Simplifying Equations of the Form ax+b=cx+d

Wagner, William F.                       Hyde Park Career Academy
                                         947-7233
                           

Objective(s):

(1) Students will be able to solve equations of the form ax+b=cx+d.

(2) Students will be able to visualize the concept of "balancing" an 
equation.

Apparatus Needed:

Balance scale (easily obtainable if you have an understanding and 
               cooperative science department)
Assorted weights (I adapted a box of Cuisenaire rods)
Yard stick (optional)
Small stuffed animal or soft rubber toy (optional)
Assorted "unknowns" (any objects that are of an unknown weight but can 
                    be weighed as a combination of rods)

Recommended Strategy

     As a link between this lesson and previous lessons, I started with 
a review of the four basic types of equations, i.e. equations that can 
be solved by adding, subtracting, multiplying or dividing. I 
differentiate between the four types so that the students can remember 
them more easily by associating them with the four basic operations. 
Others have argued that there is only two or three basic types. I leave 
the final decision up to you. 
     After this short review and discussion of methods for solving 
these types of equations, we show a more complicated equation of the 
ax+b=cx+d form and ask the students for suggestions on how to solve it 
based on present knowledge. After students have had an opportunity to 
give input on the solution, we demonstrate how equations can be 
displayed using a balance scale. 
     Balance Scale Demonstration:
     Using the Cuisenaire rods, set up a situation you know is equal. 
For example, 4 light green rods with 4 white cubes weighs the same as 2 
light green with 10 white cubes. Place these different combinations on 
opposite sides of the balance scale and watch as the scale balances 
(experimenting before is highly recommended to avoid an embarassing 
situation). Then ask the students what would happen if you started to 
remove pieces from different sides, (Desired response: the balance is 
upset). Next ask the students, "What should I do to put the balance 
back?" (Desired response: Whatever you did to the first side, now do to 
the other). Continue in this manner until the following situation is 
reached: 2 light green rods are balancing 6 white cubes. Now say, "If 2 
green rods balance 6 white rods, what would happen if I took out half 
of the green rods?" (Desired response: Balance upset). Next say, "What 
should I do to restore the balance?" (Desired response: Take out half 
the white cubes). Upon doing this, the solution is now evident: 1 light 
green rod equals 3 white rods. Try this again with different 
combinations of rods (Again, experiment beforehand). As you go through 
each step, write on the board what the situation on the scale is, but 
in equation form, i.e. 4 light green with 4 white cubes balances 2 
light green with 10 white cubes would become 4g+4=2g+10. 
     After a few attempts, show that they are merely doing the same 
manipulations that they did on the basic 4 types with the following 
conditions: Adding or subtracting is done first with multiplication or 
division done last, each step should make the problem simpler than 
before and the ultimate end is to have all the unknowns on one side and 
all the constants on the other. Once this condition is met, the final 
step is to multiply or divide by some constant. 
     This demonstration can be made more dynamic by allowing individual 
students to try their hand at the manipulations to produce equivalent 
situations. Other objects may be used as unknowns as students 
experiment to find other true statements. 
     At the end of the lesson and as a taste of what is to come, I 
might use the yard stick and the stuffed animal to set up a situation 
to demonstrate inequalities. Place the yard stick so that half is on 
the desk (Use the stuffed animal as a weight by placing it on the end 
of the yardstick on the desk) and the other half off the desk. Then 
proceed to strike the free end of the yardstick sending the stuffed 
animal flying (experiment beforehand to be certain that the yardstick 
is strong enough and the stuffed animal light enough to make the 
flight). I then ask, just as the bell rings, "Why did the animal fly?". 
Without answering, class is dismissed and as they are leaving I tell 
them we will be discussing what they saw in our next class, which will 
be on inequalities.