*Editor’s Note: In the author’s words, “Although graphing calculators are mainstays of most secondary math classrooms, it is important for all students to understand the concept of graphing on a coordinate plane before they move to the graphing calculator.” This is especially important for visually impaired students, and Susan Osterhaus, math teacher at TSBVI, ensures that her students learn to do so manually. They must physically plot points, graph lines, and find slope. Below are her answers to questions about how to teach this skill, and her suggestions for students, teachers, and parents.*

**1. How can blind students graph linear equations, inequalities, and systems of inequalities independently and efficiently? Or is this the time when the VI student doesn’t participate because of the visual nature of the task?**

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!” There are several tools they can use to do so:

**The Graphic Aid for Mathematics**, from the American Printing House for the Blind (APH), is excellent for graphing algebraic equations. It can also 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. Two perpendicular rubber bands, held down by thumbtacks, can create the x- and y-axes. 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). I like for my students to graph extensively, and they can do so incredibly fast on the APH Graphic Aid. In fact, many print students also like using it because it is fast, fun, and allows them to learn graphing skills in another modality. You can make your own graph board by affixing a piece of raised line graph paper (also available from APH) to a **cork board **and proceeding as described for the Graphic Aid.

If a student needs to turn in copies of graphs for homework, he can use **Wikki Stix **and **High Dots **on **APH graph paper**. This method can be quite expensive, however, and is very time consuming. It also tends to be more of a test of artistic ability than a demonstration of understanding of graphing concepts.

The **ORION TI-84**+ talking graphing calculator (from Orbit Research) and the **Desmos graphing calculator** are examples of more high tech solutions for graphing activities. However, I strongly recommend that students be able to graph manually as well. It is important for visually impaired students to be able to use a variety of tools, and know when to use each of them. For example, a former student decided to graph a quadratic function manually because it was “too easy to bother with the computer.” Yet, he will use the Orion TI-84+ or Desmos to graph an exponential function.

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

The APH Graphic Aid described above works well. Plot the points with pushpins and connect them with a rubber band when the boundary line is to be included in the solution (a solid line in print). Leave off the rubber band when the boundary line is not included in the solution (dotted or dashed line in print).

**3. How can VI students show shaded parts on a graph?**

When graphing one inequality in two variables, I simply have my students place their hand on the shaded side. I check each graph as my students complete them. 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 (the solution) is located.

**4. Is there a way for them to do multiple problems on a piece of paper? What if they need to be turned in to another teacher, or you can’t check each graph as it’s completed?**

If a student-made, manually produced paper copy is required, the student could use APH graph paper attached to a corkboard. She could plot her points using stick-on high dots, puff paint, etc. and could form solid lines using Wikki Stix. She could actually use a colored pen, pencil, or crayon to color the shaded area of the solution. This would take much longer, however, and would be very labor intensive. It will be important to know the purpose of the assignment and the concept(s) being taught. A paper copy can be generated from either the Orion TI-84+ or Desmos. A sighted person (teacher, peer, parent, teaching assistant) could make a print copy of the student’s graph. That is, the visually impaired student graphs on the Graphic Aid and someone copies it exactly onto a piece of paper to turn in. (A math teacher made me do that one year when I was supporting a student in her class. The student was so much faster than I was, and all I was doing was copying her work onto graph paper!! We had to use several graph boards since I couldn’t keep up with her.) You can also divide the Graphic Aid into 4 to 6 small, separate coordinate planes for multiple problems. However, you could take a picture of each graph with a digital camera or your cell phone. Then you could just e-mail or print a picture of the student’s graphs. Better yet, have the student take her own photos! It is really easy for them to do that, and they feel so independent! You can purchase index cards on a spiral and add large print and braille numbers to each index card. The student flips to the problem number associated with each of her graphs and places the card file in the bottom right hand corner (or wherever there is space), so the sighted math teacher knows which homework problem goes with each graph.

Please be sure that visually impaired students are allowed to participate in all kinds of graphing activities and that they are supplied with the proper tools. I would rather see them become proficient using a rubber graph board because they will learn so much more with this method, and they can do so independently. Creative exploration should begin in the early grades and 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 graphic than one created by those very same fingertips.

**Sources for materials described in this article:**

- APH Graphic Aid for Mathematics and APH Graph Paper: www.aph.org
- Wikki Stix: www.wikkistix.com

- High Dots: www.exceptionalteaching.com

- ORION TI-84+ Talking Graphing Calculator: www.orbitresearch.com

- Desmos Graphing Calculator: https://www.desmos.com/calculator

*This article was originally published in the See/Hear Newsletter 8(1),19-21 by Texas School for the Blind and Visually Impaired (TSBVI) and* *is reprinted here with permission.* * Updated in 2024.*