Download Slopes of Perpendicular Lines

Grade: 9
Lesson Title: Relationship Between the Slopes of
Date: Nov. 19
Perpendicular Lines
Strand / Curriculum Expectations
 identify, through investigation, properties of the slopes of lines and line segments (e.g.
direction, positive or negative rate of change, steepness, parallelism, perpendicularity), using
graphing technology or facilitate investigations, where appropriate.
 determine the equation of a line from information about the line (e.g., the slope and y-intercept;
the slope and a point; two points)
What do students need to know and be able to do? (consider curriculum mapping expectations)
 how to graph linear relations using a table of values and the slope-intercept method
 calculate and recognize the slope and y-intercept of a graph or equation
 terms such as direct variation, partial variation, independent and dependent variables.
Learning Goals
Content:
The students will...
Recognize the relationship between the slopes of
perpendicular lines.
Process:
Lesson Components
Action!
- Students will be given sets of ordered pairs and
plot them to create two perpendicular lines
- They will measure the angle between the lines,
calculate the two slopes and look for the
relationship between the two slopes
- Possible pairs of lines
Lines with fraction slopes (3/2, -2/3, A(-4,
-2), B(0, 4), C(-6, 1), D(3, -5), plot A, B, C, D,
draw the line passing through AB and the line
through CD
- (-4/3, 3/4, A(0, -4), B(6, 4), C(-4, 5), D(4, -1)
Lines with whole number slopes (2, -1/2,
A(-1, -3), B(2, 3), C(4, -3), D(-6, 2)
(Vertical / Horizontal lines)
Task:
Plot the points A(-4, -2), B(0, 4), C(-6, 1), D(3, -5)
Draw the line passing through AB and the line through
CD. Measure the angle between the lines.
Calculate the slope of each line.
Graph the lines y = (4/5)x – 4 and 5x + 4y - 2 = 0.
Measure the angle between the lines.
Determine the slope of each line.
Plot the points J(-1, -3), K(2, 3), L(4, -3), M(-6, 2)
Draw the line passing through JK and the line through
LM. Measure the angle between the lines.
Calculate the slope of each line.
Anticipated Student Responses and
Teacher Prompts / Questions
Scaffolding Questions
How else can you represent this?
How are these ___the same or different?
If I do ____, what will happen?
How can you prove your answer or verify your estimate?
How do you know?
Have you found all the possibilities?
How could you arrive at the same answer in a different
way?
Some anticipated misconceptions or areas
of difficulty:
*the labels on the diagram need to be
correct
*need to connect the correct points to draw
the lines
*x and y intercept confusion (which is which
in a set of co-ordinates, axis)
*might need support in how to find slope
and that all strategies are valuable
*know to isolate y in equation
What appears to be the relationship between slopes of
perpendicular lines?
Determine the equation of the line through the point (2,
-1) and perpendicular to 3x – 2y + 3 = 0
Students worked in heterogeneous groups of three.
Minds-on
Students will be given:
 A graph with a line that has coordinates that are
easy to identify on it and will be asked to
determine the slope
 a table of values (TOV) to use to calculate the
slope of the line related to the TOV.
Lesson Component
Looking to activate prior knowledge about
how to calculate slope from a line or from a
table of values.
Anticipated Student Responses and
Teacher Prompts / Questions
After / Consolidation / Reflecting and Connecting
1. Is slope best communicated using a
fraction or a decimal? What works best?
Why?
 Look at 2 pieces student work
(Liam’s Group used decimals
number for slope in the first task.
Use this work and the work of
another group that used decimals to
compare.
2. Articulate the relationship between the
slope of perpendicular lines.
 Look at work from Erin, Brittany and
Roman’s group. What did they
discover about the slopes of
perpendicular lines?
What did Erin discover? (the intercept can
be a fraction – ½ - how did she count that
on her graph?)
What did Brittany discover? (the negative
reciprocal “the rise and the run are
opposite” and later shared that “one is
positive and one is negative”
3. Take a deeper look at 4th task. Look at
various answers and determine how it
could be solved.
Areas of Focus:
 Y intercept numbers a little “ugly”
 What was the equation? How did
you get to it?
Consolidation
Independent Practice/Exit Card/Reflection
Student Next Steps
(Large Group/Small Group/Individual)
Another task similar to # 4 from this Lesson’s Action.
Or
Given 4 co-ordinates, have students prove that the
shape is a rectangle, when you connect the 4 points.
This will be determined after the
independent practice.