Unit 1: Slope and Rate of Change. - St John Brebeuf

Apprenticeship and workplace Math 11
St.John Brebeuf
Unit 1:
Slope and Rate of
Change.
Mathematics
Department
Goals:
In this chapter you will use familiar mathematical concepts such as
rate, ratio, trigonometry and graphing to:
Develop an understanding of slope
Create and Interpret graphs
Key Terms: You will be able to define and use the following terms:
Dependent Variable
Run
Grade
Slope
Independent Variable
Undefined slope
Rate of Change
Zero slope
Rise
1.1 Slope
‣ Rise: is the vertical distance between 2 points.
‣ Run: is the horizontal distance between 2 points.
Slope =
Slope =
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Apprenticeship and workplace Math 11
St.John Brebeuf
Example 1:
Charlene is working on a project in her Art Metal
class. She wants to make a brass cookbook holder
like the one in the diagram. She does not have a
protractor to measure the angle, but the diagram
is drawn on graph paper.
What is the slope of the cookbook holder
a) as a simplified fraction b) as a decimal
Solution:
Example 2:
Duncan and Casey are using hand trucks to move small boxes from a
house to a garage. They lay a loading ramp against the house steps
which are 18’’ high. The slope of the steps is 0.2.
What is the horizontal distance in feet from the base of the ramp to
the point x .
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Apprenticeship and workplace Math 11
St.John Brebeuf
base
x
Solution:
Example 3:
a) Calculate the slopes of all the lines in the diagram below.
b) Compare your answers and write down what you notice.
A
B
C
D E F G
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Apprenticeship and workplace Math 11
St.John Brebeuf
Solution:
a)
b)
Sloping up Positive
Sloping down Negative
Complete notebook assignment page 19 # 1-10
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Apprenticeship and workplace Math 11
St.John Brebeuf
1.2: Grade,Angle of Elevation and Distance:
Grade: The slope of a physical feature such as a
road or hill.
Usually measured as a %
The steeper the slope the higher its %
To convert from a slope to a % grade x 100
% grade = x 100
Tangent Ratio:
Hypotenuse
Opposite
Angle of Elevation
Adjacent
Tan = =
Angle of Elevation:The angle formed by a horizontal line
segment and an inclined line segment
% grade = tan x 100
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Apprenticeship and workplace Math 11
St.John Brebeuf
Pythagorean Theorem:
a 2 c 2
Ma
b 2
+
a 2 + b 2 = c 2
Example1:
Brad needs to unload a quad from the box of his pickup truck. He
places an aluminum ramp against the truck bed at a slope of 7 : 40.
What is the angle of elevation of the ramp
Solution:
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Apprenticeship and workplace Math 11
St.John Brebeuf
Angle of depression: The angle formed between the horizontal
and the line of sight looking downwards.
Angle of depression
Drop: The difference in height between one end of an object and
the other end; equivalent to the rise.
Example 2:
The slope of a driveway must have a minimum angle of depression of 1
to allow the surface water to drain away from the house. If the end of
a driveway is 8m from the house, how many cm does the driveway need
to drop in order to maintain proper drainage
Round your answer to 2 d.p.
Solution:
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Apprenticeship and workplace Math 11
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Example 3:
Josette wants to build a skateboard ramp with a 20% grade so that
the top of the ramp is level with a rail that is 30cm high. How long
does the ramp need to be
Round your answer to the nearest cm.
Solution:
30cm
Complete notebook assignment page 30 # 1-8
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Apprenticeship and workplace Math 11
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1.3 Rate of Change.
Rate of Change:
The rate at which one variable changes with
another variable.
As rate compares 2 variables, the change
in one depends on the amount of change in the other.
Dependent Variable:
A variable whose value relies on the value of another
Independent Variable:
A variable whose values may be freely chosen
When the relationship remains constant between variables
is a linear relationship.
The graph of a linear relationship is a straight line
y –axis Independent Variable
x –axis Dependent Variable
Zero slope:
A line with a slope of zero is horizontal.
Undefined slope:
A line with a slope that cannot be calculated is vertical.
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Apprenticeship and workplace Math 11
St.John Brebeuf
Example 1
Karli and Josef drive delivery trucks. Karli gets paid $20 per hour and
Josef gets paid $16 per hour plus a $20 gas allowance at the beginning
of each workday for using his own vehicle.
a) Write an equation to calculate each person’s earnings.
Use p for pay and h for hours. Graph 5 points of data for each
person on the same graph.
b) Who makes more money after 3 hours of work and how much more
c) When will they make the same amount of money
d) Who makes more after 9 hours of work and how much more
e) Calculate the slope for each graph. What can you conclude from
these values
Solution:
a)
Karli
Josef
Hours Pay Hours pay
1 20 1 36
2 40 2 52
4 4
6 6
8 8
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Apprenticeship and workplace Math 11
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Equation for Karli:
Equation for Josef:
p
Pay ( $)
b)
Time (h)
h
c)
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Apprenticeship and workplace Math 11
St.John Brebeuf
d)
e)
Note: The larger the rate of change the steeper the slope.
Formula for slope:
(x 2 ,y 2 )
(x 1 ,y 1 )
Slope =
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Apprenticeship and workplace Math 11
St.John Brebeuf
Example 2:
Doug wants to install shelves in a display case. He needs to cut a
groove in his wood so he can attach his shelve to the case.
His groove should start at A(4,12) and finish at the point B(4,28)
What is the slope and length of the groove
Solution:
Draw a good diagram of the shelf…………….
Calculate the slope using the formula
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Calculate the length using Pythagorean Theorem
Note:
Slope =
Length =
Complete notebook assignment page 46# 1-9
Complete Unit 1 Review page 52 # 1-8
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Apprenticeship and workplace Math 11
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Reflect on your learning
Now that you have completed this unit check the box that applies
to you
RED AMBER GREEN
I understand all the key terms.
I understand the relationship
between rise, run and slope.
I can calculate slope.
I can express slope as a ratio,
angle or %.
I can create and interpret line
graphs.
I can calculate rate of change
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Apprenticeship and workplace Math 11
St.John Brebeuf
I can create a slope that meets
safety and functionality
requirements.
I have completed all
homework assignments.
I have attended tutorials
for extra help.
I am ready to sit my
unit 1 test.
Target:
In my Unit Test I hope to achieve
%
Student’s Signature ____________________
Date__________
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