counting that there are five items.
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In general, all the entries in the above addition or sum table
follow from counting. Once that is understood, we may observe 2 + 4
is one more than 2 + 3 = 5. So it is 6. That pattern explains why
the entries increase by 1 when one of the addends increased by 1.
Nuance: Addition is Commutative:
Now a set of two tally marks and three tally marks may be counted
forwards and backwards. That explains why 2 + 3 = 3 +
2. That explains in general why pairs of whole number can be
added in any order.
Another Nuance:
The philosophical question of why 1 + 1 = 2 is answered by counting
tally marks
In Sum, Addition via Counting leads to the addition table
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1
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2
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3
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5
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6
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7
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8
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9
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1
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2
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3
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4
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5
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6
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7
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8
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9
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10
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2
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3
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4
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5
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6
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7
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8
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9
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10
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11
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3
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4
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5
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6
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7
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8
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9
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10
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11
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12
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4
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5
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6
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7
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8
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9
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10
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11
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12
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13
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5
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6
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7
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8
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9
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10
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11
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12
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13
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14
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6
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7
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8
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9
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10
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11
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12
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13
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14
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15
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7
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8
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9
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10
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11
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12
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13
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14
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15
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16
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8
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9
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10
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11
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12
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13
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14
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15
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16
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17
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9
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10
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11
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12
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13
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14
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15
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16
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17
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18
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If you do or review it with a student, observe how
adding n+1 to number is one more than adding n to the number. That
provides a mechanism for filling in the table by rows (horizontally
from right to left) and by columns (vertically from top to
bottom).
Students should know the sums of all pairs of digits from 1 to 9, as well
as how to add 0 and 10 to a single digit number. Filling in the addition
table is a good exercise.
Addition with Decimals - How to Justify
Besides column methods for addition shown below, there are column
methods for subtraction, multiplication and even long division. All
methods, some more directly than others, take advantage of decimal
place value.
Single to Triple Digit Examples - with carries
Provide the following experiences (and similar ones).
-
By counting 3 ones added to 4 ones is 7 ones. On paper this
can be written in the shorthand form
3
+ 4
----------
7
----------
-
By counting 8 ones added to 7 ones yields 15 ones. On paper
this can be written in the shorthand form
8
+ 7
----------
15
----------
-
By counting, 12 ones added to 25 ones is 37 ones or three
tens and 7 ones. On paper this can be written in the
shorthand form
12 (read backwards: 2 ones plus 1 ten )
+ 25 (read backwards: 5 ones plus 2 tens)
------ added together gives
37 7 ones plus 3 tens
------
244 ( 4 ones plus 4 tens plus 2 hundreds )
85 ( 5 ones plus 8 tens)
----- added together gives
9 ones plus 12 tens plus 2 hundreds
-----
But the 12 tens is the same as 2 tens plus 1 hundred. Thus
replacement suggests the result
____________________________________\_
/
244 (read 2 ones plus 4 tens plus 2 hundreds ) |
85 (read 5 ones plus 8 tens) |
---------- added together gives |
329 9 ones plus 12 tens plus 3 hundred |
--------- or 9 ones plus 2 tens plus 2+1 hundreds |
1 or 9 ones plus 2 tens plus 3 hundreds \|/
/____________________________________________
\
shorthand + Longhand Representations, respectively,
of the addition
The one in the last row denotes a carry. The on the left and the
words on the right represent the same number. Here the shorthand
expressions requires less work to write, but its justification
requires a knowledge of the longhand form. In the following
examples, we write the shorthand form, then do the calculation
with the help of the longhand representation and lastly return to
the shorthand form. With practice, the long representation and
explanation of the shorthand will be understood, and it need not
be written down. The calculation is then done and represented via
its shorthand representation.
Examples:
Addition without Conversions (Carries)
Steps
- add units
- add tens
- add hundreds
Conclusion: 243 + 452 = 695.
Addition with Conversions (Carries)
Steps
- Add units: 9 and 6 give 15 = 5 + 10 or a 5 in the unit column
with one ten to carry over into the ten's column.
- Add tens: 5 + 2 + the 1 from the carry give 8.
- Add hundreds: 4 + 8 hundreds give 10 + 2 hundreds or 2 in the
hundred column with a one thousand to carry over to into the
thousands column.
- Add thousands. the carry of 1 + 2 + 4 = 7. Done
- Conclusion: 2459 + 4826 = 7285.
Reference: The site section Decimal Arith -
Video Based includes a full development of
arithmetic with decimals for whole numbers.
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Teachers & Tutors: Site pages offer better or best practices for providing skills -
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