Making Sense of
Two-digit Multiplication and Beyond
24
x 35
Take a moment to think about how you would perform the
computation. How you probably learned to perform this computation involved
using an algorithm driven by speed and accuracy. Most of us never were asked by our teachers or considered
why the algorithm worked. We simply computed. Our goals are different now. If
we want students to perform the computation with speed and accuracy we can
simply hand them a calculator. Our desire now is for something more from our
students than performing steps to a memorized algorithm. We want them to understand the purpose behind
the steps and the meaning associated with the steps of the algorithm. We want them
to understand the structure of multiplication. We also want students to
understand the relationship between place value and the computations performed
to complete a two-digit (or more) by two-digit multiplication algorithm. The
goal is to make a clear connection between the computations performed with a visual,
concrete representation and the abstract representation. Base-Ten Blocks are an
effective visual, concrete representation for modeling a two-digit
multiplication problem such as 35 * 24. Making a clear link to place value
ideas, the numeral ‘35’ represents 3 tens and 5 ones and the numeral 24
represents 2 tens and 4 ones. These two numerals are represented by the outer
boundaries shown in red. The computation shown on the right is the Partial
Product Algorithm whose results are also shown in the Base-Ten Block
representation.
The top Partial Product Algorithm computations are written
in terms of the units shown in the four quadrants of the Base 10 representation
{20 ones, 10 tens, 12 tens, and 6 hundreds}. Three of the four computations can
be re-expressed (i.e. 20 ones is also 2 tens, 10 tens is also 1 hundred, and 12
tens is also 1 hundred and 2 tens) yielding an answer of 8 hundreds and 4 tens
(840). Interestingly, when you write the numerals representing each of these
partial products as shown in the bottom Partial Product Algorithm (20, 100,
120, 600) both interpretations are visible. For example, the numeral ‘20’ can be interpreted as either 20 ones or 2 tens and the numeral '120' can be interpreted as either 1 hundred, 2 tens, or 12 tens.
A less concrete, visual model than Base 10 Blocks is an area
model of multiplication. Each numeral is separated by the place value of each
of the numbers. The area model representing 35 * 24 is shown with 35 as 30 (3
tens) and 5 (5 ones) and 24 as 20 (2 tens) and 4 (4 ones).
One of the key points of making sense of multiplication
through visual representations is the reasoning and sense-making potential
these representations afford. Although not an exhaustive list of the potential
reasoning and sense-making possibilities the three examples shared exemplify the
potential of these models to help students prove conjectures generated from
analyzing number pattern relationships, to explain why a mental arithmetic
strategy works and to make sense of algebraic relationships.
Proving conjectures
generated from analyzing number pattern relationships
Suppose that you introduced the following pattern to
students and asked them to reason about how one could use the given information
to determine the value of 852.
152 = 225
252 = 625
352 = 1225
452 = 2025
A typical response from students is that 852 is
7225 because you take the tens digit {8} and multiply it by the next
consecutive tens digit {8*9} to find the last two digits and drop 25 down to represent
the first two digits {7225}.
Reasonable follow-up questions are 1) Does the algorithm always work when the
units digit is a 5? 2) Are there other multiplication facts in which the
algorithm will work?, and 3) Why does the algorithm work? Understanding the
structure of the multiplication in terms of visual representations might enhance
students’ ability to reason about each of these questions. Using an area model
one of the instances in the pattern, 252, is shown.
Recall that the student’s algorithm involved multiplying the
tens digit {2} and multiplying by the next consecutive tens digit {2 * 3 = 6}
and placing this value to determine the number of hundreds {6__ ___}. It is
important to note that mathematically the student was actually multiplying 20 *
30 or 20 groups of 30, or 2 tens and 3 tens.
Moving the rectangle
representing 20 groups of 5 to the right of the figure it can be seen that the
two rectangles together represent 20 groups of 10 which means there will be a
total of 20 groups of 30 {600} and 5 groups 5 {25} or 625.
Modeling multiplication using an area model provides the
means for students to reason and make sense of why not only this particular
student’s algorithm works but also why other potential algorithms generated by
students might work. The area model also provides the means to reason about why
the last two digits are always a 2 and a 5 and why problems such as 23 * 27
{621} can be computed using the student’s algorithm but other problems such as
squares not ending in 5 (e.g. 272) or those in which the units
digits do not add up to ten {e.g. 23 * 26} will not work.
There is also the potential to make a connection between the
concrete representation shown with the area model and the abstract
representation that shows in general why the student’s algorithm works. The
abstract representation of the length of each side of any square can be
represented by the expression 10n + 5
in which n represents the number of
tens. For example, 252 can be written as 10(2) + 5 in which the 2
represents the number of tens. Squaring the expression 10n + 5 results in the
expression (10n + 5)2 = 100n2 + 100n + 25 = 100(n(n+1)) +
25. The first term 100(n(n+1)) yields the number of hundreds and is analogous
to the student’s step of taking the numeral representing the number of tens and
multiplying it by the numeral representing the next consecutive tens digit
(e.g. 100(2(2 +1) = 100(2 * 3)) = 600). The abstract representation 100(n(n+1))
+ 25 also reveals why the last two digits are a 2 and a 5 since the first term
reveals the number of hundreds and the last term reveals the number of tens and
ones (e.g. 100(2*3) + 25 = 600 + 25 = 625).
Making sense of why a mental arithmetic strategy works
A mental arithmetic strategy for computing a multiplication
problem such as 18 * 22 is to square the number that is in the middle of the
two numbers {202 = 400}. Subtract that number {400} from the
difference each number is from the middle number squared {22}. In
other words, or symbols, 18 * 22 can be computed by taking 202 and
subtracting 22 {400 – 4 =
396}. Check it out! Pretty cool! Try it for 33 * 27! Remember in an earlier
problem it was discussed that 252 = 625. We can apply that knowledge
and the mental arithmetic strategy just discussed to also find 24 * 26 or 23 *
27. The big question is, why does this mental arithmetic strategy work? As
before, refer to an area model for 18 * 22, but this time the area model is
represented using a Cartesian plane.
The model is representing 18 groups of 22 or 18 * 22.
Consider cutting the last two columns (18 groups of 2) and moving the piece to
the location shown to make a larger square with a smaller square (shown in
black) missing. The area of the larger square is 400 {20 * 20} and the area of
the smaller square is 4 {2 * 2}. This results in an area of 396.
This mental arithmetic strategy can be connected to an
important algebraic idea, difference of squares. The generalized version of
difference of squares is (x + y)(x – y) = x2 – y2. Now think about this idea in terms of
computing 18 * 22. We can represent 18 * 22 equivalently as (20 – 2)(20 + 2) =
202 – 22 = 400 – 4 = 396. Once again making a connection
between what we are doing arithmetically with an important algebraic ideas.
Making sense of an
algebraic relationship
A
mistake that many algebra students make in expanding the expression (x + y)2
is to square each term resulting in the expression x2 + y2. Using
an area model to represent this action not only exemplifies the meaning of “squaring”
but also why the expression (x + y)2
is not equivalent to x2
+ y2 but is equivalent to x2 + 2xy + y2.
Findings
from brain research suggests that vision is the primary way that most of us
receive stimuli and that analogous thinking is a strong learning mechanism. (Almarode,
2016, personal conversation). It makes sense then that teachers can significantly
impact student learning by providing opportunities for students to reason about
important mathematical ideas, such as multiplication, using visual representations.
It also makes sense that student learning can be greatly impacted by providing
them opportunities to reason about the relationship between what he/she “sees”
in the visual, concrete representations of those ideas and what he/she “sees”
in the abstract symbols, as well as making connections between these two
representational forms.