Everyday Math Algorithms by xiaoyounan

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									    Computation
     Algorithms

Everyday Mathematics
    Computation Algorithms in
      Everyday Mathematics
       Instead of learning a prescribed (and limited)
set of algorithms, Everyday Mathematics encourages
students to be flexible in their thinking about
numbers and arithmetic. Students begin to realize
that problems can be solved in more than one way.
They also improve their understanding of place value
and sharpen their estimation and mental-
computation skills.
       The following slides are offered as an
extension to the parent communication from your
child’s teacher. We encourage you to value the
thinking that is evident when children use such
algorithms—there really is more than one way to
solve a problem!
Before selecting an algorithm, consider
how you would solve the following
problem.
                     48 + 799
We are trying to develop flexible thinkers who recognize that
this problem can be readily computed in their heads!
One way to approach it is to notice that 48 can be renamed
as 1 + 47 and then
48 + 799 = 47 + 1 + 799 = 47 + 800 = 847
         What was your thinking?
An algorithm consists of a precisely specified sequence of steps
that will lead to a complete solution for a certain class of problems.

       Important Qualities of Algorithms
• Accuracy
   –    Does it always lead to a right answer if you do it right?
• Generality
   –    For what kinds of numbers does this work? (The larger the set of
        numbers the better.)
• Efficiency
   –    How quick is it? Do students persist?
• Ease of correct use
   –    Does it minimize errors?
• Transparency (versus opacity)
   –    Can you SEE the mathematical ideas behind the algorithm?
Hyman Bass. “Computational Fluency, Algorithms, and Mathematical Proficiency: One
Mathematician’s Perspective.” Teaching Children Mathematics. February, 2003.
 Table of Contents
          Partial Sums
          Partial Products
          Partial Differences
          Trade First
          Partial Quotients
          Lattice Multiplication
Click on the algorithm you’d like to see!
          Click to proceed
            at your own
               speed!             735
                                + 246
Add the hundreds (700 + 200)
                                  900
Add the tens        (30 + 40)
                                   70
Add the ones (5 + 6)
                                  +11
Add the partial sums
       (900 + 70 + 11)            981
                                  356
                                + 247
Add the hundreds (300 + 200)
                                  500
Add the tens        (50 + 40)
                                   90
Add the ones (6 + 7)              +13
Add the partial sums
       (500 + 90 + 13)
                                  603
                      429
                    + 989
                    1300
                      100
                    + 18
 Click here to go
                     1418
back to the menu.
      Click to proceed
        at your own
                             56
           speed!
                           × 82
  50 X 80                 4,000
  50 X 2                    100
  6 X 80                    480
   6X2                   +   12
Add the partial
products                  4,592
How flexible is your
thinking? Did you notice
that we chose to multiply      52
in a different order this    × 76
time?
   70 X 50                   3,500
   70 X 2                      140
    6 X 50                     300
    6X2
                            +   12
 Add the partial
 products                    3,952
A Geometrical Representation
     of Partial Products         52
        (Area Model)

                   50   2
                               × 46
                               2,000
40              2000    80       300
                                  80
6                 300   12        12
     Click here to go
    back to the menu.
                               2,392
Students complete all regrouping
before doing the subtraction. This
                                        11 13
can be done from left to right. In    6 12
this case, we need to regroup a
100 into 10 tens. The 7 hundreds is   72 3
                                      45 9
now 6 hundreds and the 2 tens is
now 12 tens.


                                      2 64
Next, we need to regroup a 10 into
10 ones. The 12 tens is now 11
tens and the 3 ones is now 13
ones.
Now, we complete the subtraction. We have 6
hundreds minus 4 hundreds, 11 tens minus 5 tens,
and 13 ones minus 9 ones.
           9 12       13 16
        7 10        8 14
       80 2         94 6
       27 4         56 8
       5 28         3 78
 Click here to go
back to the menu.
                                 73 6
                               – 2 45
Subtract the hundreds
                (700 – 200)      500
Subtract the tens                  10
                                    1
              (30 – 40)
Subtract the ones
              (6 – 5)

Add the partial differences
           (500 + (-10) + 1)     491
                                          41 2
                                        – 3 35
Subtract the hundreds
                       (400 – 300)        100
Subtract the tens                           20
                                             3
                     (10 – 30)
Subtract the ones
                      (2 – 5)

Add the partial differences
                 (100 + (-20) + (-3))       77
  Click here to go
 back to the menu.
                       19 R3
                      I know
                    10 x 12 will
Click to proceed
  at your own
     speed!
                  12 2 31
                      work…



                     1 20 10
 Add the partial
                     1 11
  Students begin
  quotients, and
   by choosing
                       60 5
    record the
 partial quotients
  quotient along
     that they
                       51
      with the
     recognize!
    remainder.
                       48 4
                        3 19
                      85 R6
               32 27 26
                  1 6 0 0 50
   Compare the    11 26
partial quotients
used here to the
                    80 0 25
  ones that you     32 6
      chose!
 Click here to go
                    3 20 10
back to the menu.
                        6 85
     Click to proceed
       at your own
                             53
      5    3
          speed!

                           × 72
    3× 7 2 × 7 7         3500
 3 55     3
             1          Compare
                           100
    1 × 2 0× 2 2
     5    3                210
                        to partial
 8 0         6                 6
                        products!
                         +
    the
Add 1 numbers
          6
on the diagonals.
                         3816
                    16
        1    6    × 23
       0    1 2   200
          2    2    30
       0    1 3   120
      3 3      8 + 18
       6    8     368
 Click here to go
back to the menu.

								
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