Transcript Section 3.5

Chapter 3
Whole Numbers
Section 3.5
Algorithms for Whole-Number Addition
and Subtraction
In this section we look at some algorithms for adding and subtracting whole
numbers. An algorithm is a method to do something, in this case addition and
subtraction. We will look at the standard way you learned to add and subtract and
some alternatives.
What is common to all these arithmetic algorithms is that they involve breaking
numbers up into parts (digits), doing something with each part (digit), then putting
it together to get the overall answer.
Addition Algorithms
+
0
1
2
3
4
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7
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0
0
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2
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4
5
6
7
8
9
1
1
2
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4
5
6
7
8
9
10
2
2
3
4
5
6
7
8
9
10
11
3
3
4
5
6
7
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9
10
11
12
4
4
5
6
7
8
9
10
11
12
13
5
5
6
7
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10
11
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13
14
6
6
7
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12
13
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15
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9
10
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12
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16
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8
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10
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13
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9
9
10
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12
13
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The addition problems shown to the
right are the basic addition facts.
Those are the addition problems
that show how all the single digit
numbers are added to all other
single digit numbers. These facts
are eventually memorized through
frequent use and from some
techniques such as flash cards.
The addition algorithm you remember from school is a breaking apart type of
method. It is known as the partial sums algorithm and is based on knowing the
basic addition facts and the concept of carrying.
45
=
4  10 +
5
Write in expanded form (break apart)
+ 38
=
3  10 +
8
=
7  10 +
13
Use the basic addition facts
=
7  10 + 1  10+ 3
Write 13 in expanded form
carrying
=
(7+1)10 +
3
Distributive property of mult over addition
=
8  10 +
3
Basic addition Fact
83
This can also be represented in terms of Dienes blocks by making a series of
trades.
45
38
83
4 longs
3 longs
7 longs
5 units
8 units
13 units

=
Trade 10
units for
1 long
8 longs
3 units
=
Error Patterns in Addition
As teachers you would be expected to analyze a series of problems that students
have done and be able to identify the mistake they are making. Look at the
addition problems below and see if you can spot the mistake that is being made.
Show what the child would get if they did the next problem and continue to make
the same mistake.
What mistake is made here?
127
+ 54
12754
49
+ 78
4978
453
+247
453247
They are writing the numbers
together one after the other.
This is called an amalgamation
error.
What mistake is made here?
127
+ 54
49
+ 78
453
+247
1711
1117
6910
They are not carrying the tens
digit up to the next place value.
Subtraction Algorithms
The standard subtraction algorithm that you learned is known as the partial
differences algorithm. It also breaks apart the number to do smaller number
computations.
56
=
5  10 +
6
write in expanded form
- 32
=
3  10 +
2
(5-3)  10 +
6–2
subtract corresponding digits
2  10 +
4
subtraction facts
24
This can also be represented using Dienes blocks.
56
subtract 3
long and 2
units
leaves 24
The difficulty for subtraction is when you want to take a larger digit away from a
smaller digit. (i.e. the digit on the bottom is larger than the digit on the top) This
is when we need to introduce the concept of borrowing.
Subtraction With Borrowing
We look at when we want
56
=
5  10
- 27
=
2  10
= (4 + 1)  10
=
2  10
=
4  10
=
2  10
=
4  10
=
2  10
= (4 – 2)  10
=
2  10
29
to do the following
+
6
+
7
+
6
+
7
+ 1 10+6
+
7
+
16
+
7
+ (16 – 7)
+
9
type of subtraction problem.
write in expanded form
basic addition fact
distributive property
expanded form in reverse
subtraction fact
Using Dienes blocks we get:
trade 1 long
for 10 units
56
borrow
subtract 2
longs and 7
units
leaves 2 longs
and 9 units
Equal Addends Subtraction Method
The equal addends subtraction method is an alternate method to the standard
subtraction that eliminates the need for borrowing. It adds the same amount to
both numbers so that the need to borrow is eliminated.
The idea is when you have a digit in the number you are subtracting that is larger
than the number you are subtracting from add the correct number to both to get
the digit in the number being subtracted to be 0. This is slightly different than the
way that is demonstrated in the book, but I think it is easier.
Example:
246
- 158
+2=
+2=
248
- 160
+ 40 =
+ 40 =
288
- 200
88
Lets do the following problems with the equal addends subtraction method.
5164
- 327
+3=
+3=
2629
- 845
+ 60 =
+ 60 =
5167
- 330
2689
- 905
+ 700 =
+ 700 =
+ 100 =
+ 100 =
5867
- 1030
4837
2789
- 1005
1784