Discrete Math - Day 11

Today we worked on another flag problem. Before working on this problem I passed back their quizzes and talked about what I was looking for in responses. I emphasized that they needed to do more than simply write down a correct answer, which would receive an incomplete grade. There were a few students unhappy about this but in the grand scheme of things one score of 6 out of 10 in a category that accounts for only 10% of their grade will not make any significant difference in their grade.

I also had several students ask about making quiz corrections. My school has instituted quiz and test corrections in lower grades in the hope that students will learn and hopefully master content that they did not grasp at the time of taking the assessment. I have mixed feelings on this practice, since it tends to have students not study and prepare as much as they should ahead of time. Because my classes are composed primarily of seniors and since colleges and universities (I teach night courses at Metropolitan State University at Denver) do not typically have this policy, I do not allow quiz or test corrections and I do not provide extra credit.

A student's grade should reflect their knowledge of the content: concepts, skills, application, etc. At the same time, the tasks and assignments that students complete are scored in such a way that it is difficult for students to not receive a passing grade if they are doing the work they should. On the other side, students who receive an A grade should clearly demonstrate knowledge and performance well above average. Those students who receive an A grade should earn that grade.

The flag problem was a variation of the first problem. This problem asks how many flags can be formed with at least 2 blue stripes on it. We briefly revisited the results of the problem for at least 6 blue stripes and then students dove into the problem. I was encouraged to see students looking for patterns or making lists and other representations.

Some groups tried to focus on looking at flags with 5, 4, 3, and 2 blue stripes. Others initially started trying to figure out what was happening with the 2 blue stripe flag. These later groups quickly found themselves stuck. They correctly calculated the number of options for 2 blue stripes on the flag as 8C2 = 28. Some groups even used the successive values of combinations to determine how many ways 3, 4, and 5 blue stripes could be placed on the flag. What they struggled with was the remaining stripes.

I asked them questions about the remaining stripes. How could they be configured. What color options were available, etc. When they started listing these the students quickly gave up as the number of options seemed immense. I suggested that they reduce the size of the problem to get a feel for what might be happening. What if you only had a 3 or 4 stripe flag? Only one group pursued this investigation.

The two groups who started looking at flags with 5 blue stripes were much more successful. One group started listing what the other 3 stripes could look like and came up with 8 configurations. When they looked at flags with 4 blue stripes they listed out 16 configurations for the other 4 stripes. As they started listing out the options for 5 non-blue stripes they realized the options were doubling every time. Looking back they saw this doubling pattern start from the very beginning. Using this pattern they quickly found the remaining values and calculated the total flag options correctly.

The second group to start using the 5 blue-striped flag noticed a connection to the flag options and Pascal's triangle. They also realized that the options were doubling and calculated a correct total. Both groups came to their answers almost at the exact same time and were excited that the other had arrived at the same answer.

We had students discuss what they were trying, patterns that they saw, where they got stuck and why. The two groups that arrived at correct solutions then presented their results and the class discussed why these worked. When the patterns and connections to Pascal's triangle were presented several students were saying, "I knew there had to be a pattern connection" or "I told you there would be a connection."

Even though most groups got stuck I was pleased to see that students were attempting to look for these patterns and connections. It was also interesting to see how changing the starting point so radically changed the ability of students to complete the task. I'll need to remember that "try starting from the other end" should be another suggestion I give to students as they work through sticking points.

We'll be working on card-based problems next class. I think the students are ready to move on from the flags.

Visit the class summary to read a student's perspective on the class and view lesson slides.