Authors and Consultants First Nations, Métis

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Authors and Consultants
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Cathy Canavan-McGrath, Hay River, Northwest Territories
Serge Desrochers, Calgary, Alberta
Hugh MacDonald, Edmonton, Alberta
Carolyn Martin, Edmonton, Alberta
Michael Pruner, Vancouver, British Columbia
Hank Reinbold, St. Albert, Alberta
Rupi Samra-Gynane, Vancouver, British Columbia
Carol Shaw, Winnipeg, Manitoba
Roger Teshima, Calgary, Alberta
Darin Trufyn, Edmonton, Alberta
First Nations, Métis, and Inuit Cultural Reviewers
e-
Joanna Landry, Regina, Saskatchewan
Darlene Olson-St. Pierre, Edmonton, Alberta
Sarah Wade, Arviat, Nunavut
Cultural Reviewer
Pr
Karen Iversen, Edmonton, Alberta
Assessment Reviewer
Gerry Varty, Wolf Creek, Alberta
principles of Mathematics 11
editorial director
Linda Allison
director, Content and Media
production
Linh Vu
publisher, Mathematics
Colin Garnham
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Erynn Marcus
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product Manager
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ISBN-13: 978-0-17-650412-0
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Printed and bound in Canada
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For more information contact
Nelson Education Ltd.,
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Nelson Education Ltd.
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Authors
Cathy Canavan-McGrath
Michael Pruner
Carol Shaw
Darin Trufyn
Hank Reinbold
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n
lead educator
Chris Kirkpatrick
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Reviewers and Advisory Panel
Karen Buro
Instructor, Department of Mathematics and Statistics
MacEwan University
Edmonton, AB
Sean Chorney
Mathematics Department Head
Magee Secondary School
School District 39
Vancouver, BC
Carolyn Martin
Department Head of Mathematics
Archbishop MacDonald High School
Edmonton Catholic School Board
Edmonton, AB
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Hank Reinbold
Retired Educator
Morinville, AB
Rupi Samra-Gynane
Vice-Principal
Magee Secondary School
Vancouver School Board (#39)
Vancouver, BC
Steven Erickson
Mathematics Teacher
Sisler High School
Winnipeg School Division
Winnipeg, MB
Carol Shaw
Mathematics Department Head
Elmwood High School
Winnipeg School Division
Winnipeg, MB
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Serge Desrochers
Bishop McNally High School
Calgary Catholic School District
Calgary, AB
Roger Teshima
Learning Leader of Mathematics
Bowness High School
Calgary Board of Education
Calgary, AB
George Lin
Mathematics Department Head
Pinetree Secondary School
School District 43
Coquitlam, BC
Sarah Wade
Curriculum Developer
Nunavut Board of Education
Ottawa, ON
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Joanna Landry
Coordinator, First Nations, Inuit, and Métis Education
Services
Regina Catholic Schools (#81)
Regina, SK
Pr
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Hugh MacDonald
Principal
Austin O’Brien Catholic High School
Edmonton Catholic School District
Edmonton, AB
NEL
Reviewers and Advisory Panel
iii
table of Contents
Features of Nelson Principles
of Mathematics 11
viii
2
Getting Started: The Mystery of the
Mary Celeste
4
1.1 Making Conjectures: Inductive Reasoning
Math in Action: Oops! What Happened?
1.2 Exploring the Validity of Conjectures
1.3 Using Reasoning to Find a Counterexample
to a Conjecture
6
15
Chapter 2: Properties of Angles
and Triangles
66
Getting Started: Geometric Art
68
2.1 Exploring Parallel Lines
70
2.2 Angles Formed by Parallel Lines
73
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Chapter 1: Inductive and Deductive
Reasoning
Applying Problem-Solving Strategies:
Checkerboard Quadrilaterals
16
83
Mid-Chapter Review
84
2.3 Angle Properties in Triangles
86
2.4 Angle Properties in Polygons
Applying Problem-Solving Strategies:
Analyzing a Number Puzzle
26
Math in Action: “Circular” Homes
94
100
1.4 Proving Conjectures: Deductive Reasoning
27
Mid-Chapter Review
34
History Connection: Buckyballs—
Polygons in 3-D
103
1.5 Proofs That Are Not Valid
36
Chapter Self-Test
Chapter Review
104
45
2.6 Proving Congruent Triangles
107
52
Chapter Self-Test
116
58
Chapter Review
117
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1.7 Analyzing Puzzles and Games
2.5 Exploring Congruent Triangles
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1.6 Reasoning to Solve Problems
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History Connection: Reasoning in Science
18
24
59
63
Project Connection: Creating an
Action Plan
64
121
Project Connection: Selecting
Your Research Topic
122
Chapters 1–2 Cumulative Review
124
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Chapter Task: How Many Sisters
and Brothers?
Chapter Task: Designing Logos
iv
Table of Contents
NEL
126
Getting Started: Lacrosse Trigonometry
128
3.1 Exploring Side–Angle Relationships in
Acute Triangles
130
3.2 Proving and Applying the Sine Law
132
Mid-Chapter Review
142
3.3 Proving and Applying the Cosine Law
144
Math in Action: How Good Is Your
Peripheral Vision?
History Connection: Fire Towers and
Lookouts
Applying Problem-Solving Strategies:
Analyzing a Trigonometry Puzzle
154
172
Getting Started: Photography
174
4.1 Mixed and Entire Radicals
176
4.2 Adding and Subtracting Radicals
History Connection: It’s Radical
4.3 Multiplying and Dividing Radicals
Applying Problem-Solving Strategies:
Defining a Fractal
163
164
184
190
191
201
Mid-Chapter Review
202
4.4 Simplifying Algebraic Expressions
Involving Radicals
204
4.5 Exploring Radical Equations
214
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3.4 Solving Problems Using Acute
Triangles
Chapter 4: Radicals
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Chapter 3: Acute Triangle
Trigonometry
4.6 Solving Radical Equations
216
Math in Action: How Far Is the Horizon? 224
Chapter Self-Test
225
166
Chapter Review
226
167
Chapter Task: Acute Triangles in
First Nations and Métis Cultures
Chapter Task: Designing a
Radical Math Game
229
169
Project Connection: Creating
Your Research Question or
Statement
Project Connection: Carrying Out
Your Research
230
170
Chapter Self-Test
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Chapter Review
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165
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Table of Contents
v
234
Chapter 6: Quadratic Functions
318
Getting Started: Comparing Salaries
236
Getting Started: String Art
320
5.1 Exploring Data
238
6.1 Exploring Quadratic Relations
322
6.2 Properties of Graphs of Quadratic
Functions
325
5.2 Frequency Tables, Histograms, and
Frequency Polygons
241
253
History Connection: Duff’s Ditch
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Chapter 5: Statistical Reasoning
6.3 Factored Form of a Quadratic Function
337
347
5.3 Standard Deviation
254
Mid-Chapter Review
266
Mid-Chapter Review
351
5.4 The Normal Distribution
269
6.4 Vertex Form of a Quadratic Function
354
282
6.5 Solving Problems Using Quadratic
Function Models
Applying Problem-Solving Strategies:
Predicting Possible Pathways
Math in Action: Paper Parabolas
283
History Connection: Trap Shooting
5.6 Confidence Intervals
295
Applying Problem-Solving Strategies:
Curious Counting Puzzles
382
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5.5 Z-Scores
368
381
Math in Action: Finding and Interpreting
304
Data in the Media
305
Chapter Review
Chapter Task: True-False Tests
383
Chapter Review
384
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Chapter Self-Test
Chapter Self-Test
306
Chapter Task: Parabolas in Inuit Culture
389
311
Project Connection: Identifying
Controversial Issues
390
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Project Connection: Analyzing Your Data 312
315
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Chapters 3–5 Cumulative Review
vi
Table of Contents
NEL
Chapter 7: Quadratic Equations
392
Chapter 8: Proportional Reasoning
440
Getting Started: Factoring Design
394
Getting Started: Interpreting
the Cold Lake Region
442
396
7.2 Solving Quadratic Equations by
Factoring
405
7.3 Solving Quadratic Equations Using the
Quadratic Formula
History Connection: The Golden Ratio
414
422
423
7.4 Solving Problems Using Quadratic
Equations
425
Applying Problem-Solving Strategies:
Determining Quadratic Patterns
Chapter Self-Test
432
433
Chapter Review
434
8.2 Solving Problems That Involve Rates
454
Applying Problem-Solving Strategies:
Analyzing a Rate Puzzle
437
438
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Project Connection: The Final Product
and Presentation
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462
Mid-Chapter Review
463
8.3 Scale Diagrams
466
8.4 Scale Factors and Areas of 2-D Shapes
475
8.5 Similar Objects: Scale Models and
Scale Diagrams
483
8.6 Scale Factors and 3-D Objects
494
Math in Action: Are Prices of TVs Scaled
Appropriately?
503
Chapter Self-Test
504
Chapter Review
505
Chapter Task: Mice To Be Here
509
Project Connection: Peer Critiquing
of Research Projects
510
Chapters 6–8 Cumulative Review
512
Glossary
514
Answers
519
Index
575
Z-Score Table
580
Credits
582
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Chapter Task: A Teaching Tool
History Connection: Ivy Granstrom
444
453
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Mid-Chapter Review
8.1 Comparing and Interpreting Rates
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7.1 Solving Quadratic Equations by
Graphing
Table of Contents
vii
Features of Nelson Principles
of Mathematics 11
Lessons
Chapters begin with two-page activities
designed to activate concepts and skills
you have already learned that will be
useful in the coming chapter.
Some lessons help to develop your understanding
of mathematics through examples and questions.
Most lessons give you a chance to explore or
investigate a problem. Many of the problems are
in a real-life context.
3
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Getting Started
Getting Started
2.1
Lacrosse Trigonometry
Daniel is about to take a shot at a field lacrosse net. He estimates his
current position, as shown below.
YOU WILL NEED
• calculator
YOU WILL NEED
• dynamic geometry software
GOAL
Identify relationships among the measures of angles formed
by intersecting lines.
ic
a
OR ruler and protractor
Exploring Parallel Lines
EXPLORE the Math
A sports equipment manufacturer builds portable basketball systems, like
those shown here. These systems can be adjusted to different heights.
goal
6'
65°
transversal
bl
Daniel
?
How can you use these estimates to determine the width
of the net?
A.
Does Daniel’s position form a right triangle with the goalposts?
B.
A primary trigonometric ratio cannot be used to determine the width
of the net directly. Explain why.
C.
Copy the triangle that includes Daniel’s position in the diagram above.
Add a line segment so that you can determine a height of the triangle
using trigonometry.
D.
Determine the height of the triangle using a primary trigonometric ratio.
E.
Create a plan that will allow you to determine the width of the lacrosse
net using the two right triangles you created.
F.
Carry out your plan to determine the width of the net.
Chapter 3 Acute Triangle Trigonometry
transversal
A line that intersects two or more
other lines at distinct points.
transversal
Pr
When a transversal intersects two parallel lines, how are the
angle measures related?
Reflecting
a
Use the relationships you observed to predict
the measures of as many of the angles a to g in
this diagram as you can. Explain each of your
predictions.
140°
b
c
d
f
70
Chapter 2 Properties of Angles and Triangles
This team from Caughnawaga
toured England in 1867,
demonstrating the game. As a
result of the tour, lacrosse clubs
were established in England. The
game then spread around the
world.
The Canadian team won the
bronze medal at the 2009
Women‘s Lacrosse World Cup in
Prague, Czech Republic.
e
g
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Reflecting
These questions give
you a chance to reflect
on the lesson so far and
to summarize what you
have learned.
WHAT DO You Think?
Decide whether you agree or disagree with each statement. Explain your
decision.
1. When using proportional reasoning, if you apply the same operation,
using the same number, to all the terms, the ratios remain equivalent.
2. If you know the measures of two angles and the length of any side in
an acute triangle, you can determine all the other measurements.
NEL
viii
?
When a system is adjusted,
the backboard stays
perpendicular to the ground
and parallel to the supporting
post.
NEL
What Do You Think?
Statements about mathematical
concepts are presented and you
must decide whether you agree
or disagree. Use your existing
knowledge and speculations
to justify your decisions. At the
end of the chapter, revisit these
statements to decide whether
you still agree or disagree.
When the adjusting
arm is moved, the
measures of the
angles formed with
the backboard and
the supporting
post change. The
adjusting arm forms
a transversal .
A.
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128
5'
Pu
Field lacrosse, Canada‘s national
sport, originated with First
Nations peoples, probably
in central North America.
Hundreds of years ago, both the
number of players and the size
of the field were much greater
than in the modern game. The
Iroquois peoples of what is now
southern Ontario and western
New York may have been the
first to limit the number of
players to 12 or 15.
Features of Nelson Principles of Mathematics 11
Getting Started
129
NEL
Explore
You Will Need
Try these at the
beginning of each lesson.
They may help you think
in ways that will be
useful in the lesson.
YOU WILL NEED
t calculator
t graph paper
t ruler
EXPLORE…
t World-class sprinters can
run 100 m in about 9.8 s. If
they could run at this rate
for a longer period of time,
estimate how far they could
run in a minute, in an hour,
and in a day.
Communication Tip
The time 01:20:34.7, or
80:34.7, can also be written
as 1 h 20 min 34.7 s or as
80 min 34.7 s.
Comparing and
Interpreting Rates
GOAL
Represent, interpret, and compare rates.
APPLY the Math
INVESTIGATE the Math
A triathlon consists of three different races: a 1.5 km swim, a 40 km bike
ride, and a 10 km run. The time it takes to “transition” from one race
to the next is given as “Trans” in the official records. The setup for the
transition area is different in different triathlons.
EXAMPLE
Simon Whitfield of Victoria, British Columbia, participated in the
triathlon in both the 2000 and 2008 Olympic Games. His results for both
Olympics are shown below.
Cycling
Rank Athlete Country Time Trans Time Trans
1
Whitfield Canada
17:56 0:21 58:54 0:17
Cycling
Time
Total Time
30:52
1:48:24.02
Running
Rank Athlete Country Time Trans Time Trans
A comparison of two amounts
that are measured in different
units; for example, keying
240 words/8 min
Communication Tip
The word “per” means
“to each” or “for each.”
It is written in units using a
slash (/).
30:48
Total Time
1:48:58.47
How can you compare Simon’s speeds in his two Olympic
medal-winning triathlons?
A.
Compare Simon’s times for each race segment in the two triathlons,
ignoring the transition times. Which race segment had the greatest
difference in Simon’s times?
B.
Speed, the ratio of distance to time, is an example of a rate . Create
a distance versus time graph to compare Simon’s swimming speeds in
these triathlons. Express time in hours and distance in kilometres.
C.
A polygon in which each interior
angle measures less than 180°.
Viktor’s Solution
Zfem\o
efe$Zfem\o
ZfeZXm\
Determine the slope of each line segment on your distance versus time
graph. What do these slopes represent?
E.
Do the slopes you determined in part D support Carmen’s claim? Explain.
Chapter 8 Proportional Reasoning
:-
:)
8
:(
:.
The sum of the measures of the angles
in n triangles is n(180°).
:+
:*
:,
:-
:)
8
:.
:(
:e
:/
I drew an n-sided polygon. I
represented the nth side using a
broken line. I selected a point in
the interior of the polygon and
then drew line segments from
this point to each vertex of the
polygon. The polygon is now
separated into n triangles.
The sum of the measures of the
angles in each triangle is 180°.
:/
:e
Carmen claims that Simon swam faster in the 2000 Sydney Olympic
Games Triathlon than in Beijing in 2008. Explain how the data in the
table and in the graph you created in part B support her claim.
D.
:,
Two angles in each triangle
combine with angles in the
adjacent triangles to form two
interior angles of the polygon.
Each triangle also has an angle at
vertex A. The sum of the measures
of the angles at A is 360° because
these angles make up a complete
rotation. These angles do not
contribute to the sum of the
interior angles of the polygon.
The sum of the measures of the interior angles of the polygon, S(n), where n is
the number of sides of the polygon, can be expressed as:
S n 180°n 360°
S n 180° n 2
The sum of the measures of the interior angles of a convex polygon can be
expressed as 180° n 2 .
NEL
Your Turn
0-
Explain why Viktor’s solution cannot be used to show whether the
expression 180° n 2 applies to non-convex polygons.
Chapter 2 Properties of Angles and Triangles
NEL
e-
+++
18:18 0:27 58:56 0:29
Time
Pu
rate
Whitfield Canada
convex polygon
bl
2
?
Reasoning about the sum of the interior
angles of a polygon
:*
Running
2008 Beijing Olympic Games Triathlon: Men
Swimming
1
Prove that the sum of the measures of the interior angles of any n-sided
convex polygon can be expressed as 180° n 2 .
:+
2000 Sydney Olympic Games Triathlon: Men
Swimming
New terms and ideas are highlighted
in the text. In the margin nearby,
you’ll find a definition.
ic
a
8.1
Definitions
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Items you will need to
carry out the lesson, such
as a calculator or graph
paper, will be listed here.
Communication Tips
Pr
These tips give explanations
for why or when a specific
mathematical convention
or notation is used.
NEL
Your Turn
These questions appear
after each Example.
They may require you to
think about the strategy
presented in the example,
or to solve a similar
problem.
Examples
A complete solution to the example
problem is given on the left. Explanations
for steps in the solution are included in
green boxes on the right. Use the examples
to develop your understanding, and for
reference while practising.
Features of Nelson Principles of Mathematics 11
ix
Practising
The In Summary box presents the Key
Ideas for the lesson, and summarizes other
useful information under Need to Know.
These problems and questions give you a chance
to practise what you have just learned. There are a
variety of types of questions, from simple practice
questions that emphasize skill and knowledge
development to problems that require you to make
connections, visualize, and reason.
In Summary
Key Idea
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UPNBLFBHFOFSBMDPODMVTJPOXIJDIZPVDBOTUBUFBTBDPOKFDUVSF
Need to Know
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Closing
15. Create a real-life problem that can be modelled by an acute triangle.
Exchange the problem you created with a classmate. Sketch the situation
in your classmate’s problem, and explain what must be done to solve it.
CHECK Your Understanding
Extending
1. Troy works at a ski shop in Whistler, British Columbia, where three
))(%)d
Egypt has a square base with
sides that are 232.6 m long. The
:
;
distance from the top of the
>
=
pyramid to each corner of the
8
9
)*)%-d
base was originally 221.2 m.
a) Determine the apex angle of a face of the pyramid (for example,
AEB to the nearest degree.
b) Determine the angle that each face makes with the base
(for example, EGF to the nearest degree.
2. Tomas gathered the following evidence and noticed a pattern.
23 11 253
17 11 187
62 11 682
41 11 451
Tomas made this conjecture: When you multiply a two-digit number
by 11, the first and last digits of the product are the digits of the
original number. Is Tomas’s conjecture reasonable? Develop evidence
to test his conjecture and determine whether it is reasonable.
ic
a
17. Cut out two paper strips, each 5 cm wide.
Lay them across each other as shown at the
right. Determine the area of the overlapping
region. Round your answer to the nearest
tenth of a square centimetre.
PRACTISING
3. Make a conjecture about the sum of two even integers. Develop
evidence to test your conjecture.
*'—
History Connection
4. Make a conjecture about the meaning of the symbols and colours in
Fire Towers and Lookouts
the Fransaskois flag, at the left. Consider French-Canadian history and
the province of Saskatchewan.
5. Marie studied the sum of the angles in quadrilaterals and made a
conjecture. What conjecture could she have made?
Closing
19. Lou says that conjectures are like inferences in literature and
hypotheses in science. Sasha says that conjectures are related only
to reasoning. With a partner, discuss these two opinions. Explain
NEL
how both may be valid.
Chapter 1 Inductive and Deductive Reasoning
Extending
For about 100 years, observers have been watching for forest
fires. Perched in lookouts on high ground or on tall towers (there
are three fire towers in Alberta that are 120 m high), the observers
watch for smoke in the surrounding forest. When the observers see
signs of a fire, they report the sighting to the Fire Centre.
The first observers may have had as little as a map, binoculars,
and a horse to ride to the nearest station to make a report. Today,
observers have instruments called alidades and report using radios.
Alidades consist of a local map, fixed in place, with the tower or
lookout at the centre of the map. Compass directions are marked in
degrees around the edge of the map. A range finder rotates on an arc above the map. The observer rotates the
range finder to get the smoke in both sights, notes the direction, and reports it to the Fire Centre. A computer at
the Fire Centre can use the locations of the fire towers and directions from two or more observers to fix the
location of the fire. The latitude and longitude are then entered into a helicopter’s flight GPS system. The
helicopter’s crew flies to the fire, records a more accurate location, and reports on the fire.
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16. The Great Pyramid at Giza in
types of downhill skis are available: parabolic, twin tip, and powder.
The manager of the store has ordered 100 pairs of each type, in
various lengths, for the upcoming ski season. What conjecture did the
manager make? Explain.
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In Summary
A.
B.
20. Photographs lead to conjectures about what was happening
around the time that the images were captured. With a partner,
develop at least three different conjectures about what could have
led to the situation in this photograph.
Analyzing a Number Puzzle
Arithmagons are number puzzles that have addition as the central
operation. They are based on polygons, with a circle at each vertex and a
box on each side.
The Puzzle
The number in each box of an arithmagon is the sum of the two numbers
in the circles adjacent to the box.
(
)
Chapter 3 Acute Triangle Trigonometry
NEL
A.
dentists recommend it.” Discuss this statement with a partner,
and decide whether it is a conjecture. Justify your decision.
22. From a news source, retrieve evidence about an athlete’s current
Pu
B.
Oops! What Happened?
estop
sign
Pr
NEL
1.1 Making Conjectures: Inductive Reasoning
()
(,
c)
*/
+/
-/
Create your own arithmagon. Exchange arithmagons with a partner,
and solve your partner’s arithmagon.
C.
D.
E.
F.
)-
What patterns did you notice?
What relationship exists between the numbers in the circles and the
numbers in the opposite boxes?
Describe the strategy you used to solve your partner’s arithmagon.
Of the arithmagons you’ve encountered, which was easiest to
solve? Explain.
Chapter 1 Inductive and Deductive Reasoning
NEL
(,
Math in Action
History Connection
These tasks give you a chance to apply the
math concepts you have learned in a realworld context.
Find out how mathematics
has been used throughout
history, and think about
some implications.
x
0
The Strategy
Investigator: ent: 14:15
-Time of accidcovered
Driver of yellow car:
-Roads snow nt and
-Came to full stop,
-Stop sign evide
clear of snow s evidentt checked for cars, and
-No brake markbecause proceeded through the
for either car
in
r c i n
intersection
of snow cover
Passenger
Driver of blue car:
r
r:
-DidnÕt see of blue car:
-Driving 60 km/h
looking at anything, was
the map
(speed limit)
ap
-Had lights on
-Saw yellow car Witness at stop sign:
but didnÕt stop -Was waiting to cross
road
-Yellow car didnÕt
completely stop before
starting to go through
the intersection
-Blue car driver was
speeding
(/
(0
Math in Action
*
,
Solve the three triangular arithmagons below.
a)
b)
(.
performance or a team’s current performance. Study the
evidence, and make a conjecture about the athlete’s or team’s
performance over the next month. Justify your conjecture, and discuss
the complexity of conjectures about sports.
t *EFOUJGZUIFQJFDFTPGHJWFOFWJEFODFUIBUBSFDPOKFDUVSFT
t .BLFBDPOKFDUVSFBCPVUXIBUDBVTFEUIFBDDJEFOU
t 8IBUFWJEFODFTVQQPSUTZPVSDPOKFDUVSF
t *GZPVDPVMEBTLUISFFRVFTUJPOTPGUIFESJWFST
PSUIFXJUOFTTXIBUXPVMEUIFZCF
t $BOUIFDBVTFPGBOBDDJEFOUTVDIBTUIJT
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+
*
How could a Fire Centre use trigonometry to determine the location of a fire?
How could an observer use trigonometry to estimate the size of a fire?
(-+
21. In advertising, we often see statements such as “four out of five
Applying Problem-Solving Strategies
Features of Nelson Principles of Mathematics 11
Applying ProblemSolving Strategies
These games or puzzles
develop your problemsolving skills by allowing
you to create and analyze
strategies.
NEL
Mid-Chapter Review
Chapter Self-Test
Chapter Task
See the Mid-Chapter Review for
Frequently Asked Questions.
Try the suggested Practising
questions if you feel you need
to test your understanding.
Try the Self-Test at the end of the
chapter to discover if you have met
the learning goals of the chapter.
If you are uncertain about how to
address any of the questions, look
to the FAQs in the Mid-Chapter
and Chapter Reviews for support.
Have you developed a deep
understanding of the mathematics
presented in the chapter? Find out
by trying this performance task.
1
Mid-Chapter Review
Chapter Self-Test
Frequently Asked Questions
Q:
What are the relationships among the angles formed when
a transversal intersects two parallel lines?
A:
When two lines are parallel, the following angle relationships hold:
Questions 1 to 5.
b
a
a
b
Corresponding
angles are equal.
Study Aid
• See Lesson 2.2, Example 2.
• Try Mid-Chapter Review
Question 6.
Study Aid
• See Lesson 2.2, Example 3.
• Try Mid-Chapter Review
Questions 6 and 8.
Q:
A:
e
c
Alternate interior
angles are equal.
2. Frank tosses a coin five times, and each time it comes up tails. He
Figure 2
makes the following conjecture: The coin will come up tails on every
toss. Is his conjecture reasonable? Explain.
e
Alternate exterior
angles are equal.
3. Koby claims that the perimeter of a pentagon with natural
Interior angles on
the same side of
the transversal are
supplementary.
number dimensions will always be an odd number. Search for a
counterexample to his claim.
Draw a diagram that shows parallel lines and a transversal. Label your
diagram with any information you know about the lines and angles.
State what you know and what you are trying to prove. Make a plan.
Use other conjectures that have already been proven to complete each
step of your proof.
Q:
How can you use angles to prove that two lines
are parallel?
A:
Draw a transversal that intersects the two lines, if the diagram does
not include a transversal. Then measure, or determine the measure of,
a pair of angles formed by the transversal and the two lines. If
corresponding angles, alternate interior angles, or alternate exterior
angles are equal, or if interior angles on
P
B
the same side of the transversal are
supplementary, the lines are parallel.
cream. Determine the order in which they are lined up, using these
clues:
• CandiceisbetweenAndyandBonnie.
• DarleneisnexttoAndy.
• Bonnieisnotfirst.
Explain.
Let a 5 9.9.
10a 5 99.9
10a 2 a 5 90
9a 5 90
a 5 10
Q
3
Chapter 2 Properties of Angles and Triangles
?
Multiply by 10.
Subtract a.
Divide by 9.
NEL
Chapter Review58
8
Chapter 1 Inductive and Deductive Reasoning
NEL
FREQUENTLY ASKED Questions
Q:
To use the cosine law, what do you need to know about
a triangle?
A:
You need to know two sides and the contained angle, or three sides
in the triangle.
For example, you can use the cosine law to determine
the length of p.
P
R
8 cm
72°
6 cm
to 3.
• Try Chapter Review
Questions 6 to 9.
You know:
• the lengths of two sides.
• the measure of the contained angle. You can use the cosine law to
determine the length of the side
opposite the contained angle, p.
p2 5 82 1 62 2 2 182 162 cos 72°
Solving for p will determine the length.
Pu
p
Study Aid
• See Lesson 3.3, Examples 1
Q
What are the measures of all the sides and angles in each
triangle in the frame of a tipi?
A.
Design a tipi. Choose the number of poles, the interior height, and the
diameter of the base.
B.
What regular polygon forms the base of your tipi? What type of
triangle is formed by the ground and supporting poles? Explain how
you know.
Simplify.
WHAT DO You Think Now? Revisit What Do You Think? on page 5.
How have your answers and explanations changed?
S
C
The frame of a tipi consists of wooden poles supported by an inner tripod.
Additional poles are laid in and tied with rope. When the frame is complete,
the covering is drawn over the frame using two additional poles. The interior
of the tipi creates a roughly conical shape. The number of poles used is
indicative of geographic area and people, ranging from 13 to 21 poles.
7. The following proof seems to show that 10 5 9.9. Is this proof valid?
D
60°
Structures made from acute triangles are used in most cultures. The First
Nations and Métis peoples who lived on the Prairies used acute-triangle
structures to cook, dry meat, and transport goods and people. Acute
triangles are also the support for tipis. Tipis suit a life based on the migration
of the buffalo, because they are easily assembled and taken down. Tipis are
still used for shelter on camping trips and for ceremonial purposes.
Choose a natural number. Double it. Add 20. Divide by 2. Subtract
the original number.
6. Andy, Bonnie, Candice, and Darlene are standing in line to buy ice
R
60°
A
Measure alternate interior angles /ARQ
and /PSD. If these angles are equal,
then AB i CD.
5. Prove that the following number trick always results in 10:
Figure 3
Blackfoot tipis
Inside a Dakota tipi
4. Prove that the product of two consecutive odd integers is always odd.
How can you prove that a conjecture involving parallel lines
is valid?
For example, determine if AB is parallel
to CD. First draw transversal PQ.
84
d
180° � d
c
a) What would the 4th and 5th structures look like? How many
cubes would Danielle need to build each of these structures?
b) Make a conjecture about the relationship between the nth
structure and the number of cubes needed to build it.
c) How many cubes would be needed to build the 25th structure?
Explain how you know.
Figure 1
Chapter Task
Acute Triangles in First Nations and
Métis Cultures
ic
a
Lesson 2.2, Example 1.
• Try Mid-Chapter Review
3
bl
Study Aid
• See Lesson 2.1 and
1. Danielle made the cube structures to the left.
tio
n
2
C.
Determine the side lengths of the frame triangles.
D.
Determine the interior angles of the frame triangles.
E.
Explain why your design is functional.
Project Connection
frame
triangle
Task Checklist
✔ Did you draw labelled
diagrams for the problem?
✔ Did you show your work?
✔ Did you provide appropriate
reasoning?
✔ Did you explain your
thinking clearly?
Chapter Task
NEL
169
Peer Critiquing of Research Projects
Now that you have completed your research for your question/topic,
prepared your report and presentation, and assessed your own work,
you are ready to see and hear the research projects developed by your
classmates. You will not be a passive observer, however. You will have an
important role—to provide constructive feedback to your peers about their
projects and their presentations.
Critiquing a project does not involve commenting on what might have
been or should have been. It involves reacting to what you see and hear.
For example, pay attention to
• strengthsandweaknessesofthepresentation.
• problemsorconcernswiththepresentation.
Think of suggestions you could make to help your classmate improve
future presentations.
You can also use the cosine law to determine the measure of /Y .
X
2 cm
Z
4 cm
5 cm
Y
You know:
• the lengths of all three sides.
You can use the cosine law to
determine the measure of /Y .
22 5 42 1 52 2 2 142 152 cos Y
2 2 2 42 2 52
5 cos Y
22 142 152
Solving for Y will determine
the measure of /Y .
When solving a problem that can be modelled by an acute
triangle, how do you decide whether to use the primary
trigonometric ratios, the sine law, or the cosine law?
A:
Draw a clearly labelled diagram of the situation to record what
you know.
• You may be able to use a primary trigonometric ratio if the diagram involves a right triangle.
• Use the sine law if you know two sides and one opposite angle, or two angles and one opposite side.
• Use the cosine law if you know all three sides, or two sides and the angle between them.
e-
Q:
Study Aid
• See Lesson 3.4,
Examples 1 to 3.
• Try Chapter Review
Questions 10 to 12.
While observing each presentation, consider the content, the organization,
and the delivery. Take notes during the presentation, using the following
rating scales as a guide. You can also use these scales to evaluate the
presentation.
You may need to use more than one strategy to solve some problems.
Pr
NEL
NEL
Chapter Review
167
510
Chapter 8 Proportional Reasoning
NEL
Chapter Review
Project Connection
Like the FAQ in the Mid-Chapter Review, the FAQ
covers the immediately preceding lessons. Practising
covers the whole chapter. There are also three
Cumulative Reviews (not shown), which you can use
to practise skills and use concepts you’ve learned in
the preceding chapters.
You will be working on a project throughout
the year. Project Connections support you
as you decide on a topic, do your research,
and develop your presentation.
Features of Nelson Principles of Mathematics 11
xi