*Most of my readers know that my philosophy is to challenge ALL of our students more than we do at present. The following problem should not be viewed therefore as a math contest problem for middle schoolers; rather a problem for all middle schoolers and on into high school
*

**List all 5-digit palindromes which have zero as their middle digit and are divisible by 9.**

**Comments:**

(1) Should you include a definition or example of a palindrome as is normally done on assessments or have students "look it up!"

(2) Is it necessary to clarify that we are only considering positive integers when we refer to a 5-digit number

**?**

(3) What is the content knowledge needed? Skills? Strategies? Logic? Reasoning? Do these questions develop the mind while reviewing the mathematics? In other words, are they worth the time?

(4) BTW, there are ten numbers in the list. Sorry to ruin the surprise!

(5) How would this question be worded if it were an SAT problem? Multiple-choice vs. grid-in?

## 6 comments:

1) I'd ask who could explain what it meant. "Look it up" is something I'd beg off on as something I "don't have time for".

2)I don't think so

3) Content knowledge: divisibility by 9 rule

4) Wait, really? I have 9. Are you letting the number begin with 0? That typically wouldn't count as a five-digit number.

5) Not sure about the SAT but I'd ask how many rather than asking them to make a list. I would assume most of them *would* make a list, but knowing *to* make the list is part of what I want them figuring out.

Mathmom,

Thanks for your thoughts about instructional decision-making.

Did you remember to include both 90009 and 99099?

I designed this problem so that it would be inviting to many students. On the surface it's fairly straightforward but I like to add a little twist for the super-quick student who might jump too quickly. Also, considering how many careless errors I make, I am gun-shy about giving my answer!

Ah, 99099 is the one I missed :)

I didn't answer the meaty part (

"Do these questions develop the mind while reviewing the mathematics? In other words, are they worth the time?") because you already know my answer to this.I may add this to my list of questions for my older group on Friday. In fact I need to review recent posts because there were a few problems I wanted to try with those guys...

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So what if we lift the restriction on the middle digit? (major extension. Changes focus)

Or if we ask for palindromes under 100,000? (small but important extension. Forces dealing with "middle," a brief additional search, and whether a 1-digit number is a palindrome, and whether 0 is a 1-digit number (or a "number" at all).

I think I will try to find space to use both extensions.

Jonathan

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