Transcript Everyday Math and Algorithms

Everyday Math
and
Algorithms
A Look at the Steps in
Completing the Focus
Algorithms
Partial Sums
An Addition
Algorithm
Add the hundreds (200 + 400)
Add the tens (60 +80)
Add the ones (8 + 3)
Add the partial sums
(600 + 140 + 11)
268
+ 483
600
140
+ 11
751
Add the hundreds (700 + 600)
Add the tens (80 +40)
Add the ones (5 + 1)
Add the partial sums
(1300 + 120 + 6)
785
+ 641
1300
120
+
6
1426
329
+ 989
1200
100
+ 18
1318
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
12
13
12
- 3
5
6
5
7
6
When subtracting using this
algorithm, start by going from
left to right.
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.
8
9
3
2
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
11
12
15
- 4
9
8
2
2
7
Let’s try another one
together
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.
6
7
2
5
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
13
3
12
- 2
8
7
6
5
5
8
Now, do this one on
your own.
9
4
2
Last one! This
one is tricky!
9
10
13
- 4
6
9
2
3
4
6
7
0
3
Partial Products Algorithm
for Multiplication
Focus Algorithm
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 50 X 60
Calculate 50 X 7
Calculate 3 X 60
Calculate 3 X 7
Add the results
67
X 53
3,000
350
180
+ 21
3,551
Let’s try another
one.
Calculate 10 X 20
Calculate 20 X 4
Calculate 3 X 10
Calculate 3 X 4
Add the results
14
X 23
200
80
30
+ 12
322
Do this one on
your own.
Let’s see if
you’re right.
Calculate 30 X 70
Calculate 70 X 8
Calculate 9 X 30
Calculate 9 X 8
Add the results
38
X 79
2, 100
560
270
+ 72
3002
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)
There are more than three
(3 x 12 = 36), but fewer than
four (4 x 12 = 48). Record 3 as
the next guess
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 )
12
158
Subtract - 120
38
Subtract - 36
2
10 – 1st guess
3 – 2nd guess
13
Sum of guesses
Let’s try another one
36
7,891
Subtract - 3,600
4,291
Subtract - 3,600
691
- 360
331
- 324
7
100 – 1st guess
100 – 2nd guess
10 – 3rd guess
9 – 4th guess
219 R7
Sum of guesses
Now do this one on your
own.
43
Subtract
Subtract
8,572
- 4,300
4272
-3870
402
- 301
101
- 86
15
100 – 1st guess
90 – 2nd guess
7 – 3rd guess
2 – 4th guess
199 R 15
Sum of guesses
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. 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
1
2
6
0
4
2
3
8
8
2
4
0
1
2
6
0
4
2
3
8
8
2
4
1
3
5
4
1
5
1
2
9
0
1
4