Line Multiplication, Intersections and Finite Regions
After exploring the outer regions of Chinese Stick Multiplication, I started to play with line multiplication in another way. I drew two horizontal lines about two inches apart and for the example 2 x 3, placed two points on the top line and three points on the bottom line and then connected each of the top points with each of the bottom points. There were a total of six lines representing the product of 2 x 3. I tried this with a few other problems and the number of total lines generated the product each time.
Then it got interesting: I looked for patterns in the number of intersections that the connecting lines made (I excluded the intersections made by the horizontal lines). I also looked at the number of finite regions created by these lines. The concept of finite regions was new for most children but they easily saw the comparison of this type of bounded region and an unbounded or infinite region. In the 2 x 3 example, there were three intersections and eight finite regions. I then began to try other multiplication problems and recorded my results on a spreadsheet (see the pdf which ends with “(Record Results)”).
For the kindergarteners and some of the first graders, I created a separate document that already had the lines drawn since it is difficult to use a ruler for many children. See the pdf that ends in “(Lines Provided)”; an example is on page one here. Here the children only were required to count the total lines (the product), the number of intersections and the number of finite regions. I encouraged the children to use only their pencil point as a counting device (fingers and erasers only obscure their view and almost always result in errors). I asked them to touch each line as they count. When counting the intersections, I encouraged them to make small dots on each intersection as they count so they can see what they have included. Finally, I would like them to number each of the finite regions to make this process easy. Be careful not to miss those very small triangular regions.
The older children (2nd through 6th grade) where tasked with drawing the lines on the pdf ending in “Blank”. This is almost impossible to accomplish if you do not use a ruler. I taught them my best practice for ruler use (your pencil is the tree, hold your ruler under thumb and fore and middle finger with fingers spread wide with the center of gravity in the middle of the ruler and next to the line they are drawing. They should push the ruler agains the already placed pencil and then swivel the other end toward the second point; push down and lightly make the line.
The kindergarteners and first graders do not have to record the results in the table on Record Results pdf., but they should record their results next to each line figure. The 2nd through 6th graders should record their results on the four column spread sheet as well as the Line Intersection Multiplication Table.
They will likely see a most fantastic pattern emerge. First they may see the relationship between the Product, Intersections and Finite Regions. In every case, the sum of the Product and Intersections minus one equaled the number of Finite Regions. In our example, 6 + 3 - 1 = 8. I recorded each of the intersection results in a special multiplication table on the last page of the above referenced Record Results.pdf. On a normal multiplication table, 2 x 3 would yield a result of 6, but on my table since we were recording only intersections, we would record a 3.
If they see the pattern they can find out the number of intersections and finite regions for every multiplication problem. For example, 10 x 5 would yield 50 lines, 450 intersections and 499 finite regions. I would never attempt this mechanically as it would be impossible to see each region an intersection. Here is a hint to find this algorithm: look at the row and column with the number 2. The numbers are 0, 1, 3, 6, 10, 15, ..... We will be exploring this sequence of numbers next week. Those numbers actually create its own real multiplication table within our intersection table.
