1. Alternative Algorithms
Presented at SAG 2009
Presented by: Marlene Letkeman
David Milley

2. Agenda
We will be going through all 4 operations and showing some alternative algorithms.

3.
How many ways can you add these two numbers?

4. 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)

5. Addition

6. Alternative Algorithms for Addition
horizontal addition
base-ten blocks
Kagan method
partial sums
empty number line
compensation

7. Horizontal Addition 264 + 397 = 200 + 60 + 4 + 300 + 90 + 7= = = =
= 661
talk about grouping like terms
it is important to do the long way first
do:

8. Base Ten Blocks
this uses the traditional algorithm but it is modeled concretely
Hundreds Tens Ones
346
+ 233
579 do do

9. Kagan Method (1) 23 18 (1) 15 17 (3) 11 ______ 84
this method requires the student knows that they are carrying a 10 over, not a 1
do:
(1) 47
26 (3) 13 62 21
169

10. Finding Partial Sums 345 +138 400 (add the hundreds) 13 (add the ones)
70 (add the tens)
+13 (add the ones)
483 (add the partial sums)
13 (add the ones)
+400 (add the hundreds)
similar to horizontal method in that you are breaking the number up, but it remains vertical
students need to have a good idea of place value before using this one
do:

11. Empty Number Line 47 + 25 (Jump method) +10 +10 +3 +2 47 57 67 70 72
this is the jump method
split method would be going from
do: using both jump and split methods

12. Compensation
( )

13. Subtraction

14. Alternative Algorithms for Subtraction
base-ten blocks
empty number line
horizontal subtraction (not as friendly)
“Add-up-to”

15. – this uses the traditional algorithm but it is modeled concretely
Base Ten Blocks
– this uses the traditional algorithm but it is modeled concretely
Hundreds Tens Ones
346
- 233
113
Use pieces of paper for the subtrahend
do: 275 – 143
do: 432 – 256

16. Yes Virginia! There is an easier way!
Those Zero’s!
Yes Virginia! There is an easier way!
1000 – 763
763
= 237

17. Empty Number Line 47 – 25 (jump method) -5 -10 -10 22 27 37 47
this is once again the jump method
split would be going from 40 – 20 add 7 – 5- not as convenient – I usually only use jump method for subtraction

18. Horizontal Subtraction (Split-to-fit Method)
528
- 192
= 336
in order for this algorithm to work, students needs to have a pretty solid concept of place value
do: 439 – 217
do:

19. “Add-up-to”
6391
1662
this comes from Europe
students start with the subtrahend and add up to the minuend
they try to make friendly numbers to work with

20. Multiplication

21. Multiplication Problem
Make up a word problem for the following:
14 x 12

22. Alternative Algorithms for Multiplication
Distributive property
area model
lattice
Grid
Straight line
before showing slide, have them make a word problem for 14 x 12. Put aside
Distributive property
do: 14 x 27 ( 10 x x x 27)

23. Area Model
32 x 24 10 1
100
do: 21 x 43

24. 7 x 8 5
+ 2 25 10 4
- Multiplication dot sheets

25. 32 x 24 30
+ 2 20
600 40
+ 4
120 8
Connect to alge-tiles

26. x
100
+ 4
10 000
+ 400 7
700
- 28

27. Lattice Multiplication3 2 6 4 1 8 7
do 21 x 43

28. Straight Line Multiplication
32 x 24 16
32 x 24 = 768

29. Chart Multiplication 3 2 4 8 4 x 2 1 4 x 30 20 x 2 6 20 x 30 7
4 x 30
20 x 2 6
20 x 30 7
do 21 x 43
after going through these algorithms, would your word problem be different? maybe area model problem?

30. Division

31. Symbols 9 ~ 3 means 9 “divided by” 3 3w9 means 3 “divided into” 9
important for students to realize what question is asking them to do

32. The “Gozinta” Method 51
5e255 25 05 5

33. 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
curriculum sheets are available in the old K-4 Implementation guide

34. Base Ten Blocks: 342 ~ 6
Hundreds Tens Ones
do 471 ~ 9
do 238 ~ 7

35. Repeated Subtraction Algorithm-7
119
112
105 98

36. Trevor Brown’s Method 7 7r126 1 7 70 (10 groups of 7) 2 14 56
(5 groups of 7)
(2 groups of 7) 7
(1 group of 7)
(18 groups of 7)
you can extend your T-table to include 20 groups, 50 groups, 100 groups for when doing larger numbers

37. Conclusion