Monthly Archives: November 2012

Look Through the Eyes of Your Students

Here in Pennsylvania, many high schools are gearing up for the upcoming Keystone Exam in Algebra 1.  In this first year of Keystones, the Algebra 1 exam is being given not only to students as they complete Algebra 1, but also to 10th and 11th graders who have passed the course in the past.  The state has provided a number of sample items, which we have been using in math classes to help our students prepare.

A discussion of one of these released items not only revealed a common algebra misconception, but also generated thoughts of how teachers may see problems differently that their students.  Here is the question:


So, how could teachers and students view this problem differently?


Teacher Problem

My teacher eyes tell me immediately that this is a factoring problem.  I’m not sure how it is going to factor yet, but I am pretty confident that the answer will be C or D.  Choices A and B are not really even up for debate here.

My radar perked up when a colleague shared that a handful of students in one of her classes chose B.  B???  How the heck did they get B????


Problem Student

My old nemesis….cancelling across addition and subtraction signs, how nice to see you again!

So, while I immediately see the problem as two expressions which will separately require factoring, I need to remember that students don’t always view problems the same way.  Being able to identify, discuss, and personalize these common errors are all part of the wonderful art of teaching.

And while illegal cancelling will be a struggle for students long after I retire, I often use the exercise below to generate discussion.



So, either I have made a math error here, or I have a great method for generating some side income here (and why am I sharing it with you, anyway???).

The Take-Away Game

A recent visit to a 6th-grade classroom gave me a chance to introduce a simple game I have used in the past as an-going challenge.  Even after a few pop-ins to this 6th grade class, I am still undefeated, and don’t plan on giving up my championship belt anytime soon!


On a board, or piece of paper, draw 23 X’s.  Players will alternate turns, and on each turn a player must erase 1, 2 or 3 X’s.  The winner is the player who erases the last X.

It’s an easy game to understand.  An example is given here:

With a class, I will give students a chance to use dry-erase boards and play against each other.  Then, as students begin to understand the game, they are allowed to challenge me.  This usually ends badly (for them), as I know the tricks to the game.  I start by asking the player if they would like to go first, or allow me to go first.  Since kids are usually nice, they will allow me to go first, and this sets them up for certain doom.  Also, I will use my best poker skills to agonize over my moves, though I know exactly where I want to go with my moves.

Eventually, students will gather around to suggest moves.  Their first realization is that if I get the board down to 4 Xs, I will win.  This will then extend to 8 remaining.  With some classes, I have placed a fist behind my back, and done a thumbs-up to signal those watching when I know I have the game won.  Shoot me an e-mail if you need thorough instructions on how to win.

As students master the game, we can ask some extension questions:

  • Does the number of X’s we draw change the game?  What if we use 25, 35, or 50 X’s?
  • What if we could erase 4, 5, 6 or n number of X’s?  How would the strategy change?

For now, play the game with your students, and I look forward to retaining my Inter-Galactic Take-Away Game BeltChampionship Belt!

Shuffle Up and Deal, and Deal, and Deal….

Take a look at the video below, where Stephen Fry, host of the British panel show QI, alleges to do something never done before:

EDIT:  Seems as though the YouTube folk removed the video clip.  Try this link instead, and let’s hope it lasts:

QI Card Shuffling Clip

What a great “hook” for a probability or counting principles unit. Some thoughts about how to use this in your class.

1.  The result given in the video can be expressed as

If we were to shuffle the cards once every second, with each arrangement occurring once, how long would it take for use to go through every possible arrangement?  A neat example of something “big”, which is accessible and easy to discuss.

2.  The online poker site PokerStars is celebrating it’s 10th anniversary, and is offering a prize to the players who participate in their 100 billionth hand (assumed to occur around the 10th anniversary).  At this rate, how long should it take PokerStars to go through all possible arrangements?

3.  As an extension, challenge your class to find the number of possible arrangements of a deck of Pinochle cards. Cards The main differences with a Pincohle deck are that there are only 48 cards, and each card (like the 9 of diamonds) appears twice in the deck.  This problem introduce the idea of permutations with duplicate items.  In this case, we start with 48!, but then must divide out the double-count which occur with the repeat items.  We divide by two for each instance of a repeat item, and the number of permutations is given by:

4.  Let’s evaluate Mr. Fry’s conjecture:

Were you to imagine if every star in our galaxy had a trillion planets, each with a trillion people living on them, and each of these people had a trillion packs of cards, and somehow they managed to shuffle them all a thousand times a second and they had been doing that since the Big Bang, they would just now begin to repeat shuffles

To summarize, we are looking at this many shuffles per second:

Dividing by the number of possible shuffles yields:

The number of seconds in each year is given by:

Which implies we would have to shuffle for this many years:

Great exercises in laws of exponents for your students.  Share your thoughts and ideas about this fascinating video!

How to “Break” Your Graphing Calculator

A conversation with a colleague on function operations reminded me of one of my favorite “Oh Wow!” moments from math class.

You’ll need a TI-83 or 84.  This is a case where the Nspires are too nice for our use.

Enter functions as shown below.  We are interested in the rational function which comes from dividing the two previous functions:


Since the denominator is a factor of the numerator, we can simplify the expression, resulting in a linear function when we graph.  But what about x=-3?  What happens there?  Let’s explore a bit:

Screen 2

Zoom in even more….doesn’t seem all that interesting…..


Pretty boring.  But tell your students to keep zooming.  And then….

….wait for it……


Screen 5

Zoom in even more….

Screen 6

This is a nice reminder that, while we may simplify a function, we are still looking at a quotient and need to consider the parent functions used in the division.  And the “noise” we get here is a result of the algorithms TI calculators utilize to plot the points.  Actually, if somebody has a better explanation for the noise, I’d love to hear it.  Some “new” calculators will now show the hole in this function, like my new man-crush, the Desmos calculator:


But the noise is more fun.

One-Variable Equations – Let’s Look at Visual Methods

This past week, I worked with 2 Algebra 1A classes, which were working through solving one-variable equations.  By the time of my visit, students had worked through solving equations which had a variable on both sides, some which required students to combine like terms, and some where the distributive property was required.

The students in the class had been exposed to the traditional series of worksheets, drills, and various games to keep the class motivated.  Homework the previous night consisted of problems like the one below:

My challenge to the class was to work with me to think about these one-variable equations differently.  Rather than thinking of the equation as a whole, we have two expressions, one on the left and one on the right.  While by this point in algebra 1 the students do not know how to graph a linear equation, this lesson provides a good opportunity to open that door.  By this point, students do understand that:

  • Expressions can be evaluated, given a value for x, and this provides an output (y).
  • Function tables provide ordered pairs which can be plotted
  • While we often are asked to “plug in” a specific value for x, we can plug any value into most expressions and obtain an output.

So, let’s use this background to our advantage and bring consider a graphical method for solving one-variable equations.  The class I worked with used laptops, and worked through some examples using the Desmos Calculator.  


Students generally understood that we had strong interest in that point of intersection.  But what does the x represent?  What is the y?  Which one represents the solution we seek?  And what a great review of ordered pairs, more work with functions, and a preview of linear functions.  Students worked through function table with me on this worksheet, and used the grapher to work through problems on their own.

class 1

Class 2

Working with this class produced some rich discussions, particularly when the class was faced with a problem which had no solution, or a problem which had a “messy” solution.  Discussing the parallel lines in teams strengthened the students understanding of “no solution” problems, and problems with messy fractional solutions became possible.

The activity also provided a nice segue to inequalities.  By problem 6 on the worksheet given above, a number of students noted that they could look at when one line was “above” the other.  And some nice foreshadowing of linear systems has been built into early units.

Should students be proficient in traditional algorithms for solving equations?  Of course!  But looking at algebraic ideas using multiple representations often lets students personalize their learning by assessing methods and making connections between tables, equations and graphs.