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Everyday Math and Algorithms A Look at the Steps in Completing the Focus Algorithms

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Partial Sums An Addition Algorithm

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

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785+ 641 1300 Add the hundreds ( 700 + 600) Add the tens (80 +40) 120 Add the ones (5 + 1) Add the partial sums (1300 + 120 + 6) + 6 1426

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329+ 989 1200 100 + 18 1318

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The partial sums algorithm for addition is particularly useful for adding multi-digit numbers. The partial sums are easier numbers to work with, and students feel empowered when they discover that, with practice, they can use this algorithm to add number mentally. The partial sums algorithm for addition is particularly useful for adding multi-digit numbers. The partial sums are easier numbers to work with, and students feel empowered when they discover that, with practice, they can use this algorithm to add number mentally.

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An alternative subtraction algorithm

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When subtracting using this algorithm, start by going from left to right. 9 3 2 - 3 5 6 Ask yourself, “Do I have enough to subtract the bottom number from the top in the hundreds column?” In this problem, 9 - 3 does not require regrouping. 12 13 Move to the tens column. I cannot subtract 5 from 3, so I need to regroup. 12 8 Now subtract column by column in any order 5 6 7 Move to the ones column. I cannot subtract 6 from 2, so I need to regroup.

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Let’s try another one together 7 2 5 - 4 9 8 15 12 11 6 Now subtract column by column in any order 2 7 2 Start by going left to right. Ask yourself, “Do I have enough to take away the bottom number?” In the hundreds column, 7- 4 does not need regrouping. Move to the tens column. I cannot subtract 9 from 2, so I need to regroup. Move to the ones column. I cannot subtract 8 from 5, so I need to trade.

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Now, do this one on your own. 9 4 2 - 2 8 7 12 3 13 8 6 5 5

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Last one! This one is tricky! 7 0 3 - 4 6 9 13 9 6 2 4 3 10

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Partial Products Algorithm for Multiplication Focus Algorithm

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Calculate 50 X 60 67 X 53 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

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Calculate 10 X 20 14 X 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.

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Calculate 30 X 70 38 X 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.

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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. 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 - 120 38 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 – 2 nd guess - 36 2 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 )

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Let’s try another one 36 7,891 100 – 1st guess - 3,600 4,291 Subtract 100 – 2 nd guess - 3,600 7 219 R7 Sum of guesses Subtract 691 10 – 3 rd guess - 360 331 9 – 4th guess - 324

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Now do this one on your own. 43 8,572 100 – 1st guess - 4,300 4272 Subtract 90 – 2 nd guess -3870 15 199 R 15 Sum of guesses Subtract 402 7 – 3 rd guess - 301 101 2 – 4th guess - 86

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1. Create a grid. Write one factor along the top, one digit per cell. 2. Draw diagonals across the cells. 3.Multiply each digit in the top factor by each digit in the side factor. Record each answer in its own cell, placing the tens digit in the upper half of the cell and the ones digit in the bottom half of the cell. 4. Add along each diagonal and record any regroupings in the next diagonal 0 6 2 4 1 8 0 8 3 2 2 4 Write the other factor along the outer right side, one digit per cell.

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0 6 2 4 1 8 0 8 3 2 2 4

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3 5 1 5 1 0 4 9 2 1 1 4

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The lattice algorithm for multiplication has been traced to India, where it was in use before A.D.1100. Many Everyday Mathematics students find this particular multiplication algorithm to be one of their favorites. It helps them keep track of all the partial products without having to write extra zeros – and it helps them practice their multiplication facts

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Addition and Subtraction 29 Addition and Subtraction Algorithms 2 ones plus 9 ones equal 11 ones. 11 is 1 ten and 1 one. Record the 1 one in the ones column.

Addition and Subtraction 29 Addition and Subtraction Algorithms 2 ones plus 9 ones equal 11 ones. 11 is 1 ten and 1 one. Record the 1 one in the ones column.

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