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Q&A #6297


Simultaneous equations

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From: Marielouise (for Teacher2Teacher Service)
Date: May 08, 2001 at 17:39:53
Subject: Re: Simultaneous equations

Hi, Sue,

Simulataneous equations are two or more equations with a common solution.   
That means that a value of (x,y) = (a,b) where x = a and y = b will be true 
for all of the equations.   There are different methods for solving them:
1. by graphing on the same grid
2. by elimination of one or more of the equations by multiplication/addition/
subtraction for equivalent equations.   For example,   x + y = -3  is the same 
equation as -2x - 2y = 6 
3. by solving one equation for one variable and substituting the expression 
for the variable in the other equations.

Another associate wrote the following:

>being a teacher i would like to know the different ways of teaching
algebraic
expression and equations also i would appreciate activities for learning and
teaching simultaneous equations.
> thankyou for giving opportunity to learn from you.

That is a pretty big order!

One thing in teaching algebraic expressions, is to use algebra tiles or
algeblocks?? I don't know if you are familiar with the process. For some of
the students who tend to be more visual learners, it allows them a way to
abstract to the algebraic expressions a little easier.  There is a wonderful
book published by NCTM (National Council of Teachers of Mathematics) called
ALGEBRA FOR EVERYONE. I recall reading in the book about the students
struggling from going from 23 to be a number composed of 2*10+3*1...then we
automatically throw ab at them, and expect them to understand that the
expression means a times b.   It was a moment of reflection for me.

The best idea I have ever heard about teaching simultaneous equations,
begins with what a friend of mine calls "spaghetti systems".  Lets take a
simple coin problem.  Jeni has 20 coins consisting of nickels and dimes
which total $1.40. How many of each coin?  Most students can pick two
numbers that will give a sum of 20. Give each student a sheet of graph paper
and two pieces of uncooked spaghetti (Linguini maybe better because it
doesn't roll as easily).  Say the student says 10 + 10  or 15 + 5.  Decide
which axis is nickels and which are dimes.  Put one piece of spaghetti as
the line between those two points. (Review geometry...two points determine a
line)  Then have them pick two solutions for the $1.40... 14 dimes and 0
nickels  or 10 dimes and 8 nickels....use the other piece of spaghetti.
Where the two pieces of pasta intersect is the solution.

This is a nice non threatening way of introducing systems to students,
especially those afraid of word problems.  Then you go to other methods,
since the spaghetti might be hard to estimate fraction or decimal values, or
not long enough for large number solutions..........

You might also want to check out the www.enc.org web site. THe Eisenhower
National Clearinghouse has lots of projects, activities and lessons for all
topics in math and science.


 -Claudia, for the Teacher2Teacher service-Marielouise, for the T2T service

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