This lesson covers perpendicular lines.
Perpendicular lines are lines that intersect to form right angles. Also, if each of the four angles formed at the intersection of two lines measures 90⁰, the lines are perpendicular.
Let’s take a plain sheet of paper shaped like a rectangle that is not a square and carefully fold it in half lengthwise or vertically in this case. Make a good crease, and then unfold the paper. Now, carefully fold it in half widthwise or horizontally in this case. Make a good crease, and then unfold the paper. The creases that you just formed intersected and created four right angles in the center of the paper.
Let’s take another similar plain sheet of paper and look at the corners. Each corner is formed by one of the long sides and one of the short sides, and it measures 90⁰. Therefore, each side and its adjacent side are segments of perpendicular lines. Wow! Perpendicular lines are everywhere!
We are going to take a third piece of paper and Construct a Perpendicular to a Line from a Point on the Line, using a compass and a straightedge.
Step 1:
Draw a line l horizontally across the page using a straightedge and pen or pencil.
Place a point P on line l somewhere approximately in the middle.
Place the compass point on P.
Open the compass or extend the compass wide enough to be able to draw arcs that intersect l at two points, Q and R equidistant from P.
Step 2:
Open or extend the compass to about the length of QR.
Place the compass point on Q and make a small arc above the line and above point P.
Place the compass point on R and make another small arc that crosses the first small arc, intersecting at point S.
Step 3:
Use a straightedge to join Points S and P. Line SP is perpendicular to l at P.
You could adapt this construction for a student with low vision by simply ensuring that you placed your piece of paper on a well-lit flat table with good contrast on top of some type of drawing board, using a compass that has an expandable leg to hold a black-line marker such as a Sharpie, 20-20, or Flair pen on a white piece of paper. You could use any straightedge. I am sure that you noticed I already used this technique to perform the initial construction.
You could adapt this construction for a student who is blind by using:
1. A Howe Press Compass or an APH Tactile Compass for Math and Art on top of a piece of braille paper placed on a rubber pad, a clipboard with craft foam, or a Sewell Raised Line Drawing Board – basically a clipboard with a rubberized surface already attached. You might want to flip the paper after Step 1 and flip it back after Step 2 to get the best effect, since this technique raises the area on the opposite side of the paper. I would not take the time to label all the points, assuming the student was able to do so without. You would need to be a bit more verbal when describing the points. You could use any straightedge.
OR
2. A piece of special film placed on either an APH Draftsman or an EASY inTACT Sketchpad and the accompanying stylus placed inside a compass with an expandable leg. The drawing will appear raised on the drawing surface of the film. I would not take the time to label all the points, assuming the student was able to do so without. I would also use any straightedge.
OR
3. Any other method you may devise to accomplish a similar effect.
Two other constructions that involve perpendicular lines are: A Perpendicular from a Point not on a Line and the Perpendicular Bisector of a Line. These and other constructions can be completed by a student with low vision or a student with no vision similar to the techniques listed above.
In this figure line t intersects line u, and we know they are perpendicular because of the special right angle marking, that looks like a little box. We can also draw this box tactually with a tracing wheel for a student who is blind.
We have a special symbol in mathematics to indicate that they are perpendicular. It is one line segment drawn perpendicular to another ( | ), which is very appropriate in print. We also have a symbol for students who read braille. In the Nemeth code it is $p (dots 1,2,4,6 followed by dots 1,2,3,4). In UEB it is #- (dots 3,4,5,6 followed by dots 3,6).
We can extend our work with perpendicular lines to the Cartesian coordinate plane. So, let’s look at two perpendicular lines in the coordinate plane. While parallel lines have the same slope, lines that are perpendicular to each other have opposite reciprocal slopes. We can determine perpendicularity just by looking at the equations of lines just as we did with parallel lines. One line has a slope of -4 and the other has a slope of ¼.
You could adapt this figure for a student with low vision by simply ensuring that you had good contrast between the lines, the axes, and the gridlines of the coordinate plane and placing them both on a well-lit and contrasting additional drawing surface. You could use a black-line marker such as a Sharpie, 20-20 pen, or Flair pen. I would also use a straightedge to ensure that the two lines are drawn as straight as possible. You could purchase already prepared bold-line graph paper or create it yourself.
You could adapt this figure for a student who is blind by using:
1. The APH Graphic Aid for Mathematics (rubber graph board), push pins, thumb tacks, and rubber bands.
OR
2. Raised line graph paper on a cork board, push pins, thumb tacks, and rubber bands.
OR
3. Any other method you may devise to accomplish a similar effect.
The student would be able to create this same figure independently using the above methods and should be encouraged to do so, as creating their own graphics (in addition to reading them) reinforces the math concept.
If your student still has difficulties with understanding the concept of perpendicular lines, you might wish to describe certain real life examples of perpendicular lines such as railroad ties to train tracks or perpendicular streets in a city grid pattern. Have the student examine a real train track if possible or at least a wooden toy train on tracks. Have the student’s Orientation and Mobility Specialist reinforce the concept of perpendicular streets during a mobility run or at least use something like the APH Picture Maker Wheatley Tactile Diagramming Kit to construct perpendicular streets on a tactile map.
In this video, we have covered the definition of perpendicular lines, how to construct them, graph them on a coordinate plane, how to write perpendicular in mathematical notation, and how to create the right-angle marking.