MATHEMATICS 10C
UNDERSTANDING SLOPE
High School collaborative venture with
Jasper Place, Ross Sheppard and Victoria Schools
Jasper Place: Martin Fechner, Elisha Pinter, Nic Ryan, Suzan Saad
Ross Sheppard: Tim Gartke, Jeremy Klassen, Don Symes
Victoria: Kevin Bissoon
Facilitators: Greg McInulty (Consulting Services), Gail Drouin (Alberta Education)
Writer: Rosalie Mazurok (Ross Sheppard High School)
Spring, 2009
Mathematics 10C
Understanding Slope
Page 2 of 67
TABLE OF CONTENTS
STAGE 1 DESIRED RESULTS
PAGE
Big Idea
4
Enduring Understandings
4
Essential Questions
5
Knowledge
6
Skills
7
Stage 2 ASSESSMENT EVIDENCE
Teacher Notes For Transfer Tasks
.
8
Transfer Tasks
The Miracle At McInulty Summit
Teacher Notes for The Miracle At McInulty Summit and Rubric
Transfer Task
Rubric
Possible Solution
9
10 - 11
12 - 13
14 - 15
Logo Assignment
Teacher Notes for Logo Assignment and Rubric
Transfer Task
Rubric
Possible Solution
16
17 - 18
19 - 20
21 - 23
Stage 3 LEARNING PLANS
Lesson #1
Slope, But Not The Equation
24 - 28
Lesson #2
Slope With A Value
29 - 36
Lesson #3
The Slope Formula
37 - 41
Lesson #4
Applications Of Slope
42 - 44
Lesson #5
Linear Vs. Nonlinear
45 - 47
Lesson #6
Rate of Change
48 - 52
Lesson #7
Parallel Lines
53 - 55
Lesson #8
Perpendicular Lines
56 - 59
APPENDIX – Handouts
Grid Paper
Mathematics 10C
61
Understanding Slope
Page 3 of 67
Mathematics 10C
Understanding Slope
STAGE 1
Desired Results
Big Idea:
An understanding of slope will lead to the ability to interpret rate of change in real
world applications.
Implementation note:
Post the BIG IDEA in a prominent
place in your classroom and refer to it
often.
Enduring Understandings:
The student will understand that…

The slope represents the orientation of a line segment.

The value of a slope can be determined in a variety of ways.

The slope represents a rate of change.

All line segments on a given line will have the same slope.

Lines that do not intersect have the same slope, and lines that intersect have
different slopes.
Mathematics 10C
Understanding Slope
Page 4 of 67
Essential Questions:

What is the meaning of slope?

How do we interpret the differences between the magnitudes and signs of the
slopes, and the orientation of the lines?
o Why is the product of the slopes of oblique perpendicular lines equal to
-1?
o How can you show that horizontal and vertical lines are perpendicular?
o What relationships are there between parallel lines?
o What relationships are there between perpendicular lines?

What are the connections between trigonometry and slope?
o What is the relationship between the tangent ratio and the slope?
Implementation note:
Ask students to consider one of the
essential questions every lesson or two.
Has their thinking changed or evolved?
Mathematics 10C
Understanding Slope
Page 5 of 67
Knowledge:
Enduring Understanding
Specific
Outcomes
The student will understand that…
Knowledge that applies to
this Enduring Understanding
Students will know…
*RF3

the slope represents the orientation
of a line segment.

the definition of slope as
rise
.
run
Students will know…
The student will understand that…

*RF3

the value of a slope can be
determined in a variety of ways.
the definition of slope as
m
y2  y1
y
,
, m
x
x2  x1
rise
,
run
percentages, grade, gradient,
direct variation, etc…
The student will understand that…
Students will know…
*RF3


the slope represents a rate of
change.

The student will understand that…

*RF3
Students will know…

all line segments on a given line will
have the same slope.
the definition of linear and nonlinear relations.
To connect rate of change to
slope.
the definitions of slope, line
segment, line and collinear.
Students will know…

The student will understand that…

lines which do not intersect have the
same slope; lines which intersect
have different slopes.
*RF3
the definitions of parallel and
perpendicular lines.
 that parallel lines have equal
slopes.
 that perpendicular oblique lines
have slopes which are negative
reciprocals and the product of
their slopes is -1.
 the intersection of horizontal and
vertical lines forms a special case.
*RF = Relations and Functions
Mathematics 10C
Understanding Slope
Page 6 of 67
Skills:
Enduring
Understanding
Specific
Outcomes
Skills that apply to this
Enduring Understanding
Students will be able to…
The student will understand that…
*RF3

the slope represents the
orientation of a line segment.
The student will understand that…




create a line given a slope.
determine the slope of a given line.
compare and interpret slope values.
plot points on a coordinate plane.
Students will be able to…
*RF3


the value of a slope can be
determined in a variety of ways.
The student will understand that…
Students will be able to…
*RF3

calculate slopes using rise and run,
slope formula and by visual
inspection of a graphed line.

the slope represents a rate of
change.

The student will understand that…
compare and interpret slopes –
greater values result in a steeper
rise.
compare units when finding rate of
change.
Students will be able to…
*RF3



all line segments on a given line
will have the same slope.
The student will understand that…
determine the slope of a line.
find another point on a line given one
point on the line and the slope.
Students will be able to…
*RF3

lines which do not intersect have
the same slope; lines which
intersect have different slopes.


compare and interpret slope values.
predict whether two lines intersect
based on their slopes.
*RF = Relations and Functions
Implementation note:
Teachers need to continually ask
themselves, if their students are
acquiring the knowledge and skills
needed for the unit.
Mathematics 10C
Understanding Slope
Page 7 of 67
STAGE 2
1
2
Assessment Evidence
Desired ResultsDesiredd Results
The Miracle at McInulty Summit or Logo Assignment
Teacher Notes
There are two transfer tasks to evaluate student understanding of the concepts relating to
slope. The teacher (or the student) will select one for completion. Photocopy-ready versions
of the two transfer tasks and rubric are included in this section.
Implementation note:
Students must be given the transfer task & rubric at
the beginning of the unit. They need to know how
they will be assessed and what they are working
toward.
Each student will:



Be able to calculate slopes of line segments.
Interpret the results of calculations.
Identify parallel segments and perpendicular line segments.
Mathematics 10C
Understanding Slope
Page 8 of 67
Teacher Notes for The Miracle at McInulty Summit Transfer Task
Part A
Students need to use the slope formula to determine the slopes of each segment of the journey,
AB, BC, CD, etc. They need to compare slopes of different magnitudes and signs and identify
those that are parallel or perpendicular. Part of this analysis requires students to interpret
this information as a rate of change (where the sign is irrelevant).
Part B
Students are encouraged to go online and find their own pictures to illustrate the trip or in the absence
of internet connectivity, students can use the pictures provided below. (Links to the pictures on
page 11 are given on pages 62 – 64.)
They should be able to defend why they chose certain pictures and the order in which they
placed them. A completed project should have a slope calculation, a picture and an
explanation for each segment.
Students are encouraged to use their creativity in the plausible presentation of the material.
Teacher Notes for Rubric

No score is awarded for the Insufficient/Blank column , because there is no evidence of
student performance.

Limited is considered a pass. The only failures come from Insufficient/Blank.

When work is judged to be Limited or Insufficient/Blank, the teacher makes decisions
about appropriate intervention to help the student improve.
Implementation note:
Teachers need to consider what performances and
products will reveal evidence of understanding?
What other evidence will be collected to reflect
the desired results?
Mathematics 10C
Understanding Slope
Page 9 of 67
The Miracle at McInulty Summit - Student Assessment Task
Your friend has just returned from a hiking adventure. He has kept a record of his locations
along his path using a Chronological GPS locator. He has presented you with a list of coordinates that
indicate his elevation in terms of meters from his starting point and the time elapsed. He fell and now
has amnesia and developed amnesia and wants you to help him remember what happened on his trip.
He gives you his pictures and now you need to put them in order based on the information that he has
given you. Can you recreate his trip for him so that his pictures make sense?
Here is his list of points.
Point
A
Time (hours)
0
Elevation (m)
0
B
2
100
C
3
300
D
6
400
E
8
300
F
11
200
G
11.2
200
H
16
400
I
17
100
J
20
100
K
22
300
L
23
200
M
26
400
N
28
300
O
30
500
P
30
0
…and this is where things got foggy. (Nobody expected him to live. This was the Miracle at McInulty
Summit!)
Part A
You need to use the slope formula to determine the slopes of each segment of the journey, AB, BC,
CD, etc. Compare slopes of different magnitudes and signs and identify those that are parallel or
perpendicular. Part of your analysis requires interpretation of this information as a rate of change
(where the sign is irrelevant).
Part B
You are encouraged to go online and find pictures to illustrate the trip or in the absence of internet
connectivity, you can use the pictures provided below. You should be able to defend why certain
pictures were chosen and the order in which they are placed. A completed project should have a slope
calculation, a picture and an explanation for each segment.
Use your creativity in the plausible presentation of the material.
Optional Pictures
1.
2.
5.
9.
6.
8.
11.
14.
17.
12.
15.
18.
22.
4.
7.
10.
13.
21.
3.
16.
19.
23.
20.
24.
25.
Assessment
MATHEMATICS 10C
Understanding Slope
Rubric
Level
Criteria
Excellent
4
Proficient
3
Adequate
2
Limited*
1
Insufficient /
Blank*
No score is
awarded
because there
is no evidence
of student
performance.
No data is
presented.
Performs
Calculations
Performs
precise and
explicit
calculations.
Performs
focused and
accurate
calculations.
Performs
appropriate
and generally
accurate
calculations.
Performs
superficial
and irrelevant
calculations.
Presents Data
Presentation of
data is
insightful and
astute.
Presentation
of data is
logical and
credible.
Presentation of
data is
simplistic and
plausible.
Presentation of
data is vague
and
inaccurate.
Explains
Choice
Shows a
solution for the
problem;
provides an
insightful
explanation.
Shows a
solution for
the problem;
provides a
logical
explanation.
Communicates
findings
Develops a
compelling and
precise
presentation
that fully
considers
purpose and
audience; uses
appropriate
mathematical
vocabulary,
notation and
symbolism.
Develops a
convincing
and logical
presentation
that mostly
considers
purpose and
audience;
uses
appropriate
mathematical
vocabulary,
notation and
symbolism.
Shows a
solution for the
problem;
provides
explanations
that are
complete but
vague.
Develops a
predictable
presentation
that partially
considers
purpose and
audience; uses
some
appropriate
mathematical
vocabulary,
notation and
symbolism.
Shows a
solution for the
problem;
provides
explanations
that are
incomplete or
confusing.
Develops an
unclear
presentation
with little
consideration
of purpose and
audience; uses
inappropriate
mathematical
vocabulary,
notation and
symbolism.
No explanation
is provided.
No findings are
communicated.
Glossary
accurate – free from errors
astute – shrewd and discerning
appropriate – suitable for the circumstances
compelling – convincing and persuasive
complete – including every necessary part
convincing – impressively clear or definite
credible – believable
explicit – expressing all details in a clear and obvious way
focused – concentrated on a particular thing
incomplete – partial
inaccurate – not correct
inappropriate – not suitable
insightful – a clear perception of something
irrelevant – not relevant or important
logical - based on facts, clear rational thought, and sensible reasoning
precise - detailed and specific
plausible – believable
predictable - happening or turning out in the way that might have been expected
simplistic – lacking detail
superficial - having little significance or substance
unclear – ambiguous or imprecise
vague - not clear in meaning or intention
Partial Possible Solution to the Miracle at McInulty Summit
1. Calculation of Slopes.
Segment Substitution
AB
 y  yB   0  100  100
m A
2
 x A  xB  0  2
BC
CD
DE
EF
FG
GH
HI
IJ
JK
KL
LM
MN
NO
OP
 y A  yB   100  300  200
23
1
 x A  xB 
 y  yB   300  400  100
m A
36
3
 x A  xB 
 y  yB   400  300  100
m A
68
2
 x A  xB 
 y  yB   300  200  100
m A
8  11
3
 x A  xB 
 y  yB   200  200  0
m A
 xA  xB  11  11.2 0.2
 y  yB   200  400  200
m A
 xA  xB  11.2  16 4.8
 y  yB   400  100  300
m A
1
 xA  xB  16  17
 y  yB   100  100  0
m A
 xA  xB  17  20 3
 y  yB   100  300  200
m A
2
 xA  xB  20  22
 y  yB   300  200  100
m A
 xA  xB  22  23 1
 y  yB   200  400  200
m A
3
 xA  xB  23  26
 y  yB   400  300  100
m A
 xA  xB  26  28 2
 y  yB   300  500  200
m A
2
 xA  xB  28  30
 y  yB   500  0  500
m A
 xA  xB  30  30 0
m
Mathematics 10C
Understanding Slope
Slope
= 50
= 200

100
3
= 50

100
3
=0

125
3
= –300
=0
= 100
= –100

200
3
= –50
= 100
Undefined slope
Page 14 of 67
2. Slope Analysis
The following segments are truly parallel:
FG and IJ (both flat: slope = 0)
JK and NO (m = 100)
The following segments are truly perpendicular:
FG and OP, and IJ and OP (vertical and horizontal lines)
The following segments have the same steepness (but different direction):
AB and DE (50,-50)
AB and MN (50,-50)
CD and EF (100/3,-100/3)
JK and KL (100,-100
-100)
3. JPEG Journal of Journey (incomplete at this point)
AB
Moderate hike up
DE
Moderate hike downhill
Mathematics 10C
1. BC
2. CD
Steepest Climb: met the goats
on the way up.
4. EF
Easy walk up
5. FG
Easy walk downhill; didn’t
realize we’d gone so far!
Understanding Slope
3.
6.
Perfectly flat
Page 15 of 67
Teacher Notes for Logo Assignment Transfer Task
The student is asked to design or recreate a logo on graph paper. The completed logo would
presumably be used to cut out a screen print template from a piece of steel. It would be more
economical to use straight line segments.
Differentiated Instruction

Calculate the y-intercept of at least 4 lines.

Find the “equation of the line” ( y  mx  b ) for your lines (one that is vertical and one
that is horizontal)

Calculate the length of 4 line segments in your drawing. Express the length both as an
exact radical and as a decimal approximation.

Communicate which lines belong to which slope, y-intercept, equation, and length.
Consider creating a legend that associates a set of calculations with a given line or
segment.

Calculate manufacturing costs given a cost per mm for cutting the steel logo template.
Different costs could be used for parallel lines, perpendicular lines etc. A given budget
could be set.
Teacher Notes for Rubric

No score is awarded for the Insufficient/Blank column , because there is no evidence of
student performance.

Limited is considered a pass. The only failures come from Insufficient/Blank.

When work is judged to be Limited or Insufficient/Blank, the teacher makes decisions
about appropriate intervention to help the student improve.
Implementation note:
Teachers need to consider what performances and
products will reveal evidence of understanding?
What other evidence will be collected to reflect
the desired results?
Mathematics 10C
Understanding Slope
Page 16 of 67
Logo Assignment - Student Assessment Task
Background:
Logos are used everyday as an icon to promote and market products, companies and ideas.
Logos such as the Nike “swoosh” and the "golden arches" are known universally. Logo
design has even become an art form in its own right, with artists reproducing impressions of
landscapes, portraits and other established works of art.
Your task:
Design or recreate a logo on graph paper to create your own piece of art. Your completed
logo will be used to cut out a screen print template from a piece of steel. For this reason, it will
be more economical to use straight lines.
Note: You will need to modify your design to suit the requirement of the stencil cutter
 The stencil cutting machine requires the slope of each line to be cut.
 Parallel and Perpendicular lines are more economical to cut. You should include at
least 2 pairs of each.
Procedure:
1. Draw a rough sketch of your logo.
2. Transfer your initial design to a blueprint on the graph paper.
3. Label each point where a line segment changes direction.
4. Calculate the slope of each line segment.
5. Label each line segment with its slope.
6. Identify line segments which are parallel or perpendicular.
Alternate source for graph paper
http://www.printfreegraphpaper.com/
Assessment
MATHEMATICS 10C
Understanding Slope
Rubric
Level
Criteria
Excellent
4
Proficient
3
Adequate
2
Limited*
1
Insufficient /
Blank*
No score is
awarded
because there
is no evidence
of student
performance.
No data is
presented.
Performs
Calculations
Performs
precise and
explicit
calculations.
Performs
focused and
accurate
calculations.
Performs
appropriate
and generally
accurate
calculations.
Performs
superficial
and irrelevant
calculations.
Presents Data
Presentation of
data is
insightful and
astute.
Presentation
of data is
logical and
credible.
Presentation of
data is
simplistic and
plausible.
Presentation of
data is vague
and
inaccurate.
Explains
Choice
Shows a
solution for the
problem;
provides an
insightful
explanation.
Shows a
solution for
the problem;
provides a
logical
explanation.
Communicates
findings
Develops a
compelling and
precise
presentation
that fully
considers
purpose and
audience; uses
appropriate
mathematical
vocabulary,
notation and
symbolism.
Develops a
convincing
and logical
presentation
that mostly
considers
purpose and
audience;
uses
appropriate
mathematical
vocabulary,
notation and
symbolism.
Shows a
solution for the
problem;
provides
explanations
that are
complete but
vague.
Develops a
predictable
presentation
that partially
considers
purpose and
audience; uses
some
appropriate
mathematical
vocabulary,
notation and
symbolism.
Shows a
solution for the
problem;
provides
explanations
that are
incomplete or
confusing.
Develops an
unclear
presentation
with little
consideration
of purpose and
audience; uses
inappropriate
mathematical
vocabulary,
notation and
symbolism.
No explanation
is provided.
No findings are
communicated.
Glossary
accurate – free from errors
astute – shrewd and discerning
appropriate – suitable for the circumstances
compelling – convincing and persuasive
complete – including every necessary part
convincing – impressively clear or definite
credible – believable
explicit – expressing all details in a clear and obvious way
focused – concentrated on a particular thing
incomplete – partial
inaccurate – not correct
inappropriate – not suitable
insightful – a clear perception of something
irrelevant – not relevant or important
logical - based on facts, clear rational thought, and sensible reasoning
precise - detailed and specific
plausible – believable
predictable - happening or turning out in the way that might have been expected
simplistic – lacking detail
superficial - having little significance or substance
unclear – ambiguous or imprecise
vague - not clear in meaning or intention
Mathematics 10C
Understanding Slope
Page 20 of 67
Possible Solution to Logo Design Project
Logo Design Blueprint
D (8, 7)
d(-8.6)
e (-4, 6)
C (6, 6)
A (0, 5)
c (-10, 5)
B (3, 5)
E (8, 5)
Z (-4, 4)
F (7, 4)
b (-9 , 3)
P (2,
3)
Q (-1, 3)
a(-7,3)
G (4, 3)
H (4, 1)
O (2,
0)
R (-3, 0)
Y (-6,-2)
N (1,-3)
S (-4,-3)
I (3,-3)
X (-7,-5)
T (-4,-6)
M (0,-6)
W (-8,-7)
J (4,-7)
L (0,-8)
U(-5,-8)
V (-7,-8)
K (2,-9)
LOGO DESIGN CALCULATIONS
Horizontal Line Segments

AB 
55 0
  0 - horizontal line
30 3

PQ 
33
0
  0 - horizontal line
2   1 3

UV 
8   8  0
  0 - horizontal line
5   7  2
Mathematics 10C
Understanding Slope
Page 21 of 67

ab 
33
0
  0 - horizontal line
7   9  2

de 
66
0

 0 - horizontal line
8   4  4
Vertical Line Segments

DE 
75 2

- undefined - vertical line
88 0

GH 
3 1 2

- undefined - vertical line
44 0

LM 
8   6  2

- undefined - vertical line
00
0

OP 
0  3 3

- undefined - vertical line
22 0

ST 
3   6  3

- undefined - vertical line
4   4  0
Remaining Line Segments

BC 
65 1

63 3

RS 
0   3  3
 3
3   4  1

CD 
76 1

86 2

TU 
6    8  2
 2
4    5  1

EF 
54 1
 1
87 1

VW 
8   7  1

 1
7   8  1

43 1
FG 

74 3

WX 
7   5  2

 2
8   7  1

1   3 4
HI 
 4
43
1

XY 
5   2  3

3
7   6  1

IJ 
3   7  4
 2
42
2
Mathematics 10C
Understanding Slope
Page 22 of 67

JK 
7   9  2
 1
42
2

KL 
9   8 1

20
2

MN 
6   3 3

3
0 1
1

NO 
3  0 3

3
1 2
1

QR 
30
3

1   3 2

YZ 
2  4
6

3
6   4  2

Za 
43
1

4   7  3

be 
35
2

 2
9   10  1

cd 
56
1 1


10   8 2 2

eA 
65
1

4  0 4
Parallel Line Segments
All horizontal line segments are parallel.
All vertical line segments are parallel.
Line segments with the same slope are parallel:
BC and FG and ca
EF and JK
RS and XY and MN and YZ and NO
TU and IJ
FG and Za
CD and cd
WX and be
Perpendicular Line Segments
All vertical - horizontal pairs are perpendicular.
Line Segments with negative reciprocal slopes are perpendicular:
CD and be orWX
KL and IJ or TU
VW and JK or EF
HI and eA
Mathematics 10C
Understanding Slope
Page 23 of 67
STAGE 3
Learning Plans
Lesson 1
Slope, But Not the Equation
STAGE 1
BIG IDEA:
An understanding of slope will lead to the ability to interpret rate of change in real world applications.
ENDURING UNDERSTANDINGS:
ESSENTIAL QUESTIONS:
The student will understand that…

the slope represents the orientation of a
line segment.

the slope represents a rate of change.

all line segments on a given line will have
the same slope.

What is the meaning of slope?

How do we interpret the differences
between the magnitudes and signs of the
slopes, and the orientation of the lines?

What are the connections between
trigonometry and slope?
o
What is the relationship between the
tangent ratio and the slope?
.
KNOWLEDGE:
SKILLS:
Students will know…
Students will be able to…


rise
.
run

the definition of slope as

to connect rate of change to slope.
compare and interpret slope values.
compare and interpret slopes – greater
values result in a steeper rise.
Implementation note:
Each lesson is a conceptual unit and is not intended to
be taught on a one lesson per block basis. Each
represents a concept to be covered and can take
anywhere from part of a class to several classes to
complete.
Mathematics 10C
Understanding Slope
Page 24 of 67
Lesson Summary
Slope describes the steepness of a line. Students will understand that the larger the
slope the steeper the line.
Lesson Plan
Lesson Goal
Students will recognize and make a connection between the magnitude of the slope
and the steepness of a line. The larger the magnitude, the steeper the line is.
Activating prior knowledge
In grade 9 students plotted points and graphed linear relations from a table of values.
Have students look at different tables of values and describe the patterns they see.
Give several sets of data. Give a table of values where the x-values go from 1 to 5
and then create the y values to match the pattern you want. One set can go up by 1,
the other goes up by 4, the other goes down by 1 and another that goes down 3. You
might want to include a table which creates a slope of ½.
Lesson
Challenge:
How would you describe the difference between the following three lines?
Mathematics 10C
Understanding Slope
Page 25 of 67
Have students look at the tables and describe the pattern using words underneath the
tables. Plot the points on grids with the same scale and ask how the rule which was
described is visible in the graph. (We should be getting to the conclusion that the
larger the absolute value of the number, the steeper the line.) Then label the slope on
each.
Examples
x
1
2
3
4
5
x
1
2
3
4
5
y
2
3
4
5
6
.
The pattern: y-values go up by one.
x
1
2
3
4
5
y
2
6
10
14
18
The pattern: y-values go up by four. (Students should
notice that, when graphed, this line is steeper)
y
6
4
2
0
-2
The pattern: y-values go down by two.
(Students should notice that, when graphed, this line is
steeper that the first, not as steep as the second, but it is
falling – not rising.)
Teachers should use as many tables as required.
Assessment
Show a variety of photos with different slopes of lines like pitches of roofs, ramps,
mountain sides etc. Discuss the difference between the lines irrespective of their
directions and lengths. Do not expect exact answers, only comparisons here. Then,
the students can share several results and put them on the board. Discuss the
reasonableness of the values and the comparisons. Students should know that the
word we use to discuss steepness is slope and the greater the slope the steeper the
line. This is not referring to a specific numerical value at this point. You might choose
to include a line that has a negative slope to see how students deal with it.
Mathematics 10C
Understanding Slope
Page 26 of 67
Or, split the class into two groups and give half the students a set of 5 pictures and the
other half a different set of 5 pictures. Everyone assigns approximate values to the
slopes of the lines (include at least one with negative slope for enrichment). The
group should be given time to discuss their values and come to a consensus. Then
put the results of each group on the board and the other group gets to agree or
disagree with what decided.
2.
1.
3.
..
..
4.
5.
..
..
7.
..
..
6.
..
8.
..
Going Beyond
Have students go out with their cell phones and take their own pictures depicting a
variety of slopes. Have them share their pictures with a partner or you could project
them for the class to compare. You could project the images and superimpose a grid
over the picture. You could also use a document camera to project images with
various lines.
You could introduce the idea of horizontal and vertical lines in their pictures.
Mathematics 10C
Understanding Slope
Page 27 of 67
Resources
Foundations and Pre-calculus Mathematics 10 (Pearson: sec 6.1)
Math 10 (McGraw Hill: sec 6.5)
Slope Applet
http://www.ronblond.com/M10/slopes.APPLET/index.html
cell phones (if allowed)
cameras
digital projector or an interactive whiteboard to display photos
a set of photos that can be handed out
graph paper
Supporting
Use an applet to illustrate how the value of a slope changes when you change the
steepness of a line and direction.
You could show pictures of ski slopes and discuss the steepness and the speed that a
skier would be going down. Make a direct correlation between the two.
Glossary
slope – incline or grade or
rise
run
Glossary hyperlinks redirect you to the Learn Alberta Mathematics Glossary
(http://www.learnalberta.ca/content/memg/index.html). Some terms can be found in
more than one division. Some terms have animations to illustrate meanings.
Mathematics 10C
Understanding Slope
Page 28 of 67
Lesson 2
Slope With a Value
STAGE 1
BIG IDEA:
An understanding of slope will lead to the ability to interpret rate of change in real world applications.
ENDURING UNDERSTANDINGS:
ESSENTIAL QUESTIONS:
The student will understand that…

the slope represents the orientation of a
line segment.

the value of a slope can be determined in
a variety of ways.

the slope represents a rate of change.

What is the meaning of slope?

How do we interpret the differences
between the magnitudes and signs of the
slopes, and the orientation of the lines?
KNOWLEDGE:
SKILLS:
Students will know…
Students will be able to…

rise
the definition of slope as
.
run

compare and interpret slope values.

to connect rate of change to slope.

compare and interpret slopes – greater
values result in a steeper rise.
Lesson Summary
Students know that the numerical value of a slope can be determined by using the
ratio, rise over run. They should also be able to produce a line with a specific slope.
Mathematics 10C
Understanding Slope
Page 29 of 67
Lesson Plan
Lesson Goal
Get the students to make connections between slope and the numeric value of slope
rise
using the ratio
. Students should produce a line with a given slope.
run
Activating prior knowledge
Previously students associated values with the slopes of lines, but the use of
these numbers was comparative. There was no standard approach to slope. Show
another picture and ask the students to suggest a value for the slope. There should
be a bit of disagreement because at this point they have no standard method for
determining slope.
Lesson
Now we superimpose a grid on a picture.
Mathematics 10C
Understanding Slope
Page 30 of 67
Ask the students how they might assign a numeric value for the slope of this line.
rise
Students should be guided towards slope as the ratio
.
run
(Notice the slope of the above is less than 1.)
Now determine the slope of the frame from the pedal to the handle bar.
Here the slope of the bar is exactly one.
Mathematics 10C
Understanding Slope
Page 31 of 67
Find the slope of the steeple.
Notice that the slope here is greater than one.
Students should now be able to find the slope of line segments on the grid without
reference to a specific object. Examples of horizontal and vertical lines should be
included. (The slope of a horizontal line is zero, and the slope of a vertical line is
undefined).
Students should draw a line with a given slope. This can be used to assess a
student’s understanding of the slope. (Have students keep the sheet for future
reference – we will be making a point about the location of line segments).
Mathematics 10C
Understanding Slope
Page 32 of 67
To see if students can go beyond what has been discussed, some students may be
given the challenge of drawing a line with a negative slope. Students are not
expected to know what is meant by this yet.
Applets that may help:
http://www.members.shaw.ca/ron.blond/slopes.APPLET/index.html
http://www.analyzemath.com/Slope/Slope.html
Challenge:
Given pictures, a student should be able to reasonably estimate a slope value,
regardless of direction, including vertical and horizontal lines.
Look at two lines on a grid and give students the opportunity to discuss how to
calculate the slope. Identify that we have established that one line has a greater slope
than the other, and ask students to derive a method for illustrating this property
numerically (segments should be the same length so as not to introduce another
rise
variable). Students should be guided toward the ratio
.
run
Have students generalize that a slope of one generates a line with an angle of
elevation of 45°.
Use this generalization to estimate, and then use the previous relationship to
determine the slopes of various lines, including vertical and horizontal lines, and those
of varying lengths.
Going Beyond
Students can relate the slope of line to the tangent of the angle of elevation.
Mathematics 10C
Understanding Slope
Page 33 of 67
Resources
Math 10 (McGraw Hill: sec 6.5)
Foundations and Pre-calculus Mathematics 10 (Pearson: sec 6.1)
slope comparison applet
grid paper
digital projector or an interactive whiteboard
graph paper with photos provided
grids on transparencies to be placed over photos
Supporting
Math 10 (McGraw Hill: “Key Ideas” pg. 324)
slope comparison applet
Assessment
This is a formative assessment. Give students a photo and ask them to discuss the
characteristics of the slope of the line and approximate numerical value for the slope.
Mathematics 10C
Understanding Slope
Page 34 of 67
For example,
Give another photo superimposed on a grid and have students provide an exact
numerical value of the slope of a line segment (i.e. the banister).
Mathematics 10C
Understanding Slope
Page 35 of 67
Have the students compare and contrast the slopes (students should be using words
like “steepness” and “greater than” or “less than”, they should also be referring to the
slope value).
Glossary
horizontal – a line or segment parallel to the horizon
rise – change in the vertical
run – change in the horizontal
vertical – a line or segment perpendicular to a horizontal line or segment
Mathematics 10C
Understanding Slope
Page 36 of 67
Lesson 3
The Slope Formula
STAGE 1
BIG IDEA:
An understanding of slope will lead to the ability to interpret rate of change in real world applications.
ENDURING UNDERSTANDINGS:
ESSENTIAL QUESTIONS:
The student will understand that…

the slope represents the orientation of a
line segment.

the value of a slope can be determined in
a variety of ways.

the slope represents a rate of change.

all line segments on a given line will have
the same slope.

What is the meaning of slope?

How do we interpret the differences
between the magnitudes and signs of the
slopes, and the orientation of the lines?

What are the connections between
trigonometry and slope?
o
What is the relationship between the
tangent ratio and the slope?
.
KNOWLEDGE:
SKILLS:
Students will know…
Students will be able to…

the definitions of slope, line segment, line
and collinear.

determine the slope of a given line.

find another point on a line given one
point on the line and the slope.
Lesson Summary
The formula, m 
y y2  y1

can be used to determine slope. The letter m denotes
x x2  x
the slope.
Lines that rise from left to right have a positive slope, and lines that fall from left to
right have a negative slope.
Horizontal lines have a slope of zero, and vertical lines have an undefined slope.
Mathematics 10C
Understanding Slope
Page 37 of 67
Lesson Plan
Learning Goal
Students will develop the slope formula and use it to develop slope. They will also
understand the difference between positive and negative slopes.
Activating Previous Knowledge
rise
. Give
run
students a grid and have them determine the slope of a given line segment.
Earlier, students determined the slope of a line segment in terms of
y
x
O
Mathematics 10C
Understanding Slope
Page 38 of 67
Lesson
Determine the coordinates of the endpoints of the given line segment. Identify the
coordinates of the right angle in the corresponding triangle.
Use an example with two points A & B in quadrant I.

Have student extend a vertical line segment from A until it intersects a
horizontal line segment through B at point C (the right angle).

Have students determine the coordinates for C.

Have students determine the slope of AB, and then connect to the formula for
slope. Next, the students should be exposed to an example using all
quadrants, as well as examples involving various slopes.
y
B (10, 8)
4 1
slope  
8 2
C(10, 4)
C (10,
4)
A (2, 4)
x
O
Lead the class in a discussion about how to use the coordinates to determine the rise
and the run of the line segment. Develop a formula to represent the slope.
Note that the rise is the difference in the y-coordinates of the vertical line segment and
that the run is the difference in the x-coordinates of the horizontal line segment.
Mathematics 10C
Understanding Slope
Page 39 of 67
Consider the line segment on the grid below.
y
(2, 9)
(10, 5)
x
O
Apply the formula you developed previously to the above graph. Students should
notice that the value of the slope is the same, but the sign is different. Students
should discuss what the change in signs means and the significance of the same
1
value applied to the slopes. The first should have a slope of
and the second has a
2
1
slope of . They have the same steepness, but the first rises and the second falls as
2
we move to the right.
Connect the idea of positive and negative slope to the graph and then direct students
to the importance of order when using the slope equation.
Slope  m =
y y2  y1

x x2  x1
At this point students should be able to extend their application of the slope formula to
line segments in other quadrants, and examples with horizontal and vertical lines.
Present students with four line segments, including lines with positive, negative, zero
and undefined slope.
Have students determine the slope of each line segment.
Mathematics 10C
Understanding Slope
Page 40 of 67
Resources
Foundations and Pre-calculus Mathematics 10 (Pearson: sec 6.1)
Math 10 (McGraw Hill: sec 6.5)
calculating slopes applet
http://www.ronblond.com/M10/sl.APPLET/index.html
grid paper
Mathematics 10C
Understanding Slope
Page 41 of 67
Lesson 4
Applications of Slope
STAGE 1
BIG IDEA:
An understanding of slope will lead to the ability to interpret rate of change in real world applications.
ENDURING UNDERSTANDINGS:
ESSENTIAL QUESTIONS:
The student will understand that…

the slope represents the orientation of a
line segment.

the value of a slope can be determined in
a variety of ways.


What is the meaning of slope?

How do we interpret the differences
between the magnitudes and signs of the
slopes, and the orientation of the lines?

What are the connections between
trigonometry and slope?
the slope represents a rate of change.
o

all line segments on a given line will have
the same slope.
What is the relationship between the
tangent ratio and the slope?
.
KNOWLEDGE:
SKILLS:
Students will know…
Students will be able to…

the definitions of slope, line segment, line
and collinear.

determine the slope of a line.

find another point on a line given one
point on the line and the slope.
Lesson Summary
A line can be produced given any point on the line and the slope of the line.
Mathematics 10C
Understanding Slope
Page 42 of 67
Lesson Plan
Lesson Goal
A line can be produced given any point on a line and the slope of the line.
Activating prior knowledge
Students should have the following knowledge: the slope formula, what positive and
negative slopes imply, and what zero and undefined slopes imply. Students should be
3
able to draw a line given a slope. Have students draw a line with a slope of .
4
Students should identify two points on the line, then demonstrate that the slope
formula produces the proper slope.
Lesson
Ask students to draw a line with a given slope through a particular point. This can be
modeled using the computer and an applet or on an interactive whiteboard. Students
should then be asked to identify another point on the line.
Give students the coordinates of a point and a slope value. Have students provide the
coordinates at any other point that would be on the line, and have them connect the
two points to create a line.
Next, give students a point and a slope, and just one coordinate of another point.
Students will find the missing coordinate.
Draw a line through A (-4,-1) with a slope of 2/3.
y
•
The second point has the coordinates (x, 5).
Find the value of x.
x
We can find the missing coordinate using
the formula.
2 5  (1)

3 x  (4)
2  x  4   3  5  1
2 x  8  18
2 x  10
x5
Mathematics 10C
Understanding Slope
Page 43 of 67
Resources
Math 10 (McGraw Hill: sec 6.5)
Foundations and Pre-calculus Mathematics 10 (Pearson: sec 6.1)
graph paper
a digital projector or an interactive whiteboard to view applets
Assessment
B (4, 3.5)
A (0, 0)
C (x, 0)
A roof has the coordinates A (0, 0) and B (4, 3.5) and C (C is not known).
1. Determine the slope of AB
2. Given that BC has a slope at 1.75 , find the coordinates of point C.
Mathematics 10C
Understanding Slope
Page 44 of 67
Lesson 5
Linear vs. Nonlinear
STAGE 1
BIG IDEA:
An understanding of slope will lead to the ability to interpret rate of change in real world applications.
ENDURING UNDERSTANDINGS:
ESSENTIAL QUESTIONS:
The student will understand that…

the slope represents the orientation of a
line segment.

the value of a slope can be determined in
a variety of ways.

the slope represents a rate of change.

all line segments on a given line will have
the same slope.

What is the meaning of slope?

What is the impact of constant vs. variable
rates of change concerning different
graphical representations?
.
KNOWLEDGE:
SKILLS:
Students will know…
Students will be able to…

the definition of linear and non-linear
relations.

to connect rate of change to slope.

determine the slope of a given line.

compare and interpret slope values.
Lesson Summary
If a series of line segments have a constant slope value, then they form a line (linear).
If a series of line segments does not have a constant slope value, then they do not
form a line (nonlinear).
Any two points on a line can be used to determine slope.
Mathematics 10C
Understanding Slope
Page 45 of 67
Lesson Plan
Activate prior knowledge

Using diagram #1, determine the slope of the line segments AB, BC, CD and
DE. Determine the slope of the line segments AF, FG, GH and HI.
Lesson

What do you notice about the first set of slopes? The 2nd set?

What is the difference between the two sets of line segments?
y
I
H
G
E
D
C
F
A
B
x
Discuss with students what they expect the slope from A to D to be, then calculate it.
The slopes of every line segment along a line will have the same slope.
Mathematics 10C
Understanding Slope
Page 46 of 67
Resources
Foundations and Pre-calculus Mathematics 10 (Pearson: sec 6.1)
Math 10 (McGraw Hill: sec 6.5)
slope applet
Supporting
Use an applet to explore the properties of linear and nonlinear relations. Connect this
to the activity from the Explore section.
OR
Recreate the Explore activity on a giant Cartesian coordinate system on the floor of
the classroom or gym. Have students play the role of the points A through I. Have
students move into the coordinate system in pairs while other students determine the
slope between those two points. Ensure students make the connection that constant
slope  linear, non-constant slope  nonlinear, and that for lines, any two points
can be used to determine the slope.
Glossary
constant – a term that does not change
line segment – part of a line between two endpoints on the line
linear – capable of being represented by a straight line
non-linear – not capable of being represented by a straight line
Mathematics 10C
Understanding Slope
Page 47 of 67
Lesson 6
Rate of Change
STAGE 1
BIG IDEA:
An understanding of slope will lead to the ability to interpret rate of change in real world applications.
ENDURING UNDERSTANDINGS:
ESSENTIAL QUESTIONS:
The student will understand that…

the slope represents the orientation of a
line segment.

the value of a slope can be determined in
a variety of ways.

What is the meaning of slope?

How do we interpret the differences
between the magnitudes and signs of the
slopes, and the orientation of the lines?
o
Why is the product of the slopes of
oblique perpendicular lines equal to
-1?

the slope represents a rate of change.

all line segments on a given line will have
the same slope.
o
How can you show that horizontal and
vertical lines are perpendicular?

lines which do not intersect have the
same slope, and lines which intersect
have different slopes.
o
What relationships are there between
parallel lines?
o
What relationships are there between
perpendicular lines?

What are the connections between
trigonometry and slope?
o

What is the relationship between
the tangent ratio and the slope?
What is the impact of constant vs. variable
rates of change concerning different
graphical representations?
.
KNOWLEDGE:
SKILLS:
Students will know…
Students will be able to…

to connect rate of change to slope.

the definitions of parallel and
perpendicular lines.

that parallel lines have equal slopes.

that perpendicular oblique lines have
slopes which are negative reciprocals and
the product of their slopes is -1.

that the intersection of horizontal and
vertical lines forms a special case.
Mathematics 10C

compare and interpret slope values.

predict whether two lines intersect based
on slopes.
Understanding Slope
Page 48 of 67
Lesson Summary
Students are interpreting slopes in the context of the given units to describe slopes as
rates of change.
Lesson Plan
Activating Prior Knowledge
Look at the challenge question below. We activate prior knowledge using the first
question given. Students should be able to calculate the slopes of the line segments.
Challenge
Students should be able to work through a problem similar to the following:
This graph illustrates the distance that someone is away from their home at a given
time.
t
(0, 10)
(1, 10)
(8, 8)
km
(4, 8)
(9, 0)
O
hours
d
a) Calculate the slope of each segment.
b) Interpret the meaning of the slope of each segment, providing a plausible
explanation or description of the motion.
Mathematics 10C
Understanding Slope
Page 49 of 67
This is a good opportunity to do an experiment with rates of change.
One example would be a student’s pay cheque at their part-time job. Discuss how
much the student gets paid over a work day. Build a chart and graph the line. Then
offer the student a raise and build the chart and graph the line again. Discuss the
difference between the lines and relate it to the scenario.
Number of
Hours
Worked
0
12
20
28
36
40
Pay Earned
($)
0
105
175
245
315
350
Plot this data on the grid below.
360
Pay Earned ($)
300
240
180
120
60
20
40
Number of Hours Worked (h)
Determine the slope of the line, and using the units provided; determine the
significance of the slope. Students should realize that the slope ($/h) represents the
worker’s hourly wage.
Mathematics 10C
Understanding Slope
Page 50 of 67
Provide students with various straight line graphs and include the units. Have them
interpret the graphs keeping the units in mind (the inclusion of the units in their
discussion should be emphasized). Their explanations do not necessarily need to
include specific values at this point. They should understand that the slope of the line
is a relationship between two values, such as speed and time.
Another example would be a graph of earnings vs. sales. The units represent a
percent commission for the slope.
Earnings ($)
For Example
Sales ($)
The slope is a rate of commission. This also allows for an opportunity to
discuss with the class how someone can make money without any sales.
Students could be encouraged to do research online
and look for other examples of a rate of change.
They should specifically be encouraged to find
examples that are not related to time.
Mathematics 10C
Understanding Slope
Page 51 of 67
Resources
Math 10 (McGraw Hill: sec 6.5)
Foundations and Pre-calculus Mathematics 10 (Pearson: sec 6.1)
a worksheet with several good questions near its end
http://www.teacherweb.com/NY/Arlington/AlgebraProject/2L3SlopeasRateofChange.p
df
grid paper
digital projector or an interactive whiteboard
Glossary
parallel – lines in the same plane that do not intersect
perpendicular – lines which intersect at right angles (90°) to each other
Mathematics 10C
Understanding Slope
Page 52 of 67
Lesson 7
Parallel Lines
STAGE 1
BIG IDEA:
An understanding of slope will lead to the ability to interpret rate of change in real world applications.
ENDURING UNDERSTANDINGS:
ESSENTIAL QUESTIONS:
The student will understand that…

What is the meaning of slope?

How do we interpret the differences
between the magnitudes and signs of the
slopes, and the orientation of the lines?

the slope represents the orientation of a
line segment.

the value of a slope can be determined in
a variety of ways.

the slope represents a rate of change.
o Why the product of the slopes of
oblique perpendicular is lines equal to
-1?

all line segments on a given line will have
the same slope.
o How can you show that horizontal and
vertical lines are perpendicular?

lines which do not intersect have the
same slope, and lines which intersect
have different slopes.
o What relationships are there between
parallel lines?
o What relationships are there between
perpendicular lines?

What are the connections between
trigonometry and slope?
o What is the relationship between the
tangent ratio and the slope?

What is the impact of constant vs. variable
rates of change concerning different
graphical representations?
KNOWLEDGE:
SKILLS:
Students will know…
Students will be able to…

the definition of parallel lines.

to compare and interpret slope values.

that parallel lines have equal slopes.

predict whether two lines intersect based
on slopes.
Mathematics 10C
Understanding Slope
Page 53 of 67
Lesson Summary
Students understand that the slopes of parallel lines are equal. They are able to draw
lines parallel to a given line through given points, and they are also able to indicate
coordinates of some points on the lines.
Lesson Plan
Activating prior knowledge
Have students graph two horizontal lines. They can either identify or calculate the
slopes of the lines.
Challenge
Given two slopes, determine whether the lines segments are parallel.
Given a slope and a point determine another point on the line with the given slope.
Students should also be able to draw this situation.
Give the students grid paper and have them draw any two line segments. Discuss
whether or not they are parallel and link this to their slopes. Students can then take
one endpoint of both line segments (like the right endpoint) and vertically translate
them the same number of units. The lines seem to remain parallel and the students
can then calculate the slopes. Do this for a few different vertical translations to verify
the pattern.
y
y
x
Mathematics 10C
Understanding Slope
x
Page 54 of 67
Conclusions: Parallel lines have equal slopes, and if two lines have equal slopes,
then they are parallel.
After students understand the relationship between the slopes of parallel lines give
them a line and a point not on the line. They should be able to draw a line parallel to
the given line through the given point. Then students could be challenged to
algebraically determine another point on that parallel line.
Be sure to include a discussion of vertical and horizontal lines.
Resources
Foundations and Pre-calculus Mathematics 10 (Pearson: sec 6.2)
Math 10 (McGraw Hill: sec 7.4)
http://www.ronblond.com/M10/slopes.APPLET/index.html
Grid paper
Digital projector or an interactive whiteboard
GeoGebra
Supporting
Students could access some applets online that explore the relationships between
parallel lines. They can be created in GeoGebra and their slopes can be measured.
The lines can then be adjusted and their slopes will change appropriately. This can
also be done in Geometer’s Sketchpad and on the TI-84 using Cabri Jr.
Glossary
direction – the way in which a line or segment goes or points
magnitude – numerical value that describes the size of something
Mathematics 10C
Understanding Slope
Page 55 of 67
Lesson 8
Perpendicular Lines
STAGE 1
BIG IDEA:
An understanding of slope will lead to the ability to interpret rate of change in real world applications.
ENDURING UNDERSTANDINGS:
ESSENTIAL QUESTIONS:
The student will understand that…

What is the meaning of slope?

How do we interpret the differences
between the magnitudes and signs of the
slopes, and the orientation of the lines?

the slope represents the orientation of a
line segment.

the value of a slope can be determined in
a variety of ways.

the slope represents a rate of change.
o Why is the product of the slopes of
oblique perpendicular lines equal to
-1?

all line segments on a given line will have
the same slope.
o How can you show that horizontal and
vertical lines are perpendicular?

lines which do not intersect have the
same slope, and lines which intersect
have different slopes.
o What relationships are there between
parallel lines?
o What relationships are there between
perpendicular lines?

What are the connections between
trigonometry and slope?
o What is the relationship between the
tangent ratio and the slope?

What is the impact of constant vs. variable
rates of change concerning different
graphical representations?
.
KNOWLEDGE:
SKILLS:
Students will know…
Students will be able to…

the definition of perpendicular lines.


that perpendicular oblique lines have
slopes which are negative reciprocals and
the product of their slopes is -1.
calculate slopes using rise and run, slope
equation and by visual inspection of a
graphed line.

compare and interpret slope values.

predict whether two lines intersect based
on slopes.

that the intersection of horizontal and
vertical lines forms a special case.
Mathematics 10C
Understanding Slope
Page 56 of 67
Lesson Summary
Students understand that slopes of perpendicular lines are the negative reciprocals of
each other and the product of their slopes is -1. They are able to determine points on
lines perpendicular to a given line through a given point.
Lesson Plan
Activating prior knowledge
Have students draw two lines that are perpendicular. The most straightforward
example would be to draw a vertical and a horizontal line together on the grid. They
should identify the slopes of these two lines. Then let the next part of the example
show the endpoints of the lines equidistant from the point of intersection.
Challenge
Given two slopes, determine whether the lines segments are perpendicular.
Given a slope and a point, determine algebraically another point on the line with
a slope perpendicular to the one given. Students should also be able to draw this
situation.
Students can rotate the line segments by moving the endpoints. Take the top
endpoint of the vertical line and translate it horizontally to the right a specified number
of spaces. Then take the right endpoint of the horizontal line and translate it vertically
down the same number of units. As long as they are moved in the same relative
direction this should keep them perpendicular.
y
y
x
Mathematics 10C
Understanding Slope
x
Page 57 of 67
This can also be done using a set perpendicular rays and then rotate them as shown.
y
y
x
x
Another option would be to have students create an L shape on a grid and then cut it
out. They can then place it on another grid and calculate the slopes of the line
segments. Then they can rotate the L and calculate the new slopes.
After calculating the slopes for several rotations of the shape which the students know
to be perpendicular, the students should be encouraged to multiply the slopes
together.
They should notice that the slopes are the negative reciprocals of each other and that
the product of the two slope values is -1.
This can be modeled using GeoGebra or Geometer’s Sketchpad or an applet.
Students can also be told to draw the most obvious perpendicular lines – horizontal
and vertical lines – specifically the axes. Then rotate the lines around the origin.
Choose a point on the y-axis and a point on the x-axis and move them appropriately.
Now we can explore the relationship between perpendicular lines.
After they understand the relationship between the slopes of perpendicular lines give
them a line and a point not on the line. They should be able to draw a perpendicular
line through the given point. Then determine another point on the perpendicular line.
Note: Be sure to include a discussion of vertical and horizontal lines. This is an
exception to the rule. The slopes are not the negative reciprocals of each other
and their product is not -1.
Mathematics 10C
Understanding Slope
Page 58 of 67
Resources
Math 10 (McGraw Hill: sec 7.4)
Foundations and Pre-calculus Mathematics 10 (Pearson: sec 6.2)
negative reciprocal slopes (perpendicular lines) applet
http://www.ronblond.com/M10/perp.APPLET/index.html
grid paper
digital projector or a an interactive whiteboard
GeoGebra
Supporting
Students could access some applets online that explore the relationships between
perpendicular lines. They can be created in GeoGebra and their slopes can be
measured. The lines can then be adjusted and their slopes will change appropriately.
This can also be done in Geometer’s Sketchpad and on the TI-84 using Cabri Jr.
Mathematics 10C
Understanding Slope
Page 59 of 67
Appendix
Handouts
Mathematics 10C
Understanding Slope
Page 60 of 67
Alternate source for graph paper
http://www.printfreegraphpaper.com/
ACKNOWLEDGEMENTS
Pictures or Digital Images
Page 11
1.
http://images.google.ca/imgres?imgurl=http://image62.webshots.com/62/3/35/0/413933500aUnjhO_fs.j
pg&imgrefurl=http://travel.webshots.com/photo/1413933500046516165aUnjhO&usg=___0jtb7WZ6AVkxNXwGXBZ8y3yU0=&h=1536&w=2048&sz=661&hl=en&start=10&tbnid=9aq8Oh78WqIM3M:
&tbnh=113&tbnw=150&prev=/images%3Fq%3Dhiking%2Bassinaboine%26gbv%3D2%26hl%3Den%26
safe%3Dactive%26sa%3DG
2.
http://images.google.ca/imgres?imgurl=http://www.canadianrockieshiking.com/images/mount_assiniboi
ne_hike_6.jpg&imgrefurl=http://www.canadianrockieshiking.com/backpacking-trips/mountassiniboine/&usg=__alBBxwGwY8Vz05gYQ0cWe1QRKTA=&h=290&w=480&sz=27&hl=en&start=8&tb
nid=OWD_CEbXPGi91M:&tbnh=78&tbnw=129&prev=/images%3Fq%3Dhiking%2Bassinaboine%26gbv
%3D2%26hl%3Den%26safe%3Dactive%26sa%3DG
3.
http://images.google.ca/imgres?imgurl=http://www.canadianrockieshiking.com/images/mount_assiniboi
ne_hike_6.jpg&imgrefurl=http://www.canadianrockieshiking.com/backpacking-trips/mountassiniboine/&usg=__alBBxwGwY8Vz05gYQ0cWe1QRKTA=&h=290&w=480&sz=27&hl=en&start=8&tb
nid=OWD_CEbXPGi91M:&tbnh=78&tbnw=129&prev=/images%3Fq%3Dhiking%2Bassinaboine%26gbv
%3D2%26hl%3Den%26safe%3Dactive%26sa%3DG
4.
http://images.google.ca/imgres?imgurl=http://besthike.com/blog/wp-content/uploads/2007/12/mountaingoats.jpg&imgrefurl=http://besthike.com/blog/2007/12/23/3-best-hiking-region-in-the-world-is-the%25E2%2580%25A6/&usg=__G8C5fFs34mQsSKMifh_6YbhjBU=&h=354&w=500&sz=97&hl=en&start=19&tbnid=hWKbfu53RoqKM:&tbnh=92&tbnw=130&prev=/images%3Fq%3Dhiking%2Bassinaboine%26gbv%3D2%
26ndsp%3D18%26hl%3Den%26safe%3Dactive%26sa%3DN%26start%3D18
5.
http://images.google.ca/imgres?imgurl=http://www.ineedtoknow.com/assiniboine/images/2003_mccharles/images/027nub_et.jpg&imgrefurl=http://www.ineedtoknow.com/assiniboine/details/index.html&usg=__oAxrzxOLO6DCz4NsqbxcGS3mr44=&h=300&
w=400&sz=17&hl=en&start=21&tbnid=kdmFIYV7MSqhqM:&tbnh=93&tbnw=124&prev=/images%3Fq%
3Dhiking%2Bassinaboine%26gbv%3D2%26ndsp%3D18%26hl%3Den%26safe%3Dactive%26sa%3DN
%26start%3D18
6.
http://images.google.ca/imgres?imgurl=http://www.ineedtoknow.com/assiniboine/images/2005_long/bus2.jpg&imgrefurl=http://www.ineedtoknow.com/assiniboine/details/03.html&usg=__OIWhEt8KfSNBfA38QiZlQsDPR9g=&h=300&w=4
00&sz=29&hl=en&start=23&tbnid=wD2kOK1P0W0ZJM:&tbnh=93&tbnw=124&prev=/images%3Fq%3D
hiking%2Bassinaboine%26gbv%3D2%26ndsp%3D18%26hl%3Den%26safe%3Dactive%26sa%3DN%
26start%3D18
7.
http://images.google.ca/imgres?imgurl=http://www.ineedtoknow.com/assiniboine/images/2005_long/bus2.jpg&imgrefurl=http://www.ineedtoknow.com/assiniboine/details/03.html&usg=__OIWhEt8KfSNBfA38QiZlQsDPR9g=&h=300&w=4
00&sz=29&hl=en&start=23&tbnid=wD2kOK1P0W0ZJM:&tbnh=93&tbnw=124&prev=/images%3Fq%3D
hiking%2Bassinaboine%26gbv%3D2%26ndsp%3D18%26hl%3Den%26safe%3Dactive%26sa%3DN%
26start%3D18
Mathematics 10C
Understanding Slope
Page 62 of 67
8.
http://images.google.ca/imgres?imgurl=http://www.mchalepacks.com/gallery/customer%2520gallery/Cin
dy%2520with%2520flowers.jpg&imgrefurl=http://www.mchalepacks.com/gallery/detail/Red%2520Helmu
t%2520Whitney.htm&usg=__v1P78VwQD_rhbUxj2vjYPvnRyo=&h=698&w=465&sz=132&hl=en&start=32&tbnid=TQIn4r5psARljM:&tbnh
=139&tbnw=93&prev=/images%3Fq%3Dhiking%2Bassinaboine%26gbv%3D2%26ndsp%3D18%26hl%
3Den%26safe%3Dactive%26sa%3DN%26start%3D18
9.
http://images.google.ca/imgres?imgurl=http://image44.webshots.com/45/4/25/98/256542598004916792
9KTjuPs_ph.jpg&imgrefurl=http://outdoors.webshots.com/photo/2565425980049167929KTjuPs&usg=_
_Oi374vjQEuAJPGQGpZA7_HA6U1g=&h=640&w=480&sz=110&hl=en&start=109&tbnid=Gctt4rSizOg
NcM:&tbnh=137&tbnw=103&prev=/images%3Fq%3Dhiking%2Bassinaboine%26gbv%3D2%26ndsp%3
D18%26hl%3Den%26safe%3Dactive%26sa%3DN%26start%3D108
10.
http://images.google.ca/imgres?imgurl=http://image18.webshots.com/19/4/69/64/197346964bOWnSi_fs
.jpg&imgrefurl=http://travel.webshots.com/photo/1197346964030289480bOWnSi&usg=__uxz60MycWlx
g-97C-APRd3YIIbw=&h=960&w=1280&sz=194&hl=en&start=113&tbnid=t1uQ-g48FowAM:&tbnh=113&tbnw=150&prev=/images%3Fq%3Dhiking%2Bassinaboine%26gbv%3D2%26ndsp%3D
18%26hl%3Den%26safe%3Dactive%26sa%3DN%26start%3D108
11.
http://images.google.ca/imgres?imgurl=http://image18.webshots.com/19/4/69/64/197346964bOWnSi_fs
.jpg&imgrefurl=http://travel.webshots.com/photo/1197346964030289480bOWnSi&usg=__uxz60MycWlx
g-97C-APRd3YIIbw=&h=960&w=1280&sz=194&hl=en&start=113&tbnid=t1uQ-g48FowAM:&tbnh=113&tbnw=150&prev=/images%3Fq%3Dhiking%2Bassinaboine%26gbv%3D2%26ndsp%3D
18%26hl%3Den%26safe%3Dactive%26sa%3DN%26start%3D108
12.
http://images.google.ca/imgres?imgurl=http://farm1.static.flickr.com/21/30147636_aa17d27340.jpg%3F
v%3D0&imgrefurl=http://flickr.com/photos/refrozenseabass/30147636/&usg=__zzvekwF5VR0mCrywMI
bPCqgjC1I=&h=333&w=500&sz=57&hl=en&start=122&tbnid=dpZuNvcF_KQSEM:&tbnh=87&tbnw=130
&prev=/images%3Fq%3Dhiking%2Bassinaboine%26gbv%3D2%26ndsp%3D18%26hl%3Den%26safe
%3Dactive%26sa%3DN%26start%3D108
13.
http://images.google.ca/imgres?imgurl=http://www.diadav.com/travel/canadian-rockies/images/d12marveltrail.jpg&imgrefurl=http://www.diadav.com/travel/canadianrockies/chapter05.htm&usg=__j7sYPfdVL1ev9Z1L1LJxMHWxnDU=&h=322&w=500&sz=33&hl=en&sta
rt=2&tbnid=WKHhg_fIGZ3mM:&tbnh=84&tbnw=130&prev=/images%3Fq%3Dmarvel%2Blake%2Bpass%26gbv%3D2%26h
l%3Den%26safe%3Dactive
14.
http://images.google.ca/imgres?imgurl=http://lh4.ggpht.com/_aQYEj8tugVU/SLSxlbvuaPI/AAAAAAAAC
Mw/PeNGVqfvQg/P1010806.JPG&imgrefurl=http://picasaweb.google.com/lh/photo/O2X5VKDiAzCdrkph68Bvw&usg=__tcpMFI_EBBDkPHa03DwxKF49BY=&h=1200&w=1600&sz=18&hl=en&start=29&tbnid=wwxQQdZq5ZhqM:&tbnh=113&tbnw=150&prev=/images%3Fq%3Dmarvel%2Blake%2Bpass%26gbv%3D2%26ndsp
%3D18%26hl%3Den%26safe%3Dactive%26sa%3DN%26start%3D18
15.
http://images.google.ca/imgres?imgurl=http://lh4.ggpht.com/_aQYEj8tugVU/SLSxlbvuaPI/AAAAAAAAC
Mw/PeNGVqfvQg/P1010806.JPG&imgrefurl=http://picasaweb.google.com/lh/photo/O2X5VKDiAzCdrkph68Bvw&usg=__tcpMFI_EBBDkPHa03DwxKF49BY=&h=1200&w=1600&sz=18&hl=en&start=29&tbnid=wwxQQdZq5ZhqM:&tbnh=113&tbnw=150&prev=/images%3Fq%3Dmarvel%2Blake%2Bpass%26gbv%3D2%26ndsp
%3D18%26hl%3Den%26safe%3Dactive%26sa%3DN%26start%3D18
Mathematics 10C
Understanding Slope
Page 63 of 67
16.
http://images.google.ca/imgres?imgurl=http://www.ontariowildflower.com/images/andy_fyon_photograp
her2.jpg&imgrefurl=http://www.ontariowildflower.com/andyfyon.htm&usg=__RmqD70eghPNsvfxFH_N8
s0gaXo4=&h=525&w=700&sz=242&hl=en&start=2&tbnid=8w2a8oRIcDKchM:&tbnh=105&tbnw=140&pr
ev=/images%3Fq%3Dtable%2Bmoutain%2Balberat%26gbv%3D2%26hl%3Den%26safe%3Dactive
17.
http://images.google.ca/imgres?imgurl=http://www.rpmgroundworks.co.uk/images/steps_1.jpg&imgrefur
l=http://www.rpmgroundworks.co.uk/groundworks.htm&usg=__EKv6Vef5O_UpvoMsvqmJZx0RFXU=&h
=300&w=400&sz=86&hl=en&start=26&tbnid=dTH1_kPDo4BhjM:&tbnh=93&tbnw=124&prev=/images%
3Fq%3Dsteep%2Bincline%26gbv%3D2%26ndsp%3D18%26hl%3Den%26safe%3Dactive%26sa%3DN
%26start%3D18
18.
http://images.google.ca/imgres?imgurl=http://www.rpmgroundworks.co.uk/images/steps_1.jpg&imgrefur
l=http://www.rpmgroundworks.co.uk/groundworks.htm&usg=__EKv6Vef5O_UpvoMsvqmJZx0RFXU=&h
=300&w=400&sz=86&hl=en&start=26&tbnid=dTH1_kPDo4BhjM:&tbnh=93&tbnw=124&prev=/images%
3Fq%3Dsteep%2Bincline%26gbv%3D2%26ndsp%3D18%26hl%3Den%26safe%3Dactive%26sa%3DN
%26start%3D18
19.
http://images.google.ca/imgres?imgurl=http://www.pilotguides.com/images/content/destination_guide/n
orth_america/american_rockies/americanrockies_teton_incline.jpg&imgrefurl=http://www.pilotguides.co
m/destination_guide/north-america/americanrockies/climbing_the_grand_tetons.php&usg=__3fKWzUndOlARaOOVRnGW6818pW8=&h=169&w=26
0&sz=8&hl=en&start=24&tbnid=q36y6BcImf2WYM:&tbnh=73&tbnw=112&prev=/images%3Fq%3Dstee
p%2Bincline%2Bhiking%2Bpicture%26gbv%3D2%26ndsp%3D18%26hl%3Den%26safe%3Dactive%26
sa%3DN%26start%3D18
20.
http://images.google.ca/imgres?imgurl=http://www.perryzip.com/hawk_rock_2006/here_we_go.jpg&img
refurl=http://www.perryzip.com/hawk_rock_2006/hawk_rock_2006.htm&usg=__OLaXbgrVJyvy0rgTcCja
3OzwJcc=&h=1704&w=2272&sz=477&hl=en&start=67&tbnid=px9BPHNdS73fwM:&tbnh=113&tbnw=15
0&prev=/images%3Fq%3Dsteep%2Bincline%2Bhiking%2Bpicture%26gbv%3D2%26ndsp%3D18%26hl
%3Den%26safe%3Dactive%26sa%3DN%26start%3D54
21.
http://images.google.ca/imgres?imgurl=http://wanderus3.files.wordpress.com/2008/04/halfdome.jpg&imgrefurl=http://wanderus.com/2008/04/16/3-things-to-do-to-climb-halfdome/&usg=__Gkqax0WHXTfrjoiy51DwSDaf8eE=&h=500&w=333&sz=158&hl=en&start=70&tbnid=DZ
C29Xu8HEHdmM:&tbnh=130&tbnw=87&prev=/images%3Fq%3Dsteep%2Bincline%2Bhiking%2Bpictur
e%26gbv%3D2%26ndsp%3D18%26hl%3Den%26safe%3Dactive%26sa%3DN%26start%3D54
22.
http://images.google.ca/imgres?imgurl=http://1.bp.blogspot.com/_uXDMPKN0Bjo/STrW1ZqfFI/AAAAAAAABq0/j8pBhJ_m060/s400/Scree%2Bslope.jpg&imgrefurl=http://cookjmex.blogspot.com/20
08/12/climbing-mexicos-nevado-decolima.html&usg=__xWYs1At6ix6Yj9EQdby7PAw5CaI=&h=300&w=400&sz=39&hl=en&start=71&tbnid
=HHadrqjPtxOfBM:&tbnh=93&tbnw=124&prev=/images%3Fq%3Dsteep%2Bincline%2Bhiking%2Bpictu
re%26gbv%3D2%26ndsp%3D18%26hl%3Den%26safe%3Dactive%26sa%3DN%26start%3D54
23.
http://images.google.ca/imgres?imgurl=http://www.ultrarob.com/blog/uploaded_images/ManitouInclineP
4221457764791.JPG&imgrefurl=http://www.ultrarob.com/blog/labels/Colorado%2520Springs.php&usg=__6so3s
XOV7ZUbjo0fbc6gOJ9TMfo=&h=1600&w=1200&sz=505&hl=en&start=79&tbnid=bBkxJLsZW7S8kM:&t
bnh=150&tbnw=113&prev=/images%3Fq%3Dsteep%2Bincline%2Bhiking%2Bpicture%26gbv%3D2%2
6ndsp%3D18%26hl%3Den%26safe%3Dactive%26sa%3DN%26start%3D72
Mathematics 10C
Understanding Slope
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24.
http://images.google.ca/imgres?imgurl=http://www.hallscountryhouse.com/images/ThukelaFalls.jpg&im
grefurl=http://www.hallscountryhouse.com/drakensberg.html&usg=__i1tlr1K0ohcdCGTfhDJv0BeoiQA=
&h=452&w=350&sz=61&hl=en&start=98&tbnid=7Q0Q9bKtJVJrBM:&tbnh=127&tbnw=98&prev=/images
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%3Dactive%26sa%3DN%26start%3D90
25.
http://www.destination360.com/asia/china/great-wall-of-china
Page 15
1.
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2.
http://images.google.ca/imgres?imgurl=http://besthike.com/blog/wp-content/uploads/2007/12/mountaingoats.jpg&imgrefurl=http://besthike.com/blog/2007/12/23/3-best-hiking-region-in-the-world-is-the%25E2%2580%25A6/&usg=__G8C5fFs34mQsSKMifh_6YbhjBU=&h=354&w=500&sz=97&hl=en&start=19&tbnid=hWKbfu53RoqKM:&tbnh=92&tbnw=130&prev=/images%3Fq%3Dhiking%2Bassinaboine%26gbv%3D2%
26ndsp%3D18%26hl%3Den%26safe%3Dactive%26sa%3DN%26start%3D18
3.
http://images.google.ca/imgres?imgurl=http://farm1.static.flickr.com/21/30147636_aa17d27340.jpg%3F
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4.
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5.
http://www.destination360.com/asia/china/great-wall-of-china
6.
http://www.terragalleria.com/pictures-subjects/pastures/picture.pastures.usco44179.html
Page 27
1.
http://images.google.com/imgres?imgurl=http://www.propools.com/images/40271a.jpg&imgrefurl=http://www.propools.com/family.php/40271&usg=__U0bf8zswxhxhUyUUkxjDG7C02Gg=&h=300&w=386&sz=53&hl=en&start=1&um=1&itbs=1
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microsoft:en-US&sa=N&um=1
Mathematics 10C
Understanding Slope
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2.
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m=1&itbs=1&tbnid=irKgTctHDMqIM:&tbnh=93&tbnw=124&prev=/images?q=slope&hl=en&safe=active&rls=com.microsoft:en
-US&sa=N&um=1
3.
http://images.google.com/imgres?imgurl=http://www.dudh.gov.bt/Thimphustructural/images/slope_view.
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4.
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ki/File:Toyota_Land_Cruiser_at_a_80Percent_slope_at_the_IAA_2005.jpg&usg=__5nfukZAlzHsVvFqj
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oft:en-US&sa=N&start=21&um=1
5.
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6.
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7.
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lope&ndsp=21&hl=en&safe=active&rls=com.microsoft:en-US&sa=N&start=147&um=1
8.
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n-US&sa=N&start=168&um=1
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http://marciabonta.wordpress.com/2007/12/01/golden-eagle-days-part-2/
Mathematics 10C
Understanding Slope
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Page 31
http://www.bendelectricvehicles.com/yahoo_site_admin/assets/images/Sprint_slope.327223715.gif
Page 32
http://traylork.home.comcast.net/~traylork/slopepix/pages/YellowChurch2.html
Page 35
Snow scene
http://images.google.ca/imgres?imgurl=http://www.background-wallpapers.com/d/88256/-/snowslope_1152_x_864.jpg&imgrefurl=http://www.background-wallpapers.com/naturewallpapers/snow/snowslope.html&usg=__8fmxheQO72d6lrVGDBSUBXTc8z0=&h=864&w=1152&sz=247&hl=en&start=422&u
m=1&itbs=1&tbnid=fn5LhSPzjAbBM:&tbnh=113&tbnw=150&prev=/images%3Fq%3Dslope%26start%3D420%26um%3D1%26hl%3De
n%26safe%3Dactive%26sa%3DN%26rls%3Dcom.microsoft:en-US%26ndsp%3D21%26tbs%3Disch:1
Stair
http://imagesme.net/homedosh/systema-modula-staircase.jpg
Mathematics 10C
Understanding Slope
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