A Parents’ Guide to Alternative Algorithms

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

Draw vertical lines to separate the ones, tens, and hundreds. Column Addition 3 6 7 Draw vertical lines to separate the ones, tens, and hundreds. +1 4 7 4 10 14 Add each column separately in any order. 5 14 5 1 4 3. Adjust by making trades as needed.

- 267 200 20 + -4 Partial Differences 216 483 Subtract the hundreds (400 - 200) 200 Subtract the tens (80 -60) 20 Subtract the ones (3 - 7) + -4 Compute the differences (200 + 20 - 4) 216

Trade First Subtraction 12 8 12 13 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. 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

6 7 X 5 3 3,000 350 180 21 + 3,551 Partial Products Add the results 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 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

Lattice Multiplication Lattice Multiplication is rooted in place value. The diagonals in the lattice correspond to place value columns. The far right-hand diagonal is the ones place, the next diagonal to the left is the tens place, and so on. Lattice Multiplication 316 x 68 = 21488 3 1 6 1. Draw the lattice, including the diagonals. 3 6 2. Write one factor along the top of the lattice and the other along the side, one digit for each row or column. 1 8 6 6 8 2 1 3. Multiply each digit in one factor by each digit in the other factor. Write the products in the cells where the corresponding rows and columns meet. Write the tens digit of these products above the diagonal and the ones digit below the diagonal. 8 4 8 2 4 4 1 8 1 8 4. Add the numbers inside the lattice along each diagonal, beginning with the bottom right diagonal. Write these sums along the bottom and left sides of the lattice. If the sum on a diagonal exceeds 9, carry the tens digit to the next diagonal.

158 - 120 Partial Quotients 38 - 36 2 13 R 2 12 10 – 1st guess 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. Partial Quotients 13 R 2 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 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 )