Course: Grade 9 Applied Mathematics (MFM1P)
Unit 1:
Measurement (2D & 3D)
Unit 1
Measurement (2D & 3D)
Section
1.1.1
1.1.2
1.2.1
1.2.J
1.2.2
1.2.P
1.3.2
1.3.P
1.4.2
1.4.3
1.4.P
1.4.4
1.4.5
1.5.1
1.5.3
1.5.5
1.5.P
1.5.6
1.6.1
1.6.2
1.6.P
1.6.J
1.W
1.S
1.R
1.RLS
Activity
Investigation - Comparing Volumes
Pair Share – Volume
Melting Ice Cream
Journal Activity
Applications of Volume
Practice
Scale Drawing Details
Practice
Composite Figures Notes
Exploring Composite Shapes
Practice
Use What You Know
Frayer Model
The Rope Stretchers
Right or Not?
The Pythagorean Theorem Notes
Practice
Pythagorean Theorem Puzzle
Coach or Be Coached
The Container!
Practice
Journal Activity
Definition Page
Unit Summary Page
Reflecting on My Learning (3, 2, 1)
Reflecting on Learning Skills
Page
3
4
6
7
8
11
13
16
17
19
20
23
24
25
26
28
30
31
32
34
36
38
39
41
42
43
MFM 1P - Grade 9 Applied Mathematics – Unit 1: Measurement 2D & 3D (DPCDSB Dec 2008)
1-2
1.1.1: Investigation – Comparing Volumes
Purpose
Compare volumes of shapes that have the same base and height.
Hypothesis
I think that...
1.
× _________ =
2.
× _________ =
3.
× _________ =
Investigate
How many times will the volume of the shape on the left fill the shape on the right?
1. Vcone × ___ = Vcylinder
2. Vsquare pyramid × _____ = Vsquare prism
3. Vtriangular pyramid × _____ = Vtriangular prism
Conclusion
MFM 1P - Grade 9 Applied Mathematics – Unit 1: Measurement 2D & 3D (DPCDSB Dec 2008)
1-3
1.1.2: Pair Share – Volume
A – Prisms
Circle the shapes that are prisms.
Volume of a prism = area of __________ × __________.
B – Pyramids
Circle the shapes that are pyramids.
Volume of a pyramid = volume of a prism ÷__________.
MFM 1P - Grade 9 Applied Mathematics – Unit 1: Measurement 2D & 3D (DPCDSB Dec 2008)
1-4
1.1.2: Pair Share – Volume (continued)
Calculate the volume of the following figures. Show your work.
A
Calculate the volume of the following figures. Show your work.
A
B
8 mm
15 mm
12 cm
5 cm
MFM 1P - Grade 9 Applied Mathematics – Unit 1: Measurement 2D & 3D (DPCDSB Dec 2008)
1-5
1.2.1: Melting Ice Cream
•
•
•
•
Heather purchased an ice cream cone.
The scoop of ice cream was a sphere.
The height of the cone equals the diameter of both the
cone and the sphere.
If the ice cream melts, how much will overflow the cone?
In Pairs
A. What do we already know?
1.
=
Volumecone
÷ _______
=
Volume ______
÷ _______
2. Volumecylinder
=
(Area of Base) × (
)
3. Volumecone
=
(Area of Base) × (
) ÷ _____
B. Using the two 3-D relational solids of the cone and sphere, compare.
1. Heightcone and Heightsphere
Comparison:
2. Radiuscone and Radiussphere
Comparison:
3. Volumecone and Volumesphere
Comparison:
C. Conclusion
_______ ×
=
D. How much ice cream will overflow? Write a relationship statement.
MFM 1P - Grade 9 Applied Mathematics – Unit 1: Measurement 2D & 3D (DPCDSB Dec 2008)
1-6
1.2.J: Journal Activity
1.
Using words, pictures, numbers, and symbols, describe the relationships you
discovered today.
2.
Use the 3-D relational sets and record as many paired relationships as you can.
For example, the small triangular prism is half the volume of the small squarebased prism.
MFM 1P - Grade 9 Applied Mathematics – Unit 1: Measurement 2D & 3D (DPCDSB Dec 2008)
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1.2.3: Applications of Volume
1. Calculate the volume of the perfume bottle.
5 cm
5 cm
15 cm
3 cm
8 cm
2. How much soup can this container hold?
7 cm
10 cm
3. Calculate the volume of the rectangular prism.
5 cm.
4 cm.
6 cm.
MFM 1P - Grade 9 Applied Mathematics – Unit 1: Measurement 2D & 3D (DPCDSB Dec 2008)
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1.2.3: Applications of Volume (continued)
4. Which popcorn container will hold more?
16 cm
8 cm
16 cm
15 cm
15 cm
5. How many times bigger is the second sphere?
5m
MFM 1P - Grade 9 Applied Mathematics – Unit 1: Measurement 2D & 3D (DPCDSB Dec 2008)
15 m
1-9
1.2.3: Applications of Volume (continued)
6. How much more expensive should the large aquarium be than the small aquarium if the cost
is based on the volume?
12 cm.
16 cm.
8 cm.
24 cm.
32 cm.
16 cm.
7. Determine the volume of the cabin. Show your work.
2m
3m
6m
6m
MFM 1P - Grade 9 Applied Mathematics – Unit 1: Measurement 2D & 3D (DPCDSB Dec 2008)
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1.2.P: Practice
1.
Using the relationships you have discovered, calculate the volume of each of the
following 3-D figures.
(a)
4 mm
6 mm
14 mm
5mm
(b)
3m
88 m
m
(c)
4m
2m
MFM 1P - Grade 9 Applied Mathematics – Unit 1: Measurement 2D & 3D (DPCDSB Dec 2008)
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1.2.P: Practice (continued)
2.
A movie theatre wants to compare the volumes of popcorn in two containers, a
cube with edge length 8.1 cm and a cylinder with radius 4.5 cm and height 8.0
cm.
Which container holds more popcorn?
Draw diagrams to support your solution.
3. An entertainment room is 10.5 m long by 7.5 m wide by 3.5 m high. If 5 m 3 of air
is needed for each person, how many people can use the room at one time?
MFM 1P - Grade 9 Applied Mathematics – Unit 1: Measurement 2D & 3D (DPCDSB Dec 2008)
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1.3.2: Scale Drawing Details
1. Consider the two composite figures.
(a) Identify the geometric shapes in each. Write the names on the diagrams.
Figure 1
Figure 2
(b) Area
For determining the area of the shaded regions, describe the features and calculations
that are:
i)
the same
ii) different
(c) Which of the two figures is larger?
By how much?
Justify your answer using pictures, symbols, and words.
(d) Perimeter
Use a coloured pencil to outline the perimeters of the two figures. How do these two
perimeters compare?
MFM 1P - Grade 9 Applied Mathematics – Unit 1: Measurement 2D & 3D (DPCDSB Dec 2008)
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1.3.2: Scale Drawing Details (continued)
2. Use the diagram above
a) What dimensions are needed to determine the perimeter?
b) What dimensions are needed to determine the area?
c) Calculate:
Area
Perimeter
3. Provide an example in daily life of a figure that involves more than one geometric shape.
MFM 1P - Grade 9 Applied Mathematics – Unit 1: Measurement 2D & 3D (DPCDSB Dec 2008)
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1.3.2: Scale Drawing Details (continued)
5 cm
40 cm
2. Use the diagram above
a) What dimensions are needed to determine the area of the square?
b) What dimensions are needed to determine the area of the circle?
c) Calculate:
Area of square
Area of circle
Area of shaded area
3. Provide an example in daily life of a figure that involves more than one geometric shape.
Example: a church window
MFM 1P - Grade 9 Applied Mathematics – Unit 1: Measurement 2D & 3D (DPCDSB Dec 2008)
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1.3.P: Practice
1. (a) Design a logo of your first and last initial, made of two or more simple geometric
shapes.
Make the appropriate measurements and calculate the total area and perimeter of
your logo.
MFM 1P - Grade 9 Applied Mathematics – Unit 1: Measurement 2D & 3D (DPCDSB Dec 2008)
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1.4.2: Composite Figures Notes
MFM 1P - Grade 9 Applied Mathematics – Unit 1: Measurement 2D & 3D (DPCDSB Dec 2008)
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1.4.2: Composite Figures Notes (continued)
MFM 1P - Grade 9 Applied Mathematics – Unit 1: Measurement 2D & 3D (DPCDSB Dec 2008)
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1.4.3: Exploring Composite Shapes
3 cm
Perimeter Calculation
7 cm
7 cm
4 cm
4 cm
3 cm
8 cm
14 cm
Shape Divisions
Area Calculations
Option A
Option B
?
Option C
MFM 1P - Grade 9 Applied Mathematics – Unit 1: Measurement 2D & 3D (DPCDSB Dec 2008)
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1.4.P: Practice
1. Carpeting costs $12.50/m2. How much would it cost to carpet the room below
including GST and PST?
5m
6m
10 m
7m
MFM 1P - Grade 9 Applied Mathematics – Unit 1: Measurement 2D & 3D (DPCDSB Dec 2008)
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1.4.P: Practice (continued)
2. Calculate the area and perimeter of each figure.
12 cm
(a)
8 cm
4 cm
4.6 cm
(b)
5 cm
6.7 m
(c)
7m
4m
6m
MFM 1P - Grade 9 Applied Mathematics – Unit 1: Measurement 2D & 3D (DPCDSB Dec 2008)
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1.4.P: Practice (continued)
3. Find the area of the shaded regions only.
a)
150 cm
30 cm
200 cm
b)
MFM 1P - Grade 9 Applied Mathematics – Unit 1: Measurement 2D & 3D (DPCDSB Dec 2008)
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1.4.4: Use What You Know
Knowledge and Skills
Reasoning and Proving
Calculate the area of the given circle.
Westview School has a track.
Show your work.
r = 2.5 cm
You want to run 2 km every day. Determine
how many times you have to go around the
track.
Show your work.
Hint: A = πr
2
Communicating
Connecting
Gemma wants to tile her bathroom counter
with mini tiles. She needs to determine the
area of her counter space. Explain with
words, diagrams, and symbols how she
should determine the area.
This figure has a radius of r units.
r
Which of the following formulas could be
used to determine the perimeter?
a)
2πr −
1
+r +r
4
b)
0.75πr 2
c)
3
(2πr ) + r + r
4
d)
2πr −
Give reasons for your answer.
MFM 1P - Grade 9 Applied Mathematics – Unit 1: Measurement 2D & 3D (DPCDSB Dec 2008)
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1
4
1.4.5: Frayer Model
Definition
Facts/Characteristics
Composite
Figures
Examples
NonExamples
MFM 1P - Grade 9 Applied Mathematics – Unit 1: Measurement 2D & 3D (DPCDSB Dec 2008)
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1.5.1: The Rope Stretchers
In ancient Egypt, mathematicians developed many useful ideas for everyday living. One
example was used by Egyptian farmers. Each year the Nile River flooded, leaving behind a
stretch of fertile land where the Egyptians grew their crops of barley and emmer wheat. But,
when the river flooded, the boundaries of the fields
were lost and had to be accurately “redrawn.”
Egyptian surveyors or “rope stretchers” used
lengths of ropes with equally spaced knots tied in
them to measure land boundaries. When two fields
bordered one another, the rope stretchers had to
measure a right angle to form the corners of the
fields. The establishment of boundaries was also important because the area of the land
determined the amount of taxes, and the scribes kept the accounts for taxation.
Excerpted from http://www.edhelper.com/ReadingComprehension_35_193.html
July 26, 2005
Literacy Connections
1.
Why do the rope stretchers need to redraw the boundaries every year.
________________________________________________________________________
________________________________________________________________________
2.
Why is it important to have boundaries?
________________________________________________________________________
________________________________________________________________________
3.
Why is it important that the boundaries are 90˚?
________________________________________________________________________
________________________________________________________________________
________________________________________________________________________
4.
Provide examples where you could use this technique today.
________________________________________________________________________
________________________________________________________________________
________________________________________________________________________
________________________________________________________________________
MFM 1P - Grade 9 Applied Mathematics – Unit 1: Measurement 2D & 3D (DPCDSB Dec 2008)
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1.5.3: Right or Not?
Part A: Group Work
1) Record the three side lengths:
Card A _____________________
Card B ___________________
2) Follow the instructions on the overhead to construct your two triangles.
Draw a labelled diagram of the triangle created by the three squares. Show side lengths.
Card A
Card B
3) Complete the table:
Card
Square with
side a
Square with
side b
Largest Square
with side c
Type of Triangle
Right or Not
4) Share your data with another group. What observations can you make about the data?
MFM 1P - Grade 9 Applied Mathematics – Unit 1: Measurement 2D & 3D (DPCDSB Dec 2008)
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1.5.3: Right or Not? (continued)
Part B: Class Work
5) Complete the data table:
Group
Area of Square
with side a
Area of Square
with side b
Area of Largest
Square with side c
Type of Triangle
Right or Not
6) What other observations can you make about the class data set?
Verbal Model:
Visual Model:
Algebraic Model:
MFM 1P - Grade 9 Applied Mathematics – Unit 1: Measurement 2D & 3D (DPCDSB Dec 2008)
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1.5.5: The Pythagorean Theorem Notes
MFM 1P - Grade 9 Applied Mathematics – Unit 1: Measurement 2D & 3D (DPCDSB Dec 2008)
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1.5.5: The Pythagorean Theorem Notes (Continued)
MFM 1P - Grade 9 Applied Mathematics – Unit 1: Measurement 2D & 3D (DPCDSB Dec 2008)
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1.5.P: Practice
1.
Imagine you are a rope stretcher. Find two different combinations, not in the
class list, which would create right-angled triangles. Explain how you know that
the triangles are right-angled.
Hint:
Class Triangle 1:
3, 4, 5
Class Triangle 2:
6, 8, 10
What do you notice about these two triangles?
MFM 1P - Grade 9 Applied Mathematics – Unit 1: Measurement 2D & 3D (DPCDSB Dec 2008)
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1.5.6: Pythagorean Theorem Puzzle
Instructions:
1. Colour the 5 pieces of the puzzle above and the 5 pieces you received from your
teacher.
2. Cut out pieces 1-5 only from copy your teacher provided.
3. Fit the 5 pieces of the puzzle into the large square above. Be sure that the pieces fit
exactly into the square (There should be no paper overlapping, and no blank spaces)
MFM 1P - Grade 9 Applied Mathematics – Unit 1: Measurement 2D & 3D (DPCDSB Dec 2008)
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1.6.1: Coach and Be Coached
A coaches B
x
5 mm
B coaches A
10 cm
12 mm
y
14 cm
12 m
9m
x
2 cm
z
2 cm
3m
12 m
8m
p
8m
m
MFM 1P - Grade 9 Applied Mathematics – Unit 1: Measurement 2D & 3D (DPCDSB Dec 2008)
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1.6.1: Coach and Be Coached (continued)
A coaches B
B coaches A
55 cm
60 cm
130 cm
w
A hydro pole casts a shadow that is 10
m long. A technician measures the wire
that runs from the top of the pole to the
end of the shadow and finds it to be 26
m. How tall is the pole?
44 cm
h
Don is building a loft in his garage. The
ladder he is using extends to 10 metres.
The loft is
8 m from the floor. How far away from the
wall should he anchor the ladder?
MFM 1P - Grade 9 Applied Mathematics – Unit 1: Measurement 2D & 3D (DPCDSB Dec 2008)
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1.6.2: The Container!
The Geobellies company wishes to make a new type of container for their product. The designer
has created two containers: one the shape of a square-based pyramid, and the other a cone.
Your job is to determine which container holds more.
1) Label the dimensions from your group card.
Pyramid
Cone
slant height _____
slant height ______
diameter ________
base length ______
2) Discuss with your partner how you could find the height of each container.
3) Record your ideas using words, pictures, and symbols.
4) Determine the height of each container. Show all of your work.
Pyramid
Cone
MFM 1P - Grade 9 Applied Mathematics – Unit 1: Measurement 2D & 3D (DPCDSB Dec 2008)
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1.6.2: The Container! (Continued)
5) How much can each container hold? Show your work.
Pyramid
Cone
6) If the geobellies cost $0.005/cm3, how much will it cost to fill each container?
7) Identify the shape with the greater volume.
8) Make a recommendation for the preferred design shape. Provide at least two reasons for
your choice.
MFM 1P - Grade 9 Applied Mathematics – Unit 1: Measurement 2D & 3D (DPCDSB Dec 2008)
1-35
1.6.P : Practice
1. Calculate the length of the missing side in each triangle.
(a)
c
10 cm
13 cm
(b)
9 cm
b
15 cm
(c)
y
12 m
5m
MFM 1P - Grade 9 Applied Mathematics – Unit 1: Measurement 2D & 3D (DPCDSB Dec 2008)
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1.6.P : Practice (continued)
2.
A 5-m ladder is leaning against a house. Draw a diagram and add in any
given measurements.
The foot of the ladder is 3 m from the base of the wall.
How high up the wall does the ladder reach?
3.
Adam has made a picture frame. Draw a diagram and add in any given
measurements.
The frame is 60 cm long and 25 cm wide.
To check that the frame has square corners, Adam measures the diagonal.
How long should the diagonal be?
Sketch a diagram to support your answer.
MFM 1P - Grade 9 Applied Mathematics – Unit 1: Measurement 2D & 3D (DPCDSB Dec 2008)
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1.6.J: Journal Activity
Your friend was away for today’s lesson.
Write an email describing how to find the height of a pyramid or cone when given the
base length and slant height or diameter and slant height.
Provide your own example.
MFM 1P - Grade 9 Applied Mathematics – Unit 1: Measurement 2D & 3D (DPCDSB Dec 2008)
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1.W: Definition Page
Term
Picture / Sketch
Definition
Perimeter
Length
Width
Radius
Diameter
Circumference
Circle
Area
Rectangle
Square
Rhombus
Parallelogram
MFM 1P - Grade 9 Applied Mathematics – Unit 1: Measurement 2D & 3D (DPCDSB Dec 2008)
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1.W: Definition Page (Continued)
Term
Picture / Sketch
Definition
Triangle
Right-Angled Triangle
Hypotenuse
Composite Figure
Volume
Prism
Rectangular Prism
Cylinder
Sphere
Hemisphere
Cone
Pyramid
MFM 1P - Grade 9 Applied Mathematics – Unit 1: Measurement 2D & 3D (DPCDSB Dec 2008)
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1.S: Unit Summary Page
Complete the following table to summarize the formulas you used in this unit.
2D Shape
Sketch / Description Area Formula
Example
Rectangle / Square
Triangle
Circle
Parallelogram
Trapezoid
3D Solid
Sketch / Description
Volume Formula
Example
Rectangular Prism
Cylinder
Sphere
Cone
Pyramid
MFM 1P - Grade 9 Applied Mathematics – Unit 1: Measurement 2D & 3D (DPCDSB Dec 2008)
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1.R Reflecting on My Learning (3, 2, 1)
3 Things I know well from this unit
2 Things I need explained more
1 Question I still have
MFM 1P - Grade 9 Applied Mathematics – Unit 1: Measurement 2D & 3D (DPCDSB Dec 2008)
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1.RLS: Reflecting on Learning Skills
Students should be aware of the importance that these skills have on your performance.
After receiving your marked assessment, answer the following questions. Be honest
with yourself. Good Learning Skills will help you now, in other courses and in the future.
E – Always
G – Sometimes
S – Need Improvement
N – Never
Organization
E G S N
E G S N
E G S N
I came prepared for class with all materials
My work is submitted on time
I keep my notebook organized.
Work Habits
E G S N
E G S N
E G S N
E G S N
E G S N
E G S N
I attempt all of my homework
I use my class time efficiently
I limit my talking to the math topic on hand
I am on time
If I am away, I ask someone what I missed,
I complete the work from the day that I missed.
Team Work
E G S N
E G S N
E G S N
I am an active participant in pairs/group work
I co-operate with others within my group
I respect the opinions of others
Initiative
E G S
E G S
E G S
E G S
I participate in class discussion/lessons
When I have difficulty I seek extra help
After I resolve my difficulties, I reattempt the problem
I review the daily lesson/ideas/concepts
N
N
N
N
Works Independently
E G S N
I attempt the work on my own
E G S N
I try before seeking help
E G S N
If I have difficulties I ask others but I stay on task
E G S N
I am committed to tasks at hand
Yes No
I know all the different ways available in my school, where I can seek extra help.
Yes No
I tried my best.
What will I do differently in the next unit to improve?
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
MFM 1P - Grade 9 Applied Mathematics – Unit 1: Measurement 2D & 3D (DPCDSB Dec 2008)
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MFM 1P - Grade 9 Applied Mathematics – Unit 1: Measurement 2D & 3D (DPCDSB Dec 2008)
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Course: Grade 9 Applied Mathematics (MFM1P)
Unit 2:
Plane Geometry
Unit 2
Plane Geometry
Section
2.1.1
2.1.2
2.1.4
2.1.P
2.5.1
2.5.2
2.6.1
2.7.1
2.7.2
2.7.J
2.7.P
2.S
2.R
2.RLS
Activity
I Remember
Plane Geometry Record Sheet
Parallel Lines Exploration – Optional
Practice
What’s So Special Guide Sheet
What’s So Special Record Sheet
Learn the Lingo
Exterior and Interior Angles of a Polygon
Interior Angle Sums
Journal Activity
Practice
Unit Summary Page
Reflecting on My Learning (3, 2, 1)
Reflecting on Learning Skills
MFM 1P - Grade 9 Applied Mathematics – Unit 2: Plane Geometry (DPCDSB July 2008)
Page
3
6
9
10
17
18
20
23
25
27
28
30
31
32
2-2
2.1.1: I remember….
Work with a partner to complete the definitions below that you know. Leave the ones you are unsure of
and come back to them throughout this unit as you learn more about them.
Word/Term
Definition
Diagram
Supplementary Angles
Complimentary Angles
Opposite Angles
Corresponding Angles
Alternate Angles
Co-Interior Angles
Parallel Lines
Transversal
MFM 1P - Grade 9 Applied Mathematics – Unit 2: Plane Geometry (DPCDSB July 2008)
2-3
2.1.1: I remember….(continued)
Word/Term
Definition
Diagram
Triangle
Isosceles Triangle
Equilateral Triangle
Right Triangle
Acute Triangle
Obtuse Triangle
Scalene Triangle
MFM 1P - Grade 9 Applied Mathematics – Unit 2: Plane Geometry (DPCDSB July 2008)
2-4
2.1.1: I remember….(continued)
Word/Term
Definition
Diagram
Quadrilateral
Parallelogram
Rhombus
Trapezoid
Square
Rectangle
Hexagon
Polygon
MFM 1P - Grade 9 Applied Mathematics – Unit 2: Plane Geometry (DPCDSB July 2008)
2-5
2.1.2: Plane Geometry Record Sheet
Use this page to record your observations and conclusions from the Plane Geometry GSP®4
file. Determine the unknown angle in the right column. Give reasons for your answer.
MFM 1P - Grade 9 Applied Mathematics – Unit 2: Plane Geometry (DPCDSB July 2008)
2-6
2.1.2: Plane Geometry Record Sheet (continued)
MFM 1P - Grade 9 Applied Mathematics – Unit 2: Plane Geometry (DPCDSB July 2008)
2-7
2.1.2: Plane Geometry Record Sheet (continued)
MFM 1P - Grade 9 Applied Mathematics – Unit 2: Plane Geometry (DPCDSB July 2008)
2-8
2.1.4: Parallel Lines Exploration - Optional
Explore and Reflect
1. How did you know when Line 1 and Line 2 were
parallel?
Sketch
Angle Relationships
2. Find one pair of equal angles. Explain how you
know they are equal.
3. Find another pair of equal angles. Explain how you
know they are equal.
4. Find as many pairs of angles that are
supplementary (add to 180°) as you can. Explain
how you know.
Summary (to be completed as a whole class)
MFM 1P - Grade 9 Applied Mathematics – Unit 2: Plane Geometry (DPCDSB July 2008)
2-9
2.1.P: Practice
1.
Look at this diagram.
(a)
Name two parallel line segments.
(b)
Name two transversals.
(c)
Name two corresponding angles.
(d)
Name two alternate angles.
(e)
Find the measures of ∠ ECD, ∠ ACE, and ∠ BCA.
A
E
•
65º
50º
B
C
•D
MFM 1P - Grade 9 Applied Mathematics – Unit 2: Plane Geometry (DPCDSB July 2008)
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2.1.P: Practice (continued)
2.
(a)
Draw parallelogram ABCD with ∠ A = 51º.
(b)
How can you use what you know about parallel line segments and a transversal
to find the measures of the other 3 angles in the parallelogram? Explain your
work.
(c)
When is a quadrilateral a parallelogram? Explain.
MFM 1P - Grade 9 Applied Mathematics – Unit 2: Plane Geometry (DPCDSB July 2008)
2-11
2.1.P: Practice Sheet (continued)
Define each principle and determine the unknown angles.
1.
xo =
Reason:
=
x° 85°
2.
70°
r°
ro =
=
Reason:
3.
mo =
=
Reason:
m°
30°
4.
q°
50°
55°
qo =
=
Reason:
MFM 1P - Grade 9 Applied Mathematics – Unit 2: Plane Geometry (DPCDSB July 2008)
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2.1.P: Practice Sheet (continued)
5.
bo =
60°
Reason:
=
b°
45°
6.
ao =
=
75°
Reason:
a°
7.
xo =
=
Reason:
x°
MFM 1P - Grade 9 Applied Mathematics – Unit 2: Plane Geometry (DPCDSB July 2008)
2-13
2.1.P: Practice Sheet (continued)
8.
x°
xo =
Reason:
=
72°
68°
mo =
Reason:
wo =
Reason:
m°
83°
w°
MFM 1P - Grade 9 Applied Mathematics – Unit 2: Plane Geometry (DPCDSB July 2008)
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2.1.P: Practice (continued)
9.
Find the measure of a.
Give reasons.
5.
Find the value of x.
Give reasons.
68º
aº
xº
71º
10.
49º
46º
Find x. Give reasons.
7.
Find the values of the missing angles.
Give reasons.
yº
89º
44º
xº
42º
11.
42º
96º
xº
The diagram shows two parallel lines cut by a transversal. The measure of a + b is
_____. Give reasons.
aº
bº
MFM 1P - Grade 9 Applied Mathematics – Unit 2: Plane Geometry (DPCDSB July 2008)
2-15
2.1.P: Practice (continued)
12.
For the following diagram, list as many examples of each Angle Theorem as possible.
a° b°
c° d°
Z – pattern
C – pattern
F - pattern
e.g. ∠c = ∠ g
e° f°
g°
h°
13.
Solve for x and y
a)
b)
MFM 1P - Grade 9 Applied Mathematics – Unit 2: Plane Geometry (DPCDSB July 2008)
2-16
2.5.1: “What’s So Special?” Guide Sheet
Explore!
Drag each vertex in the figure.
As you drag vertices, look for some of the following:
• measurements that always seem to be equal to each other
• measurements that never seem to change
• measurements that might have a constant ratio (proportional)
• lines that always seem to be parallel or perpendicular
• line segments that always seem to be bisected
• figures that always seem to be congruent
• objects that don’t seem to be connected, yet they move together when something is
dragged
Make an Hypothesis
Decide which measurements you need to test your hypothesis.
Drag each vertex again while you pay close attention to the way the object moves and to the
way the measurements change.
Test Your Hypothesis
Collect and record evidence to test your hypothesis.
What can you measure?
• angles
• lengths
• areas
• perimeters
• slopes
•
•
What can you calculate?
• sums
• ratios
• formulas
•
•
•
•
MFM 1P - Grade 9 Applied Mathematics – Unit 2: Plane Geometry (DPCDSB July 2008)
2-17
2.5.2: What’s So Special? Record sheet
Investigation 1. Special Triangles
Explore: Move the vertices of the triangles around and examine how the angles and side
lengths change.
Hypothesis: Make a hypothesis about what type of triangle each figure is and record it in the
chart below.
Test your Hypothesis: Make any measurements that will help test your hypothesis.
Hypothesis:
Type of Triangle
Conclusion:
Type of Triangle
Evidence:
(what measurements support your
conclusion)
ΔABC
ΔDEF
ΔGHI
ΔKLM
Investigation 2. Parallel or Perpendicular?
Explore: Drag the endpoints of the line segments.
Hypothesis: Make a hypothesis about which lines are parallel, which are perpendicular, and
which are neither.
Test your Hypothesis: Make any measurements that will help test your hypothesis.
Conclusions: Make a statement about which lines are parallel and which are perpendicular
and provide evidence (which measurements support your claim)
MFM 1P - Grade 9 Applied Mathematics – Unit 2: Plane Geometry (DPCDSB July 2008)
2-18
2.5.2: What’s So Special? Record sheet
Investigation 5: Special Quadrilaterals?
Explore: Drag each vertex of each figure.
Hypothesis: Make a hypothesis about what type of quadrilateral each figure is and record
your hypothesis in the chart below.
Test your Hypothesis: Make any measurements that will help test your hypothesis.
Conclusions:
Evidence
Quadrilateral
Hypothesis
Conclusions
(What measurements prove your
conclusions?)
A
B
C
D
E
F
G
MFM 1P - Grade 9 Applied Mathematics – Unit 2: Plane Geometry (DPCDSB July 2008)
2-19
2.6.1: Learn the Lingo
1. Part a) shows an example of how to complete a word chart.
Complete the remaining word charts.
a)
Term:
Visual
Representation:
b)
Term:
Equilateral
Triangle
Definition:
An equilateral
triangle is a triangle
for which all sides
have the same
length.
Visual
Representation:
Triangle
Association:
A Yield sign
c)
Definition:
Association:
d)
Term:
Visual
Representation:
Exterior
Angle
Definition:
Term:
Visual
Representation:
Interior
Angle
Association:
Definition:
Association:
MFM 1P - Grade 9 Applied Mathematics – Unit 2: Plane Geometry (DPCDSB July 2008)
2-20
2.6.1: Learn the Lingo (continued)
e)
Term:
Visual
Representation:
Parallel Lines
Definition:
Association:
Definition:
Association:
h)
Visual
Representation:
Perpendicular
Bisector
Definition:
Visual
Representation:
Transversal
g)
Term:
f)
Term:
Term:
Visual
Representation:
Diagonal
Association:
Definition:
MFM 1P - Grade 9 Applied Mathematics – Unit 2: Plane Geometry (DPCDSB July 2008)
Association:
2-21
2.6.1: Learn the Lingo (continued)
2. Determine the unknown angle. Give reasons for your answer.
b)
a)
c)
T
D
A
U
°
108
B
E
G
C
F
A
∠DEG = ____
W
Z
AB = AC = BC
∠ACB = ________
X
V
Y
TZ UY
∠TWX = 75o
∠UVW = __
d)
e)
C
f)
M
F
N
C
O
O
D
D
E
S
∠COD = 64
∠FOE = ___
h)
W X
°
42
S
T
∠BOC = 43o
∠COE = ____
∠EOD = ____
∠NQR = 115 o
∠MRQ = ____
i)
W
X
Y
R
Z
O
MP NO
∠COF = ___
V
E
R Q
P
o
g)
B
Create your own question!
Z
A
U
WX = WY
∠VRW = 42
RT = RU
o
∠YWX = 118o
∠WXZ = _____
∠SRT = 19o
∠RSZ =
MFM 1P - Grade 9 Applied Mathematics – Unit 2: Plane Geometry (DPCDSB July 2008)
2-22
2.7.1: Exterior and Interior Angles of a Polygon
Part A – Exterior Angles of a Triangle
Using the GSP files: Angles Triangles. gsp
1. Click the “Show Measurements” Tab.
2. Drag vertices A, B, and C.
3. What do you notice?
4. Click the “Reset the triangle” Tab.
5. Click the “Make the triangle smaller” Tab.
6. If we decrease the size of the triangle, what
do you notice about the sum of the exterior
angles?
Part B – Exterior Angles of a Quadrilateral
Similarly with the quadrilateral,
1. Click the “Make the quadrilateral smaller”
Tab.
2. Describe what just happened.
3. If we decrease the size of the
quadrilateral, what do you notice about the
sum of the exterior angles?
MFM 1P - Grade 9 Applied Mathematics – Unit 2: Plane Geometry (DPCDSB July 2008)
2-23
2.7.1: Exterior and Interior Angles of a Polygon (continued)
Part C – Exterior Angles of Any Polygon
Similarly with any polygon,
1. Click on the tab “Show Sum”.
2. Drag vertices A, B, and C.
3. What do you notice?
4. Click the “Reset” Tab.
5. Click the “Shrink polygon” Tab.
6. If we decrease the size of the polygon, what
do you notice about the sum of the exterior
angles?
7. Click the “One less side” Tab. What shape do
you have now? What is the sum of the exterior
angles?
8. Click the “Another side less” Tab. What shape
do you have now? What is the sum of the
exterior angles?
Compare the conclusions you reached in Part A and Part B.
Write your final conclusion about the sum of the exterior angles of any polygon.
MFM 1P - Grade 9 Applied Mathematics – Unit 2: Plane Geometry (DPCDSB July 2008)
2-24
2.7.2: Interior Angle Sums
1. Complete the chart.
Diagram
Number
of sides
Sum of
interior angles
Understanding
The sum of the angles in any triangle is
180o.
3
180°
4
5
n
2. a) Determine the sum of the interior angles in a polygon with 15 sides. Show your work.
b) Determine the number of sides in a polygon if the sum of the interior angles is 5400°.
Show your work.
MFM 1P - Grade 9 Applied Mathematics – Unit 2: Plane Geometry (DPCDSB July 2008)
2-25
2.7.2: Interior Angle Sums (continued)
3. Derek is building a deck for his summer job in the shape of a regular octagon.
a) Define: regular octagon
?
b) Determine the measure of the interior angles of the deck.
Show your work.
4. A Canadian $1 coin, known as a loonie, is a regular polygon with 11 sides, called an
undecagon.
a) Define a regular polygon with 11 sides.
b) Determine the sum of the interior angles of the loonie.
c) What is the size of one of the interior angles?
MFM 1P - Grade 9 Applied Mathematics – Unit 2: Plane Geometry (DPCDSB July 2008)
2-26
2.7.J: Journal Activity
Write a letter to Abe, who missed Math class, explaining how he can determine the sum of the
interior and exterior angles in a decagon (10-sided polygon).
MFM 1P - Grade 9 Applied Mathematics – Unit 2: Plane Geometry (DPCDSB July 2008)
2-27
2.7.P: Practice
1. Determine the measure of each indicated angle and state reasons.
a)
b)
c)
107º
41º
100º
104º
xº
xº
2.
70º
49º
Determine the values of x, y, and z. Give reasons.
a)
b)
xº
zº
3.
yº
xº
xº
108º
yº
c)
84º
64º
96º
47º
yº
132º
zº
zº
Determine the measures of a and b. Give reasons.
a)
b)
55º
105º
bº
25º
aº
115º
aº
83º
bº
MFM 1P - Grade 9 Applied Mathematics – Unit 2: Plane Geometry (DPCDSB July 2008)
2-28
57º
yº
2.7.P: Practice (continued)
4.
Find the measure of x in the following pentagon. Give reasons.
100º
xº
100º
5.
100º
100º
Find the measures of a, b, and c. Give reasons.
135º
bº
aº
cº
MFM 1P - Grade 9 Applied Mathematics – Unit 2: Plane Geometry (DPCDSB July 2008)
2-29
2.S: Unit Summary Page
Complete the following concept map to relate all the terms from this unit.
PLANE GEOMETRY
MFM 1P - Grade 9 Applied Mathematics – Unit 2: Plane Geometry (DPCDSB July 2008)
2-30
2.R: Reflecting on My Learning (3, 2, 1)
3 Things I know well from this unit
2 Things I need explained more
1 Question I still have
MFM 1P - Grade 9 Applied Mathematics – Unit 2: Plane Geometry (DPCDSB July 2008)
2-31
2.RLS: Reflecting on Learning Skills
Students should be aware of the importance that these skills have on your performance. After
receiving your marked assessment, answer the following questions. Be honest with yourself.
Good Learning Skills will help you now, in other courses and in the future.
•
•
•
•
E – Always
G – Sometimes
S – Need Improvement
N – Never
Organization
• E G S N
• E G S N
• E G S N
I came prepared for class with all materials
My work is submitted on time
I keep my notebook organized.
Work Habits
• E G S N
• E G S N
• E G S N
• E G S N
• E G S N
• E G S N
I attempt all of my homework
I use my class time efficiently
I limit my talking to the math topic on hand
I am on time
If I am away, I ask someone what I missed,
I complete the work from the day that I missed.
Team Work
• E G S N
• E G S N
• E G S N
I am an active participant in pairs/group work
I co-operate with others within my group
I respect the opinions of others
Initiative
• E G S
• E G S
• E G S
• E G S
I participate in class discussion/lessons
When I have difficulty I seek extra help
After I resolve my difficulties, I reattempt the problem
I review the daily lesson/ideas/concepts
N
N
N
N
Works Independently
I attempt the work on my own
• E G S N
• E G S N
I try before seeking help
• E G S N
If I have difficulties I ask others but I stay on task
• E G S N
I am committed to tasks at hand
Yes No
I know all the different ways available in my school, where I can seek extra help.
Yes No
I tried my best.
What will I do differently in the next unit to improve?
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
MFM 1P - Grade 9 Applied Mathematics – Unit 2: Plane Geometry (DPCDSB July 2008)
2-32
Course: Grade 9 Applied Mathematics (MFM1P)
Unit 3:
Looking for Relationships, Lines
and Curves of Best Fit
Unit 3
Exploring Relationships: Lines and Curves of Best Fit
Section
Activity
3.1.1 Graphing Review
3.1.2 Relationships
3.1.3 Data Collection – Is there a Relationship
Here?
3.1.4 Class Data Sheet
3.1.5 Graphing the Data
3.2.1 Plotted Points
3.2.2 Scatter Plots – Types of Correlation
3.2.3 Line of Best Fit
3.2.P Practice
3.2.4 Relationships Summary
3.3.1 Could I Be a Forensic Scientist?
3.3.2a Introduction to FATHOM
3.3.2b Introduction to TI-83 TI-84
3.3.2c Introduction to TI-Nspire CAS
3.3.3 Choosing the Best Model
3.3.P Practice
3.4.1 Creating Scatter Plots and Lines of Best Fit
3.4.3 Forensic Analysis
3.4.J Journal Activity
3.5.1 Investigations
3.7.1 First Differences
3.7.2 Using What You Have Discovered
3.8
Unit 3 Review (Practice)
3.W
Definition Page
3.S
Unit Summary Page
3.R
Reflecting on My Learning
3.RLS Reflecting on Learning Skills
Page
3
4
5
MFM 1P - Grade 9 Applied Mathematics – Unit 3: Lines and Curves of Best Fit (DPCDSB Dec 2008)
6
7
8
9
10
11
12
14
15
16
17
20
22
23
24
25
26
36
41
42
47
49
50
51
3-2
3.1.1: Graphing Review
Plot the wingspan data for these.18 birds on the grid below.
Source: Faculty of Mathematics, University of Waterloo, “Linear Relations: Graphing and Analyzing”
(Wingspread of Birds)
1. Describe the pattern of the dots on the scatter plot.
2. Describe the relationship between bird length and wingspan.
MFM 1P - Grade 9 Applied Mathematics – Unit 3: Lines and Curves of Best Fit (DPCDSB Dec 2008)
3-3
3.1.2: Relationships
Complete the following statements by yourself, then share your
answers with your partner. Explain the reasons for your choice.
Indicate if you and your partner agree or disagree.
Is There a Relationship?
As a person gets taller their armspan ______________________.
(gets wider, gets smaller, stays the same)
The longer a person's legs are ______________________ they run.
(the faster, the slower, will make no difference to how fast)
My Partner and I:
__ agree
__ disagree
__ agree
__ disagree
As a person's foot size increases, their walking stride
_____________________.
__ agree
(gets longer, gets shorter, stays the same)
__ disagree
As a person's forearm gets longer, their armspan _______________.
__ agree
(gets longer, gets shorter, stays the same length)
__ disagree
The longer a person's thumb is ______________________ their
index finger.
__ agree
(the longer, the shorter, will make no difference to the length of)
__ disagree
As a person gets taller, their foot size ______________________.
__ agree
(gets longer, gets shorter, is not affected)
__ disagree
MFM 1P - Grade 9 Applied Mathematics – Unit 3: Lines and Curves of Best Fit (DPCDSB Dec 2008)
3-4
3.1.3: Data Collection – Is There a Relationship Here?
With a partner, measure and record each measurement to the nearest centimetre. Enter your
data into the class data collection chart.
a) total height ____________ cm
b) forearm ____________ cm
c) arm span from fingertips to fingertips ____________ cm
d) foot length _______________ cm
e) walking stride length _______________ cm
f) hand span ____________________cm
MFM 1P - Grade 9 Applied Mathematics – Unit 3: Lines and Curves of Best Fit (DPCDSB Dec 2008)
3-5
3.1.4: Class Data Sheet
Maggie
134
22
130
20
120
Hand
span
(cm)
13
Homer
162
26
160
24
140
18
Stefan
169
27
170
26
145
19
Teniesha
150
24
149
23
132
16
Debbie
143
23
143
22
122
15
Kevin
167
26
161
23
145
17
Dwight
178
28
178
27
155
19
Shahad
165
25
163
24
142
18
Name
Height
(cm)
Forearm
(cm)
Arm
span
(cm)
Foot
length
(cm)
Walking
Stride
length (cm)
MFM 1P - Grade 9 Applied Mathematics – Unit 3: Lines and Curves of Best Fit (DPCDSB Dec 2008)
3-6
3.1.5: Graphing the Data
Using class data from Day 1, choose two measurements that you would like to investigate.
Create a scatter plot of your chosen relationship on grid paper.
Using your graph answer the following questions:
1. Which phrase describes the direction of the plotted points in the graph?
a) The plotted points rise upward to the right.
b) The plotted points fall downward to the right.
c) The plotted points are scattered across the graph.
2. Describe the relationship between the two quantities.
3. How could you use this graph to predict additional measurements? Explain.
MFM 1P - Grade 9 Applied Mathematics – Unit 3: Lines and Curves of Best Fit (DPCDSB Dec 2008)
3-7
Leg
Length
(cm)
Tibia Length (cm)
Number of Baskets
3.2.1: Plotted Points
Distance from the Basket
1. The graph shows the plotted points rising
upwards to the right.
• Agree
• Disagree
• Pass
1. The graph shows the plotted points falling
to the right.
• Agree
• Disagree
• Pass
2. As the length of the tibia increases the
length of the leg increases.
• Agree
• Disagree
• Pass
2. As the distance from the net increases
the number of baskets made decreases.
• Agree
• Disagree
• Pass
3. The graph can be used to determine the
length of a person's leg if you know the
length of the tibia bone.
• Agree
• Disagree
• Pass
3. The graph can be used to determine the
number of baskets you will make if you
know the distance from the basket.
• Agree
• Disagree
• Pass
2. As the age of the house increases the
price of the house is either large or small.
• Agree
• Disagree
• Pass
3. The graph can't be used to determine the
price of the house if you know how old it
is.
• Agree
• Disagree
• Pass
House Price ($)
1. The graph shows the plotted points
scattered.
• Agree
• Disagree
• Pass
Age of House
MFM 1P - Grade 9 Applied Mathematics – Unit 3: Lines and Curves of Best Fit (DPCDSB Dec 2008)
3-8
3.2.2: Scatter Plots - Types of Correlation
Correlation helps to describe the relationship between 2 quantities in a graph.
Correlation can be described as positive or negative, strong or weak or none.
Positive or Negative Correlation
A scatter plot shows a ____________ correlation when the
pattern rises up to the right.
This means that the two quantities increase together.
A scatter plot shows a ____________ correlation when the
pattern falls down to the right.
This means that as one quantity increases the other
decreases.
Strong or Weak Correlation
If the points nearly form a line, then the correlation is
__________________.
To visualize this, enclose the plotted points in an oval. If the oval
is narrow, then the correlation is strong.
If the points are dispersed more widely, but still form a rough
line, then the correlation is ___________________.
If the points are dispersed even more widely, but still form a
rough pattern of a line, then the correlation is
___________________.
If the oval is wide, then the correlation is weak.
No Correlation
A scatter plot shows _______________ correlation when no
pattern appears.
Hint:
If the points are roughly enclosed by a circle, then there is no
correlation.
MFM 1P - Grade 9 Applied Mathematics – Unit 3: Lines and Curves of Best Fit (DPCDSB Dec 2008)
3-9
3.2.3: Line of Best Fit
Line of Best Fit
To be able to make predictions, we need to model the data with a line or a curve of best fit.
Rules for drawing a line of best fit:
1. The line must follow the _____________________.
2. The line should __________ through as many points as possible.
3. There should be ____________________________ of points above and below the line.
4. The line should pass through points all along the line, not just at the ends.
Use the information below to draw a scatter plot. Describe the correlation and draw the line of
best fit.
The teachers at Holy Mary high school took a survey in their classes to determine if there is a
relationship between the student’s mark on a test and the number of hours watching T.V. the
night before.
Mark %
Number
of Hours
75
1
70
2
68
3
73
2
59
4
57
4.5
80
1
65
3
63
3.5
55
4
85
1
MFM 1P - Grade 9 Applied Mathematics – Unit 3: Lines and Curves of Best Fit (DPCDSB Dec 2008)
70
2.5
3-10
55
4
3.2.P: Practice
For each of the graphs below:
1) Draw a line of best fit if possible. If a straight line can be drawn, label the graph as linear.
If a straight line cannot be drawn, label the graph as non-linear.
2) Label each graph as showing a relationship or no relationship.
3) The following instructions are for the linear graphs only.
a) Describe the correlation of each scatter plot as positive or negative.
b) Describe the correlation as weak or strong.
a
d
g
Linear / Non-linear
Relationship / No Relationship
Positive / Negative
Strong / Moderate / Weak
Linear / Non-linear
Relationship / No Relationship
Positive / Negative
Strong / Moderate / Weak
Linear / Non-linear
Relationship / No Relationship
Positive / Negative
Strong / Moderate / Weak
b
e
h
Linear / Non-linear
Relationship / No Relationship
Positive / Negative
Strong / Moderate / Weak
Linear / Non-linear
Relationship / No Relationship
Positive / Negative
Strong / Moderate / Weak
Linear / Non-linear
Relationship / No Relationship
Positive / Negative
Strong / Moderate / Weak
c
f
i
Linear / Non-linear
Relationship / No Relationship
Positive / Negative
Strong / Moderate / Weak
Linear / Non-linear
Relationship / No Relationship
Positive / Negative
Strong / Moderate / Weak
Linear / Non-linear
Relationship / No Relationship
Positive / Negative
Strong / Moderate / Weak
MFM 1P - Grade 9 Applied Mathematics – Unit 3: Lines and Curves of Best Fit (DPCDSB Dec 2008)
3-11
3.2.4: Relationships Summary
A scatter plot is a graph that shows the ____________ between two variables.
The points in a scatter plot often show a pattern, or ____________.
From the pattern or trend you can describe the ________________.
Example:
Julie gathered information about her age and height from the markings on the wall in her house.
Age (years)
1
2
3
4
5
6
7
8
Height (cm)
70
82
93
98
106
118
127
135
a) Label the vertical axis.
b) Describe the trend in the data.
c) Describe the relationship.
Variables
The independent variable is located on the ___________ axis.
This variable does not depend on the other variable.
The dependent variable is located on the ____________ axis.
This variable depends on the other variable.
Independent variable: _______________
Note:
The independent
variable comes first in
the table of values.
Dependent variable: _____________
MFM 1P - Grade 9 Applied Mathematics – Unit 3: Lines and Curves of Best Fit (DPCDSB Dec 2008)
3-12
3.2.4: Relationships Summary (continued)
Making Predictions
Use your line of best fit to estimate the following:
Question
Answer
Method of Prediction
How tall was Julie when she was
5 years old?
How tall will Julie be when she is
9 years old?
How old was Julie at 100 cm tall?
How tall was Julie when she was born?
Interpolate
When you interpolate, you are making a prediction __________ the data.
These predictions are usually _________.
Hint:
You are interpolating when
the value you are finding is
somewhere between the
first point and the last point.
Extrapolate
When you extrapolate, you are making a prediction _____________ the data.
It often requires you to ____________the line.
These predictions are less reliable.
You are extrapolating when the
value you are finding is before
the first point or after the last
point. This means you may need
to extend the line.
MFM 1P - Grade 9 Applied Mathematics – Unit 3: Lines and Curves of Best Fit (DPCDSB Dec 2008)
3-13
3.3.1: Could I Be a Forensic Scientist?
Name: _________________________________________
Date: __________________
Exploring the Problem
Remnants of a human skeleton were found at an archaeological dig that is thought to be the
ruins of an ancient civilization. From the bones discovered, the scientists have determined the
following:
• length of the forearm is 23 cm
• armspan is 185 cm
• handspan is 23 cm
• foot length is 24 cm
The scientists call you in as an expert in anthropology who is currently researching relationships
between body measurements to help them determine an estimated height of the skeleton in
question.
As the expert, your job will be to:
• estimate the height of the skeleton;
• explain the procedure you used to determine the height of the skeleton;
• include evidence (tables, graphs, and other models) to support your conclusion;
• explain the limitations of your method or discuss a different way to conduct your
investigation.
Clarifying the Problem
Review the problem and highlight any
important information.
•
•
•
What are the variables?
What exactly are you being asked to find?
Are there certain variables that would be more
useful than others?
Formulating an Hypothesis
•
•
Decide which pairs of variables you think could show a relationship that would aid the
scientists in their predictions.
Explain your reasoning.
MFM 1P - Grade 9 Applied Mathematics – Unit 3: Lines and Curves of Best Fit (DPCDSB Dec 2008)
3-14
3.3.2a: Introduction to FATHOM Æ Creating Scatter Plots
Method
Step 1) Pull down a case table. Enter the heading YEAR in the <new> column and enter
Height in the next column. Enter the data as shown.
Step 2) Pull down an empty scatter plot. Grab and drag the YEAR heading to the horizontal
axis, and the HEIGHT heading to the vertical axis. Your scatter plot is done!
Step 3) Pull down the GRAPH menu and choose MOVABLE LINE. This will be your line of
best fit when you move it to its best position.
Step 4) Pull down the GRAPH menu and choose SHOW SQUARES. Try to position the line
such that the sum of the squares is a MINIMUM. Watch the SUM change as you
reposition the line.
Step 5) Change the horizontal and vertical scales by grabbing and dragging them towards
zero. This will change the scale of the scatter plot and allow you to make predictions
beyond the data collected.
Step 6) Predict the Height for Year 10 Æ
Predict the Height for Year 20 Æ
Predict the Year when the Height will be 30 Æ
Step 7) Add a text box to record your description of the scatter plot and the predictions by
pulling down the INSERT menu and choosing TEXT.
Type "Scatter Plot” by "your name."
Describe your scatter plot with three sentences.
* One sentence will describe the correlation.
* The next sentence will describe the relationship and how strong it is.
* The third sentence will use examples to support your conclusion.
Use the line of best fit to make your predictions.
Collection 1
Year
Height (cm)
1
2
1
2
4
2
3
5
3
4
7
4
5
10
5
MFM 1P - Grade 9 Applied Mathematics – Unit 3: Lines and Curves of Best Fit (DPCDSB Dec 2008)
3-15
3.3.2b: Introduction to Calculator TI-83 or TI-84 Æ Creating a
Line of Best Fit Method
Step 1)
Press STAT. Press 1:Edit… Enter the data for Year in L1 and Height in L2..
Step 2)
Press STATPLOT. Press 1:Plot1…Off. Press ENTER to turn on the statplot. Set
the following options as shown.
Step 3)
Press ZOOM. Press 9:ZoomStat.
Step 4)
Press APPS. Choose TRANSFRM. Press ENTER. Press Y=. Type the equation
AX + B for Y1.
Step 5)
Press Window then press up arrow then Æ to get to SETTINGS. Scroll down and
enter values for A, B and STEP from your teacher.
Step 6)
Press GRAPH. Use up and down arrows to move between A or B. Use Å and Æ
to decrease or increase the value. Continue until the line becomes a line of best fit.
Record your equation:
_______________________________
Step 7)
Press APPS. Choose TRANSFRM. Press 1:UNINSTALL.
Step 8)
Press STAT. Arrow over to CALC. Press 4:LinReg (ax+b). Press 2ND then 1.
Press 2nd then 2. Press ENTER. Record the Linear Regression values below
(Round the values of A and B to one decimal place):
_______________________________
Step 8)
Press Y=. Type in the equation given above for Y2. Press Graph.
Step 9)
Press TRACE and use Æ and Å to move the cursor until you get the X value you
want.
Predict the Height for Year 10 :
_________
Predict the Height for Year 20:
_________
Predict the Year when the Height will be 30:
_________
Step 10)
Make a sketch your graph.
Describe your scatter plot with three sentences.
* One sentence will describe the correlation.
* The next sentence will describe the relationship
and how strong it is.
* The third sentence will use examples to support
your conclusion.
MFM 1P - Grade 9 Applied Mathematics – Unit 3: Lines and Curves of Best Fit (DPCDSB Dec 2008)
3-16
3.3.2c: Introduction to Calculator TI-Nspire CASÆ Creating a
Line of Best Fit Method
Getting to know your calculator
Green Letters
White buttons
- Alphabet
- Numbers
Grey Buttons
- Math Functions
Dark Grey Buttons - Math Operations
- home – add a new page
Menu - menu options available – similar to a computer menu
- menu items – use the arrows and then press enter OR press the number only (this is
how the instructions are given)
Mouse (NavPad)
up
left
right
show all pages
CTRL + mouse
page left
down
page right
show one page
Esc – go back a step
Ctrl z – undo the last step
Entering Data
1.
2.
3.
4.
Press HOME
Choose 6:New Document
Press Æ to Select No to Save Changes and press Enter.
Choose 3:Add Lists and Spreadsheets
5.
6.
7.
8.
Move the cursor up to the spot beside A
Enter the column heading eg: Height (minimum 4 characters, no spaces)
Press Enter
To widen the column press Menu
a. 1:Actions
b. 2:Resize
c. 1:Resize Column Width
d. move the mouse left or right to widen the column
e. press Enter
9. Move to cell A1
10. Enter your data – Type in each value then press Enter
11. Move the cursor up to the top beside B
12. Repeat the above steps for all your data
MFM 1P - Grade 9 Applied Mathematics – Unit 3: Lines and Curves of Best Fit (DPCDSB Dec 2008)
3-17
3.3.2c: Introduction to Calculator TI-Nspire CASÆ Creating a
Line of Best Fit Method (Continued)
Graphing the Data
1. Press Home
2. Choose 5:Data and Statistics
3. A blank graph will be displayed. You will need to specify the independent and dependent
variables.
4. Move the cursor to the bottom of the screen (Click to add variable) and click the Mouse
Button and choose the independent variable (Armspan) from the list. Press Enter.
5. Move the cursor to the left side of the screen (Click to add variable) and click the Mouse
Button and choose the dependent variable (Height) from the list. Press Enter.
6. Repeat this section for all other Height vs _______________ graphs.
To move between the different graphs – Press Ctrl + the mouse arrow
Drawing a line of Best Fit – Manually
1. Press Menu
2. Choose 3:Actions
3: Choose Add Movable Line
4. A line will now appear
pointing to this edge, the cursor changes to this symbol
pointing to the centre, the cursor changes to this symbol
pointing to this edge, the cursor changes to this symbol
To move your line
– point to the part of the line you wish to move
- press the mouse button down until the cursor changes to the closed hand and
stays that way.
- move your line using the cursor keys
- press Enter when your line becomes the line of best fit
(Hint: You may need to move all 3 parts of the line)
NOTE: An equation appeared with your line. Point at the equation and move the equation to
the bottom to see it.
MFM 1P - Grade 9 Applied Mathematics – Unit 3: Lines and Curves of Best Fit (DPCDSB Dec 2008)
3-18
3.3.2c: Introduction to Calculator TI-Nspire CASÆ Creating a
Line of Best Fit Method (Continued)
Draw a line of Best Fit – Regression Linear
1.
2.
3:
4.
Press Menu
Choose 3: Actions
Choose Regression
Choose 1: Show Linear (mx + b)
Compare your line with the Linear Regression
Place lines on each graph.
MFM 1P - Grade 9 Applied Mathematics – Unit 3: Lines and Curves of Best Fit (DPCDSB Dec 2008)
3-19
3.3.3: Choosing the Best Model - Could I Be a Forensic
Scientist?
Use the graph to examine each pair of variables from your data set. Sketch or print your scatter
plots and place your graphs in the boxes below and describe the correlation.
Height vs. Length of Forearm
Height vs. Armspan
Height vs. Handspan
Height vs. Foot Length
MFM 1P - Grade 9 Applied Mathematics – Unit 3: Lines and Curves of Best Fit (DPCDSB Dec 2008)
3-20
3.3.3: Choosing the Best Model - Could I Be a Forensic
Scientist? (continued)
Inferring and Concluding
1. Describe the relationships in the graphs.
2. Which relationship shows the strongest correlation?
3. Do any of the graphs show no relationship?
4. Which model is the best predictor of the height? Give reasons for your answers.
5. What is the height of the skeleton? Give evidence to support your answer.
MFM 1P - Grade 9 Applied Mathematics – Unit 3: Lines and Curves of Best Fit (DPCDSB Dec 2008)
3-21
3.3.P - Practice
1.
2.
3.
List two variables that will show a positive correlation.
Eg.
As the population of a city increases, garbage increases
i.
_________________________
ii.
_________________________
List two variables that will show a negative correlation.
Eg.
As the temperature increases, the amount of snow decreases
i.
_________________________
ii.
_________________________
List two variables that will show no correlation.
i.
_________________________
ii.
_________________________
MFM 1P - Grade 9 Applied Mathematics – Unit 3: Lines and Curves of Best Fit (DPCDSB Dec 2008)
3-22
3.4.1: Creating Scatter Plots and Lines of Best Fit
Test the hypothesis: The older you are, the more money you earn.
Plot the data on the scatter plot below, choosing appropriate scales and labels.
Age
Earnings ($)
25
22000
30
26500
35
29500
37
29000
38
30000
40
32000
41
35000
45
36000
55
41000
60
41000
62
42500
65
43000
70
37000
75
37500
Note: The symbol _______ is used to signal a “break” in the axis when the scale does not start
at zero to avoid a large empty space in one corner of the graph.
1) Draw a line of best fit. Describe the trend in the data.
2) Does the data support the hypothesis? Give reasons to support your answer.
(Refer to the scatter plot.)
3) Explain why the data for ages over 65 do not correspond with the hypothesis.
4) Explain what the point (41, 35000) represents.
MFM 1P - Grade 9 Applied Mathematics – Unit 3: Lines and Curves of Best Fit (DPCDSB Dec 2008)
3-23
3.4.3: Forensic Analysis
Anthropologists and forensic scientists use data to determine information about people.
Scientists can make predictions about the height, age, and sex of the person they are
examining by looking for relationships in large amounts of data.
1. Construct a graph of the length of the humerus bone vs. the length of the radius.
Length of
Radius
(cm)
25.0
22.0
23.5
22.5
23.0
22.6
21.4
21.9
23.5
24.3
24.0
Length of
Humerus
(cm)
29.7
26.5
27.1
26.0
28.0
25.2
24.0
23.8
26.7
29.0
27.0
2. Circle the point on the graph that represents the data for a radius that is 21.9 cm long.
How long is the humerus? _____________.
3. Put a box around the point on the graph that represents the data for a humerus that
is 27.1 cm long. How long is the radius? ______________.
4. Describe the trend.
5. Describe the relationship: As the length of the radius gets longer, the humerus
____________________________.
6. a) Draw a line of best fit.
b) Use the line of best fit to predict the length of the humerus, if the radius is 24.5 cm long.
Did you interpolate or extrapolate?
c) Use the line of best fit to predict the length of the radius, if the humerus is 25 cm long.
Did you interpolate or extrapolate?
MFM 1P - Grade 9 Applied Mathematics – Unit 3: Lines and Curves of Best Fit (DPCDSB Dec 2008)
3-24
3.4.J: Journal Activity
Write a response to:
What have you learned about how scientists use data in their jobs?
MFM 1P - Grade 9 Applied Mathematics – Unit 3: Lines and Curves of Best Fit (DPCDSB Dec 2008)
3-25
3.5.1: Investigations
Investigation 1 – Ball Bounce
Purpose
To determine if there is a relationship between the drop height of a ball and its
rebound height.
Hypothesis
I think that as the drop height increases, the rebound height ________________ because…
____________________________________________________________________________
Procedure
•
•
•
Attach measuring tape to the wall so you can measure the heights.
Drop the ball from various heights and record the rebound height.
Always drop the ball so that the bottom of the ball is just over the drop height.
Models
Conclusion
•
Describe the relationship in your own words
•
Was your hypothesis correct?
•
Describe any factors that may have affected your results.
MFM 1P - Grade 9 Applied Mathematics – Unit 3: Lines and Curves of Best Fit (DPCDSB Dec 2008)
3-26
3.5.1: Investigations (Continued)
Going Further: Investigation 1 – Ball Bounce
Answer the questions for your investigation.
1.
Use your graph to determine the rebound height if the ball was dropped from a height of
120 cm.
2.
How high was the ball dropped from if it rebounded 40 cm?
3.
Sketch the line of best fit for your ball on the graph below. Sketch and label a new line
representing a:
a)
super ball (more bouncy)
b)
beach ball (less bouncy)
Give reasons for your answers.
MFM 1P - Grade 9 Applied Mathematics – Unit 3: Lines and Curves of Best Fit (DPCDSB Dec 2008)
3-27
3.5.1: Investigations (continued)
Investigation 2 – Pendulum Swing
Purpose
To determine if there is a relationship between the mass of the swinging object and
the time it takes for it to make five complete swings.
Hypothesis
I think that as the mass increases, the time to complete five swings will _____________
because…
____________________________________________________________________________
Procedure
•
•
•
•
Attach one weight (paper clip) to end of the pendulum string.
Release the pendulum from a 35° angle and start the timer.
Measure and record the length of time for five complete swings.
Repeat, after increasing the mass at the end of the pendulum by one paper clip each time.
Models
Conclusion
•
Describe the relationship in your own words
•
Was your hypothesis correct?
•
Describe any factors that may have affected your results.
MFM 1P - Grade 9 Applied Mathematics – Unit 3: Lines and Curves of Best Fit (DPCDSB Dec 2008)
3-28
3.5.1: Investigations (Continued)
Going Further: Investigation 2 – Pendulum Swing
Answer the questions for your investigation.
1.
What do you notice that is different about this graph than the others?
2.
Use your graph to find the length of time it takes if 8 paper clips are used.
MFM 1P - Grade 9 Applied Mathematics – Unit 3: Lines and Curves of Best Fit (DPCDSB Dec 2008)
3-29
3.5.1: Investigations (continued)
Investigation 3 – Cylinder Size
Purpose
To determine if there is a relationship between the height of various cylindrical containers and
their diameter.
Hypothesis
I think that as the height increases, the diameter ___________________ because…
____________________________________________________________________________
Procedure
Measure and record the height and the diameter of the cylinders.
Models
Conclusion
•
Describe the relationship in your own words
•
Was your hypothesis correct?
•
Describe any factors that may have affected your results.
•
Going Further: Do you think there would be a relationship between the diameter and the
circumference of the cylinder? Explain.
MFM 1P - Grade 9 Applied Mathematics – Unit 3: Lines and Curves of Best Fit (DPCDSB Dec 2008)
3-30
MFM 1P - Grade 9 Applied Mathematics – Unit 3: Lines and Curves of Best Fit (DPCDSB Dec 2008)
3-31
3.5.1: Investigations (continued)
Investigation 4 – Bag Stretch
Purpose
To determine if there is a relationship between the height of a bag suspended
by elastics over the floor and the number of books in the bag.
Hypothesis
I think that as the number of books increases, the distance of the bag from the floor
___________________because…________________________________________________
Procedure
•
•
•
•
Hang the shopping bag from elastics so that the bottom is about 1 m above the floor.
Measure and record the distance from the bottom of the bag to the floor.
Add one book to the bag. Measure and record the distance from the bottom of the bag
to the floor.
Repeat, adding one book at a time until all the books are in the bag.
Models
Conclusion
•
Describe the relationship in your own words
•
Was your hypothesis correct?
•
Describe any factors that may have affected your results.
MFM 1P - Grade 9 Applied Mathematics – Unit 3: Lines and Curves of Best Fit (DPCDSB Dec 2008)
3-32
3.5.1: Investigations (Continued)
Going Further: Investigation 4 – Bag Stretch
Answer the questions for your investigation.
1.
How many books will it take for the bag to touch the floor?
2.
If a stretchier rubber band was used, how would this affect the graph? Sketch the line of
best fit for your investigation on the graph below. Sketch and label a new line
representing a stretchier rubber band.
Give reasons for your answer.
MFM 1P - Grade 9 Applied Mathematics – Unit 3: Lines and Curves of Best Fit (DPCDSB Dec 2008)
3-33
3.5.1: Investigations (continued)
Investigation 5 – Water Drains
Purpose
To determine if there is a relationship between the height of water in a container and the time it
takes to drain the container.
Hypothesis
I think that as time increases, the height of the water in the container ___________________
because…_______________________________________________
Procedure
•
•
•
•
Fill the bottle with water up to the point where the container slopes towards the top.
Record the height of the water in the bottle.
When your timer is ready, poke a hole in the bottom and top of the bottle.
Record the height of water every 20 seconds until the water is past the cylindrical part of the
bottle.
Models
Conclusion
•
Describe the relationship in your own words
•
Was your hypothesis correct?
•
Describe any factors that may have affected your results.
MFM 1P - Grade 9 Applied Mathematics – Unit 3: Lines and Curves of Best Fit (DPCDSB Dec 2008)
3-34
3.5.1: Investigations (Continued)
Going Further: Investigation 5 – Water Drains
Answer the questions for your investigation.
1.
How long would it take for the water to drain completely from the bottle?
2.
Estimate the height of the water after 30 seconds.
3.
If the container had a larger diameter, but was still the same height, how would this
affect the graph if the hole in the bottom stayed the same? Sketch the line or curve of
best fit for your investigation on the graph below. Sketch and label a new line or curve
representing a container with a larger diameter.
Give reasons for your answer.
MFM 1P - Grade 9 Applied Mathematics – Unit 3: Lines and Curves of Best Fit (DPCDSB Dec 2008)
3-35
3.7.1: First Differences
Problem 1
A. Jody works at a factory that produces square tiles for bathrooms and kitchens. She helps
determine shipping costs by calculating the perimeter of each tile.
i)
Calculate the perimeter and record your answers in the Perimeter column of the table.
Side Length
(cm)
Perimeter
(cm)
First Differences
1
2
3
4
5
ii) Describe what happens to the perimeter of each tile when the side length increases by one
centimetre. _______________________________________________________________
iii) Construct a graph of the perimeter vs. the side length. Include labels and titles.
a) Which variable is the independent variable?
b) Which variable is the dependent variable?
c) Use the graph to describe the relationship
between the perimeter and side length of a tile.
d) Describe the shape of the graph.
iv) Calculate the first differences in the First
Differences column of the table. What do you
notice about the first differences?
v) Summarize your observations.
a) When the side length increases by one
centimetre, the perimeter increases by ________.
b) The plotted points suggest a…
c) The first differences are…
MFM 1P - Grade 9 Applied Mathematics – Unit 3: Lines and Curves of Best Fit (DPCDSB Dec 2008)
3-36
3.7.1: First Differences (continued)
B. Jody is paid $8.50/hour to calculate perimeters.
i)
Calculate her pay and record your answers in the Pay column of the table.
Number of
Hours
Pay ($)
First Differences
1
2
3
4
5
ii) Describe what happens to her pay when the number of hours she works increases by one
hour. ___________________________________________________________________
iii) Construct a graph of her pay vs. the number of hours she works. Include labels and titles.
a) Which variable is the independent variable?
b) Which variable is the dependent variable?
c) Use the graph to describe the relationship
between her pay and the number of hours
she works.
d) Describe the shape of the graph.
iv) Calculate the first differences in the First
Differences column of the table. What do you
notice about the first differences?
v) Summarize your observations.
a) When the number of hours worked increases
by one, the pay increases by _________.
b) The plotted points suggest a…
c) The first differences are…
MFM 1P - Grade 9 Applied Mathematics – Unit 3: Lines and Curves of Best Fit (DPCDSB Dec 2008)
3-37
3.7.1: First Differences (continued)
C. Raj, another employee at the factory, also works with the tiles. He helps to determine the
shipping costs by calculating the area of each tile.
i) Calculate the area and record your answers in the Area column of the table.
Length of
sides (cm)
Area (cm2)
First Differences
1
2
3
4
5
ii) Describe what happens to the area of each tile when the side length of a tile increases by
one centimetre. ____________________________________________________________
iii) Construct a graph of the area vs. the length of the
sides of the tiles. Include labels and titles.
a) Which variable is the independent variable?
b) Which variable is the dependent variable?
c) Use the graph to describe the relationship
between the area and the side length of the tile.
d) Describe the shape of the graph.
iv) Calculate the first differences in the First
Differences column of the table. What do you
notice about the first differences?
v) Summarize your observations.
a) When the side length increases by one centimetre,
the area increases by __________________.
b) The plotted points suggest a…
c) The first differences are…
MFM 1P - Grade 9 Applied Mathematics – Unit 3: Lines and Curves of Best Fit (DPCDSB Dec 2008)
3-38
3.7.1: First Differences (continued)
Problem 2
Chuck works on commission for sales. He earns $12.00 for each of the first 3 boxes he sells.
He earns $24.00 each for boxes 4, 5, and 6, and $36.00 each for selling boxes 7, 8, 9, and 10.
i)
Calculate Chuck’s earnings for the following numbers of boxes of files and record your
answers in the Earnings column of the table.
Number of
Boxes
Earnings ($)
First Differences
1
2
3
4
5
6
7
8
9
10
ii) Describe what happens to his earnings when the number of boxes he sells increases
by one box.
MFM 1P - Grade 9 Applied Mathematics – Unit 3: Lines and Curves of Best Fit (DPCDSB Dec 2008)
3-39
3.7.1: First Differences (continued)
iii) Construct a graph of his earnings vs. the number
of boxes he sells. Include labels and titles.
a) Which variable is the independent variable?
b) Which variable is the dependent variable?
c) Use the graph to describe the relationship
between his earnings and the number of
boxes he sells.
d) Describe the shape of the graph.
iv) Calculate the first differences in the third column of the table. What do you notice about the
first differences?
v) Summarize your observations.
a) When the number of boxes he sells increases by one box, his earnings increase by…
b) The plotted points suggest a…
c) The first differences are…
MFM 1P - Grade 9 Applied Mathematics – Unit 3: Lines and Curves of Best Fit (DPCDSB Dec 2008)
3-40
3.7.2: Using What You Have Discovered
Deep Sea Divers
The table below shows data collected as divers descend below sea level. Calculate the first
differences. Use the first differences to determine if the relationship is linear or non-linear.
Check your solution by graphing. Include labels and titles.
Time
(min)
Depth
(m)
0
-2
1
-4
2
-6
3
-8
4
-10
First
Differences
The relationship is:
Hot Air Ballooning
The table shows data collected as a hot air balloon leaves the ground. Calculate the first
differences. Use the first differences to determine if the relationship is linear or non-linear.
Check your solution by graphing. Include labels and titles.
Time
(sec)
Height
(m)
0
2
1
4
2
6
3
8
4
10
First
Differences
The relationship is:
MFM 1P - Grade 9 Applied Mathematics – Unit 3: Lines and Curves of Best Fit (DPCDSB Dec 2008)
3-41
3.8: Unit 3 Review
Scatter Plots
1. David went for a bike ride. The table shows his distance from home at different times.
a)Graph
the data. Plot Time (min) along the horizontal axis and Distance (km) along the vertical axis.
b)How
far from home was David after each time?
c)After
how many minutes was David each distance from home? i) 4 km
i)
15 min _____
_____
ii)
55 min _____
ii)
13 km ______
2. The relationship between the Fahrenheit and Celsius scales can be seen in the table below.
a) Graph the data. Plot Temperature (°C) along the
horizontal axis and Temperature (F) along the y- axis.
b) Determine the temperature in °C for each of the
following:
i) 100°F ______ ii) 150°F _____
c) Determine the temperature in °F for each of the
following:
i) 100°C ______ ii) 150°C _____
Temperature (°C)
Temperature (F)
0
32
20
68
40
104
60
140
80
176
MFM 1P - Grade 9 Applied Mathematics – Unit 3: Lines and Curves of Best Fit (DPCDSB Dec 2008)
100
212
3-42
3.8: Unit 3 Review (Continued)
3. Below is a comparison of the number of oil changes in a year and the cost of auto repairs.
Oil Changes
Annual Repair Cost
(per year)
($)
0
5
5
7
5
8
6
8
5
0
1
10
3
2
1
3
850
310
270
125
400
110
150
95
300
1290
560
0
400
650
750
450
a) Create a scatter plot. Plot Oil Changes on the x-axis and Annual Repair Cost on the y-axis.
b) Draw a line of best fit.
c) Use your line of best fit to:
i.
Estimate the annual repair cost if someone had 2 oil changes. _________
ii.
Estimate the number of oil changes for an annual repair cost of $200. _________
4. The graph shows the number of people that visited the Long Island Aquatic Club outdoor
pool each day from July 14th to July 27th.
a) How many people were at the pool each day?
July 17th _________
July 23rd _________
b) On what day(s) were each number of people at the pool?
20 _________
45 _________
c) Which day had the highest attendance? _________
d) Which day had the lowest attendance? _________
MFM 1P - Grade 9 Applied Mathematics – Unit 3: Lines and Curves of Best Fit (DPCDSB Dec 2008)
3-43
3.8: Unit 3 Review (Continued)
5. The table shows, for 10 students, the number of hours their spent studying for their final
exam and the mark they received on the exam.
Study Time
(hours)
10
0
1
3
5
4
7
9
8
5
Exam Mark (%)
100
50
60
70
65
80
95
80
90
55
a) Create a scatter plot for this data. Plot Study Time on the horizontal axis and Exam Mark
on the vertical axis.
b) Draw a line of best fit.
c) Use your line of best fit to answer the questions:
i.
How many hours would someone have to study
to get a mark of 70%?
ii.
6.
What mark would someone get if they studied
for 6 hours?
The following scatter plot shows the number of bacteria living in a culture at various
temperatures.
a) Draw a line of best fit.
b) How many bacteria will live in a temperature of:
40 degrees Celsius? _________
80 degrees Celsuis? _________
c) Predict the temperature if:
8 bacteria are in the culture ________
14 bacteria are in the culture ________
MFM 1P - Grade 9 Applied Mathematics – Unit 3: Lines and Curves of Best Fit (DPCDSB Dec 2008)
3-44
3.8: Unit 3 Review (Continued)
7.
The following graph represents the long term trends in smoking by teenagers aged 15 – 19.
a) Draw the line of best fit.
b) What is the average number of cigarettes
smoked per day by 15-19 year old in 1989?
c) In what year were 15-19 year olds smoking
an average of 14 cigarettes/day?
8.
The table of values below shows life expectancy and year of birth.
Year of Birth
1900
1910
1920
1930
1940
1950
1960
1970
1980
1990
Life Expectancy
(years)
47.3
50.0
54.1
59.7
62.9
68.2
69.7
70.8
73.7
75.4
a) Create a scatter plot. Plot year of birth on the x-axis and life expectancy on the y-axis.
b) Draw a line of best fit.
c) What is the average life expectancy for
someone born in:
1945? _________
1965? _________
d) Predict the year of birth of someone with a life
expectancy of:
55 years. _________
65 years. _________
MFM 1P - Grade 9 Applied Mathematics – Unit 3: Lines and Curves of Best Fit (DPCDSB Dec 2008)
3-45
3.8: Unit 3 Review (Continued)
9. The table below shows the height of a certain type of tree over several years.
4
10
6
12 11
8
13 13 10
8
3
Age (years)
6
4
5
6
3
5
8
4
5
3
Height (metres) 2
6
3
a) Create a scatter plot.
Plot Age on the x-axis and Height on the
y-axis.
b) Draw a line of best fit.
c) What is the height of this tree after:
5 years? _________
9 years? _________
d) Predict the age of a tree that is:
1 m tall. _________
7 m tall. _________
10. The table shows the number of successful shots Alex made a various distances from the
basket.
Distance from
3
5
7
8
9
10
the basket (m)
Number of shots
22 17 16 10 8
3
made
a) Create a scatter plot.
Plot Distance on the x-axis and
Shots Made on the y-axis.
b) Draw a line of best fit.
c) Predict the number of shots Alex will make
from 4 m?
d) Predict Alex’s distance if she makes 15 shots.
MFM 1P - Grade 9 Applied Mathematics – Unit 3: Lines and Curves of Best Fit (DPCDSB Dec 2008)
3-46
3.W: Definition Page
Term
Picture / Sketch / Examples
Definition
Scatter Plot
Dependent Variable
Independent Variable
Continuous Data
Discrete Data
Variable
Linear
Correlation
Non-Linear
Finite Differences
First Differences
MFM 1P - Grade 9 Applied Mathematics – Unit 3: Lines and Curves of Best Fit (DPCDSB Dec 2008)
3-47
3.W: Definition Page (continued)
Term
Picture / Sketch / Examples
Definition
Trend
Interpolate
Extrapolate
Algebraic Model
Graphical Model
Numerical Model
Hypothesis
MFM 1P - Grade 9 Applied Mathematics – Unit 3: Lines and Curves of Best Fit (DPCDSB Dec 2008)
3-48
3.S Unit Summary Page
Create a Mind Map for the following graphing calculator process of:
•
•
•
•
entering data in lists
making a scatter plot
setting the window
making a line of best fit
MFM 1P - Grade 9 Applied Mathematics – Unit 3: Lines and Curves of Best Fit (DPCDSB Dec 2008)
3-49
3.R Reflecting on My Learning (3, 2, 1)
3 Things I know well from this unit
2 Things I need explained more
1 Question I still have
MFM 1P - Grade 9 Applied Mathematics – Unit 3: Lines and Curves of Best Fit (DPCDSB Dec 2008)
3-50
3.RLS: Reflecting on Learning Skills
Students should be aware of the importance that these skills have on your performance. After
receiving your marked assessment, answer the following questions. Be honest with yourself.
Good Learning Skills will help you now, in other courses and in the future.
•
•
•
•
E – Always
G – Sometimes
S – Need Improvement
N – Never
Organization
• E G S N
• E G S N
• E G S N
I came prepared for class with all materials
My work is submitted on time
I keep my notebook organized.
Work Habits
• E G S N
• E G S N
• E G S N
• E G S N
• E G S N
• E G S N
I attempt all of my homework
I use my class time efficiently
I limit my talking to the math topic on hand
I am on time
If I am away, I ask someone what I missed,
I complete the work from the day that I missed.
Team Work
• E G S N
• E G S N
• E G S N
I am an active participant in pairs/group work
I co-operate with others within my group
I respect the opinions of others
Initiative
• E G S
• E G S
• E G S
• E G S
I participate in class discussion/lessons
When I have difficulty I seek extra help
After I resolve my difficulties, I reattempt the problem
I review the daily lesson/ideas/concepts
N
N
N
N
Works Independently
• E G S N
I attempt the work on my own
• E G S N
I try before seeking help
• E G S N
If I have difficulties I ask others but I stay on task
• E G S N
I am committed to tasks at hand
Yes No
I know all the different ways available in my school, where I can seek extra help.
Yes No
I tried my best.
What will I do differently in the next unit to improve?
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
MFM 1P - Grade 9 Applied Mathematics – Unit 3: Lines and Curves of Best Fit (DPCDSB Dec 2008)
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MFM 1P - Grade 9 Applied Mathematics – Unit 3: Lines and Curves of Best Fit (DPCDSB Dec 2008)
3-52
Course: Grade 9 Applied Mathematics (MFM1P)
Unit 4:
Proportional Reasoning: Ratio, Rate and
Proportion
Unit 4
Proportional Reasoning: Ratio, Rate and Proportion
Section
4.1.2
4.1.3
4.1.4
4.1.J
4.2.1
4.2.2
4.2.J
4.3.1
4.3.2
4.3.P
4.4.1
4.4.2
4.4.3
4.4.4
4.4.P
4.5.1
4.5.2
4.5.P
4.5.J
4.6.2
4.6.P
4.W
4.S
4.R
4.RLS
Activity
Who Eats More? Worksheet
What’s in the Bag? Worksheet
Middle Mania (optional)
Journal Activity
Anticipation Guide
Growing Dilemma Investigation
Journal Activity
Television Viewing
Television Dimensions
Practice
Estimating Crowd Size
Techniques for Estimating Crowd Size
I’d Rather Be Scaling
More Scaling Problems
Practice
Elastic Meter and Percent
Types of Percent Problems
Practice
Journal Activity
Review Relay
Practice
Definitions
Unit Summary
Reflecting on My Learning (3, 2, 1)
Reflecting on Learning Skills
MFM 1P - Grade 9 Applied Mathematics – Unit 4: Proportional Reasoning (DPCDSB Dec 2008)
Page
3
6
7
9
10
11
16
17
18
19
21
22
26
27
29
30
31
33
36
37
39
43
45
46
47
4-2
4.1.2: Who Eats More? Worksheet
Task 1
Individually
Using the cards in Envelope 1:
a) Arrange them in order of which animal you believe eats more, from most to least.
Most
Least
In Pairs
b) Explain the reason why you placed the animals in this order.
Task 2
Pairs
Using the cards in Envelope 2:
a) Arrange them in order of which animal you believe eats more, from most to least.
Most
Least
b) Explain your reasons for this arrangement if it was different from the arrangement in Task 1.
MFM 1P - Grade 9 Applied Mathematics – Unit 4: Proportional Reasoning (DPCDSB Dec 2008)
4-3
4.1.2: Who Eats More? Worksheet (continued)
Task 3
Pairs
Using the cards in Envelope 3:
a) Arrange them in order of which animal you believe eats more, from most to least.
Most
Least
b) Explain your reasons for this arrangement if it was different from the arrangement in Task 2.
Task 4
Groups of 4
a) Explain the reasoning used in Task 3.
Using the cards from Envelope 3:
b) Arrange them in a different order of which animal you believe eats more, by using the data
in another way.
Most
Least
c) Explain your reasons for this arrangement.
MFM 1P - Grade 9 Applied Mathematics – Unit 4: Proportional Reasoning (DPCDSB Dec 2008)
4-4
4.1.2: Who Eats More? Worksheet (continued)
Task 5
Pairs
a) Compare the arrangements created in Task 3 and Task 4. Select the arrangement that you
believe best illustrates who eats more.
Task 3
Most
Least
Task 4
Most
Least
Justify your choice.
Task 6
Individually
a) Pick three other animals.
b) Predict their placement relative to the arrangement selected in Task 5.
Most
Least
c) Explain how you determined their placement.
d) Gather evidence to prove or disprove your prediction.
MFM 1P - Grade 9 Applied Mathematics – Unit 4: Proportional Reasoning (DPCDSB Dec 2008)
4-5
4.1.3: What’s in the Bag? Worksheet
Station: What’s in the Bag?
At your station you have a bag with two different-coloured tiles.
1. Without looking, pull a tile out of the bag. Make a tally mark in the appropriate column in the
table below.
2. Put the tile back into the bag and shake it up.
3. Repeat steps 1 and 2 a total of 20 times.
Colour 1:
Colour 2:
Tally
Total
4. What appears to be the ratio of colour 1 to colour 2 in your bag?
5. Answer the following questions using the information you have collected.
Justify your answers.
a) If you had 30 of colour 1 in your bag, how many of colour 2 would you expect to have?
b) If you had 20 of colour 2 in your bag, how many of colour 1 would you expect to have?
c) If you had a total of 80 tiles in your bag, how many of each colour would you expect to
have?
d) If you had 40 of colour 1 in your bag, how many tiles in total would you expect to have?
MFM 1P - Grade 9 Applied Mathematics – Unit 4: Proportional Reasoning (DPCDSB Dec 2008)
4-6
4.1.4: Middle Mania Worksheet - Optional
Station: GSP®4 Middle Mania
Launch GSP®4 Middle Mania on the computer and you should see the following. Follow
the instructions on the screen and complete the worksheet below.
Midpoint Trianlges
Midpoint Segments
MFM 1P - Grade 9 Applied Mathematics – Unit 4: Proportional Reasoning (DPCDSB Dec 2008)
4-7
4.1.4: Middle Mania Worksheet – Optional (continued)
1. Follow the Midpoint Triangle instructions and complete the following chart.
Area Δ ABC
24.98 cm²
Area Δ DEF
6.25 cm²
Ratio ABC/DEF
4.00:1
2. What do you notice about the ratio of the areas?
3. If Area ΔABC = 64 cm², what is the area of ΔDEF? Explain.
4. If Area ΔDEF = 15 cm², what is the area of ΔABC? Explain.
5. Follow the Midpoint Segments instructions and complete the following chart.
BC
DE
Ratio BC/DE
6. What do you notice about the ratio of the length of the line segments?
7. If the length of BC = 17 cm, what is the length of DE? Explain.
8. If length of DE = 22 cm, what is the length of BC? Explain.
MFM 1P - Grade 9 Applied Mathematics – Unit 4: Proportional Reasoning (DPCDSB Dec 2008)
4-8
4.1.J: Journal Activity
List four examples of ratio, rate, and unit rate from your environment.
Example
Type
(Ratio, Rate, Unit Rate)
1.
2.
3.
4.
MFM 1P - Grade 9 Applied Mathematics – Unit 4: Proportional Reasoning (DPCDSB Dec 2008)
4-9
4.2.1: Anticipation Guide
Instructions
•
Check Agree or Disagree, in ink, in the Before category beside each statement before you
start the Growing Dilemma task.
•
Compare your choice with your partner.
•
Revisit your choices at the end of the investigation.
Before
Agree
Disagree
Statement
After
Agree
Disagree
1. If you double the length of a square,
then the perimeter also doubles.
2. If you double the length of a square,
then the area also doubles.
3. If you double the length of a square,
then the length of the diagonal also
doubles.
4. If you double the sides of a cube, then
the volume also doubles.
MFM 1P - Grade 9 Applied Mathematics – Unit 4: Proportional Reasoning (DPCDSB Dec 2008)
4-10
4.2.2: Growing Dilemma Investigation
Investigation 1: Perimeter Ratios
Use the colour tiles to create squares with the indicated side length.
1. Determine the perimeter for each side length.
2. Complete the chart.
3. Graph Perimeter vs. Side Length on the grid provided.
Side
Length
(S)
Perimeter
(P)
First
Differences
Ratio
(S:P)
Ratio in
Lowest
Terms
1
2
3
4
5
4. State the characteristics of this relationship:
a) first differences
b) ratios
c) graph
MFM 1P - Grade 9 Applied Mathematics – Unit 4: Proportional Reasoning (DPCDSB Dec 2008)
4-11
4.2.2: Growing Dilemma Investigation (continued)
Investigation 2: Area Ratios
Use the colour tiles to create squares with the indicated side length.
1. Determine the area for each side length.
2. Complete the chart.
3. Graph Area vs. Side Length on the grid provided.
Side
Length
(S)
Area
(A)
First
Differences
Ratio
(S:A)
Ratio in
Lowest
Terms
1
2
3
4
5
4. State 3 characteristics of this relationship:
a) first differences
b) ratios
c) graph
MFM 1P - Grade 9 Applied Mathematics – Unit 4: Proportional Reasoning (DPCDSB Dec 2008)
4-12
4.2.2: Growing Dilemma Investigation (continued)
Investigation 3: Diagonal Length Ratios
Use the colour tiles to create squares with the indicated side length.
1. Determine the length of the diagonal for each side length. Use Pythagorean Theorem.
a 2 + b2 = c 2
2. Complete the chart.
3. Graph Diagonal Length vs. Side Length on the grid provided.
Side
Length
(S)
Diagonal
(D)
First
Differences
Ratio
(S:D)
Ratio in
Lowest
Terms
1
2
3
4
5
4. State 3 characteristics of this relationship:
a) first differences
b) ratios
c) graph
MFM 1P - Grade 9 Applied Mathematics – Unit 4: Proportional Reasoning (DPCDSB Dec 2008)
4-13
4.2.2: Growing Dilemma Investigation (continued)
Investigation 4: Volume Ratios
Use the linking cubes or tiles to create cubes with the indicated side length.
1. Determine the volume of the cube for each side length.
2. Complete the chart.
3. Graph Volume vs. Side Length on the grid provided.
Side
Length
(S)
Volume
(V)
First
Differences
Ratio
(S:V)
Ratio in
Lowest
Terms
1
2
3
4
5
4. State 3 characteristics of this relationship:
a) first differences
b) ratios
c) graph
MFM 1P - Grade 9 Applied Mathematics – Unit 4: Proportional Reasoning (DPCDSB Dec 2008)
4-14
4.2.2: Growing Dilemma Investigation (continued)
A proportion is a statement of two equal ratios.
Conclusion
a) Which of the 4 relationships that you have investigated are proportional?
b) What else can you conclude about relationships that are proportional?
MFM 1P - Grade 9 Applied Mathematics – Unit 4: Proportional Reasoning (DPCDSB Dec 2008)
4-15
4.2.J: Journal Activity
1. Give a personal example of proportional reasoning.
________________________________________________________________
________________________________________________________________
________________________________________________________________
________________________________________________________________
2. Using the scenarios below, check for proportionality and justify your response.
(a)
You are paid an hourly wage. If you work 3 times the number of hours, does your
pay triple?
________________________________________________________________
________________________________________________________________
________________________________________________________________
________________________________________________________________
(b)
Student council raffle tickets cost $0.50/each or 3 for $1. If you buy twice as many
tickets, does your cost double?
________________________________________________________________
________________________________________________________________
________________________________________________________________
________________________________________________________________
MFM 1P - Grade 9 Applied Mathematics – Unit 4: Proportional Reasoning (DPCDSB Dec 2008)
4-16
4.3.1: Television Viewing
Use a different method to complete each part of the question.
You should be prepared to explain your methods to the class.
Did you know that there is an optimal distance for a person to be from a television for ideal
viewing?
The ratio of the size of the television screen to the distance a person should sit from it
is 1:6.
a) How far away should a person sit from a 20-inch television?
b) If the room is 17 feet long, can a person sit at an optimal distance from a 27-inch television?
Explain your reasoning.
c) What is the largest television that can be used in the 17-foot room for a person to sit an
optimal distance from it?
MFM 1P - Grade 9 Applied Mathematics – Unit 4: Proportional Reasoning (DPCDSB Dec 2008)
4-17
4.3.2: Television Dimensions
3
Basic Television Information
4
•
Traditional televisions have a ratio of width to height of 4:3.
•
High definition televisions (HDTV) have a ratio of width to height of 16:9.
•
Television sizes are given as the length of the diagonal of the screen, i.e., a 27-inch
television is 27 inches from one corner to the diagonally opposite corner.
9
16
Problem 1
Darren wants to buy a new television. He finds a traditional television at the store and measures
the width of it to make sure it fits in his home. He measures the width to be 24 inches but he
forgets to measure the height and the diagonal.
a) Draw a diagram.
b) What is the height of the television?
c) What is the size of the television? (the length of the diagonal)
Problem 2
Sasha is buying a new HDTV. She finds one and measures the width to be about 35 inches.
a) Draw a diagram.
b) What is the height of the television?
b) What is the size of the television?
c) What is the optimal viewing distance for Sasha’s new HDTV?
MFM 1P - Grade 9 Applied Mathematics – Unit 4: Proportional Reasoning (DPCDSB Dec 2008)
4-18
4.3.P: Practice
1. For safety reasons, a wheelchair ramp must be 1 m high for every 12 metres in
horizontal length.
The ratio of the height of the ramp to the length of the ramp is 1:12.
Draw a diagram.
(a) What is the horizontal length of a ramp that is 2 m tall?
(b) A ramp has a height of 2.6 m and a sloping length of 30 m. Is this wheelchair
ramp safe?
(c) Another wheel chair ramp is being built. It must be 4.8m in horizontal length.
Determine the sloping length of this new ramp.
MFM 1P - Grade 9 Applied Mathematics – Unit 4: Proportional Reasoning (DPCDSB Dec 2008)
4-19
4.3.P: Practice (continued)
2. A ladder is leaning against a house. To be safe the ratio of the ladder’s length to the
distance of the ladder’s base from the house must be 5:3.
The ratio of the size of the ladder to the distance from the house is 5:3.
Draw a diagram.
(a) Determine how far the base of a 6.0 m ladder is from the house if it is being used
safely.
(b) How high up a wall does a 4.5 m ladder reach if it is being used safely?
(c) Is a 5.6m ladder is being used safely if its base is 3.3 m from the wall? Explain.
MFM 1P - Grade 9 Applied Mathematics – Unit 4: Proportional Reasoning (DPCDSB Dec 2008)
4-20
4.4.1 Estimating Crowd Size
Exercise 1: Estimating the Size of a Crowd from an Aerial Diagram
Aerial Diagram of a political rally
1. Using the diagram above, choose a section to count the number of people; circle and
label this section. Complete the following table
Location (code)
Number of people
Total
2. How can you use the chart to estimate the total number of people at the rally?
MFM 1P - Grade 9 Applied Mathematics – Unit 4: Proportional Reasoning (DPCDSB Dec 2008)
4-21
4.4.2 Techniques for Estimating Crowd Size
Caribou on Amethyst Island
Each year a fall caribou hunt is planned for Amethyst Island. It is important to determine how
many caribou live on the island before the Ministry of Natural Resources (MNR) issues hunting
licenses. The diagram below was made from the air of the herd on Amethyst Island. Determine
the number of caribou that are on Amethyst Island.
Scale :
Each square
represents 1km2
1. Find the total area of the grid above.
2. Complete the following chart by counting the caribou five boxes.
Location (code)
Number of caribou
Total
MFM 1P - Grade 9 Applied Mathematics – Unit 4: Proportional Reasoning (DPCDSB Dec 2008)
4-22
4.4.2 Techniques for Estimating Crowd Size (Continued)
total
3. a) Average number of caribou per square =
number of squares counted
b) Estimate the total number of caribou on the island.
Estimated number of caribou = average number of caribou per square x total number of squares
4. If the MNR has decided that one out of every six caribou can be hunted this year and each
hunter can only take one caribou, how many hunting licenses should be issued?
5. If the MNR has decided that 2 out of every 9 caribou can be hunted this year and each
hunter can only take one caribou, how many hunting licenses should be issued?
MFM 1P - Grade 9 Applied Mathematics – Unit 4: Proportional Reasoning (DPCDSB Dec 2008)
4-23
4.4.2 Techniques for Estimating Crowd Size (Continued)
Manitouwadge Lake Sailing Regatta
A sailing regatta is a sailing competition. Competitive sailing is a sport with small teams of eight
per sailing vessel and boats 32-48 foot in length. After an initial sailing skills training, the teams
will sail their boats in a series of sprint training races to practice their skills in preparation for the
final regatta. Below is an aerial diagram of the Manitouwadge Lake Sailing Regatta.
Each small boat (no mast)
has two people on board.
Each large boat (with mast)
has six people on board.
1. Using the diagram complete the following chart.
Grid
Square
Number
Number of
small boats
# of people
on small
boats(2
people/boat)
Number of
large boats
# of people
on large
boats (6
people/boat)
MFM 1P - Grade 9 Applied Mathematics – Unit 4: Proportional Reasoning (DPCDSB Dec 2008)
total # of
people in
the grid
square
4-24
4.4.2 Techniques for Estimating Crowd Size (Continued)
1. What is the average number of people per grid square?
2. The Manitouwadge Lake Sailing Association charges $2.00 per person attending the
regatta. Based on the diagram how much money have they earned? Show your work.
3. The association is thinking of changing the fee to $5.00 per small boat and $10 per large
boat. If they made these changes for this year’s competition, how much could have been
earned? Show your work.
4. Should The Manitouwadge Lake Sailing Association base their fees per person or per boat?
Justify your answer.
MFM 1P - Grade 9 Applied Mathematics – Unit 4: Proportional Reasoning (DPCDSB Dec 2008)
4-25
4.4.3: I'd Rather Be Scaling
MFM 1P - Grade 9 Applied Mathematics – Unit 4: Proportional Reasoning (DPCDSB Dec 2008)
4-26
4.4.4: More Scaling Problems
1. Dandelion Inquiry
In testing a new product for its effectiveness in killing dandelions,
it is necessary to find an area containing many dandelions,
count them, apply the product, and count the dandelions
again at a later time. How might this be accomplished
without counting every single dandelion? Design a
technique different from the one used in class.
2. Interpreting Scale Diagrams
Recall that scale = diagram measurement : actual measurement
Usa a ruler to measure the line.
a) Finding the scale
The actual length of this cell is 0.32 mm across.
What scale was used to draw this diagram?
b) Using the scale
This diagram was drawn using a scale of 1:7.
What is the actual height of this penguin?
MFM 1P - Grade 9 Applied Mathematics – Unit 4: Proportional Reasoning (DPCDSB Dec 2008)
4-27
4.4.4: More Scaling Problems (continued)
3.
Complete the table.
i.
Scale
Diagram : Actual
1:400
ii.
12000:1
iii.
iv.
Diagram Measurement
6 cm
0.00375 mm
7.2 cm
1:250000
Actual Measurement
0.6 mm
8 cm
4.
The prices of a baseball glove and a tennis racket are in the ratio 7:12. If the price of the
racket is $62.40, determine the price of the glove.
5.
A particular mortar mix contains cement, water, and sand in the ratio 2:1:6. How much
cement and water should be in a batch of mortar containing 11.4 kg of sand?
6.
A picture of an ant is given below, determine the scale if its actual length is
2.4 mm
58 mm
MFM 1P - Grade 9 Applied Mathematics – Unit 4: Proportional Reasoning (DPCDSB Dec 2008)
4-28
4.4.P: Practice
1.
A Canadian football field is 110 yards long and 65 yards wide. Often the end zones
extend another 20 yards beyond the goal posts at each end of the field. Draw a scale
diagram of this playing field and the end zones using a scale of 1 square = 20 yards.
2.
The wheelbase of a vehicle is the distance between the front and back axles. Determine
the actual wheelbase of the vehicle in this scale drawing.
2.9 cm
Scale 1:50
MFM 1P - Grade 9 Applied Mathematics – Unit 4: Proportional Reasoning (DPCDSB Dec 2008)
4-29
elastic
4.5.1: Elastic Meter and Percent
Part A: Make the elastic meter
1. Take a piece of elastic 32 cm long. Mark a line at 1 cm from one end.
From this point make 10 marks every 3 cm. There will be 1 cm left.
(The centimetre at each end of the elastic provides a way to hold and stretch the
elastic ruler.)
2. On the first line write 10%; 2nd line, 20%; 3rd line, 30% (…up to 100%).
Part B: Use the elastic meter
3. Estimate from the bottom to the top where 60% of the right edge of your desk would be.
Put a very small pencil mark here. (Please erase it after the experiment.)
4. Stretch out the elastic meter from the bottom to the top, with 0% at the floor, and 100%
at the desk surface.
Use the 60% mark on the elastic meter to correct your estimate.
5. Use a measuring tape to measure this length. Record it in the appropriate place in the
following chart.
Percent %
0%
10%
33%
45%
50%
60%
75%
90%
100%
20%
10%
Measure (cm)
Use your elastic
meter to complete
the chart.
6. Graph your data on the grid below. Be sure to label your axes. Choose an appropriate scale.
7. On your graph draw a line of best fit.
Interpolate: Use line of best fit to estimate the lengths of the following percents:
a) 85%
b) 65%
c) 43%
d) 58%
Extrapolate: Estimate the following lengths:
a) 120%
b) 135%
MFM 1P - Grade 9 Applied Mathematics – Unit 4: Proportional Reasoning (DPCDSB Dec 2008)
4-30
4.5.2: Types of Percent Problems – Guided Lesson
1. Determining an unknown part:
a) Do together: HDTVs are on sale for 25% off. What is the discount on a television that
normally costs $885?
•
set up and solve a ratio
•
solve an equation 0.25 × 885 =
b) Do on your own: If you purchase a CD for $18.99, how much tax would you pay?
(both GST and PST)
2. Determining an unknown percent:
a) Do together: Shuva purchased a new MP3 player on sale. It was $219.50 originally, but
she paid $142.68, not including tax. What was the percent discount on the MP3 player?
•
set up and solve a ratio
•
solve an equation
× 219.50 = (219.50 – 142.68)
b) Do on your own: David was shopping for a new pair of shoes. He found a pair that was
$89.99 on sale for $22.50 off. What was the percent discount on the shoes?
MFM 1P - Grade 9 Applied Mathematics – Unit 4: Proportional Reasoning (DPCDSB Dec 2008)
4-31
4.5.2: Types of Percent Problems – Guided Lesson (continued)
3. Determining the unknown whole:
a) Do together:
Cayla wanted to return a defective calculator, but her dog Buster had chewed up the
receipt. She could still see that the 13% tax came to $2.25. What was the cost of Cayla’s
calculator?
•
$2.25
set up and solve a ratio
•
b) Do on your own:
Himay was very happy because his new cell phone was on sale for 40% off and was
only $65.00. What was the original price of Himay’s phone?
MFM 1P - Grade 9 Applied Mathematics – Unit 4: Proportional Reasoning (DPCDSB Dec 2008)
4-32
4.5.P: Practice
1.
2.
Express each of the following as a percent.
(a)
12 out of 16
(b)
one fifth
(c)
14
35
(d)
80:50
Find the number for each of the following.
(a)
3.
25% of a number is 5.
(b)
120% of a number is 48.
Find the amount for each of the following.
(a)
15% of 125g
(b)
9% of 45 cm
MFM 1P - Grade 9 Applied Mathematics – Unit 4: Proportional Reasoning (DPCDSB Dec 2008)
4-33
4.5.P: Practice (continued)
4.
5.
A skateboard is priced at $92.00 and is reduced by 20%.
(a)
What is the amount of the discount on the skateboard?
(b)
Calculate the new price of the skateboard.
In 2004, the cost to join a gym was $169.00.
In 2005, the cost was 7% more.
How much did it cost to join the gym in 2005?
MFM 1P - Grade 9 Applied Mathematics – Unit 4: Proportional Reasoning (DPCDSB Dec 2008)
4-34
4.5.P: Practice (continued)
6.
On average, a girl will reach 90% of her final height by the time she is 11 years
old and 98% of her final height when she is 17 years old.
(a)
Beth is 11 years old. She is 150cm tall.
Estimate her height when she is 20 years old.
(b)
Lena is 17 years old. She is 176 cm tall.
Estimate her height when she is 30 years old.
(c)
Jodi is 35 years old. She is 165 cm tall.
(i)
Estimate her height at 11 years old.
(ii)
Estimate her height at 17 years old.
MFM 1P - Grade 9 Applied Mathematics – Unit 4: Proportional Reasoning (DPCDSB Dec 2008)
4-35
4.5.J: Journal Activity
In the space provided, create your own percent reference sheet, showing examples of
the various types of percent problems.
MFM 1P - Grade 9 Applied Mathematics – Unit 4: Proportional Reasoning (DPCDSB Dec 2008)
4-36
4.6.2: Review Relay
1. Reduce the ratios to lowest terms:
2. Calculate the following percents:
15:35 =
45% of 220 =
18
=
6
120% × 555 =
1.5% × 1400 =
144:72 =
3. The driving distance from Thunder Bay to
Vancouver is approximately 2500 km.
How long would it take you to drive from
Thunder Bay to Vancouver at
90 km/hour without making any stops?
4. If the ratio of the Canadian dollar to the US
dollar is $1.04:$1.00, how much Canadian
money is equivalent to US$250?
MFM 1P - Grade 9 Applied Mathematics – Unit 4: Proportional Reasoning (DPCDSB Dec 2008)
4-37
4.6.2: Review Relay (continued)
5. Measure the length indicated in
centimetres. What is the actual length of
the shark, in metres?
Scale Diagram
6. You want to purchase a new shirt that
costs $22.50.
a) How much tax will you have to pay
including GST and PST?
b) What is the total cost of your shirt?
1:70
7. You are shopping for DVDs at the video
store with a $30.00 gift certificate that you
received from a friend. You find a great
DVD that was $34.50 on sale for 25% off.
Do you have enough money to buy the
DVD including GST and PST?
8. You are working at Tecky Television
Sales. Recall that the HDTV’s width:height
ratio is 16:9. A customer wants to know:
a) If he has an entertainment centre that
has an opening that is 48 inches wide,
how high will the cabinet opening have
to be?
b) If the cabinet opening is 48 inches by
32 inches, will a 50-inch HDTV fit
inside?
MFM 1P - Grade 9 Applied Mathematics – Unit 4: Proportional Reasoning (DPCDSB Dec 2008)
4-38
4.6.P: Practice
1.
Express each of the following as a unit rate. Round answers to two decimal
places, if necessary, and state the units for each answer.
(a)
40 mm of rain in 4 h ______________________________
(b)
$6.49 for 24 cans of pop ___________________________
(c)
$58.00 for 8 h of work _____________________________
2.
A newborn child usually triples its birth weight in a year. If a baby weighed 3.35 kg
at birth, what is the baby likely to weigh on her first birthday?
3.
A single bus fare costs $2.10. A monthly bus pass costs $50.00. Katelyn estimates
that she will ride the bus 25 times this month. Boris estimates that he will ride the
bus 16 times. Should they each buy a monthly pass? Explain.
4.
The price for gold is usually given in US dollars per ounce. Find the cost in
Canadian dollars for an ounce of gold selling at US$559.00 when the exchange
rate is $1 USD = $1.17 CDN.
MFM 1P - Grade 9 Applied Mathematics – Unit 4: Proportional Reasoning (DPCDSB Dec 2008)
4-39
4.6.P: Practice (continued)
5.
Determine the better buy for each of these school supplies. Show your calculations.
(a) A box of 12 pens for $2.59 or a box of 15 pens for $3.35.
(b) 250 sheets of graph paper for $2.39 or 120 sheets for $1.15.
6. Kelly ran 8 laps of the track in 18 minutes. Jack ran 6 laps in 10 minutes. Who had the
greater average speed? Explain.
7.
Carl bought a football jersey with a regular price of $129.49. The jersey was on
sale for 30% off, and the taxes were 14%. Determine each amount.
(a)
the discount
(b)
the sale price
(c)
the taxes
(d)
the total amount Carl paid
MFM 1P - Grade 9 Applied Mathematics – Unit 4: Proportional Reasoning (DPCDSB Dec 2008)
4-40
4.6.P: Practice (continued)
8.
9.
Determine the missing value for each of the following.
(a)
64 = ______%
72
(b)
15% of _______ = 6
(c)
32% of 65 = ______
(d)
0.08% of 25 000 000 = ________
(e)
124 = ______%
96
(f)
135% of _____ is 108.
10. A stop sign has an actual width of 60 cm. Determine the scale of the diagram
below.
3.6 cm
MFM 1P - Grade 9 Applied Mathematics – Unit 4: Proportional Reasoning (DPCDSB Dec 2008)
4-41
4.6.P: Practice (continued)
11. The Canadian Flag has a width to height ratio of 2:1. On a 1:50 scale
drawing of a flag its width is 13.0 cm. What is the actual size of this flag?
Actual height:__________
Actual width:_____________
MFM 1P - Grade 9 Applied Mathematics – Unit 4: Proportional Reasoning (DPCDSB Dec 2008)
4-42
4.W: Definition Page
Term
Picture / Sketch /
Examples
Definition
Algebraic Reasoning
Constant of Proportionality
Cross Product
Equivalent Ratios
Fraction
Lowest Terms
Percent
Probability
Qualitative
Quantitative
Rate
MFM 1P - Grade 9 Applied Mathematics – Unit 4: Proportional Reasoning (DPCDSB Dec 2008)
4-43
4.W: Definition Page (continued)
Term
Picture / Sketch /
Examples
Definition
Ratio
Scale
Scale Diagram
Scaling
Unit Price
Unit Rate
MFM 1P - Grade 9 Applied Mathematics – Unit 4: Proportional Reasoning (DPCDSB Dec 2008)
4-44
4.S: Unit Summary Page
Rate
Percent
Ratio
Complete the concepts circles for Rate, Ratio and Percent.
MFM 1P - Grade 9 Applied Mathematics – Unit 4: Proportional Reasoning (DPCDSB Dec 2008)
4-45
4.R: Reflecting on My Learning (3, 2, 1)
3 Things I know well from this unit
2 Things I need explained more
1 Question I still have
MFM 1P - Grade 9 Applied Mathematics – Unit 4: Proportional Reasoning (DPCDSB Dec 2008)
4-46
4.RLS: Reflecting on Learning Skills
Students should be aware of the importance that these skills have on your performance.
After receiving your marked assessment, answer the following questions. Be honest
with yourself. Good Learning Skills will help you now, in other courses and in the future.
E – Always
G – Sometimes
S – Need Improvement
N – Never
Organization
E G S N
E G S N
E G S N
I came prepared for class with all materials
My work is submitted on time
I keep my notebook organized.
Work Habits
E G S N
E G S N
E G S N
E G S N
E G S N
E G S N
I attempt all of my homework
I use my class time efficiently
I limit my talking to the math topic on hand
I am on time
If I am away, I ask someone what I missed,
I complete the work from the day that I missed.
Team Work
E G S N
E G S N
E G S N
I am an active participant in pairs/group work
I co-operate with others within my group
I respect the opinions of others
Initiative
E G S
E G S
E G S
E G S
I participate in class discussion/lessons
When I have difficulty I seek extra help
After I resolve my difficulties, I reattempt the problem
I review the daily lesson/ideas/concepts
N
N
N
N
Works Independently
E G S N
I attempt the work on my own
E G S N
I try before seeking help
E G S N
If I have difficulties I ask others but I stay on task
E G S N
I am committed to tasks at hand
Yes No
I know all the different ways available in my school, where I can seek extra help.
Yes No
I tried my best.
What will I do differently in the next unit to improve?
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
MFM 1P - Grade 9 Applied Mathematics – Unit 4: Proportional Reasoning (DPCDSB Dec 2008)
4-47
MFM 1P - Grade 9 Applied Mathematics – Unit 4: Proportional Reasoning (DPCDSB Dec 2008)
4-48
Course: Grade 9 Applied Mathematics (MFM1P)
Unit 5:
Linear Relations: Constant Rate of Change,
Initial Condition, Direct and Partial Variation
Unit 5
Linear Relations: Constant Rate of Change, Initial Condition,
Direct and Partial Variation
Section
5.1.1
5.1.2
5.1.3
5.1.4
5.2.1
5.2.2
5.2.P
5.3.1
5.3.3
5.3.4
5.3.P
5.4.1
5.4.2
5.4.3
5.4.P
5.6.1
5.6.2
5.7.1
5.8.1
5.9.1
5.9.2
5.9.3
5.W
5.S
5.R
5.RLS
Activity
Walk This Way: Setup Instructions
Walk This Way
CBR: DIST MATCH Setup Instructions
Distance Time Graph
Graphical Stories
Writing Stories Related to a Graph
Interpretation of Graphs
Practice
Rate of Change Notes
Ramps, Roofs, and Roads
Practice
A Runner’s Run
Models of Movement
The Blue Car and the Red Car
Practice
Outfitters
Descriptions, Tables of Values, Equations, Graphs
Walk the Line: Setup Instructions
Modelling Linear Relations with Equations
Graphing Linear Relations
The Speedy Way to Graph
Relationships: Graphs and Equations
Definitions
Unit Summary
Reflecting on My Learning (3, 2, 1)
Reflecting on Learning Skills
MFM 1P - Grade 9 Applied Mathematics – Unit 5: Direct and Partial Variation (DPCDSB Dec 2008)
Page
3
4
6
7
8
9
10
11
12
14
17
19
20
22
23
25
29
33
40
43
44
46
47
49
50
51
5-2
5.1.1: Walk This Way: Setup Instructions
You will need:
• 1 CBR™
• 1 graphing calculator
• 1 ruler
Connect your calculator to the CBR™ with the Link cable and follow these instructions:
Setting up the RANGER Program
Press the APPS key
Select 2: CBL/CBR
Press ENTER
Select 3: RANGER
Press ENTER
You are at the MAIN MENU.
Select 1: SETUP/SAMPLE
Use the cursor → and ↓ keys and the ENTER key to
set-up the CBR:
MAIN MENU
START NOW
REAL TIME:
No
TIME(S):
10
DISPLAY:
DIST
BEGIN ON:
[ENTER]
SMOOTHING:
none
UNITS:
METERS
Cursor up to START NOW
Press ENTER to start collecting data
1. Walk away at a steady pace.
2. Press ENTER then 5: REPEAT SAMPLE if necessary.
3. Press ENTER then 7: QUIT when you are satisfied with the graph.
4. Press GRAPH. This is the graph you will analyse.
MFM 1P - Grade 9 Applied Mathematics – Unit 5: Direct and Partial Variation (DPCDSB Dec 2008)
5-3
5.1.2: Walk This Way
1. Student walks away from CBR™ (slowly).
2. Student walks towards CBR™ (slowly).
3. Student walks very quickly towards CBR™.
MFM 1P - Grade 9 Applied Mathematics – Unit 5: Direct and Partial Variation (DPCDSB Dec 2008)
5-4
5.1.2: Walk This Way (continued)
4. Student increases speed while walking towards the CBR™.
5. Student decreases speed while walking away from the CBR™.
6. Student walks away from ranger, at 2 metres stops for 5 seconds, then returns at the same
pace.
MFM 1P - Grade 9 Applied Mathematics – Unit 5: Direct and Partial Variation (DPCDSB Dec 2008)
5-5
5.1.3: CBR™: DIST MATCH Setup Instructions
Work with your partner/group to complete this activity.
•
•
You will need:
1 CBR™ (motion detector) with linking cable
1 graphing calculator
Setting up the calculator and motion detector
Set up your calculator as follows:
•
•
•
•
•
Insert one end of linking cable FIRMLY into CBR™ and the other end FIRMLY into
graphing calculator.
Press the Akey and select 2: CBL/CBR
Press e
Select 3: RANGER
Press e
You should now be at the MAIN MENU
•
•
•
Select 3: APPLICATIONS
Select 1: METERS
Select 1: DIST MATCH and press e
Activity
Think about how you will need to walk to match the graph shown on the calculator.
When you are ready, press e and try to match it.
If you are not happy with your graph,
Press e, Select 1: SAME MATCH to try again
Now give someone else a turn. Press e and select 2: NEW MATCH to try a different
graph.
Try to match as many graphs as possible.
MFM 1P - Grade 9 Applied Mathematics – Unit 5: Direct and Partial Variation (DPCDSB Dec 2008)
5-6
5.1.4 Distance – Time Graphs
1. Which letters of the alphabet could you not create by walking in front of the motion
detector? Explain why.
2. Draw a graph to match the following description:
A student stands 4 metres from the CBR and walks at a constant rate towards the
CBR for 5 seconds. They then stand still for 3 seconds, and run back to the starting
position.
Distanct from CBR (m)
Distance vs. Time
Time(s)
3. Create your own graph and write a
description to match it.
MFM 1P - Grade 9 Applied Mathematics – Unit 5: Direct and Partial Variation (DPCDSB Dec 2008)
5-7
5.2.1: Graphical Stories
Below the following graphs are three stories about walking from your locker to your class.
Three of the stories correspond to the graphs. Match the graphs and the stories. Write stories
for the other graph. Draw a graph that matches the forth story.
1. I started to walk to class, but I realized I had forgotten my notebook, so I went back to my
locker and then I went quickly at a constant rate to class.
2. I was rushing to get to class when I realized I wasn’t really late, so I slowed down a bit.
3. I walked at a steady, slow, constant rate to my class.
4. I walked to my friend’s locker, and stopped to talk to her for a few minutes. After
she had collected all of her books, we walked (a little faster this time) to class together.
Graphs which compare DISTANCE FROM A POINT and TIME are called Distance-Time
graphs. These graphs can be used to indicate direction, speed and total length of trip (from
starting point to ending point). A story can be made from a distance-time graph.
MFM 1P - Grade 9 Applied Mathematics – Unit 5: Direct and Partial Variation (DPCDSB Dec 2008)
5-8
5.2.2: Writing Stories Related to a Graph
Names:
As you create your story: Focus on the rate of change of each section of the graph and
determine whether the rate of change is constant, varying from fast to slower or slow to faster
or zero.
Criteria
Does your story include:
•
the description of an action? (e.g., distance travelled by bicycle, change of
height of water in a container, the change of height of a flag on a pole)
•
the starting position of the action?
•
the ending position of the action?
•
the total time taken for the action?
•
the direction or change for each section of the action?
•
the time(s) of any changes in direction or changes in the action?
•
the amount of change and time taken for each section of the action?
•
an interesting story that ties all sections of the graph together?
Yes
9
Scale your graph, and label each axis!
MFM 1P - Grade 9 Applied Mathematics – Unit 5: Direct and Partial Variation (DPCDSB Dec 2008)
5-9
5.2.P: Interpretations of Graphs
Sunflower Seed Graphs
Ian and his friends were sitting on a deck and eating sunflower seeds. Each person had a bowl
with the same amount of seeds. The graphs below all show the amount of sunflower seeds
remaining in the person’s bowl over a period of time.
Write sentences that describe what may have happened for each person.
a)
b)
c)
d)
Multiple Choice
Indicate which graph matches the statement. Give reasons for your answer.
1. A bicycle valve’s distance from the ground as a boy rides at a constant speed.
a)
b)
c)
d)
2. A child swings on a swing, as a parent watches from the front of the swing.
MFM 1P - Grade 9 Applied Mathematics – Unit 5: Direct and Partial Variation (DPCDSB Dec 2008)
5-10
5.3.1: Practice
Recall: Converting fractions ↔ decimals ↔ percents.
1.
2.
3.
Write as a decimal. Round to two decimal places where necessary.
(a)
13
15
(b)
45
30
(c)
36%
(d)
127.5%
Write as a fraction in lowest terms.
(a)
0.14
(b)
0.06
(c)
25%
(d)
62.5%
Write as a percent. Round to two decimal places where necessary.
(a)
12
20
(b)
18
15
(c)
0.34
(d)
1.05
MFM 1P - Grade 9 Applied Mathematics – Unit 5: Direct and Partial Variation (DPCDSB Dec 2008)
5-11
5.3.3: Rate of Change Notes
Rate of Change of a Linear
Relationship
run
Rate of Change
rise
Rate of Change =
rise
run
The rate of change of a linear relationship is
the steepness of the line.
Rates of change are seen
everywhere.
The steepness of
the roof of a house
is referred to as the
pitch of the roof by
home builders.
Give one reason
why some homes
have roofs which
have a greater
pitch.
There is less snow buildup in the
wintertime.
Engineers refer to the
rate of change of a road as
the grade.
MFM 1P - Grade 9 Applied Mathematics – Unit 5: Direct and Partial Variation (DPCDSB Dec 2008)
5-12
5.3.3: Rate of Change Notes (continued)
100
8
A grade of 8% would mean for every
rise of 8 units there is a run of 100 units.
8
100
Rate of change =
They often represent the rate of
change as a percentage.
The steepness of wheelchair ramps is of
great importance for safety.
1
Determine the rate of
change (pitch) of the
roof.
3m
5m
12
Rate of change of wheelchair ramp =
1
12
If the rise is 1.5 m, what is the run?
Answer: 18 m
because
1 15
.
=
12 18
Determine the rate of change of each
staircase.
rate of change
3
=
= 8%
2
3
rate of change
3
=
3
=1
2
3
3
rate of change =
5
3
Determine the rate of change.
E
a
r
n
i
n
g
s
Which points will
you use to
determine rise and
run?
rate of change
4
20
=
rise
run
=
$ 20
4 hr
= $5/hr
What does this rate of
Number of Hours Worked
change represent?
The hourly wage
MFM 1P - Grade 9 Applied Mathematics – Unit 5: Direct and Partial Variation (DPCDSB Dec 2008)
5-13
5.3.4: Ramps, Roofs, and Roads
Ramps
Types of inclines and recommendations by
rehabilitation specialists
Rise
(Vertical
Distance)
Run
(Horizontal
Distance)
Rate of
Change
The recommended incline for wheelchair uses is 1:12.
For exterior ramps in climates where ice and snow are
common, the incline should be more gradual, at 1:20.
For unusually strong wheelchair users or for motorized
chairs, the ramp can have an incline of 1:10.
The steepest ramp should not have an incline
exceeding 1:8.
Building Ramps - Calculate the rate of change to find which of four ramps could be
built for each of the clients below?
1.
2.
3.
4.
Clients
Choice of Ramp
and Reason
Client A lives in a split-level town house. He owns a very
powerful motorized chair. He wishes to build a ramp that leads
from his sunken living room to his kitchen on the next level.
Client B requires a ramp that leads from her back deck to a
patio. She is of average strength and operates a manual
wheelchair.
Client C lives in Sudbury where ice and snow are a factor. She
is healthy, but not particularly strong. Her house is a single
level bungalow but the front door is above ground level.
Client D will not get approval because the design of his ramp
is too dangerous.
MFM 1P - Grade 9 Applied Mathematics – Unit 5: Direct and Partial Variation (DPCDSB Dec 2008)
5-14
5.3.4: Ramps, Roofs, and Roads (continued)
Roofs
Calculate the rate of change (pitch) of each roof. Answer the questions that follow the diagrams.
1. If all four roofs were placed on the same-sized foundation, which roof would be the most
expensive to build?
Hint: Steeper roofs require more building materials.
2. Why do you think apartment buildings have flat roofs? What is the rate of change of a flat
roof?
3. In the winter snow builds up on the roof. Sometimes, if the snow builds up too high, the roof
becomes damaged. Which roof would be the best for areas that have a large amount of
snowfall? Why?
MFM 1P - Grade 9 Applied Mathematics – Unit 5: Direct and Partial Variation (DPCDSB Dec 2008)
5-15
5.3.4: Ramps, Roofs, and Roads (continued)
Roads
The inclination of a road is called “percent grade.” Severe grades (greater than 6%) are difficult
to drive on for extended amounts of time. The normal grade of a road is between 0% and 2%.
Warning signs are posted in all areas where the grades are severe.
Percent grade
A
Fraction
Rise
Run
1
50
Rate of change
(decimal form)
1%
B
0.035
C
D
4%
525
E
10 000
3
50
F
0.1
G
1
H
2
0.75
I
1
J
2
5
K
L
3
8.25%
Which of the roads, A–L, would require a warning sign?
Some of the values in the table are fictional. There are no roads that have grades that are that
severe. Which roads, A–L, could not exist? Explain your reasoning.
Describe a road with a 0% grade.
MFM 1P - Grade 9 Applied Mathematics – Unit 5: Direct and Partial Variation (DPCDSB Dec 2008)
5-16
5.3.P: Practice
1.
Determine the rate of change for each object.
(a)
The pitch of the roof is the rate of
change.
Rate of change =
7.2 m
The pitch is
5.6 m
(b)
The steepness of the ramp is the rate
of change.
Rate of change =
1.2 m
4.8 m
The rate of change is
(c)
The steepness of the staircase is the
rate of change.
Rate of change =
4m
The rate of change is
4m
MFM 1P - Grade 9 Applied Mathematics – Unit 5: Direct and Partial Variation (DPCDSB Dec 2008)
5-17
5.3.P: Practice (continued)
2.
If a wheelchair ramp must have a rate of change of
1
, determine the horizontal
12
distance required for a ramp that has a vertical distance of 5.2m.
3.
The grade of a road is often given as a percent. If the road rises 15 m over a horizontal
distance of 180 m, determine the grade as a percent.
4.
The pitch of a roof of a house is given by a rate of change of
5
. If the horizontal
6
distance is actually 10.5 m, determine the vertical distance of the roof.
MFM 1P - Grade 9 Applied Mathematics – Unit 5: Direct and Partial Variation (DPCDSB Dec 2008)
5-18
5.4.1: A Runner’s Run
Chris runs each day as part of his daily exercise. The graph shows his distance from home as
he runs his route.
100
Calculate his rate of change (speed) for each segment of the graph.
Rate of change AB =
Rate of change BC =
Rate of change CD =
Rate of change DE =
Rate of change EF =
Rate of change FG =
MFM 1P - Grade 9 Applied Mathematics – Unit 5: Direct and Partial Variation (DPCDSB Dec 2008)
5-19
5.4.2: Models of Movement
Distance vs. Time
700
Distance from Home (m)
At 11 o’clock, Micha’s mother sends him to
the corner store for milk and tells him to be
back in 30 minutes. Examine the graph.
D E
600
F
500
C
400
300
200
B
100
G
A
4
8
12
16
20
24
28
32
36
Time (min)
1. Why are some line segments on the graph steeper than others?
2. Calculate the rate of change (speed) of each of the line segments:
MFM 1P - Grade 9 Applied Mathematics – Unit 5: Direct and Partial Variation (DPCDSB Dec 2008)
5-20
40
44
48
5.4.2: Models of Movement (continued)
3. Over what interval(s) of time is Micha travelling the fastest?
the slowest?
Compare steepness, not direction.
4. How long did it take Micha to reach the store? How do you know?
5. How long did Micha stay at the store?
6. How long did it take Micha to get home from the store?
7. How can you use the graph to tell which direction Micha is travelling?
8. Did Micha make it home in 30 minutes? How do you know?
9. Using the information the graph provides, write a story that describes Micha’s trip to the
store and back.
MFM 1P - Grade 9 Applied Mathematics – Unit 5: Direct and Partial Variation (DPCDSB Dec 2008)
5-21
5.4.3: The Blue Car and the Red Car
Distance from parking lot (km)
Two friends are leaving a parking lot at the same time. They agree to meet later at the home of
a friend who lives 400 km from the parking lot. One friend drives a blue car and the other a red
car. The blue car is labelled B and the red car, R. Answer the questions below using the
following graph.
400
B
300
R
200
100
1
2
3
4
5
6
Time (h)
1. At what time do the cars pass each other? How far are they from the parking lot?
2. Which car stopped and for how long? How far from the parking lot did the car stop?
3. Suggest reasons for the car stopping.
4. Which car got to the final destination first? Explain.
5. The posted speed limit was 80 km/h. If you were a police officer, could you stop either of the
cars for speeding? Explain.
MFM 1P - Grade 9 Applied Mathematics – Unit 5: Direct and Partial Variation (DPCDSB Dec 2008)
5-22
5.4.P: Practice
1.
A bakery delivery truck spends 3 hours driving the morning run. There are two
deliveries made and then the truck returns to the bakery.
Distance (km)
90
60
30
0
0.5
1.0
1.5
2.0
2.5
3.0
Time (h)
(a)
What is the speed of the truck as it approaches its first delivery?
(b)
When does the truck leave to make the second delivery?
(c)
What is the speed of the truck as it approaches its second delivery?
(d)
How long does it take to complete the second delivery?
(e)
What is the speed of the truck as it approaches the bakery at the end of the run?
MFM 1P - Grade 9 Applied Mathematics – Unit 5: Direct and Partial Variation (DPCDSB Dec 2008)
5-23
5.4.P: Practice (continued)
2.
Amanjot takes the bus to school. Lucky for her, she is the third last stop on the way to
school. The bus arrives to pick up Amanjot and it drives at a constant speed for 5
minutes to the next stop 3 km away. It takes 1 minute for the students to get on the bus.
The bus then travels 50km/h to a stop that is 5 km away. 2 minutes later, the bus is on
its way to the school. It takes 8 minutes to reach the school which is 6 km away.
Distance (km)
Draw a distance-time graph of Amanjot’s bus ride to school.
Time (min)
MFM 1P - Grade 9 Applied Mathematics – Unit 5: Direct and Partial Variation (DPCDSB Dec 2008)
5-24
5.6.1: Outfitters
Jaraad wants to rent a canoe for a day trip. He gathers this information from two places and
decides to make a table of values and graph each of these relationships.
• Big Pine Outfitters charges a base fee of $40 and $10 per hour of use.
• Hemlock Bluff Adventure Store does not charge a base fee, but charges $30 per hour to use
the canoe.
Jaraad’s Working Sheet
1. a) What is the cost of each canoe if Jaraad cancels his reservation?
b) Compare the rate of change of cost for Big Pine and for Hemlock Bluff to the cost per
hour for each outfitter.
MFM 1P - Grade 9 Applied Mathematics – Unit 5: Direct and Partial Variation (DPCDSB Dec 2008)
5-25
5.6.1: Outfitters (continued)
2. Which graph illustrates a proportional relation? How do you know? This is called a direct
variation.
3. Which graph has an initial value other than zero? This is called a partial variation.
4. Which outfitter company should Jaraad choose if he estimates he will canoe for
0.5 h?…1.5 h?…2.5 h?
Time (h)
Big Pine Cost ($)
Hemlock Bluff Cost ($)
0.5
1.5
2.5
Explain how you determined your answers.
MFM 1P - Grade 9 Applied Mathematics – Unit 5: Direct and Partial Variation (DPCDSB Dec 2008)
5-26
5.6.1: Outfitters (continued)
NOTE: Linear equations follow this format:
Dependent Variable = Initial Value + Rate of change x Independent Variable
OR
Dependent Variable = Rate of change x Independent Variable + Initial Value
5. Write an equation to model the cost for each outfitter.
Let C represent the cost in dollars and h represent the time in hours.
Big Pine
C=
Hemlock Bluff
C=
6. If Big Pine Outfitters decided to change its base fee to $50 and charge $10 per hour, what
effect would this have on the graph?
a) Draw a sketch of the original cost and show the changes on the same sketch.
b) Write an equation to model the new cost.
7. If Hemlock Bluff Adventure Store decided to change its hourly rate to $40, what effect would
this have on the graph?
a) Draw a sketch of the original cost and show the changes on the same sketch.
b) Write an equation to model the new cost.
MFM 1P - Grade 9 Applied Mathematics – Unit 5: Direct and Partial Variation (DPCDSB Dec 2008)
5-27
5.6.1: Outfitters (continued)
8. For Big Pine Outfitters, how are the pattern in the table of values, the description, the graph,
and the equation related?
Description
Big Pine Outfitters charges a base fee of $40 to deliver the canoe to the launch site and $10 per
hour of use.
Table of Values
Graph
Time (h) Cost ($)
0
40
1
50
2
60
3
70
4
80
Equation
C = 40 + 10h
9. For Hemlock Bluff, how are the pattern in the table of values, the description, the graph, and
the equation related?
Description
Hemlock Bluff charges $30 per hour.
Table of Values
Graph
Time (h) Cost ($)
0
0
1
30
2
60
3
90
4
120
Equation
C = 30h
MFM 1P - Grade 9 Applied Mathematics – Unit 5: Direct and Partial Variation (DPCDSB Dec 2008)
5-28
5.6.2: Descriptions, Tables of Values, Equations, Graphs
1. A rental car costs $50 per day plus $0.20 for each kilometre it is driven.
a) What is the dependent variable? ______________
b) Make a table of values for the rental fee up to 1000 km.
c) Graph the relationship.
Number of
Kilometres
Cost ($)
Cost vs. Number of Kilometres
0
260
200
240
220
400
200
Cost ($)
180
160
140
120
100
80
60
40
20
100 200 300 400 500 600 700 800 900 1000 1100 1200 1300 1400
Number of Kilometres
d) Write an equation to model
the relationship. C is the
cost and n is the number of kilometres.
Dependent Variable = Initial Value + Rate of Change x Independent Variable
________________ = _________ + _________________ x _____________
e) Does this relation represent a partial or direct variation? Explain.
f)
Determine the rental fee for 85 km. Show your work.
MFM 1P - Grade 9 Applied Mathematics – Unit 5: Direct and Partial Variation (DPCDSB Dec 2008)
5-29
5.6.2: Descriptions, Tables of Values, Equations, Graphs
(continued)
2. There is $500 in Holly’s bank account. She takes out $50 from her account
each month but doesn’t put any back in.
a) Make a table of values for up to 6 months.
b) Graph the relationship.
Balance vs. Number of Months
0
600
500
Balance ($)
400
300
200
100
1
2
3
4
5
6
Number of Months
c) Write an equation to model the relationship.
Dependent Variable = Initial Value + Rate of Change x Independent Variable
________________ = _________ + _________________ x _____________
d) Does this relation represent a partial or direct variation? Explain.
e) How much will Holly have in her account after 8 months?
f)
How many months will have passed when Holly has $0 in her account? Explain how
you got your answer.
MFM 1P - Grade 9 Applied Mathematics – Unit 5: Direct and Partial Variation (DPCDSB Dec 2008)
5-30
7
5.6.2: Descriptions, Tables of Values, Equations, Graphs
(continued)
3. Nisha is just learning how to snowboard. White Mountain charges $10/hour
for lessons and $40 for the lift ticket and snowboard rental.
a) Make a table of values for up to 6 hours.
b) Graph the relationship.
150
100
50
2
4
6
8
10
12
c) Write an equation to model the relationship.
___ = _________________
d) Does this relation represent a partial or direct variation? Explain.
e) How much will it cost in total for Nisha to take 2.5 hours of lessons?
f)
If Nisha paid $75, how long was she at the White Mountain getting lessons?
MFM 1P - Grade 9 Applied Mathematics – Unit 5: Direct and Partial Variation (DPCDSB Dec 2008)
5-31
14
5.6.2: Descriptions, Tables of Values, Equations,
Graphs (continued)
4. Ishmal sells high-definition televisions. He is paid a weekly salary of 20%
commission of his total weekly sales.
a) Complete the table of values.
b) Graph the relationship.
Weekly
Sales ($)
Total Pay ($)
0
1000
2000
1800
2000
1600
1400
3000
1200
1000
800
4000
600
400
5000
200
2000
4000
6000
8000
10000
c) Write an equation to model the relationship.
___ = _________________
d) Does this relation represent a partial or direct variation? Explain.
e) Determine Ishmal’s pay if his sales for the week were $8000.
f)
Ishmal made $900. How much were his weekly sales?
MFM 1P - Grade 9 Applied Mathematics – Unit 5: Direct and Partial Variation (DPCDSB Dec 2008)
5-32
12000
5.7.1: Walk the Line: Setup Instructions
You will need:
• 1 CBR™
• 1 graphing calculator
• 1 ruler
Connect your calculator to the CBR™ with the Link cable and follow these instructions:
Setting up the RANGER Program
Press the APPS key
Select 2: CBL/CBR
Press ENTER
Select 3: RANGER
Press ENTER
You are at the MAIN MENU.
Select 1: SETUP/SAMPLE
Use the cursor → and ↓ keys and the ENTER key to
set-up the CBR:
MAIN MENU
START NOW
REAL TIME:
no
TIME(S):
10
DISPLAY:
DIST
BEGIN ON:
[ENTER]
SMOOTHING:
none
UNITS:
METERS
Cursor up to START NOW
Press ENTER to start collecting data
1. Walk away at a steady pace.
2. Press ENTER then 5: REPEAT SAMPLE if necessary.
3. Press ENTER then 7: QUIT when you are satisfied with the graph.
4. Press GRAPH. This is the graph you will analyse.
MFM 1P - Grade 9 Applied Mathematics – Unit 5: Direct and Partial Variation (DPCDSB Dec 2008)
5-33
5.7.1: Walk the Line: Setup Instructions (continued)
Part One: Draw your graph.
Stand about 0.5 metres from the CBR™. Walk slowly away from the CBR™ at a steady pace.
• Copy the scale markings on the distance and time axes from your calculator.
• Mark your start and finish position on the graph using the coordinates Time and Distance.
• Connect the start and finish position with a line made with your ruler.
________________________’s Walk
Calculate the rate of change of the graph (speed of your walk).
•
Draw a right-angled triangle under the graph and label it with the rise and run values.
•
Calculate the rate of change of your walk using the formula rate of change = rise .
run
•
Complete the following:
a) The rate of change of my walk is ________________.
b) The speed of my walk is ________________ m/s away from the CBR™.
MFM 1P - Grade 9 Applied Mathematics – Unit 5: Direct and Partial Variation (DPCDSB Dec 2008)
5-34
5.7.1: Walk the Line: Setup Instructions (continued)
Describe your walk.
Use your starting position and rate of change to write a walking description statement:
I started ____ metres from the CBR™ and walked away from it at a
speed of ____ metres per second.
After 10 seconds, I was ____ __ from the motion detector.
At this rate, how far would you have walked after 30 seconds?
Construct an equation to model your walk.
Read this walking statement:
A student started 0.52 metres from the CBR™ and walked away at a speed
of 0.19 metres/second.
The equation D = 0.52 + 0.19t models the student’s position from the CBR™.
To graph it on the graphing calculator use: Y = 0.52 + 0.19x.
Write a walking statement and equation for your walk:
_____________ started _____ from the CBR™ and walked away at a speed of _____
metres/sec.
The equation __________________________ models my distance from the CBR™. The
graphing calculator equation is ____________________.
MFM 1P - Grade 9 Applied Mathematics – Unit 5: Direct and Partial Variation (DPCDSB Dec 2008)
5-35
5.7.1: Walk the Line: Setup Instructions (continued)
Verify your equation with your walk using the graphing calculator.
Turn off the STATPLOT.
Type your equation into the Y= editor
Graph your equation (Press: GRAPH)
Turn on the STATPLOT. Press GRAPH again.
Change the numbers in your Y = equation until you get the best possible match for the graph
you walked.
The best equation that matches your walk is: ___________________.
MFM 1P - Grade 9 Applied Mathematics – Unit 5: Direct and Partial Variation (DPCDSB Dec 2008)
5-36
5.7.1: Walk the Line: Setup Instructions (continued)
Use the equation to solve problems.
The equation D = 0.52 + 0.19t models the student’s distance away from the CBR™, over time.
We can calculate the student's distance from the CBR™ after 30 seconds:
D = 0.52 + 0.19t
D = 0.52 + (0.19)(30)
D = 0.52 + 5.7
D = 6.22
The student will be 6.22 metres from the CBR™ after 30 seconds.
Now, calculate your distance from the CBR™ after 30 seconds:
(Use the best equation that matches your walk.)
a) The equation ____________________ models your distance from the CBR™.
b) Calculate your distance from the CBR™ after 30 seconds:
Check your answer with your graph.
First, turn off the STATPLOT
Next, press: GRAPH
Then press: TRACE
Arrow right until you reach 30 seconds.
Record the distance the CBR™ displays for 30 seconds _________.
How does this compare with your answer using the equation?
How does this answer compare with your estimate at the beginning of the activity?
Use your equation to calculate how long it will take to walk 1 km from the CBR™.
MFM 1P - Grade 9 Applied Mathematics – Unit 5: Direct and Partial Variation (DPCDSB Dec 2008)
5-37
5.7.1: Walk the Line: Setup Instructions (continued)
Part Two: Draw your graph.
Stand about 3 metres from the CBR™. Walk slowly towards the CBR™ at a steady pace.
• Copy the scale markings on the distance and time axes from your calculator.
• Mark your start and finish position on the graph using the coordinates Time and Distance.
• Connect the start and finish position with a line made with your ruler.
________________________’s Walk
Calculate the rate of change of the graph (speed of your walk).
Draw a large right-angled triangle under the graph and label it with the rise and run values.
Calculate the rate of change using the formula: rate of change = rise .
run
The rate of change of my walk is ________________.
Hint: The rise will be a negative number! Why?
The speed of my walk is ________________ m/s away from the CBR™.
MFM 1P - Grade 9 Applied Mathematics – Unit 5: Direct and Partial Variation (DPCDSB Dec 2008)
5-38
5.7.1: Walk the Line: Setup Instructions (continued)
Describe your walk.
Use your initial position and rate of change to write a walking description statement:
I started ______metres from the CBR™ and walked towards it at speed
of _____metres per second.
After 10 seconds, I was ______ from the motion detector.
At this rate, how far would you have walked after 30 seconds?
Construct an equation to model your walk.
Read this walking statement:
A student started 4 metres from the CBR™ and walked towards it at a speed
of 0.32 metres/second.
The equation D = 4 – 0.32t models the students position from the CBR™.
To graph it on the graphing calculator use: Y = 4 – 0.32x.
Write a walking statement and equation for your walk:
_____________started ____ metres from the CBR™ and walked towards it at a speed of
_____ metres per second.
The equation ___________________________ models my distance from the CBR™. To graph
it on the graphing calculator use: ________________________.
Verify your equation with your walk using the graphing calculator.
Remember that you can change the numbers in your Y = equation until you get the best
possible match for the graph you walked.
The best equation that matches your walk is: ___________________
MFM 1P - Grade 9 Applied Mathematics – Unit 5: Direct and Partial Variation (DPCDSB Dec 2008)
5-39
5.8.1: Modelling Linear Relations with Equations
NOTE: Linear equations follow this format:
Dependent Variable = Initial Value + Rate of change x Independent Variable
OR
Dependent Variable = Rate of change x Independent Variable + Initial Value
Write the equation for each relationship in the space provided. Show any calculations you
made. Indicate if the relation is a partial or direct variation and whether the line modelling the
relationship is solid or dashed.
A coaches B
1. A family meal deal at Chicken Deluxe
costs $26, plus $1.50 for every extra
piece of chicken added to the bucket.
B coaches A
2. A Chinese food restaurant has a special
price for groups. Dinner for two costs $24
plus $11 for each additional person.
3.
4.
5.
Number of
Toppings
0
1
2
3
4
Cost of a Large
Pizza ($)
9.40
11.50
13.60
15.70
17.80
6.
Number of
Scoops
Cost of Ice
Cream with
Sugar Cone ($)
0
1
2
3
4
1.25
2.00
2.75
3.50
4.25
MFM 1P - Grade 9 Applied Mathematics – Unit 5: Direct and Partial Variation (DPCDSB Dec 2008)
5-40
5.8.1: Modelling Linear Relations with Equations (continued)
Write the equation for each relationship in the space provided. Show any calculations you
made. Indicate if the relation is a partial or direct variation and describe why these variables are
discrete.
A coaches B
1. A banquet hall charges $100 for the hall
and $20 per person for dinner.
B coaches A
2. The country club charges a $270 for their
facilities plus $29 per guest.
3.
4.
5.
Number of
Athletes
Cost of
Attending a
Hockey
Tournament
0
1
2
3
4
0
225
450
675
900
6.
Number of
People
Cost of
Holding an
Athletic
Banquet
0
20
40
60
80
75
275
475
675
875
MFM 1P - Grade 9 Applied Mathematics – Unit 5: Direct and Partial Variation (DPCDSB Dec 2008)
5-41
5.8.1: Modelling Linear Relations with Equations (continued)
Write the equation for each relationship in the space provided. Show any calculations you
made. Indicate if the relation is a partial or direct variation and whether the line modelling the
relationship is solid or dashed.
A coaches B
1. Rent a car for the weekend costs $50
plus $0.16/km.
B coaches A
2. A race car travels at a constant speed of
220km/h. Write an equation for the total
distance travelled over time.
3.
4.
5.
Distance
(km)
0
10
20
30
40
Cost of a Taxi
Fare ($)
3.50
6.50
9.50
12.50
15.50
6.
Distance
(km)
0
100
200
300
400
Cost of Bus
Charter ($)
170
210
250
290
330
MFM 1P - Grade 9 Applied Mathematics – Unit 5: Direct and Partial Variation (DPCDSB Dec 2008)
5-42
5.9.1: Graphing Linear Relations
NOTE: Linear equations follow this format:
Dependent Variable = Initial Value + Rate of Change x Independent Variable
OR
Dependent Variable = Rate of Change x Independent Variable + Initial Value
A tennis club charges $25 initial membership fee plus $5 per day. The equation of this relation is
C = 25 + 5d, where C is the cost and d is the number of days.
Total Cost vs. Number of Day Passes
65
60
55
50
45
40
Total Cost ($)
35
30
25
20
15
10
5
0
1
2
3
4
5
6
7
8
Number of Day Passes
a)
Indicate where the rate of change is displayed on the graph.
b)
If the initial membership fee is changed to $15 and daily cost to $10, graph the new
relation on the same grid.
Indicate the procedure you followed to graph the line.
c)
Write the equation of the new line.
MFM 1P - Grade 9 Applied Mathematics – Unit 5: Direct and Partial Variation (DPCDSB Dec 2008)
5-43
5.9.2: The Speedy Way to Graph
NOTE: Linear equations follow this format:
Dependent Variable = Initial Value + Rate of Change x Independent Variable
OR
Dependent Variable = Rate of Change x Independent Variable + Initial Value
Write the equation for the relationship and graph the relationship.
1. Movie House charges $5 to rent each DVD.
2. Repair-It charges $60 for a service call plus $25/h
to repair the appliance.
Equation:
Equation:
3. A golf club charges an annual membership fee of 4. A kite is 15 m above the ground when it
$1000 plus $100 for a green fee to play golf.
descends at a steady rate of 1.5 m/s.
Equation:
Equation:
MFM 1P - Grade 9 Applied Mathematics – Unit 5: Direct and Partial Variation (DPCDSB Dec 2008)
5-44
5.9.2: The Speedy Way to Graph (continued)
Partner A ___________________________
Partner B___________________________
Write the equation for the relationship and graph the relationship.
1. The Recreation Centre charges a monthly
membership fee of $20 plus $5 per class.
Show the relationship for one month.
2. Repair Window charges a $20 service fee
plus $10/h to fix the window pane.
Equation:
Equation:
3. Yum-Yum Ice Cream Shop charges $0.50
for the cone plus $1 per scoop of ice
cream.
4. A submarine model starts 6.5 m above the
bottom of the pool. It gradually descends
at a rate of 0.25 m/s.
Equation:
Equation:
MFM 1P - Grade 9 Applied Mathematics – Unit 5: Direct and Partial Variation (DPCDSB Dec 2008)
5-45
5.9.3: Relationships: Graphs and Equations
Write the equation for the relationship and graph the relationship.
1. A taxi cab company charges $3.50 plus
2. Shelly has $250 in her bank account. She
$0.50/km.
spends $10/week on snacks.
Equation:
Equation:
3. Dino’s Pizza charges $17 for a party-sized 4. Katie sells programs at the Omi Arena.
pizza plus $2 per topping.
She is paid 50 cents for every program she
sells.
Equation:
Equation
MFM 1P - Grade 9 Applied Mathematics – Unit 5: Direct and Partial Variation (DPCDSB Dec 2008)
5-46
5.W: Definition Page
Term
Picture / Sketch / Examples
Definition
Rate of Change
Constant Rate of Change
Increasing Rapidly
Increasing Slowly
Decreasing Rapidly
Decreasing Slowly
Pitch
Grade
Ramp Incline
Direct Variation
Partial Variation
MFM 1P - Grade 9 Applied Mathematics – Unit 5: Direct and Partial Variation (DPCDSB Dec 2008)
5-47
5.W: Definition Page (Continued)
Initial Value
Base Fee
MFM 1P - Grade 9 Applied Mathematics – Unit 5: Direct and Partial Variation (DPCDSB Dec 2008)
5-48
5.S: Unit Summary Page
Unit Name: ____________________________________________
Using a graphic organizer of your choice create a unit summary.
MFM 1P - Grade 9 Applied Mathematics – Unit 5: Direct and Partial Variation (DPCDSB Dec 2008)
5-49
5.R: Reflecting on My Learning (3, 2, 1)
3 Things I know well from this unit
2 Things I need explained more
1 Question I still have
MFM 1P - Grade 9 Applied Mathematics – Unit 5: Direct and Partial Variation (DPCDSB Dec 2008)
5-50
5.RLS: Reflecting on Learning Skills
Students should be aware of the importance that these skills have on your performance. After
receiving your marked assessment, answer the following questions. Be honest with yourself.
Good Learning Skills will help you now, in other courses and in the future.
•
•
•
•
E – Always
G – Sometimes
S – Need Improvement
N – Never
Organization
• E G S N
• E G S N
• E G S N
I came prepared for class with all materials
My work is submitted on time
I keep my notebook organized.
Work Habits
• E G S N
• E G S N
• E G S N
• E G S N
• E G S N
• E G S N
I attempt all of my homework
I use my class time efficiently
I limit my talking to the math topic on hand
I am on time
If I am away, I ask someone what I missed,
I complete the work from the day that I missed.
Team Work
• E G S N
• E G S N
• E G S N
I am an active participant in pairs/group work
I co-operate with others within my group
I respect the opinions of others
Initiative
• E G S
• E G S
• E G S
• E G S
I participate in class discussion/lessons
When I have difficulty I seek extra help
After I resolve my difficulties, I reattempt the problem
I review the daily lesson/ideas/concepts
N
N
N
N
Works Independently
• E G S N
I attempt the work on my own
• E G S N
I try before seeking help
• E G S N
If I have difficulties I ask others but I stay on task
• E G S N
I am committed to tasks at hand
Yes No
I know all the different ways available in my school, where I can seek extra help.
Yes No
I tried my best.
What will I do differently in the next unit to improve?
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
MFM 1P - Grade 9 Applied Mathematics – Unit 5: Direct and Partial Variation (DPCDSB Dec 2008)
5-51
MFM 1P - Grade 9 Applied Mathematics – Unit 5: Direct and Partial Variation (DPCDSB Dec 2008)
5-52
Course: Grade 9 Applied Mathematics (MFM1P)
Unit 6:
Multiple Representations: Using Linear Relations
and their Multiple Representations
Unit 6
Multiple Representations: Using Linear Relations
and their Multiple Representations
Section
6.1.1
6.1.2
6.1.3
6.1.5
6.1.6
6.1.P
6.3.1
6.3.2
6.3.P
6.4.1
6.4.2
6.5.1
6.5.2
6.6.1
6.6.J
6.W
6.S
6.R
6.RLS
Activity
Working with Equations
Concept Circles - Equations
Frayer Model – Expressions & Equations
The Equation Game: One Step Equations
The Equation Game: Two Step Equations
Practice
Mathematical Models
Solving Equations Using Substitution
Real World Mathematical Models
Planning a Special Event
Cell Phone Problem
An Environmental Project
Cooling It!
Linear and Non-Linear Investigations
Journal Activity
Definitions
Unit Summary
Reflecting on My Learning (3, 2, 1)
Reflecting on Learning Skills
Page
3
5
6
7
9
11
13
16
18
20
24
27
30
32
40
41
42
43
44
MFM 1P - Grade 9 Applied Mathematics – Unit 6: Multiple Representations (DPCDSB Dec 2008)
6-2
6.1.1: Working with Equations
Jenise has inquired about the cost of renting a facility for her wedding. She used the data she
received to draw the graph below.
Cost of Holding a Wedding at a Facility
3500
3000
Cost ($)
2500
2000
1500
1000
500
20
40
60
80
100
120
140
Number of Guests
1. Jenise said the graph shows a linear relationship. Justify Jenise’s answer.
2. Does this relation represent a direct or partial variation? Explain your answer.
3. State the initial value and calculate the rate of change of this relation.
MFM 1P - Grade 9 Applied Mathematics – Unit 6: Multiple Representations (DPCDSB Dec 2008)
6-3
6.1.1: Working with Equations (continued)
4.
Use the graph to complete the chart:
a)
b)
c)
d)
e)
f)
g)
Number of
Guests
10
Cost ($)
1250
110
2500
0
3500
30
5. Determine an equation for the relationship.
6. Solve the above equation to determine the number of guests Jenise could have for $1750.
Verify your answer using the graph.
7. Solve the equation to determine the cost for 175 guests. Show your work.
We will learn how to use
algebra to solve this
question later in this unit!
MFM 1P - Grade 9 Applied Mathematics – Unit 6: Multiple Representations (DPCDSB Dec 2008)
6-4
6.1.2: Concept Circles – Equations
1. Draw an “X” through the example that does not belong. Justify your answer.
a)
b)
x+ 4 = 8
2+x=8
3x
x–4=3
2x = 8
-2x = 4
y = 3x + 1
C = 10t + 1
2y + 3x
P = 2 l + 2w
x+ 4
c)
3x = 9
d)
3x – 3 = 3
2x – 1 = 5
4x – 2
-2x = 4
2. Answer True (T) or False (F). Be prepared to justify your answer.
a) Every equation has exactly two sides.
____
b) Every equation has one equal sign. ____
c) Every equation has one variable.
____
MFM 1P - Grade 9 Applied Mathematics – Unit 6: Multiple Representations (DPCDSB Dec 2008)
6-5
6.1.3: Equations and Expressions
Complete the following Frayer models.
Definition:
Facts/Characteristics:
Expression
Examples:
Definition:
Non-examples:
Facts/Characteristics:
Equation
Examples:
Non-examples:
MFM 1P - Grade 9 Applied Mathematics – Unit 6: Multiple Representations (DPCDSB Dec 2008)
6-6
6.1.5: The Equation Game
One Step Equations
Solve each equation using algebra tiles. Have your partner check your answers.
x–2=4
g + 1 = -7
-4 = 2 + a
3 – b = -2
x + 1 = -3
t+6=9
MFM 1P - Grade 9 Applied Mathematics – Unit 6: Multiple Representations (DPCDSB Dec 2008)
6-7
6.1.5: The Equation Game (continued)
One Step Equations
Solve each equation using algebra. Have your partner check your answers.
p – 8 = 10
m + 3 = 15
-5=-2+y
k+6=9
4 + h = -2
9 – w = -2
MFM 1P - Grade 9 Applied Mathematics – Unit 6: Multiple Representations (DPCDSB Dec 2008)
6-8
6.1.6The Equation Game (continued)
Two Step Equations
Solve each equation using algebra tiles. Have your partner check your answers.
3x – 2 = 4
4n + 1 = -7
-4 = 2 + 2a
3 – 5b = -2
-4x + 1 = -3
3t + 6 = 9
MFM 1P - Grade 9 Applied Mathematics – Unit 6: Multiple Representations (DPCDSB Dec 2008)
6-9
6.1.6: The Equation Game (continued)
Two Step Equations
Solve each equation using algebra. Have your partner check your answers.
3p – 8 = 10
- 6m + 3 = 15
- 5 = 2 + 14y
3k + 6 = - 9
4 = -2 + 3h
7 – 3w = -2
MFM 1P - Grade 9 Applied Mathematics – Unit 6: Multiple Representations (DPCDSB Dec 2008)
6-10
6.1.P: Practice
Solve the following equations using algebra. Check every second equation..
a.
s + 5 = 14
b.
u – 5 = - 14
c.
-5 = v – 14
d.
7x = 14
e.
-7 = -14y
f.
3m + 1 = 10
g.
2h + 7 = 15
h.
4 – 2d = -2
i.
5y – 3 = 12
j.
6 = 4w -6
k.
4 = 3t - 8
l.
3c + 12 = 36
MFM 1P - Grade 9 Applied Mathematics – Unit 6: Multiple Representations (DPCDSB Dec 2008)
6-11
6.1.P: Practice (continued)
Solve the following equations using algebra. Check every second equation.
a.
t–2=7
b.
4d = -16
c.
2m – 4 = 10
d.
-3 = 7 – 5p
e.
4x + 28 = 16
f.
5y – 12 = 13
g.
2g – 1 = 7
h.
-3 = 4 - 7m
i.
-3f – 2 = 7
j.
5k – 6 = 24
k.
12 = 12 – 3b
l.
-2 – 5b = -12
MFM 1P - Grade 9 Applied Mathematics – Unit 6: Multiple Representations (DPCDSB Dec 2008)
6-12
6.3.1: Mathematical Models
Each situation has a graphical model (graph), an algebraic model (equation) and a numerical
model (table of values). Choose either the graphical model or the algebraic model to complete
the table of values. Show your work and justify your choice of model.
1. Big Pine Outfitters charges a base fee of $40 and $10 per hour of use.
C represents the total cost ($) and t represents the numbers of hours the canoe is used.
Algebraic Model:
C = 40 + 10t
Graphical Model:
Cost ($)
Big Pine
90
80
70
60
50
40
30
20
10
0
0
1
2
3
4
Time (h)
t (h)
a)
Numerical Model:
Solutions:
a)
b)
C ($)
0
b)
70
c)
230
c)
MFM 1P - Grade 9 Applied Mathematics – Unit 6: Multiple Representations (DPCDSB Dec 2008)
6-13
6.3.1: Mathematical Models (continued)
2. A rental car costs $50 per day plus $0.20 for each kilometre it is driven.
C represents the total cost ($) and d represents the distance (km).
Algebraic Model:
C = 50 + 0.2d
Car Rental
Cost ($)
150
Graphical Model:
125
100
75
50
25
0
0
100
200
300
400
500
600
distance (km)
d (km)
Numerical Model:
a)
250
b)
1000
c)
Solutions:
a)
b)
C ($)
300
c)
Justify your choice.
MFM 1P - Grade 9 Applied Mathematics – Unit 6: Multiple Representations (DPCDSB Dec 2008)
6-14
6.3.1: Mathematical Models (continued)
3.
Algebraic Model:
y = -3x + 5
(label the axes)
30
20
10
Graphical Model:
-2
0
-1
0
-10
1
2
3
4
5
6
7
8
9
10
-20
-30
x
Numerical Model:
a)
0
b)
6
c)
y
-55
Solutions:
a)
b)
c)
Justify your choice.
Challenge
Describe a situation that could be modelled with the given graph or equation.
MFM 1P - Grade 9 Applied Mathematics – Unit 6: Multiple Representations (DPCDSB Dec 2008)
6-15
6.3.2: Solving Equations Using Substitution
1. Solve the following equations for y if x = 4:
a. y = 2x – 6
b.
2x + y = 3
c.
4x = 12 - y
2. Fiona has 300 m of fencing to surround a vegetable garden. If the width of the garden is 10
m, what is the length?
Hint: P = 2l + 2w
3. A carpenter is making a circular tabletop with circumference 4.5 m. What is the radius of the
tabletop in centimetres?
Hint: C=πd
MFM 1P - Grade 9 Applied Mathematics – Unit 6: Multiple Representations (DPCDSB Dec 2008)
6-16
6.3.2: Solving Equations Using Substitution (continued)
4a.
The formula for the final amount, A, in an investment with principal, P, and Interest, I, is A
= P + I.
Determine the principal if A is $6000 and I is $750.
4b.
The interest, I, is calculated by I = Prt, where P is the amount of the principal from (a), r is
the interest rate and t is the number of years the principal was invested.
Determine the number of years the principal was invested if the interest earned is $750
when the interest rate is 6%.
5.
Extend your thinking. Find the missing value:
a. A = π r2
where A = 63.585 cm2
b.
A=
bh
1
bh or A =
2
2
where A = 27.3 and b =6.5
MFM 1P - Grade 9 Applied Mathematics – Unit 6: Multiple Representations (DPCDSB Dec 2008)
6-17
6.3.P: Real World Mathematical Models
1. To fix a car, Joe’s Garage charges a base fee of $25 and $40/h.
a. Make a table of values of the cost of fixing a car for each hour up to 4 hours.
Number of hours
(h)
Cost to Fix the Car
(C)
b. Using your table from (a), calculate the first differences and the rate of change.
Number of hours
(h)
Cost to Fix the Car
(C)
Rate of Change =
Difference in C
Difference in h
First Difference
MFM 1P - Grade 9 Applied Mathematics – Unit 6: Multiple Representations (DPCDSB Dec 2008)
6-18
6.3.P: Real World Mathematical Models (continued)
c. Graph the cost of fixing a car for up to 4 hours.
d. Identify the rate of change and the initial value. What do they mean in this problem?
e. Determine an equation to model the graph.
f.
Determine the cost of a 2.5 hour repair job.
Show your work using the equation from part (e).
Check your answer using your graph from part (d).
g. What does the point (6, 265) represent?
h.
If it costs $155, how long was spent working on the car?
MFM 1P - Grade 9 Applied Mathematics – Unit 6: Multiple Representations (DPCDSB Dec 2008)
6-19
6.4.1: Planning a Special Event
Work through Menu 1 with your teacher as a class.
Maxwell’s Catering Company prepares and serves food for large gatherings. They charge a
base fee of $200 for renting the facility, plus a cost per person based on the menu chosen.
Menu 1 is a buffet that costs $10 per person.
Menu 2 is a three-course meal that costs $14 per person.
Menu 3 is a five-course meal that costs $18 per person.
1. Complete the table of values for each relation: [*Note: n must go up by equal increments]
Menu 1: C = 10n + 200
n
No. of
people
C
Cost
($)
First
Difference
Menu 2: C = 14n + 200
n
No. of
people
C
Cost
($)
First
Difference
25
0
50
50
75
100
100
150
125
200
*n goes up by 25
*n goes up by 50
Menu 3: C = 18n + 200
n
No. of
people
C
Cost
($)
First
Difference
*n goes up by ____
2. a) Graph the 3 relations on
the same set of axes.
Use an appropriate scale,
labels, and title.
b) Explain whether to use
dashed or solid lines to
draw these graphs.
MFM 1P - Grade 9 Applied Mathematics – Unit 6: Multiple Representations (DPCDSB Dec 2008)
6-20
6.4.1: Planning a Special Event (continued)
3. a) Identify the rate of change and the initial amount of the Menu 1 line. How do these relate
to the total cost?
What does it mean in this problem?
Rate of change:
Initial amount:
b) Identify the rate of change and the initial amount of the Menu 2 and 3 lines.
Line
Rate of change
Initial amount
2
3
4. a) Examine the first differences and the increment in n.
Line
Increment in n
1
25
2
50
First Differences
First Differences
Increment in n
3
b) How do they relate to the graph and the equation?
MFM 1P - Grade 9 Applied Mathematics – Unit 6: Multiple Representations (DPCDSB Dec 2008)
6-21
6.4.1: Planning a Special Event (continued)
5. Compare the three graphs. How are the graphs the same? different?
Same
Different
6. a) For Menu 2, what does the ordered pair (120, 1780) mean?
b) For Menu 3, what does the ordered pair (80, 1540) mean?
MFM 1P - Grade 9 Applied Mathematics – Unit 6: Multiple Representations (DPCDSB Dec 2008)
6-22
6.4.1: Planning a Special Event (continued)
7. Seventy people are expected to attend a school event. How much will it cost for each
menu?
Menu
Cost (show your work)
1
2
3
8. Vadim and Sheila are planning a celebration. They have $3000 to spend on dinner. They
would like to have Menu 3. What is the greatest number of guests they can have?
9. Logan’s Plastics employs 50 people. Each year the company plans a party for its
employees.
a) Find the cost for Menu 2 and write your answer as the ordered pair (50, C).
b) Find the cost for Menu 3 and write your answer as the ordered pair (50, C).
c) How many more dollars will Logan’s Plastics have to pay if they choose Menu 3 instead
of Menu 2?
MFM 1P - Grade 9 Applied Mathematics – Unit 6: Multiple Representations (DPCDSB Dec 2008)
6-23
6.4.2: The Cellular Phone Problem Practice
Two cellular phone companies have a monthly payment plan. They charge a flat fee plus a fee
for each minute used.
Call-A-Lot plan
Talk-More plan
C = 0.50t +20
C = 0.25t +25
Where C represents the total monthly cost and
t represents the number of minutes.
1. Create a table of values showing the total charges for a month for up to 30 minutes.
(Remember to make time go up by the same amount for each interval.)
Call-A-Lot
t
(time in
minutes)
C
(cost in $)
Talk-More
First
Difference
t
(time in
minutes)
C
(cost in $)
First
Difference
2. a) Graph the relations on the
same set of axes.
Use an appropriate scale.
b) Independent variable:
Dependent variable:
MFM 1P - Grade 9 Applied Mathematics – Unit 6: Multiple Representations (DPCDSB Dec 2008)
6-24
6.4.2: The Cellular Phone Problem Practice (continued)
3. Identify the rate of change and the initial value of the Call-A-Lot line. Explain what each
means in this problem.
What does it mean in this problem?
Rate of change:
Initial value:
4. Examine the differences. How do they relate to the graph and the equation?
(Hint: calculate C differences ).
t differences
5. Compare the graphs. How are the graphs…
a) the same?
b) different?
6. For Talk-More, what does the ordered pair (8, 27) mean?
7. One month, Leslie used 13 minutes on the Talk-More plan. How much did it cost her?
MFM 1P - Grade 9 Applied Mathematics – Unit 6: Multiple Representations (DPCDSB Dec 2008)
6-25
6.4.2: The Cellular Phone Problem Practice (continued)
8. Arjun had a bill of $29 last month on the Call-A-Lot plan. How many minutes did he use the
phone?
9. Marsha thinks that she will use an average of 12 minutes each month.
a) Find the cost for the Call-A-Lot plan and write as the ordered pair (12, C).
b) Find the cost for the Talk-More plan and write as the ordered pair (12, C).
c) Which plan is better for Marsha and how much will she save with this plan?
MFM 1P - Grade 9 Applied Mathematics – Unit 6: Multiple Representations (DPCDSB Dec 2008)
6-26
6.5.1: An Environmental Project
A coaches B
For a project on the environment, you have decided to gather data on two similar types of
vehicles – an SUV and a minivan. Compare the distance that the vehicles can travel on a full
tank of gasoline. For each kilometre a vehicle is driven, the gasoline is used at the given rate.
SUV
G = 80 – 0.20d, where G represents the amount of gasoline remaining in
litres and d represents the number of kilometres driven
Minivan
G = 65 – 0.15d, where G represents the amount of gasoline remaining in
litres and d represents the distance travelled in kilometres
1. Create a table of values showing the amount of gasoline remaining for up to 400 km.
Note: d must go up by the same amount each time.
SUV
d
(distance
in km)
G
(gasoline
remaining
in litres)
Minivan
First
Difference
d
(distance
in km)
G
(gasoline
remaining
First
in litres) Difference
0
100
200
300
400
Independent variable:
Dependent variable:
2. a) Graph the relations on the
same set of axes.
Use an appropriate scale,
labels, and a title.
b) Explain how you know that
this data is continuous.
MFM 1P - Grade 9 Applied Mathematics – Unit 6: Multiple Representations (DPCDSB Dec 2008)
6-27
6.5.1: An Environmental Project (continued)
B coaches A
3. Identify the rate of change and the initial value of the SUV.
What does it mean in this problem?
Rate of change:
Initial value:
4. Examine the differences. How do they relate to the graph and the equation?
(Hint: calculate G differences .)
d differences
5. Compare the graphs. How are the graphs…
a) the same?
b) different?
6. For the minivan, what does the ordered pair (100, 50) mean?
7. If the SUV is driven 250 km, how much gasoline is left?
MFM 1P - Grade 9 Applied Mathematics – Unit 6: Multiple Representations (DPCDSB Dec 2008)
6-28
6.5.1: An Environmental Project (continued)
A coaches B
8. If the minivan has 35 L of gasoline left, how far has it been driven since fill-up?
9. A vehicle has a full tank of gasoline and is driven 250 km.
a) Find the amount of gasoline remaining in the SUV and write the answer as the ordered
pair (250, G).
b) Find the amount of gasoline remaining in the minivan and write the answer as the
ordered pair (250, G).
c) Which vehicle has more gasoline remaining? How much more?
d) Explain what this problem tells you about the two vehicles.
MFM 1P - Grade 9 Applied Mathematics – Unit 6: Multiple Representations (DPCDSB Dec 2008)
6-29
6.5.2: Cooling It!
Denis measured the temperature of a cup of hot water as it cooled. He then made the graph on
the right. Complete the scale, and then answer the following questions about the graph.
a) One of the points on the graph is (6, 35).
Explain the meaning of this point, in the
context of Denis’ measurements.
b) Independent variable:
Dependent variable:
c) Explain why this is continuous data.
d) Use your graph to determine the temperature after 3.5 minutes.
e) Identify the rate of change and the initial value and explain what they mean in this problem.
What do they mean in this problem?
Rate of change:
Initial value:
MFM 1P - Grade 9 Applied Mathematics – Unit 6: Multiple Representations (DPCDSB Dec 2008)
6-30
6.5.2: Cooling It! (continued)
f)
Write an equation to model Denis’ data. Use T for temperature and t for time.
g) Use your equation to determine the temperature of the water after:
i) 3.5 minutes
ii) 20 minutes
h) Your results for 20 minutes may conflict with what you know about cooling water. Explain.
What does this tell you about the limitations of this linear model?
i)
Use your equation to predict when the temperature will be 39°C.
MFM 1P - Grade 9 Applied Mathematics – Unit 6: Multiple Representations (DPCDSB Dec 2008)
6-31
6.6.1: Linear and Non-Linear Investigations
Investigation 1 – Building Crosses
Purpose
Find the relationship between the figure number and the total number of cubes.
Procedure
Using linking cubes, make two more figures by adding a cube to each end of the cross.
Hypothesis
Write your hypothesis on the Record Sheet.
• We think that as the figure number increases, the total number of cubes will
increase or decrease
because ______________________________________ .
•
We think that the relationship will be
•
The data is
continuous or discrete
linear or non -linear
.
.
Mathematical Models
•
•
•
Record your observations in the table provided and calculate the first differences.
Make a scatter plot and draw the line (or curve) of best fit.
Determine the algebraic model or describe the relationship in words.
Numerical:
Complete the table of values and calculate the differences.
Figure #
Cubes #
First Differences
MFM 1P - Grade 9 Applied Mathematics – Unit 6: Multiple Representations (DPCDSB Dec 2008)
6-32
6.6.1: Linear and Non-Linear Investigations
Graphical:
Make a scatter plot and draw the line of best fit.
Algebraic Model: (or
a description of the relationship
in words)
1. How many cubes are required to make model number 10? Show your work.
2. What figure number will have 25 cubes?
3. How would adding two blocks to each end of the cross rather than one affect the graph and
the equation?
Conclusion
Make a conclusion. Refer to your hypothesis.
MFM 1P - Grade 9 Applied Mathematics – Unit 6: Multiple Representations (DPCDSB Dec 2008)
6-33
6.6.1: Linear and Non-Linear Investigations (continued)
Investigation 2 – Pass the Chocolate Bar
Purpose
Find the relationship between the number of pieces of “chocolate bar” remaining
and the total number of times the chocolate bar was passed around.
Procedure
Every time the chocolate bar is passed, you “eat” half ( 21 ) of what remains.
Hypothesis
Write your hypothesis on the Record Sheet.
•
We think that the more times the chocolate bar is passed, the number of pieces remaining
will
increase or decrease
because ____________________________________________.
•
We think that the relationship will be
•
The data is
continuous or discrete
linear or non -linear
.
.
Mathematical Models
•
•
•
•
Record the number of pieces of the chocolate bar that remain after 0 passes, 1 pass,
2 passes (up to 4 passes).
Calculate the first differences.
Make a scatter plot and draw the line (or curve) of best fit.
Determine the algebraic model or describe the relationship in words.
Numerical:
Complete the table of values and calculate the differences.
# Passes
# Pieces
First Differences
MFM 1P - Grade 9 Applied Mathematics – Unit 6: Multiple Representations (DPCDSB Dec 2008)
6-34
6.6.1: Linear and Non-Linear Investigations (continued)
Graphical:
Make a scatter plot and draw the line of best fit.
Algebraic Model: (or
a description of the relationship
in words)
1. How many pieces of chocolate bar will remain after 6 passes? Show your work.
2. Using this method of eating the chocolate bar, when will it be fully “eaten”? Explain.
3. If the chocolate bar began with 32 pieces instead of 16, how would the graph be different?
Include a sketch of the original graph and the new graph on the same set of axes.
Give reasons for your answer.
Conclusion
Make a conclusion. Refer back to your hypothesis.
MFM 1P - Grade 9 Applied Mathematics – Unit 6: Multiple Representations (DPCDSB Dec 2008)
6-35
6.6.1: Linear and Non-Linear Investigations (continued)
Investigation 3 – Area vs. Length of a Square
Purpose
Find the relationship between the area and the length of a side of a square.
Procedure
•
•
On grid paper, draw squares with side lengths of 1 cm, 2 cm, 3 cm, and 4 cm.
Draw and calculate the area of squares with sides measuring 1 cm, 2 cm, 3 cm, and 4 cm.
Hypothesis
Write your hypothesis on the Record Sheet.
•
We think that as the side length increases, the area will
increase or decrease
because ____________________________________________ .
•
We think that the relationship will be
•
The data is
continuous or discrete
linear or non -linear
.
.
Mathematical Models
•
•
•
Record your observations in the table provided and calculate the first differences.
Make a scatter plot and draw the line (or curve) of best fit.
Determine the algebraic model or describe the relationship in words.
Numerical:
Complete the table of values and calculate the differences.
Length
Area
First Differences
MFM 1P - Grade 9 Applied Mathematics – Unit 6: Multiple Representations (DPCDSB Dec 2008)
6-36
6.6.1: Linear and Non-Linear Investigations (continued)
Graphical:
Make a scatter plot and draw the line of best fit.
Algebraic Model: (or
a description of the relationship
in words)
1. What is the area of a square with a side length of 9 cm?
2. What side length does a square with an area of 100 cm2 have?
3. Describe the pattern in the first differences.
Conclusion
Make a conclusion. Refer back to your hypothesis.
MFM 1P - Grade 9 Applied Mathematics – Unit 6: Multiple Representations (DPCDSB Dec 2008)
6-37
6.6.1: Linear and Non-Linear Investigations (continued)
Investigation 4 – Burning the Candle at Both Ends
Purpose
Find the relationship between the number of blocks and the figure number.
Procedure
•
•
•
Using cube links, build a long chain with 20 blocks.
To create the next figure, remove 1 block from each end.
Record the number of blocks remaining.
Repeat this process four more times.
Hypothesis
Write your hypothesis on the Record Sheet.
• We think that as the figure number increases, the total number of blocks will
increase or decrease
because ________________________________________.
•
We think that the relationship will be
•
The data is
continuous or discrete
linear or non -linear
.
.
Mathematical Models
•
•
•
Record your observations in the table provided and calculate the first differences.
Make a scatter plot and draw the line (or curve) of best fit.
Determine the algebraic model or describe the relationship in words.
Numerical:
Complete the table of values and calculate the differences.
Figure #
# of Cubes
First Differences
MFM 1P - Grade 9 Applied Mathematics – Unit 6: Multiple Representations (DPCDSB Dec 2008)
6-38
6.6.1: Linear and Non-Linear Investigations (continued)
Graphical:
Make a scatter plot and draw the line of best fit.
Algebraic Model: (or
a description of the relationship
in words)
1. How many cubes are required to make figure number 7? Show your work.
2. What figure number will have 4 cubes?
3. How would removing 2 blocks from each end of the "candle" rather than 1 affect the graph
and the equation?
4. If 5 more blocks were added to the original model, how would that affect the graph and the
equation?
Conclusion
Form a conclusion. Refer back to your hypothesis.
MFM 1P - Grade 9 Applied Mathematics – Unit 6: Multiple Representations (DPCDSB Dec 2008)
6-39
6.6.J: Journal Activity
Sally was not in class today. She doesn’t know how to use differences to determine if a
relationship is linear or non-linear. Use words, pictures, and symbols to explain it to her.
MFM 1P - Grade 9 Applied Mathematics – Unit 6: Multiple Representations (DPCDSB Dec 2008)
6-40
6.W: Definition Page
Term
Picture / Sketch
Definition
Expression
Equation
Algebra Tiles
One Step Equations
Two Step Equations
Algebraic Model
Graphical Model
Numerical Model
Subsitution
First Difference
Non-linear Relation
MFM 1P - Grade 9 Applied Mathematics – Unit 6: Multiple Representations (DPCDSB Dec 2008)
6-41
6.S: Unit Summary Page
Unit Name: ____________________________________________
Using a graphic organizer of your choice create a unit summary.
MFM 1P - Grade 9 Applied Mathematics – Unit 6: Multiple Representations (DPCDSB Dec 2008)
6-42
6.R Reflecting on My Learning (3, 2, 1)
3 Things I know well from this unit
2 Things I need explained more
1 Question I still have
MFM 1P - Grade 9 Applied Mathematics – Unit 6: Multiple Representations (DPCDSB Dec 2008)
6-43
6.RLS: Reflecting on Learning Skills
Students should be aware of the importance that these skills have on your performance. After
receiving your marked assessment, answer the following questions. Be honest with yourself.
Good Learning Skills will help you now, in other courses and in the future.
•
•
•
•
E – Always
G – Sometimes
S – Need Improvement
N – Never
Organization
•
E G S N
•
E G S N
•
E G S N
I came prepared for class with all materials
My work is submitted on time
I keep my notebook organized.
Work Habits
•
E G S N
•
E G S N
•
E G S N
•
E G S N
•
E G S N
•
E G S N
I attempt all of my homework
I use my class time efficiently
I limit my talking to the math topic on hand
I am on time
If I am away, I ask someone what I missed,
I complete the work from the day that I missed.
Team Work
•
E G S N
•
E G S N
•
E G S N
I am an active participant in pairs/group work
I co-operate with others within my group
I respect the opinions of others
Initiative
•
E G S
•
E G S
•
E G S
•
E G S
I participate in class discussion/lessons
When I have difficulty I seek extra help
After I resolve my difficulties, I reattempt the problem
I review the daily lesson/ideas/concepts
N
N
N
N
Works Independently
•
E G S N
I attempt the work on my own
•
E G S N
I try before seeking help
•
E G S N
If I have difficulties I ask others but I stay on task
•
E G S N
I am committed to tasks at hand
Yes No
I know all the different ways available in my school, where I can seek extra help.
Yes No
I tried my best.
What will I do differently in the next unit to improve?
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
MFM 1P - Grade 9 Applied Mathematics – Unit 6: Multiple Representations (DPCDSB Dec 2008)
6-44
Course: Grade 9 Applied Mathematics (MFM1P)
Unit 7:
Algebraic Models: Making Connections
Unit 7
Algebraic Models: Making Connections
Section
7.1.1
7.1.2
7.1.3
7.2.1
7.2.2
7.3.3
7.3.P
7.4.3
7.4.P
7.5.1
7.5.2
7.5.3
7.5.4
7.6.2
7.6.P
7.7.1
7.7.2
7.7.P
7.7.J
4.W
4.S
4.R
4.RLS
Activity
Linear and Non-Linear Investigations
We’re All Correct
Feeding Frenzy
Equivalent Algebraic Expressions
Exploring The Distributive Property
Adding and Subtracting Polynomials
Practice
Algebraic Expressions
Practice
Practice
We’re All Correct Using Algebra
The Frame Problem
The Walkway Problem
Powers with Variable Bases
Practice
Solving Measurement Problems
Connecting Algebra to Geometry
Practice
Journal Activity
Definitions
Unit Summary
Reflecting on My Learning (3, 2, 1)
Reflecting on Learning Skills
MFM 1P - Grade 9 Applied Mathematics – Unit 1: Measurement 2D & 3D (DPCDSB July 2008)
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3
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9
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29
34
36
37
38
39
40
7-2
7.1.1: Linear or Non-Linear
Complete the tables of values and determine if the relationship is linear or non-linear. Give reasons for
your answers.
Figure Number
Number of Shaded Circles
First Differences
1
2
3
4
5
This relationship is
linear or non-linear
because ________________________________________
If Linear, write equation: ____________________________
Figure Number
Number of Unshaded Circles
First Differences
1
2
3
4
5
This relationship is
linear or non-linear
because ________________________________________
If Linear, write equation: _________________________________
Figure Number
Total Number of Circles
First Differences
1
2
3
4
5
This relationship is
linear or non-linear
because ________________________________________
If Linear, write equation: _________________________________
MFM 1P - Grade 9 Applied Mathematics – Unit 1: Measurement 2D & 3D (DPCDSB July 2008)
7-3
7.1.1: Linear or Non-Linear
Complete the tables of values and determine if the
relationship is linear or non-linear.
Give reasons for your answers.
Figure Number
Number of Shaded Squares
First Differences
1
2
3
4
5
This relationship is
linear or non-linear
because ________________________________________
If Linear, write equation: _________________________________
Figure Number
Number of Unshaded
Squares
First Differences
1
2
3
4
5
This relationship is
linear or non-linear
because ________________________________________
If Linear, write equation: _________________________________
Figure Number
Total Number of Squares
First Differences
1
2
3
4
5
This relationship is
linear or non-linear
because ________________________________________
If Linear, write equation: _________________________________
MFM 1P - Grade 9 Applied Mathematics – Unit 1: Measurement 2D & 3D (DPCDSB July 2008)
7-4
7.1.2: We’re All Correct!
Reconciling Equivalent Algebraic Expressions
How many toothpicks are needed for n squares?
Show a picture of each student’s thinking. Explain why each solution is correct.
Anju’s Solution
Erin’s Solution
“If T is the number of toothpicks
and n is the number of squares,
then the number of toothpicks is
equal to 1 plus three times the
number of squares.”
“If T is the number of toothpicks
and n is the number of squares,
then the number of toothpicks is
equal to 4 plus three times one
less than the number of squares.”
My equation is T = 1 + 3n.
My equation is T = 4 + 3(n –1).
n
1
2
3
4
5
6
T
n
1
2
3
4
5
6
T
Silva’s Solution
Bijuan’s Solution
“If T is the number of toothpicks
and n is the number of squares,
then the number of toothpicks is
equal to 2 times the number of
squares plus one more than the
number of squares.”
“If T is the number of toothpicks
and n is the number of squares,
then the number of toothpicks is
equal to 4 times the number of
squares minus one less than the
number of squares.”
My equation is T = 2n + (n + 1).
My equation is T = 4n – (n –1).
n
1
2
3
4
5
6
T
n
1
2
3
4
5
6
T
MFM 1P - Grade 9 Applied Mathematics – Unit 1: Measurement 2D & 3D (DPCDSB July 2008)
7-5
7.1.3: Feeding Frenzy – Patterning to Algebraic Modelling
Part A
Frieda runs a catering business. She often has to set up table arrangements like the ones
shown below.
Help her determine the number of chairs and/or tables that she needs.
1. Start by completing the Number of Chairs column.
(You may wish to use the algebra tiles to create physical models for terms 4, 5, and 6.)
Term
Number
1
Picture
Number of
Chairs
Expression
#1
#2
6
2
3
4
5
6
2. a) Build a number pattern in the last column.
b) Use the number pattern to find an expression for the Number of Chairs if the term
number is n.
3. Find a different but equivalent algebraic model. Explain how it relates to the picture model.
MFM 1P - Grade 9 Applied Mathematics – Unit 1: Measurement 2D & 3D (DPCDSB July 2008)
7-6
7.1.3: Feeding Frenzy (continued)
Part B
Frieda runs a catering business. She often has to set up table arrangements like the ones
shown below. Frieda sometimes uses trapezoidal tables.
Help her determine the number of chairs and/or tables that she needs.
1. Start by completing the Number of Chairs column. (You may wish to use pieces from the
pattern blocks set to create physical models for terms 4 and 5.)
Term
Number
1
Picture
Number of
Chairs
Expression
#1
#2
5
2
3
4
5
2. a) Build a number pattern in the last column.
b) Use the number pattern to find an expression for the Number of Chairs for n tables.
c) Explain how your answer to part (b) relates to the picture model.
3. Find a different but equivalent algebraic model. Explain how it relates to the picture model.
MFM 1P - Grade 9 Applied Mathematics – Unit 1: Measurement 2D & 3D (DPCDSB July 2008)
7-7
7.2.1: Equivalent Algebraic Expressions
1. Complete the following tables of values:
a) y = 3(x − 1)
x
b) y = 3x − 3
y
x
0
0
1
1
2
2
3
3
4
4
y
2. How do the tables compare? ___________________________________________
3. Graph the two relations below.
4. What do you notice about the lines? Do you think the lines are the same or different? (Make
sure you make the scale the same)
MFM 1P - Grade 9 Applied Mathematics – Unit 1: Measurement 2D & 3D (DPCDSB July 2008)
7-8
7.2.2: Exploring the Distributive Property (Using CAS or Algebra Tiles)
Goal: Using the Nspire CAS Handheld, expand 2(x + 1)
Instructions
•
•
•
•
Screenshot
Turn on the Nspire CAS handheld
Open a New Document (HOME + 6: New Document)
Choose NO to “Do you want to save this document?”
Insert a 1. Calculator page.
• Using the green keys, type expand(2(x+1))
• Press Enter
• An equivalent expanded expression is displayed on the right.
Source: http://www.ti-nspire.com/tools/nspire/index.html
MFM 1P - Grade 9 Applied Mathematics – Unit 1: Measurement 2D & 3D (DPCDSB July 2008)
7-9
7.2.2: Exploring the Distributive Property (Using CAS or Algebra Tiles)
Using Nspire CAS or Algebra Tiles, complete the table by expanding each expression:
2 (3)
2 (x + 1)
2 (2x – 1)
2 (3x + 4)
2 (5x – 10)
2 (4m + 3n)
2 (x2 – 2x)
2 (2m2 – 3m + 5)
2 (4g – 5h + 3k – 2l)
What happens when you multiply the
monomial 2 by each bracket?
3 (4)
3 (x + 1)
3 (2x – 1)
3 (3x + 4)
3 (6x – 12)
3 (5d + 4g)
3 (x3 – 2x)
3 (2m2 – 3m + 6)
3 (4g – 5h + 3k – 2l)
What happens when you multiply the
monomial 3 by each bracket?
4 (3)
4 (x + 1)
4 (2x – 1)
4 (3x + 4)
4 (5x – 10)
4 (4m + 3n)
4 (x2 – 2x)
4 (2m2 – 3m + 5)
4 (4g – 5h + 3k – 2l)
What happens when you multiply the
monomial 4 by each bracket?
5 (4)
5 (x + 1)
5 (2x – 1)
5 (3x + 4)
5 (6x – 12)
5 (5d + 4g)
5 (x3 – 2x)
5 (2m2 – 3m + 6)
5 (4g – 5h + 3k – 2l)
What happens when you multiply the
monomial 5 by each bracket?
Describe how to multiply a monomial by a polynomial.
MFM 1P - Grade 9 Applied Mathematics – Unit 1: Measurement 2D & 3D (DPCDSB July 2008)
7-10
7.2.2: Exploring the Distributive Property (Using CAS or Algebra Tiles)
Expand the following expressions to write an equivalent algebraic expression for each. Then
verify using Nspire CAS or Algebra Tiles.
a) 2(x – 5)
b)
5(x + 1)
c)
4(3x – 1)
d) −3(2x + 4)
e)
2(4x – 5)
f)
−5(x + 4)
g) 6(3x2 – 2x + 4)
h)
2(5 – 5m + 6n)
i)
-3(3x – 4y + 5z)
MFM 1P - Grade 9 Applied Mathematics – Unit 1: Measurement 2D & 3D (DPCDSB July 2008)
7-11
7.3.3: Adding and Subtracting Polynomials
Part A: Picture Representation
1.
+
+
+
What do you have? Explain in words.
2.
++
What do you have? Explain in words.
3.
Huang Li's Order
Rena's Order
Mohammed's Order
Pavel's Order
Saroge's Order
What do you have to order? Explain in words.
4.
What do you have to order? Explain in words.
MFM 1P - Grade 9 Applied Mathematics – Unit 1: Measurement 2D & 3D (DPCDSB July 2008)
7-12
7.3.3: Adding and Subtracting Polynomials (Continued)
Part B: Algebra Tiles
1.
+
+
+
+
+
What do you have? Explain in words, and explain using algebra tiles.
2.
2x2 + 5x – 1 + 3x + 4. Using algebra tiles or draw pictures to represent this algebraic
expression.
3.
(x2 - 3x + 2) + (2x2 - 3x - 4). Using algebra tiles or draw pictures to represent this
algebraic expression.
4.
(2x2 - 5x + 3) - (x2 - 3x + 2). Using algebra tiles or draw pictures to represent this
algebraic expression.
5.
(3x2 - 3x + 2) + (2x2 - 3x - 4). Using algebra tiles or draw pictures to represent this
algebraic expression.
MFM 1P - Grade 9 Applied Mathematics – Unit 1: Measurement 2D & 3D (DPCDSB July 2008)
7-13
7.3.3: Adding and Subtracting Polynomials (Continued)
Part C: Algebraic Model
A Coaches B
1.
(x + 1) + (2x + 3)
B Coaches A
2.
(4x – 5) - (2x + 3)
3.
(x2 + 5x + 3) + (x2 + 6x – 2)
4.
(x2 + x + 2) + (x2 + x + 1)
5.
2x3 – 5x2 + 6x – 8 + 3x2 – 8x + 2
6.
2x – 3y + 5x2 – 6y2 – 3y + 2x – 2x2
7.
12a – 15b + 22a – 16b – 2a – 6b
8.
2a – 21a + 32b – 6b – 12b – 16a
MFM 1P - Grade 9 Applied Mathematics – Unit 1: Measurement 2D & 3D (DPCDSB July 2008)
7-14
7.3.P: Practice
1.
Use algebra tiles to show that the given three expressions are equivalent:
(a)
2.
3.
2 + 4n
(b)
1 + 2n + 2n + 1
(c)
6 + (n − 1)( 4)
Simplify each of the following using algebra:
(a)
3 x + 2x + 4 x
(b)
3 x + 4 + 2x + 1
(c)
2x + 5 − 2x − 3
(d)
− 3x + 6 − x − 8
Simplify first then evaluate where x = 2 and y = -1.
(a)
9 x + 2y − 4 x + 3 y
(b)
8 y − 4 y + 3 x + 2y
MFM 1P - Grade 9 Applied Mathematics – Unit 1: Measurement 2D & 3D (DPCDSB July 2008)
7-15
7.4.3: Algebraic Expressions
Simplify each algebraic expression.
Create a word statement for your answer.
The first question is completed as an example.
Algebraic Expression
1. 4x + 20 – 3x + 6
= x + 26
Word Statement
twenty-six more than a number
2. 3(2x – 4)
3. 2(x + 4)
4. 5x – 3 + 2(x + 1)
5. 3(2x + 3) – 2(2x + 3)
6. (3x2 + 4x – 3) + (2x2 – 2x + 1)
MFM 1P - Grade 9 Applied Mathematics – Unit 1: Measurement 2D & 3D (DPCDSB July 2008)
7-16
7.4.P: Practice
1.
2.
Simplify.
(a)
4 x + 8 − 2x + 3
(b)
2(3 x − 5)
(c)
2(x − 3) + 4 x
(d)
3 x − 2(x − 4)
(e)
4(2x − 3) − 3(x + 5)
(f)
(2x
(g)
(3 x
2
) (
+ 5 x − 6 − x 2 + 2x − 8
2
) (
− 3x + 1 + x 2 + 5x + 3
)
)
Write an algebraic expression for each of the following:
(a)
three more than a number
_________________
(b)
eight less than twice a number
_________________
(c)
a number increased by six times a different number
_________________
MFM 1P - Grade 9 Applied Mathematics – Unit 1: Measurement 2D & 3D (DPCDSB July 2008)
7-17
7.5.1: Practice
1.
Simplify.
(a)
3x + 2 − 5 + 4x
(b)
y − 6 + 2 − 3y
(c)
2( x + 10 )
(d)
2(5m + 3) − 3(2m − 6)
2. a) Describe the pattern in the Output column.
Input
1
2
3
4
5
n
Output
8
11
14
17
20
?
b) Determine two equations to represent the relationship between the input, n, and the
output.
c) If the input value is 20, what is the output value? Use the equations from part (b).
d) If the output value is 41, determine the input value.
MFM 1P - Grade 9 Applied Mathematics – Unit 1: Measurement 2D & 3D (DPCDSB July 2008)
7-18
7.5.2: We’re All Correct! Using Algebra
Reconciling Equivalent Algebraic Expressions
How many toothpicks are needed for n squares?
Expand and simplify to show why each solution is correct.
Anju’s Solution
Erin’s Solution
“If T is the number of toothpicks
and n is the number of squares,
then the number of toothpicks is
equal to 1 plus three times the
number of squares.”
“If T is the number of toothpicks
and n is the number of squares,
then the number of toothpicks is
equal to 4 plus three times one
less than the number of squares.”
My equation is T = 1 + 3n.
My equation is T = 4 + 3(n –1).
Silva’s Solution
Bijuan’s Solution
“If T is the number of toothpicks
and n is the number of squares,
then the number of toothpicks is
equal to 2 times the number of
squares plus one more than the
number of squares.”
“If T is the number of toothpicks
and n is the number of squares,
then the number of toothpicks is
equal to 4 times the number of
squares minus one less than the
number of squares.”
My equation is T = 2n + (n + 1).
My equation is T = 4n – (n –1).
MFM 1P - Grade 9 Applied Mathematics – Unit 1: Measurement 2D & 3D (DPCDSB July 2008)
7-19
7.5.3: The Frame Problem
Problem
The Capture-It Company makes picture frames.
Tiles are used for the border of the frames.
The light area represents the square space for the picture.
Procedure
Marla and Tim work together to find an algebraic model to represent this problem.
They build models with colour tiles and count the number of dark tiles needed on pictures of
different sizes.
Frame
Number
(n)
1
2
Number of
Dark Tiles (d)
Marla and Tim determined different equations to
represent the relationship between the frame number (n)
and the number of dark tiles (d).
Marla’s equation:
Tim’s equation:
d = 2(n + 2) + 2n
d = 4(n + 1)
3
4
5
6
1) Determine your own equation to represent the relationship between the frame (n) and the
number of dark tiles (d).
2) Compare your equation with Marla’s and Tim’s to determine that they represent equivalent
algebraic models.
MFM 1P - Grade 9 Applied Mathematics – Unit 1: Measurement 2D & 3D (DPCDSB July 2008)
7-20
7.5.4: The Walkway Problem
Problem
The Larry’s Landscaping Company makes walkways. One walkway starts with a
hexagonal piece of concrete. To make the walkway longer, square pieces are added.
This walkway begins with a hexagon and has three square pieces added. The length
of each side is the same. The perimeter of this walkway is 12.
Procedure
Cara and Cal work together to find an algebraic model to represent this problem. They build a
model with pattern blocks and determine the perimeter of the walkway. The perimeter only
includes sides on the outer edge of the walkway.
Number of
Squares
(n)
1
2
Perimeter
(P)
Cara and Cal determined different equations to
represent the relationship between the number of
squares (n) and the perimeter (P).
Cara’s equation:
P = 5 + 2n + 1
Cal’s equation:
P = 6 + 4n – 2n
3
4
5
6
1) Determine your own equation to represent the relationship between the number of squares
(n) and the perimeter (P)?
2) Compare your equation with Cara’s and Cal’s to determine that they represent equivalent
algebraic models.
MFM 1P - Grade 9 Applied Mathematics – Unit 1: Measurement 2D & 3D (DPCDSB July 2008)
7-21
7.6.2: Powers with Variable Bases
(Numeric, Graphical, and Algebraic Models)
Part 1
a) Complete the table of values for:
y = x(x + 3)
x
y
-2
-1
0
1
2
y = x2 + 3x
x
y
-2
-1
0
1
2
b) How do the tables compare?
c) Graph both relations on the grids below. Graph y = x(x + 3) in blue and graph y = x2 + 3x in
red.
d) How do the graphs compare?
e) The tables of values in a) and the graphs in b) are _______________.
f)
What must this mean about the expressions x(x + 3) and x2 + 3x?
g) What process would transform x(x + 3) into x2 + 3x?
MFM 1P - Grade 9 Applied Mathematics – Unit 1: Measurement 2D & 3D (DPCDSB July 2008)
7-22
7.6.2: Powers with Variable Bases
(Numeric, Graphical, and Algebraic Models) (continued)
Part 2
a) Complete tables of values and compare them for:
y = x(x2 + 2)
y = x3 + 2x
Y = x(x2 + 2)
x
y
-2
-1
0
1
2
y = x3 + 2x
x
y
-2
-1
0
1
2
b) What process would transform x(x2 + 2) into x3 + 2x?
c) Graph y = 2x(x – 2) and y = 2x2 – 4x on the same axes and compare the graphs.
What process would transform 2x(x – 2) into 2x2 – 4x?
10
10
8
8
6
6
4
4
2
2
y
y
5
5
-2
-2
d) Explain why y = x(x)(x) and y = x(x2) and y = x3 have identical graphs.
MFM 1P - Grade 9 Applied Mathematics – Unit 1: Measurement 2D & 3D (DPCDSB July 2008)
7-23
7.6.2: Powers with Variable Bases
(Numeric, Graphical, and Algebraic Models) (continued)
The process of distributing through the brackets is called “expansion” or “distribution.”
Expand the following:
1. 2x(x + 4)
2.
3x(x2 + 2x)
3. 4x(3x2 + 2x – 5)
4.
-3a(a2 – 4a)
5. 5x2(3x – 4)
Check your understanding
Three students were asked to expand this expression: x(x2 – 2x + 3x)
Kevin’s answer
2
x(x – 2x + 3x)
= x3 – 2x2 + 3x2
Sal’s answer
2
x(x – 2x + 3x)
= x(x2 + x)
= x3 + x2
Ari’s answer
x(x2 – 2x + 3x)
= x3 – 2x2 + 3x2
= x3 + x2
Which solution is the most efficient?
Explain your choice.
MFM 1P - Grade 9 Applied Mathematics – Unit 1: Measurement 2D & 3D (DPCDSB July 2008)
7-24
7.6.P: Practice
Expand the following:
1.
Expand and simplify.
(a)
r (r + 2)
(c)
3 x 2x 2 − 4 x − 3
(e)
d d 2 − 2d − d 2 (d − 5 )
(
(
)
)
(b)
2c (c 2 − 5c + 6)
(d)
2x( x 2 + 3 x ) + x 3 x − x 2
(f)
3 x 2 − 5 x − 6 − 2x x 2 − 8
(
(
)
)
(
MFM 1P - Grade 9 Applied Mathematics – Unit 1: Measurement 2D & 3D (DPCDSB July 2008)
)
7-25
7.7.1: Solving Measurement Problems
Problem A: Sam makes rectangular paving stones that are 10 cm longer than they are wide.
w + 10
w
1. Determine a formula for the perimeter in terms of w.
(Hint: formula for finding the perimeter of a rectangle is P =2(l + w))
2. Use this formula to calculate the perimeter when the width is 6.75 cm.
3. Use a graphing calculator to graph the equation describing the perimeter.
a) Write the equation you entered:
Y = _____________
b) Sketch a graph in the space provided below.
c) Trace to locate the (width, perimeter) corresponding to the calculation in question 2.
X = _______
Y = ______
4. Use the formula to calculate the width when the perimeter is 60 cm.
MFM 1P - Grade 9 Applied Mathematics – Unit 1: Measurement 2D & 3D (DPCDSB July 2008)
7-26
7.7.1: Solving Measurement Problems (continued)
Problem B: This diagram shows the size of the sides in terms of x.
x+3
2x
3x + 2
1. Determine a formula for the perimeter in terms of x.
(Hint: formula for finding the perimeter of a triangle is P = a + b + c)
2. Use this formula to calculate
a) the perimeter when x is 3 cm
b) the length of each of the sides when x = 3.
3. Use the graphing calculator to graph the equation describing the perimeter.
a) Write the equation you entered:
Y = _____________
b) Sketch a graph in the space provided below.
c) Use Trace to locate the point (x, perimeter) corresponding to the calculation in question
2a). You can also use [2nd] TABLE (over the GRAPH key) to see the table of values.
X = _______
Y = ______
4. Use the formula to calculate the value of x when the perimeter is 41 cm.
MFM 1P - Grade 9 Applied Mathematics – Unit 1: Measurement 2D & 3D (DPCDSB July 2008)
7-27
7.7.1: Solving Measurement Problems (continued)
Problem C: Explain how this model shows that the length is 2 times the width and the height is
3 times the width.
3x
2x
x
1. Determine a formula for the volume.
2. Use this formula to calculate the volume when the width is 225 m.
Challenge
Can you find the width of the shape that has a volume of 162 cm³?
MFM 1P - Grade 9 Applied Mathematics – Unit 1: Measurement 2D & 3D (DPCDSB July 2008)
7-28
7.7.2: Connecting Algebra to Geometry
1. Write an equation and solve for the unknown. State the theorem used to make the equation.
a)
b)
c)
d)
MFM 1P - Grade 9 Applied Mathematics – Unit 1: Measurement 2D & 3D (DPCDSB July 2008)
7-29
7.7.2: Connecting Algebra to Geometry
e)
f)
g)
h)
(continued)
MFM 1P - Grade 9 Applied Mathematics – Unit 1: Measurement 2D & 3D (DPCDSB July 2008)
7-30
7.7.2: Connecting Algebra to Geometry
(continued)
j)
i)
t°
j°
2t°
75°
l)
k)
2t° + 20°
2k°
2t°
k°
t°
MFM 1P - Grade 9 Applied Mathematics – Unit 1: Measurement 2D & 3D (DPCDSB July 2008)
7-31
7.7.2: Connecting Algebra to Geometry
m)
(continued)
n)
2n°
30°
80°
2k°
120°
3n°
MFM 1P - Grade 9 Applied Mathematics – Unit 1: Measurement 2D & 3D (DPCDSB July 2008)
7-32
7.7.2: Connecting Algebra to Geometry (continued)
2. a) The sum of the interior angles in
a triangle is:
b) An equation that models the sum
of the interior angles in this
triangle is:
c) Solve the equation to determine the value of x.
d) Use the value of x to calculate the size of:
∠W:
∠Y:
∠Z:
3. a) The sum of the angles in a right angle is:
b) Write 2 equations to model the sums of
the 2 sets of angles that add to 90º:
(i)
(ii)
c) Solve these equations to determine the values.
(i) solve for xº
(ii) solve for yº
d) Use the values of x and y to calculate the size of:
∠CBP:
∠ABQ:
MFM 1P - Grade 9 Applied Mathematics – Unit 1: Measurement 2D & 3D (DPCDSB July 2008)
7-33
7.7.P: Practice
1.
2.
Find the value of a.
Find the value of ∠ AXB .
B
Y
aº
70º
40º
A
3.
Find the value of each angle.
4.
X
Solve for x and y.
2xº
2xº
2xº
3xº
xº
xº
xº
MFM 1P - Grade 9 Applied Mathematics – Unit 1: Measurement 2D & 3D (DPCDSB July 2008)
yº
7-34
7.7.P: Practice (continued)
5.
Find the values of a and b.
6.
Find the value of the missing angles.
dº
bº
eº
67º
aº
7.
Solve for f.
110º
8.
Determine the value of g.
60º
gº
MFM 1P - Grade 9 Applied Mathematics – Unit 1: Measurement 2D & 3D (DPCDSB July 2008)
7-35
7.7.J: Journal Activity
Respond to the following question:
What jobs might use algebra to model measurement relationships?
MFM 1P - Grade 9 Applied Mathematics – Unit 1: Measurement 2D & 3D (DPCDSB July 2008)
7-36
7.W: Definition Page
Term
Picture / Sketch / Examples
Definition
Equivalent
Expand
Like Terms
Simplify
Polynomials
Base
Exponent
Power
MFM 1P - Grade 9 Applied Mathematics – Unit 1: Measurement 2D & 3D (DPCDSB July 2008)
7-37
7.S: Unit Summary Page
Unit Name: ____________________________________________
Using a graphic organizer of your choice create a unit summary.
MFM 1P - Grade 9 Applied Mathematics – Unit 1: Measurement 2D & 3D (DPCDSB July 2008)
7-38
7.R: Reflecting on My Learning (3, 2, 1)
3 Things I know well from this unit
2 Things I need explained more
1 Question I still have
MFM 1P - Grade 9 Applied Mathematics – Unit 1: Measurement 2D & 3D (DPCDSB July 2008)
7-39
7.RLS: Reflecting on Learning Skills
Students should be aware of the importance that these skills have on your performance. After
receiving your marked assessment, answer the following questions. Be honest with yourself.
Good Learning Skills will help you now, in other courses and in the future.
•
•
•
•
E – Always
G – Sometimes
S – Need Improvement
N – Never
Organization
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E G S N
•
E G S N
•
E G S N
I came prepared for class with all materials
My work is submitted on time
I keep my notebook organized.
Work Habits
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E G S N
•
E G S N
•
E G S N
•
E G S N
•
E G S N
•
E G S N
I attempt all of my homework
I use my class time efficiently
I limit my talking to the math topic on hand
I am on time
If I am away, I ask someone what I missed,
I complete the work from the day that I missed.
Team Work
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E G S N
•
E G S N
•
E G S N
I am an active participant in pairs/group work
I co-operate with others within my group
I respect the opinions of others
Initiative
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E G S
•
E G S
•
E G S
•
E G S
I participate in class discussion/lessons
When I have difficulty I seek extra help
After I resolve my difficulties, I reattempt the problem
I review the daily lesson/ideas/concepts
N
N
N
N
Works Independently
•
E G S N
I attempt the work on my own
•
E G S N
I try before seeking help
•
E G S N
If I have difficulties I ask others but I stay on task
•
E G S N
I am committed to tasks at hand
Yes No
I know all the different ways available in my school, where I can seek extra help.
Yes No
I tried my best.
What will I do differently in the next unit to improve?
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
MFM 1P - Grade 9 Applied Mathematics – Unit 1: Measurement 2D & 3D (DPCDSB July 2008)
7-40
Course: Grade 9 Applied Mathematics (MFM1P)
Unit 8:
Measurement Optimization
UNIT 8
MEASUREMENT OPTIMIZATION
Section
8.1.1
8.1.2
8.1.P
8.2.1
8.2.P
8.2.J
8.4.1
Activity
8.4.J
8.5.1
8.5.P
8.S
8.R
The Garden Fence
What is The Largest Rectangle?
Practice
Down by the Bay
Practice
Journal Activity
The Kittens with Mittens Come to Math Class:
Story
Scatter Plots on the Graphing Calculator
The Kittens with Mittens Come to Math Class:
Activity
Journal Activity
Greenhouse Commission
Practice
Unit Summary Page
Reflecting on My Learning (3, 2, 1)
8.RLS
Reflecting on Learning Skills
8.4.2
8.4.3
MFM 1P - Grade 9 Applied Mathematics – Unit 8: Measurement Optimization (DPCDSB July 2008)
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8-2
8.1.1: The Garden Fence
Problem
Your neighbour has asked for your advice about building his garden.
He wants to fence the largest rectangular garden with
20 metres of fencing.
Clarify the Problem
What are you being asked to determine?
What information is useful?
Explore
Use a geoboard or draw a diagram to show a model of one possible rectangular garden.
Hypothesize
What do you think the largest rectangular garden will look like? Sketch a picture of it with the
dimensions. Calculate the area and perimeter.
MFM 1P - Grade 9 Applied Mathematics – Unit 8: Measurement Optimization (DPCDSB July 2008)
8-3
8.1.2: What Is the Largest Rectangle?
Your neighbour has asked for your advice about building his garden.
He wants to fence the largest rectangular garden possible with
_____ metres of fencing.
Hypothesize
What do you think the largest rectangular garden will look like?
Explore
You can use manipulative, chart grid paper, markers, string, and rulers. Brainstorm strategies
you could use to determine the largest area.
Model
Choose a strategy and model your results. Try it out to determine the largest rectangle.
Conclude
Present your solution to the problem, checking that it satisfies all of the conditions and makes
sense.
MFM 1P - Grade 9 Applied Mathematics – Unit 8: Measurement Optimization (DPCDSB July 2008)
8-4
8.1.2: The Garden Fence (continued)
Model
Use the geoboard to help you complete the table of values for the garden.
Perimeter (m)
Width (m)
20
1
2
Area (m2)
l×w
Length (m)
Describe what happens to the area when the width of the garden increases.
Construct a scatter plot of area vs. width.
Area vs. Width
26
24
22
20
18
Area (m2)
16
14
12
10
8
6
4
2
1
2
3
4
5
6
7
8
9
10
11
Width (m)
MFM 1P - Grade 9 Applied Mathematics – Unit 8: Measurement Optimization (DPCDSB July 2008)
8-5
8.1.2: The Garden Fence (continued)
Manipulate
Look at the scatter plot.
Circle the region on the scatter plot where the area of the garden is the largest.
Construct two more sketches of garden areas with lengths and areas in this region.
Add these points to the scatter plot.
Conclude
What are the best dimensions for the garden? Justify your choice. Include a sketch and the area
of the garden that you are recommending.
MFM 1P - Grade 9 Applied Mathematics – Unit 8: Measurement Optimization (DPCDSB July 2008)
8-6
8.1.P: Practice
Solve the following problems. Draw diagrams for each problem.
1.
If the perimeter of a rectangle is 72 m, what is the largest area?
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MFM 1P - Grade 9 Applied Mathematics – Unit 8: Measurement Optimization (DPCDSB July 2008)
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8-7
8.1.P: Practice (continued)
2.
If the perimeter of a rectangle is 90 m, what is the largest area?
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MFM 1P - Grade 9 Applied Mathematics – Unit 8: Measurement Optimization (DPCDSB July 2008)
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8.1.P: Practice (continued)
Using the strategies you have discovered today, complete the following problems.
1.
(a) Calculate the perimeter of each of the following rectangles.
(i)
2.0 m
5.0 m
(ii)
4.2 m
2.3 m
(iii)
3.5 m
3.5 m
MFM 1P - Grade 9 Applied Mathematics – Unit 8: Measurement Optimization (DPCDSB July 2008)
8-9
8.1.P: Practice (continued)
(b)
Predict which rectangle will have the largest area.
List the rectangles in order from largest to smallest.
(c)
Find out if your prediction was true. Calculate the area of each of the rectangles.
MFM 1P - Grade 9 Applied Mathematics – Unit 8: Measurement Optimization (DPCDSB July 2008)
8-10
8.2.1: Down by the Bay
The city planners would like you to design a swimming area at a local beach. There is
100 m of rope available to enclose the swimming area. The shore will be one side of the
swimming area; so only three sides of the rectangle will
be roped off. It is your job to design the largest
rectangular swimming area.
Explore
It is possible to build a long, narrow swimming area.
90 m
5m
5m
Area = length × width
Area = 90 x 5
Area = 450 m2
Sketch three more swimming areas that have a larger area than this swimming area.
Label the dimensions on the sketch and calculate the area, as shown above.
Hypothesize
Predict the dimensions of the largest rectangular swimming area. _________
MFM 1P - Grade 9 Applied Mathematics – Unit 8: Measurement Optimization (DPCDSB July 2008)
8-11
8.2.1: Down by the Bay (continued)
Model
Complete the table with possible combinations of width and length for the swimming pools.
Calculate the area.
Perimeter (m)
Width, w, (m)
100
100
100
100
100
100
100
100
0
5
Length, l, (m)
Area, A, (m2)
l×w
Describe what happens to the area when the width of the swimming area increases.
Construct a scatter plot of area vs. width. Choose appropriate scales.
MFM 1P - Grade 9 Applied Mathematics – Unit 8: Measurement Optimization (DPCDSB July 2008)
8-12
8.2.1: Down by the Bay (continued)
Manipulate
Circle the region on the scatter plot where the area of the swimming area is the largest.
Construct two more sketches of swimming areas with widths and areas in the circled region.
Add these points to the scatter plot.
Conclude
Write a report to the town advising them of the dimensions that would be best for the new
swimming area. Justify your choice. Include a sketch and the area of the swimming area that
you are recommending.
MFM 1P - Grade 9 Applied Mathematics – Unit 8: Measurement Optimization (DPCDSB July 2008)
8-13
8.2.P: Practice
Using the strategies you have discovered today complete the following problems.
1.
You have 120 m of fence to enclose a rectangular area to be used for a snow sculpture
competition.
One side of the area is bounded by the school, so the fence is required for only three
sides of the rectangle.
Determine the dimensions of the maximum area that can be enclosed.
2.
The Peel Parks and Recreation Committee is discussing about enclosing a beach area
for swimmers in the summer. If they have 90 m of buoy rope, what is the largest area
they can enclose.
Remember that they will only put the buoy rope in the water and will not close off the
beach.
MFM 1P - Grade 9 Applied Mathematics – Unit 8: Measurement Optimization (DPCDSB July 2008)
8-14
8.2.P: Practice (continued)
Using the strategies you have discovered today, complete the following problems.
1.
(a) Calculate the perimeter of each of the following rectangles. Note: the dashed line
should not be included in your perimeter.
(i)
2.5 m
5.0 m
(ii)
3.8 m
2.4 m
(iii)
3.3 m
3.4 m
MFM 1P - Grade 9 Applied Mathematics – Unit 8: Measurement Optimization (DPCDSB July 2008)
8-15
8.2.P: Practice (continued)
(b)
Predict which rectangle will have the largest area.
List the rectangles in order from largest to smallest.
(c)
Find out if your prediction was true. Calculate the area of each of the rectangles.
MFM 1P - Grade 9 Applied Mathematics – Unit 8: Measurement Optimization (DPCDSB July 2008)
8-16
8.2.J: Journal Activity
Write a journal response:
Jessica wants to build a corral for her horses.
She has 65 m of fencing.
She wants the corral to be rectangular.
a)
What dimensions do you think Jessica should make the corral?
Use pictures, words and numbers to explain.
b)
She would like to build the coral next to a barn. What dimensions do you think Jessica
should make the corral? Use pictures, words and numbers to explain.
MFM 1P - Grade 9 Applied Mathematics – Unit 8: Measurement Optimization (DPCDSB July 2008)
8-17
8.4.1: The Kittens with Mittens Come to Math Class
The clock must have stopped as we sat in math class that day.
We'd never get out, I was certain there's no way.
Hally and I, we sat there, it's true.
Perimeter and area, we weren't sure what to do.
It was an investigation the teacher wanted us to complete.
We were to be very careful and especially neat.
"Look at me!” said the teacher, "Look at me now!
Regular shapes with the same perimeter, you have to know how!"
But as hard as I tried I could not stay awake,
Until a big BUMP caused me to shake.
I opened my eyes and our teacher was gone,
And the Kittens with Mittens were out on the lawn.
They strolled into our room with a box over their heads.
"Get ready to have fun!" is what they both said.
"In this box you will find something fickle!
Two little variables to get you out of this pickle.”
They jumped on the box and opened the lock,
And both of us were too excited to talk.
But slowly out of the box came Variable Two and Variable One.
They looked rather sad. They didn't want to have fun.
They explained to us that they were in a real bind.
They hadn't done their homework and they were really behind.
Their problem was in math as you could probably guess.
It was area and perimeter. What a coincidence? Yes?
They had 50 m of rope with which to enclose a rectangular ground for play.
Whoever enclosed the biggest area was champion for the day.
Two designs were required to be written down with our pen, this was not cool,
A 4-sided enclosure and also a 3-sided enclosure attached to the school.
Hally and I knew they needed our help, but what could we do?
We didn't listen to the area and perimeter lesson, too.
But then Hally jumped up and started to shout.
"We’ll do the investigation so we can figure it out!"
That's what we did for Variable One and Variable Two.
We found the answer, can you find it too?
MFM 1P - Grade 9 Applied Mathematics – Unit 8: Measurement Optimization (DPCDSB July 2008)
8-18
8.4.3: Scatter Plots on the Graphing Calculator
Scatter Plots.ppt
(Presentation software file)
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9
e.g. 0 enter, 2 enter
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MFM 1P - Grade 9 Applied Mathematics – Unit 8: Measurement Optimization (DPCDSB July 2008)
8-19
8.4.2: The Kittens with Mittens Investigations
Create graphical models on the graphing calculator to solve the Kittens with Mittens problems.
Understanding the Problem
1. In the last paragraph, highlight the key phrases that identify the two problems in the poem.
Entering the Data
2. To begin entering data, press STAT, then choose 1: Edit. Press ENTER.
3. Enter the width data into L1 (0, Press ENTER, 2, 4, 6, 8…24)
OR [Press 2nd STAT, choose 5:seq, Press (, press Alpha A, press ,, press Alpha A, press
1 (your starting value), press 24 (your ending value), press 2
(your step value), press )]
“ allows you keep
the formula in the
4. Move the cursor to the top of L2 (on top of the letters) and
calculator
press ENTER. Enter the formula for length. (Remember that
nd
2 , 1 gives you L1.)
Hint: Length = [50 – 2(width)]/2, so
for INVESTIGATION 1 you must enter Æ “(50 – 2 * L1)/2” (four-sided enclosure)
for INVESTIGATION 2 you must enter Æ “(50 – 2 * L1)” (three-sided enclosure)
5. Move the cursor to the top of L3 (on top of the letters) and press ENTER. Enter the formula
for area. (Remember that 2nd, 1 gives you L1 and 2nd, 2 gives you L2)
Hint: Area = Length x Width
so, you must enter Æ “L1 * L2”
6. To plot the data, press 2nd, Y= for [STATPLOT]. Select 1: Plot 1…Off and press ENTER.
Using the arrow keys < and > and the ENTER key:
Turn the graph on by setting On-Off to On.
Set the Type to a Line Graph (second picture on top row)
Check that the Xlist is L1.
Change the Ylist to L3 using 2nd, 3.
Set the Mark to
.
7. To set the viewing window for your graph, press ZOOM and use the arrow keys to select
9: ZoomStat.
8. To view the graph press ENTER.
9. Use the Trace feature to view the coordinate values of each point. Press TRACE. When you
press the arrow keys, you will be able to see the x and y values for each point.
MFM 1P - Grade 9 Applied Mathematics – Unit 8: Measurement Optimization (DPCDSB July 2008)
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8.4.2: The Kittens with Mittens Investigations (continued)
Investigation 1: The Four-sided Enclosure
1. Copy your data from the graphing calculator for the four-sided enclosure in the table below.
(To view the data, Press STAT, ENTER)
Perimeter (m)
50
50
50
50
50
50
50
50
50
50
50
50
50
L1
L2
Width, w, (m)
Length, l, (m)
L3
Area, A, (m2)
l×w
0
2
4
6
8
10
12
14
16
18
20
22
24
2. Draw a sketch of the graph shown on the screen of the calculator.
3. What variable is represented on the
horizontal axis?
600
500
4. What variable is represented on the
vertical axis?
400
5. Which variable is:
independent?
200
300
100
2
4
6
8
10
12
14
16
18
20
22
dependent?
6. Describe what happened to the area as the width increased.
MFM 1P - Grade 9 Applied Mathematics – Unit 8: Measurement Optimization (DPCDSB July 2008)
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24
8.4.2: The Kittens with Mittens Investigations (continued)
Investigation 2: The Three-sided Enclosure
1. Enter data for the three-sided enclosure in the table below.
Perimeter (m)
50
50
50
50
50
50
50
50
50
50
50
50
50
L1
L2
Width, w, (m)
Length, l, (m)
L3
Area, A, (m2)
l×w
0
2
4
6
8
10
12
14
16
18
20
22
24
2. Graph the area vs. width data on the grid.
600
500
3. What appears to be the relationship between
the area and the width?
400
300
200
100
2
4
6
8
10
12
14
16
18
20
22
4. Make a scatter plot of the same data using the graphing calculator. To do this, follow steps
2 to 6 from the calculator instructions. This time set the Type to a Scatter Plot (first picture
on top row). Continue with the rest of steps 7 to 10.
MFM 1P - Grade 9 Applied Mathematics – Unit 8: Measurement Optimization (DPCDSB July 2008)
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24
8.4.2: The Kittens with Mittens Investigations (continued)
5. How does this scatter plot compare to the graph that you drew?
6. Should the points be joined by a solid or a dashed line? Explain.
7. What recommendation would you make for the four-sided enclosure?
8. What recommendation would you make for the three-sided enclosure?
9. Refer back to the Kittens with Mittens Come to Math Class poem to decide if you will be the
“champion of the day?” Explain.
MFM 1P - Grade 9 Applied Mathematics – Unit 8: Measurement Optimization (DPCDSB July 2008)
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8.4.J: Journal Activity
Write a journal response to:
Your friend, Khalid who has his own graphing calculator, missed the class on how to make a
scatter plot and has asked you for help. Explain to him the steps for the graphing calculator
needed to:
• enter data into lists
• make scatter plot
• set the window settings
You may use words and pictures in your explanation.
MFM 1P - Grade 9 Applied Mathematics – Unit 8: Measurement Optimization (DPCDSB July 2008)
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8.5.1: Greenhouse Commission
Elaine and Daniel are building a rectangular greenhouse. They want the area of the floor to be
36 m2. Since the glass walls are expensive, they want to minimize the amount of glass wall they
use. They have commissioned you to design a greenhouse which minimizes the cost of the
glass walls.
Explore
It is possible to build a long,
1m
narrow greenhouse.
36 m
Area = 36 m2
Perimeter = 2l + 2w
= 2(36) + 2(1)
= 74 m
Sketch three more greenhouses that have a perimeter smaller than this greenhouse. Label the
dimensions on the sketch and calculate the perimeter.
Hypothesize
Based on your exploration, predict the length and the width of the greenhouse with the least
perimeter.
Model
Complete as much of the table as required to determine the dimensions that result in the least
perimeter. You may not need to fill in the whole table.
Area, A, (m2)
Width, w, (m)
Length, l, (m)
36
1
36
Perimeter (m) (P = 2l + 2w)
2(36) + 2(1) = 74
36
2
18
2(18) + 2(2) =
36
3
36
36
36
36
36
36
What happens to the perimeter of the greenhouse as the width increases?
MFM 1P - Grade 9 Applied Mathematics – Unit 8: Measurement Optimization (DPCDSB July 2008)
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8.5.1: Greenhouse Commission (continued)
Construct a graph of perimeter vs. width.
Conclude
Write a report for Elaine and Daniel, advising them of the dimensions that would be the best for
their greenhouse. Justify your recommendation using both the table and the graph. Include a
sketch and the perimeter of the greenhouse that you are recommending.
MFM 1P - Grade 9 Applied Mathematics – Unit 8: Measurement Optimization (DPCDSB July 2008)
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8.5.1: Greenhouse Commission (continued)
Apply
1. If the greenhouse is to have a height of 2 m and the price of the glass from Clear View
Glass is $46.75/m2, what will it cost to purchase the glass for the walls of the greenhouse?
Show all of your work.
2. Translucent Inc. charges $50/m2 for the first 30 m2 and then they give a 20% reduction on
the rest of the glass. From which company should Elaine and Daniel purchase the glass?
Explain fully and show all of your calculations.
MFM 1P - Grade 9 Applied Mathematics – Unit 8: Measurement Optimization (DPCDSB July 2008)
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8.5.P: Practice
1.
A daycare centre is required to have 0.25 m2 of floor space for each preschool age child.
(a)
If there are 22 children attending the centre, determine the total amount of floor
space needed.
(b)
A play area at the daycare centre has an area of 12.5 m2. How many preschool
age children can be in the play area at one time?
(c)
If the play area in (b) is rectangular with a length of 2.5 m, determine the width of
the play area, to the nearest tenth of a metre.
MFM 1P - Grade 9 Applied Mathematics – Unit 8: Measurement Optimization (DPCDSB July 2008)
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8.5.P: Practice (continued)
(d)
Area, A, (m2)
Find three other possible rectangles for the play area in (b). Be sure your
rectangles make sense. Remember preschool age children like to run and jump!
Width, w,
(m)
Length, l,
(m)
Perimeter (m)
(P=2l + 2w)
Diagram of
Rectangle
12.5
12.5
12.5
(e)
You are giving your idea to the manager of the daycare centre. Write a report
outlining which of the above rectangles you would recommend for the play area
at the daycare centre.
MFM 1P - Grade 9 Applied Mathematics – Unit 8: Measurement Optimization (DPCDSB July 2008)
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8.S Unit Summary Page
Unit Name: ____________________________________________
Using a graphic organizer of your choice create a unit summary.
MFM 1P - Grade 9 Applied Mathematics – Unit 8: Measurement Optimization (DPCDSB July 2008)
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8.R Reflecting on My Learning (3, 2, 1)
3 Things I know well from this unit
2 Things I need explained more
1 Question I still have
MFM 1P - Grade 9 Applied Mathematics – Unit 8: Measurement Optimization (DPCDSB July 2008)
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8.RLS: Reflecting on Learning Skills
Students should be aware of the importance that these skills have on your performance. After
receiving your marked assessment, answer the following questions. Be honest with yourself.
Good Learning Skills will help you now, in other courses and in the future.
E – Always
G – Sometimes
S – Need Improvement
N – Never
Organization
E G S N
E G S N
E G S N
I came prepared for class with all materials
My work is submitted on time
I keep my notebook organized.
Work Habits
E G S N
E G S N
E G S N
E G S N
E G S N
E G S N
I attempt all of my homework
I use my class time efficiently
I limit my talking to the math topic on hand
I am on time
If I am away, I ask someone what I missed,
I complete the work from the day that I missed.
Team Work
E G S N
E G S N
E G S N
I am an active participant in pairs/group work
I co-operate with others within my group
I respect the opinions of others
Initiative
E G S
E G S
E G S
E G S
I participate in class discussion/lessons
When I have difficulty I seek extra help
After I resolve my difficulties, I reattempt the problem
I review the daily lesson/ideas/concepts
N
N
N
N
Works Independently
E G S N
I attempt the work on my own
E G S N
I try before seeking help
E G S N
If I have difficulties I ask others but I stay on task
E G S N
I am committed to tasks at hand
Yes No
I know all the different ways available in my school, where I can seek extra help.
Yes No
I tried my best.
What will I do differently in the next unit to improve?
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
MFM 1P - Grade 9 Applied Mathematics – Unit 8: Measurement Optimization (DPCDSB July 2008)
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