# Multiplication Puzzle for the Very Young

I just read a recent post on NRICH Mathematics that asked readers or students to list four consecutive whole numbers and compare the products of the outer pair of numbers in the list to the product of the inner pair.  For example, if you used the list {4, 5, 6, 7}, you would have $4\cdot 7=28$ and $5\cdot 6=30$.  Nothing particularly exciting seems to be here, but try another list of four consecutive whole numbers.  Grab a calculator if you want to be particularly daring or obnoxious with the members in your list.  Do you notice anything now?

I argue the beauty of mathematics as the “science of patterns” kicks in after you find these products for a few different lists.

LEVEL 1:  For the very young who are just learning to multiply, I think this is a GRAND problem.  No proof required.  It’s just crazy cool that those two products always have the same relationship.  Allowing calculators to permit young explorers to try lists beyond their ability to hand or mentally compute enhances the mystery, in my opinion.

I just played this with my eldest daughter.  She first wrote {19, 20, 21, 22} when I asked her for a list of consecutive numbers.  When I then asked her for the products, she asked if she could use a smaller list.  She opted for {3, 4, 5, 6} and {1, 2, 3, 4} without seeing the pattern.  When I offered a calculator for her original list, she got 418 & 420.  Surprised that they were so close, she said, “Wow, they’re only 2 apart!” I asked if that happened other times.  She looked at her simpler two lists and exclaimed, “Cool!”  I asked if that always happened.  She said, “No.  It couldn’t.”  When I asked for a list where it wouldn’t, she suggested {401, 402, 403, 404}.  The outer product was 162004.  You should have seen her face after she pressed enter on the inner product to get 162006.  “Maybe it does always work!”  Then she asked if she could move on to clean her desk.  Game over … for now.

Part of the power and beauty of mathematics lies in showing that patterns are universal and aren’t limited to numbers we can manipulate quickly in our heads.  I think calculators added to my daughter’s wonder.  I’d love to see my daughter going up to one of her teachers, posing the problem, and predicting the answer without ever knowing the numbers the teacher (or anyone else) had picked.  I think I’d smile even bigger if she had a calculator at hand to offer the adult some “help” if needed!  Math is magical.  Play it up!

LEVEL 2a:  Extend to all integers.  NRICH suggests that the lists need to be whole numbers.  That just isn’t true.  You can start with any integer.  My eldest has been playing with adding negative numbers lately, so I may see if she’s interested in multiplication of negatives.  I’ll think about how to make that idea make sense to her.  At some point in the future, I’ll bring this problem up again and she’ll get an even bigger kick out of seeing that it doesn’t just apply to ordinary and ridiculously large numbers, but negatives, too.

LEVEL 2b:  Proof for the very young.  The NRICH site offers two solutions from “students”.  Whether she’s real or fictional, the approach “Alison” uses is one that I think some sophisticated young learners could grasp long before they learn what a variable is.  Granted, the geometric understanding of multiplication technically works only for specific (not generic) products, but if you set up a few of these, your young one might start to see how the areas grow as the list numbers grow, but the differences in the areas remain constant.

NOTE:  LEVELS 2a and 2b, in my mind, are pretty interchangeable, depending on the readiness and interest of your young learners.  As with all things for young people, throw out the line.  If the interest isn’t there, save the idea for another day.  If you get a nibble, prepare to play!!!

LEVEL 3:  Extend to any arithmetic sequence.  The suggestions NRICH makes for extending the problem all dance around the idea that this property works for any list of four consecutive elements of any arithmetic sequence.  The difference between the two products depends solely on the common difference of the sequence and is completely independent of the initial term in the sequence.  Try {1.1, 1.2, 1.3, 1.4}.  The difference in the outer and inner pair products will be the same as for {98.8, 98.9, 99.0, 99.1} simply because both lists increase by 0.1.

LEVEL 4:  Algebra.  Those who remember their algebra classes may have jumped right to an algebraic justification.  That’s what I did, and that’s the solution “Charlie” gives on the original NRICH post.  In a way, I think I cheated myself out of seeking the pattern as my daughter discovered it.  Whenever your young ones are ready to deal with the magic and power of variables, try out proving this for integers.  When they’re ready for more, prove it for all arithmetic sequences with any initial term.  You’ll know they’re strong when they can argue on their own why the initial term is irrelevant.

LEVEL 5:  More Algebra.  This “trick” extends to to any arithmetic sequence of any length.  With algebra, one can determine a formula for the difference between the products of the last terms and the next-to-last terms.  I think a talented middle school student or young high school student who knows how to handle very generic cases could find that formula.

And it all starts with playing with some little numbers.