1. Alternative Algorithms
Everyday Mathematics
Alternative Algorithms

2. Partial Sums
An Addition Algorithm

3. + 483 600 140 + 11 Partial Sums 751 268 Add the hundreds (200 + 400)
Add the tens (60 +80)
140
+ 11
Add the ones (8 + 3)
Add the partial sums
( )
751

4. Let's try another one
785
+ 641
1300
Add the hundreds ( )
Add the tens (80 +40)
120
Add the ones (5 + 1)
Add the partial sums
( )
1426

5. + 989 1200 100 + 18 1318 Do this one on your own
329
+ 989
Do this one on your own
Let's see if you're right.
1200
100
+ 18
1318
Well Done!

6. An alternative subtraction algorithm
Trade-First Subtraction
An alternative subtraction algorithm

7. 12 8 12
In order to subtract, the top number must be larger than the bottom number 2
To make the top number in the ones column larger than the bottom number, borrow 1 ten. The top number become 12 and the top number in the tens column becomes 2. 5 7 6
To make the top number in the tens column larger than the bottom number, borrow 1 hundred. The top number in the tens column becomes 12 and the top number in the hundreds column becomes 8.
Now subtract column by column in any order

8. 2 2 7 7 2 5 - 4 9 8 11 Let’s try another one together 15 1 6
To make the top number in the ones column larger than the bottom number, borrow 1 ten. The top number become 15 and the top number in the tens column becomes 1. 2 2 7
To make the top number in the tens column larger than the bottom number, borrow 1 hundred. The top number in the tens column becomes 11 and the top number in the hundreds column becomes 6.
Now subtract column by column in any order

9. 6 5 5 9 4 2 - 2 8 7 Let's see if you're right. Congratulations! 13 12
Now, do this one on your own. 6 5 5
Let's see if you're right.
Congratulations!

10. Let's trade from the hundreds column9
Last one! This one is tricky! 6 13 10 2 3 4
Oh, no! What do we do now?
Let's trade from the hundreds column
Congratulations!
Let's see if you're right.

11. Partial Products Algorithm for Multiplication

12. 6 7 x 5 3 3,000 350 180 21 + 3,551 Add the results Calculate 50 X 60
To find 67 x 53, think of 67 as and 53 as Then multiply each part of one sum by each part of the other, and add the results 6 7 x 5 3
3,000
Calculate 50 X 60
350
Calculate 50 X 7
180
Calculate 3 X 60 21 +
Calculate 3 X 7
3,551
Add the results

13. 1 4 x 2 3 200 80 30 12 + 322 Let’s try another one. Add the results
Calculate 10 X 20 80
Calculate 20 X 4 30
Calculate 3 X 10 12 +
Calculate 3 X 4
322
Add the results

14. 3 8 x 7 9 2,100 560 270 72 + 3002 Do this one on your own.
Let’s see if you’re right.
2,100
Calculate 30 X 70
560
Calculate 70 X 8
270
Calculate 9 X 30 72 +
Calculate 9 X 8
3002
Add the results

15. Lattice Method of Multiplication
Another Multiplication Algorithm

16. The lattice method of multiplication has been used for hundreds of years. It is very easy to use if you know basic multiplication facts. It becomes a favorite algorithm of students learning double digit multiplication
45 x 3 4 5
Draw a box with squares and diagonals, this is called a lattice.
Write 45 above the lattice.
Write 3 on the right side of the lattice. 1 1 1 3 2 5 3 5
Multiply 3 x 5.
Write the number as shown.
45 x 3 = 135
Multiply 3 x 4.
Write the number as shown.
Add the numbers along each diagonal.

17. Multiply 7 x 89. 7 x 89 = 623 Let’s Try Another One! Way To Go! 8 9 61 6 5 7 6 3 6 2 3
7 x 89 = 623
Way To Go!

18. 2-Digit by 2-Digit Multiplication
34 x 26 3 4 1 2 6 8 1 2 6 8 8 4 8 4
34 x 26 = 884

19. Pretty cool...huh!

20. Partial Quotients
A Division Algorithm

21. The Partial Quotients Algorithm uses a series of “at least, but less than” estimates of how many b’s in a. You might begin with multiples of 10 – they’re easiest.
13 R2
There are at least ten 12’s in 158 (10 x 12=120), but fewer than twenty.
(20 x 12 = 240) 12
158
- 120
10 – 1st guess
Subtract 38
There are more than three (3 x 12 = 36), but fewer than four (4 x 12 = 48). Record 3 as the next guess
- 36
3 – 2nd guess
Subtract 2 13
Sum of guesses
Since 2 is less than 12, you can stop estimating. The final result is the sum of the guesses ( = 13) plus what is left over (remainder of 2 )

22. Let’s try another one 219 R7 7,891 - 3,600 4,291 - 3,600 691 - 360 331
100 – 1st guess
Subtract
4,291
- 3,600
100 – 2nd guess
Subtract
691
10 – 3rd guess
- 360
331
- 324
9 – 4th guess 7
219 R7
Sum of guesses

23. Now do this one on your own.43
8,572
- 4,300
100 – 1st guess
Subtract
4,272
-3,870
90 – 2nd guess
Subtract
402
7 – 3rd guess
- 301
101
2 – 4th guess 15
199 R 15
Sum of guesses

24. Congratulations on a job well done!