YOUNGSTOWN CITY SCHOOLS
MATH: PRECALCULUS
UNIT 10: SOLVING A SYSTEM OF EQUATIONS USING MATRICES (3 WEEKS) 2013-2014
Synopsis: In this unit students will solve a system of equations by representing them as a matrix equation in a
vector variables and then solving them by using the inverse of the matrix.
STANDARDS
A.REI.8 (+) Represent a system of linear equations as a single matrix equation in a vector variable.
A.REI.9 (+) Find the inverse of a matrix if it exists and use it to solve systems of linear equations (using technology for matrices
of dimension 3 x 3 or greater).
G.MD.2 (+) Give an informal argument using Cavalieri’s principle for the formulas for volume of a sphere and other solid figures
MATH PRACTICES
1. Make sense of problems and persevere in solving them.
2.
Reason abstractly and quantitatively.
3.
Construct viable arguments and critique the reasoning of others.
4.
Model with mathematics.
5.
Use appropriate tools strategically.
6.
Attend to precision.
7.
Look for and make use of structure.
8.
Look for and express regularity in repeated reasoning
LITERACY STANDARDS
L.2 Communicate using correct mathematical terminology
L.6 Represent and interpret data with an without technology
MOTIVATION
TEACHER NOTES
1. Cryptography uses matrices to code messages. One simple way to do this is to
assign the letters of the alphabet numbers 1 through 26 with 0 representing a space.
Take your message and convert it to numbers, then form an m x 2 matrix and multiply it
by a 2 x 2 coding matrix. For example, to code the message “GO PENQUINS,” assign
the letters to numbers
G O P E N Q U I N S with coding matrix
7 15 0 16 5 14 17 21 9 14 19
7 15
0 16
5 14
17 21
9 14
19 0
=
*
67 89
64 80
61 80
101 139
65 88
19 38
So the coded message is 67| 89| 64| 80| 61| 80|101| 139| 65| 88| 19| 38. Later in the
unit, we will discuss how to decode the message.
2. Preview expectations for end of Unit.
3. Have students set both personal and academic goals for this Unit.
7/1/2013 YCS PRE-CALC UNIT 10: SOLVING A SYSTEM OF EQUATIONS USING MATRICES 2013-2014 1
TEACHING-LEARNING
Vocabulary:
Matrix
Dimensions
TEACHER NOTES
Vector Variable
Inverse Matrix
Matrix Equation
Cavalieri’s Principle
Identity Matrix
Determinant
1. Consider the problem: The sides of an angle are parts of two lines whose equations are 3x
+ 2y = -7 and 2x – 3y = -9. The angle’s vertex is the point where the two sides meet. Find
the coordinates of the vertex of the angle. To solve the system of equations, we set up a
matrix equation. A * x = B where A is an m x n (2x2) matrix, x is the column vector
variable with n (2) entries and b is the column vector with m (2) entries
A
* x = B
[
[
]
To solve this equation, multiply both sides of the equation by the inverse matrix of A. If A =
Remember the determinant of
=
To check if the inverse is correct, we know that a multiplicative inverse * the element is
equal to the identity element. That is A*A-1 = A-1 * A = I where I =
which is the
identity matrix.
Have students apply the definition of the inverse of A to solve the original system of
equations. Reinforce with additional problems in the textbook on pages 103-104. If the
matrices are larger than a 2x2, use the TI calculator to solve. At this time it would be good
to review that matrix multiplication is not commutative and the rules for multiplying matrices
involving the dimensions. (A.REI.8, A.REI.9, MP.1, MP.2, MP.4, MP.5, MP.8, L.2,L.6)
2. Going back to the code problem in the motivation section, to decode the message, multiply
the coded matrix by the inverse of the coding matrix and the original message will appear
after you replace the numbers by the letters.
(A.REI.8, A.REI.9, MP.1, MP.2, MP.4, MP.5, MP.8, L-2,L-6)
3. Bonaventura Cavalieri (1598-1647) stated that if you have two solids that are placed
between two parallel planes and you drew other planes parallel to the original two lines
through the shapes, if their cross-sections that are the same distance from the bases have
equal areas, then the two solids have equal volumes. This is also true in the two
dimensional case. That is, if there are two shapes that lie between two parallel lines and
additional lines are drawn parallel to the given lines such that they intersect the two
shapes, then if the portions of the parallel lines that intersect the shapes are equal, then
the shapes have equal areas. After reading Cavalieri’s principle, have students draw
diagrams to illustrate the principle. To reinforce, give students area and volume problems.
The following links may be of help.
http://www.tutorgigpedia.com/ed/Method_of_indivisibles
http://education.ti.com/en/timath/us/detail?id=DEA761964EB640B7A90CD198F42EF964&
sa=71A40A9FD9E84937B8C6A8A4B4195B58 TI 84 or TI NSpire are needed for this
activity
http://education.ti.com/en/timath/us/detail?id=DEA761964EB640B7A90CD198F42EF964&
sa=71A40A9FD9E84937B8C6A8A4B4195B58 Three problems involving volume.
7/1/2013 YCS PRE-CALC UNIT 10: SOLVING A SYSTEM OF EQUATIONS USING MATRICES 2013-2014 2
TRADITIONAL ASSESSMENT
TEACHER NOTES
1. Unit Test: Multiple-Choice Questions
TEACHER NOTES
TEACHER CLASSROOM ASSESSMENT
1. Teacher Classroom Assessments
2. Smaller authentic assessments as you go along
AUTHENTIC ASSESSMENT
TEACHER NOTES
1. Have students evaluate goals for the unit.
2. Students create a real-life problem involving two unknowns with two equations. They must
state the problem, state the variables and what they represent, write the equations, set up
the matrix equation, find the inverse matrix, and solve the problem. (A.REI.8, A.REI.9, MP.1,
MP.2, MP.4, MP.6, MP.7, L.2, L.6)
AUTHENTIC ASSESSMENT RUBRIC
ELEMENTS OF THE
PROJECT
State the problem
0
Did not state the
problem
Did not state the
variables
State the variables
Write the equations
Set up matrix equation
Find the inverse matrix
Found values for each variable
Did not write the
equations
Did not set up matrix
equation
Did not find the
inverse matrix
Did not solve the matrix
equation
1
2
Stated the problem, however
there is only one unknown
Stated the variables, however
did not state what they
represent
Wrote one equation
Stated the problem containing two
unknowns
Stated the variables and what they
represent
Set up matrix equation
incorrectly
Found the inverse matrix with
errors
Solved the matrix equation
with errors
Set up matrix equation correctly
Wrote both equations
Found the inverse matrix correctly
Solved the matrix equation
correctly
7/1/2013 YCS PRE-CALC UNIT 10: SOLVING A SYSTEM OF EQUATIONS USING MATRICES 2013-2014 3