Tag Archives: polynomials

Hitting the Home Stretch: Exponents, GCF’s and LCM’s

This is a busy stretch in my school year.  My 2 Prob/Stat classes are nearing the end of new material with PA Keystone Exams in Algebra 1 looming. For my College Prep class, about half have not yet taken the Keystone while the rest took it last year as 8th graders. In Academic, all students will take the Keystone in May.  Combine this with my AP Stats class taking their final exam this week, with the AP Exam next week, and my track and field meet responsibilities building as the season reaches its peak; it’s a hectic time of year.

In both of my Prob/Stat classes, we are beginning unit on polynomials.  The Prob/Stat class is a course we offer between Algebra 1 and 2. While the course contains much Prob/Stat material, we also clean up some concepts from algebra.  Unlike other algebraic units like systems of equations where there are many rich examples and opportunities to differentiate, the start of a polynomials unit often feels static. Here are two activities I have used this week for Laws of Exponents and GCFs/LCM’s.


This activity worked equally well in my college-prep group (for whom this was review material), and my academic group (where this was mostly new).  The file below contains 16 cards with numeric statements.  Break your class into teams of 2, 3 or 4.  The job of the group is to identify the true statements and the false statements.  For this activity I banned all calculators.

The letters on the cards are not anything the kids need to worry about, but allow me to easily check progress. The cards with vowels are all the false statements.

I was surprised at how much trouble my college-prep group had with building the correct false pile.  To reach some consensus on the false pile, I asked every group to hold up one card they knew to be absolutely, positively false.  Many groups mistakenly agreed that any number raised to the zero power was worth zero, which led to a class argument on who was right.

Tomorrow, we will look more closely at the falses.  In the file above, note that the cards are arranged in groups of 4. In the first group, we will review the addition rules for exponents; then the subtraction rule in the next group of 4; then the multiplication rule for the next group.  In the end, this felt much more satisfying, with increased engagement and peer discussion than simply listing rules on the board.


The speed dating concept is one many math teachers have stolen from the great Kate Nowack, and it worked perfectly in my Academic class to work through greatest common factors and least common multiples.  After doing just one example on the board, desks were arranged  into a pairs facing each other, down one long row.

Speed Dating

All students were given a card with a monomial.  They then worked with their partner facing them, and found the GCF of the two monomials.  The first time around, my co-teacher and I provided help to just about all groups.  After teams found their GCF’s, all students on the right-hand side stood and moved down one seat and worked with their new partner.  There were so many plusses to this activity:

  • all students were repsonsible for their own monomial
  • all students were engaged: no hiding behind a worksheet
  • students worked together, and with different partners each time

Some of the cards I handed out are shown here.  I tried to have a variety of cards which clearly shared factors, with different powers of x and y.


I was very impressed with how my class performed on this activity, and we moved onto a second round where LCM’s were found. This time I had students trade cards, and the left-hand side shifted down each time.

Let your kids work together, discuss and find patterns – the notes then write themselves.


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:


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!




3 Phrases from Math Class we Need to Expunge.

A brief twitter exchange last night between myself and the great NY math educator Mike Pershan caused me to get off my rear to assemble a post which I had kicking around my head for some time now, a list of terms and shortcuts we use in math class which, while well-intentioned and used everyday by many math teachers, aren’t necessaily helpful in causing kids to understand their underlying math concepts.


In a recent in-service with middle-school math teachers, I used a video by Phil Daro (one of the authors of the Common Core math standards) to have colleagues reflect upon the practice of “answer getting”, short-term strategies employed by teachers to get students through their immediate math assessment, but with little long-term value in math understanding.  Click on the “Against Answer-Getting” tab for the video.

So, here is my first list of nominees for elimination, and some strategies for helping students develop underlying algebraic ideas.  It probably won’t be my only list, and I welcome your candidates and thoughts.


This is a device I often see in pre-algebra classrooms, often times as a poster for easy reference, other times as a mantra for the students to help complete worksheet problems.  From the site Algebra-Class.com:

TIP: For subtracting integers only, remember the phrase

“Keep – change – change
So, we have a short and snappy device which helps us with just one specific type of integer problem.  It’s not wrong, just too specific, and do students understand why it works?
What to do instead:
Let students develop their own summaries of integer problems, and create their own posters which describe their findings.  Use integer zero-pair chips or online applets, like from the National Library of Virtual Manipulatives (search for “chips”).  Number line applets can also help students visualize addition and sibtraction problems.  Have students write stories about given integer and subtraction problems, and have students peer-assess work for proper use of math terms.  Eventually, have students debate the possible equivalence of integer pairs:
  • 5 – (-2) and 5 + 2
  • a + (-b) and a – b
  • a – b and b – a


The ad-laden math site Coolmath gives its own snazzy description of foil:

We’ve got a cool little trick called “FOIL” for multiplying binomials….it’s really just an easy way to do the distributive property twice, which would be really messy and confusing to do.

YEY!  You mean I can multiply stuff without that nasty and scary distributive property, without actually talking about the distributive property!  Yey shortcuts!  I’m in! {insert sad face}

Folks, ditch FOIL, and use the opportunity to talk about the double-distributive property.  Re-write the binomials as an equivalent expression and multiply.  Set the stage for factoring and note how much more understanding factoring by parts takes on.  And, now we can tackle those “messy” trinomials too.


Try this exercise tomorrow: take a class tht has been through Algebra 1, and as an opener tomorrow ask them to explain what the phrase “Cancel Like Terms” means when dealing with a rational expression.  Or, if that is a bit too scary, simply ask your students what it means to recude a fraction.  This is a nice activity to do as a Google form, and have students assess the explanations.  Many students will give an example as a definition, which is not what we are looking for here.  How many students discuss factors, GCF’s, numerators or denominators?

Reducing a rational expression means to divide both the numerator and the denominator by the greatest common factor of both numerator and denominator.  (Incidentally, also try having your students provide steps for finding a GCF.  This one also reveals what your students understand.)  The great part about this procedure for reducing is that it works equally well for each of the following expressions:

To many of our students, cancel is digested as “cross-out stuff”.  We have better vocabulary for it, so let’s encourage its use.