Multiplying Numbers
Parents’ Guide
Teaching Mathematics That Makes Sense
Multiplication
Students often learn how to multiply numbers one way. Yet,
there are many ways to think about how to multiply numbers.
Think about how you learned to multiply:
!
23
!
× 17
What is the “traditional” way of finding the answer? You follow
these steps:
1)
2)
3)
4)
5)
6)
Multiply 7 x 3, get 21, and put the 1 under the 7 and “carry”
the 2
Multiply 7 x 2, get 14, add the 2 above the 2, get 16 and
write this in front of the 1
Put a zero in the ones place under the 1
Multiply 1 x 3, get 3, and write this under the 6
Multiply 1 x 2, get 2, and write this under the 1
Add 161 and 230, getting 391
Here’s what your work might look like:
!
2
23
× 17
161
+ 230
!
391
Now, two questions:
1)
2)
When you multiply 7 x 3, why does the 1 go down below,
but the 2 goes up above?
Why, exactly, do you put a zero in the one’s place before
multiplying 1 x 3?
You may be able to follow the steps reliably, but do you really
know what’s going on, and why these steps work?
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CMDC
11/26/12
CMDC
Mathematics
Multiplying
Numbers
CMDC Mathematics
Parents’
Guide
Parents’
Guide
Parents’
Guide
Teaching Mathematics That Makes Sense
Using Area
USING AREA
Models
as a
USING AS
AREA
MODELS
A
Way
to
MODELS
AS A
WAY
TO
Understand
WAY TO
UNDERSTAND
UNDERSTAND
MMultiplication
ULTIPLICATION
MULTIPLICATION
Teaching Mathematics That Makes Sense
Teaching Mathematics That Makes Sense
Youof
can
also visualize
multiplying
numbers.
Instead
following
mechanics
we don’t
fully Instead of
following
mechanics
wemechanics
don’t
fully
let’s think about
understand
let’s
think about
things
in understand
a
more
intuitive,
Instead
of following
we
don’t
fully
things
in way.
a more
intuitive,
concrete
way.
Think of
concrete
Think
of multiplying
two
asmultiplying two
understand
let’s
think
about
things
innumbers
a more
intuitive,
numbers
as
calculating
the
area
of
a
rectangle,
the two
calculating
area
of of
a rectangle,
the twowhere
concretethe
way.
Think
multiplyingwhere
two numbers
as
numbers
are
the
lengths
the
rectangle’s
sides
–two
see diagram
numbers
are
the
lengths
ofof
rectangle’s
sides
– see
calculating
the
area
of
athe
rectangle,
where
the
below.
diagram
below.
numbers
are the lengths of the rectangle’s sides
– see
!
!
diagram below.
23
23
Consider
!
Considerthe
theoriginal
originalproblem:
problem:
×× 17 23
Consider the original problem:
× 17 !
!
When we are asked
to multiply 23 by 17,
When we are asked
we are being asked
to multiply
23are
by asked
17,
Whentowe
calculate
the
we to
are
being 23
asked
multiply
by 17,
area of a
rectangle
towe
calculate
are
asked
thatbeing
is 23the
units
high
areato
ofand
a
rectangle
calculate
17 unitsthe
wide.
A picture is worth a
thousand
A picturewords.
is worth a
thousand words.
that
is 23ofunits
high
area
a rectangle
and
17
units
wide.
that is 23 units high
and 17 units wide.
Partial
Products
focuses on
place value
Now, let’s calculate our answer again, this time using
Now, let’s calculate our answer again, this time using the
the picture
above
to guideour
us:answer again, this time using
Now,
let’s calculate
picture above
to guide us:
the picture above to guide us:
Partial Products
Method
– focuses
on
Partial
Products
place
value.
Method – focuses on
place value.
7×3
purple 1 x 1 squares
purple 1 x 1 squares
 7 × 3 orange
1 x 10 rectangles
 7 × 20
orange 1 x 10 rectangles
 7 × 20 green
10 x 1 rectangles
 10 × 3
green 10 x 1 rectangles
 10 × 3
blue 10 x 10 squares
 10 × 20
blue 10 x 10 squares
 10 × 20

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Multiplying Numbers
Parents’ Guide
Teaching Mathematics That Makes Sense
Multiplication:
Generic Area
Model
You can also multiply numbers by using what we call the
generic area model — in other words, the sides of the rectangle
!
are not proportional.
!
!
Consider the original problem:
23
!
× 17
!
! a 2-digit number,
Since we are multiplying a 2-digit number by
create a 2 X 2 rectangle and think of it like a multiplication chart:
20
3
10
200
30
7
140
21
200 + 30 + 140 + 21 = 391
Notice we get the same numbers as we did with partial
products and the area model methods.
Multiplication:
Distributive
Property
There is yet another way to think about multiplying numbers —
we can multiply numbers using the distributive property.
23× 17
= ( 20 + 3) × (10 + 7 )
First distributing the 20 to (10 + 7) and distributing the 3 to (10 + 7)
= 20 (10 + 7 ) + 3(10 + 7 )
Now distributing the 20 to 10 and 7 and distributing the 3 to 10 and 7
= 200 + 140 + 30 + 21
= 391
Understanding the connections among all of these methods will
help students in Algebra and higher mathematics courses.
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