The Distributive Property

Students should have a deep understanding of the distributive property. By understanding the distributive property, students will be able to manipulate the traditional algorithm and understand what is happening when using it.

The distributive property states:
When a number is multiplied by the sum of two other numbers, the first number can be handed out or distributed to both of those two numbers and multiplied by each of them separately. Here’s the distributive property in symbols:
a * (b + c) = a * b + a * c.

By approaching multi-digit multiplication in this way, one can assume that you would help to eliminate the confusion regarding where to line the partial products up in the traditional algorithm. Let’s walk through a problem together.

dist-prop-1.jpg

As you can see, students should be able to see the connection between the distributive property and the traditional algorithm. When using the distributive property students are also working on place value, multiplication by 10, as well as strengthening their knowledge of basic facts. Once students have become comfortable with the distributive property and move into the traditional algorithm, educators can show them why the “zeros” can be eliminated.

dist-prop-2.jpg

But where do we begin? Third grade is where we formally begin multiplication. Students are developing the concept of multiplication and mastering their “facts.” Fourth grade is where students will move into 1-digit by 2-digit multiplication. This is a great time to begin to use the distributive property. Beginning the year with problems such as 14 x 6, students can be working on the structure of the distributive property (14 x 6 = (10 x 6) + (4 x 6)) as well as their mental math. As the year progresses, and students become more facile with the ideas being developed, they can work with more difficult numbers.

Whether or not it is fourth or fifth grade, students should then move to 2-digit by 2-digit multiplication. This is likely to take more time to develop then one-digit by two-digit multiplication. I also thinks that once students comprehend the 2 by 2 digit multiplication, they will easily be able to apply their knowledge to 3 by 3-digit multiplication. Like the above illustrations show, work with the distributive property first, then relating it to the traditional algorithm demonstrates what actual happens in the traditional algorithm. These types of experiences help to bridge the distributive property and the traditional algorithm.

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2 thoughts on “The Distributive Property

  1. I really like how you pointed out the connection between the distributive property and the traditional method of multiplication.

  2. @loop2dsto@gmail.com-Thank you. A lot of these ideas came from Liping Ma in her book, Knowing and Teaching Mathematics. I feel that this not only develops an understanding of multiplication, but number sense and moves them along in their understanding of algebra.

    Dan

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