Everyday Math and Algorithms

Document Sample
Everyday Math and Algorithms Powered By Docstoc
					Getting to Know Everyday Mathematics
              Andy Barlow
             Chrissy Thonet
             Math Specialists
             Jenkins School
Everyday Math
     and
 Algorithms

  Taking a journey
  in completing the
  Focus Algorithms
            Did You Know?
   40% of adults hated math in school

   84% of middle schoolers would
    rather do “anything” other than math
    homework
    Everyday Mathematics in the
            Classroom
   Developed by the University of Chicago
    School Mathematics Project

   Based on research about how children
    learn and develop mathematical power

   Provides the broad mathematical
    background needed in the 21st century
            Everyday Math Lesson
1.Getting Started-Mental Math, Math Message

2. Introduce Content

3.Ongoing Learning & Practice

4. Differentiation Options
         ~Readiness
         ~Enrichment
         ~Extra Practice
What are the main messages of

Everyday Mathematics
   regarding computation?
  Everyday Mathematics
   takes a moderate position,
combining elements from both the

         child-centered,
  invented-algorithm approach

             and the
 Traditional-algorithm approach.
        During the early phases
       of learning an operation,
       Everyday Mathematics
        encourages students to
     invent their own procedures.

 Students are asked to solve problems
          in real-life contexts
before they learn systematic procedures
      for solving such problems.
  After students have had plenty of
   opportunities to experiment with
computational strategies of their own,
EDM introduces several algorithms for
            each operation.

   EDM also designates one of the
     alternative algorithms as the
          Focus Algorithm.
 Important Qualities of Algorithms
  An algorithm consists of a precisely specified sequence of
   steps that lead to a complete solution for a certain class of
                           problems.
Accuracy
   • Does it always lead to a right answer if you do
     it right?
Efficiency
   • Does the student get bogged down with many
     steps or lose track of logic?
Flexibility
   • Is the student able to choose an appropriate
     strategy for the problem? Can the student use
     one method to solve and another to check?
    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?
Focus Algorithms
    + Partial Sums
     -Trade First
  x Partial Products
  ÷ Partial Quotients

      We will also address

   Partial Differences
  Lattice Multiplication
Partial Sums
  An Addition
   Algorithm
                                 268
                               + 483
Add the hundreds (200 + 400)     600
Add the tens (60 +80)            140
Add the ones (8 + 3)           + 11
Add the partial sums
       (600 + 140 + 11)          751
                                 785
                               + 641
Add the hundreds (700 + 600)    1300
Add the tens (80 +40)            120
Add the ones (5 + 1)           +    6
Add the partial sums
       (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
  Trade First                                  8
                                                   12
                                                        12
                                                   2
In order to subtract, the top
number must be larger than                     9   3    2
the bottom number.
                                           - 3     5    6
To make the top number in the ones
column larger than the bottom number
trade 1 ten for 10 ones. You now have 2
tens and 12 ones.
                                               5   7    6
Move to the tens column. I cannot subtract 5
from 2. To make the top number in the tens
column larger than the bottom number, trade
1 hundred for 10 tens. You now have 8
hundreds and 12 tens.


 Now subtract column by column in any order
Let’s try another one                              11
                                               6   12   15
together.
                                               7   2    5
This time we will trade from left to
right.                                       - 4   9    8
Move to the tens column. I cannot subtract 9
from 2. To make the top number in the tens     2   2    7
column larger than the bottom number, trade
1 hundred for 10 tens. You now have 6
hundreds and 12 tens.

Move to the ones column. To make the top
number in the ones column larger than the
bottom number trade 1 ten for 10 ones. You
now have 11 tens and 15 ones.


Now subtract column by column in any order
                            13
                       8    14   12
Now, do this one on
your own.              9    4    2
                      - 2   8    7
                       6    5    5
Be careful! It’s tricky!          9
                            6    10   13
Watch out for the O!        7    0    3
                           - 4   6    9
                            2    3    4
  Partial Differences
                                 736
                               – 245
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
  Try Another One
                                   412
                                 – 335
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
Partial Products Algorithm
     for Multiplication
        Focus Algorithm
                       Partial Products
To find 24 x 13, think of 24 as 20 + 4
and 13 as 10 + 3. Then multiply each
part of one sum by each part of the           Build an array
other, and add the results
                                              showing
        24                                    24 x 13 !
      x 13                                       20            4

        12           3x4
        60           3 x 20              10
                                                 200       40

        40           10 x 4
      +200
      ____           10 x 20             3        60       12


        312
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                 67
                                         X 53
          Calculate 50 X 60
                                    3,000
          Calculate 50 X 7
                                      350
          Calculate 3 X 60
                                      180
          Calculate 3 X 7
                                     + 21
         Add the results
                                     3,551
Let’s try another
one.                   14
                     X 23
 Calculate 10 X 20    200
Calculate 20 X 4       80
Calculate 3 X 10       30
Calculate 3 X 4
                     + 12
Add the results
                      322
Do this one on
your own.              38
Let’s see if
you’re right.        X 79
 Calculate 30 X 70   2,100
 Calculate 70 X 8      560
 Calculate 9 X 30      270
 Calculate 9 X 8
                     + 72
 Add the results       3002
 Partial        Standard
Products        Algorithm
    21              21
  X 23           X 23
     3 3x1
    60 3 x 20      63
    20 20 x 1
                  420
                +____
  ___ 20 x 20
 +400
  483             483
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.             0       2       1
3.Multiply each digit
in the top factor by
each digit in the side
                                  6       4       8
factor. Record each
answer in its own
cell, placing the tens
digit in the upper half       0       3       2
of the cell and the
ones digit in the
bottom half of the                8       2       4
cell.

4. Add along each
diagonal and record
any regroupings in the
next diagonal
0       2       1
    6       4       8
0       3       2
    8       2       4
3       1       1
    5       5       0

4       2       1
    9       1       4
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
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                12       158
than twenty. (20 x 12 = 240)                               10 – 1st guess
                                          Subtract - 120
There are more than three                             38
(3 x 12 = 36), but fewer than                               3 – 2nd guess
four (4 x 12 = 48). Record 3 as           Subtract - 36
the next guess

                                                       2     13
Since 2 is less than 12, you can stop
estimating. The final result is the sum
                                                           Sum of guesses
of the guesses (10 + 3 = 13) plus what
is left over (remainder of 2 )
Let’s try another one

               36       7,891
            Subtract - 3,600    100 – 1st guess
                        4,291
            Subtract - 3,600    100 – 2nd guess
                         691
                       - 360    10 – 3rd guess
                          331
                       - 324    9 – 4th guess

                            7      219 R7
                                Sum of guesses
Now do this one on your
own.
               43         8,572
           Subtract    - 4,300    100 – 1st guess
                           4272
            Subtract     -3870    90 – 2nd guess
                            402
                          - 301    7 – 3rd guess
                            101
                         - 86      2 – 4th guess
                                     199 R 15
                             15
                                  Sum of guesses
Everyday Mathematics Games

  •Provide frequent practice


  •Fun and flexible


  •Played regularly
              Name That Number!



                                        Target Number



            8 x 2 = 16
 Possible
Solutions     10 + 8 – 2 = 16
Include…

              7 x 2 + 10 – 8 = 16
              8 / 2 + 10 + 7 – 5 = 16
Math Projects and Math Word Wall
Set of Data…………… Minimum &Maximum
Range
Mode   Median
Graphing the Data
      Parental Involvement
Classroom Volunteering
•

Parent Letters
•

Assist with Home Links and Study Links
•

Play games at home
•

Practice fact triangle cards
•

Read math literature books
•

Apply math practice to every day situations
•
         Jenkins Math Website

http://www.scituate.k12.ma.us/schools-jenkins-
elementary-mathematics.htm



           Study Island
           http://www.studyisland.com


      Symphony Math
      http://symphonylearning.com/
Thank You for being here for your child!

   Please email or call with questions!
   abarlow@scit.org
   cthonet@scit.org
   781-545-4910 (Jenkins School)


   Visit www.WrightGroup.com
   for more information.
    Thanks
For joining us!




on the journey!

				
DOCUMENT INFO
Shared By:
Categories:
Tags:
Stats:
views:22
posted:2/13/2012
language:
pages:52