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Topic: Other Issues for Grades 4 and 5
Replies: 131   Last Post: Oct 25, 2006 7:55 PM

 Messages: [ Previous | Next ]
 Wendy Gulley Posts: 39 Registered: 12/3/04
Posted: Jul 14, 2000 9:09 AM

Mike,
I think Susan Jo is probably thinking of five steps in the traditional
algorithm for 62x36 as follows:
6x2, 6x60
30x2, 30x60

It's like the five steps used in multiplying binomials (four
multiplications and then adding them all up):
(a+b)(c+d) = ac + ad + bc + bd
(60+2)(30+6)= 60(30) + 60(6)+ 2(30) + 2(6) = 2232

But the traditional algorithm is more efficient than the line above in
the way that it records those 5 steps. It makes the final addition
easier by combining two of the products at a time, taking advantage of
place value:
6x2=12 and 6x60= 360 is recorded in a vertical format as 372
and
30x2=60 and 30x60=1800 is recorded as 1860
then you only have to add those two numbers at the end.

So we've avoided writing a lot of zeros
and we only have to add two numbers at the end,
but at what price? Unfortunately, the price
of this efficiency is that many students don't
get that they are really multiplying 30x60 not
3x6, and that the answer for that part of the
problem is really 1800, not 18. And many students
don't know why the heck they have to put a zero
at the right end of the second line, which could
be considered a sixth step in itself.

I guess Susan Jo's method in her example could also
be considered 5 steps, if you count 62x30 as two steps, though
she might be doing this in her head and writing down

Anyway, I don't think counting up steps is a perfect way to measure
efficiency of an algorithm, because some steps take longer than
others, either in thinking or in recording. The traditional algorithms
are certainly efficient in recording, but by avoiding writing zeros
that show place value they put up a barrier to understanding.

Date Subject Author
11/24/98 wendy
4/21/99 Janice Odom
4/25/99 Mary Beth LaHaye
4/26/99 Wendy Gulley
5/9/99 Nancy B.
5/11/99 Cindy Stogsdill
5/18/99 Sherry Rosenberg
5/11/99 Cindy Stogsdill
5/24/99 Rachelle Tome
5/19/99 Sherry Rosenberg
6/3/00 E.A. Miller
1/16/04 Laura
1/27/04 Mary
6/7/99 Carnie Burns
6/8/99 Sam Morris
5/24/04 Linda Wicker
6/9/99 Cindy Stogsdill
6/9/99 Sam Morris
7/14/00 Megan Murray
11/27/02 Laurie Sholinsky
7/8/03 Peter Avirom
9/9/99 Becky Peterson
9/12/99 Sam Morris
10/31/99 Nancy B.
10/13/99 connie
10/17/99 Sam Morris
10/14/99 Maggie Charron
10/19/99 Sam Morris
10/28/99 Wendy Gulley
11/2/99 Gary Woodford
11/3/99 Wendy Gulley
11/3/99 Wendy Gulley
9/16/00 E.A. Miller
9/18/00 Megan Murray
3/13/00 Joan Schaffer
3/13/00 Sam Morris
7/14/00 Megan Murray
6/7/00 jansa
7/11/00 Mike Swaine
7/14/00 Wendy Gulley
11/29/00 debbie
12/5/00 Sam Morris
11/2/00 Jacki Riffey
11/9/00 Linda Dodge
11/11/00 Jacki Riffey
11/9/00 Nancy Carnevale
11/12/00 Sam Morris
11/17/00 Susan Weiss
11/19/00 Sam Morris
11/21/00 Wendy Gulley
11/21/00 Wayne
11/21/00 Susan Weiss
11/21/00 Susan Weiss
11/27/00 Jacki Riffey
4/6/01 Kathy Horal
4/6/01 Cynthia Garland-Dore
4/9/01 Brenda
10/25/06 Teresa Holyoke
5/5/01 Sara Smith
5/5/01 Cynthia Garland-Dore
5/5/01 Wayne
5/12/01 Sara Smith
10/23/01 jfoley
3/19/02 Jules
3/19/02 Jacki Riffey
3/20/02 Bob Meyer
4/8/02 Mary
4/23/02 Debra
4/24/02 Cynthia Garland-Dore
4/25/02 Wayne
4/27/02 Debra
4/27/02 Debra
5/4/02 cathie
6/2/03 Chris Natale
7/5/02 Yolanda Farmer
7/6/02 Wendy Gulley
8/12/02 Yolanda Farmer
8/23/02 Mark
8/25/02 Cynthia Garland-Dore
8/26/02 Bob
8/27/02 Michela
8/27/02 Susan Weiss
8/27/02 jfoley
8/29/02 Mark
9/30/02 Nancy
9/11/02 Debra
9/29/02 Mark
9/30/02 Lola
9/30/02 Mark
9/30/02 Roberta
9/30/02 Nancy Low
9/30/02 david
11/8/02 Mark
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11/9/02 Mark
11/4/02 Kathy Horal
11/20/02 Nancy Low
7/8/03 Peter Avirom
12/6/02 Alice MacDonald
1/27/03 Nancy Buell
1/27/03 Cynthia Garland-Dore
1/25/03 Susan Taber
5/22/03 Chris Natale
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5/23/03 Mary Ellen Krueger
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5/28/03 Karen Brannon
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6/23/03 Sandy
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10/1/03 Paul
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2/14/04 Angela
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2/14/04 Angela
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5/18/04 Lynn
7/30/04 Bree Metzger
5/20/04 Wayne
7/10/04 Patty
6/22/04 Linda Boland
6/25/04 Patty
7/9/04 Susan
7/26/04 Pam S