|
Points
of Intersection
Designed by: Melanie B. Moorer
School: Dreher High School
Grade
Level: 9-10 Subject:
Mathematics (Geometry or Algebra 11)
Core
Curriculum Objective: Find the
intersection, if it exists, of two lines using
various methods.
Overview: Students will work
independently and in small groups to develop
strategies to fmd intersections and apply
their knowledge in everyday situations.
Students will use various instructional
strategies such as: demonstrations,
presentations, critical thinking, and
deduction. As a culminating assessment,
students will create three sets of equations
of lines that intersect in one point, many
points, or no points at all.
Purpose/Essential
Question(s): How do you find the
intersection of two lines if it exists?
Time Frame: This lesson is designed for
one ninety-minute class period.
Resources
and Instructional Materials:
Geometry:
An Integrated Approach
D. C. Heath and Company
Lexington, Massachusetts/Toronto, Ontario
Pages 111-116
Transparencies
Overhead
Projector
Graph
paper
Rulers
Bell-Work: (work to be done when the
student enters the class before the tardy
bell)
Have
students draw lines on a piece of paper. The
teacher will walk around the room to find the
three types of intersecting or nonintersecting
lines.
Motivating the lesson:
Describe to the students how to read a map.
The streets are like lines, and you want to
know if they intersect or not so you'll know
which way to go. Explain to the students this
is what you plan to find out: Do lines
intersect or not?
Work through the following teacher-led
examples using all three methods (graphing,
substitution, and linear combinations):
| 1. |
Line
1: x - 2y = -7 |
|
Line
2: 3x + 4y = 9 |
| 2. |
Line
3: y = -3x + 2 |
|
Line
4: y = -3x + 7 |
| 3. |
Line
5: 3x + 2y = -7 |
|
Line
6: -6x + -4y = 14 |
**Each one of these examples should have a
different solution:
#1
has one solution,
#2 has no solution, and
#3 has infinitely many solutions.
Activity
One:
Have students work independently on Worksheet
A for 15 minutes. Then let the students get
with their partner (already selected by the
teacher when designating collaborative pairs)
compare work and reach a consensus on what the
correct answers are. After another ten
minutes, randomly select student pairs to come
up and find distances on the overhead for the
class to see.
For
homework, give the students some problems
similar to those on Worksheet A.
Activity
Two:
Pair students with a partner. Students will
receive graph paper in order to create three
sets of equations. One set needs to have one
point of intersection. Another set needs to
have infinitely many points of intersections.
The third set needs to have no points of
intersection. The students will create the
lines, find the intersection(s) if they exist
by using all three methods (including graphing
- hence the graph paper), then label them as
having one, many, or no points of
intersection. The teacher will evaluate the
culminating assessment with the following
rubric:
| Culminating
Activity |
Excellent
25-18 |
Acceptable
17-14 |
Unacceptable
13-0 |
| Correct
Lines Selected |
All
three sets of lines are correctly
selected |
Two
sets of lines are correctly selected |
One
or no sets of lines are correctly
selected |
| Lines
are Labeled Correctly |
All
three sets of lines labeled correctly |
Two
sets of lines are labeled correctly |
One
or no sets of lines are labeled
correctly |
| Graphs |
All
three sets of lines are graphed neatly |
Two
sets of lines are graphed neatly |
One
or no sets of lines are graphed neatly |
| Calculations
Using Substitutions and Linear
Combinations |
All
calculations are correct and work is
shown |
Calculations
are correct but no work is shown |
Calculations
are incorrect |
Worksheet
A
Finding Points of Intersection
(Click
here for a printable Adobe Acrobat version of
this worksheet)
Name
________________________________________________________
Date
__________________________________________
Determine if the two lines have one, many, or
no points of intersection using the method of
your choice. State the point of intersection
if one exists.
1. Y = -2x + 1 and y = 1/2x - 2
2.
2x + y = -2 and 3x + y = -1
3. x - y = 0 and 5x - 2y = 6
4.
2x + 3y = 5 and 3x + y = 11
5.
x - y = 1 and -3x + 3y = 18
6.
y + 2x = 1 and y - 1/2x = 1
7.
x + y = -1 and x + y = 2
8.
y = 2x + 5 and y = 2x - 4
Complete
the following statements.
9.
A system of linear equations (always,
sometimes, never) has a solution.
10.
If a system of linear equations in x and y has
no solutions, then their graphs are
(coincident, intersecting, parallel) lines. |
