Using Graphing Programs Properly

For years, the graphing calculator was the gatekeeper of high school mathematics—a expensive, handheld device with a pixelated screen that students often used more for "button-pushing" than for actual discovery. Today, the landscape has shifted. Browser-based graphing programs like Desmos and GeoGebra have democratized math, turning abstract equations into vibrant, interactive playgrounds.

However, simply putting a laptop in front of a student doesn't guarantee learning. To help students move from "playing with the software" to "exploring the math," educators must use these programs as tools for conjecture and visualization, rather than just answer-checkers.

The true power of modern graphing programs lies in dynamic sliders. In a traditional textbook, a student sees three separate graphs for $y=x^2$, $y=2x^2$, and $y=0.5x^2$. They are expected to notice a pattern from static images.

In a dynamic program, the student creates a single equation: $y=a{x}^2$. By attaching a slider to the variable an and sliding it back and forth, the parabola breathes. It widens, narrows, flips, and flattens. It leads to an "Aha" moment.  The student isn't just told that a affects the vertical stretch; they feel the relationship between the number and the shape. This builds a spatial intuition that rote memorization cannot touch.

Proper use of graphing software starts with a prompt, not a procedure. Instead of saying, "Graph this circle," ask: "What happens to the circle if we change the constant at the end of the equation to a negative number?"  Ask students to predict the outcome on paper first. Then, let them use the program to test their hypothesis. If the graph disappears or does something unexpected, they have an immediate, non-punitive feedback loop to refine their thinking.

Graphing programs are peerless when it comes to teaching systems of inequalities or linear programming. Students can overlay multiple shaded regions to find the "feasible region" of a real-world problem, such as maximizing profit for a small business.  In addition, by dragging the boundary lines, they can see how changing a single constraint (like labor hours or material costs) shifts the entire solution set. This turns a dry algebra problem into a lesson in decision-making and optimization.

Some of the best practices include using sliders to show cause and effect rather than having students use the program to verify a hand-drawn graph.  Consider hiding the equation and ask students to "guess" the rule based on shape.  Avoid providing the equation and asking for a point-by-point plot.  Have students compare multiple graphs using ne screen to see intersections rather than asking students to clear the screen between problems.

Programs like GeoGebra allow for a "dual view" where geometry and algebra live side-by-side. If a student draws a circle and drags a point on its circumference, they can watch the (x,h,k) values in the equation update in real-time. This bridge between the visual and the symbolic is where true mathematical fluency is born. It removes the "mystery" of where the numbers come from.

When used properly, graphing programs act as a cognitive prosthetic. They offload the tedious task of plotting dozens of individual points, freeing the student's brain to focus on high-level patterns, transformations, and relationships.

By framing these programs as "discovery labs" rather than "digital paper," we empower students to treat mathematics not as a list of rules to follow, but as a world of patterns waiting to be explored.  Let me know what you think, I'd love to hear.  Have a great weekend.

More on Descartes Dots and Graphing

I reviewed Descartes Dots earlier in this blog but I've discovered that some students have difficulty using it if they are classified as ELL.  I've been working with my Pre-Algebra class which is 75% designated ELL students.  I can't always go as fast as I would like.

If you haven't run into it, it is a program designed to help students practice graphing on a coordinate plane by creating pictures using the provided coordinates.  You have to set point 1, set point 2, set the line, set the first point again at the end of the line.  We spent about 15 minutes on it but they will need more practice.

We ended up sitting on the floor with my students sitting around me.  I brought up the app and showed them how to do it step by step.  When I paused, I had them do the move before I demonstrated the next step.  A couple of the students got it but most of them are going to need additional practice.

I started the students on the Coordinate Plane app which relies on students finding the point by taping on the coordinate. At the end the app connected all the dots but it only uses the first quadrant.  So once they were comfortable, I moved them to Descartes Dots which uses all four quadrants.  Once they are comfortable using all four quadrants, I'll have them do an extention.

I am going to have students create a drawing on graph paper and write down the coordinates of the drawing  so other people can recreate the pictures.  Once, I've collected the drawings with coordinates, I'm going to make copies and then pass them out to other students to "test" how well they did.

Back to Descartes Dots.  There is both a free version and paid version but I use the free version and I really love it.  I find it fun and I love the pictures that are created and you never know what it will be.  Have a great time and enjoy.

Websites with real life type problems for graphing.

Math warehouse has two sets that look pretty good.  The first set is for linear equations and has the students write the equation associated with problems involving cabs.  Out where I am, the only cabs use a flat rate system such as $6.00 from the airport to the hotel and they often take 3 to 5 passengers in the cab to various locations.  These problems would help build a knowledge base for my students who are not accustomed to the way normal cabs charge.  After writing the equation, I can have them either build a table in a spread sheet and then graph it or I can have them create a graph on a graphing utility and then create the table by reading the graph.
The second set that the math warehouse has deals with dropping things from a certain height based on parabolas .  I like this because you can adjust some of the problems to discuss dropping supplies into an area that was hit by a tidal wave such as in Indonesia a few years ago.  The ones that deal with dropping something off a building, just change it to someone who is mischievous and then have them do the calculations.  Again, I can add in the spreadsheet, point graph or line graph.
I found this linear equation activity using bank deposits.  This activity focuses on straight deposits like in a piggy bank but could easily be extended to look at various interest rates to see how those effect the balance and could be done on a spread sheet.  This activity could easily be modified to discuss pay, theater or movie tickets, plane tickets, car rental, etc.
I am about to loose the internet for the afternoon so I will continue this tomorrow with one or two lesson plans/ideas tomorrow.

Quandary

I have been having students take notes on their iPads and now one class has a test coming up.  In the past, I have allowed students to use their notes during a test but the notes had to be kept across the room, only one person could be up at any time and they could not take a paper or pencil to the notes. They had to remember the material.

So where the quandary comes in is that I have been encouraging students to use the graphing app for families of graphs, and inequalities.  If I let them use their notes on the iPads, then they do not get a chance to get up, move around and clear their brains.  I know that computer based state testing begins in another year or two and that will have a calculator available to use during the test.

I think for now, I will go with the handwritten notes so they get one more chance to learn the material.

inequalities and graphing apps.

In algebra 2, we are learning to graph linear and absolute value inequalities using a graphing app.  I like the free graph calc for graphing as it has so many choices and resembles a regular graphing calculator.  It does not seem to support graphing inequalities which is fine with me because the students can graph the base linear or absolute value equation, then do the analysis to determine if they need a dotted or solid line and which part is shaded in.  Too often my students just copy down the result from the graphing calculators or a regular calculator without checking to see if its reasonable. 
Yesterday, all the high school teachers gave students a directions test during study hall. The test had them writing down things like the letter D in the middle of the page, writing down their favorite musical group in a corner, etc.  We did this because the majority of our students do not read directions.  As expected, most students failed the test but the kids got such a charge out of the test.  What they don't know is that they will get another one next month.
Finally, the tech dept and I are still working to figure out how I can have students either send the material to me using e-mail or get it to me via a drop box type of situation.  I am not allowed to use google docs, regular e-mail accounts, drop box or any other type of app like that.  So it is a challenge.  I am doing some research and have a couple possibilities that I need to explore hopefully over the weekend.  I hope by January to greatly reduce the paper flow.

Updates, etc

I just updated the iPads to iOS 7 this past Tuesday mostly because students were trying to download the new operating system. Of the three, one downloaded successfully and the other two generated the plug screen of death.  Friday, the students are telling me about the update but agreed to let me do it later today, so during my prep period and after school, I'll be downloading the newest version of the operating system.  I can say, my kids like the new look and feel of the latest operating system.
I had the students in Algebra 1 use popplet to create a map of relations, functions, domain and range, and the vertical line test.  Several of the students understood the difference between relations and functions at the end of the exercise but popplet won't do what I wanted in terms of being a notebook so I found another program to use.  So far, having a different notebook app for each period is working well on the shared iPads.  It is interesting to see the students creating notes for themselves. Some just shove something together while others take the time to make some awesome notes and if I had had to predict everyone who would make great notes, there were a few I would have been wrong on.
Later today, I am going to have to track down another graphing program that will allow me to graph conic sections without rewriting problems into the y = format.  I used it last year and it has a small tutoring section so the students can read up on the topic if they forget.  Actually, I need to go through past blogs to find the name of it but I use it when I teach conic sections.

Construction lite

Today in two different classes we used apps to help the students learn.
In the college prep math class we used free graphing calculator to learn what the graphs of sin and cos looked like and what happens if you add this or that to the equation.  We discussed it, played with equations and the kids really had a blast.
We started with y = sinx.  I took time to have them read the unit circle so that we could connect the unit circle to the sin graph.   They graphed y = 3sinx on a different line so they could have both equations on the screen at once.  This way they were able to tell me what happened to the graph by adding the 3.  I had them graph y = sinx + 2,  y = sin(x+pi/2) and y = 3sin(x + pi/2)+2.  After comparing and contrasting the graphs, I was able to lead them to seeing how the transformations from earlier math classes applied to trig graphs.  It was awesome.
I downloaded and had my geometry students use construction lite which simulates using the compass and straightedge.  The first two sections are unlocked in this free version of the app.  The sections are lines and angles.  We looked at copying a segment and constructing a perpendicular bisector.  The app shows the student how to do each activity and then allows them to do it themselves using the digital compass and straightedge. 
I just let my students loose to try the sections themselves and as one figured out how to do something, I would have him or her go help others so those who finished early were working as peer tutors with the others.  Although only two topics are available, there is a sketch function that will allow me to have students follow the construction directions in the books and I don't have to keep track of all the compasses, pencils etc.  The kids enjoyed themselves.

NASA space math problems

I forgot about NASA space math problems until I got one of their regular updates in my e-mail this morning.  The latest problem is Problem 607: The Launch of LADEE to the Moon.
The description is :
Students plot the altitude, range and speed of the LADEE rocket launch and investigate rates of change including acceleration by graphing the tabular data and determining the slope of the graph using the definition of the slope of a line between two points.
This is a wonderful example of rate of change in a real situation where students can try it themselves.  The material NASA offers covers grades 6 to 12 and has all the past problems available for use along with some other items such as a space math book for grades 6 to 8 which is a standards based with multi-media math resources. 
The information is in PDF form so I can bring it into an app like subtext to interject questions, tags, activities, etc.  Most will require use of a calculator/graphing app, a drawing or note app and they could place their answers into a QR code to turn in.  Hmmmm the possibilities.