Open Math Problem Solving

 I decided this month that I am going to create open math problems for Teachers Pay Teachers, because they work fantastically with my gifted students. Open math problems are questions that have more than one right answer.  Therefore, my students can explore many solutions and can come up with ways to generate multiple solutions to a problem rather than one limited way.   

For example, rather than give my 4th graders the question: What is 25 X 17?  I ask them how many  ways can you solve 25 X 17?  

I then follow up with questions like:

• How can you represent that visually? 
• Can you create your own model?  
• How is this strategy connected to another strategy you used? 

Open problems, especially ones completed on white boards, lend students to push their thinking farther and give me avenues for higher level questioning.  Also, a bonus is that almost all my students can access the questions at some level but then its easy to give students a nudge farther along the continuum of understanding. 

Another open question I tried was one from Youcubed and mentioned again in the Building Thinking Classroom tasks.  It is the painted cube task that you can find here: Painted Cube.  In the problem, it asks if a 4 X 4 X 4 cube (made up of little cubes) was dipped in paint, how many smaller cubes would have 3 faces painted, 2 faces painted, 1 face painted and no faces painted.

I liked this problem because all my students were able access it and many of my students built the model with cubes in my classroom.  I was able to extend it for my gifted students fairly easily by asking what would happen in a 5 x 5 x 5 cube and a 6 x 6 x6 cube and then creating rules about the patterns. 

One key teaching technique that helped me was actually putting all their information into a graphic organizer, that I created   This organizer provided the structure around the patterns so we could have a math discussion about what we noticed and wondered.  We talked about why patterns emerged as well. 

As I reflect on this lesson, I am reminded of how graphic organizers, when used correctly, can aid in student understand and deepen thinking.  It can clarify ideas, solidify understandings and push kids to see things in new ways.  For example, my students noticed that there were always going to be 8 smaller squares that had three faces painted because there are always 8 corners on cubes and we were not changing the shape dipped in paint.  They found connections between the inner cubes with zero faces painted and the smaller cubes that they already examined.  It was a great activity and I highly recommend this one for anyone studying volume, geometry, and visual learning. 

I believe in the power of open questions, which is why I am going to create a resource for other people to explore.  Wish me luck on this crazy endeavor!