[Blindmath] Solving systems by Graphing

[Susan Jolly via Blindmath]

Just because someone is able to push the buttons on a calculator or to use 
math software does not mean they understand math.  The new way of teaching 
math that relies on the use of these tools makes it easier for sighted 
students and harder for blind students to get good grades in a math class 
but it also makes it harder for everyone to gain an understanding of math. 
There is definitely something serously wrong about the way math is being 
taught now in the United States and it sounds as if we aren't the only ones 
with a problem.  It seems to me that with proper teaching blind students 
should be able to understand math just as well as sighted ones.

Now I want to write about the subject of this email.  I am going to try to 
give an introductory explanation that I think blind students should be able 
to understand.  I would appreciate comments about anything that is hard to 
understand in order for me to get a better appreciation of the problems some 
of you are experiencing.

When studying for a test it is good to memorize definitions.  I like this 
definition of a system of equations from a 1945 Intermediate Algebra book. 
"When two or more equations are considered together, and a common solution 
for them is desired, this is called a system of equations."

This book goes on to state that a set or system of equations which make 
contradictory statements is called an inconsistent system. An example would 
be if the system has two equations and the first one states that x plus y 
equals zero and the second one states that x plus y equals one.  Since both 
equations cannot be true at the same time they are inconsistent.  In other 
words, they don't have a common solution.

Whether or not you understand the reason why you can at least  memorize the 
statement that one way to tell if equations are inconsistent is to graph 
them.  If the graphs don't cross yes they are inconsistent. So you could at 
least get this fact correct on a true false test.

Now let's back up and try to understand this statement about graphs crossing 
with some simple examples.  In algebra you first start with solving simple 
linear equations in one unknown.  Solving means finding the value of the 
unknown. An example is x plus three equals zero where x represents the 
unknown.  A standard way to solve such an equation is to rewrite it so the x 
is on one side of the equals mark and the numbers are on the other.  So x 
plus three equals zero can be rewritten as x equals minus three. You can 
check this result by going back to the first equation and substituting the 
minus three for x.  Minus three plus three does equal zero!

Now lets consider a simple equation with two unknowns. x plus y plus three 
equals zero. There is not just one right answer. Typically this type of 
equation is rewritten with the y on one side of the equals mark and 
everything else on the other side. In this case we would get y equals 
negative x minus three. The value of y depends on the value we choose for x. 
If we choose x equals zero then y = -3. If we choose x = 1 then y = -4. An 
alternative to graphing that everyone should be able to do is to make a 
table or chart. You could do this in Excel or other math software but you 
should be able to do this by hand.  You can make a simple braille chart 
using a slate and stylus. Put some choices for x in one column and then what 
y turns out to be for that particular x in another column in the same row. 
The chart is easier to read if you choose sequential values for x such as 
minus one, zero, one, two, three and so on in the column for x.

You can use this chart method as an alternative to graphing  All a graph 
does is uses markers to represent each pair of values of x and y that are in 
the same row of the chart.

Now lets see how to use a  brailled chart with three columns to illustrate 
our earlier example of two inconsistent equations.  The first equation is x 
plus y equals zero and the second  equation is x plus y equals 1.  Now make 
a chart where you put a value for x in the first column, the corresponding 
value for y according to the first equation in the same row of the second 
column and the corresponding value for y according to the second equation in 
the same row of the third column.  Then add some more rows with different 
values of x. Now see if you can find a row where both values of y are the 
same for the same x.  If you can't that means that if you had graphed these 
two equations they wouldn't cross.  Of course you need to put enough rows in 
the chart to see the pattern to be sure you will never get to a row where 
both values of y are the same. No matter how many rows you add for this 
example you won't find a row with the same values for x because these two 
equations are inconsistent.

I am hoping that if you have trouble understanding what I've written here 
you will take the time to figure out where you got lost.  Then maybe we can 
work together to begin developing some suggestions for teachers who want to 
do a better job helping all their students really understand math. One of 
the wonderful things about math is that once you understand it no one can 
take it away from you by changing the rules.

Best wishes,
SusanJ