Getting Started with Algebra 2
Balance Me – ID: XXXX
By Pat Flynn
Time required
60 minutes
Activity Overview
Students will balance chemical equations using systems of equations, inverse matrices, and
determinants.
Concepts
Inverse matrix method for solving systems of equations
Teacher Preparation
This activity is intended for use with students who are enrolled or were enrolled in high school
chemistry. If this is not the case, then some introductory work needs to be done with the
students in terms of chemical equations and the balancing of them. Seek the help of your
school’s chemistry instructor for help in this.
In chemistry, many students encounter difficulties balancing chemical equations—even when
they are following steps outlined by their instructor. The purpose of this activity is to provide the
students a method to use when all else fails. Students should already be familiar with matrix
representation for a system of equations as well as the concept of the determinant.
Mathematical prerequisites from Algebra 2 include familiarity with different methods of solving
systems of equations, including a brief exposure to the use of matrices in doing so.
The screenshots on pages 2–5 demonstrate expected student results. Refer to the
screenshots on pages 6 and 7 for a preview of the student TI-Nspire document (.tns file).
To download the student .tns file and student worksheet, go to
education.ti.com/exchange and enter “XXXX” in the quick search box.
Classroom Management
This activity is intended to be completed individually, with the teacher
facilitating instruction. Use the following pages as needed to assist the class
in completing the activity.
If time is limited, you may choose to have students complete Problems 1 and 2 in
class, assigning the exercises from Problem 3 as homework.
There is no student worksheet for this activity. The student TI-Nspire document
provides a place for students to record their answers. Alternatively, you may wish to
have the class record their answers on separate sheets of paper, or just use the
questions posed to engage a class discussion.
The TI-Nspire solution document Alg2Act22_BalanceMe_Soln_EN.tns shows
the expected results of working through the activity.
TI-Nspire™ Applications
Calculator, Notes
©2008 Texas Instruments Incorporated
Page 1
Getting Started with Algebra 2
Problem 1 – Balancing Al O2
Al2O3
Problem 1 walks students through how to balance the chemical equation
Al O2
Al2O3 . There are no actions to be performed by students for the first
three steps; they should just be sure they read and understand each step.
Step 1: Assign a variable coefficient to each of the
c Al2O3
three terms in the equation: a Al b O2
Step 2: Translate the chemical equation into a
mathematical system of equations, as shown
in the screenshot to the right. There should
be an equation for each element in the
chemical equation. If needed, remind
students that the coefficients in these two
equations are derived from the number of
atoms of the element in each chemical
compound.
Step 3: Write the system of equations as a matrix
equation. For students who are comfortable
with using matrices to solve equations, this
matrix equation should look familiar, except
for the appearance of the scalar, c. Tell
students that a value will be chosen for c in
the next step.
Step 4: Page 1.6 explains that the absolute value of
the determinant of the coefficient matrix
should be used as the value of c.
Using the determinant ensures that the values found for a and b in
the next step are integers. Ask students why it makes sense to use the
absolute value of the determinant (c must be a positive integer; it
represents a number of molecules.)
Students may recall that the determinant of a
a b
2×2 matrix A
is given by
c d
det( A) ad bc , but students can use the
mathematical capabilities of the Notes
application to evaluate the determinant.
Instruct them to hold down the SHIFT key
(g) while they arrow over the expression to
select it, and choose Evaluate Selection from
the Actions menu. Pressing · will return the
value of c; here, 2.
©2008 Texas Instruments Incorporated
Page 2
Getting Started with Algebra 2
Step 5: To solve this equation for
a
, both sides of
b
the equation must be multiplied by the
inverse of the coefficient matrix. Remind
students that matrix multiplication is not
commutative, and must be done in the order
shown. Students should substitute 2 for c
and then they can use Evaluate Selection
on the selection shown to the right to
a
4
determine
.
b
3
Step 6: Now, students need to substitute the values they obtained for a, b, and c,
and check that the values indeed balance the chemical equation.
4 Al 3 O2
2 Al2O3
According to these values, there are 4 aluminum atoms on the left-hand
side and 2 × 2 = 4 aluminum atoms on the right-hand side, so the
aluminum is balanced. Similarly, there are 3 × 2 = 6 oxygen atoms on
the left and 2 × 3 = 6 oxygen atoms on the right. Therefore, the chemical
equation is balanced.
Problem 2 – Balancing C2H6
O2
CO2
H2O
Problem 2 asks students to balance the chemical
CO2 H2O . Note that this
equation C2H6 O2
equation has four terms and three elements, so it will
require a system of three equations, rather than two.
For this example, students will complete more of the
required steps themselves.
Step 1: Assign a variable coefficient to each of the
four terms in the equation.
Step 2: Translate the chemical equation into a
mathematical system of equations, as shown.
On page 2.3, students will find there is an extra step required here. The “c”
terms should be subtracted from both sides in order to isolate the “d” term on
the right-side of the equation. Students can copy (/ + C) and paste
(/ + V) their equations from page 2.2 to page 2.3 instead of retyping them.)
After manipulation, the equations should be as shown below.
C: 2a 0b 1c 0d
H: 6a 0b 0c 2d
O: 0a 2b 2c 1d
©2008 Texas Instruments Incorporated
Page 3
Getting Started with Algebra 2
Step 3: Write the system as a matrix equation. Here,
the coefficient matrix is a 3×3 matrix, as
shown to the right.
Step 4: Find the value of d. This coefficient matrix
has a determinant of –12, so this is an
example of why you should use the absolute
value of the determinant. Again, students
can use copy/paste and the mathematical
capabilities of the Notes application to find
this value.
Step 5: The values for a, b, and c may then be found as shown below.
a
b
2 0
6 0
1
0
c
0 2
2
1
0
12 2
4
14
1
8
Step 6: Now, the values of a, b, c, and d may be
substituted back into the original chemical
equation. However, students are not ready to
check their solutions.
Be sure students understand the information
on page 2.6 regarding reducing the
coefficients in a chemical by any common
factors.
Students should then check these reduced
values, as before, on page 2.7.
Problem 3 – More practice
Problem 3 provides students with eight exercises to practice balancing chemical
equations using inverse matrices and determinants. Students may answer
directly in the student TI-Nspire document (see TI-Nspire solution document) or
on separate sheets of paper. Solutions to each problem and given below.
1. Mg P4
Mg3P2
2. H2
aMg bP4
cMg3P2
Mg: 1a 0b
3c
1 0
a
P: 0a 4b
2c
0 4
b
c
4;
a
12
b
2
12Mg 2P4
aH2
c
©2008 Texas Instruments Incorporated
bN2
cNH3
H: 2a 0b
3c
2 0
2
N: 0a 2b
1c
0 2 b
c
2Mg3P2
NH3
3
4Mg3P2 , which reduces to:
6Mg P4
N2
6H2
4;
a
6
b
2
2N2
3H2
a
c
3
1
4NH3 , which reduces to:
N2
2NH3
Page 4
Getting Started with Algebra 2
3. aP4
bO2
cP2O5
P: 4a 0b
2c
4 0
O: 0a 2b
5c
0 2 b
c
8;
4P4
a
4
b
20
5. aKO2
a
c
Sb: 1a 0b
4c
1 0
a
5
O: 0a 2b
6c
0 2
b
c
5O2
2;
a
8
b
6
8Sb 6O2
cK 2CO3
dO2
6. aFe2O3
c
4
6
2Sb4O6 , which reduces to:
4Sb 3O2
2P2O5
bCO2
cSb4O6
2
8P2O5 , which reduces to:
20O2
P4
bCO
Sb4O6
cFe dCO2
K: 1a 0b 2c
0d
1 0
2
a
0
Fe: 2a 0b 1c
0d
2 0
O: 2a 2b 3c
2d
2 2
3
b
d 2
O: 3a 1b 0c
2d
3 1 0
b
d 2
C: 0a 1b 1c
0d
0 1
1 c
0
C: 0a 1b 0c
1d
0 1 0
c
1
d
3;
a
4
b
2
c
2
4KO2
7.
4. aSb bO2
aC3H8
d
2CO2
bO2
C: 3a 0b 1c
2K2CO3
cCO2
0d
1
b
3
c
2
Fe2O3
3O2
dH2O
3 0
3;
a
3CO
8. aC2H3Cl bO2
1 a
cCO2
0
C: 2a 0b 1c 0d
0e
1e
H: 8a 0b 0c
2d
8 0 0
b
d 2
O: 0a 2b 2c
1d
0 2
c
1
Cl: 1a 0b 0c 0d
1e
O: 0a 2b 2c 1d
0e
d
16;
4C3H8
a
4
b
20
c
12
20O2
5O2
16H2O ,
3CO2
a
b
1 0 0 0 c
0 2 2 1 d
which reduces to:
C3H8
dH2O eHCl
2 0 1 0
3 0 0
2
12CO2
4H2O
e
4;
a
b
4
10
c
d
8
4
4C2H3Cl 10O2
0
2Fe 3CO2
H: 3a 0b 0c 2d
2
1 a
8CO2
e
0
1
1
0
4H2O 4HCl ,
which reduces to:
2C2H3Cl 5O2
©2008 Texas Instruments Incorporated
4CO2
2H2O 2HCl
Page 5
Getting Started with Algebra 2
Balance Me – ID: XXXX
(Student)TI-Nspire File: Alg2Act22_BalanceMe_EN.tns
©2008 Texas Instruments Incorporated
Page 6
Getting Started with Algebra 2
©2008 Texas Instruments Incorporated
Page 7