Transcript 3. Place Value Addition

Taking the Fear
out of Math
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#5
Adding
Whole Numbers
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Addition Through the Eyes of Place Value
The idea of numbers being viewed as
adjectives not only provides a clear
conceptual foundation for addition, but when
combined with the ideas of place value
yields a powerful computational technique.
In fact, with only a knowledge of the
0 through 9 addition tables (i.e. addition of
single digit numbers), our “adjective/noun”
theme and our other rules allow us to easily
add any collection of whole numbers.
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Addition Through the Eyes of Place Value
The main idea is that in our place value
system, numerals in the same column
modify the same noun; therefore, we just
add the adjectives and “keep” the noun
that specifies the place value column.
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To illustrate the idea, let’s carefully
analyze the “traditional” way for how we
add the two numbers 342 and 517.
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According to our knowledge of the
place value representation of numbers,
we set up the problem as follows…
hundreds
tens
ones
3
4
2
5
1
7
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hundreds
3
5
tens
4
1
ones
2
7
In each column, we use the addition
table for single digits. We then solve the
above problem by treating it as if it were
three single digit addition problems.
adjective noun
adjective
3
hundreds
4
5
hundreds
1
8
hundreds
5
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noun
tens
tens
tens
adjective
2
7
9
noun
ones
ones
ones
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Of course, in everyday usage we do
not have to write out the names of the
nouns explicitly since the digits
themselves hold the place of the nouns.
The numbers in the same column modify
the same noun. Thus, we usually write the
solution in the following succinct form…
3 4 2
+ 5 1 7
8 5 9
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Since the nouns are not visible in the
customary format for doing place value
addition, it is important for a student to keep
the nouns for each column in mind.
For example, in reading the leftmost
column of our solution out loud
a student should be saying…
“3 hundred + 5 hundred + 8 hundred”
…rather than just using the adjectives,
as in “3 + 5 + 8.”
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In that way, one reads the answer as…
“8 hundreds, 5 tens, and 9 ones.”
Of course, in more common terminology
(since we usually say “fifty” rather than
“5 tens”), we read the solution as…
“eight hundred fifty-nine.”
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Using the Properties of
Whole Number Addition
3 4 2
+ 5 1 7
In using the traditional format to perform
the above addition problem, you may not
have noticed our subtle use of the
associative and commutative properties
of addition.
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If we use the words “hundreds”, “tens”,
and “ones”, 342 + 517 is an abbreviation
for writing…
(3 hundreds + 4 tens + 2 ones) + (5 hundreds + 1 ten + 7 ones)
However, in using the vertical form of
addition, we actually used the
rearrangement…
(3 hundreds + 5 hundreds) + (4 tens + 1 ten) + (2 ones + 7 ones)
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(3 hundreds + 4 tens + 2 ones) + (5 hundreds + 1 ten + 7 ones)
So whether we did it consciously or not,
the fact is that the vertical format for
doing addition of whole numbers is
justified by the associative and
commutative properties of addition.
(3 hundreds + 5 hundreds) + (4 tens + 1 ten) + (2 ones + 7 ones)
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For example, suppose you have
3 hundred dollar bills, 4 ten dollar bills,
and 2 one dollar bills…
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$100
$100
$100
$10
$10
$10
$10
$1
$1
…and add 5 more hundred dollar bills,
1 more ten dollar bill and 7 more one dollar
bills…
$100
$100
$100
$100
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$100
$10
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$1
$1
$1
$1
$1
$1
$1
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Most likely you would compute the total
sum in the following way…
$100
$100
$100
$100
$100
$100
$100
$10
$10
$100
$10
$10
$10
$1
$1
$1
$1
$1
$1
$1
$1
$1
If you did this, you are using the
commutative and associative properties of
addition.
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Note
Notice the difference between a job being
“difficult” or just “tedious.”
For example, we see from the computation
that follows, it is no more difficult to add, say,
12-digit numbers than 3-digit numbers, it is
just more tedious (actually, more repetitious).
Instead of carrying out three simple
single-digit addition procedures,
we have to carry out twelve.
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In general, no matter how many digits
are in the numbers that are being added,
the process remains the same, but as the
number of digits increases, the process
becomes more and more tedious.
For example, the numbers 234,267,580,294
and 352,312,219,602 are added as follows…
2 3 4, 2 6 7, 5 8 0, 2 9 4
+ 3 5 2, 3 1 2, 2 1 9, 6 0 2
5 8 6, 5 7 9, 7 9 9, 8 9 6
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What Ever Happened to “Carrying”?
Earlier generations used a technique for
adding that was referred to as “carrying”.
Nowadays the technique is more visually
referred to as “regrouping”.
Whichever way we refer to it, the idea
behind it is best explained by our
adjective/noun theme.
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Notice that given a problem such as
finding the sum of 35 and 29, a young
student who just learned how to add two
single digit numbers will often write the
problem in vertical form and treat it
as if it involved two separate single digit
addition problems.
For example, they would add
3 and 2 to obtain 5 and then
add 5 and 9 to obtain 14; and
thus write…
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3 5
+ 2 9
5 14
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This gives the appearance
of obtaining the incorrect
answer, 514. Yet if the
adjective/noun theme is
understood, it is not difficult to
see that 5141 could also be the
correct answer.
3 5
+ 2 9
5 14
note
1
If we wanted to use grouping symbols, we could write 5(14) to indicate that there
are 14 ones and 5 tens but this would quickly become very cumbersome as the
number of digits increases.
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Line 1 and Line 2 in the chart
below provide two different ways to
represent the same amount of money.
Line 1
Line 2
$10 bills
5
6
$1 bills
14
4
However, if the nouns are now omitted,
and all we see is Line 1 in the form
514, there is no way of telling whether we
are naming 5 hundreds 1 ten and 4 ones
or 5 tens and 14 ones.
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The point is that as long as the nouns
are visible it is okay to have more than 9 of
any denomination. However, if we wish, we
may exchange 10 $1 bills for 1 $10 bill.
Thus, when we said such things as…
“5 + 9 = 14,
so we bring down the 4 and carry the 1”…
…we were merely saying that the statement
“5 ones + 9 ones = 14 ones”
means the same thing as the statement
“5 ones + 9 ones = 1 ten and 4 ones”.
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To avoid such ambiguities in which
5 tens and 14 ones can be confused with
5 hundreds, 1 ten and 4 ones, we adopt the
following convention (or agreement) for
writing a number in place value notation…
We never use more than one digit
per place value column.
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Counting on Your Fingers
Myth
As teachers, we often tend to discourage
students from “counting on their fingers”.
We often say such things as “What would
you do if you didn’t have enough fingers?”
The point is that
in place value,
we always have
enough fingers!
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Consider, for example, the
following addition problem…
5286
2959
+1673
9918
Notice that this result could be obtained
even if we had forgotten the addition
tables, provided that we understood place
value and knew how to count.
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5286
Remembering that numbers
2959
in the same column modify
the same noun and using the + 1 6 7 3
associative property of
addition, 3
…we could start with the 6 in the ones
place and on our fingers add on nine
more to obtain 15.
note
3 Up
to now we have talked about the sum of two numbers. However, no matter how
many numbers we are adding, we never add more than two numbers at a time.
For example, to form the sum 2 + 3 + 4, we can first add 2 and 3 to obtain 5 and then
add 5 and 4 to obtain 9. We would obtain the same result if we had first added
3 and 4 to obtain 7 and then add 2 to obtain 9.
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Then starting with 15 we
could count three more
to get 18; after which we
would exchange ten 1’s for
one 10 by saying “bring down
the 8 and carry the 1”.
1
5286
2959
+1673
8
We may then continue in this way, column
by column, until the final sum is obtained.
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More explicitly…
5, 2 8 6
2, 9 5 9
+ 1, 6 7 3
18
20
6 + 9 + 3 ones = 18 ones
8 + 5 + 7 tens
= 20 tens
1700
2 + 9 + 6 hundreds = 17 hundreds
8000
9, 9 1 8
5 + 2 + 1 thousands = 8 thousands
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However, the point we wanted to
illustrate in the previous example is the
following…
Even though there is a tendency to tell
youngsters that “grown ups don’t count
on their fingers”, the fact remains that
with a proper understanding of place
value and knowing only how to count on
our fingers, we can solve any whole
number addition problem.
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In particular at any stage of the
addition process, we are always
adding two numbers,
one of which is a single digit.
One goal of critical thinking is to
reduce complicated problems to a
sequence of equivalent but simpler
ones. Here we have a perfect
example of the genius that goes
into making things simple!
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Same Sum Technique
For students who find it difficult to regroup
(as well as for students who like to see
alternative approaches to problem solving)
the “same sum” technique might pique
students’ interest. It is based on the fact
that the sum of two numbers remains
unaltered if we add a certain amount to
one of the numbers and subtract the same
amount from the other number.
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More concretely, suppose that John
and Mary have a total of 100 marbles and
that John gives Mary 3 of his marbles.
Even though Mary now has 3 more and
John has 3 less, they still have a total of
100 marbles.
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Suppose we want to compute the sum
679 + 298. The problem would have been
much less difficult if it had been 679 + 300.
So what we can do is add 2 to 298 and
subtract 2 from 679. 4
This will not change the sum.
679 + 298 = (679 – 2) + (298 + 2)
677 + 300 = 997
note
4
You might want to postpone this method until after the students have studied
subtraction.
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By using the “same sum” technique and
then adding the numbers in their original
form, students get extra practice with
addition as well as a good opportunity to
internalize the structure of addition.
For example, they could perform the
computation below in the traditional way…
457 + 296 = 753
Then they could add 4 to 296 and subtract
4 from 457, rewriting it in the form…
453 + 300 = 753
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addition
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In the next presentation
we will talk about
various ways to
estimate sums,
especially when many
large numbers are
involved.
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