1. Unaddition (Subtraction)
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Taking the Fear
out of Math
Unaddition (Subtraction) #3
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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 5 – 3 2
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7. When students check their
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Our Point of View
When students check their
answer, the check parallels the mathematical definition of subtraction. 5
– 3 2 5 3
To check their answer they add the bottom number (difference) 2
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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9. the opposite of “even” is “uneven”;
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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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11. In fact people studying English as
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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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13. Consider the following…
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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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14. #2 You need $5 to buy an item, and all you have is
Below are three different problems, all of which are related to addition, but two of which are usually expressed in terms of subtraction.
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#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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15. The answer is given in the form of an addition problem.
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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.
$3 + $2 = $52
In this problem you were given $3 and $2 and asked to find the sum ($5).
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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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16. The answer is usually given in the form of a subtraction problem.
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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. 5 4 3
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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18. The reasoning is similar to that done in the previous problem.
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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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19. Let’s review and show how this discussion relates to how we used
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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 = 5.
One way is to word the question in the form…
3 + 2 = _____
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20. = In this case if students have memorized the addition tables,
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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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21. Another way to write 5 – 3 = 2 would
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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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22. Too often students “hear” the problem as if it had been…
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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
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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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23. to the tiles on the left to display a correct addition problem.
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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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25. You can now see that we have added 2 tiles to the set of 3 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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26. topics later in the curriculum.
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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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