The Second Problem of 2015

Students have turned in some astonishing work on the Bathtub problem.  Almost all teams came up with correct answers, and provided compelling arguments explaining why the answer would be 20 minutes.  A few submissions had unexpectedly creative ways of envisioning the problem.  The problem can become quite simple if we just "see" it correctly.

Time for the second problem:

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The Sums

of Odd Numbers

Here is a pattern that seems to emerge from observing square numbers.  The sums of odd numbers produce square numbers:

1 = 1  (a square number)

1 + 3 = 4  (another one)

1 + 3 + 5 = 9 (yup!)

1 + 3 + 5 + 7 = 16  (hmmm…….  interesting…….)

1 + 3 + 5 + 7 + 9 = ___________  (well?)

Check a few more.  Will this always be true?

If you don’t think this will always work, come up with a counterexample (in other words, find a case where it does not work).

If you think it will always be true, find a way to show that it will always work (in other words, find a way to convince a skeptic.)

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This problem comes from Proofs Without Words: Exercises in Visual Thinking by Roger B. Nelsen.  We hope that students will get a visual understanding of what square numbers are, where they come from, and how they grow.  

While students are having fun with this problem, they will hardly notice the serious math content they are learning:

square numbers

square roots

exponents

quadratic growth

the distributive law

binomial expansion

for any given square (81 for example), where does the square root come from?

--There are at least three distinct ways to think about this.  Can you think of three different ways that the square root 9 shows up?  is there a fourth?  (I'll give you one interesting answer:  the number of terms in the addition problem.  The sum of the first 9 odd numbers is 81!  Interesting!  Is this an accident?)

rich connections between arithmetic, algebra and geometry

the importance of seeing numbers and relationships visually

the concept of "proof"

pattern recognition

One hint:  Mathematicians draw pictures.  Think of different ways to draw 1 + 3 + 5.  Is there a way to organize your pictures so that you begin to see squares?  If you are stuck, ask your child to help you think!

Enjoy the problem!