Transcript Mathematics as a Second Language

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Mathematics
as a
Second Language
Arithmetic Revisited
© 2010 Herb I. Gross
Developed by
Herb I. Gross and Richard A. Medeiros
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Lesson 2 Part 3.1
Whole Number
Arithmetic
Multiplication
© 2010 Herb I. Gross
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Multiplication of Whole Numbers
Understanding the Algorithm
When dealing with whole numbers, we have
already seen that subtraction is a form of
addition that we can view as “unadding”. In a
similar vein, we may also view multiplication
of whole numbers as a form of addition.
More specifically, using place value and our
“adjective/noun” theme, we can show that in
place value format the multiplication
algorithm is rapid, repeated addition.
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We will assume that most students know
the tables and the traditional algorithm(s)
for multiplication.
However, if students memorize the
algorithms without properly understanding
them some very serious errors involving
critical thinking can occur.
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Note
There is a raging debate about
whether calculators hamper
students’ efforts to understand mathematics.
We should keep two points in mind.
For one thing, if the traditional methods had
been as successful as its advocates would
have us believe, there might not have been
a need for new standards and educational
“reform”.
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Secondly, it is possible that students, if
they do not fully understand what
multiplication means, will not recognize
when the answers they obtain are
unreasonable. Moreover, this applies in
equal measure to those students who use
calculators as well as to those who learn
algorithms by rote.1
note
1Prior
to the advent of calculators, it was crucial for students to learn the arithmetic
algorithms in order to compute. It was not as great a priority for students to
understand the logic behind the algorithm. In other words, accuracy was more
important than understanding. Nowadays, however, calculators can do the required
arithmetic and as a result more emphasis is now being placed on understanding and
applications.
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Thus, we will accept the fact that most
students can perceive that they
have correctly assimilated the various
multiplication algorithms by rote; yet
because of subtleties that they overlook,
they often make serious errors by not
understanding each step in the process.
For example, in computing a product
such as 415 × 101, they often
disregard the 0 because it is “nothing”.
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In this context, they might write…
415
×101
415
+415
4565
2
415
× 11
415
+415
4565
note
that even though the problem is written as 415 × 101, the computation was
done as if the problem had been 415 × 11 (which is 4,565). Namely, in terms of our
adjective/noun theme by placing the 5 under the 1, we multiplied 415 by 10 rather
than by 100.
2Notice
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By understanding the multiplication
algorithm students should realize that
415 × 101 must be greater than 415 × 100;
and since 415 × 100 = 41,500 it is clear that
415 × 101 > 41,500.
There are many numbers that are greater
than 41,500 but 4,565 isn’t one of them.3
note
is an important learning device, especially when we don’t know the correct
answer, to realize that some incorrect answers are less plausible than others.
Thus, recognizing what can’t be the correct answer can often serve as a clue for
finding the correct answer.
3It
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Notes on Multiplying by a Power of 10
In Roman numerals, we multiply by ten by
changing each I to an X; each X to a C; and
each C to an M etc.
Thus, to multiply XXXII by ten, we would
change each X to a C and each I to an X,
thus obtaining CCCXX…
C
X C
XC
X XI X
I
(32)
(320)
…which corresponds to 10 × 32 = 320.
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Notes on Multiplying by a Power of 10
We obtain the same result in place value by
placing a 0 after the 2 to obtain the fact that
10 × 32 ( = 32 × 10) = 320. More specifically,
the 0 moves the 2 from the ones place to the
tens place and the 3 from the tens place to
the hundreds place.4
note
4Notice
in this situation that 0 is not “nothing”. Rather it is used as a place holder
that changes the nouns that 3 and 2 modify. That is by annexing the zero, the 2 now
modifies tens instead of ones, and 3 now modifies hundreds instead of tens.
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Notes on Multiplying by a Power of 10
In terms of our adjective/noun theme…
10 × 32 = 32 × 10 = 32 tens = 320
100 × 32 = 32 × 100 = 32 hundreds = 3,200
1,000 × 32 = 32 × 1,000 = 32 thousands = 32,000
And in this same vein…
100 × 415 = 415 hundreds = 41,500
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The Origin of the “>” and “<” Signs
Because the equal sign (=) consists of two
parallel lines, the spaces at each end are the
same. Thus the symbolism was that since
the spaces are equal the numbers at either
end are also equal.
To indicate that 3 is less than 4,
the two lines in the equal sign were pinched
closer together at one end, with the
understanding that the smaller number was
placed next to the smaller space.
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The Origin of the “>” and “<” Signs
Because the equal sign was often written
quickly, it was difficult to distinguish
between a “sloppy” equal sign and the “less
than” sign. To avoid confusion, the two
lines at the smaller end were pinched
together so that there was no space between
them. Thus, 3 < 4 is an abbreviation for “3
is less than 4”.
The thing to remember is that the smaller
space (the closed end) is next to the
lesser number.
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The Origin of the “>” and “<” Signs
In this way we have two choices for writing
that 3 is less than 4; namely, either 3 < 4 or
4 > 3. In either case the closed end is next
to the lesser number; and when written in
the form 4 > 3, we usually we usually read it
as “4 is greater than 3”.
A common memory device (but hardly a
logical device) is to memorize that the
“arrow head” points to the lesser number.
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Note
Using a calculator does not
make us immune from
making the error described
previously.
More specifically, even with a calculator we
can strike a key too lightly to have it
register, and we can also type a number
incorrectly. So even when we use a
calculator to compute 415 × 101, we should
still be aware of such “advance information”
as 415 × 101 > 41, 500.
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An easier but not as accurate
estimate is to observe that since
400 < 415 and 100 < 101,
400 × 100 < 415 × 101.
And since
400 × 100 = 40,000,
it means that
415 × 101 > 40,000.
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There are more “concrete” explanations
that can be helpful to the more visually
oriented students.
For example, in terms of money (and
this is something all students can relate to),
the person who has 101 checks each worth
$415 has $415 more than the person who
has only 100 checks that are worth
$415 each. 5
note
make the mistake of saying that since 101 is 1 more than 100; 415 ×101 is
1 more than 415 × 100. Remember that in terms of the adjective/noun theme, he only
has 1 more check but 415 more dollars.
5Don't
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As a way to utilize a little geometry,
we can think in terms of the area
of a rectangle.
© 2010 Herb I. Gross
101 ft
415 feet
101 ft
415 ft
100 ft
For example, the rectangle whose
dimensions are 415 feet by 101 feet has a
greater area than the rectangle whose
dimensions are 415 feet by 100 feet.
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Multiplication of Whole Numbers
or Rapid, Repeated Addition
In order to understand the traditional
whole number multiplication algorithm,
students should be nurtured to
understand the concept of rapid,
repeated addition.
Here is a possible simple starting point…
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Suppose you are buying 4 boxes of candy
that cost $7 each. We could think of asking
two questions based on this information…
(1) “How many boxes of candy did you buy?” In
this case we can see directly that the adjective 4
is modifying the noun phrase “boxes of candy”.
(2) “How much did the 4 boxes of candy cost?”
Explicitly, the 4 is still modifying “boxes of
candy”, but to answer the question we see
that it is being used to tell us how many times
we are spending $7.
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The mathematical way of writing “$7 four
times is to write 4 × $7. 4 × $7 is called the
4th multiple of 7 (dollars).6
More generally, no matter what number 7
modifies, 4 × 7 is called the 4th multiple of 7.
note
6It
is important to have students internalize the concept of multiple. As a preliminary step
it might help to define the 4th multiple of 7 as the fourth number we come to if we are
“skip counting” by 7’s. In this context, even if we do not know the answer in place value
notation, 127 × 378 is the 127th number we come to if we are skip counting by 378's while
378 × 127 is 378th number we come to if we are skip counting by 127’s.
More generally, the product of any two numbers is a common multiple
of the two numbers.
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In the expression 7 × 4…
4 and 7 are called the factors;
and 4 × 7 is called the product of 4 and 7.
Note
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We tend to confuse 4 × 7 with
7 × 4. The fact is that while the
product in both case is 28, the
concepts are quite different.
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Note
For example, we are viewing 4 × 7
as the 4th multiple of 7; that is,
7 + 7 + 7 + 7; and we are viewing
7 × 4 as the 7th multiple of 4; that
is, 4 + 4 + 4 + 4 + 4 + 4 + 4.
Clearly these two sums look different!
In fact, it might be an interesting
experiment to ask students questions such
as the following one…
Which names the greater sum…
4 + 4 + 4 + 4 + 4 + 4 + 4 or 7 + 7 + 7 + 7?
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Note
For “self evident” reasons,
students are willing to accept the
fact that 4 × 7 = 7 × 4.
However, there is a conceptual
difference between buying 7 pencils at $4
each and buying 4 pencils at $7 each (even
though the cost is the same in both cases).
$4
each
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$7
each
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$4
each
4+4+4+4+4+4+4
$7
each
7+7+7+7
In fact, the previous question addresses
this issue. Notice that the first group is the
answer to the cost of buying seven pencils
at $4 each; and that the second group is
the answer to the cost of buying four
pencils at $7 each. Certainly, the two events
are quite different even though the total
cost is the same in both cases.
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In terms of our “adjective/noun” format,
when we write 4 + 4 + 4 + 4 + 4 + 4 + 4,
we may view 4 as the noun (because that’s
the digit we see) and 7 as the adjective
(because that’s the number of times
4 appears).
However, when we write 7 + 7 + 7 + 7,
it is 7 that plays the role of the noun
(because that’s the digit we see), and 4 plays
the role of the adjective (because that’s the
number of times 7 appears).
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We shall adopt the notation 4 × 7 to mean
the 4th multiple of 7; that is, 7 + 7 + 7 + 7.
If we mean 4 + 4 + 4 + 4 + 4 + 4 + 4,
we will write 7 × 4.7
The reason is somewhat related to the fact ‚
that we usually say such things as “four
apples” rather than “apples four”.8
note
7This
agreement is used in algebra as well. For example, if we want to solve the
equation 7x = 21, we divide both sides by 7 to obtain x = 3. In essence we were treating
7 as the adjective and x as the noun. That is we were saying
that if seven x's are worth 21, each x is worth 3.
note
it's important to distinguish the difference in meaning between 7 × 4 and 4 × 7;
as far as adjectives are concerned, there is no harm done in confusing one notation
with the other because as adjectives, 4 × 7 = 7 × 4 = 28.
8While
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The Commutative Property
of Multiplication
Under the heading of “a picture is worth
a thousand words”, notice how easy it is
by rearranging tally marks (which we’ve
written as “dots” for aesthetic reasons)
to see why 7 × 4 = 4 × 7.
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Namely, we may view 4 × 7 as a rectangular
array consisting of 4 rows, each of which
contains 7 “dots”. That is…
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
However, if we look at the columns rather
than at the rows, we see 7 columns, each of
which contains 4 “dots”.
In other words 4 rows of 7 “dots” is the same
number of “dots” as 7 columns of 4 “dots” (28).
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Youngsters sometimes visualize
square tiles more readily than they do
dots or tally marks.
Hence, by using a rectangular array of
tiles we could indicate why 4 × 7 = 7 × 4.
7×4
4×7
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The Area Model
height
height
4
7
base 7
base 4
Here we have another application to geometry
(area). Namely, the two rectangles above have
the same area. To find the area we multiply
the base by the height. In the first rectangle,
the base is 7 and the height is 4, and in the
second rectangle the base is 4, and the height
is 7. Hence, 7 × 4 = 4 × 7.
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More generally, the product of two
numbers does not depend on the order in
which the two numbers are written.
This is known as…
The Commutative Property
for Multiplication.
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Classroom Activity
With respect to our last
observation, the
following type of
question
might make a good
classroom activity.
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Classroom Activity
Ask the students to compute,
the sum of ten 2’s, written as…
2+2+2+2+2+2+2+2+2+2
and see how long it takes them to discover
that all they had to do was to “annex” a 0
after the 2; that is, 2 × 10 = 10 × 2 = 20.9
note
have to make sure that students do not just “blindly” annex a 0. Otherwise in the
study of decimals they may make such errors as saying that 0.832 × 10 = 0.8320 rather
than realizing that 0.832 × 10 = 8.32.
9 We
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Classroom Activity
Then ask them to compute a sum such as…
2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2…+ 2
100 terms
…and see how long it takes them to discover
that the sum is equal to 2 ×100 or 200.
Such exercises should help students see
why when we multiply a whole number by
100, we simply have to annex two 0’s.
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Continuing with Multiplication
It is not a particularly noteworthy saving
of time to write, for example, 4 × 7
in place of 7 + 7 + 7 + 7. However, with
respect to our “boxes of candy”
situation, suppose we wanted to buy
400 boxes at a cost of $7 per box.
It would indeed be very tedious to write
explicitly the sum of four hundred 7’s;
that is....
7 + 7 + 7... + 7
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400 terms
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Note
The above problem, when stated
as a multiplication problem,
should be written as 400 × 7.
Writing the problem as 7 × 400
gives us the equivalent but
simpler addition problem…
400 + 400 + 400 + 400 + 400 + 400 + 400
7 × 400
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Note
However, this obscures the fact
that we want the sum of
four hundred 7’s; not the sum of
seven 400’s.
That is, while the answer is the same,
the “mental image” is quite different.10
note
10Again
notice the difference between buying 400 items at $7 each and buying 7 items
at $400 each. The total cost is the same in both cases, but the event is different.
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Using the adjective/noun theme there is
another way to visualize the “quick way” of
multiplying by 400. Namely, once we know
the “number fact” that 4 × 7 = 28, we also
know such facts as…
4 × 7 apples = 28 apples,
4 × 7 lawyers = 28 lawyers,
4 × 7 hundred = 28 hundred.
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The latter result, stated in the language of
place value (namely we replace the noun
“hundred” by annexing two 0’s) says that
4 × 700 = 2,800.
In other words,
once we know that 4 × $7 is $28,
we also know that 400 × $7 is $2,800.
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This observation gives us an insight to
rapid addition. As an example, let’s
make a multiplication table for a number
such as 13 which isn’t usually
included as part of the traditional
multiplication tables.
The idea is that we can think of 13
as being an abbreviation for the sum of
1 ten and 3 ones.
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Thus, a “quick” way to add thirteen is to
add 1 in the tens place and then
3 in the ones place.
For example, starting with 13, we add 10 to
get 23 and then 3 ones to get 26.
Starting with 26 we add 10 to get 36 and
then 3 to get 39. Adding 10 to 39 gives us
49 and then adding 3 gives us 52.
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Continuing in this way, it is easy to see
that…
1 × 13 = 13
2 × 13 = 13 + 10 + 3 = 26
3 × 13 = 26 + 10 + 3 = 39
4 × 13 = 39 + 10 + 3 = 52
5 × 13 = 52 + 10 + 3 = 65
6 × 13 = 65 + 10 + 3 = 78
7 × 13 = 78 + 10 + 3 = 91
8 × 13 = 91 + 10 + 3 = 104
9 × 13 = 104 + 10 + 3 = 117
10 × 13 = 117 + 10 + 3 = 130
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From the chart we see that at $13 each,
9 items would cost $117.
The problem with this approach is that it
would be cumbersome, to say the least, to
continue to go row-by-row to find the cost
of, say, 234 items that cost $13 each.
In the next part of this lesson, we will
demonstrate how our adjective/noun theme
allows us to compute such products as
234 × 13 just by knowing the first nine
multiples of 13.
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Classroom Application
Students often have trouble
learning the multiplication
tables. One reason for this is
that the process is not much
fun for them. However, there
is a bit of mathematical
humor that often shows
students a “fun” way to learn
the “9’s table”.
This method appears on the next slide.
© 2010 Herb I. Gross
Johnny is being tested on the 9’s
table…
1×9=
He knows that 9 × 1 = 9
2×9=
Not knowing the other
3×9=
answers, he feels he
4×9=
should help the teacher
5×9=
correct his paper. So
6×9=
he counts the number
7×9=
he has wrong.
8×9=
To check his work he
9×9=
now counts starting
from the bottom of the
list.
next
9
18
27
36
45
54
63
72
81
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Obviously,
Johnny is amazed
when he gets his
paper back with a
grade of 100.
A Possible Teaching Moment
Our experience shows that students enjoy
seeing this “trick”. It might be nice to have
them explain why this trick works, and why
it doesn’t work for the other digits in the
multiplication tables.
© 2010 Herb I. Gross