Everyday Mathematics

Alternative Algorithms

Partial Sums

An Addition Algorithm

268+ 483

600Add the hundreds (200 + 400)

140Add the tens (60 +80)

Add the ones (8 + 3)

Add the partial sums(600 + 140 + 11)

+ 11751

785+ 6411300Add the hundreds (700 + 600)

120Add the tens (80 +40)

Add the ones (5 + 1)

Add the partial sums(1300 + 120 + 6)

+ 6 1426

329+ 9891200 100

+ 18

1318

An alternative subtraction algorithm

In order to subtract, the top number must be larger than the bottom number 9 3 2

- 3 5 6 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.

12

2

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.

12 8

Now subtract column by column in any order

5 6 7

Let’s try another one together 7 2 5

- 4 9 8 To make the top number in the ones column larger than the bottom number, borrow 1 ten. The top number become 1515 and the top number in the tens column becomes 1.

15

1

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.

11 6

Now subtract column by column in any order

2 7 2

Now, do this one on your own. 9 4 2

- 2 8 7

12

313 8

6 5 5

Last one! This one is tricky! 7 0 3

- 4 6 9

13

9 6

2 4 3

10

Partial Products Algorithm for Multiplication

Calculate 50 X 60

67x53

Calculate 50 X 7

3,000 350 180 21

Calculate 3 X 60

Calculate 3 X 7 +Add the results 3,551

To find 67 x 53, think of 67 as 60 + 7 and 53 as 50 + 3. Then multiply each part of one sum by each part of the other, and add the results

Calculate 10 X 20

14x 23

Calculate 20 X 4

200 80 30 12

Calculate 3 X 10

Calculate 3 X 4 +Add the results 322

Let’s try another one.

Calculate 30 X 70

38x 79

Calculate 70 X 8

2,100 560 270 72

Calculate 9 X 30

Calculate 9 X 8 +Add the results

Do this one on your own.

3002

Let’s see if you’re right.

Lattice Method of Multiplication

Another Multiplication Algorithm

Add the numbers along each diagonal.

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

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.

4 5

3

Multiply 3 x 5.Write the number as shown.

Multiply 3 x 4.Write the number as shown.

1

3 5

45 x 3 = 135

45 x 3

1

2

1

5

Let’s Try Another One!

Multiply 7 x 89.9

3 756

66

8

32

1

7 x 89 = 623

2-Digit by 2-Digit Multiplication

34 x 26

3 4

2

6

0

8

0

6

1

8

2

4

48

1

8

0

34 x 26 = 884

Partial QuotientsA Division Algorithm

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.

12 158

There are at least ten 12’s in 158 (10 x 12=120), but fewer than twenty.

(20 x 12 = 240) 10 – 1st guess

- 12038

Subtract

There are more than three (3 x 12 = 36), but fewer than four (4 x 12 = 48). Record 3 as the next guess

3 – 2nd guess- 362 13

Sum of guesses

Subtract

Since 2 is less than 12, you can stop estimating. The final result is the sum of the guesses (10 + 3 = 13) plus what is left over (remainder of 2 )

Let’s try another one

36 7,891100 – 1st guess

- 3,6004,291

Subtract

100 – 2nd guess

- 3,600

7 219 R7

Sum of guesses

Subtract

69110 – 3rd guess

- 360 331

9 – 4th guess

- 324

Now do this one on your own.

43 8,572100 – 1st guess

- 4,3004,272

Subtract

90 – 2nd guess

-3,870

15 199 R 15

Sum of guesses

Subtract

4027 – 3rd guess- 301

101 2 – 4th guess

- 86