Download (Division) Using Tiles

Taking the Fear
out of Math
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#3
Unmultiplication
(Division)
Using Tiles
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Our Point of View
In the same way that the term
“unadding” leads to a better
understanding of the relationship
between adding and subtracting, the
term “unmultiplying” leads to a better
understanding of the relationship
between multiplying and
dividing.
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Our Point of View
To find the answer to a “question” such
as…
Find the product of 4 and 3
(or in terms of fill-in-the-blank 4 × 3 = ____).
…we multiply 4 and 3 to obtain 4 × 3 =
12.1
note
1
When taught to look for “key” words, students are often told to interpret “and” as
indicating addition. However, “and” is also a conjunction, and when we say
“Find the product of 4 and 3, the term “product” is telling us to multiply 4 and
3. So while it is correct to say that 4 + 3 = 7, it is incorrect to say that the
product of 4 and 3 is 7 when in fact it is 12.
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Our Point of View
On the other hand to find the answer to a
question such as…
4 × ____ = 12
…we unmultiply 12 by 4 to obtain 12 ÷ 4 =
3.
In words, 4 × ____ = 12 is asking us to
find the number we must multiply by 4 to
obtain 12 as the answer.
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Our Point of View
Using more user-friendly vocabulary,
the question is asking us “How many
4’s do we have to add to
obtain 12 as the sum?”,
or, in slightly more formal language,
“12 is what multiple of 4? 2”
note
2
A multiple is the product of a given number and any whole number.
The multiples of 4 would be 4, 8, 12, 16, 20, 24, 28, 32, 36, 40…
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If we represent the numbers by tiles
and use our idea of rectangular arrays, we
count by 4’s until we get to 12.
1 2 3 4
5 6 7 8
9 10 11 12
In other words, each row of tiles
represents 4, and 4 + 4 + 4 = 12.
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1 2 3 4
5 6 7 8
9 10 11 12
On the other hand, looking at the columns
in the above array, we see that there are
4 columns, each with 3 tiles.
In other words, we have to add four 3’s in
order to obtain 12 as the sum.
In the language of unmulitplying (dividing),
this means that 12 ÷ 4 = 3.
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The Mathematical Definition of Division
12 ÷ 3 is the number (the quotient) by
which we must multiply 3 (the divisor) in
order to obtain 12 (the dividend) as the
product.
In other words,
12 ÷ 3 is the answer to 3 x _____ = 12.
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More generally, if b and c are numbers,
b ÷ c is the number by which we must
multiply c in order to obtain b as the
product. In other words, b ÷ c is the
answer to _____ × c = b.1
note
is an excellent way to motivate the “invention” of fractions.
For example, it is possible to divide $12 equally among 5 people, but the answer is
not a whole number. Since 2 × 5 = 10, 2 is too small to be the correct
answer, and since 3 × 5 = 15, 3 is too great to be the correct answer. And
since there are no whole numbers between 2 and 3, the correct answer must
be a fractional number of dollars (and, in fact, the exact answer is $2.40).
1 This
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In summary, as shown below,
each multiplication fact leads
to two division facts.
For example…
3 rows each with 4 tiles
or
3 × 4 = 12 or 12 ÷ 3 =
4
4 columns each with 3 tiles
or
4 × 3 = 12 or 12 ÷ 4 = 3
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1 2 3 4
1 2 3 4
1 2 3 4
1 1 1 1
2 2 2 2
3 3 3 3
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In a similar way the diagram below
illustrates the “chain”…
5 × 6 = 30
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30 ÷ 5 =
6
1 2 3
7 8 9
2
13
3 14 15
19
4 20 21
25
5 26 27
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30 ÷ 6 = 5
4
10
16
22
28
5
11
17
23
29
6
12
18
24
30
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1
1
2
1
3
1
4
1
5
1
2
2
2
3
2
4
2
5
1
3
3
2
3
3
4
3
5
1
4
4
2
4
3
4
4
5
1
5
5
2
5
3
5
4
5
1
6
6
2
6
3
6
4
6
5
5 rows each with 6 tiles or
5 × 6 = 30 or 30 ÷ 5 = 6
6 columns each with 5 tiles or
6 × 5 = 30 or 30 ÷ 6 = 5
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All of the discussion in this
presentation can be illustrated in terms of
four questions based on the following
information…
You buy 6 pens, each of which cost $5.
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Question #1
You buy 6 pens, each of which cost $5.
How much did you pay for the pens?
In this case, we are spending $5 six times.
So we multiply to obtain…
6 × 5 dollars = 30 dollars.
In this question, we were given both
numbers and multiplied them to find
their product.
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Question #2
You spend $30 to buy 6 equally-priced pens.
How much did each pen cost?
In this case, one way to represent the
answer is by…
30 (dollars) ÷ 6 (pens) = 5 (dollars per pen)
In terms of unmultiplying, we wanted to
find out what we had to multiply
6 by in order to get 30 as the product.
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Question #3
You spend $30 to buy pens that cost $5 each.
How many pens didyou buy?
In this case, we can represent the answer
in the form…
30 (dollars) ÷ 5 (dollars per pen) = 6 (pens)
In terms of unmultiplying, we wanted to
find out what we had to multiply
5 by in order to get 30 as the product.
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Question #4
You spend $30 to buy equally-priced pens.
How many pens did you buy, and how much
did each pen cost?
In this case, we can represent the answer
in the form…
30 (dollars) ÷ ? (dollars per pen) = ?
(pens)
This is the situation in which
factorization is needed. There are as
many answers as there are divisors
(factors) of 30.
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Question #4
More specifically, the factors of 30 are
1, 2, 3, 5, 6, 10, 15 and 30.
1 pen @ $30
The possible 2 pens @ $15
answers are…
3 pens @ $10
5 pens @ $6
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6 pens @ $5
10 pens @ $3
15 pens @ $2
30 pens @ $1
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A Closing Note
In later grades, the children will represent
12 ÷ 3 in the form…
3 12
Some students run into trouble by reading
the above division problem as 3 ÷ 12
because in the above array, the 3 is to the
left of 12 and they tend to read from left to
right. Thus, they might write such things
as 3 ÷ 12 = 4.
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A Closing Note
To avoid this type of misinterpretation, in
some cultures rather than write…
4
3 12
they write it in the form…
4
12
3
thus, preserving the same order as in 12 ÷ 3.
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A Closing Note
The main point is that it is crucial for the
students to understand that 3 ÷ 6
is not the same as 6 ÷ 3.
Namely, it makes a difference
in how much pie each person gets,
if we divide 3 pies equally among
6 people or 6 pies equally among 3 people.
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15 ÷ 3
This completes our present
discussion. In our next
presentation, we will
show how the tiles can be
used to introduce some
elementary number
theory.
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3 × _ = 15