Using "The Letter Game" to Introduce Deductive Systems and Proof in Geometry

A new trimester will begin in just a couple of weeks, and I will have a new bunch of geometry students.  Before long, we'll be digging in to geometric proof, which is a concept/activity/process that we will develop and use throughout most of the course.  It is always a tough concept to develop with students, and I am always wondering how I can make it easier for them to grasp.

One really nice, fairly accessible activity that we have used with students at my school is "The Letter Game", which was published by Don Gernes in his article The Rules of the Game (full citation below).  The game amounts to a very simple deductive system, in which students are asked to "prove" various statements.  It gives students an opportunity to practice working within the bounds a deductive system, and to get a sense for the structure of proofs and the process of proving something.  This comes before they have to confront the added complexity of geometry, which is the big deductive system they ultimately will be learning to work within.

Before getting to the "The Letter Game", Gernes suggests having students think about games that they are already familiar with: monopoly, basektball, soccer, etc.  Each of these games has undefined terms, defined terms, postulates, and maybe even some theorems.  These are the makings of a deductive system.  Gernes presents basketball as an example, and here's what he lists in each of these categories:
______________
Undefined Terms
Ball
Player
Court
Baskets

Defined Terms
Field Goal
Foul
Free Throw
Traveling

Postulates
If a player is fouled, then the player gets to shoot a free throw.
If a player travels, then the other team gets possession of the ball.
If a player makes a field goal, then the player's team gets two points.

Theorem
The referee objectively applies the rules of the game to each play.

This obviously doesn't represent a complete list of all of the terms and rules that make up the game of basketball, but it is enough to have a discussion about what the undefined terms, etc. are, and how each of them contributes to the structure of a deductive system.

The Letter Game is where students get a chance to try their hand at some proofs.  Gernes keeps the system simple, so students don't have too much to keep track of as they begin proving "theorems".  Here's the deductive system he establishes:
______________
Undefined Terms
Letter M, I, and U

Definition
x means any string of I's and U's.

Postulates
1. If a string of letters ends in I, you may add a U at the end.
2. If you have Mx, then you may add x to get Mxx
3. If 3 I's occur, that is, III, then you may substitute U in their place.
4. If UU occurs, you drop it.

With these four postulates, students take given strings of letters and then "prove" another string of letters, using one postulate at a time to manipulate the string.  For example - Given: MIII   Prove: M.

Even though many of my students end up groaning about geometric proofs, most of them end up enjoying the letter game.  It is accessible enough that most of them are able to make some headway, and after a quick example or two, most of the class is able to work independently.  I usually don't get any students reverting to "shut down" mode because they can't get it.  Sometimes I even have students finished early who create their own new letter "theorems" and challenging one another to prove them.  While the activity doesn't make them masters of geometric proof, it does reinforce the concept of a deductive system, and what it is to work in and prove new theorems in a deductive system.  Every step that is taken or statement that is made needs to be backed up by a definition, postulate, or theorem.

Opening up the much more broad deductive system of geometry, with all of its definitions, postulates, and theorems, always makes geometric proof more difficult for my students.  The Letter Game hasn't eliminated those difficulties for my geometry students, but it has been a good first experience with proof for them.  Gernes' activities build a nice segue from concepts that students are familiar and comfortable with to the way geometry works as a system.  His article is definitely worth a look:

Gernes, D. (1999).  The rules of the game.  The Mathematics Teacher, 92(5), 424-429.

An Introductory GeoGebra Activity for Calculus Students

My calculus students had never used GeoGebra before this activity, so I wanted to do something fairly simple with them.  This activity takes about five steps on GeoGebra, so it wasn't too demanding for first time users.  At the same time, it provided some nice visual support for the concepts we had been learning in class, and provided them with a tool to explore those concepts further.  I've tried to make the instructions below friendly for first time GeoGebra users, and I'm a novice anyway.

Setting Within the Course

When I brought my class to the computer lab for this activity, my calculus students had already been using the limit definition of a derivative for three days.  We had also been discussing the derivative at a point as the function's instantaneous rate of change, and the slope of a line tangent to the function at that point.  When we had opportunities to do so, we were taking special notice of times when the derivative was zero and the tangent line was horizontal.  We had also spent just a little bit of time more generally discussing the relationships between graphs of functions and the graphs of their derivatives, including a few mentions of concavity.  Furthermore, after working with the limit definition for several days, my students had noticed and we had informally discussed the power rule.

The Activity

With these concepts in mind, we did the following together on GeoGebra:

1) First, we entered a function in the Input bar at the bottom of the window.  The function we started with was y = x^3+3x^2-1.  In the algebra pane, GeoGebra relabels this as f(x).  This function provided opportunities to talk about horizontal tangent lines (critical points) as well as changes in concavity (inflection points).  At this point I also showed my students how to use the Move Graphics tool to center their graph in the window, and to adjust the scale on each axis to get a good picture of the function.

2) With our function graphed, we added a new point on the curve by selecting the Point tool and clicking on the curve.  GeoGebra automatically labels this point A.

3) We then put a tangent line on the curve at point A.  GeoGebra has a tool for this under the fourth box from the left.  Once the Tangents tool is selected, click on the curve and click on the point of tangency (point A for us), in no particular order.  This is a good time to show students how to use the Move tool if they haven't discovered it already.  It's the cursor at the upper left of the tool bar (or pressing the Esc key will select this tool).  If students click and hold point A with the move tool, they can slide it along the curve and watch the tangent line change.

4) Now we'll have GeoGebra measure the slope of the tangent line (the value of which is that of the derivative).  GeoGebra has a variety of measuring tools.  Once the Slope tool is selected, click on the the tangent line and GeoGebra will add a rise/run triangle and display the slope.  The value of the slope gets added to the algebra pane as m.

5) We finally add one more point to the sketch, using the Input bar again.  Type in (x(A),m) to define this last point, which GeoGebra will call point B.  This tells GeoGebra to use the x-value from point A (the point on the curve) and to plot the slope of the curve as the y-value.  This point displays the value of the graph's derivative at any x-value.  By right-clicking on point B, Trace On can be selected.

With trace turned on, the path of this point will be traced out on the sketch as point A is moved along the curve, revealing the shape of the derivative function.  (I also changed the color of point B using the pull-downs at the top of the graphics view.)

Discussion

The completed sketch opens up a lot of good talking points.  Students should be able to see that the local maximum and local minimum of  f(x) line up with the zeros of the derivative function.  If inflection points have been discussed, it's also nice to see that this point lines up with the vertex of the derivative.  As point A is moved back and forth across the inflection point, the tangent line makes a nice rocking motion that shows the slope is beginning to increase/decrease.  In general, the tangent line in the sketch provides a good visual connection between the curve of  f(x) and the sign of the derivative.

Of course, students can also try to predict the shape of the derivative function, or find the derivative function algebraically and plot it on the sketch as well.  Wherever possible, it seems good to keep students' visual and algebraic understandings closely intertwined. (Check out Aspinwall and Shaw's (2002) article, When Visualization is a Barrier to Mathematical Understanding, in Mathematics Teacher vol. 95, no. 9)

This GeoGebra sketch is also a good tool for students to explore derivatives for functions that they don't yet know how to differentiate algebraically.  Helping my students work through this activity took less time than I had anticipated, so they had about ten minutes of class time in the computer lab after we were done with the first function.  After I explained that they could change f(x) by double-clicking it in the algebra pane without altering any of the other features of the sketch, most of them started exploring other functions.  Students were discovering on their own that cosine is the derivative of sine, and one of them was looking at the graph of tangent, and noticing similarities (but important differences) between the sketched derivative and the secant function.  Having a tool available to explore derivatives increased their curiosity.  I may pull up the same sketch again in class as we learn about the derivatives of specific types of functions.

If you would like to go to the completed sketch, here is the GeoGebra Tube link.