 # Alternative Algorithms

## Presentation on theme: "Alternative Algorithms"— Presentation transcript:

Alternative Algorithms
Everyday Mathematics Alternative Algorithms

+ 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

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

+ 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!

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

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

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

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!

Let's trade from the hundreds column
9 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.

Partial Products Algorithm for Multiplication

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

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

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

Lattice Method of Multiplication
Another Multiplication Algorithm

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.

Multiply 7 x 89. 7 x 89 = 623 Let’s Try Another One! Way To Go! 8 9 6
1 6 5 7 6 3 6 2 3 7 x 89 = 623 Way To Go!

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

Pretty cool...huh!

Partial Quotients A 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. 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 )

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

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

Congratulations on a job well done!