Line multiplication is a nice activity for teaching multiplication especially for more than one-digit numbers. The method is shown in the figure. The horizontal line represents the number 13 where the top line represents the tens digit and the lines below it represents the ones digit. The lines are grouped according to their place value. The same is true for the number 22. To find the final answer, count the number of intersections and add them diagonally. Dr. James Tanton produced a video about line multiplication. Click the link to view.

James Tanton related this procedure to rectangle multiplication. For example, the problem 13 x 22 in rectangle multiplication is

If this is done in class I would suggest that before you show the rectangle multiplication as explanation to the process of line multiplication it would be great to connect it first to counting problem. Instead of counting the points at each cluster by one by one, you can ask the class to find for a more efficient way of counting the points of intersections. It will not take long for students to think of multiplying the array of points in each cluster. Given time I’m sure students could even ‘invent’ the rectangle multiplication themselves. Inventing and generalizing procedures are very important math habits of mind.

Line multiplication or counting intersections of sets of parallel lines is generalizable. You can ask your students to show the product of (a+b)(c+d) using this technique. The answer is shown in the figure below. Note that like rectangle multiplication this can be extended to more than two terms in each factor also. This is much better than the FOIL method which is restricted to binomials. I’m not a fan of FOIL method especially if it is taught and not discovered by the students themselves. Through this line multiplication activity I think they can discover that shortcut.

“The lines are grouped according to their place value.” Yes, but this isn’t emphasized, or easily seen, when either line or rectangle multiplication are taught in elementary school – so it becomes another meaningless algorithm.

I do like the transition from line to rectangle multiplication though.

You are right. That’s why I tried to make sense of it and made it a little more meaningful mathematically by connecting it to area/rectangle multiplication, counting problem and multiplication of algebraic expressions. Line multiplication activity should not be an end in itself.