Math Mondays: From Problem Solving to Formal Steps

Do you ever give those "zoo" problems?  I don't really know what they are called but they are the ones where the zookeeper has some flamingos and some elephants. There are 15 animals in total and 38 legs. How many of each animal are there? (Or if you are feeling other worldly maybe you use cyclops and fairies and you ask how many eyes there are.) 

Anyway I love these problems when we are working on problem solving strategies because there are many ways to approach - make a table, guess and check, solve a simpler problem, and draw it are just a few that come to mind.  But in my more advanced math, students learn to go beyond these strategies and use more formal algebraic approaches. 

Yet, I like to start with problem solving approaches particularly when teaching systems of equations. Although students know what equations are, this whole systems idea can seem kind of abstract, particularly when working primarily with variables.

So when a student gets to systems of equations, they start with an inquiry that is familiar - a problem solving question that just happens to also be a systems of equations problem. 

  • A student solves an inquiry problem first using any method they choose.  
  • Next, the student sets up equations and problem solve how they might use the equations together to find a solution.
  • At the next lesson when the student is exploring formal strategies for systems of equations such as substitution and elimination, the strategies are a little more logical because students really have already used these strategies.

And although this post was specifically related to inquiry with systems of equations the basic teaching strategies can be used across the curriculum:

  • Start with something familiar
  • Create an opportunity for inquiry
  • Give students the chance to problem solve (and this is not about getting the answer now)
  • Provide formal strategies related to the inquiry activity.
What kind of inquiry or approach do you use when teaching new math concepts?  Join the discussion in the comments below and checkout some inquiry resources for your classroom.



  1. I, too, like to include a balance of discovery-based learning and traditional notes. I think it's important for students to wrap their head around the big ideas and make connections between the new topic and previously learned material. It is also helpful to then summarize the steps and procedures so that students can effectively practice at home on their own. Great post!

  2. Yes! Inquiry and discovery-based lessons challenge students to apply concepts they may already know and/or identify patterns. I think some teachers hear the word "inquiry" and it makes them nervous that their students will not understand, but I find quite the opposite. It is more challenging then spoon feeding them the material, but the students take more ownership over their learning and have a deeper understanding of it.

  3. While it can be challenging to give students the time and space to engage their own curiosity and creativity, I think it leads to deeper learning. Your description of the process shows how much agency you give your students as they explore a scenario they can understand and creatively experiment to try and solve the problem. This not only contextualizes and incentivizes engagement, it also leads them to a deeper understanding of why the algebraic tools exist. Great post!