Lattice Multiplication - OrangeCountyCCCC

Lattice Multiplication
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Lattice multiplication

Lattice multiplication is a method of multiplying large numbers using a grid. This method breaks the
multiplication process into smaller steps, which some students find easier. Digits to be carried are written within
the grid, making them harder to miss.

By David Walbert

About this article

This article was adapted from a presentation at the 2005 conference of the North Carolina Council of Teachers
of Mathematics by Vicki Thomas, a fifth-grade math and social studies teacher at River Dell Elementary in
Johnston County, North Carolina.

Learn more

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Lattice multiplication is a method of multiplying large numbers using a grid. It is algorithmically equivalent to
regular long multiplication, but the lattice method breaks the multiplication process into smaller steps, which
some students find easier. Digits to be carried are written within the grid, making them harder to miss.

Multiplying on a lattice

After the lattice is constructed, a round of multiplication is followed by a round of addition, much as in
traditional multiplication of large numbers.

Setting up the lattice

First, draw a grid that has as many rows and columns as the multiplicand and the multiplier. The grid shown
here is for multiplying a 4-digit number by a 3-digit number.

http://www.learnnc.org/lp/pages/4458

Hoosiers Against Common Core

http://hoosiersagainstcommoncore.com/why-does-common-core-mandate-fuzzy-math/
Note: It’s fun to read the comments on this website.

Why does Common Core mandate fuzzy math?

Erin Tuttle • June 10, 2013

Below are the standards for multiplication and division for fourth, fifth and sixth grade under Common
Core. While the Common Core tests, like New York’s, require students to be able to do multi digit
multiplication and division in fourth grade, the Common Core standards do not require the mastery of the
standard algorithm until later. This creates a mandate that students learn “alternative strategies.” Here are the
standards for multiplication:

EXAMPLES OF HOW TO MEET THE FOURTH GRADE ALGORITHM WITHOUT THE STANDARD
ALGORITHM:

THE RECTANGLE METHOD (DIRECTLY BELOW) WAS ON THE SMARTER BALANCED SAMPLE
TEST:

OR

THE CRAZY LATTICE METHOD FOR MULTIPLICATION:

Partial Products Method 83
27
In the Partial Products Method one takes the base-ten decomposition ----
of each factor and forms the products of all pairs of terms. Then these 80*20 -> 1600
partial products are added together. The student text does not 80* 7 -> 560
recommend any particular addition algorithm for this second stage. In 3*20 -> 60
the example at right I’ve assumed traditional addition with carries done

mentally, but an Everyday Mathematics pupil may well do that addition 3* 7 -> 21
problem by the Partial Sums or the Column Addition method. Observe ----
that the number of terms in the addition problem is the product of the 2241
numbers of digits in the factors.
83
Lattice Method +---+---+
|1 /|0 /|
The lattice method employs a grid of squares. One factor is written |/|/|2
along the top, left to right, and the other factor is written along the right 1 |/ 6|/ 6|
edge, top to bottom. In the example at right the factors are 83 and 27. +---+---+
Each square of the grid defined by the two factors is divided by a |5 /|2 /|
diagonal. The digits of the factors are multiplied pairwise and the two- |/|/|7
digit result written down in the corresponding square in the manner 11 |/ 6|/ 1|
shown. The result of the multiplication is then obtained by addition +---+---+
down the diagonals.
14 1

1,11,14,1
-> 1,12,4,1
-> 2,2,4,1

= 2241

THE COMMON CORE STANDARDS DO NOT REQUIRE THE STANDARD ALGORITHM (BELOW)
UNTIL 5TH GRADE. THE IDEA BEHIND THIS IS THAT IT WILL STUNT THEIR UNDERSTANDING
OF MULTIPLICATION. I’M NOT JOKING. THEY ACTUALLY CALL IT THE SHORT CUT METHOD,

NOT THE STANDARD ALGORITHM??

How about division? This what the Common Core standards include:

SO, WE DON’T REQUIRE OR TEST OVER THE STANDARD ALGORITHM UNTIL SIXTH GRADE,
BUT REQUIRE DIVISION OF UP TO FOUR DIGITS BY ONE DIGIT DIVISORS IN FOURTH GRADE,

AND DIGIT DIVISORS IN FIFTH?? HERE’S HOW TO DO DIVISION WITHOUT THE STANDARD
ALGORITHM:

OR

Column Division Method 0|1|7|4
-----+---+---+--
The column division method appears to be the traditional method, but 7) 1|2|2|0
implemented in a rather verbose manner, and presented in a way that
constitutes a viable student algorithm only when the divisor is a single- -0
digit number. The Everyday Mathematics 5th and 6th grade student ----+---
reference books present it via a visualization as money sharing. In the
example at right we divide 1220 by 7. We have one $1000 note, which 1|
can not be shared by 7 people. We change it to 10 $100 notes, giving -> |12
us 12 $100 notes in all. With 7 people each gets 1 such note, and we
mark the 1 above the dividend. This removes 7 $100 notes, leaving us -7
with 5, which we convert into 50 $10 notes, giving us 52 such notes in ----+---
all. With 7 people each can get 7 such notes, which we mark above
the dividend. This removes 49 $10 notes, leaving us with 3, which we 5|
convert into 30 $1 notes, giving us indeed 30 $1 notes in all. With 7 -> |52
people each can get 4 such notes, which we mark above the dividend.
This removes 28 such notes, leaving us with 2. That 2 is our -49
remainder, and the integer result of the division is read above the line. ----+---

3|
-> |30

-28
---
2

ans: 174 R2

OR

Partial Quotients Method ------ |
16 ) 1220 |
The Partial Quotients Method, the Everyday Mathematics focus
algorithm for division, might be described as successive - 800 | 50
approximation. It is suggested that a pupil will find it helpful to prepare ---- |
first a table of some easy multiples of the divisor; say twice and five 420 |
times the divisor. Then we work up towards the answer from below. In
the example at right, 1220 divided by 16, we may have made a note - 320 | 20
first that 2*16=32 and 5*16=80. Then we work up towards 1220. --- |
50*16=800 subtract from 1220, leaves 420; 20*16=320; etc.. 100 |

- 80 | 5
--- |
20 |

- 16 | 1
-- | --
4 | 76

ans: 76 R4

THIS IS THE STANDARD ALGORITHM (BELOW) FOR DIVISION-NOT TAUGHT UNTIL SIXTH
GRADE WHILE DIVISION OF MULTI-DIGIT NUMBERS IS REQUIRED ON COMMON CORE TESTS

STARTING IN FOURTH GRADE.

Barry Garelick says:
June 10, 2013 at 11:24 am
Defenders of Common Core math standards (like Bill McCallum, the lead writer of the standards) will say that
the standards do not prohibit teaching the standard algorithms earlier than the specified grade. For example, the
standard algorithm for multi-digit addition and subtraction is required to be learned by 4th grade. He posits that
teachers are free to teach it earlier. But who is going to do that when the standards for 2nd and 3rd grade require
it to be taught via various place value “strategies”. Further, programs like EM and Investigations readily provide
such alternative strategies in those earlier grades.

Michelle says:
June 11, 2013 at 1:28 am
Years ago, I assisted in a fourth-grade classroom using Everyday Mathematics as the curriculum. At first, I was
“sold” because it was developed by the University of Chicago and deemed to be “cutting edge.” Who wouldn’t
want such “advanced” learning for their children? In reality, it was a disaster. Students were regularly lost and
confused and had not a clue as to how and why they arrived at their answers–if they arrived at them at all. The
only students who “got it” were worked with intensely one-on-one and few, if any, had mastery of simple
computation by year’s end. Since parents hadn’t been taught using this method, they were powerless to help at
home. Most classroom days were spent watching students boil over in frustration. This type of math instruction
does not work in large groups because it is developed by “educational experts” who have not the first clue as to
the realities of the classroom environment.

Patrick says:
June 14, 2013 at 11:16 pm
Encouraging children to think, not just calculate, isn’t necessarily a bad thing is it? You can buy a calculator for
only a few bucks now so the need to be proficient at calculating has been somewhat diminished. An alarming
number of high school kids I tutor are completely incapable of comprehending the mathematical relationships
defined here. This type of mathmatical thinking, not just calculating, is essential in technical fields. Instead of
ensuring we can save our children the 3 bucks it takes to buy calculator, maybe we can help them get their
minds to make these relationships on their own and save me from having so many kids to tutor.

Erin Tuttle says:

June 17, 2013 at 3:32 pm
I think most people would expect a well educated child in math not to need a calculator. Many ideas in math
stem from these simple calculations which math textbooks have ignored in favor of concepts at the expense of
many children having difficulties with higher level math. Learning calculations helps children to think and
shape mathematical thinking.

Brendan says:
August 14, 2013 at 10:09 am
I’m a PhD Engineer. Math is a central component of everything I do – if I’m not using directly, I need to
understand what is happening. The understanding these people are looking for won’t come until the children
begin to mature. Teach ‘em the basics, and even if they don’t “understand” they can cope later. Don’t teach
them the basics, and the ones who aren’t interested in the basics will never learn.

This is a sad state for education… I’ve already begun to encounter this idiocy with my kids. I do my best to
counter program them, for their sakes.

Jim says:
August 14, 2013 at 11:26 am
I have a Ph.D. in Physics, and I’ll be damned if I can figure out the last steps in the “lattice” method of
multiplication. I’m sure if I spent more than 5 minutes, maybe I could, but really . . . this is multiplication we’re
talking about! That lattice thing is more confusing than anything I have ever seen. The division methods aren’t
quite as awful; at least they made sense to me rather quickly. Stick with the older, working algorithms, and
leave this other nonsense to the educrats.

Scott says:
August 20, 2013 at 9:46 am
This is exactly why my children are being homeschooled. For example my daughter is now in 7th grade and is
being homeschooled for the second year now because she was #1 bullied and the school did nothing(thats a
different matter all together) and #2 she was so behind in math because nothing was explain to her and when
she brought her work home it was “foreign” to us. Now that she is home she has come into her own and is
excelling in math because she is learning the basics that should be taught not this new crap that most adults
don’t understand. I learned math using the basic way and I turned out just fine.

Pulling up internet resources about “Everyday Math”, it looks as if lattices are taught in solving
multiplication problems. If this is the way students are assessed on the Common Core assessments,
the reader’s plea to her principal is useless. If the student doesn’t understand the process of a
problem, the student will fail. If the student fails, the teacher’s accountability rating goes down. What
teacher wants to teach math problems in a manner that will affect his/her job?
If “Singapore Math” is not the consortia’s assessment decision for your state, it’s a safe bet your local
school district won’t teach it, even if the taxpayers demand it. Is this really the best way to teach math
to children? This is the lattice method for the multiplication problem 567 x 432. And you thought
word problems were bad?

It’s clear as mud to me.

http://beforeitsnews.com/libertarian/2012/07/common-core-lattice-math-problems-and-a-parents-
frustration-2443810.html