I tried a problem today in three classes, two 9th grade and on 7th grade, which I consider successful. I found the problem from Alicia Burdess (aliciaburdess.com) in a collection called Teaching Through Problem Solving. Also credited are Daniel Student, Geri Ann Lafleur, Dawn Morris Blackburn, Doris Duret, and Jonathan Scott.
The basic setup involves talking about number palindromes. I get into that with sentence palindromes, and my current favorite (which I first heard from my son) is, "No sir! Away! A papaya war is on!" I stretched this part more with the 7th graders, but eventually get into number palindromes, and then introduce an idea called palindrome depth.
If you take a number like 84, which clearly is not a palindrome and reverse the digits you get 48.
Add the two, 84 + 48 = 132, which is still not a palindrome. Reverse the digits and add again.
132 + 231 = 363, a palindrome.
Because it took us two operations to get a palindrome, we say that 84 has a palindrome depth of 2. Some numbers, like multiples of 11, have a depth of 0.
The task: find the palindrome depth of all two-digit numbers.
It was great that all students are able to access the problem. I definitely consider it low threshold. Students of all levels were excitedly exploring. I use visibly random groupings of three, and almost all groups were dividing labor and sharing results, and both making and testing conjectures.
One of the most interesting extensions a student with colored pencils came up with was to make a 9 x 10 table with each two-digit number, and they color-coded each number palindrome depth. Wow!
I will definitely use this one again.
