Good Questions Lead To Good Questions

Hitting the Bullseye

During a recent scan through my Twitter feed, I came across a tweet from Megan Hayes-Golding (@mgolding) about a problem she had shared:

The original problem came from LC Dawson (@CDawson) that I had missed when it was originally shared on April 10:

I loved the simplicity of the problem and thought it was very accessible mathematically but still promoted thinking, problem-solving and mathematical reasoning.  This problem can easily be adapted to allow for more challenging or easier versions and students can make their own to share with a friend.  The other thing I liked about this is that though students might have the mathematical tools to solve the problem, they probably had never seen anything like this before.  Solving non-routine problems in mathematics class is important because students can very easily fall into the trap of being good "problem performers" and not good "problem solvers".  More about that here from the NCTM article Fostering Mathematical Thinking and Problem Solving:

The dilemma is that the instructional approaches advocated by many commercially available problem-solving resources and curricula encourage teachers to train their students with “how to” approaches to problem solving. They develop students as problem performers, students focusing on an end or completion of a problem. Teachers using these materials may—

• present problems that can be solved without much cognitive effort;
• supply or expect the use of a specific heuristic or strategy for solving a problem; or
• provide their students with specific formats for their problem response or write-up (e.g., restate the problem, explain your thinking, check your work).

When students experience these learning opportunities, they develop a narrowly defined view of mathematics and problem solving. These instructional approaches can leave students dependent on prescribed processes and unable to readily face problems without an immediately obvious strategy. The short-term goal of developing students who complete the problem (problem performers) may be attained, but the more important long-term goal of developing flexible and fluent mathematical thinkers (problem solvers) may not be.

I also made an immediate connection with the Numberphile "Chicken Nugget" video which makes the rounds from time to time.  Numberphile's Chicken Nugget problem goes like this:

McDonalds sells chicken nuggets in 6, 9, and 20 pieces.  What is the biggest number of chicken nuggets that you cannot buy?

I usually start this by asking easy questions such as "Can I buy 26 nuggets?  How about 24?  Can I buy 10?" to get students thinking.  The thing with the Chicken Nugget problem is that the numbers can get quite cumbersome and I've seen adults trying to solve it get frustrated because the numbers grow quickly and they do not see patterns emerge quickly enough to keep their attention.

I knew I wanted to try this problem with a group of students so I emailed a friend, Ms. Mohammed at St. Andre Bessette because I knew she had a keen group of grade 5 problem solvers who are always up to working on a good problem.  I also knew they had been doing a lot of work on mathematical problem solving so this would put their learning to the test.  My colleague Mike Jacobs (@msbjabobs) joined me to be another set of eyes and ears in the classroom because I didn't want to miss anything.  Coincidentally, he had some laminated hundreds charts on his desk that were used for another activity and we brought those along just in case.

Mike and I went to the school were greeted by 30 enthusiastic 10 and 11 year olds who were eager to get to work.  Here is the original problem we asked them:

If you want to work on this yourself, stop reading here and give it a go!

I decided to go with a little bit more complex problem than the original 5, 2 dartboard because I wanted them to get a little bit stuck into the problem before getting an answer.  The first thing students said were answers like "a million!" or "infinity!" but we prompted them to prove their answers were correct.  Bringing the laminated hundreds charts proved to be a great idea as students gravitated towards them to keep track of their thinking.  Using dry-erase markers on them allowed for easy correction of any mistakes or missed counting.

Most groups began their thinking by looking at what numbers they can and cannot make using 4, 7, and 10:

After some initial thinking, some generalizations began to occur.  One group noticed that they had a 10 to work with so once they found a number that was possible, they could also get any other number in that column of the hundreds chart.  When asked, they told us that they could also get any numbers beyond the hundreds chart because the pattern would carry on:

I was very impressed by that level of reasoning and proving from a group of grade 5s and the hundreds chart really supported their thinking.  Other groups came to the same conclusions and worked towards finding the biggest number that was not possible:

Eventually groups began to hone in on numbers that were not possible:

So when asked "did we get it?" they were asked to prove their answer to their classmates and share their thinking.  As groups began to near the end of the first dartboard, we created another to work on:

And then another:

Some students continued with the laminated hundreds charts while others used the iPad app CountingBoard2:

The final dartboard I shared was a little bit different.  Can you figure out why?  Maybe you can try it too:

To avoid spoiling your thinking I won't share the answer,  but these grade 5s were able to make amazing generalizations about this dartboard and numbers you can and cannot make.  

The mathematics behind all of this is called Frobenius Numbers or The Coin Problem.  More on that can be found here, here and here.

A huge thank you to Ms. Mohammed and the grade 5s at St. Andre Bessette Catholic School. They have worked very hard on being great math problem solvers and that effort is reflected in their work on this task!!!