Alternative Algorithms
Presented at SAG 2009
Presented by: Marlene Letkeman
David Milley
Agenda
We will be going through all 4 operations
and showing some alternative
algorithms.
59
+ 16
• How many ways can you add these two
numbers?
Why Alternatives?
• Research shows that once the traditional
algorithm is taught, thinking stops.
• “The traditional algorithm limits our thought
process. As teachers we have been
taught, and we teach this way. As a
result, we as teachers have our own limits
imposed upon us. We can’t think ‘outside
the box.’ ” (Trevor Brown)
Addition
Alternative Algorithms for
Addition
•
•
•
•
•
•
horizontal addition
base-ten blocks
Kagan method
partial sums
empty number line
compensation
Horizontal Addition
• 264 + 397
= 200 + 60 + 4 + 300 + 90 + 7
= 200 + 300 + 60 + 90 + 4 + 7
= 500 + 150 + 11
= 500 + 100 + 50 + 10 + 1
= 600 + 60 + 1
= 661
Base Ten Blocks
this uses the traditional algorithm but it is modeled
concretely
Hundreds
346
+ 233
579
Tens
Ones
Kagan Method
(1)
(1)
23
(1)
18
15
(3)
17
11
______
84
Finding Partial Sums
345
345
+138
+138
400 (add the hundreds)
13 (add the ones)
70 (add the tens)
70 (add the tens)
+13 (add the ones)
+400 (add the hundreds)
483 (add the partial sums) 483 (add the partial sums)
Empty Number Line
47 + 25 (Jump method)
+10
47
+10
57
+3
67
+2
70
72
Compensation
• 497 + 235
(500 + 232)
Subtraction
Alternative Algorithms for
Subtraction
•
•
•
•
base-ten blocks
empty number line
horizontal subtraction (not as friendly)
“Add-up-to”
Base Ten Blocks
– this uses the traditional algorithm but it is
modeled concretely
Hundreds
346
- 233
113
Tens
Ones
Those *%^#$@ Zero’s!
Yes Virginia! There is an easier way!
1000 – 763
999 + 1
- 763
236 + 1 = 237
Empty Number Line
47 – 25 (jump method)
-5
22
-10
27
-10
37
47
Horizontal Subtraction
(Split-to-fit Method)
528
- 192
400 + 120 + 8
- 100 + 90 + 2
300 + 30 + 6 = 336
“Add-up-to”
852
- 268
+
2
270 + 30
300 + 500
800 + 52
584
6391
- 4729
4730
4800
5000
6000
+
1
+ 70
+ 200
+ 1000
+ 391
1662
Multiplication
Multiplication Problem
• Make up a word problem for the
following:
14 x 12
Alternative Algorithms for
Multiplication
•
•
•
•
•
Distributive property
area model
lattice
Grid
Straight line
Area Model
32 x 24
10
10
10
1
1
10
100
100
100
10 10
10
100
100
100
10 10
1
10
10
10
1
1
1
10
10
10
1
1
1
10
10
10
1
1
1
10
10
10
1
1
7x8
5
+2
5
25
10
+2
10
4
32 x 24
30
+2
20
600
40
+4
120
8
x
100
+4
100
10 000
+ 400
-7
-700
- 28
Lattice Multiplication
3
2
6
4
1
2
7
6
8
8
2
4
Straight Line Multiplication
32 x 24
6
16
8
32 x 24 = 768
Chart Multiplication
3
2
2
4
8
4x2
2
0
4 x 30
4
0
20 x 2
6
0
0
20 x 30
7
6
8
1
Division
Symbols
• 9~ 3
• 3w9
•
9#
means 9 “divided by” 3
means 3 “divided into” 9
means 9 “divided by” 3
The “Gozinta” Method
51
5e255
25
05
5
0
Alternative Algorithms for
Division
•
•
•
•
curriculum sheets
base-ten blocks
repeated subtraction
Trevor Brown method
**in WNCP curriculum, anything beyond a 1 digit divisor, use a
calculator
Base Ten Blocks: 342 ~ 6
Hundreds
Tens
5
Ones
7
Repeated Subtraction Algorithm
7r 126
-7
119
-7
112
-7
105
-7
98
Trevor Brown’s Method
7
1
2
5
10
7
14
35
70
7r126
70
56
35
21
14
7
7
0
(10 groups of 7)
(5 groups of 7)
(2 groups of 7)
(1 group of 7)
(18 groups of 7)
Conclusion