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

Partial Sums An Addition Algorithm

600 Add the tens (60 +80) 140 Add the ones (8 + 3) 751 Partial Sums 2 6 8 + 4 8 3 Add the hundreds ( 200 + 400) Add the partial sums (600 + 140 + 11) + 11

1300 Add the tens (80 +40) 120 Add the ones (5 + 1) 1426 Let's try another one 7 8 5 + 6 4 1 Add the hundreds ( 700 + 600) Add the partial sums (1300 + 120 + 6) + 6

1200 100 1318 Do this one on your own Let's see if you're right. Well Done! 3 2 9 + 9 8 9 + 18

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. 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.

An alternative subtraction algorithm Trade-First Subtraction

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.

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.

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

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

Partial Products Algorithm for Multiplication Focus Algorithm

Calculate 50 X 60 6 7 X 5 3 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 +

Calculate 10 X 20 1 4 X 2 3 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. +

Calculate 30 X 70 3 8 X 7 9 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. +

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. 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 ) 13 R2 12 158

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

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

Lattice Method of Multiplication

286 X 34

2 8 6 3 4 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. 4 2 1 7 1 9 0

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

732 X 57

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

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 Lattice Multiplication