unaddition (subtraction) © math as a second language all rights reserved next #3 taking the fear...

Unaddition

(Subtraction)

Unaddition

(Subtraction)

© Math As A Second Language All Rights Reserved

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#3

Taking the Fearout of Math

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“In grades K – 2 , students should understand subtraction as

taking apart and taking from”.


The following is a direct quote taken from the Common Core

Standards concerning subtraction.

Our Point of View

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This tends to have students being taught to read 5 – 3 = 2 as…

Our Point of View

“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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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.

Our Point of View

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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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.

Our Point of View

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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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

Our Point of View

5

3

–

2

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When students check their answer, the check parallels the mathematical definition of subtraction.

Our Point of View

In this way they verify that 2 is the number we add to 3 to obtain 5 as the sum.

5 – 3

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53 To check their answer

they add the bottom number (difference) to the middle number (subtrahend), and the sum should equal the top number (minuend).

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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.

Key Point

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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”

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If that pattern always held, the opposite of “taller” would

be “untaller” instead of “shorter”.


The Prefix “un”

However, in English, knowing the word “taller” does not

mean that you automatically know the meaning of the

word “shorter”.

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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.


The Prefix “un”

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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.

The Prefix “un”

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.

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Let’s look at this in a way that should be relatively easy for students to internalize.

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Consider the following…

You have 3 dollars and your friend gives you 2 dollars.

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next 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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The answer is given in the form of an addition problem.


$3 + $2 = $52

#1 You have $3 and your friend gives you 2 more dollars. How much money do you have now?

note2 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.

In this problem you were given $3 and $2 and asked to find the sum ($5).

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The answer is usually given in the form of a subtraction problem.


$5 – $3 = $2

#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?

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.


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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The answer, this time, is again given in the form of a subtraction problem.


$5 – $2 = $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 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).

next Let’s review and show how this discussion relates to how we used

tiles to perform addition and unaddition.

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Suppose that you want to use a fill-in-the-blank type of question to test

whether students know that 3 + 2 = 5.


3 + 2 = _____

One way is to word the question in the form…

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next In this case if students have memorized the addition tables, they will immediately replace

the blank by the numeral 5

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In terms of tiles, the solution would appear as…


3 + 2 = _____

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=

5

next Another way to write 5 – 3 = 2 would

be to paraphrase the question in the form…

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3 + _____ = 5

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“What number must we add to 3 in order to obtain

5 as the sum?”

If this problem were stated in words, the wording would be…

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Too often students “hear” the problem as if it had been…

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It is important for them to grasp the idea that 5 was the sum, not one of the terms.3


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note3 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.

“What is the sum when we add 3 and 5?”

next 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 acorrect addition problem.

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In words, the problem is asking us to find the number of tiles we must add to


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=

to obtain as the total number of tiles.

next 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.

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=

We then place additional tiles in the bottom row until the two rows have an

equal number of tiles.

next You can now see that we have added

2 tiles to the set of 3 tiles to give us a set of 5 tiles.

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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 advancedtopics later in the

curriculum.


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In any event, this concludes our discussion

of why we prefer to think ofsubtraction 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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unaddition (subtraction) © math as a second language all rights reserved next #3 taking the fear...

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