Interpretations of Subtraction
Subtraction Properties and Algorithms
Mental Subtraction Methods
MATH 112
Section 3.2: Understanding Subtraction
Prof. Jonathan Duncan
Walla Walla College
Fall Quarter, 2006
Conclusion
Interpretations of Subtraction
Subtraction Properties and Algorithms
Outline
1
Interpretations of Subtraction
2
Subtraction Properties and Algorithms
Subtraction Algorithms
3
Mental Subtraction Methods
4
Conclusion
Mental Subtraction Methods
Conclusion
Interpretations of Subtraction
Subtraction Properties and Algorithms
Mental Subtraction Methods
Conclusion
Set Model of Subtraction
Just as with addition, subtraction can be modeled in several
different ways. Again, sets provide one of the most basic models.
Example
Jane had 7 oranges. She gave 3 of them to Billy. How many
oranges does she have left?
The “take away” model is easy to see and works well when the
problem is phrased in a particular way.
Example
Jane has 7 oranges. Billy has 3 oranges. How many more oranges
does Jane have than Billy?
Interpretations of Subtraction
Subtraction Properties and Algorithms
Mental Subtraction Methods
Conclusion
Number Line Model of Subtraction
Another model of subtraction which works well for certain
applications is the number line model.
Example
Sam walked 6 miles home from school and passed a candy store on
the way. After raiding his piggy bank, Sam walked one mile back
to the candy store. How far is the candy store from Sam’s school?
In this problem, it is perhaps easier to see that we can define subtraction in terms of addition. Subtraction “undoes” addition.
Subtraction and Addition
For whole numbers a, b, and c the following are equivalent.
c −b =a
a+b =c
Interpretations of Subtraction
Subtraction Properties and Algorithms
Mental Subtraction Methods
Bringing the Models Together
So if subtraction and addition are tied together, their models
should be tied together as well.
Modeling Addition
In general, a + b is modeled by combining two parts to make a
whole as shown below.
Modeling Subtraction
In general, a + b is modeled by combining two parts to make a
whole as shown below.
Conclusion
Interpretations of Subtraction
Subtraction Properties and Algorithms
Mental Subtraction Methods
Conclusion
Properties of Subtraction
In the last section, we used algebra to express several properties of
addition for whole numbers. What about subtraction?
Properties of Subtraction
For whole numbers a, b, and c which of the following properties
which were true for subtraction are also true for addition?
Identity Property: 0 − a = 0 − a = a
Commutative Property: a − b = b − a
Associative Property: (a − b) − c = a − (b − c)
Closure: a − b is another whole number
Counting: a − 1 is the previous whole number
Solutions
None but the last of the properties above are true of subtraction.
Interpretations of Subtraction
Subtraction Properties and Algorithms
Mental Subtraction Methods
Modeling Base 10 Subtraction
As you have seen in your lab work, we can use manipulatives to
model numerals (think of flats, longs, and units) and operations
such as subtraction.
Example
Use base 10 blocks to model each subtraction problem.
-
2
1
4
7
7
1
+
0
3
7
2
7
6
Conclusion
Interpretations of Subtraction
Subtraction Properties and Algorithms
Mental Subtraction Methods
Conclusion
Subtraction Algorithms
The Standard Algorithm
As with addition, there are several different algorithms for
subtraction. Most of you have probably learned the “standard
algorithm” in school.
Standard Subtraction Algorithm
In the standard algorithm, subtraction is done column-by-column starting
with the right-most column. If the bottom digit is larger than the top
digit, then we “borrow” one from the digit at the top of the next column
to the left and add 10 (or whatever the base) to the top digit in the
current column.
Example
Use the standard addition algorithm to find the following
difference.
-
4
1
8
8
2
4
Interpretations of Subtraction
Subtraction Properties and Algorithms
Mental Subtraction Methods
Conclusion
Subtraction Algorithms
Pay-Back Subtraction
Another popular method for subtracting is the “pay-back” method.
Instead of borrowing from the top digit in the next column, we pay
back the bottom digit in the next column.
Pay-Back Subtraction Algorithm
In the pay-back algorithm for subtraction, we start with the right-most
column and subtract the bottom digit from the top. If the bottom digit
is larger than the top, we add 10 (or our base) to the top and continue
the subtraction. Then, we pay-back our use of 10 by adding one to the
bottom digit in the column to the left.
Example
Use the pay-back subtraction algorithm to find the following
difference.
-
4
1
8
8
2
4
Interpretations of Subtraction
Subtraction Properties and Algorithms
Mental Subtraction Methods
Conclusion
Subtraction Algorithms
Indian Subtraction Algorithm
The Indian algorithm uses borrowing just as does the standard
algorithm. However, there are two major differences–we subtract
from left-to-right and “borrow” from the answer.
Indian Subtraction Algorithm
In the Indian subtraction algorithm subtraction is performed from right to
left. If the bottom digit of a column is larger than the top digit, then the
one is is “borrowed” from the solution to the previous column and 10 (or
whatever the base) is added to the top digit in the current column.
Example
Use the Indian subtraction algorithm to find the following
difference.
-
4
1
8
8
2
4
Interpretations of Subtraction
Subtraction Properties and Algorithms
Mental Subtraction Methods
Conclusion
Subtraction Algorithms
The Multicultural Subtraction Method
The multicultural subtraction method has been used in a wide
variety of places and times. According to your text this method
was used in 15th century Italy and is currently used in Japan.
Multicultural Subtraction Algorithm
In the multicultural subtraction algorithm subtraction is performed in
columns from right-to-left. If the bottom digit is smaller than the top,
then subtraction is performed as normal. If the bottom digit is larger
than the top digit, then the bottom digit is subtracted from 10 (or
whatever the base) and that difference is added to the top digit. Finally,
the top digit of the next column over is reduced by one.
Example
Use the multicultural algorithm to find the following.
-
4
1
8
8
2
4
Interpretations of Subtraction
Subtraction Properties and Algorithms
Mental Subtraction Methods
Conclusion
Algorithms for Mental Addition
As with addition, many of the algorithms we use to perform
subtraction with pencil and paper do not work as well when
subtracting without these tools. In the next few slides we will look
at several ways to subtract mentally.
Example
Find each of the differences in your head as quickly and accurately
as possible.
1
65 − 28
2
42 − 19
3
184 − 125
Interpretations of Subtraction
Subtraction Properties and Algorithms
Mental Subtraction Methods
Conclusion
Adding Up Method
Since addition and subtraction are closely related, it makes sense
that we might be able to use addition to make a subtraction
problem easier.
Example
To subtract 65 − 28 using the adding up method, follow these
steps:
We want to know “what do we add to 28 to get 65?”
Adding 28 + 40 yields 68 which is close to 65.
Since 8 − 5 is 3, we need to add 40 − 3 = 37 to 28 to get 65.
Example
Try this method with the other differences: 42 − 19 and 184 − 125.
Interpretations of Subtraction
Subtraction Properties and Algorithms
Mental Subtraction Methods
Conclusion
Add to Nearest 10
Since our system of numeration is base 10, multiples of 10 are
particularly easy to work with in addition and subtraction problems.
Example
To subtract 65 − 28 using the add to the nearest 10 method,
follow these steps:
First note that 28 is 2 less than 30 (the nearest 10).
Next, 30 + 35 yields 65 which is the number we are
subtracting from.
Finally, 35 + 2 = 37 giving the amount we must add to 28 to
get 65.
Example
Use the nearest 10 method to find the other differences: 42 − 19
and 184 − 125.
Interpretations of Subtraction
Subtraction Properties and Algorithms
Mental Subtraction Methods
Conclusion
Piecemeal Subtraction
As with the break and bridge addition method, the piecemeal
subtraction method emphasizes breaking numbers up to make the
process easier.
Example
To subtract 65 − 28 using the piecemeal method, follow these
steps:
First break 28 into 20 and 8.
Next subtract 20 from 65 to get 45.
Finally, subtract the remaining 8 from 45 yielding 37.
Example
Use piecemeal subtraction method to find the other differences:
42 − 19 and 184 − 125.
Interpretations of Subtraction
Subtraction Properties and Algorithms
Mental Subtraction Methods
Conclusion
Using Negative Numbers
So far we have only been working with whole numbers. However,
negative numbers can allow us to more quickly carry out
subtraction problems.
Example
To subtract 65 − 28 using negative numbers, follow these steps:
Subtract the first digits, 6 − 2 = 4.
Next subtract the second digits, 5 − 8 = −3.
Finally, put 40 and -3 together to get 37.
Example
Use negative numbers to find the other differences: 42 − 19 and
184 − 125.
Interpretations of Subtraction
Subtraction Properties and Algorithms
Mental Subtraction Methods
Conclusion
Estimating Addition and Subtraction
In many instances we do not necessarily need the exact answer to
an addition or subtraction process–an estimate is close enough.
Example
Use strategies seen in the mental addition/subtraction sections we
have covered to estimate each sum or difference.
1
5321 − 4152
2
75, 145 + 34, 135 + 55, 124
3
38 + 72 + 89 + 65 + 27
Interpretations of Subtraction
Subtraction Properties and Algorithms
Mental Subtraction Methods
Important Concepts
Things to Remember from Section 3.2
1
Ways to model subtraction
2
Properties of subtraction
3
Alternative subtraction algorithms
4
Mental subtraction algorithms and estimations
Conclusion