LearnBop Community Blog

Rich Problems Yield Engaged Students and Insightful Data

Posted by Cindy Bryant

Mar 7, 2014 6:00:00 AM

We all have our favorites!  Whether it is a favorite food, favorite song, or favorite animal, we all have something that resonates with us and draws us to it. And I would venture to guess that all math teachers have favorite problems - or at least I do!

When it comes to my favorite math problems, it’s the richness of the following problem that makes it one of my picks.


King Arthur realized it was time to marry off his beautiful daughter to one of his knights.  He knew that all of the knights were brave, but he wanted his daughter to also marry the most brilliant knight in the court. He used the following method to determine the brightest knight in his court. 

He gathered all 12 knights to the round table and told them to choose a seat to sit in.  After they were seated, he pointed at the knight in chair number one and said, “You stay.”  He pointed to the knight in chair number two and said, “Back to dragon slaying for you!”  He continued this process until only one knight was left to marry his daughter.

Fortunately, the brave and brilliant Sir Cole had heard of the king’s plan and knew exactly which chair to choose in order to be the last night sitting to win the hand of the beautiful princess.

Discussion Questions:                 

  • How did Sir Cole know exactly where to sit?
  • What if there were a different number of chairs, where would Sir Cole need to sit?
  • What chair(s) should Sir Cole never choose to sit in?
  • Can you find a general rule to determine the chair Sir Cole would need to sit in if there were any number of chairs?

It’s the perfect problem to engage students and a source of great fun!  But, it’s also rich in that it provides a springboard for a variety of problem solving strategies and practices ranging from acting out a problem and looking for patterns to reasoning abstractly and quantitatively, attending to precision, and looking for and making use of structure.  The problem supports and conveys perseverance in problem solving as students make conjectures, find solutions, and look at different approaches to solving it.  

 Allowing students to approach solving problems in different ways using different strategies helps them to better understand mathematics, develop mathematical fluency, and see mathematics as worthwhile and doable. Doing so sets the stage for transferring conceptual knowledge to new situations (Many, Fyfe, Lewis, & Mitchell, 1996).

So what about your favorite?  What’s it about a problem that draws you to choose it to use with your students?  More than likely, it’s several things.  Maybe it’s the context or scenario, the mathematics required for the task, or the processes and practices involved in solving the problem that lead you to favor it. 

We know that you want and need rich problems and resources that will give you the most “bang for the buck” when it comes to student learning.  LearnBop provides an engaging and challenging learning experience for each student through rich content, processes, and practices that support students in learning concepts and skills they may not have yet mastered as they are guided through the step-by-step process in solving the problem.

But it’s also rich in the data that’s provided in helping you identify student learning gaps.




For example, the performance data for the Bop (problem) shown above aligned to 8.EE.C.8b-1, shows that students apparently have some wide gaps in prerequisite skills. Only about 12% were able to use the distributive property (6.EE.A.3) and only about 15% were able to use substitution (6.EE.B.5-2).   In addition, 38% could not write/solve equations (6.EE.B.7) and another 25% were not able to evaluate formulas (6.EE.A.2c-1). 

Knowing which concepts students are struggling with is incredibly valuable information which can save you time in identifying individual student knowledge gaps.  It also enables you to plan small group or individualized instruction for those students who are struggling with certain concepts, be they prerequisites or current concepts, using LearnBop’s recommended resources and interventions.

For a gold mine of rich 5th – 9th grade problems that provide you with data for informing and guiding instruction, sign up for a free 30 day trial of LearnBop for your class(es) here – or sign up your school and get 60 days of free access. 

We value your feedback so we invite you to share your questions, comments, and suggestions along with your favorite math problems.  Thanks!

Topics: Teachers Using LearnBop, Differentiating Instruction, Implementing the Common Core, Resources, Teaching & Learning, Algebra & Algebra Readiness

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