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**Alternative Algorithms**

Everyday Mathematics Alternative Algorithms

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Partial Sums An Addition Algorithm

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**+ 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

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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

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**+ 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!

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**An alternative subtraction algorithm**

Trade-First Subtraction An alternative subtraction algorithm

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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

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**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

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**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!

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**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.

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**Partial Products Algorithm for Multiplication**

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**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

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**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

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**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

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**Lattice Method of Multiplication**

Another Multiplication Algorithm

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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.

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**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!

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**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

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Pretty cool...huh!

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Partial Quotients A Division Algorithm

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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 )

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**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

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**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

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**Congratulations on a job well done!**

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Welcome to Everyday Mathematics University of Chicago School Mathematics Project.

Welcome to Everyday Mathematics University of Chicago School Mathematics Project.

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