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 x 8 3 x 8 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

2 0 20 30 x 20=600 x 2 = 4 0

30 x 7=210 2x 7 7 = 1 4

Final Product : 600 + 210 + 40 + 14 = 864

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:

20 X2 20 20 30 x 20 = 60

2 7 30 x 7 = 210 X7 14

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 2 600 40

2 0 210 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 = So 32 x 27 =

(10x27) + (10x27)+(10x27) + (1x27)= (10 x 32) + (10x32) + (7x32) =

270 + 270 + 270 + 54 = 864 320 + 320 + 224 = 864

32 x 27 = 32 x 27 =

(30x27) + (2x27) = (32 x 20) + (32 x 7)=

810 54 = 864 640 + 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.