Download Systems of Equations

Systems of Equations
An Opening, Work Session, and
Closing Using the TI-84 and
Navigator
The Opening
LearningCheck
Warm-Up Results
• Are there any misconceptions?
Georgia Performance Standards
• M8A5. Students will understand
systems of linear equations and
inequalities and use them to solve
problems.
– a. Given a problem context, write an
appropriate system of linear equations or
inequalities.
– b. Solve systems of equations graphically
and algebraically, using technology as
appropriate.
– c. Graph the solution set of a system of
linear inequalities in two variables.
– d. Interpret solutions in problem contexts.
Essential Questions
•
How can I interpret the meaning of a “system of equations” algebraically and
geometrically?
•
What does it mean to solve a system of linear equations?
•
How can the solution to a system be interpreted geometrically?
•
Why is graphing a system of inequalities a good way to show the solution set?
•
How can I translate a problem situation into a system of equations or
inequalities?
•
What does the solution to a system tell me about the answer to a problem
situation?
The Work Session
Creating Triangles, Using TI-84
and the Navigator
Activity: “Try”-angles
• If your calculator number is odd,
move to the left side of the room.
You will be the RED group.
• If your calculator number is even,
move to the right side of the room.
You will be the BLUE group.
“Try”-angles
• RED Group
– Using the red paper,
straight edge, and
compass in front of
you, create right
triangles where one of
the angles, x is twice
as large as the other
angle, y. Each
member must have
unique angle measures
for x and y.
– Cut out your triangles
and place them on the
grid paper in the
appropriate place.
• Blue Group
– Using the blue paper,
straight edge, and
compass in front of
you, create triangles
where two of the
angles, x and y, are
complementary. Each
member must have
unique angle measures
for x and y.
– Cut out your triangles
and place them on grid
paper in the
appropriate place.
“Try”-angles
• Using your keyboards and
complete sentences, make some
observations using mathematical
lingo about what you see on the
grid paper.
Try-angles
• Using your TI-84, logon to
NavNet.
• Plot your triangle’s “point”.
• Input an equation that would
accurately represent the points
your group plotted.
Try-angles
• Going back to your keyboards, answer the
following:
– 1) What do you notice about the equation, the
points, and the initial directions given to your
group?
– 2) Are there any other triangles that could have
been “plotted” that are not already on the grid
paper? How do you know?
– 3) Create a triangle that could be placed in the
fourth quadrant. What do you notice?
– 4) Create a triangle that could be plotted in the
second quadrant. What do you notice?
– 5) What is significant about the the red and blue
triangles that overlap?
System of Equations
• Each of the initial instructions for the red and
blue group can be translated into equations:
– x = 2y (Left or Red Group)
– x + y = 90 (Right or Blue Group)
• You noticed after “plotting” the two points that
one of the red triangles and one of the blue
triangles overlapped.
• This overlapping is consequently where the
two triangles have degree measures of 30°
and 60°, called the solution.
• Because both equations are true for x = 30°
and y = 60° at the same time, they are
considered to be a system of equations.
Substitution Method
• Another method for solving a system of equations is by
substituting one variable in one of the equations for the
variable expression in the other.
– x = 2y
– x + y = 90
• Because the first equation is solved for x, you can
substitute the expression 2y in the second equation for
x.
• (2y) + y = 90
• Notice now that the expression on the left can be
simplified and y can easily be solved for.
• 3y = 90
• y = 30
• Now that y has been determined, to solve for x,
substitute the value for y into either equation.
Substitution Method
• X = 2y
• X = 2(30)
• X = 60
• X + y = 90
• X + 30 = 90
•
X = 60
Notice that the result for x is the same in both equations.
The Closing
LearningCheck
Cool Down
• Are there any misconceptions?