# Multiplication Puzzle 2 for the Very Young

A former student of mine, Paul Sperduto, is currently teaching 5th grade math in Houston as part of Teach for America.  After reading yesterday’s post on inspiring multiplication in young learners, he responded with the following story.  I reprint it here (with Paul’s permission) for two reasons:  to give another fun math game for parents and teachers to use with eager young learners, AND to show what can happen when young people have the freedom to think and explore (whether explicitly designed or personally re-claimed as in Paul’s case).

Here’s Paul’s story in his own words with occasional commentary from me in brackets.

I use a similar multiplication “trick” with my students, and I think it is setting them up very well to learn and understand applications for the difference of squares down the road. It’s actually something I randomly happened into in about 3rd grade (I was one of those elementary school kids who knew his times tables before I even started 1st grade, so I had a lot of math class time to think about stuff like this…), but I didn’t really understand why until I was much, much older.

[As I’ve noted earlier, it is far more important for younger learners to play the game.  Finding a pattern you think others have overlooked is far more motivating for young people than knowing why the trick works.]

I told my students that I could mentally multiply any two numbers between 1 and 31 in under five seconds as long as the numbers have an even difference. For example, I can multiply $23\cdot 27$ in under five seconds since $27-23=4$, but I can’t do $23\cdot 26$ that quickly because $26-23=3$.

[This is a glorious hook!  From experience, many students will jump at a claim like this.  Part of the game becomes trying to find two numbers the teacher can’t handle.  Learning = game = fun.]

When they asked me how I do it, I told them that if they memorize their squares (I have mine memorized through 30), they can do it too. This was surprisingly motivational, and I quickly had many students with a lot of squares memorized, eager to learn how to use them.

[This is what happens when you make learning FUN.  Even seemingly dry topics like memorizing squares of numbers becomes a worthwhile endeavor when it has an entertaining purpose.  Students are willing to engage in what they perceive to be drudgery if there is a payoff.]

The trick lies in the difference of two squares formula, $A^2 - B^2 = (A+B)*(A-B)$. Given the example of multiplying 23 and 27, it is easy to see that 25 would be the mean of the two numbers–each is 2 away from 25. So if A is the mean of the numbers, and B is the distance to the mean from each, the difference of squares formula gives us the answer to the problem: $27\cdot 23 = (25+2)\cdot (25-2) = 25^2 - 2^2 = 625 - 4 = 621$

That last step is easy once you’ve memorized the squares.  So as long as you do this and are good with generally simple subtraction, this trick is very easy.

It’s been fun trying to help them discover this one. We started by noticing patterns near the squares on a standard multiplication chart. They noticed that all squares have another multiplication that gives one less than the square. For example $6\cdot 8=7^2-1$ and $10\cdot 12=11^2-1$. Working our way along the diagonals of the chart, they also discovered that each square has a multiplication that gives 4 less than the square ( $5\cdot 9 =7^2-4$), 9 less than the square ( $4\cdot 10= 7^2-9$) and 16 less than the square ( $3\cdot 11=7^2-16$). It took a little prodding, but they figured out that the differences are just the squares, and it was all downhill from there.

[Here’s another thought:  I suspect participants typically make these problems easier on the “performer” while thinking they’re doing just the opposite.  Many likely think giving two bigger numbers to multiply would be a harder task because the algorithms they typically are required to use in school become longer, if not more challenging, when longer numbers are employed.  But using two larger numbers under Paul’s approach probably makes the value of B smaller in most cases, simplifying the final subtraction.

Thank you, Paul, for sharing.]