Differentiating homework with rich tasks

Differentiation is an ongoing challenge for me.  I teach math to gifted 6th graders, but with our district’s broad definition of “gifted,”  the range of my students’ mathematical knowledge and general aptitude is pretty wide.  I have to be honest and admit that I don’t do the best job at differentiating assignments and lessons for my students.  It is so is hard!  I’m happy to  have a success story to share from this week, though.  Just in time for Julie Reulbach’s #msSunFun assignment to blog about differentiating homework!

Using rich tasks is a method of differentiation that really works for me.  If you give students a meaty problem with multiple access points, lower-performing students can still get into the problem while high-achieving students can really dig into the problem and take it to the next level!  One of my favorite resources for rich problems is nrich.maths.org, which is where I found the problems I used this week.  (I professed my love for nrich at the #GlobalMath meeting last Tuesday, February 5th, so if you missed it, check out my own Favorite — plus favorites from other teachers — by watching the recorded session at Global Math Department at BigMarker.com).

Back to the problems and how I differentiated for homework.  We’ve been working on mean, median, mode, and range, but I didn’t want to just hand my students sets of numbers and ask them to find the mean, median, mode, and range of each set.  Enter nrich’s problem “M, M, and M“:

There are several sets of five positive whole numbers with the following properties:

• Mean = 4
• Median = 3
• Mode = 3

Can you find all the different sets of five positive whole numbers that satisfy these conditions?

Can you convince us you have found them all?

If I also tell you that the range is 10, can you identify my numbers?

I assigned this problem for homework, knowing that I wanted to spend class time the next day discussing the idea of how you’d know that you have found all possible sets.  I wanted every student to come to the next class being able to participate in the discussion and having something to contribute. But for some of my students, just finding a few sets of numbers that satisfied the mean, median, and mode requirements was enough to challenge them significantly.  On the other end of the spectrum, I expected that a handful of students would be able to tackle the full problem and identify EVERY possible set.  So for homework I gave a minimum number of sets that students had to find (5), but I challenged them to find more than that, and I told them I’d really love it if they found all possible sets.  Of course they all wanted to know how many sets were possible, and of course I wouldn’t tell them! (We’re over halfway into the school year and some students still think I’d answer a question like that?!)

The next day they came into class BEGGING to know how many sets were possible. I had some students come in with the required 5 sets, many come in with more than that, and a few come in with every possible set.  I was able to quickly pull out the students who’d found all (or most, or in a couple of cases, too many) sets so that they could work together and not spoil the solution for the other students.  Students then worked in groups of 3 or 4 students to come up with a list of all of the sets possible, and nearly every group came up with all correct sets in the end!  For the groups that started with most, or all, of the correct solutions, I asked them to spend their time working to prove that their lists were complete.

For the next night’s homework, students tackled the Final Challenge that goes with the M, M, and M problem.  It really was a challenge, and even after our “how do you know if you’ve gotten all of the possibilities” discussion, many students had trouble wrapping their minds around what the challenge question was even asking.  I asked students to find at least 3 sets of numbers that would work.  Just like before, many of them found more than that, and a few found all of them

How many sets of five positive whole numbers are there with the following property?

Mean = Median = Mode = Range = a single digit number

When I give one of these “Do at least this much, but I’d love it if you did more” homework assignments, it’s always interesting to see who will try to go beyond the minimum requirement.  Of course I have the usual bunch of high-achievers, but every once in a while one of the other students will really surprise me with the effort that they’ve put into a non-required assignment.  Those are the ones who really make my day!