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Everyday Math and Algorithms

Everyday Math and Algorithms. A Look at the Steps in Completing the Focus Algorithms. Partial Sums. An Addition Algorithm. 2 6 8. Add the hundreds ( 200 + 400). + 4 8 3. + 11. Add the partial sums (600 + 140 + 11). Partial Sums. 600. Add the tens (60 +80). 140.

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Everyday Math and Algorithms

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

  2. Partial Sums An Addition Algorithm

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

  4. 785 Add the hundreds (700 + 600) + 641 + 6 Add the partial sums (1300 + 120 + 6) Let's try another one 1300 Add the tens (80 +40) 120 Add the ones (5 + 1) 1426

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

  6. Partial Sums 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.

  7. Trade-First Subtraction An alternative subtraction algorithm

  8. 12 When subtracting using this algorithm, start by going from left to right. 8 12 13 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. 5 7 6 Move to the tens column. I cannot subtract 5 from 3, so I need to regroup. Move to the ones column. I cannot subtract 6 from 2, so I need to regroup. Now subtract column by column in any order

  9. 11 Let’s try another one together 6 15 12 7 2 5 - 4 9 8 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. 2 2 7 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. Now subtract column by column in any order

  10. 13 8 12 3 9 4 2 - 2 8 7 Now, do this one on your own. 6 5 5 Let's see if you're right. Congratulations!

  11. 9 Last one! This one is tricky! 6 13 10 7 0 3 - 4 6 9 2 3 4 Let's see if you're right. Congratulations!

  12. Partial Products Algorithm for Multiplication Focus Algorithm

  13. + 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 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

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

  15. + Do this one on your own. 3 8 Let’s see if you’re right. X 7 9 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

  16. Partial Quotients A Division Algorithm

  17. 12 158 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) - 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 3 – 2nd guess - 36 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 (10 + 3 = 13) plus what is left over (remainder of 2 )

  18. 36 7,891 Let’s try another one 219 R7 - 3,600 100 – 1st guess Subtract 4,291 - 3,600 100 – 2nd guess Subtract 691 - 360 10 – 3rd guess 331 - 324 9 – 4th guess 7 219 R7 Sum of guesses

  19. 43 8,572 Now do this one on your own. 199 R 15 - 4,300 100 – 1st guess Subtract 4272 -3870 90 – 2nd guess Subtract 402 - 301 7 – 3rd guess 101 - 86 2 – 4th guess 199 R 15 Sum of guesses 15

  20. Lattice Method of Multiplication

  21. 286 X 34

  22. 1. Create a grid. Write one factor along the top, one digit per cell. Write the other factor along the outer right side, one digit per cell. 2 8 6 1 0 2. Draw diagonals across the cells. 2 0 3 6 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 8 2 0 3 9 4 8 2 4 4. Add along each diagonal and record any regroupings in the next diagonal 1 7 1 2 4

  23. Answer 2 8 6 1 1 1 0 2 0 3 6 4 8 2 0 3 9 4 8 2 4 7 2 4 286 X 34 = 9 7 2 4

  24. 732 X 57

  25. 7 3 2 1 1 3 5 4 5 5 0 4 1 2 7 1 1 9 1 4 1 7 2 4 732 X 57 = 1, 7 2 4 4

  26. Lattice Multiplication 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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