Math Mondays: Teaching Permutations vs Combinations

Probability as a general concept is not that difficult.  If there are 6 green marbles and 1 white marble in a bag, the probability of choosing a green marble is higher than the probability of selecting a white marble.  Students see this with ease.  But when we get to the difference between permutations and combinations as part of our probability unit in Algebra 2, it isn't as obvious.  Here's how I teach these sometimes confusing concepts:

  • Inquiry:  We start with problem solving (see problems to the right).  Students work in small groups to solve these using a method of their choosing.  We share our strategies and discuss.  One of the key discussion points here is to determine what makes these problems different.  They seem remarkably similar, but the key difference is that order matters (permutations) in the second problem and does not matter (combinations) in the first problem.  This step is focused on identifying the differences between the questions and strategies for solving.
  • Formalizing strategies:  Next step is to start to formalize these ideas.  Students use graphic organizers to formalize how to identify and solve permutations vs combinations.  Students sometimes ask why formalize if they can solve a problem by just writing out the possibilities.  To this I ask, well what if there were 25 or 50 or 100 desserts to choose from and 8 people choosing desserts.  The formulas are to make it easier.  We prove the formulas with our inquiry to make sure that they make sense.  Students retain the information better if they can prove the formulas rather than just memorize them. 
    The organizers include: definitions, how it works, examples, formulas, and practice problems which sometimes are in class and sometimes are homework.  I use different colors for parts of the formula to help in visualizing how the numbers correlate to the variables.  I find this particularly helpful for my visual learners. In addition to the notes in class, I encourage students to check out videos such as the difference between permutations and combinations (all visual, no lecture) or Combinations(permutations) from Kolumath.  
  • Trying it out: permutations and combinations are the basis of so much of probability that I want to make sure my students have a solid foundation before we continue, so we test our skills with the paper chain activity.  I use this in different ways.  It all depends on the personality of my class.
    I have had students build parts of a chain individually and then find their missing links by connecting to other students' chains, race in groups to complete the chains, or even set up links as a scavenger hunt to keep students up and moving. 
  • Assessment: we have to move fast, so we don't have a lot of time for assessment. Sometimes the assessment of permutations and combinations comes as part of our unit project, but depending on which projects we do, they do not fit neatly into the project.  When this happens, I have my students teach me permutations and combinations.  They prepare short presentations demonstrating the difference between the two, applications of each, and how to solve.  Students present to me directly during our one-on-one time.  In this way I can assess quickly their understanding and ask questions to know if they have mastered the material or not and provide additional support where needed.
I hope you find this helpful in approaching permutations and combinations in your own classroom.  Stop by for more tips, any time.  

Math Mondays is a bi-weekly blog post (2nd and 4th Monday of each month) sharing tips, ideas, resources, and products for teaching math.  If you have questions or think there is something I should include, you can leave me a message in the comments section below or at the store in the question and answer section.