Tag Archives: algebra 2

Student-Created Polynomial Digital Notebooks

Over half-way done my first time teaching Algebra 2 in over 8 years (under block scheduling), and it’s amazing how much technology has changed many of my former approaches.  Nearing a chapter on polynomial functions, I was somewhat dreading the experience.  This is a pretty dry chapter…synthetic division, rational root theorem, complex conjugate roots…there’s a lot of rules and regulations here, and not much room for engagement.  Here’s what I settled upon to avoid dry lectures, promote student ownership, and encourage a serious review of resources:

DIGITAL NOTEBOOKS FOR POLYNOMIALS

On day 1 of the unit, I gave students their assignment for the chapter, which you are welcome to download: Polynomial Notebooks.  The link describes the assignment:

This chapter contains many landmark theorems and ideas for analyzing polynomial functions.  Your job is to create a digital notebook of the ideas below, using a Google Document to house your information, and eventually share with the class.  You may use online resources, or examples you create, to serve as examples for each idea.  The goal of this document is to serve as a resource which demonstrates an understanding of the ideas, and which helps you study their meaning and usage.

To be honest, this did not start off as I expected, as my students were simply not accustomed to not being fed material.  The idea of seeking outside resources, being able to weigh their merits, and summarize them for use was foreign.  But our second day in the computer lab was fruitful, as students had great questions about the Intermediate Value Theorem and Descartes’ Rule of Signs; what was really fasicnating were the student attempts to think about the ideas in language which made sense to them: “Does this mean that…?”, “Does this example work here…”.

Not all class time was spent doing research.  We started class with sythetic division problems done on boards around the room.  Also, I developed 3 “food for thought” questions each day during the unit, which helped drive discussions.

  • Name a polynomial with roots  2i and -3 (multiplicity 2).
  • How many possible negative real zeroes does the following polynomial have?
  • Show that polynomial given must have a real zero between 2 and 3.

Enjoy all of these Food for Thought questions: food_for_thought_daily_qs2

It’s also interesting how different groups have chosen to format their notebooks.  While some are going through my list in a linear fashion, others have developed connections and examples to help make sense of the rules.  One ambitious young man has developed an outline of the topics, rearranging the ideas, providing examples, and organizing links.

Tomorrow we will have a group quiz on the material, and I feel like this experience has helped students personalize their needs, and think about their gaps.  I also feel more confident in the aiblity of the class to think about these  polynomial rules as a whole, rathern than as a set of disconnected ideas.  Hoping for a good day!

 

 

 

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Visualizing Shared Work Problems

Fred can paint a room in 5 hours, working alone.  His friend, Joe, can paint the same room in 7 hours.  How long will it take for them to paint the room, working together?

It’s a shared-work party, people!  Get your party hats on and let’s look at a visual method for exploring these often mundane problems.  This past summer at Twitter Math Camp, I participated in an algebra 2 group where part of our time was spend considering methods to re-think the traditionl approach to rational functions and their applications.  Thanks to John Berray for the great conversations, which led to some changes in how I appoached shared work problems this year.

My approach this year started similarly to previous years: guiding a dicussion with the class, with the goal of developing models for the amount of work done by each painter.  I find that quesitons like “How much of the job will fred have complete after 1 hour? 2 hours…etc” will usually lead to the models we seek.  What I did differently this year was graph the two work functions.  Using the Desmos calculator works nicely, and allowed for a discussion of the problem much richer than if the expressions had been just jotted down on the board.  Many students followed along on their TI calculators.

SW1

From here, we can make connections betweem the functions, their graphs, and make conjectures about the sum of these functions.

SW2

In my class, students certainly completed similar problems (including distance / rate / time), with the graphs serving as a check and visual affirmation.  With the graphs, we could also look at adaptations to the theme, such as “what happens if one of the painters shows up 2 hours late?”

SW3

Also, problems where the combined time was given, with the goal of finding a missing individual rate, were explored and discussed.

SW4

Click the icon below to play with this model on your own.  This is a great opportunity to let students observe function behavior and communicate results from a graph.

UPDATE: The Desmos folks flew with this one, and added a whole bunch of bells and whistles.  Click the graph below to experience their shared-work extravaganza.