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 talk about grouping like terms it is important to do the long way first do: 427 + 284

Base Ten Blocks this uses the traditional algorithm but it is modeled concretely Hundreds Tens Ones 346 + 233 579 do 324 + 145 do 186 + 246

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: 47 + 26 + 13 + 62 + 21 (1) 47 26 (3) 13 62 21 169

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: 583 + 356

Empty Number Line 47 + 25 (Jump method) +10 +10 +3 +2 47 57 67 70 72 +10 +10 +3 +2 47 57 67 70 72 this is the jump method split method would be going from 40 + 20 + 7 + 5 do: 63 + 41 using both jump and split methods

Compensation 497 + 235 (500 + 232) 59 + 16

Subtraction

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

– 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 1000 - 763

Yes Virginia! There is an easier way! 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 -10 -10 22 27 37 47 -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

Horizontal Subtraction (Split-to-fit Method) 528 - 192 400 + 120 + 8 - 100 + 90 + 2 300 + 30 + 6 = 336 in order for this algorithm to work, students needs to have a pretty solid concept of place value do: 439 – 217 do: 326 - 147

“Add-up-to” 852 - 268 + 2 270 + 30 300 + 500 800 + 52 584 6391 4729 + 1 4730 + 70 4800 + 200 5000 + 1000 6000 + 391 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

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 before showing slide, have them make a word problem for 14 x 12. Put aside Distributive property do: 14 x 27 ( 10 x 27 + 2 x 27 + 2 x 27)

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

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

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

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

Lattice Multiplication 3 2 6 4 1 8 7 do 21 x 43

Straight Line Multiplication 32 x 24 16 6 8 32 x 24 = 768

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?

Division

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

The “Gozinta” Method 51 5e255 25 05 5

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

Base Ten Blocks: 342 ~ 6 Hundreds Tens Ones 5 7 do 471 ~ 9 do 238 ~ 7

Repeated Subtraction Algorithm -7 119 112 105 98

Trevor Brown’s Method 7 7r126 1 7 70 (10 groups of 7) 2 14 56 2 14 56 5 35 35 (5 groups of 7) 70 21 14 (2 groups of 7) 7 7 (1 group of 7) 0 (18 groups of 7) you can extend your T-table to include 20 groups, 50 groups, 100 groups for when doing larger numbers

Conclusion