[Blindmath] Solving systems by Graphing

Sun May 25 00:38:19 UTC 2014

I know it may seem as though these are specific to those who are vi/blind but they are not.  Those things benefit all students. As previously mentioned sighted students have the advantage of more easily faking it.  That says nothing about mathematical knowledge or ability.  

Jon Yaggie
EYH Volunteer Coordinator
UIC Mathematics

> On May 24, 2014, at 19:15, sabra1023 via Blindmath <blindmath at nfbnet.org> wrote:
> 
> I understood that fine. I wish more text books were written that way. I usually go from -5 deposit of five when I am making a table. Another thing is that you could give a blind student to simple graphs with two equations, and the blind person could be asked to tell whether the equations are consistent. The blind person would have to know before taking the test to look and make sure of whether the graphs cross or not. However, the graphs never helped me understand very well. Solving the system algebraically helped me understand the best. Then, I was able to learn about the graphs. You should also mention systems with multiple solutions, such as two lines that have different equations but are on top of each other. I liked how you explained verbally what was going to happen first and then explained numerically. I also say that and the younger grades, a blind student should use a graphing board to make simple graphs. Any older grades, this will be repetitive and time-consuming, 
> but in the younger grades it might help to understand what agraffe is and the concept behind it. It will also help in being able to read a graph if you actually know the concept behind making one.
> 
>> On May 24, 2014, at 3:37 PM, Susan Jolly via Blindmath <blindmath at nfbnet.org> wrote:
>> 
>> 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 
>> 
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