INTRODUCTION TO SOLVING EQUATIONS

Mary Lynn Bochenek             Central Junior High
                               17248 S. 67th Ave.
                               Tinley Park, IL 60477
                               1-708-532-1771

OBJECTIVE:
    To arrive at an equation of the form X = [ ] in which the variable is isolated 
and the solution is then obvious through use of equivalent equations. 

MATERIALS:
    -film cans (with appropriate weights inside and labeled)
    -balance scales (at least 2)
    -cards with equations written on them
    -handouts (equations to solve, twelve ball problem, the hundred 
     artisans, and various others)
    -overhead projector
    -card with equal symbol and not equal symbol

STRATEGIES:

    Each film can weighs approximately 5 grams so I let that be my 1. A individual 
film can with cover is approximately 7 grams so I filled each can with sand, 
nails, money, etc. to have 10 grams be a 2, 15 grams be a 3, etc.  I let x = 4, y 
=3, z =8, w = 15 and v = 2 for my equations.  You also need cans labeled 3x-2, 2y, 
x-1, z-2, 3x, 5y, z/2, w/3 and 2y+1. 
    Equations used v=v, 2+3=5, y+4=7, 6=2+x, 4-1=3, x-1<>x, x-1=3, 6=z-2, 2*3=6, 
3x=12, 15=5y, 6/3=2, z/2=4, 5=w/3, 2y+1=7 and 3x-2=10. 
    Prior knowledge:  variables and expressions, order of operations, 
evaluating expressions, properties of operations and inverse operations. 
    Discussion at beginning:
    Solving equations:  "The basic idea is to find out what x is.  The catch is, 
that as soon as you do, they change it to something else!" 
    Equal symbol states that two expressions name exactly the same number. An 
equation is a number sentence which states that two expressions are equal.  
Therefore, an equation must have an equal symbol.  We discussed a not equal to 
symbol and defined a variable as a symbol, usually a letter, that can represent 
any number.  Remember when the solution of an equation is found, it must be 
checked.  Symmetric property states that x-4=9 is the same as 9=x-4 and 
commutative property states that 113 + x is the same as x+ 113. 
    I will use the pan-balance scale to demonstrate how to solve simple equations 
incorporating the concepts of our properties and inverse operations.  Each step is 
an equivalent equation that is easier to solve.  Although the equation changes in 
the process, the solution remains the same.  In your last step, the equation is so 
simple that it tells you the solution. 
    Okay, let's start with our first equation v=v.  I demonstrated on a balance 
scale how any number equals itself.  Also that if I add the same number to both 
sides of the balance scale, I will have an equivalent equation.  Then we proved 
2+3=5 on the scale.  Now two equations with variables are solved and checked.  
Discussion of what we did should lead to the Subtraction Property of Equality 
which states that subtracting the same number from both sides of an equation does 
not change the equality.  For all real nos. a, b, and c, if a=b, then a-c=b-c.  
    Our fourth equation 4-1=3 lead to a discussion of subtraction on the balance 
scale.  We must break 4 down to 3+1 and then take 1 away or we can add 1 to both 
sides.  On the scale I showed that x-1 is not equal to x.  In order to equate them 
I must add 1 to x-1.  Therefore, I know that x-1+1=x.  Our next equation x-1=3 
2
uses the concept from the last equation that we must add 1 to both sides, then 
substitute x for x-1+1 to reach our goal x=4.  Have a student volunteer to 
demonstrate that 6=z-2.  This leads to a discussion of the Addition Property of 
Equality which states that adding the same number to both sides of an equation 
does not change the equality.  For all real nos. a, b, and c, if a=b, then 
a+c=b+c. 
    On to multiplication:  2*3=6 means 2 groups of 3 or 3 groups of 2. Both of 
these need to be demonstrated.  Therefore, 3x=12 means 3 groups of x = 12 so 3 x's 
need to be shown equal to 3x and then substituted for 3x can.  Now take away 
equals and you are left with x=4.  Make sure you check your answer.  Have a 
student demonstrate the next equation. This leads to the Division Property of 
Equality:  dividing both sides of an equation by the same nonzero number does not 
change the equality. For all real nos. a, b, and c, with c<>0, if a=b, then 
a/c=b/c. 
    Now for division:  6/3=2 is our first equation.  If we break 6 up into 3 
groups, each one will be a 2.  For z/2=4, we need to show they are equal and then 
show that z/2 + z/2 = z but z/2 + z/2 = 4 + 4.  Showing this on the scale leads to 
z=8.  Have a student demonstrate 5=w/3.  This leads to the Multiplication Property 
of Equality:  Multiplying both sides of an equation by the same nonzero number 
does not change the equality.  For all real nos. a, b, and c, with c<>o, if a=b, 
then ca=cb. 
    Finally we tried to solve 2y+1=7 on the scale using the concepts that we had 
learned in previous problems.  Break 7 into 6+1, then subtract 1 from both sides; 
break 2y into y+y and 6 into 3+3, then take away equals to have y=3.  Lastly, 3x-
2=10. 
    Students should now be given the handouts and solve the equations using the 
methods learned and the scales if needed.  When finished with their equations they 
need to try the Twelve Balls problem from the handout. This can be tried and 
demonstrated on the balance scale.  The other problems from the handouts should be 
discussed in later sessions.