Physics
Literal Equations
Manipulating Variables and Constants
About this Lesson
This activity provides significant practice for students to gain skill in manipulating literal
equations.
This lesson is included in the LTF Physics Module 2 and the LTF Chemistry Module 5.
Objectives
Students will:
• Rearrange literal equations to isolate a particular variable
• Use dimensional analysis to check the validity of rearranged relationships
Level
Middle Grades: Chemistry/Physics, Chemistry I, Physics I
T E A C H E R
Common Core State Standards for Science Content
LTF Science lessons will be aligned with the next generation of multi-state science standards
that are currently in development. These standards are said to be developed around the anchor
document, A Framework for K–12 Science Education, which was produced by the National
Research Council. Where applicable, the LTF Science lessons are also aligned to the Common
Core Standards for Mathematical Content as well as the Common Core Literacy Standards for
Science and Technical Subjects.
Code
Standard
(MATH)
A-CED.4
Rearrange formulas to highlight a quantity of
interest, using the same reasoning as in solving
equations. For example, rearrange Ohm’s law
V = IR to highlight resistance R.
(MATH)
N-Q.1
Use units as a way to understand problems and
to guide the solution of multi-step problems;
choose and interpret units consistently in
formulas; choose and interpret the scale and the
origin in graphs and data displays.
Level of
Thinking
Apply
Depth of
Knowledge
II
Apply
II
Connections to AP*
In AP Science courses, particularly Physics and Chemistry, the student is often given an equation
and asked to solve for a particular variable.
*Advanced Placement® and AP® are registered trademarks of the College Entrance Examination Board. The
College Board was not involved in the production of this product.
Copyright © 2012 Laying the Foundation®, Inc., Dallas, Texas. All rights reserved. Visit us online at www.ltftraining.org.
i
Teacher Overview – Literal Equations
Materials and Resources
Paper and pencil
Assessments
The following types of formative assessments are embedded in this lesson:
• Assessment of prior knowledge
• Visual assessment as students’ progress through the activity
Teaching Suggestions
Throughout Chemistry and Physics courses and at times in IPC, students must be able to solve
an equation for a particular variable to see how that variable depends on other variables and
constants. Manipulation of variables without the substitution of numbers is an important skill in
helping students understand that variables depend on each other in a certain way, regardless of
any particular numbers that may be substituted into the equation.
For example, in the equation Fnet = ma (Newton’s second law), the acceleration a is always
proportional to the net force Fnet regardless of the value of the mass m. In the equation P1V1 = P2V2
(Boyle’s law), pressure P and volume V are always inversely proportional to each other.
The symbols, which are not the particular variable we are interested in solving for, are called
literals and may represent variables or constants. Literal equations are solved by isolating the
unknown variable on one side of the equation, and all of the remaining literal variables on the
other side of the equation.
Sometimes the unknown variable is part of another term. A term is a combination of symbols
such as the products ma or πr2. In this case, the unknown (such as variable r in the term πr2) must
be factored out of the term before we can isolate it.
The following examples and exercises illustrate the manipulation of many of literal equations
that commonly appear in Physics and Chemistry courses. Although students may not understand
the meaning of many of the equations during this practice exercise, solving the equations will
sharpen their algebra skills. When they ultimately learn the meaning of the equations, they will
be more likely to feel comfortable with the conceptual understanding behind the equations rather
than losing the meaning of the relationships among the variables in the algebraic manipulation.
Suggested Teaching Procedure
1. Review the Procedure section with students. Emphasize keeping their equations neat and
orderly.
2. Choose several of the listed examples to work with students on the overhead or chalkboard.
3. Instruct students to complete the remaining exercises in the space provided on their student
answer pages.
Copyright © 2012 Laying the Foundation®, Inc., Dallas, Texas. All rights reserved. Visit us online at www.ltftraining.org.
ii
T E A C H E R
A literal equation is one that is expressed in terms of variable symbols (such as d, v, and a) and
constants (such as R, g, and π). Often in science and mathematics, students are given an equation
and asked to solve for a particular variable symbol or letter called the unknown.
Teacher Overview – Literal Equations
Extension Activity
After students have mastered rearranging equations, define the units for the variables used in the
first ten examples. Have students determine the units for the isolated variable or for any desired
quantity in the equation. For example, in Example 1 where F has units of (kg • m)/s2 and m has
units of kg, what are the units for a?
This extension will give students insight into dimensional analysis, which is a useful tool to help
in problem solving.
T E A C H E R
v. 2.0, 2.0
Copyright © 2012 Laying the Foundation®, Inc., Dallas, Texas. All rights reserved. Visit us online at www.ltftraining.org.
iii
Teacher Overview – Literal Equations
Answer Key
1. v  at
7. v 2  v0 2  2ax
v
t
v 2  v0 2  2ax
a
a
2. P 
A
F
P
3.  
h
p
3
k BT
2
mv
F
Gm1m2
5. U  
r
Ur  Gm1m2
Ur
Gm2
5
 F  32 
9
T
2 K avg
9. K 
3k B
1 2
mv
2
T E A C H E R
4. F t  mv
m1  
8. K avg 
2
K =k T
3 avg B
hp
6. C 
2x
F
A
PA  F
t 
v 2  v0 2
2K=mv 2
2K
 v2
m
2K
m
v
10. vrms 
vrms 2 
3RT
M
3RT
M
9
C  F  32
5
Mvrms 2  3RT
9
F  C  32
5
M
3RT
vrms 2
Copyright © 2012 Laying the Foundation®, Inc., Dallas, Texas. All rights reserved. Visit us online at www.ltftraining.org.
iv
Teacher Overview – Literal Equations
Answer Key (continued)
3k BT
11. vrms 
14.

vrms 2 
1
1
1
 
CEQ C1 C2
1
1
1


CEQ C2 C1
3k BT
ì
C2
ì vrms 2  3k BT
CEQ C2
ì vrms 2
kB 
12. F 
C1 
q1q2
40 F
si so
so  si
si so
f 
si so

CEQ C2
C2  CEQ
3V
 r3
4
40 F
si
1
C1
3
V  r 3
4
q1q2

1
C1
4
15. V   r 3
3
1 1 1
 
si so f
so


T E A C H E R
4 0 r 2 F  q1q2
13.
CEQ C2
CEQ C2
1 q1q2
4 0 r 2
r
CEQ
C2  CEQ
3T
r2 


r
3
3V
4
1
f
1
f
si so
si  so
Copyright © 2012 Laying the Foundation®, Inc., Dallas, Texas. All rights reserved. Visit us online at www.ltftraining.org.
v
Teacher Overview – Literal Equations
Answer Key (continued)
1
16. P  Dgy  Dv 2  C
2
1 

P  D  gy  v 2   C
2 

1 

C  P  D  gy  v 2 
2 

1
17. P  Dgy  Dv 2  C
2
Dv 2  2(C  P  Dgy )
v
n2
 sin  2
 n sin  
1
1

 n

2


 2  sin 1 
2  C  P  Dgy 
D
mg sin 
 M+m 
 

mg cos 
 m 
sin 
 M+m 
 

cos 
 m 
T E A C H E R
1 2
Dv  C  P  Dgy
2
v 
n1 sin 1
M m
20. mg sin    mg (cos  ) 

 m 
CP
D
1 2

 gy  v 
2 

2
19. n1 sin 1  n2 sin  2
 M+m 
tan    

 m 
  M+m  

  m 
  tan 1   
2  C  P  Dgy 
D
1
18. x  x0  v0t  at 2
2
x  x0  v0t 
1 2
at
2
2  x  x0  v0t   at 2
a

2 x  x0  v0t
t

2
Copyright © 2012 Laying the Foundation®, Inc., Dallas, Texas. All rights reserved. Visit us online at www.ltftraining.org.
vi
Foundation Lessons
Literal Equations
Manipulating Variables and Constants
A literal equation is one that is expressed in terms of variable symbols (such as d, v, and a) and
constants (such as R, g, and π). Often in science and mathematics, students are given an equation
and asked to solve it for a particular variable symbol or letter called the unknown.
The symbols that are not the particular variable we are interested in solving for are called
literals, and may represent variables or constants. Literal equations are solved by isolating the
unknown variable on one side of the equation and all of the remaining literals on the other side of
the equation.
Sometimes the unknown variable is part of another term. A term is a combination of symbols
such as the products ma or πr2. In this case, the unknown (such as r in πr2) must be factored out
of the term before we can isolate it.
The following rules, examples, and exercises will help students review and practice solving literal
equations from physics and chemistry.
Copyright © 2012 Laying the Foundation®, Inc., Dallas, Texas. All rights reserved. Visit us online at www.ltftraining.org.
1
Student Activity – Literal Equations
Procedure
In general, a literal equation is solved for a particular variable by following the basic steps
outlined here.
1. Recall the conventional order of operations, that is, the order in which the operations of
multiplication, division, addition, subtraction, and so on are performed:
a. Parentheses
b. Exponents
c. Multiplication and division
d. Addition and subtraction
This order of operations means that you should do what is possible within parentheses first,
then exponents, then multiplication and division from left to right, and then addition and
subtraction from left to right. If some parentheses are enclosed within other parentheses,
work from the inside out.
2. If the unknown is a part of a grouped expression (such as a sum inside parentheses), use the
distributive property to expand the expression.
3. By adding, subtracting, multiplying, or dividing appropriately,
a. Move all terms containing the unknown variable to one side of the equation, and
b. Move all other variables and constants to the other side of the equation. Combine like
terms when possible.
4. Factor the unknown variable out of its term by appropriately multiplying or dividing both
sides of the equation by the other literals in the term.
5. If the unknown variable is raised to an exponent (such as 2, 3, or ½), perform the appropriate
operation to leave the unknown variable raised to the first power, that is, so that it has an
exponent of 1.
Copyright © 2012 Laying the Foundation®, Inc., Dallas, Texas. All rights reserved. Visit us online at www.ltftraining.org.
2
Student Activity – Literal Equations
Examples
1. F = ma. Solve for a.
Divide both sides by m:
F
a
m
Because the unknown variable (in this case, a) is usually placed on the left side of the
equation, we can switch the two sides:
a
F
m
2. P1V1 = P2V2. Solve for V2.
Divide both sides by P2:
PV
1 1
 V2
P2
V2 
3. v 
PV
1 1
P2
d
. Solve for t.
t
Multiply each side by t:
tv = d
Divide both sides by v:
t
d
v
Copyright © 2012 Laying the Foundation®, Inc., Dallas, Texas. All rights reserved. Visit us online at www.ltftraining.org.
3
Student Activity – Literal Equations
Examples (continued)
4. PV = nRT. Solve for R.
Divide both sides by n:
PV
 RT
n
Divide both sides by T:
PV
R
nT
R
5. R 
PV
nT
L
. Solve for L.
A
Multiply both sides by A:
RA = ρL
Divide both sides by ρ:
RA

L
L
RA

Copyright © 2012 Laying the Foundation®, Inc., Dallas, Texas. All rights reserved. Visit us online at www.ltftraining.org.
4
Student Activity – Literal Equations
Examples (continued)
6. A = h(a + b). Solve for b.
7. P = P0 + rgh. Solve for g.
1
8. U  QV . Solve for Q.
2
9. U 
1 2
kx . Solve for x.
2
10. T  2
L
. Solve for L.
g
Copyright © 2012 Laying the Foundation®, Inc., Dallas, Texas. All rights reserved. Visit us online at www.ltftraining.org.
5
Student Activity – Literal Equations
Examples (continued)
11. F 
12.
Gm1m2
. Solve for r.
r2
hi
s
  i . Solve for s .
o
ho
so
1
1
1
1
13. R  R  R  R . Solve for R3.
EQ
1
2
3
14. F = qvB sin θ. Solve for θ.
15. μmg cos θ = mg sin θ. Solve for μ.
Copyright © 2012 Laying the Foundation®, Inc., Dallas, Texas. All rights reserved. Visit us online at www.ltftraining.org.
6
Student Activity – Literal Equations
Exercises
Directions: For each of the following equations, solve for the variable specified. Be sure to show
each step you take to solve the equation for the specified variable.
1. v = at. Solve for a.
2. P 
F
. Solve for A.
A
3.  
h
. Solve for h.
p
4. F(t) = mv. Solve for t.
5. U  
Gm1m2
. Solve for m1.
r
Copyright © 2012 Laying the Foundation®, Inc., Dallas, Texas. All rights reserved. Visit us online at www.ltftraining.org.
7
Student Activity – Literal Equations
Exercises (continued)
6. C 
5
 F  32  . Solve for F.
9
7. v2 = v02 + 2ax. Solve for a.
8. K avg 
9. K 
3
k BT . Solve for T.
2
1 2
mv . Solve for v.
2
10. vrms 
3RT
. Solve for M.
M
Copyright © 2012 Laying the Foundation®, Inc., Dallas, Texas. All rights reserved. Visit us online at www.ltftraining.org.
8
Student Activity – Literal Equations
Exercises (continued)
3k BT
11. vrms 
12. F 
13.

. Solve for kB.
q1q2
. Solve for r.
4 0 r 2
1
1 1 1
  . Solve for f.
si so f
1
1
1
14. C  C  C . Solve for C1.
EQ
1
2
4
15. V   r 3 . Solve for r.
3
Copyright © 2012 Laying the Foundation®, Inc., Dallas, Texas. All rights reserved. Visit us online at www.ltftraining.org.
9
Student Activity – Literal Equations
Exercises (continued)
1
16. P  Dgy  Dv 2  C . Solve for D.
2
1
17. P  Dgy  Dv 2  C . Solve for v.
2
1
18. x  x0  v0t  at 2 . Solve for a.
2
19. n2 sin 1 = n2 sin 2. Solve for 2.
M m
20. mg sin    mg cos  
 . Solve for θ.
 m 
Copyright © 2012 Laying the Foundation®, Inc., Dallas, Texas. All rights reserved. Visit us online at www.ltftraining.org.
10