Multiplication Strategies
2 digit x 1 digit
The first three strategies allow a student to break down the factors so that they can use
the multiplication facts they already know to solve the problem. We can also draw the
original array on grid paper and model how to partition the array into smaller arrays
which helps them visualize the problem.
1) Break down a factor into smaller components, multiply each part, and then add the
parts. For example:
13 X 8 = ____
13 x 8 =
(10 x 8) + (3 x 8) or
(6x8) + (7x8)
10
3
8
10 x 8
3x8
2) Recognize scenarios in which they can find the product using half of a factor and then
doubling the product. For example:
26 x 4 = _____
26 x 8 =
(26 x2) + (26 x2)
(26 x 4) + (26 x 4)
3) Add or subtract a factor from a known fact. For example:
13 x 8 =
12 x 8 = 96 + 8 = 104
4) Traditional algorithm
2 digit x 2 digit
The first three strategies are a type of partial products strategy since the students break
the problem apart into simpler equations, find the product of each, and the combine the
products.
1) Break down both factors by place value, multiply all of the parts, and add the
products. We draw arrays on graph paper to illustrate the problem and how each
product can be broken down into expanded form. While this strategy (using the
following visual array) may take longer for a student to complete, it builds directly
upon their knowledge of factors, arrays, and simple multiplication facts. For
example: 32 x 27
32 = 30 + 2 and 27 = 20 + 7
30
30 x 7=210
Final Product : 600 + 210 + 40 + 14 = 864
2x7=14
7
30 x 20=600
20 x2=40
20
2
Most of the students do not need the actual graph paper any more, but some still draw
a model of this picture to help them visualize the problem. For example, they might
draw:
30
20
2
20
X2
20
30 x 20 = 60
7
2
X7
14
30 x 7 = 210
Other students are able to break the factors apart into these four equations without the
help of the array.
2)Window pane – This strategy is similar to the one described above because it still
breaks each factor into place value. However, it does not break the whole array into
smaller arrays of representative size. It’s really just a shortcut for the previous strategy.
32 x 27 = 600 + 210 + 40 + 14
30
2
20
600
40
7
210
14
3)Break down one of the factors into multiples of 10, multiply each part, and add the
products. This strategy can also be represented visually with an array similarly to the
arrays drawn above.
32 = 10 + 10 + 10 + 2
27 = 10 + 10 + 7
So 32 x 27 =
(10x27) + (10x27)+(10x27) + (1x27)=
270 + 270 + 270 + 54 = 864
Or
32 x 27 =
(30x27) + (2x27) =
810 54 = 864
So 32 x 27 =
(10 x 32) + (10x32) + (7x32) =
320 + 320 + 224 = 864
32 x 27 =
(32 x 20) + (32 x 7)=
640 + 224 = 864
4) Traditional algorithm – After practice, many students will prefer this method because
it is shorter. When I teach this method, I will link it back to all of the work we did this
week so they can see the rationale behind each step.
First I will teach them a strategy called the Expanded Algorithm that bridges the window
pane strategy from above with the standard algorithm:
32
X 27
14 (2x7) These equations & products correspond directly to products they find
210 (30 x 7) in each square of the window pane strategy which may promote their
40 (2 x 20) understanding of each step.
+600 (20 x 30)
867
Next, we will use Expanded Algorithm strategy to lead into the standard algorithm since
we have an example to show that 14+210 = 224 and 40+600 = 640 (the subproducts
they find when regrouping). I also constantly remind them to line up their place values
and focus on multiplying first the 1’s place and then the 10’s (I even cover up the place
value I’m not using with some students).
Some of the students may still prefer one of the previous three methods because they
make more sense mathematically even though they take a bit longer to complete. My
goal is for the students to become comfortable with several strategies that they can use.
I know that I use the place value method when I calculate problems in my head, but I
use the algorithm when I work them out on paper. There really is a time and place for
all of the strategies.
Please feel free to contact me if you have any follow-up questions.