Conceptual Function Foundation

I’ve taught College Algebra for a number of years.  This course and College Trigonometry replace Pre-Calculus at our school.  I’ve struggled with helping kids with functions because of the variety of background knowledge they have on the topic.  I have tons of good activities, but never one that really built a conceptual foundation of the important features of functions in general.  It’s not that it’s impossible to create a conceptual foundation after procedures have been introduced.  It’s just really difficult to do.  (Remember this post from Christopher Danielson?)Enter New Visions for Public Schools.  Unit One of their Algebra 2 course allows kids to make sense of families of functions in their own way.

You can see the details yourself on the link, but in summary, kids sort graphs according to their own criteria and then build a definition of a key graph feature and re-sort accordingly.  Students then form new groups and share their key feature with their classmates.  Finally, the group as a whole creates statements that link the key features together.

Dan Meyer states that math is the process of confusing and unconfusing.  This progression does that perfectly.  Conceptual Understanding Achievement unlocked!


  • Students are asked to make sense of graphs based on their prior knowledge.
  • They develop a need for certain vocabulary such as “turning points” as they discuss key features of the graph. For example:
  • They need to take responsibility for their learning because they need to teach it to other students during the “jigsaw” portion.
  • They have to ask clarifying questions of each other rather than of the teacher creating student-centered discussion.

The real beauty was watching three days of making sense of graphs come together with the vocabulary.  The students are asked, with their groups, to find how the key features are related and how they are not related.   I wish I had an audio recording because it was some of the most beautiful student discussion I may have ever heard.  I captured this moment just so I could remember it:

Students discussing how function features are related and how they are not related.

Students discussing how function features are related and how they are not related.

Also, here is one of the reference graphs a student made:



  1. Megan, I just love this! I began our year with so Ss could understand why an input had to have consistent output. Suddenly a very boring topic is becoming great fun and I will definitely do something like you have proposed as we get more into graphing.

    • In past years, the only “understanding” students had of functions was 1) one output per input 2) vertical line test. I’ve been looking for a way to help them conceptually understand functions for quite some time. I look at a lot of “function family” units and this one seemed to finally be what I was looking for. I happened upon it by chance since David Wees randomly tweeted it out. I like how the unit builds from this into rates of change of different functions.
      To successfully implement, I needed to set up a culture of collaboration. Students needed to share and listen to one another.
      Did I answer the question you asked? That was my interpretation of your questions.

  2. This is great Megan. I really appreciate you taking the time to post this reflection. We are working hard to continue to develop resources for Algebra II because it seems like there is a dearth of resources available. Matthew and Sara on my team are leading this work but have limited time to do so unfortunately.

  3. Pingback: Conceptual Function Foundation Follow-up | Number Loving Beagle

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