Transcript 2. Unaddition (Subtraction) Using Tiles

Taking the Fear
out of Math
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#3
Unaddition
(Subtraction)
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Our Point of View
The following is a direct quote
taken from the Common Core
Standards concerning subtraction.
“In grades K – 2 , students should
understand subtraction as
taking apart and taking from”.
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Our Point of View
This tends to have students being taught
to read 5 – 3 = 2 as…
“5 take away 3 is 2” or “3 from 5 is 2”.
While we do not disagree with this
concept, we believe it tends to obscure
the taking apart process which is…
5 – 3 is solved by separating 5 into two
parts, 3 and the number that must be
added to 3 in order to obtain 5 as the sum.
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Our Point of View
In fact, this is basically the form in which
mathematicians define subtraction.
Namely, 5 – 3 is the number which must be
added to 3 in order to obtain 5 as the sum.
This definition works better later, when
we have to deal with such computations as
5 – -3. It makes little sense to try to take
“negative 3” away from 5; but it makes a lot
of sense to ask what number we must add
to -3 to obtain 5 as the sum.
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Our Point of View
In terms of the profit/loss model for
viewing signed numbers, 5 – -3 is asking us
what transaction is necessary to convert a
$3 loss into a $5 profit.
It is not difficult to see that one first needs
to make a $3 profit in order to break even
and then another $5 to ensure a $5 profit.
Thus, an $8 profit is needed. In other words,
using this model it is relatively easy for
students to now see that 5 – -3 = 8 and the
question of “taking away” -3 never arises.
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Our Point of View
The fact that the two points of view are
compatible can be seen when we write
subtraction in the traditional vertical form…
5–3=2
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Our Point of View
When students check their
answer, the check parallels the
5
mathematical definition of
– 3 subtraction. To check their answer
2 they add the bottom number
(difference) to the middle number
(subtrahend), and the sum should
equal the top number (minuend).
In this way they verify that 2 is the number
we add to 3 to obtain 5 as the sum.
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Key Point
Our approach is based on our
belief that students will internalize
the concept of subtraction better if
they see it defined in terms of
something they have already
learned, namely, addition.
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The Prefix “un”
By the second grade most students have
seen examples where the prefix “un”
indicates the “opposite”.
For example…
the opposite of “even” is “uneven”;
the opposite of “friendly” is “unfriendly”;
the opposite of “broken” is “unbroken”;
etc.
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The Prefix “un”
If that pattern always held,
the opposite of “taller” would
be “untaller” instead of
“shorter”.
However, in English, knowing
the word “taller” does not
mean that you automatically
know the meaning of the
word “shorter”.
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The Prefix “un”
In fact people studying English as
a second language could very easily
know the meaning of “tall” but not know
the meaning of “short” even though they
understood conceptually that if John
was taller than Bill, Bill was shorter than
John.
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The Prefix “un”
Certainly the word “untaller” suggests
the concept of being shorter much
better than the word “short” does.
In the same way “unadding” suggests
the concept of undoing addition better
than the word “subtraction” does.
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In summary, for addition you are given
two numbers and asked to find their sum,
but for subtraction (unadding) you are
given one number and the sum and asked
to find the other number.
Let’s look at this in a way that should be
relatively easy for students to internalize.
Consider the following…
You have 3 dollars and your friend
gives you 2 dollars.
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Below are three different problems, all
of which are related to addition, but two of
which are usually expressed in terms of
subtraction.
#1 You have $3 and your friend gives you 2 more dollars.
How much money do you have now?
#2 You need $5 to buy an item, and all you have is
3 dollars. How much money would your friend need to
lend you in order for you to purchase the item?
#3 Your friend lends you the $2 you need to buy a $5
item. How much money of your own do you have?
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#1 You have $3 and your friend gives you 2 more dollars.
How much money do you have now?
The answer is given in the form of an
addition problem.
2
$3 + $2 = $5
In this problem you were given $3 and $2
and asked to find the sum ($5).
note
2
In terms of our adjective/noun theme it is important to write the answer as
5 dollars. Simply writing 5 gives no hint as to what the 5 is modifying. If the
problem had asked how many dollars do you have now, it would have been
correct to write 5 because the noun is implied in the question.
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#2 You need $5 to buy an item, and all you have is
3 dollars. How much money would your friend need to
lend you in order for you to purchase the item?
The answer is usually given in the form of
a subtraction problem.
$5 – $3 = $2
When asked how they did the problem,
students often reply that they subtracted
3 from 5. Thus, they were reading
5 – 3 as “5 take away 3”.
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However, our experience tells us
that the way younger students get
the answer is that they start by
saying “$3” and then add $1 at a
time (probably counting on their
fingers) until they get to $5.
2 1
5 4
In this problem, you were given the sum
($5) and one of the terms ($3) and were
asked to find the missing term ($2).
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3
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#3 Your friend lends you the $2 you need to buy a $5
item. How much money of your own do you have?
The answer, this time, is again given in the
form of a subtraction problem.
$5 – $2 = $3
The reasoning is similar to that done in
the previous problem.
In this problem, you were given the
sum ($5) and one of the terms ($2) and
were asked to find the missing term ($3).
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Let’s review and show how this
discussion relates to how we used
tiles to perform addition and unaddition.
Suppose that you want to use
a fill-in-the-blank type of question to test
whether students know that 3 + 2 = 5.
One way is to word the question
in the form…
3 + 2 = _____
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In this case if students have
memorized the addition tables,
they will immediately replace
the blank by the numeral 5
3 + 2 = _____
5
In terms of tiles, the solution would appear
as…
=
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Another way to write 5 – 3 = 2 would
be to paraphrase the question in the form…
3 + _____ = 5
If this problem were stated in
words, the wording would be…
“What number
must we add to 3
in order to obtain
5 as the sum?”
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Too often students “hear” the problem
as if it had been…
“What is the sum when we add 3 and 5?”
It is important for them to grasp the idea
that 5 was the sum, not one of the terms.3
note
3
Too often students are told to look for “key words”. In this case they tend to focus
on the word “add” and the numbers 3 and 5. Since they see the word “add”,
there is a good chance they will add 3 and 5 to obtain 8, which is a correct
answer, but to a different problem. This error will occur even if students have
access to a calculator. In short, there is no substitute for good reading
comprehension, even in the study of mathematics.
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In terms of how we use the tiles to
present 3 + _____ = 5, the problem might
be to determine what we have to add
to the tiles on the left to display a
correct addition problem.
=
In words, the problem is asking us to find
the number of tiles we must add to
to obtain
as the total number of
tiles.
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One way to solve the problem is to start
with the 5 tiles that represent the sum and
then place the 3 tiles under those
5 tiles as shown below.
=
We then place additional tiles in the
bottom row until the two rows have an
equal number of tiles.
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You can now see that we have added
2 tiles to the set of 3 tiles
to give us a set of 5 tiles.
2
You are in the best position to judge
how much of this discussion can be
made meaningful to the students
you are teaching.
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However, whatever you
can succeed in doing
to help the students
now will be a huge help
to them when they
come to grips with
more advanced
topics later in the
curriculum.
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In any event, this
concludes our discussion
of why we prefer to think of
subtraction as being
“unaddition” and we hope
that you will try to convey
this important concept to
your students as early as
possible.
5–3
3 + __
In a subsequent presentation, we will
revisit subtraction in the traditional form.
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