Lesson #7 – Resistance Learning Task
Standard -- MA1A2. Students will simplify and operate with radical expressions, polynomials, and rational expressions.
d. Add, subtract, multiply, and divide rational expressions.
MA1A4. Students will solve quadratic equations and inequalities in one variable.
a. Solve equations graphically using appropriate technology.
b. Find real and complex solutions of equations by factoring, taking square roots, and applying the quadratic
formula.
Essential Question: How can I use algebra to understand and determine resistance in electrical circuits?
We use electricity every day to do everything from brushing our teeth to powering our
cars. Electricity results from the presence and flow of electric charges. Electrons with a
negative charge are attracted to those with a positive charge. Electrons cannot travel
through the air. They need a path to move from one charge to the other. This path is
called a circuit. A simple circuit can be seen in the connection of the negative and positive
ends of a battery.
When a circuit is created, electrons begin moving from the one charge to the other. In the
circuit on the right, a bulb is added to the circuit. The electrons pass through the filament
in the bulb heating it and causing it to glow and give off light.
Electrons try to move as quickly as possible. If a circuit is not set up carefully, too many
electrons can move across at one time causing the circuit to break. We can limit the number
of electrons crossing over a circuit to protect it. Adding objects that use electricity, such as
the bulb in the above circuit, is one way to limit the flow of electrons. This limiting of the flow
of electrons is called resistance. It is often necessary to add objects called resistors to protect
the circuit and the objects using the electricity passing through the circuit. In the circuit
on the right, R1 represents a resistor.
More than one resistor can be placed on a circuit.
The placement of the resistors determines the
total effect on the circuit. The resistors in the
diagram below are placed in parallel (this refers
to the fact that there are no resistors directly
between two resistors, not to the geometric
definition of parallel). Parallel resistors allow
multiple paths for the electricity to flow. Two
examples of parallel resistors are shown on the
right.
The resistors in the next circuit below are not parallel. These resistors are placed
in series because the electricity must travel through all three resistors as it
travels through the circuit.
Resistance is measured in units called ohms and must
always be a positive number. The omega symbol, Ω, is
used to represent ohms.
For n resistors in parallel, R1, R2, R3, etc. the total resistance, RT, across a circuit can be found using the equation:
1
1
1
1
1
=
+
+
+ …+
𝑅𝑇 𝑅1 𝑅2 𝑅3
𝑅𝑛
For n resistors is series, R1, R2, R3, etc. the total resistance, RT, across a circuit can be found using the equation:
𝑅𝑇 = 𝑅1 + 𝑅2 + 𝑅3 + … + 𝑅𝑛
1.
What is the total resistance for a circuit with three resistors in series if the resistances are 2 ohms, 5 ohms, and 4 ohms,
respectively?
2.
What is the total resistance for a circuit with two parallel resistors, one with a resistance of 3 ohms and the other with a
resistance of 7 ohms?
3.
What is the total resistance for a circuit with four resistors in parallel if the resistances are 1 ohm, 3 ohms, ohms, and
5
3
5
2
ohms, respectively?
4.
What is the total resistance for the circuit to the right?
5.
A circuit with a total resistance of has two parallel resistors. One of the resistors has a resistance of 4 ohms.
11
circuit.
a.
Let x represent the resistance of the other of the two resistors, and write an equation for the total resistance of
the circuit.
6.
28
b.
The equation in part a contains rational expressions. If you have any complex fractions, simplify them. In your
equation containing no complex fractions, what is the least common denominator of the rational expressions?
c.
Use the Multiplication Principle of Equality to obtain a new equation that has the same solutions as the equation
in part a but does not contain any rational expressions. Why do you know that x ≠ 0? How does knowing that x ≠ 0
allow you to conclude that this new equation has the same solutions as, or is equivalent to, the equation from
part a.
d.
Solve the new equation to find the resistance in the second resistor. Check your answer.
A circuit has been built using two parallel resistors.
3
a.
One resistor has twice the resistance of the other. If the total resistance of the circuit is ohms, what is the
4
resistance of each of the two resistors?
b.
One resistor has a resistance of 4 ohms. If the total resistance is one-third of that of the other parallel resistor,
what is the total resistance?
7.
A circuit has been built using two paths for the flow of the current; one of the paths has a single resistor and the other
has two resistors in series as shown in the diagram below.
a. Assume that, for the two resistors in series, the second
has a resistance that is three times the resistance of the
first one in the series. The single resistor has a resistance
that is 6 ohms more than the resistance of the first resistor
in series, and the total resistance of the circuit is 4 Ω.
Write an equation to model this situation, and solve this
equation. What is the solution set of the equation? What
is the resistance of the each of the resistors?
b. Assume that, for the two resistors in series, the second has a resistance that is 3 ohms more than twice the
resistance of the first one in the series. The single resistor has a resistance that is 1 ohm more than the resistance of
the first resistor in series, and the total resistance of the circuit is 3 Ω. Write an equation to model this situation, and
solve this equation. What is the solution set of the equation? What is the resistance of the each of the resistors?
c. Assume that, for the two resistors in series, the second has a resistance that is 2 ohms more than the first one in the
series. The single resistor has a resistance that is 3 ohms more than the resistance of the first resistor in series, and
the total resistance of the circuit is 2 Ω. Write an equation to model this situation, and solve this equation. What is
the solution set of the equation? What is the resistance of the each of the resistors?
d. Assume that, for the two resistors in series, the second has a resistance that is 4 ohms more than the first one in the
series. The single resistor has a resistance that is 3 ohms less than the resistance of the first resistor in series, and the
total resistance of the circuit is 4 Ω. . Write an equation to model this situation, and solve this equation. What is the
solution set of the equation? What is the resistance of the each of the resistors?
8.
A circuit has three resistors in parallel. The second resistor has a resistance that is 4 ohms more than the first. The third
resistor has a resistance of 8 ohms. The total resistance is one-half the resistance of the first resistor. Find each of the
unknown resistances.
Homework – Day 1
I.
Perform the following computations. Give your final answer in simplest form.
a.
II.
3
4
+
4
7
−
5
b.
6
2
3
4
+
3
8
Find the least common denominator for each of the following sets of rational expressions.
1
1
1
1
1
1
c.
+
d.
+
e.
+
+
3𝑥
𝑥−8
2𝑥+4
𝑥+2
4
4𝑥+8
1
𝑥+3
Computation with Rational Expressions
Perform the indicated operations and/or simplify. Write your answer in simplest form.
1.
2.
3.
4.
𝑥 2 + 2𝑥−3
9.
6𝑦+𝑦 2 + 8
2+𝑦
10.
−48𝑎2
8.
𝑦+1
𝑦
𝑥+4
5𝑥
𝑥 2 + 5𝑥+6
𝑥 2 + 𝑥−2
𝑥 2 + 2𝑥
𝑥2− 4
2𝑥 2 + 4𝑥
∙
2𝑥−1
2𝑥−3
-
2𝑥−1
÷
8𝑥
4
+
2
5𝑥
2𝑥 2 + 2𝑥
5𝑥 2 − 15𝑥+10
6𝑥−3𝑥 2
𝑘−7
𝑘 2 + 6𝑘+9
+
𝑘−5
𝑘 2 − 5𝑘−24
4𝑥+44
𝑥2− 1
𝑥+1
𝑥−1
4𝑥 + 1
∙(x+3)
13.
7.
+
𝑥 2 − 16
12.
6.
2
𝑦
16𝑎
11.
5.
−9𝑥5
)
7
−12𝑥 2
(
𝑥−1
14.
4𝑥 3 +8𝑥 2 + 10𝑥
2𝑥
Unit 5 – Resistance Learning – Part II
Name ___________________________
1.
What is the total resistance for a circuit with 3
resistors in series if the resistances are 5 ohms, 3
ohms, and 6 ohms?
2.
What is the total resistance for a circuit with two
parallel resistors, one with a resistance of 5 ohms
and the other with a resistance of 7 ohms?
3.
What is the total resistance for a circuit with 3
resistors in parallel if the resistances are 3 ohms,
7.
A circuit has two parallel
resistors. One resistor is
5 ohms. The total
resistance is one-third the
other resistor. Find each
of the unknown resistances.
8.
A circuit has been built using two paths for the
flow of the current; one of the paths has a single
resistor and the other has two resistors in series
as shown in the diagram at the right. Assume
that, for the two resistors in series, the second
has a resistance that is twice the first one in the
series. The single resistor has a resistance that is
4 ohms more than the resistance of the first
resistor in series. The total resistance of the
circuit is 3 ohms. What is the resistance of each
of the resistors?
1
3
6 ohms and ohms, respectively?
4.
What is the total resistance
for a circuit to the right
where R1 = 4 ohms and
R2 = 8 ohms?
5.
A circuit with a total resistance of
12
has two
7
parallel resistors. One resistor is 6 ohms. What
is the other?
6.
A circuit has been built using
two parallel resistors. One
resistor has 4 times the
resistance of the other. If
5
6
the total resistance is ohms,
what is the resistance of each of the two
resistors?