Tag Archives: integers

The NFL Draft: Shopping for Bargains!

Last week, the NFL player draft took place over 3 days in New York City, and now the annual exercise of “grading” each team based on their draft haul commences.  It’s a fun debate, with grades often based more on feel or perceived value, rather than any real analysis.

There are many ways to evaluate draft results, but from a purely mathematical standpoint, I like to look at value.  Which teams  got the best “bargains”, and which teams went out on a limb?  If you had the 20th pick in the draft, did you get the 20th best player?  Or did you draft a lower-ranked player.

I took all of this year’s 254 players drafted in the NFL draft, and compared them to their draft ranking, according to CBS Sports.  The only real reason I have for using CBS as opposed to the many other draft rankings out there, is that it was easy to pull their data out into a spreadsheet.  From there, I computed the “value” of each pick.  If a team drafted a player above his rank, this is negative value.  If a team drafted a player after his rank, this is a positive value.  Some examples:

Geno Smith was drafted with the 39th pick, but was ranked 21st by CBS Sports, so his value was +18

Meanwhile, E.J. Manuel was drafted with the 16th pick, but was ranked 40th, for a value of -24.  

Some players represented great values for the teams which picked them:

Cornelius Washington, Chicago Bears (pick 188, ranked 82, +106)

Andre Ellington, Arizona Cardinals (187, 88, +99)

Jordan Poyer, Philadelphia Eagles (218, 119, +99)

While other players could be considered “reaches”:

B.J. Daniels, SF 49ers (pick 237, ranked 818, -581)

Jon Meeks, Buffalo Bills (143, 834, -691)

Ryan Seymour, Seattle Seahawks (220, “1000”, -780).  Ryan is the only drafted player who did not appear in CBS’s top 1000, so I just assigned him #1000.

There is a bit of un-fairness here, as many teams will use later picks on “projects”, players who have little expectation of making the team, but who seem to have a particular upside, so there was much volatility in the later round values.

From there, I simply added up the value scores for the players drafted by each team, and found an overall value score.  So, which teams earn the best grades?  Only 3 teams earned overall positive scores.  This is understandable, as it is much easier to earn negative scores than positives, especially in the later rounds.

THE TOP 3:

Minnesota Vikings (+187)

Chicago Bears (+51)

Philadelphia Eagles (+25)

THE BOTTOM 3:

Buffalo Bills (-836)

SF 49ers (-1097)

Seattle Seahawks (-1571)

For math class, have your students think of other ways to measure draft success.  Is the value measure here valid?  How can the method be adjusted?  How do some of the huge negative numbers in this data influence results?  Feel free to download and toy around with the data in my draft value tracker, and let me know what you come up with!

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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.

Twitter

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.

SAME-CHANGE-CHANGE (aka KEEP-CHANGE-CHANGE):

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

FOIL

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.

CANCEL (LIKE) TERMS

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.