Download Booklet #4 - NelsonEssentialMath

Essentials of Math 11
Booklet #4 –
Slope & Rate of
Change
Instructor – Paula Nelson
ACC Adult Collegiate
Fall 2012
1
Table of Contents
1.
2.
3.
4.
5.
6.
7.
8.
9.
Learning Outcomes………………………………..
Terms to Know……………………………………
Review of Prior Knowledge……………………….
Lesson 1: Rise Over Run
Lesson 2:
Lesson 3:
Hand In Assignment………………………………
Test Overview & Review…………………………
Trigonometry Review…………………………….
Page 3
Pages 3
Pages 4-5
Pages 6-8
Pages 8-10
Pages 10-14
Page 15
Page 15
Page 16-17
Assignments:
Lesson
Number
1.
2.
3.
Topic
Assignment
Mark Value
Terms
Define the Terms
Listed on page 4
(use your text)
10
Review Questions
Rise over Run
Grade, Angle of Elevation and
Distance
Rate of Change
Page 19 #1-10
Page 30 #1-6
Review
Test
Formula
sheet
2
-
Page 46 #1-9
Hand In Assignment
15
Page 52 #1-8
10
50
5
Learning Outcomes:
By the end of this unit is it expected that students will be able to:
 Demonstrate an understanding of slope:
o as rise over run
o as rate of change
by solving problems.
 Solve problems by applying proportional reasoning and unit analysis.
 Solve problems that require the manipulation and application of formulas related
to slope and rate of change.
 Solve problems that involve scale.
 Demonstrate an understanding of linear relations by:
o recognizing patterns and trends
o graphing
o creating tables of values
o writing equations
o interpolating and extrapolating
o solving problems.
DEFINE THESE TERMS ON A SEPARATE SHEET OF PAPER
TERMS to KNOW: see text Chapter 7 for definitions
Rise
Run
Grade
Tangent ratio
Angle of elevation
Pythagorean Theorem
Angle of Depression
Drop
Rate of Change
Dependent Variable
Independent Variable
Zero Slope
Undefined Slope
3
Review
4
5
Lesson 1: Rise Over Run
Slope is a numerical value that tells you how steep something is or how slanted it is.
We can calculate how steep something is with a simple formula. Other names for slope
include pitch, gradient or incline.
Slope is calculated my measuring the amount of vertical change in distance (called rise)
and comparing it (or dividing it) by the change in horizontal distance (called run).
This comparison, or ratio, can be expressed as follows:
Slope = ∆ vertical distance
∆ hortizontal distance
In math, we use m to represent slope (this is our symbol for slope). So, therefore…
m = rise
run
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Example 1
Charlene is working on a major 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.
a) What is the slope of the cookbook holder?
b) What does your answer tell you about the relationship
between vertical change and horizontal change of the
cookbook holder?
c) Andy says the slope is 3/2. Amanda uses her calculator and
gets the answer 1.5. Jennifer says the slope is 6:4. Explain
who is correct.
Solution
a) Determine the rise and the run by counting the number of squares on the grid.
The rise is 6 units and the run is 4 units.
m = 6/4
m = 3/2
 substitute the rise and run values into the slope formula
 simplify the fraction
The slope of the cookbook holder is 3/2.
b) The slope is a comparison of vertical change (rise) over the horizontal change
(run). In this example, the slope of the cookbook holder changes vertically a
distance of 3 units for every 2 units it changes horizontally.
c) They are all correct – in order to compare and verify the answer, reduce all
representations to a decimal (3/2 = 1.5; 1.5 is already in decimal form; 6:4 is 6
divided by 4 which is 1.5).
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 step, which are 18 inches high. The slope of
the ramp is 0.2. What is the horizontal distance (run), in feet, from the base of the ramp
to point x?
** note – there
are 12 inches in
1 foot.
Solution
Substitute the know values into the slope formula.
0.2 = 18
0.2(run) = 18
run = 18
run
0.2
7
run = 90”
90/12 = 7.5 ft
ASSIGNMENT:
 Read pages 12-19 (leave out Activities) and follow the
examples carefully
 Do Building your Skills questions page 19 #1-10
Lesson 2: Grade, Angle of Elevation and Distance
If we are talking about the slope of a road – we call is the grade of the surface (road).
Grade is generally expressed as a percentage – the steeper the slope, the higher the
percent grade. To convert slope to percent grade, multiply the numerical value for slope
by 100. The rise and run must be the in the same units (i.e. both in metres) when
calculating percent grade.
Percent grade = rise
run
x 100
The slope of the line (rise/run) is the tangent (opposite/adjacent) of the angle of elevation.
When you know the tangent ratio (slope), you can use the inverse tangent function [tan-1
(x)] to find the angle of elevation.
The slope is equal to the tangent of the angle of elevation, so you can convert an angle of
elevation to a percent grade by using the following expression:
tan θ x 100 = percent grade
In addition to calculating the steepness of a slope, you will often need to calculate the
length of a line that represents a particular slope. Use can use the Pythagorean
Theorum to calculate the length of the hypotenuse.
The Pythagorean Theorem is a mathematical relationship between the sides of a
right triangle. A right triangle is any triangle that has one right (90◦) internal angle.
Pythagoras stated, if the length of the legs (smallest side) are squared and their sum
is found, the sum will be equal to the square of the hypotenuse (longest side).
Algebraically speaking, the relationship looks like...
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The legs are traditionally marked with an 'a' and a 'b,'
while the hypotenuse is marked with a 'c.' If that is
the case, then the formula is a2 + b2 = c2. The next
section will explain how the equation can be derived.
Rearranged:
c = √a2 + b2
a = √c2 - b2
b = √c2 + a2
Example 1
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?
Angle of elevation is the angle formed by a horizontal line segment and an inclined plane
segment – the angle between the horizontal and the object looking up.
Solution
Tan Ө = opposite side
Tan Ө = rise
Adjacent side
run
Tan Ө = 7
 sub the known values in
40
Tan Ө = 0.175
 convert the fraction to a decimal
Ө = tan-1 (0.175)  use the inverse tangent key on your calculator
Ө = 9.9◦
 the angle of elevation of the ramp is 9.9◦
Example 2
The slope of a driveway must have a minimum angle of depression of 1◦ to allow surface
water to drain away from the house. If the end of a driveway is 8m from the house, how
many centimetres does the driveway need to drop in order to maintain proper drainage?
Round your answer to two decimal places.
Solution Use the tangent formula to find the drop (rise) of the driveway.
Tan Ө = rise
Run
◦
Tan 1 = rise  substitute known values into the formula
8
◦
(8) tan 1 = rise  plug into your calculator
0.1396 ~ rise
0.14 m ~ rise
 to convert to centimetres, multiply by 100 = 14 cm
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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 30 cm high. How long does the ramp need to be? Round your
answer to the nearest centimetre.
Solution
First, calculate the slope of a 20% grade. Since grade is another word for slope substitute
20 in for m.
20 = rise x 100
 the 100 is in there to convert the 20 into a percent
Run
20/100 = rise/run
1/5 = rise/run  a 20% grade has a slope of 1:5
The rise of Josette’s ramp is 30 cm, so sub this value into the formula and solve for
(calculate) the run.
1 = 30  cross multiply and divide  5 x 30 = 1 x run  150 = run  run=150cm
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run
Now to solve for the length of the ramp, you must put it into Pythagoras and solve for “c”
c = √a2 + b2
c = √rise2 + run2
c = √302 + 1502
c = √900 + 22500
c = √23400
length or c = 152.97 cm
The ramp needs to be about 153 cm long.
ASSIGNMENT:
 Read pages 25-30 and follow the examples carefully
 Do Building your Skills questions page 30 #1-6
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Lesson 3: Rate of Change
http://www.regentsprep.org/Regents/math/ALGEBRA/AC1/Rate.h
tm
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A rate of change is the rate at which on variable changes compared to another variable.
Therefore, one is known as the dependent variable and the other is known as the
independent variable. You are required to be able to tell me the difference and some
students have a lot of trouble with this. The dependent variable is a variable whose
value relies on the values of the other variable (i.e. if you are planning a wedding the cost
of the food relies on the number of guests, so cost the cost depends on the number of
guests). The independent variable is a variable whose values is freely chosen (i.e. often
time is independent or in the example of above, the number of guests at the wedding).
When the relationship between the variables is constant, it has a linear relationship
(makes a straight line on a graph). Remember – a horizontal line has a zero slope and a
vertical line has an undefined slope. Slope can also be positive or negative depending
on which way the line is rising or falling (see pics of ski bird above).
Example 1
Kalirraq and Uyarak drive delivery trucks. Kalirraq gets paid $20.00 per hour and
Uyarak gets paid $16.00 an hour plus a $20 gas allowance at the beginning of each
workday for using his own vehicle.
(a) Write an equation you can use to calculate each person’s earnings, using 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? How much more?
(c) When will they make the same amount of money?
(d) Who makes more money after 9 hours of work? How much more?
(e) Find the slope of the line segments that represent Kalirraq’s and Uyarak’s pay for
hours worked. What can you conclude about these values?
Solution
(a) Make a data table for each person that includes five points.
Kalirraq
Uyarak
Hours (h) Pay (p)
Hours (h)
1
20
1
2
40
2
4
80
4
6
120
6
8
160
8
Pay (p)
20+16 = 36
36+16 = 52
52 + 84
116
148
Write an EQUATION FOR EACH PERSON:
Kalirraq:
pay = hours worked times $20/hour so an equation would be: p = 20(h)
Uyarak:
pay = hours worked times $16/hour plus $20 so the equation
would be p = 16(h) +20
Now draw a graph that compares the dependent variable (pay)
on the y axis to the independent variable (hours) on the x-axis
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(independent variable is always graphed on the x axis). Plot both sets of points on the
same graph.
(b) Find the point on the graph when Kalirraq’s line segment matches the 3 hour
value on the x-axis and match it to the corresponding value on the y axis.
Kalirraq earns $60. DO the same for Uyarak’s line; he earns $68.00
$68-60 = $8  after 3 hours of work, Uyarak earns $8 more.
(c) The two lines intersect (cross) at 5 hours, so they are making the same amount of
money when they have worked 5 hours.
(d) To find how much money they make after 9 hours, extrapolate by extending the
line and finding the point where the line segments cross the 9 hour value.
Kalirraq is making $180 and Uyarak is making $164 – Kalirraq makes $16 more.
(e) Kalirraq:
m = 120/6  m = 20  the slope is 20
Uyarak:
148 – 20 = 128
m = 128/8  m = 16  the slope is 16
The line representing Kalirraq’s pay is rising more steeply than Uyarak’s. The rate of
change is greater for Kalirraq.
Example 2
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ASSIGNMENT:
 Read pages 35-46 and follow the examples carefully
 Do Building your Skills questions page 46 #1-9
More Practice on Writing Equations
http://www.math123xyz.com/Nav/Algebra/Writing%20Linear%20Equations_Tutorial.ph
p
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/30
Hand In Assignment:
Do Activity 1.7 SLOPE DIRECTION on page 40 of your Text
23 marks for answering/completing questions 1-10
5 marks for graphs
2 marks for neatness (ruler, colour…) and presentation
Test Overview: tba

Mark breakdown
/50
o _______________________________________
o _______________________________________
o _______________________________________
o _______________________________________
YOU ARE ABLE TO HAVE A 8.5 x 11 CHEAT SHEET made up ahead of time
(5 marks)

You should be able to:
o _________________________________________________________
o _________________________________________________________
o _________________________________________________________
o _________________________________________________________
o _________________________________________________________
o _________________________________________________________
o _________________________________________________________
o _________________________________________________________
o _________________________________________________________
o _________________________________________________________
REVIEW – Practice your New Skills
text page 52 #1-8
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Trigonometry Review
Let's start with a quick review of the three trigonometric functions we already
understand. Remember that these functions work only in right
triangles.
where A represents the angle of reference.
Example: In right triangle ABC, hypotenuse length
AB=15 and angle A=35º. Find leg length BC to the
nearest tenth.
Set up the problem: Draw a right triangle. Position yourself at the given angle
as a point of reference. Label the triangle's sides as to their location: opposite,
hypotenuse, adjacent. Place the given information on the triangle. Pair up the
sides' labels with the given information. The piece with no pairing is ignored.
Set up the function based upon the pairings and solve.
Nothing is paired with side a in this
problem, so it is ignored. This problem
deals with o and h which means we will be
working with the sin function.
ANSWER: 8.6
Example: In right triangle ABC, leg length BC=15 and
leg length AC=20. Find angle A to the nearest degree.
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Set up the problem: The set up is the same as in the previous example. The
only difference is that in this problem you will be finding the angle.
Nothing is paired with side h in this
problem, so it is ignored. This problem
deals with o and a which means we will
be working with the tan function.
Calculator use: Remember you will need to activate the tan-1 key (it is located above the
tan key) when finding angles. To activate this tan-1 key on most scientific calculators,
enter 0.75, press 2nd (or shift) and then the tan key. On the graphing calculator,
activate the tan-1 first, and then enter 0.75.
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