Osterhaus, S.A. (2002). Susan's math technology corner: Teaching A Blind Student How to Graph on a Coordinate Plane: No Tech, Low Tech, and High Tech Tools. Division on Visual Impairments Quarterly, 47(3), 23-26.

# Susan's Math Technology Corner

### Background

Although the use of scientific graphing calculators is now a secondary math classroom mainstay, all students should first understand the concept of graphing on a coordinate plane manually. I REALLY insist that my students be able to physically plot points, graph lines, and find the slope as well. This ability is even more critical for blind students because most math technology is not accessible to them.

### The Problem

Recently, I have received an avalanche of requests for help from teachers of  students with visual impairments and math teachers. Question: “How can blind students graph linear equations, inequalities, and systems of inequalities independently and efficiently? Or is this the time the student doesn’t participate because of the visual nature of the task?” Answer: Most academic blind students, even those with spatial orientation problems, are quite capable of graphing, and as one of my students exclaimed, “Not only can we do it, it’s fun!”

### No Tech, Low Tech, and High Tech Solution

The Graphic Aid For Mathematics from APH is excellent for graphing algebraic equations but can be used in geometry, trigonometry, etc. It consists of a cork composition board mounted with a rubber mat, which has been embossed with a grid of 1/2-inch squares. My students use two perpendicular rubber bands held down by thumbtacks for the x- and y-axes. Then, points are plotted with pushpins at the appropriate coordinates. Points are connected with rubber bands (for lines), flat spring wires (for conic sections), or string (for polynomial functions). Sighted math teachers can easily interpret the student-made graphs correctly. You can also make your own rubber graph board by affixing a piece of raised line graph paper (also from APH) to a cork board and proceeding as outlined above.

I do mention the use of Wikki Stix and high dots on APH graph paper when the student MUST hand in copies of graphs for homework to insistent math teachers. However, this method can be quite expensive and is very time consuming and is more of a test of artistic ability. I REALLY want my students to graph extensively; and they can do so incredibly fast on the APH Graphic Aid for Mathematics. In fact, many of my print students insist on using it as well because it is faster, fun, and allows graphing skills to be learned in one more modality.

At the same time, the students are being exposed to the ORION TI-34 talking scientific calculator from Orbit Research, which allows them to perform any necessary computations to speed up the graphing process.

I introduce the AGC (Accessible Graphing Calculator from ViewPlus Technologies) when we start exploring what is and isn't a linear equation. For example, our textbook presents an exploration problem where the students are to first make an educated guess as to whether the graph of an equation will be a straight line or not. Then, they are to test their hypothesis by graphing it. The book lists about 10 equations. Well, that would take quite a long while if the students did everything manually, especially since most haven't had exposure to quadratic equations and rational functions. However, the equations can be quickly entered into the AGC, and the students can listen to the audiowave and immediately tell the differences among y=3*x+4, y=x^2, and y=3/x+2 (the way you must enter equations on the AGC). Additionally, we have a TIGER Advantage networked to each computer, so my students can also emboss each graph very quickly.

My pride and joy is a braille student who I had in Algebra 1 last year. I introduced him to graphing manually and then showed him the AGC, as indicated above. He is now in Algebra 2 and is proficient at both. I continue to show him how to solve Algebra 2 problems manually and with technology, and he analyzes which method is best for which circumstance. For example, he might graph a quadratic function manually because it was "too easy to bother with the computer." Yet, he will use the AGC to graph an exponential function.

### Specifics

#### 1. How do students represent inequalities that require a solid line or a dotted line on the graph?

Again, my students use the APH Graphic Aid for Mathematics (has raised grid lines), rubber bands held down by thumbtacks to form the x- and y-axes, and pushpins to plot points. We connect the points with a rubber band when the boundary line is to be included in the solution (solid line in print), and we leave off the rubber band when the boundary line is not included in the solution (dotted or dashed line).

#### 2. How do they show shaded parts on the graph?

When graphing one inequality in two variables, my students simply place their hand on the shaded side. When graphing a system of two inequalities, the student places one hand on the shaded side of the first inequality. Then they place the other hand on the shaded side of the second inequality. Where the two hands overlap (including the boundary lines where applicable) is the solution. Pretty soon most of my students are able to handle three or more inequalities without multiple overlapping of hands. We even progress to linear programming problems involving four or more inequalities. In these problems, a bounded area with vertices is often found, and it is pretty obvious where the shaded portion (solution) is located.

#### 3. Is there a way for them to do multiple problems on a piece of paper?

I check each graph as my students complete them. For example, during a test, they have me check each graph and write a notation on their paper before they move onto the next problem. I check to see if the boundary lines are drawn correctly (with or without rubber band) and if they place the "shading" in the correct area.

If your student does need to hand in several graphics, here are my suggestions:

When needing to graph on a coordinate plane, the student could use APH raised line graph paper attached to a corkboard. Then, he could plot his points using stick-on high dots, puff paint, etc. He could form the solid lines using Wikki Stix. He could actually use a colored pen, pencil, or crayon to color the shaded area of the solution. Of course, this all takes MUCH longer than our method, but this would be necessary if a student-made, manually produced, paper copy is required. Then again, the student could easily hand in a paper copy of any single function (can't graph multiple functions on the same graph) created on the AGC.

One year I had a student whose math teacher insisted that all graphs needed to be handed in on a two-sided piece of paper containing 9 small coordinate planes on each side. This student graphed each equation on the graph board, and I copied the work onto the “designated” sheet. The student was perturbed because I couldn’t keep up with her and was slowing her down! Nevertheless, she passed with flying colors.

I would rather see students become proficient at using the rubber graph board, as they will learn SO MUCH more with this method, and they can do so independently. As an alternative, you could divide the APH Graphic Aid for Mathematics into 4 to 6 small, separate coordinate planes. If you have a digital camera, you could even e-mail or print a picture of your student’s graphs! Better yet, have the student or his parents take the photo!

### Bottom Line

PLEASE be sure your students are allowed to participate in all kinds of graphing and are supplied with the proper tools. This creative exploration should begin in the early grades and be allowed to blossom. Remember, the beauty of a tactile graphic is found in the fingertips of the beholder. And there can be no more beautiful and meaningful a graphic than one created by those very same fingertips.

### Sources for No Tech, Low Tech, and High Tech Tools:

Susan A. Osterhaus, M.Ed.
Secondary Mathematics Teacher
Texas School for the Blind and Visually Impaired
1100 West 45th Street
Austin, TX 78756
Phone: (512) 206-9305
Fax: (512) 206-9453
E-mail:
Website: http://www.tsbvi.edu/math