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Versione delle FAQ in lingua italiana.

D1. Bisecting angles tactually.

D2. Solving quadratic equations.

D3. Subtraction problem in the brailler.

D4.Teaching him Perimeter and Area.

D1. The student I work with is a ninth grade braille reader who is in advanced classes. Since she does not like to use foil or the Sewell raised line drawing technique, I was hoping you might have information on how my student can learn to bisect angles tactually.

R1. For constructions, my students don't use foil or the "usual" Sewell raised line drawing technique either. We use some type of rubber on a flat surface - whatever you have available. Some of my students and I happen to like an old Sewell raised line drawing board which has rubber attached to a clip board so that I can clip my braille paper to this to keep it from sliding. But, others use a rubber pad on top of a regular wooden drawing board or table. Still others might like a similar rubber on wood board from Howe Press because it too has a way of clipping the paper down. Next, you will need a braille compass from Howe Press. The compass has a regular pointed end, but the other end has a small tracing wheel attached. I have not been able to find these compasses anywhere else. Should you find another source, please let me know. Next you will need a straightedge - any "print" ruler will do if you don't have a plain straightedge, since the student is a braille reader. Finally, you will need a tracing wheel. Use one from the homemaking department, or Howe Press, or the APH tactile drawing kit, or the local hardware/hobby shop. For your student to bisect an angle you would first take a piece of braille paper (not the flimsy Sewell plastic) and place it on your rubberized surface (board). Draw the angle you wish the student to bisect using a straightedge and tracing wheel. Remove it from the board. Label the angle with an "A" at the vertex using slate and stylus or your braillewriter. Return the braille paper to the board. Ask the student to bisect angle A. The student should first reverse the paper. Place the compass point on A and draw an arc, locating two points B and C on the respective rays of the angle. Reverse the paper. Place the compass point on B and draw an arc in the interior of the angle. With the same compass setting, place the compass point on C and draw an arc, locating point D - the intersection of the two arcs. Reverse the paper. Draw a ray, AD, which is the angle bisector of angle A. Voila!! Using a similar technique with only a compass and straightedge, a blind student (or anyone else) can also copy a line segment, bisect a segment, copy a triangle, copy an angle, construct the perpendicular bisector of a segment, etc. These are the same basic techniques that the math teacher would use except that the braille student would usually prefer reversing the paper so as to take the most advantage of the raised drawing on the reverse side. The end product is easily graded by the math teacher - allowing the student to stay in the regular classroom setting throughout the construction.

D2.I have a braille using student in 11th grade math. He and his class are going to be solving quadratic equations with graphing calculators next week. He has Graphit on a BNS. My question is: is there a way either using Graphit or the scientific calculator on the BNS to reveal the roots of an equation. If not, is there something you would recommend, preferably so he can do the work independently? Your help would be much appreciate.

R2.The ability to "see" the connection between a graph and its equation can be helpful to both visual and tactual learners. I still do this the old fashion way with my low vision and braille students; they manually graph selected quadratic functions on large print graph paper or graph boards. The x-intercepts are revealed to be the roots of the related quadratic equation. Then we move on to using the Accessible Graphing Calculator (AGC) from ViewPlus Software. Graphing calculators simply allow students many more opportunities to make that connection in a brief period of time. To solve a particular quadratic equation in standard form (reveal its roots), your student should be able to instruct Graph-It (or the AGC) to graph the related quadratic function. Then, the zeros will appear as the x-intercepts. In other words, the real roots of the quadratic equation will be the values of x where the function crosses the x-axis. For example: Graph y = x2-2x-3 to find the roots of 0 = x2-2x-3. The graph crosses the x-axis at x = -1 and x = 3. Therefore the roots of 0 = x2-2x-3 are -1 and 3. If the roots are not integers, you will probably not be able to determine the exact value of the roots in this manner, but solving quadratic equations graphically is still a quick way to determine the NUMBER of real roots, and this is extremely valuable information. I might add that when my braille students manually graph a quadratic function with integral zeros, they get exact answers. When a low vision student uses his TI-82 scientific graphing calculator and the trace feature, he gets decimal approximations of the correct zeros! For example, if x = 1, the graphing calculator might say x = 1.0021053. We often get similar approximations on the AGC. Since we can only find approximate solutions to quadratic functions by using the graphing method, the math teacher will next teach your student how to solve SOME quadratic equations by factoring. Finally, the teacher will introduce your student to the quadratic formula, which will allow him to solve ANY quadratic equation. With the right tools and your guidance, your student should be able to complete all of the above work independently.

D3. How important is it for our elementary kids to do a subtraction problem in the brailler (example: 3 digit subtraction with cancellation signs) if they are using the abacus? When you do a subtraction problem, on the brailler, do you have them solve the problem from right to left?

R3.I'm going to answer your question from a secondary math teacher's viewpoint. I believe that elementary students need to be exposed to working addition, subtraction, multiplication, and division problems on the braillewriter in a spatial arrangement - not necessarily using cancellation signs - until they understand the concept. They should solve the problem very similar to the way a print student would - in the case of subtraction from right to left. Although the textbook or worksheet may give examples using correct Nemeth Code including cancellation signs, you want the students' calculation procedures to be easy and quick. Your students can always use the abacus to check their work on the braillewriter. All the while, they should be learning mental math techniques as well. Once the concept is learned, speed, accuracy and flexibility are more important, and we should see the student quickly progressing to the abacus and mental math, basic calculators, and eventually scientific calculators as they begin higher mathematics. If a blind student has never had to work a math problem in a spatial arrangement (on a braillewriter, with TACK-TILES, or other manipulative) and has only used an abacus, they will most likely have difficulty in algebra with the concept of adding, subtracting, multiplying, and dividing polynomials when presented in a spatial arrangement. This is especially true with division. You just can't manipulate variables on an abacus. For more detailed information on suggested calculation procedures with the braillewriter see pages 41-54 from: Gaylen Kapperman, et al., Strategies for Developing Mathematics Skills in Students Who Use Braille, Research and Development Institute, Inc., August, 1997. This publication can be ordered from:

D4. In teaching the topic of Measurement to a blind student, I have a concern: How should I approach teaching him Perimeter and Area?

R4. I would teach linear measurement very similarly to the way one would teach a sighted student. In the United States we have two systems of units that we use to measure length. I would allow my students to measure several real world items using both customary and metric braille rulers, emphasizing the concept of precision. We would also work on several problems requiring estimation and use of the most "sensible" unit of measure within each system. In addition, we would convert from one customary unit of length to another, and from one metric unit of length to another. The student should also be exposed to raised line drawings and be required to measure these as well. From here we could move on to the concept of perimeter. For a beginning student we could define perimeter to be the distance around a shape (later, a polygon). We might have the student walk around the outside of the school building, the "perimeter" fence of the campus, or around the track and count the number of paces. A student on the track team would soon learn how many times around the "perimeter" of the track resulted in a kilometer, a mile, 100 yards, etc. Then I would present the student with a raised line drawing - perhaps of a square. Using string, we could trace the perimeter of the square and snip it to be exactly the same distance. Then the length of the string would equal the perimeter of the square. We could then examine and determine the perimeters of raised line drawings of a rectangle, triangle, trapezoid, pentagon, etc. with each side appropriately marked in braille with customary and/or metric units. Having calculated the perimeter of many different figures, the student can eventually discover the formula for the perimeter (or circumference) of a circle. When learning about area, we can say that just as we can measure distance around shapes, we can also measure how much surface (area) is enclosed by the sides of a shape (or polygon). Luckily, my classroom's floor is composed of square foot tiles, and we go about determining how many such square tiles are required to cover the surface area of this floor. Everyone is delighted when we find a much easier way to determine this by multiplying the length and width of the room. Then one can progress to various manipulatives. Paper shapes made out of raised line graph paper can be cut into pieces and reassembled to form new shapes with the same area. Rubber graph boards can be partitioned with rubber bands to form shapes, and grid squares can be counted to determine area. Wooden tiles can be assembled to form various shapes and determine area as well. This knowledge can then be transferred to raised line drawings illustrating area. The student should advance through finding the area of a square, rectangle, parallelogram, triangle, and complex shapes. Eventually, the student can investigate and use the formula for the area of a circle.

D5. My first love is braille, but I am being challenged this year, as it is my first year in over 16 years that I am co-teaching Pre-Algebra to a blind student. Next week it is factor trees; I can solve that one okay but I won't ignore any suggestions either.

R5. Now that scientific calculators are required for so much in the secondary mathematics curriculum, I am happy to suggest one kind of neat use for the abacus. (Note: Although my classroom scientific calculators are not able to prime factor a number, my Scientific Notebook computer software is able to do so.) My students use the "Osterhaus" method for prime factorization on the abacus. They tend to cringe at "factor trees;" however, I know that's the way they teach it in Pre-Algebra, or at least introduce it. Eventually, however (in my book at least) they show them the "repeated division" method. The Osterhaus method is simply this repeated division method done on the abacus. I usually just show people how to do it, and it's difficult to put it all into words. Nevertheless, I'll try. Place the whole number to be factored at the extreme right of the abacus; we'll call this the dividend. Then, beginning with 2 (the smallest prime number), check whether each prime is a factor of the dividend. If it is, place this first prime number on the extreme left of the abacus and divide the dividend by the prime. Replace the dividend with your answer (quotient) which becomes your new dividend. Continue (putting each new prime that is a factor in the column to the right of the last factor and replacing the quotient with a new dividend) until you arrive at a quotient of 1. For example, set the whole number 420 at the extreme right of the abacus (4 in the hundreds, 2 in the tens, and 0 in the ones column). Beginning with 2, you find that it is a factor of 420. Therefore, you place 2 on the extreme left of the abacus (trillions column) and replace 420 with 210 at the extreme right of the abacus. You try 2 again and find it is a factor of 210. Place another 2 directly to the right of the first 2 (hundred billions column) and replace 210 with 105 at the extreme right. 2 is not a factor of the new dividend, so you try 3. 3 is a factor of 105. Place 3 directly to the right of the 2nd 2 (10 billions column) and replace 105 with 35 at the extreme right. Is 3 a factor of 35? No, so we try the next prime of 5. Yes, 5 is a factor of 35. Therefore, we place 5 directly to the right of the 3 on the left of the abacus (billions column) and replace 35 with 7 at the extreme right. 5 is not a factor of 7, but 7 is. Therefore, place 7 directly to the right of the five on the left of the abacus (hundred millions column) and replace 7 with 1 at the extreme right. Since your quotient is now 1, you have completed the prime factorization. Reading from left to right, your factorization would be: 2 x2 x 3 x5 x 7 or 22 x3x5x7. You could have all kinds of variations. If the number to be prime factored is quite large, you may wish to use two abaci - one for the dividend and one for the factors. Some students may need to have a space between factors, which again might require two abaci. If the number is extremely large, the student may wish to use a calculator for the repeated divisions and an abacus (abaci) for recording the factors. My students use this prime factorization method when they need to determine the Greatest Common Factor (GCF) or Least Common Multiple (LCM) for rather large numbers.

D6. I have a seventh grade braille student who will soon be studying a math chapter in a regular classroom. Among the topics are the following:

- Translations (slides)
- Reflections
- Line Symmetry
- Tessellations

I have some ideas for the teacher. However, being blind myself, I know these concepts can be very difficult to grasp. I would appreciate any ideas which I might share with the classroom teacher.

R6. I usually introduce translations, reflections, and rotations (sometimes called transformations) together. As a firm believer in the use of manipulatives (for the sighted as well as the blind), I pull out my box of assorted triangles and quadrilaterals. I select two congruent non-regular polygons and place one on top of the other; two scalene triangles are my favorite. I then proceed to slide, flip, or rotate the top manipulative to demonstrate a translation, reflection, or rotation. The bottom manipulative remains in place as the original figure. This correlates well with most print textbooks which may show the original figure in red and the transformed figure in black. If you wish the student to translate a figure to a given point, rotate it to a new position, and reflect it over a given line, you could use four congruent figures. I would probably want to use magnetic manipulatives or ones with velcro in a confined space, to keep things in place. Be sure to show the student the textbook tactile graphics illustrating the same transformations, so they will become familiar with what the "average" textbook furnishes them. If these graphics are not of high quality, make your own using some type of Stereocopier and capsule/swell paper. Furthermore, I show my students examples of test questions on transformations from one of the many TAAS mathematics release tests in braille - produced by Region IV, Houston, Texas. Region IV has superb tactile foil graphics. When we reach the topic of line symmetry, I remind my students of when they were younger and made valentine hearts by cutting a folded piece of paper. Believe it or not, my high school students have fun folding a piece of braille paper and cutting out hearts or some other symmetrical design. I tell them the folded edge is a line of symmetry. Then, I get out my manipulative box again, selecting two congruent right triangles. After placing one on top of the other, I flip (reflect) the one on top over the line segment formed by one of the legs to create a larger isosceles triangle with a line of symmetry (altitude) down the middle. You can also have your student use paper folding to determine symmetry lines for figures studied so far (rectangles, hexagons, etc.). Again, be sure to show the student the textbook tactile illustrations of symmetry and/or make your own graphics as outlined above. Tessellations or tiling patterns is an arrangement of figures that fill a plane but do not overlap or leave gaps. In a pure tessellation, the same figure is used throughout. I usually begin with having my students check out my classroom floor, which is composed of square tiles. I also have a set of tables in the shape of isosceles trapezoids, which create a tessellation. Then I move to textbook or home-made tactile graphics of tessellations using rectangles, equilateral triangles, parallelograms, right triangles, regular hexagons, etc. Let the students explore to find that any triangle or quadrilateral can be used to tessellate a plane, but that only certain polygons with more than four sides tessellate a plane. Tessellations that use more than one type of polygon are called semi-pure tessellations. At this point, I get out my wooden Discovery Blocks from ETA (various and duplicate sizes of triangles, squares, rectangles, and parallelograms) and let them design their own tessellation. One young man designed an incredibly beautiful tessellation and placed the blocks inside a frame. It was quite a magnificent piece of parquetry.