Thursday, July 14th, 2011...8:37 pm

A muggle's math magic

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Tomorrow, Harry Potter and the Deathly Hallows РPart 2 opens around the country. It is also the day I return to Charlotte from the Yale National Initiative’s Intensive Session. In honor of a successful week at Yale and what will undoubtedly be a successful opening day for a movie, this blog entry is inspired by a Yale National Initiative seminar that can uncover the trick behind the apparent magic in the YouTube movie below:

Yesterday, I visited the seminar Great Ideas of Primary Mathematics led by Roger E. Howe and Amanda Folsom. A technique taught in the session was the use of a table to deepen the understanding of multiplication. For instance, suppose we multiplied 21 by 13. Then, we form the table:

  10    3

Then, each entry of the table is filled with the product of the numbers in its respective row and column, as seen below.

  10    3
20  200   60
 1 10 3

Now, to simplify the discussion below, let’s concentrate only on the entries that we were products of row and column headers in the table above. This gives us:

 200   60
10 3

Adding the entries of the table gives the desired product to our problem, 273. However, notice for a moment that we could also add along the 3 diagonals that run from the upper right to the lower left. Our first diagonal has only one entry 200. The second diagonal has the numbers 60 and 10; so the sum is 70. Finally, the last diagonal has the entry 3. As you sum along each diagonal, the numbers have the same place value. The first is hundreds, the second tens and the last ones.

Let’s use this knowledge to look at the trick again. Below is my rendering of the trick with the intersection points delineated by color.

Compare this to the table and adding along those diagonals. Notice a similarity? If not, the colors of the dots may help. Each color corresponds to a different place value. Each purple dot is a hundred, each green ten and each orange one. Further, notice how adding the dots, particularly in the middle of this picture, corresponds to adding along the diagonals in the table.

Got it? Try the method of drawing lines to find the product of 32 and 12. When you get that, try 62 times 14. You might want to go back to the table, if you struggle to think through what to do with the sums you get with the points.

Magical? Indeed, at least for mathematical Muggles!

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