Simplifying Equations of the Form ax+b=cx+d
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Wagner, William F. Hyde Park Career Academy
(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
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)
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.