Question

Dear Mitch,

I don't know if I am the only teacher who has ever written to you with a comment like this, and I hope it is not taken the wrong way, because my colleagues and I are great enthusiasts of your site. We appreciate the interesting new twists you give to topics that (quite frankly) are not always so easy to present in the kind of way that engages students the way we hope to when we start out in this field. So... now that I have tried to set a positive tone, which I want you to know is authentic, I need to tell you that two of your most recent questions are either not written correctly or are just too challenging for most students and (yes, here goes...) most TEACHERS of middle school math. And since I am withholding my name, I can speak on behalf of myself and admit that I have been stumped, and it is not for lack of trying. My wife is also stumped, and she taught middle school math for years and remains a highly-regarded teacher. We worked on the balance scale problem for an hour or two on three separate evenings, and now would very much appreciate either a correction in one or both problems or a couple of hints to set us in the right direction before school closes for the Christmas/New-Year recess.

So, to be clear, the two problems we'd like hints or corrections on are:

- The November 29th one, which you have illustrated with four different balance scales and several different figures (snowmen, Christmas trees, candy canes, and wrapped presents)
- The one about Santa Claus working on his potential travel route, which you have depicted as a triangle.

Thank you, and Happy Holidays to you, your staff, and your loved ones.

NAME AND ADDRESS WITHHELD

Answer

Dear N.A.A.W.,

Thank you for taking the time to write in, and thank you very much for your kind words during this holiday season. We always welcome comments, regardless of how critical they may be. I think it is fair to say that almost every committed teacher I have had the pleasure of knowing likes to learn any time and any way he or she can. So, from my perspective, there is no 'wrong time' or 'wrong place' to point out things we could be doing better. So...

First, with respect to the balancing scale problem, which is titled 'a question of balance', there is no error in its presentation, BUT it is CERTAINLY considered beyond the 'normal level' of challenge that one usually encounters before junior or senior year of high school. Even then, it is considered a serious challenge.

Since you already have the pictures, which can still be pulled up from the q&a 'archive' (the one dated November 29th, as you point out) and can still be printed right off the site, let's try another approach and look at it the 'old-fashioned' way.

Substituting S for each Snowman, T for each Christmas tree, C for each candy cane, P for each 'present', and an equal sign for each balancing point, we get the following three equations:

(a) T + P = S

(b) T = P + C

(c) 2S = 3C

And the question asks: How many P's = ONE T?

You asked for a hint, so here are the first two steps:

- In order to make the two left sides of equations (a) and (b) equal, add one P to each side of equation (b).
- This means that one S = 2P + C. OR, 2S = 4P+2C.

Equation (c) shows us that 2 S also equals 3C.

So, 4 P + 2 C = 3 C.

Now, there are a few more steps, and it is still considered QUITE challenging, but more to come tomorrow...

Regarding the Triangle/Santa question, you are correct in saying we could have presented it in a much clearer fashion. And we will do just that, along with an explanation of what went wrong and why it did so... Tomorrow, WE PROMISE!!

Until then, I hope this helps a little,

Mitch

P.S. Happy Holidays to you too!