Mathematics 21
Teacher and Student Support Resource
December 2013
DRAFT
Mathematics 21
Table of Contents
Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
i
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
Teaching and Learning Guidelines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
Resources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Websites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10
11
Planning. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
Sample Lesson: Reaction Time. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
Sample Lesson: Making a Paper Airplane. . . . . . . . . . . . . . . . . . . . . . . . . 15
Theme Overviews and Suggestions for Teaching and Learning . . . . . . . . . . . . .
Concept Map of Themes and Outcomes . . . . . . . . . . . . . . . . . . . . . . . . .
Outcome: Solving and Manipulating Equations. . . . . . . . . . . . . . . . . . . . .
Theme Overview: Earning and Spending Money . . . . . . . . . . . . . . . . . . .
Theme Overview: Home . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Theme Overview: Recreation and Wellness . . . . . . . . . . . . . . . . . . . . . . .
Theme Overview: Travel and Transportation . . . . . . . . . . . . . . . . . . . . . .
16
16
17
19
26
34
37
Appendices
Appendix A: Earning and Spending Money . . . . . . . . . . . . . . . . . . . . . . .
Appendix B: Home. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Appendix C: Recreation and Wellness. . . . . . . . . . . . . . . . . . . . . . . . . . . .
Appendix D: Travel and Transportation . . . . . . . . . . . . . . . . . . . . . . . . . . .
39
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134
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Mathematics 21
These materials were created by writing partnerships of school boards and the
provincial government. This document reflects the views of the developers and not
necessarily those of the Ministry of Education. Permission is given to reproduce these
materials for any purpose except profit. Teachers are also encouraged to amend,
revise, edit, cut, paste, and otherwise adapt this material for educational purposes.
Any references in this document to particular commercial resources, learning materials,
equipment, or technology reflect only the opinions of the developers of this Mathematics
21 course overview, and do not reflect any official endorsement by the Ministry of
Education or by the partnership of school boards that supported the production of the
document.
Acknowledgments
Michelle Dament
Prairie Spirit School Division
Dalmeny, Saskatchewan
Wanda Pihowich
Saskatoon Public School Division
Saskatoon, Saskatchewan
Heather Granger
Prairie South School Division
Avonlea, Saskatchewan
Kelly Russell
Lloydminster Catholic School Division
Lloydminster, Saskatchewan
Shelda Hanlan Stroh
Greater Saskatoon Catholic School Division
Saskatoon, Saskatchewan
Mathematics 21
Introduction
Recommended Prerequisite: Mathematics 11, Foundations and Pre-calculus 10,
and/or Workplace and Apprenticeship 10
This course is designed for theme-based instruction, which
should enable students to broaden their understanding of
mathematics as it is applied in important areas of
day-to-day living. There is a need for learning to be
meaningful in order to be transferable. Learning
mathematics should provide students an opportunity to
explore mathematics in their lives.
Earning
and Spending
Money
My
Life
In this course, emphasis is placed on
Travel and
making informed decisions about
Transportation
finances, home design and maintenance,
recreation and personal wellness, and
travel and transportation. All mathematics
relate to the themes: Earning and Spending
Money, Home, Recreation and Wellness, and Travel and
Transportation. Students can draw on their own or others
experiences in the workforce to develop and extend their
Recreation and
knowledge about earning and spending money. They will
Wellness
also apply mathematics for the purpose of designing,
building, and maintaining a home and yard. Students will
apply reasoning and problem solving skills to make predictions
and decisions in recreational and wellness activities. As well, they will investigate and
solve problems related to planning a trip.
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Mathematics 21
Home
Teaching and Learning Guidelines
The teacher of a Mathematics 21 course should:
 Choose themes and topics from the curriculum appropriate to student background,
interests, and motivation.
 Identify the appropriate teaching/learning and assessment/evaluation strategies to
help students achieve the outcomes.
 Use resources that best suit students’ competencies and interests, and include
both print and web-based resources.
 Plan the delivery of the themes, using the support materials as a guideline, to
provide students with a variety of learning experiences that focus on active
learning, understanding, and engagement.
Students in a modified course typically benefit from instruction that:
 Provides students with a clear overview of the course, each unit of study, and
expectations.
 Provides students with activities that involve developing critical thinking and
decision-making skills.
 Helps students organize new knowledge, understand the relationships among the
new knowledge, and connect it to knowledge already learned.
 Helps students understand where they have been, where they are now, and where
they are going in the learning process (Lenz, 2000).
 Diagnoses the students’ current understanding and skill level.
 Identifies and builds on student’s prior knowledge.
 Differentiates what students will learn in order to achieve the outcomes and
teaches the prerequisite skills if they are missing.
 Differentiates the instructional approach and instructional groups (alone, pairs,
small group, total group).
 Structures individual lessons in a systematic and organized manner, and presents
course content in a structured manner.
 Integrates technology and uses a variety of resources.
 Uses current and local information to promote relevance.
 Models and uses scaffolded instructional strategies.
 Teaches students strategies that are specific to particular learning tasks.
 Provides enough guidance and practice so that students can master the
strategies.
 Teaches students self-management, self-reflection, and self-regulation strategies
to assist students in accomplishing tasks.
 Provides timely and constructive feedback to students.
 Provides assessment criteria for tasks to students.
 Bases students’ assessment and evaluation on the knowledge, skills, and
strategies that help students achieve the outcomes.
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Mathematics 21
 Uses the information obtained from assessment and evaluation to individualize
and inform upcoming instruction.
 Shares assessment and evaluation information (e.g., rubrics, checklists, etc.) with
students before those items are used, to help students track personal growth and
set learning goals.
Strategies
Teachers use multiple teaching, learning, and assessment strategies to ensure that
students have had the opportunity to learn the curriculum content and improve skills
prior to evaluation. When deciding which strategy to use, consider the following
questions:

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Can all learners use this strategy to show thinking and learning?
Will this strategy inform my instruction and provide a way to give feedback to
students?
Will patterns of understanding or confusion emerge as a result of using this
strategy?
Is this strategy convenient to design, use, and administer?
(Cris Tovani, 2011, So What Do They Really Know?, p. 74)
The following is a partial list of strategies that could be used in the Mathematics 21
course to help students achieve the outcomes.
Strategy
Description
12 word
summary
In 12 words or less, have students summarize important aspects of a particular
chunk of instruction.
3-2-1
Students jot down 3 ideas, concepts, or issues presented.
Students jot down 2 examples or uses of the idea or concept.
Students write down 1 unresolved question or a possible misunderstanding.
60 second
think
Use in your classroom at any time as no equipment is required. Ask students to
stop, and have a 60-second think about how their learning is going right then.
Accurately “time” the 60-seconds to allow quiet thinking time.
Circular
check
In groups, students are each given a different problem with a definite answer.
The first student completes the first step without contribution from others in the
group and passes it to the next student. The second student corrects any
mistakes in the first step and completes the next step without input from the
group. The problem is passed to the next student and the process continues
until the group has the correct answer.
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Mathematics 21
Class vote
Present several possible answers or solutions to a question or problem and
have students vote on what they think is best.
Concept
circle
Ask students to quickly sketch a concept circle like this image (noting that any
number of spikes can be drawn). Students then do an “individual brainstorm”,
trying to recall the key concepts that are related to the work they are doing
now. Students then highlight or draw a box around, any concepts that they are
having trouble understanding. These concepts are then recorded by the
student in their learning logs for further examination or they can be discussed
with the teacher next time there is an opportunity to do so.
Enter/exit
slips
Ask students a specific question about the lesson (or refer to Phrases and
Prompts for ideas to respond to). Students respond on the slip and give it to
the teacher, either on their way out or on their way in the next day. Teacher
can then evaluate the need to re-teach or questions that need to be answered.
Feedback
sandwich
Good news “I did really well on … ”
Bad news “I think these parts need to be changed … because …”
Good News “Some ways I can improve it are …”
Flash cards After 10 minutes into a lecture or concept presentation, have students create a
flash card that contains the key concept or idea. Toward the end of the class,
have students work in pairs to exchange ideas and review the material.
Four
corners
Page 4
Teacher posts questions, concepts, or vocabulary words in each of the corners
of the room. Each student is assigned a corner. Once in the corner, the
students discuss the focus of the lesson in relation to the question, concept, or
words. Students may report out or move to another corner and repeat.
Mathematics 21
Frayer
model
Write a term in the middle (e.g. rational number). Complete the other four
boxes in regards to the term.
Definition
Facts
Term:
Examples
Give
one/Get
one
Non-Examples
Students are given papers and asked to list 3-5 ideas about the learning.
Students draw a line after their last idea to separate his/her ideas from their
classmate’s lists. Students get up and interact with one classmate at a time.
Exchange papers, read your partner’s list, and then ask questions about new
or confusing ideas.
Graphic
organizers
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Mathematics 21
ICE tactic
Students ask themselves:
What are the:
Ideas (basics, details, facts, terminology)?,
Connections (relationships, synthesis, patterns)?
Extensions (transfer, hypotheticals, creative adaptations, going beyond
the obvious)?
This is a simple way to keep students focused on the big picture even while
they are on the run, learning, during any lesson (Young and Wilson, 2000).
Idea wave
Each student lists 3-5 ideas about the assigned topic. One volunteer begins
the “idea wave” by sharing his idea. The student to the right of the volunteer
shares one idea; the next student to rights shares one idea. Teacher directs
the idea wave until several different ideas have been shared. At the end of the
formal idea wave, a few volunteers who were not included may contribute.
Jigsaw
Students first meet in their “expert group”, where each student has the identical
assignment. The students become a team of specialists, gathering and
synthesizing information, becoming experts on their topic, and rehearsing their
presentations. Then the students change groups to their jigsaw groups. Each
student in each group educates the whole group about her or his specialty.
Learning
cell
Students develop questions and answers on their own (possibly using the QMatrix). Working in pairs, the first student asks a question and the partner
answers and vice versa. Each student can correct the other until a satisfactory
answer is reached.
Learning
logs
Use learning logs or learning journals for students to reflect on their recent
work (perhaps at the end of their work each week). Refer to Phrases and
prompts for ideas.
Muddiest
point
Students are asked to write down the muddiest point (what was unclear) in the
lesson.
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Mathematics 21
Old school
Ipads
Give students whiteboards, paper plates, index cards, or large sheets of paper
when they enter. When asking a question have ALL students write the answer
and at your signal, have ALL students hold up the Ipad so that you can see
who/ how many got the answer. Discussion to elaborate can follow.
Phrases
and
prompts
What have I learnt?
What am I most pleased with about my work?
What did I find difficult?
How can I try to improve?
What did I learn today?
What did I do well?
What am I confused about?
What do I need help with?
What do I want to know more about?
What am I going to work on next?
(Weeden et al., 2002)
The part I liked best was…
The part I found confusing was…
Two things I learnt were…
One question I have is…
I was surprised that…
I already knew that…
One thing I know that wasn’t mentioned is…
I would like to know more about…
I would like to spend more time on…
Some questions I know how to do…
One thing I want to get better at is …
One word web card…
(Davies, 2012)
This week I have learned…
For next week I am focusing on…
I will know I am getting better when…
I feel confident when …
My strength today was …
I’m proud of this because…
I feel frustrated when …
I need to find out more about …
I need help with …
My highest priority learning goal is ..
Next time I do this I will …
When I wasn’t sure, I asked [my friend’s name] about …
When I wasn’t sure, I asked [my teacher’s name] about …
One thing I am still not sure about is …
I will work on this by …
(Office of Learning and Teaching, DE&T
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Mathematics 21
http://www.sofweb.vic.edu.au/blueprint/fs1/assessment.asp).
What is the most important point you learnt today?
What point remains least clear to you?
How is ___________ similar to/different from ____________ ?
What are the characteristics/parts of ____________ ?
In what other ways might we show/illustrate ____________ ?
How does ____________ relate to ____________ ?
Give an example of ____________ .
What approach/strategy could you use to ____________ ?
Provide three examples of ____________ and one non-example.
Explain to a student in grade X (or who was absent today) what you learned
about ____________ today.
Write about the work we did today. What was easy? What was hard? What do
you still have questions about?
If you got stuck today in solving a problem, where did you get stuck? Why do
you think you had trouble there? If you did not get stuck, what idea helped you
solve the problem?
The hardest part of this chapter so far is ….
I need help with ____________, because …
To me, ____________ (e.g. geometry) means …
____________ (e.g. measuring angles) can be useful for ….
____________ (e.g. fractions) are challenging when …
Place mat
Each group member writes ideas in a space around the centre of a large piece
of paper. Afterwards, the group compares what each member has written, and
common items are compiled in the centre of the paper.
Portfolio
In the process of selection and explanation as to why students have chosen
specific pieces for their portfolios there is already a self-assessment process in
place. However, this can be taken further by more specifically asking students
to respond to the following process and questions:
1. Arrange all your work from most to least effective
2. Reflecting on your two best works, and on a separate sheet(s) of
paper for each work, answering the following questions.
What makes this your best (second best) work?
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Mathematics 21
How did you go about it?
What problems did you encounter?
How did you solve them?
What goals did you set for yourself?
How did you go about accomplishing them?
3. Answering these two questions on a single sheet(s) of paper at the
front of your portfolio.
What makes your most effective work different from your least
effective work?
What are your goals for your future work?”
(http://www.ncrel.org/sdrs/areas/issues/students/learning/lr2port.htm)
Quick write
Students write for 2-3 minutes about what they learned or heard from the
explanation. Also it could be an open ended question from teacher (refer to
Phrases and prompts for ideas).
Studentgenerated
lists
Top 10 things I need to find out …
Questions I have about my work …
Strategies I can use to improve my work …
Think, Pair,
Share
Think about your answers and write them down, Pair with a partner to discuss
and add comments to your answers, Share your answers with the class.
Thumbs up
- thumbs
down
To check for understanding, have students hold up their thumb; thumb up
means “I got it”, thumb horizontal means “I’m not sure, maybe”, and thumb
down means “I’m lost. I have questions”.
Traffic
Lights
The traffic lights can be used in a range of different ways.
To check for understanding, during individual or group work, provide students
with a set of green, yellow, and red stacking cups. All students start with the
green cup displayed, stacked over the other two cups. As students work, they
can change the cup that is displayed to indicate to the teacher that their
progress is green (good understanding and do not need assistance), yellow
(partial understanding, getting answers, but with difficulty, minor errors, or have
a basic question), or red (no understanding, stalled, need an explanation
before moving forward).
For self-assessing their own work, students label their work green, yellow or
red according to whether they have good (“I got it”), partial (“I’m not sure,
maybe”), or little (“I’m lost, I have questions”) understanding.
(Black et al, 2003).
Examine your work and highlight where you feel
• Stopped
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Mathematics 21
• Cautious
• Going straight ahead.
Use a red marker or a pink highlighter to mark in the margins where you feel
“stopped” because you don’t understand. Write a learning goal about this. Use
an orange or yellow marker or highlighter to mark in the margins where you
feel “cautious” because you are unsure or don’t understand it very well. Use a
green marker or highlighter to mark in the margin where you feel you are
“going straight ahead” because you understand it well.
For assessing a peer’s oral presentation:
Green: better than I could have done/I learnt something from this
Yellow: about the same as I could have done/no major omissions or mistakes
Red: not as good as I could have done/some serious omission or mistakes”
Students could then go on and give their peers feedback on specific strengths
and weaknesses.
(Black et al., 2003)
Transfer
and apply
Students list what they have learned and how they might apply it to their lives.
Students list interesting ideas, strategies, concepts learned in class. They
write some possible ways to apply this learning in their lives, another class, or
in their community.
Wall
posters
Regular prompt questions can be made into wall posters. Refer to Phrases
and prompts for ideas.
Which
face?
3 boxes are labelled with:
put their work into.
and students choose which box to
(Unless otherwise referenced, the above strategies are from Office of Learning and
Teaching, DE&T http://www.sofweb.vic.edu.au/blueprint/fs1/assessment.asp).
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Mathematics 21
Resources
Each theme makes reference to the use of specific websites. Teachers need to consult
their board policies regarding use of any copyrighted materials. Before reproducing
materials for student use from printed publications, teachers need to ensure that their
board has a Can copy licence and that this licence covers the resources they wish to
use. Before screening videos/films with their students, teachers need to ensure that
their board/school has obtained the appropriate public performance licence. Teachers
are reminded that much of the material on the Internet is protected by copyright. The
copyright is usually owned by the person or organization that created the work.
Reproduction of any work or substantial part of any work on the Internet is not allowed
without the permission of the owner.
Websites
The URLs for the websites were verified by the developers prior to publication. Given
the frequency with which these designations change, teachers should always verify the
websites prior to assigning them for student use.
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Centre for Innovation in Mathematics Teaching http://www.cimt.plymouth.ac.uk/
Coolmath 4 Kids http://www.coolmath4kids.com/
Figure This! Math Challenges http://www.figurethis.org/index.html
Fun Math Lessons http://math.rice.edu/~lanius/Lessons/
Index of EARAT Manuals: The Apprenticeship Network
http://www.theapprenticeshipnetwork.com/earat/manuals/
Interactive Mathematics http://www.cut-the-knot.org/content.shtml
Intermath Online Mathematics Dictionary
http://intermath.coe.uga.edu/dictnary/homepg.asp
Math Central http://mathcentral.uregina.ca/
Math in Daily Life http://www.learner.org/interactives/dailymath/
Math is Fun http://www.mathisfun.com/
Math TV http://www.mathtv.com/
Math Worksheets http://www.math-aids.com/
Mudd Math Fun Facts http://www.math.hmc.edu/funfacts/
National Library of Virtual Manipulatives http://nlvm.usu.edu/en/nav/vlibrary.html
The Math Forum @ Drexel University http://mathforum.org/
Trades Math Workbook
http://www.hrsdc.gc.ca/eng/jobs/les/tools/support/trades_math_workbook.shtml
Virtual Math http://www.virtualmaths.org
Your Financial Toolkit http://www.fcac-acfc.gc.ca/ft-of/home-accueil-eng.html
Mathematics 21
Planning
Traditionally, teachers start unit planning with interesting activities and textbooks in
mind, rather than starting with the big ideas or concepts they want the students to
master. If learning is to be effective for the students, the teacher must begin with the
final destination in mind. Teachers should be clear about what learning outcome(s) and
goal(s) will be set for the students and what assessments will be used to provide
evidence that the students have mastered the learning outcome(s) and goal(s)
(Wiggins, G. and McTighe, J. (1998). Understanding by Design).
A concern with teaching any mathematics course is the time it takes to cover the
content. Employing a conceptual approach allows the teacher to become a facilitator or
guide to coach learners in building on what they already know. This constructivist
approach allows learners to:
 build on their prior knowledge
 place less emphasis on memorization and rote learning
 see mathematical skills as useful tools and processes
 build a depth of knowledge
 develop an understanding of the connections in mathematics
 build self-confidence and a positive disposition towards mathematics.
(ABE Level Three: Mathematics Curriculum Guide, pp. 128)
When a teacher uses a conceptual approach, instruction framed around context focuses
on concepts rather than content. According to the National Council of Teachers of
Mathematics (2000):
In planning individual lessons, instructors should strive to organize the mathematics
so that fundamental ideas form an integrated whole. Big ideas encountered in a
variety of contexts should be established carefully, with important elements such as
terminology, definitions, notation, concepts, and skills emerging in the process. (p.15)
As teachers design and plan their course, lessons should reinforce basic skills, include
a variety of instructional strategies and activities, and connect to the larger
mathematical concepts. Sample lessons have been included as examples that
incorporate overlapping outcomes, indicators, and themes and use a variety of
strategies, resources, and activities.
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Mathematics 21
Sample Lesson: Reaction Time
Outcome(s):
M21.1 Extend and apply understanding of the preservation of equality by solving
problems that involve the manipulation and application of formulae within home, money,
recreation, and travel themes.
M21.3 Extend and apply understanding of measures of central tendency to analyze
data.
M21.4 Demonstrate and extend understanding of similarity and proportional reasoning
related to scale factors, scale drawing, scale models, surface area, and volume.
Suggested Theme(s):
Travel and Transportation
Resource(s):
Reaction Time. MathLinks
9 (2009). pp. 86 – 87
Material(s):
30-cm ruler
Introduction: An important skill drivers must have is the
ability to react to obstacles that may suddenly appear in
their path. You be the driver! What types of obstacles
might you encounter? How quickly do you think you could
react to an obstacle in the road? You are going to calculate
your reaction time.
Investigate: Work with a partner.
 Your partner will hold a 30-cm ruler vertically in front of
you, with the zero mark at the bottom.
 Position your thumb and index finger on each side of
the ruler so that the zero mark can be seen just above
your thumb. Neither your thumb nor your finger should
touch the ruler.
 Your partner will drop the ruler without warning. Catch
the ruler as quickly as you can by closing your thumb
and finger.
 Read the measurement above your thumb to the
nearest tenth of a centimetre. This is your reaction
distance.
 Perform this procedure five more times, recording each
distance.
 Switch roles to determine your partner’s five reaction
distances.
Activity: Calculate your average reaction distance. The
1
formula 𝑑 = 2 𝑔𝑡 2 can be used to calculate reaction time,
where d is the reaction distance, in metres; g is the
acceleration due to gravity, which is 9.8 m/s2; t is time, in
seconds.
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Mathematics 21
Problem:
1. Imagine you are driving a car in a residential area and
a ball rolls onto the road in front of you. You move your
foot toward the brake. Based on the reaction time you
calculated, if you are driving at 40 km/h, how far will the
car travel before you step down on the brake?
2. What distance would you have travelled before
stepping down on the brake if your original speed was
100 km/h?
Discuss: What other factors might influence your reaction
time and your stopping distance?
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Mathematics 21
Sample Lesson: Making a Paper Airplane
Outcome(s):
M21.3 Extend and apply understanding of measures of central tendency to analyze
data.
M21.4 Demonstrate and extend understanding of similarity and proportional reasoning
related to scale factors, scale drawing, scale models, surface area, and volume.
Suggested Theme(s):
Activity:
Home
1. Make a paper airplane by following the folding
instructions (http://www.10paperairplanes.com/).
Travel and Transportation
Resource(s):
Making a Paper Airplane.
MathLinks 9 (2009).
pp. 40 – 41
2. Use the airplane to find the total surface area of the top
view of the two wings. Fly the airplane 5 times.
Record the average distance and direction travelled in
each flight.
3. Design and create a second airplane, which has a
different surface area. Record the new surface area
and the average distance and direction travelled in 5
trial flights.
Discuss: Which of the airplanes you constructed is the
most functional? Consider surface area when you explain
your thinking.
Material(s):
Ruler
Scissors
Different sized colored
paper
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Mathematics 21
Theme Overviews and Suggestions for Teaching and Learning
This resource was created as a teaching, learning, and assessment support to give
teachers an idea of how modified Mathematics 21 could be approached. Support
materials have been developed as a guideline and do not need to be followed precisely
or in a particular order.
Concept Map of Themes and Outcomes
The following concept map frames the themes and outcomes in Math 21.
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Mathematics 21
A Theme Overview chart for each of the themes offers a recommended clustering of
expectations and provides a starting point from which teachers can plan the course.
Following each theme overview are suggested teaching and learning experiences,
which may be used as a guideline for the teacher and may include:
Resources
Materials
Introduction
Pre-Assessment
Activities
Investigate
Assessment
Extension
Skill Building
Instruction
Practice
Questions
Interactive
Project
Problem
Connections
Game
Watch
Research
Terminology
Brainstorm
Discuss
Adaptations
Conclusion
The Solving and Manipulating Equations (M21.1) outcome overlaps in all four themes
and the intent is that this outcome may be taught in one or more of the themes.
However, if an outcome has been covered, it is not necessary to revisit it in all four
themes.
Solving and Manipulating Equations Outcome
M21.1 Extend and apply understanding of the preservation of equality by solving
problems that involve the manipulation and application of formulae within home,
money, recreation, and travel themes.
At a Glance
Solving and manipulating formulas:
 Surface area and volume
 Primary trigonometric ratios
 Mean, median, mode and range
 Leasing, renting and buying
 Simple and compound interest
 Pythagorean theorem
 Slope
Guiding Questions
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Page 17
How do you maintain equality in an equation?
Do you know the difference between an equations and an expression?
Can you read a problem and identify the given variables?
Can you isolate the unknown variable?
What are you looking for, what is the unknown value, and what are you asked to
find?
Do you prefer isolating the variable and then substituting known values for the
variable or substituting known values and then isolating for the unknown?
Mathematics 21

What are the units required?
Sample Formulas and Problems
1. A formula that estimates the stopping distance for a car on an icy road is 𝑑 =
𝑐
2
0.75𝑠 (1000) . The distance, d, is measured in metres. The speed of the car, s, is
in kilometres per hour. The mass of the car, c, is in kilograms. What is the
stopping distance for 1000-kg car travelling at 50 km/h? (Answer: 37.5 m) What is
the mass of the car if the stopping distance is 180 m when the car is travelling at 60
km/h? (Answer: 2000-kg) (MathLinks 9, p. 119)
2. A formula that approximates the distance an object falls through air in relation to
time is d = 4.9t2. The distance, d, is measured in metres, and the time, t, in
seconds. A pebble breaks loose from a cliff. What distance would it fall in 2
seconds? (Answer: 19.6 m). (MathLinks 9, p. 121)
3. A formula for estimating the volume of wood in a tree is V = 0.05hc2. The volume,
V, is measured in cubic metres. The height, h, and the trunk circumference, c, are
in metres. What is the volume of wood in a tree with a trunk circumference of 2.3 m
and a height of 32 m? (MathLinks 9, p. 123)
4. The amount of food energy required by a canoeist can be modeled by the equation
𝐶
𝑎 = 100 − 17, where a represents the person’s age and C represents the number of
calories. (MathLinks 9, p. 247)
𝑑
5. The average speed of a vehicle, s, is represented by the formula 𝑠 = where d is
𝑡
the distance driven and t is the time. If you drove at an average speed of 85 km/h
for 3.75 h, what distance did you drive? (Answer: 318.75 km). If you drove 152 km
at an average speed of 95 km/h, how much time did your trip take? (Answer: 1.6 h)
(MathLinks 9, p. 302)
6. For a fit and healthy person, the maximum safe heart rate during exercise is
4
approximately related to their age by the formula 𝑟 = 5 (220 − 𝑎). In this formula, r
is the maximum safe heart rate in beats per minute, and a is the age in years. At
what age is the maximum safe heart rate 164 beats/min? (Answer: 15 years old)
(MathLinks 9, p. 321)
Page 18
Mathematics 21
Theme Overview: Earning and Spending Money
Theme Introduction
Some students may have already entered the workforce and will have some knowledge
about earning and spending money. The intent of this theme is to develop an awareness
of financial decision making. Students will explore budgeting, financial institution
services, and leasing, renting, and buying on credit.
Outcome that overlaps in all four themes
M21.1 Extend and apply understanding of the preservation of equality by solving
problems that involve the manipulation and application of formulae within home, money,
recreation, and travel themes.
Outcomes
M21.8 Demonstrate understanding of budgets.
M21.9 Demonstrate understanding of financial institution services.
M21.10 Demonstrate understanding of financial decision making including analysis of
renting, leasing, and buying on credit.
At a Glance
Budgeting
Fixed and variable expenses
Financial institution services
Bank accounts
Bank fees
Investments
Simple interest
Compound interest
Buying on credit
Renting
Leasing
Guiding Questions
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Page 19
What is a budget?
What are fixed and variable expenses?
What are home, recreation, wellness, travel, and transportation expenses?
What types of accounts are you familiar with at a financial institution?
Do you have a banking account?
What are some ways to save money?
How can you save for a large expense?
How old do you want to be when you retire?
What is the difference between a chequing and savings account?
What is a bank statement?
What are some ways to borrow money?
Can you be charged interest on interest?
Mathematics 21

Should I buy, lease or rent a vehicle?
Career Connections
Realtor
Mortgage broker
Car sales person
TOPIC:
Rental Property owner
Banker
Financial advisor
OUTCOME:
SUGGESTED TEACHING AND LEARNING:
BUDGETS
Exploring
Terminology
M21.8
Terminology
Budgeting Key Terms
Appendix A.1.
Resource
Personal Budgets
MathWorks 11 (2011). pp. 300 – 343
Introduction to
Budgeting
M21.8
Activity
Budgeting Worksheet for Kids (Click on Excel link partway
down the page)
http://www.myliferoi.com/2009/10/budgeting-worksheetfor-kids/
Fixed and
Variable
Expenses
M21.8
Activity
Document Your Spending
http://www.financialliteracymonth.com/30steps/step21.a
spx
Create a Personal Budget with Fixed and Variable
Expenses
Appendix A.2.
Budgets
M21.8
Project
Budgeting to Live Away From Home
MathWorks 11 (2011). pp. 301, 325, 339
Budgeting to Live Away From Home Additional Questions
Appendix A.3.
Alternate Project Topics
Investigate, plan, design, and prepare a budget based on
the estimated cost from one of the other themes:
 HOME: plan a home renovation/improvement.
Page 20
Mathematics 21
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TOPIC:
OUTCOME:
Include the cost of contractors, equipment, supplies .
. . OR landscaping a property.
Sample problem: Plan, design, and prepare a
budget for the renovation of your bedroom for under
$1500. The renovations could include repainting the
walls, replacing the flooring, changing the fixtures,
and refurnishing the room.
RECREATION and WELLNESS: choose a leisure
activity or sport. Include the cost of equipment, fees,
travel, . . . OR choose a meal plan. Include the cost
of groceries, eating at restaurants, …
TRAVEL: plan a trip. Include the cost of gasoline,
accommodation, food, entertainment, car rental…OR
TRANSPORTATION: purchase and operate a
vehicle
SUGGESTED TEACHING AND LEARNING:
UNDERSTANDING
FINANCIAL
INSTITUTIONS
SERVICES
Exploring
Terminology
M21.9
Terminology
Financial Institutions Services Key Terms
Appendix A. 4.
Resource
Financial Services
MathWorks 11 (2011). pp. 252 – 275
Banking
Services and
Fees
M21.9
Activity
Researching Types of Banking Services and Fees
Appendix A.5.
Career Connection
Have students research the various banking officer
positions, including the banking service each provides.
Activity
Advantages and Disadvantages of Banking Services
Appendix A.6.
Page 21
Mathematics 21
Writing Cheques M21.9
Instruction and Practice
Writing Cheques
Appendix A.7.
Resources
Writing Cheques and Record Keeping
Mathematics 11 Workplace and Everyday Life (2007)
pp. 111 – 119
How to Write a Check – Check Writing 101
http://uninvitedwriter.hubpages.com/hub/How-to-write-acheck
Banking
Security
M21.9
Activity
Protecting Your Personal And Financial Information
MathWorks 11 (2011). pp. 262
Resource
BMO How We Protect You
http://www.bmo.com/home/about/banking/privacysecurity/how-we-protect-you
Investments
M21.9
Lesson
Investment Options
Appendix A.8.
Game
Free Stock Market Simulation Exchange Game
http://www.smartstocks.com/
Resource
Module 9 – Investing
http://www.themoneybelt.gc.ca/theCitylaZone/eng/ta/docs/html/Module_9.html
Simple and
Compound
Interest
M21.9
Practice
Comparison of Simple and Compound Interest
Appendix A.9.
Simple Interest Worksheet
http://public.clinton.k12.mi.us/CCS/CHS/brown/Senior%
20Math/Consumer%20Math/4-4_Simple_Interest.pdf
Simple and Compound Interest
http://www.kutasoftware.com/FreeWorksheets/PreAlgW
orksheets/Simple%20and%20Compound%20Interest.pd
f
Page 22
Mathematics 21
Simple and Compound Interest and The Rule of 72
http://www.teensguidetomoney.com/saving/simple-compound-interest--the-rule-of-72/compound-interest/
Resource
Simple and Compound Interest
MathWorks 11 (2011). pp. 264 – 275
TOPIC:
OUTCOME:
SUGGESTED TEACHING AND LEARNING:
FINANCIAL
DECISION MAKING
Exploring
Terminology
M21.10
Terminology
Financial Decision Making Key Terms
Appendix A.10.
Resource
Financial Services
MathWorks 11 (2011). pp. 276 – 299
Credit and
Credit Rating
M21.10
Instruction and Practice
An Introduction to Credit and Credit Rating
Appendix A.11.
Instruction
Risks and Benefits of Types of Credit
Appendix A.12.
Instruction and Practice
Credit Cards and Exploring Credit Card Use
Appendix A.13.
Activity
Credit Card Comparison
Appendix A.14.
Activity
Exploring Credit Card Use
Appendix A.15.
Resources
How Do I Find Information About Credit Cards?
http://www.creditcardflyers.com/credit-education/how-to-
Page 23
Mathematics 21
find-credit-card-information.php
Best Canadian Credit Cards
http://canada.creditcards.com/best-canadian-creditcards.php
Credit Card Selector Tool
http://www.fcac-acfc.gc.ca/iToolsiOutils/creditcardselector/CreditCardeng.aspx?lang=eng
Be Smart With Your Credit Card: Tips to Help You Use
Your Credit Card Wisely
http://www.fcacacfc.gc.ca/eng/resources/publications/paymentoptions/ts
creditshop-eng.asp
Credit Card Payment Calculator Tool
http://www.fcac-acfc.gc.ca/iToolsiOutils/CreditCardPaymentCalculator/CreditCardCalculat
orCalculate-eng.aspx
Installment
Accounts and
30 Day
Accounts
M21.10
Resources
Your Guide to Revolving Credit and Installment Credit
http://www.debthelp.com/kc/215-your-guide-revolvingcredit-and-installment-credit.html
The Types of Accounts on a Credit History
http://www.ehow.com/list_7339759_types-accountscredit-history.html
Installment Accounts
http://www.articlesbase.com/credit-articles/installmentaccounts-619406.html
What is an Accounts Payable Billing Cycle?
http://www.ehow.com/info_8405037_accounts-payablebilling-cycle.html
Loans
M21.9
M21.10
Activity
Comparing the Cost of a Loan
Appendix A.16.
Rent, Lease or
Buy a Vehicle
M21.9
M21.10
Activity
Purchasing a New Vehicle
Appendix A.17.
Resources
Rent, Lease, or Buy?
Foundations of Mathematics 12 (2012). pp. 120 - 133
Buying a Vehicle
http://www.cmcweb.ca/eic/site/cmc-
Page 24
Mathematics 21
cmc.nsf/eng/fe00108.html#buy
Loans and Savings Rates Tables: Car Loans
http://www.financialpost.com/personalfinance/rates/loans-car.html
Canadian Personal Loan Rates for Secured and
Unsecured Lines of Credit
http://www.redflagdeals.com/features/canadianmortgage-gic-rrsp-savings-rate-comparison/canadianpersonal-loan-rates-for-secured-and-unsecured-lines-ofcredit/
Insuring
Vehicles and
Insurance
Rates
M21.10
Activity
Use automobile insurance websites to investigate the
degree to which the type of car and the age and gender of
the driver affect insurance rates.
Resources
Registration and Insurance Rates
http://www.sgi.sk.ca/individuals/registration/rates/index.h
tml
Vehicle Insurance Coverage
http://www.sgi.sk.ca/individuals/registration/coverage/ind
ex.html
Basic Plate Calculator
http://www.sgi.sk.ca/online_services/rate/index.html
Page 25
Mathematics 21
Theme Overview: Home
Theme Introduction
The intent of this theme is to develop a deeper understanding of the applications of
similarity, proportional reasoning, measurement, geometry, and trigonometry for the
purpose of designing, building, and maintaining a home and yard.
Outcome that overlaps in all four themes
M21.1 Extend and apply understanding of the preservation of equality by solving
problems that involve the manipulation and application of formulae within home, money,
recreation, and travel themes.
Outcomes
M21.4 Demonstrate and extend understanding of similarity and proportional reasoning
related to scale factors, scale drawing, scale models, surface area, and volume.
M21.5 Demonstrate understanding of angles created by parallel, perpendicular, and
transversal lines and solve problems within the home theme.
M21.6 Demonstrate understanding of primary trigonometric ratios (sine, cosine, and
tangent) and slope.
At a Glance
Similarity
Proportional reasoning
Scale factor
Scale drawings
Scale models
Surface area
Volume
Angles
Transversal, parallel, and perpendicular
lines
Primary trigonometric ratios
Slope
Guiding Questions
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Page 26
How do you determine the scale factor from scale drawings?
What scale models have you seen?
How are scale models useful?
How can you ensure that the constructed object is proportional to the scale drawing?
What do you know about surface area and volume?
What is the difference between area and surface area?
What do you need to know to measure surface area?
How can you calculate surface area?
How many surfaces does an object (e.g. cereal box) have?
How many surfaces does a cylinder have?
How can you calculate volume?
Why are the units for volume cubed?
What is the difference between volume and capacity?
Mathematics 21
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
What are the units of measurement of volume? capacity?
Where have you seen units of mL? m3? cm3?
What is a parallel line?
Where have you seen a parallel line?
How do you know that lines are parallel?
What is a perpendicular line?
Where have you seen a perpendicular line?
How do you know a line is perpendicular?
What are the units of measurement in a triangle?
What do you know about the sum of the three angles in a triangle?
How can you find the measure of an angle in a right triangle if you have two side
lengths?
If you were given the length of one side is it possible to find the lengths of the other
two sides?
If given one angle, that is not the right angle measure; can you determine all three
angles in a right triangle?
If given an angle measure and one side length can you determine the other side
lengths?
What is slope?
What kinds of things have a slope or slant?
What would you need to change in the triangle to change the slope?
How can you measure slope?
What are common sense safety requirements where slope is used?
What would be a reasonable incline to push a wheelchair up if a door step is n
meters from the ground? How long would the ramp be? How could you determine
the length?
How could you apply trigonometry to solve for a real-life situation?
Career Connections
Carpenter
Concrete Mixers
Surveyors
Road construction
Architect
Masonry
Furniture designer
Web designer
Page 27
Plumbers
Truckers
Drafts person
Building Inspector
Home Inspector
Urban planner
Fashion designer
Automotive designer
Mathematics 21
TOPIC:
SCALE
FACTORS,
SCALE
DRAWINGS,
SCALE MODELS,
SURFACE AREA,
AND VOLUME
OUTCOME:
SUGGESTIONS FOR TEACHING AND LEARNING:
Measurement
M21.4
Brainstorm, Discuss and Practice
What Do You Already Know About Measurement?
Appendix B.1.
Proportional
Reasoning
and Scale
Factor
M21.4
Activity
Enlargements, Reductions, and Scale Factor
Appendix B.2
Activity
Scale Factor, Scale Drawings, and Scale Models
Appendix B.3.
Activity
Cars, Critters, and Barbie
Appendix B.4.
Teacher Resource Cars, Critters, and Barbie
Appendix B.5.
Activity
Gingerbread House
Appendix B.6.
Activity
Glowing Rectangles
Appendix B.7.
Resource
Scale Factors and Similarity
MathLinks 9 (2009). pp. 126 – 145
Surface Area
M21.4
Pre-Assessment
Geometric Shapes
Appendix B.8.
Pre-Assessment
Page 28
Mathematics 21
What is Surface Area?
Appendix B.9.
Application
How Many Sheets of Dry Wall Are Needed?
Appendix B.10.
Investigate
Heat and Frost Insulators
Appendix B.11.
Activity
You are employed by the city and responsible for determining
how much to sell new development lots for. Find information
on available lot dimensions in your area and investigate.
Determine the criteria you are going to use to set the price.
Activity
Body Surface Area Calculator
http://www.ultradrive.com/bsac.htm
Resources
Human Resources and Skills Development Canada: Trades
Math Workbook.
http://www.hrsdc.gc.ca/eng/jobs/les/tools/support/trades_m
ath_workbook.shtml#intro
Surface Area
MathLinks 9 (2009). pp. 26 - 35
Surface Area
MathWorks 10 (2010). pp. 115 – 123
Volume
M21.4
Pre-Assessment
What is Volume?
Appendix B.12.
Application
How is Volume Used?
Appendix B.13.
Watch and Investigate
You Pour, I Choose
http://threeacts.mrmeyer.com/youpourichoose/
Volume Cylinder
http://www.learner.org/interactives/geometry/area_volume2
.html
Page 29
Mathematics 21
Game
Minecraft Volume: Rectangular Prism Game
http://www.xpmath.com/forums/arcade.php?do=play&game
id=118
Project
Landscape Design
Appendix B.14.
Resource
Interactives Geometry 3D Shapes
http://www.learner.org/interactives/geometry/index.html
TOPIC:
OUTCOME:
SUGGESTIONS FOR TEACHING AND LEARNING:
ANGLES
CREATED BY
PARALLEL,
PERPENDICULA
R AND
TRANSVERSAL
LINES
Angles
Created by
Lines
M21.5
Investigate
Angles Formed by Transversals
Appendix B.15.
Diagram 1 and Beige Cards: Parallel Lines
Appendix B.16.
Diagram 2 and Pink Cards: Non-Parallel Lines
Appendix B.17.
Diagram 3
Appendix B.18.
Application
Angles in Construction
Appendix B.19.
Page 30
Mathematics 21
TOPIC:
PRIMARY
TRIGONOMETRIC
RATIOS AND
SLOPE
OUTCOME:
Triangles
M21.6
SUGGESTIONS FOR TEACHING AND LEARNING:
Pre-Assessment
Triangle Properties
Appendix B.20.
Investigate
Building Bridges Teacher Resource
Appendix B.21.
Building Bridges Student Task
Appendix B.22.
Resource
Mr. Quenneville’s Website: Unit 2 Trigonometry: 2.0 Intro to
Trigonometry
https://sites.google.com/a/hdsb.ca/mrquenneville/home/grade-10-applied-math/unit-2trigonometry
Trigonometry
M21.6
Investigate
What is the Problem? Teacher Resource
Appendix B.23.
What is the Problem? Student Task
Appendix B.24.
Watch
How to Measure a Tree
http://www.youtube.com/watch?v=F6fltSqImFM
Trigonometri
c Ratios
M21.6
Investigate
Same Shape Triangles Teacher Resource
Appendix B.25.
Practice
Mr. Quenneville’s Website: Unit 2 Trigonometry
2.4.1 What’s My Triangle
2.4 Solving for a Missing Side
2.5.2 Tangent or Something Else
https://sites.google.com/a/hdsb.ca/mrquenneville/home/grade-10-applied-math/unit-2-
Page 31
Mathematics 21
trigonometry
Activity
Going the Wrong Way
Appendix B.26.
Activity
Solving Trigonometric Problems
Appendix B.27.
Constructing a Clinometer
Appendix B.28.
Project
Who Uses Trigonometry? Teacher Resource
Appendix B.29.
Who Uses Trigonometry? Student Task
Appendix B.30.
Resources
Trigonometry of Right Triangles.
MathWorks 10 (2010). pp. 270 - 319
Trigonometry Activities
http://www.cimt.plymouth.ac.uk/projects/mepres/book9/y9s
15act1.pdf
Mr. Quenneville’s Website: Unit 2 Trigonometry
https://sites.google.com/a/hdsb.ca/mrquenneville/home/grade-10-applied-math/unit-2trigonometry
Slope
M21.6
Investigate
Staircases, Steepness, and Slope
Appendix B.31.
Staircases Handout
Appendix B.32.
Slope
Applications
M21.6
Project
Ramp It Up
Foundations and Pre-Calculus 10 (2010). p. 128
Applications
Pitch of a Roof
Appendix B.33.
Page 32
Mathematics 21
Resource
What Does the Road Sign Mean?
http://www.angelfire.com/ultra/mathproject/
Page 33
Mathematics 21
Theme Overview: Recreation and Wellness
Theme Introduction
Recreational activities such as playing games, solving puzzles, and participating in
sporting events as well as activities connected to personal wellness will be used to
teach problem solving strategies, reasoning, and budgeting skills. Students will apply an
understanding of measures of central tendency to make predictions or inform decisions
in order to effect changes in their own lives in terms of recreation and personal wellness.
Outcome that overlaps in all four themes
M21.1 Extend and apply understanding of the preservation of equality by solving
problems that involve the manipulation and application of formulae within home, money,
recreation, and travel themes.
Outcomes
M21.2 Demonstrate understanding of numerical reasoning and problem solving
strategies by analyzing puzzles and games.
M21.3 Extend and apply understanding of measures of central tendency to analyze
data.
M21.8 Demonstrate understanding of budgets.
At a Glance
Strategizing
Solving puzzles
Numerical reasoning
Inductive and deductive reasoning
Mean
Median
Mode
Budgeting
Guiding Questions
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Page 34
When you think of your favorite game, what comes to mind?
What is your strategy used to win a game?
What is an effective strategy?
Can games/puzzles be solved more than one way?
What is your favorite approach to solving a game/puzzle?
What is the difference between inductive and deductive reasoning?
How is statistics used to support an argument or a claim?
How can statistics be used to lead to different conclusions?
When is it appropriate to use the mean, median and mode?
How can the measures of central tendency be used to make informed decisions?
Are they used correctly to present information?
Mathematics 21

Should all data be included when finding measures of central tendency?
Career Connections
Chess master
Puzzle maker
Psychologist
TOPIC:
PUZZLES AND
GAMES
Sports analyst
Statistician
Public health nurse
OUTCOME:
SUGGESTED TEACHING AND LEARNING:
Inductive and
Deductive
Reasoning
M21.2
Instruction and Practice
Inductive and Deductive Reasoning
Appendix C.1.
Analyze and
Strategize
M21.2
Activity (similar to Activity Puzzles and Games in Mathematics
11)
Puzzles and Games
Appendix C.2.
Game
Golf Card Game
http://www.pagat.com/draw/golf.html#six
TOPIC:
MEASURES OF
CENTRAL
TENDENCY
Measures of
Central
Tendency
OUTCOME:
M21.3
SUGGESTED TEACHING AND LEARNING:
Activity
Measures of Central Tendency
Appendix C.3.
Activity
Use measures of central tendency to compare goals scored
by professional hockey players 10 years ago compared to
present day.
Activity
Web Quest 1 – Baseball Stats
http://www.mathgoodies.com/Webquests/sports/
Page 35
Mathematics 21
Activity
Personal Wellness
Appendix C.4.
TOPIC:
OUTCOME:
SUGGESTED TEACHING AND LEARNING:
BUDGET
Budget
Page 36
M21.8
Activity
Recreation and Personal Wellness Budget
Appendix C.5.
Mathematics 21
Theme Overview: Travel and Transportation
Theme Introduction
The Travel and Transportation theme will be used as the context of the mathematical
skills it takes to plan a trip. Students will explore map reading, budgeting, and the
mathematics involved in an area of interest. When planning a trip, students will consider
transportation, lodging, entertainment, and meals.
Outcome that overlaps in all four themes
M21.1 Extend and apply understanding of the preservation of equality by solving
problems that involve the manipulation and application of formulae within home, money,
recreation, and travel themes.
Outcomes
M21.4 Demonstrate and extend understanding of similarity and proportional reasoning
related to scale factors, scale drawing, scale models, surface area, and volume.
M21.7 Demonstrate understanding of the mathematics involved in an area of interest.
M21.8 Demonstrate understanding of budgets.
At a Glance
Similarity
Proportional reasoning
Direction
Location
Distance
Scale factor
Scale drawings
Map reading
Mathematics in an area of interest
Planning and budgeting a trip
Guiding Questions
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Page 37
What is the difference between direction and location?
How are direction, location, and distance related?
How do direction, location and distance relate to math?
Is there more than one way to get to a location?
What are the key components to giving directions?
What are the key components to giving locations?
What are the key components to determining distances?
What units are used for distance?
How do you read a map?
How do you use a map?
How do you scale diagram to create a map?
How does direction, location, and distance factor into travel plans?
What is an area of interest?
Mathematics 21
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
What makes an area of interest significant?
How is math related to an area of interest?
What fixed and variable expenses do you consider when creating a travel budget?
Career Connections
Bus driver
Taxi driver
Pilot
Traveller
TOPIC:
SCALE FACTOR
AND SCALE
DRAWINGS
OUTCOME:
Direction,
Location, and
Distance
M21.4
Pre-Assessment
Direction, Location, and Distance
Appendix D.1.
Scale Factor,
Scale
Drawings, and
Map Reading
M21.4
Activity
Map Reading
Appendix D.2.
TOPIC:
AREA OF
INTEREST
Area of
Interest
TOPIC:
BUDGETS
Budgeting for
a Trip
Page 38
OUTCOME:
M21.7
OUTCOME:
M21.8
SUGGESTED TEACHING AND LEARNING:
SUGGESTED TEACHING AND LEARNING:
Activity
Area of Interest
Appendix D.3.
SUGGESTED TEACHING AND LEARNING:
Activity
Budgeting for a Trip
Appendix D.4.
Mathematics 21
Appendices
Appendix A: Earning and Spending Money
Appendix A.1 Budgeting Key Terms
Use a learning strategy to help students familiarize and understand these terms.
Terminology:
Balanced budget
Fixed expense
Unexpected expense
Budget
Recurring expense
Utilities
Deficit
Surplus
Variable expense
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Mathematics 21
Appendix A.2 Create a Personal Budget with Fixed and Variable Expenses
Home
Mortgage
Property tax
Rent
Utilities
Cell/Home phone
Internet
Home decorating
Home repairs
Health and
Recreation
Club/Team fees
Sports equipment
Lessons
Gym membership
Prescriptions
Over-the-counter
Medications
(Tylenol, etc.)
Dental costs
ATV/Snowmobile
Leisure activities
(hunting, fishing,
etc.)
Daily Living
Groceries
Laundry
Dining out
Clothing
Gifts
Hair salon
Credit card payments
Makeup
Manicure/Pedicure
Personal care
(shampoo, deodorant,
etc.)
Vacations
Travel: bus, car, plane,
train
Accommodations
Food
Souvenirs
Activities
Transportation
Driver’s licence
Car payments
Fuel
Car plates
Package policy
Repairs
Oil changes
Tires
Car wash
Parking
Tickets
Bus pass
Savings/Investments
Entertainment
Cable/Satellite
Movies
Concerts
Books/Magazines
Music
Video games
Other
Savings accounts
Pet expenses
RESP
Piggy bank
Charities
Petty cash
Bank charges
Post-Secondary
tuition
Post-Secondary
books
1.
2.
3.
4.
5.
Identify the items that apply to you now. List them on the budget template.
What items might apply to you 3 – 5 years from now?
Identify five items in your template that you think are fixed expenses.
Identify five items in your template that you think are variable expenses.
Identify five items in your template that you think are essential expenses (not
optional when it comes to day-to-day living).
6. Identify five items in your template that you think are non-essential expenses.
7. After finishing the activity, explain what you learned about budgeting. Include
advantages and challenges.
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Mathematics 21
BUDGET TEMPLATE
This budget template can be modified by the teacher/student for other projects.
Category
INCOME:
Wages
Allowance
Babysitting
Other:
Other:
INCOME SUBTOTAL
EXPENSES
Home
Monthly Budget
Actual Amount
Estimate Your Income
Your Actual Income
Estimate Your Expenses
Your Actual Expenses
Difference
Daily Living
Transportation
Entertainment
Health/Recreation
Vacation
Saving/Investing
Other
EXPENSES SUBTOTAL
NET INCOME
(Income – Expenses)
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Mathematics 21
Appendix A.3 Budgeting to Live Away From Home Additional Questions
Questions:
1. How would you adjust your budget if you were only able to work part-time?
2. How would you adjust your budget if you had an $800 electrical problem with your
vehicle?
3. Research the cost of a major purchase (boat, motorcycle, vehicle, laptop, etc.).
Adjust your budget on a new template to demonstrate how you could keep a
balanced budget.
4. How would you adjust your budget if you wanted to save 10% each month as a
down-payment for a home?
5. What were the challenges in the creation of your budget?
6. List some advantages of working with a budget.
7. What categories would you expect to increase in 10 years?
8. What categories might you need to add to/delete from your budget in 10 years?
9. What were the top three concepts you learned in this project?
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Mathematics 21
Appendix A.4 Financial Institutions Services Key Terms
Use a learning strategy to help students familiarize and understand these terms.
Terminology:
Account
Encryption
Quarterly
Annum
Financial advisor
RESP
ATM
Full-service
RRSP
Balance
GIC
Rule of 72
Bank card
Interest
Savings account
Canada Savings Bond
Investment
Self-service
Cheque
Mobile banking
Semi-annual
Chequing account
Monthly
Simple interest
Compound interest
Monthly fee
Telephone banking
Compounding period
NSF cheque
Term
Daily
Overdraft protection
Term investment
Debit
Post-dated cheque
Transaction
Debit card
PIN
Transfer
Deposit
Principal
Withdrawal
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Mathematics 21
Appendix A.5 Researching Types of Banking Services and Fees
Use this chart to compare the banking services provided by at least two financial
institutions. Either research on the Internet or obtain brochures from various financial
institutions.
Financial Institutions
a)
b)
c)
1. What service
charges are there on
the account?
2. What are the fees
for transactions?
3. Are there
incentives or rewards
with the account?
4. Is online banking
service available? Is
there a fee for this
service?
5. Is telephone
banking service
available? Is there a
fee for this service?
6. Is mobile banking
service available? Is
there a fee for this
service?
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Mathematics 21
7. What is the
interest rate for the
savings account(s)?
8. Are cheques
available for the
accounts? What is
the cost to order
cheques? Is there a
cost for writing
cheques?
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Mathematics 21
Appendix A.6 Advantages and Disadvantages of Banking Services
Read: Advantages and Disadvantages of Savings and Checking Accounts
http://www.ehow.com/info_8093699_advantages-disadvantages-savingschecking-accounts.html
Type of Banking Service
Online Banking
Advantages
Disadvantages
Mobile Banking
Debit Card
Chequing Account
Savings Account
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Mathematics 21
Appendix A.7 Writing Cheques
Resource: Saskatchewan Learning Mathematics 21 (2007-2013).
Instruction: One way of paying for things or taking money out of your chequing account
is by writing a cheque. A cheque is a written order to the bank to pay a certain amount
of money from your account as ordered. A cheque is not money but it is used like
money. It may be cashed only by the person to whom it is written unless the person
signs it over to a second party (another person). To take money out by cheque, you
must fill in a cheque. It may be a personalized cheque or a non-personalized cheque.
It is usually better to use a personalized cheque if you have one.
Before we go over how to fill in a cheque blank, you are going to practice writing dollar
amounts in words.
1. It is important to write amounts in words correctly on checks. The words for numbers
between 20 and 100 are hyphenated when the number has two words.
Example:
45
99
is written as “Forty-five”
is written as “Ninety-nine”
2. The word “and” is reserved for the decimal point. The cents are written as a fraction
of a dollar, since the word “dollars” appears at the end of the line.
Example:
$706.10 is written:
$17.36 is written:
Seven hundred six and 10/100
Seventeen and 36/100
3. When there are no cents, the fraction is usually written in zeros.
Example:
$82.00 is written:
Eighty-two and 00/100
When you write a cheque, you must fill in five items. Notice the placement of each item
on the blank cheque shown on the next page.
1.
2.
3.
4.
5.
The date
The name of the person or company who is to receive payment (payee)
The amount written in digits
The amount written in words
Your signature
An optional line “For_______________” should be filled in to remind you what the
cheque was written for.
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Mathematics 21
Additional points to keep in mind:






Be sure not to leave any blank areas on your cheque.
Start at the beginning of the line when you are writing the cheque amount.
Write clearly and only use ink when writing your cheque to help prevent anything
from being changed on your cheque.
Your cheque is not legal until you sign it. Keep your cheques in a safe place until
you are ready to use them. Also, never sign a blank cheque.
Make sure the amount box (in numbers) and the amount line (in words), match.
If you make a mistake when writing a cheque, write “VOID” in big letters on the
cheque and then rip it up. Make sure you record the cheque in the register and
mark that it was VOID.
Practice: Write out each dollar amount in words:
1. $235.00
2. $200.39
3. $60.98
4. $1819.21
5. $607.77
6. $910.00
7. $25.86
8. $1327.56
9. $705.15
10. $384.48
11. $37.16
12. $56.00
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Mathematics 21
Fill in the four cheque blanks with the information given for each.
13.
Date
12/20
Payee
Amt. of cheque
Payer
Reader’s
Choice
$18.99
Betty L. Rain
14.
Date
Payee
Amt. of cheque
Payer
12/20
M& L Meats
$62.75
Betty L.
Rain
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For
(optional)
subscription
For
(optional)
frozen
steaks
Mathematics 21
15.
Date
Payee
6/13
Payer
For
(optional)
Fishing Spot
Amt. of cheque
(words and
numbers)
$23.00
Joe Fox
reel
16.
Date
Payee
Amt. of cheque
Payer
7/25
Mrs. Smith
$101.03
Carol Fox
For
(optional)
candles
Answers:
1. two hundred thirty-five and 00/100
2. two hundred and 39/100
3. sixty and 98/100
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Mathematics 21
4. one thousand eight hundred nineteen and 21/100
5. six hundred seven and 77/100
6. nine hundred ten and 00/100
7. twenty-five and 86/100
8. one thousand three hundred twenty-seven and 56/100
9. seven hundred five and 12/100
10. three hundred eighty-four and 48/100
11. thirty-seven and 16/100
12. fifty-six and 00/100
13.
14.
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Mathematics 21
15.
16.
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Mathematics 21
Appendix A.8 Investment Options
Resource: The CITY A Financial Life Skills Resource
Prior Knowledge:
1. List some of your short-term goals and long-term goals.
2. How much money will you need for each of the goals?
3. What is your plan for making these goals a reality?
4. How could you save for a large purchase?
5. What are savings? What do you know about investments?
Activity: Have students work through the following information using a strategy (e.g.
think-pair-share, jigsaw, concept circle, etc.)
Instruction: People can choose from a wide variety of investments. This chart shows
you some things to consider about some of the main types of investments.
Type
Savings accounts,
Guaranteed
Investment
Certificates (GICs),
term deposits:
Money deposited
with banks, trust
companies and
credit unions
Treasury bills:
Expected Return
• usually a fixed
annual rate
• principal amount is
• GICs sometimes usually insured, but
tied to performance interest rates can be
of an index or other fixed or variable
standard
• determined by
difference between
Short-term (less than purchase price and
value at maturity
1 year) debt
securities issued by
government
Equities:
• may pay regular
dividends to share
Shares in ownership holders
of a company (also
• potential return
called stocks)
may depend entirely
on changes in share
price
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Risk
• low to moderate
• very low
Liquidity
• savings may be
withdrawn at any time
• some GICs and
term deposits must
be held to maturity,
but many allow for
early redemption or
cashing out at a cost
• not redeemable, but
can usually be sold
quickly through
investment dealers
• moderate to high
• shares traded on
stock exchanges are
• depends on size and usually quite easy to
stability of company, sell
management,
competition, etc.
• shares that aren't
listed on an
exchange may be
difficult or impossible
Mathematics 21
Type
Expected Return
Risk
• risk of borrower
defaulting is very low
for government bonds
•
bonds
with
longer
but can be low to high
Government and
for corporate bonds
corporate bonds and terms will usually
pay
higher
interest
debentures
rates
• bond values go up
and down with
• high risk "junk"
changing market
bonds offer even
interest rates
higher rates
Fixed income
investments:
Mutual funds:
• interest rate is
usually fixed
• may include
interest, dividends
and capital gains (or
losses)
Units in a pool of
money that's
managed for a large
number of investors • return will depend
on manager's
by a professional
investment
money manager
decisions and on
the management
fees charged
Real estate:
Property such as
land or houses
Liquidity
to sell
• most bonds can be
bought and sold
quickly through
investment dealers
• some bonds are
traded on stock
exchanges
• low to very high,
• most mutual funds
depending on what the allow investors to
fund invests in and on cash in (redeem) their
the skill of the fund
holdings on short
manager
notice
• depends on price, • low to high •
location, real estate depends on price,
market, etc.
location, real estate
market, etc.
• may include rent
or increase in value
• takes more time to
sell than many other
investments
• hard to sell small
portions
• depends on market
Direct investment:
Investing your
money to finance a
private business
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• low to very high
• medium to very high • low
• depends on type of
• success depends business, competition, • may be very hard to
on the business
sell
skill of the business
concept, the
manager, and the
manager and on
economy
economic
conditions
Mathematics 21
Investors can choose from thousands of different investments. The investments that
offer the highest expected returns are those with the highest risk. Wise investors
diversify their investments to help manage the risk.
Some investments are very complex. Factors like commissions, sales fees and tax
levels can have a major impact on the final return. Investors usually seek expert advice
from professional advisers to be sure they fully understand their investment and that the
investment is a good choice for their investment goals. Companies that offer investment
advice must be registered (licensed) and must comply with detailed standards of
conduct.
Find out more about investment by visiting your provincial or territorial regulator's
website (see the Financial Consumer Agency of Canada for links at
www.themoneybelt.gc.ca).
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Mathematics 21
Appendix A.9 Comparison of Simple and Compound Interest
Resource: Exploring Compound Interest. Foundations of Mathematics 12 (2012). p. 19
Example: Sebastian invests $2000 at 3.5% interest, compounded annually for 4 years.
Determine the interest earned.
I=Prt
A=P+Prt
Simple Interest
Year End Investment Value
***Because we are calculating compound interest, we need to calculate the interest
every year separately.**
Determine the difference in the future value of simple interest compared to compound
interest.
Compound Interest
Year
1
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Principle
at
Beginning
of the
Year
Interest
I=Prt
2000
I=
(2000)(0.035)(1)
Year End
Investment
Value A = P
+Prt
A = 2000 +
70
Simple Interest
Principle at
Beginning
of the Year
2000
I=Prt
I=
(2000)(0.035)(1) =
Mathematics 21
= 70
2
2070
I=
(2070)(0.035)(1)
= 72.45
= 2070
A = 2070 +
72.45
= 2142.45
70
2000
70
3
4
Total:
Total:
Determine how much more interest was earned using compound interest.
Example: Levi invests $13 000 for 6 years at 2.8% interest, compounded annually.
a) Complete the following chart:
Compound Interest
Year
Principle at
Beginning
of the Year
1
13000
Interest
I=Prt
Year End Investment
Value A = P + P r t
2
3
4
5
6
b) Determine the interest earned to the nearest cent.
c) What is the maximum amount you can withdraw from the account at the end of
the investment?
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Mathematics 21
Appendix A.10 Financial Decision Making Key Terms
Use a learning strategy to help students familiarize and understand these terms.
Terminology:
Amortization period
Default
Mortgage
Cash advance
Down payment
Outstanding balance
Charge account
Financial institution
Overdraft protection
Consolidate
Fixed term
Premium
Co-signer
Installment plan
Promotion
Credit
Lease
Rent
Credit card
Line of credit
Return
Credit rating
Loan
Thirty-day account
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Mathematics 21
Appendix A.11 An Introduction to Credit and Credit Rating
Resource: Saskatchewan Learning Mathematics 21 (2007-2013).
Instruction:
The Use of Credit: Credit is defined as the advance of goods/services in exchange for a
promise to pay at some future date. If you take advantage of a “buy now, pay later”
opportunity you will pay more than the cash price. The longer you take to repay, the
greater the cost of credit will be.
Types of Credit:
1. Sales Credit: Credit extended for a purchase of an item at the time of purchase.
a) Installment Plan
For bigger items (appliances, furniture) it usually involves a down-payment
followed by regular payments of principal and interest over a period of months.
Failure to make payments could result in the item being repossessed.
b) Charge Accounts
Sales credit with a specific limit. If the entire amount billed is paid, there is no
interest charged. If the full amount is not paid, a minimum amount must be paid
with interest due in the next month on the amount not paid. For example, Sears,
Target, Petro, and Shell cards.
c) Credit Cards
A credit card is a plastic card that allows you to pay for something by charging it
and paying for it later. Each card has a credit limit. If the bill paid by the
statement date no interest is charged. Interest is charged in the next month for
the amount not paid. A minimum payment of principal and interest must be
made each month if the entire balance is not paid by the statement date. For
example MasterCard, Visa, and American Express. Interest rate is substantially
higher than a loan from a bank, because interest is compounded daily.
2. Cash Credit: Credit received by borrowing money from a bank and paying it back
later.
a) Personal Loan
The loan may be paid back in equal payments of principal and interest or in a
single payment. The quicker it is paid off, the less interest that is paid.
b) Mortgage
A long term personal loan (usually for a house) in which the house purchased is
used as security for repayment. Payments are usually PIT (principal, interest,
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Mathematics 21
tax). Interest is charged on the outstanding balance at each payment date
therefore the amount of interest paid in the early payment is a very high
percentage of the payment. As the principal is paid down, the interest portion of
the payment becomes smaller. The loan is usually paid down over a period of
15-25 years.
c) Personal Line of Credit
A one-time approved loan allowing you to borrow up to a prearranged limit by
simply writing a cheque. It can be used to purchase anything. The interest rates
are usually lower than credit cards.
d) Pay Day Loan
A small, short-term unsecured loan, not necessarily linked to the borrower’s
payday. The loans are also sometimes referred to as “cash advances” but are
different from credit card cash advances. Pay day loans rely on the consumer
having previous payroll and employment records.
3. Service Credit:
Credit extended for services provided on a daily basis but paid for only once a
month. For example, telephone, power, gas.
Interest: The amount of money paid or earned for the use of money.
Credit Rating: Credit rating is an evaluation of your past use of credit, your character,
your ability to repay and the security or collateral you have for the loan.
When being rated on character and past use of credit (will you repay what you borrow)
you will be judged on the following:
Have you used credit before?
Do bills get paid on time?
How long have you lived at your current address?
How long have you been at your current job?
Can you supply a character reference?
When being rated on your ability/capacity to repay, you will be judged on the following:
Do you have a steady job?
Do you have other loans you are paying off?
What are your living expenses?
Do you have dependents (someone depending on you for support)?
When being rated on the security or collateral you have for the loan you will be judged
on the following:
Do you have a savings account?
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Mathematics 21
Do you have any investments to use as collateral?
Do you own any property?
Building a Good Credit Rating: If you are starting out, financially speaking, the first thing
to do is build a good credit rating.
1. Open a savings account and make regular deposits. The idea is to save
money and show that you are responsible and reliable about money.
2. Pay bills promptly.
3. When borrowing, borrow only what you need and can repay back.
4. Arrange a loan repayment schedule and try to repay the loan as soon as
possible.
Good Credit Risk: You can be considered a good credit risk if you pay back loans on
time, pay back loans regularly and on time, and have a regular salary to use to pay back
loans.
 One who pays back all loans
 One who pays loans back regularly and on time
 One who has a regular salary to use to pay back loans
Practice: Indicate if the statement presents credit as an advantage or a disadvantage.
1. Credit eliminates the need to carry large quantities of cash.
2. Credit available through a store’s credit card cannot be used in other stores
therefore discouraging comparative shopping.
3. Credit allows the immediate possession of goods to be paid over a period of time.
4. Credit allows the consumer to take advantage of sales opportunity.
5. In the event of non-payment, credit may lead to the loss of property.
6. Credit helps in dealing with financial emergencies.
7. Credit encourages impulse buying.
8. Credit may increase your debt load so that you cannot save for the future.
9. Credit increases the overall price of goods. (Most credit card companies charge
retailers a 3% fee. This fee is added to the cost of all items in the store).
10. Easily available credit may encourage poor buying decisions.
11. Do you or your classmates have credit cards? How does one get a credit card?
What is meant by the limit of a credit card?
12. Do you think a person should have several credit cards? Why or why not?
13. What causes people to get into financial difficulty? Is it always the fault of the
individual?
14. At what age do you see yourself taking out a mortgage for a house? Explain.
15. The qualities you look for in a friend are the same qualities a lending institution
hopes to find in you. Do you agree or disagree with this statement? Explain.
16. Suppose you are a loans officer in a bank. List three questions you would want to
ask a customer who has come in to borrow $15 000 to buy a new boat?
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Mathematics 21
17. What is a debit card? List several tips that you should follow in order to protect
your debit card while using it?
18. Decide which of the items below you would use to credit to purchase.
When making your decisions consider:
-
Do you need the item?
Could you wait until you have saved up enough money?
Could you better use the money for some other purchase?
Does the purchase over-extend your regular budget?
Will you be able to repay the borrowed money?
Would you purchase the following items on credit? Why or why not?
a)
b)
c)
d)
e)
f)
g)
h)
A ticket to a concert?
A compact disc on sale for $16.99?
A $500.00 leather jacket on sale for $350.00?
Groceries for the family?
Taking your friend out for lunch?
A vacation to Disneyland at $500.00 each for a family of 4?
A second computer for the family?
A second television for the family?
Use the chart to put the following into one of two categories: i) good credit risk ii) high
credit risk. Be prepared to discuss your answers.
19. SaskTel employee, full time, employed for 5 years
20. Secretary, part time, employed for 10 years
21. Gas jockey, works part time on weekends, employed for 2 months
22. Student, no job
23. Lawyer, full time, employed for 20 years
24. Teacher, full time, employed for 7 years
25. Grain farmer, full time, employed for 32 years
26. Management trainee, part time, employed for 2 weeks
27. Taxi driver, full time, employed for 17 years
28. Delivery person, works part time on weekends, employed for 10 months
29. Waiter, part time, employed for 1 year
30. Waitress, full time, employed for 12 years
31. Dentist, full time, opened own practice 2 months ago
32. Seasonal worker with the city, full time, employed only for summer months
33. Unemployed person
34. Business owner, part time, employed for 5 years
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Mathematics 21
Good Credit Risk
High Credit Risk
Answers:
1. Advantage
2. Disadvantage
3. Advantage
4. Advantage
5. Disadvantage
6. Advantage
7. Disadvantage
8. Disadvantage
9. Disadvantage
10. Disadvantage
11. to 34. Answers will vary.
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Mathematics 21
Appendix A.12 Risks and Benefits of Types of Credit
Instruction:
Student Loans
Student loans can be granted by governments or by financial institutions.
Possible benefits of a student loan:
 Allows you to continue post-secondary studies.
 The government pays the interest on your loan while you are studying full-time.
You repay the loan upon completion of your studies. The interest on your loan
starts when you cease to be a full-time student.
Potential risks of a student loan:
 At the end of your studies, you may have to deal with substantial study debts.
This may delay other plans, such as travelling or buying a house.
Credit Cards
Generally, credit cards allow you to make purchases, up to a specific credit limit, for
which you will be billed at a later date. They allow you to transfer your balance from one
billing cycle to another. Nevertheless, you must pay a minimum amount every month,
and unpaid balances are subject to interest charges, based on an annual percentage
rate or APR.
Possible benefits of a credit card:
 Helps you create a credit history and earn a credit rating.
 Can be more practical than carrying cash.
 Allows you to borrow free of charge if you always pay the balance in full by the
due date.
 Can offer incentives, such as reward points that you can use towards purchasing
certain products.
 Allows you to pay conveniently for purchases made over the telephone or on the
Internet.
Potential risks of a credit card:
 Can lead you to spend more and drive you into more debt than you can handle.
 Can affect your credit rating if your monthly payments are late.
 Can carry conditions that are hard to understand.
 Is generally more expensive than other forms of credit like personal lines of credit
or personal loans.
 Chance of fraud.
Personal Line of Credit
Provided by financial institutions, this type of loan allows you to withdraw money, as
needed, up to a maximum credit limit. You are charged interest from the day you
withdraw money from your line of credit until you pay back the loan in full.
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Mathematics 21
Possible benefits of a personal line of credit:
 Gives you the convenience of borrowing money whenever you need it: you do
not have to reapply for loans.
 Offers flexible reimbursement methods.
 Offers lower interest rates than credit cards.
Potential risks of a personal line of credit:
 Some people use this loan as a source of revenue.
 Can force someone into more debt than he/she can afford.
Personal Loans
You can get a personal loan to buy a car, to buy furniture, to go on a trip, etc. You then
use the borrowed amount as you wish. The amount, the rate and the conditions of
reimbursement are fixed at the time of the contract. A personal loan is reimbursable in a
predetermined time frame through monthly payments.
Possible benefits of a personal loan:
 There are various options that allow you to obtain a loan to meet your needs.
 The loan is negotiable.
 You use the borrowed amounts as you wish.
Potential risks of a personal loan:
 Since this loan is not linked to a specific purchase, if the goods are defective or if
there is any other problem (e.g. the goods are not delivered), the loan must still be
reimbursed.
 Can drive you into more debt than you are able to pay back if the loaned amount
does not take into account your ability to repay.
 Increases your monthly obligations.
Instalment Plans
Instalment plans normally apply when you make a significant purchase at a business.
For example, you may purchase a television or a refrigerator but pay for it through
monthly instalments, usually accompanied by a certain interest rate.
For this type of contract, the seller has ownership of the goods until they are paid in full,
even though you are in possession of them. Therefore, if you miss a payment, the seller
can demand that the goods be returned. Remember that in this type of contract, the
merchant is responsible for accidental loss of the goods as long as you are not yet the
owner.
Possible benefits of instalment plans:
 The merchant is responsible for accidental loss of the goods as long as he/she is
still the owner.
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Mathematics 21
Potential risks of instalment plans:
 They increase the total price due to the interest charges.
 The seller remains the owner of the goods until they are fully paid. If you miss a
payment the store can repossess.
 They increase your monthly obligations.
Beware of “buy now, pay later” promotions; several stores offer this type of promotion.
You buy goods today, but pay nothing for one year, for example. This kind of
advertisement usually does not indicate the consequences of not making payments on
time. In fact, according to some store policies, interest starts to accrue on the date of
purchase. This interest is cancelled if the person pays within the time limit. However, if
someone pays after the time limit, he or she must pay interest for the whole period and
interest rates are usually quite high! These plans often mention there are no additional
fees and no interest; however the selling price has been increased so the retailer makes
a higher profit for the time they have to wait until it has been paid in full.
For example:
 You buy electronic equipment worth $1000.
 You do not pay within the one-year limit.
 The interest rate is 28.8%.
You will have to pay $1000 plus the year's accrued interest (from the date of purchase).
In addition, people who resort to this sort of agreement often do not have the means to
pay off the goods at the time of purchase, nor do they have the means to do so in one
year. Many possible events can change your financial status over such a long period of
time, so be careful.
Mortgage
A mortgage is a long-term loan granted to an individual in order to buy a home. The
home itself is given as a guarantee for the loan. There are different types of mortgage
loans, such as open or closed, that offer variable of fixed rates and various options
concerning the term, the payment frequency and the amortization period.
Possible benefits of a mortgage:
 Allows you to purchase a home, which would be impossible without a loan.
 Offers favorable rates.
Potential risks of a mortgage:
 Monthly payments are sometimes high.
 Because it is a long-term purchase, a change in household revenue could have a
negative effect on your ability to pay it back.
 The home is given as a guarantee of the loan, meaning that in case of
nonpayment the home could be taken.
The purchased property can easily cost double because of the length of the loan.
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Mathematics 21
Payday loans – EZ cash etc.
If consumers can’t get a loan from a bank because they are of high risk or they need
money quickly before payday they can borrow money from one of these institutions on
site or online.
Benefits – NONE
Risks – Tremendously high interest rates the most frequently posted APE was 652%
followed by 780%
How does the cost of a payday loan compare with other credit products?
Payday loans are much more expensive than other types of loans, including credit
cards. But how much are you really paying? How does the cost of a payday loan
compare with taking a cash advance on a credit card, using overdraft protection on your
bank account or borrowing on a line of credit?
Let's compare the cost of using different types of loans. We'll assume that you borrow
$300, for 14 days. Note the considerable difference in the cost of each type of loan.
Comparing the cost of a $300 loan, taken for 14 days1
Payday
loan
Cash advance
Overdraft
Borrowing from
on a credit
protection on a
a line of credit
card
bank account
—
$2.13
$2.42
$1.15
+
+
+
+
$50.00
$2.00
—2
—
=
=
=
=
Total cost of loan
$50.00
$4.13
$2.42
$1.15
Cost of the loan expressed
as a percentage of the
amount borrowed3
435%
per
year
36%
per year
21%
per year
10%
per year
Interest
Applicable fees
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Mathematics 21
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Mathematics 21
Appendix A.13 Credit and Exploring Credit Card Use
Resource: Saskatchewan Learning Mathematics 21 (2007-2013).
Instruction:
Credit Cards:
Credit Limit:
Promise to Pay:
Allow you to buy an item from a retailer or business, which accepts
the card as a method of payment. The customer can then pay the
full amount or a portion per month, but interest is charged on the
outstanding balance.
The highest amount of money one can charge to the credit card.
You must pay for all purchases you charge, and for all purchases
charged by anyone you allow to use your card.
Past Due Accounts: If you do not make a minimum payment each month, you may be
required to make immediate payment of your entire balance.
Example:
Card Name
Annual Fee
Credit Limit
$0.00
Rate of interest
%
17.9%
1. Royal Bank
Student Classic
Visa
2. TD Canada
Trust GM Card
Visa
3. CIBC
Aerogold
$0.00
18.5%
$2000.00
$120.00
19.5%
$3000.00
4. Bank of
Montreal Air
miles
MasterCard
$35.00
18.9%
$2500.00
$500.00
Benefits (Air
miles, discounts,
etc.)
None
Collect up to
$500 off a new
GM vehicle
Collect points
that can be used
for travel
For every $20
spent, collect 1
Air mile
For you, which would be the best card and why?
Which would be the worst card and why?
How Is Interest Calculated?
 An individual has until payment day to pay back the amount owing. If the amount
can’t be repaid, then interest is charged.
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Mathematics 21

Interest is paid on the full amount of the purchase until the purchase has been
entirely paid (e.g. $1000.00 stereo with $800.00 paid off and only owes $200.00.
Must still pay interest on $1000.00)
Formula: I = P r t
Principle  daily interest rate  # of days
Note: If you are given the annual percentage rate (APR), divide it by 365 to get
the daily interest rate. Now convert this to a decimal by dividing by 100.
Example 1:
APR = 18.5%  365 = 0.0506849%  100 = 0.0005068 (use this in the formula)
Example 2:
Suppose that Jill had made purchases as shown in the statement below. If the
payment due date was June 6, but her payment was not received until June 21, find
out the total amount of interest that would be charged. Daily interest rate is 0.05067%
(0.0005067). (Include purchase date and payment date when counting number of
days).
Transaction
Date
M/D
05/12
05/17
05/23
Description
Amount $
Esso
Bike Doctor
Hair Affair
$32.45
$416.72
$55.90
Number
of Days of
Interest
41
36
30
Total Interest =
Interest
Charged
$0.67
$7.60
$0.85
$9.12
Determining the Minimum Payment Required
If the new balance is:
1. Less than $10.00, then pay out that amount.
2. Less than the credit limit, then pay the greater of $10.00 or 3% of the balance
rounded up to the next highest dollar.
3. Higher than the credit limit, then pay 3% and the amount the new balance is over the
credit limit.
Using the above information, determine the minimum payment required if the credit limit
is $2000.00 on a new balance of:
Example 1 : $8.00
Solution:
$8.00
Page 70
(#1 - new balance was less than $10.00)
Mathematics 21
Example 2 : $1415.00
Solution:
0.03 x 1415 = $42.45  $43.00 (#2 – don’t forget to round)
Example 3 : $288.00
Solution:
0.03 x 288 = $8.64  $10.00
(#2 – 3% of balance is less than $10)
Example 4: $2345.00
(#3 – balance is higher than credit limit)
Solution:
0.03 x 2345 = $70.35  $71.00
$2345 - $2000 = $345 (over limit amount)
$71.00 + 345.00
= $416.00
Example 5: $3005.00
(#3 – balance is higher than credit limit)
Solution:
0.03 x 3005 = $90.15  $91.00
$3005.00 - $2000 = $1005.00
$91.00 + $1005 = $1096.00
Practice:
The credit limit is $9000.00 on a new balance of:
1. $8.50
2. $40.00
3. $550.00
4. $920.00
5. $762.95
The credit limit is $500.00 on a new balance of:
6. $450.21
7. $562.00
8. $895.00
9. $9.95
10. $49.72
The credit limit is $1500.00 on a new balance of:
11. $200.00
12. $750.00
13. $25.00
14. $1625.00
15. $1300.00
The credit limit is $750.00 on a new balance of:
16. $300.00
17. $50.00
18. $800.00
19. $450.00
20. $6.00
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Mathematics 21
If the daily interest is 0.05067% (0.0005067), calculate the amount Jill will be charged if
she has the given amount overdue for the given number of days.
21. $306.52 for 27 days
22. $54.97 for 101 days
23. $2 952.00 for 11 days
24. $1 875.26 for 127 days
25. $972.00 for 6 days
26. Suppose that Dan had made purchases as shown in the statement below. If the
payment due date was June 2, but his payment was not received until June 24, find
out the total amount of interest that would be charged. Daily interest rate is
0.05067% (0.0005067). (Include purchase date and payment date when counting
number of days).
Transaction
Date
M/D
05/03
05/11
05/11
05/13
05/17
05/17
05/21
05/22
05/24
Description
Broadway
Café
Safeway
Blockbuster
Boston Pizza
Canadian Tire
Future Shop
Boom Town
Extra Foods
Shell
Amount $
Number
of Days of
Interest
Interest
Charged
$17.45
$31.26
$9.72
$12.76
$45.86
$432.75
$236.71
$27.45
$34.96
Total =
Answers:
1. $8.50
2. $10.00
3. $17.00
4. $28.00
5. $23.00
6. $14.00
7. $79.00
8. $422.00
9. $9.95
10. $10.00
11. $10.00
12. $23.00
13. $10.00
14. $174.00
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Mathematics 21
15.
16.
17.
18.
19.
20.
21.
22.
23.
24.
25.
$39.00
$10.00
$10.00
$74.00
$14.00
$6.00
$4.19
$2.81
$16.45
$120.67
$2.96
26.
Transaction
Date
M/D
05/03
05/11
05/11
05/13
05/17
05/17
05/21
05/22
05/24
Description
Broadway
Café
Safeway
Blockbuster
Boston Pizza
Canadian Tire
Future Shop
Boom Town
Extra Foods
Shell
Amount $
Number
of Days of
Interest
Interest
Charged
$17.45
53
$0.47
$31.26
$9.72
$12.76
$45.86
$432.75
$236.71
$27.45
$34.96
45
45
43
39
39
35
34
32
$0.71
$0.22
$0.28
$0.91
$8.55
$4.20
$0.47
$0.57
Total = $16.38
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Mathematics 21
Appendix A.14 Credit Card Comparison
Resource: Saskatchewan Learning Mathematics 21 (2007-2013).
Find information about five different credit cards by asking parents, family members,
friends, etc. You can use the same card more than once if it is available from more than
one institution (e.g. GM Visa, Scotia Bank Visa). You can also include retail cards (e.g.
Sears, Target, Shell, etc.). You may get assistance on-line.
Card Name
Annual Fee
Rate of
interest
%
Credit Limit
Benefits (Air miles,
discounts, etc.)
1.
2.
3.
4.
5.
Questions:
 In your opinion, which would be the best card to use? Why?
 In your opinion, which would be the worst card to use? Why?
Page 74
Mathematics 21
Appendix A.15 Exploring Credit Card Use
Debit Cards
Using a debit card is like using cash, you may pay a transaction fee, but as long as you
have money in your account you won’t have to pay interest. Things to consider when
you have a debit card:
 Read the information you received at the time your debit card is issued, so you
know the service charges related to the use, the importance of your PIN number
and the potential liability for losses due to unauthorized transactions, what to do if
it is lost or stolen, and how to resolve complaints
 In many cases your financial institution may not send you a detailed report of your
purchases. This can make record keeping confusing and difficult. The result: it's
harder to keep on top of things. Your record keeping needs to be very accurate.
 Ask your financial institution to send you a detailed monthly report of your
purchases. See if you can receive it at a student rate or at the lowest possible
cost. Your debit card might be attached to a line of credit, which makes it very
easy to overspend — and costly too.
 If something goes wrong — say, someone gets your card and personal
identification number (PIN), and makes a fraudulent purchase or withdrawal —
you'll probably lose the money, with no recourse. If your debit card is attached to
a line of credit, the thief could clean out your line of credit too. If you have
divulged your PIN number your insurance is devoid and you are responsible for
the losses.
 If you have a debit card, keep your PIN and card in separate places
 To complete an ATM transaction form a screen with your hand or body to prevent
anyone from seeing you enter your PIN.
*when you use a private automatic
teller machine – you will pay the ATM fee (e.g. $1.75) and your bank‘s service
charge.
Credit Cards
Generally, credit cards allow you to make purchases, up to a specific credit limit, for
which you will be billed at a later date. (Exception is cash advances they are charged
interest from the date of the transaction.)They allow you to transfer your balance from
one billing cycle to another. Nevertheless, you must pay a minimum amount every
month, and unpaid balances are subject to interest charges, based on an annual
percentage rate or APR.
Your responsibilities of owning a credit card:
Borrow only what you can _________.
Read and understand the credit contract.
Pay debts _____________.
Notify creditors if you cannot meet __________.
Report lost or _________ card immediately.
Never give your card number over the phone or over the internet unless you are
certain of the identity or security of the site.
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Mathematics 21
Never leave you cards unattended work, car,….
Protect your __________________________.
Always check your card when returned after a purchase.
Sign the back as soon as you get it.
Make a list of your cards and their ____________.
Always ______________ your monthly statement.
When shopping for a credit card, compare items such as: annual interest rates, grace
period, annual fees, method of finance charge, transaction fees, credit limit, how widely
accepted, and services/features.
A credit card statement provides the following information:
1. Statement and due date
2. APR and daily interest rate
3. New and previous balance
4. Total amount of the credit line and available credit line
5. Minimum payment
6. The amount of time to repay if you only make minimum payments
Resource: Foundations of Mathematics 12 Teacher Resource (2012). p. 88
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Mathematics 21
Comparing Different Credit Cards
Jayden saw the new sound system he wanted on sale for $2623.95, including taxes.
He had to buy it on credit and had two options:

Page 77
Use his new bank credit card, which has an interest rate of 14.5%, compounded
daily. Because his credit card is new, he has no outstanding balance from the
previous month. It has an annual fee of $50, which is added to the balance at the
beginning of the year.
Mathematics 21

Apply for the store credit, which offers an immediate cash rebate of $100 on the
price but has an interest rate of 19.3%, compounded daily.
As with most credit cards, Jayden would not pay any interest if he paid off the balance
before the due date on his first statement. However, Jayden cannot afford to do this.
Both cards require a minimum monthly payment of 2.1% on the outstanding balance,
but Jayden is confident that he can make regular payments of $110.
Will a regular monthly payment of $110 enough to cover the minimum amount required?
Present Value
Present Value
Present Value
Future Value
Future Value
Future Value
Payment
Payment
Payment
Interest/Yr in
%
Periods
Interest/Yr in
%
Periods
Interest/Yr in
%
Periods
Periods/Yr
Periods/Yr
Periods/Yr
Compounds/Yr
Compounds/Yr
Compounds/Yr
Interest Paid
Interest Paid
Interest Paid
Total Cost
Total Cost
Total Cost
Cost of Borrowing on credit card __________
Cost of Borrowing paying minimum on credit card
_____________
Cost of Borrowing on store card ___________
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Mathematics 21
Appendix A.16 Comparing the Cost of a Loan
Resource: Saskatchewan Learning Mathematics 21. (2007-2013)
Choose two banks to complete the following activity:
Bank 1: ______________________________________
Bank 2: ______________________________________
1. Amount: Your teacher will assign the amount to use as your borrowing amount.
_________________
2. Purpose: Loan payments are scheduled at various rates depending on the purpose
of the loan. Your teacher will assign one of the following loan types:
 Car loan
 Mortgage loan
 Student loan
 RRSP loan
 Personal loan
3. Interest Rate: The bank will have different rates throughout the year. The interest
rate can change daily. Contact the two banks and find the current interest rate for your
loan types. You may wish to find this information on the internet, by a personal contact
with the bank or advertised rates.
Bank 1 Interest Rate: _________________________
Bank 2 Interest Rate: _________________________
4. Time: The amount of your loan can vary depending on how quickly you are able to
pay your loan off. Use both the maximum amount of time you can borrow the money
and the minimum amount of time you can borrow the money. You will need to contact
the bank regarding this.
a) Maximum Time
Calculate the monthly payment: ________________________
Calculate the total payment: ___________________________
b) Minimum Time
Calculate the monthly payment: ________________________
Calculate the total payment: ___________________________
5. What bank and loan payment was the best for you? Explain in detail.
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Mathematics 21
Appendix A.17 Purchasing a New Vehicle
Instruction: Remember that a car is not an investment. Quite the contrary — it loses
value over time. The instant you leave the dealership with your new car, it loses 10% of
its value. During the first year, it will depreciate by approximately 30%!
Here are some justifications for used cars. Thanks to regulations and standards
concerning safety and environment, among others, used cars are now more trustworthy
than they once were. Some are still protected by the guarantee of the manufacturer.
With technology, you can easily retrace a car's history to make sure that you are the
lawful owner. Many dealerships offer lower financing rates for used cars. Negotiations
are often done in a more friendly and less stressful manner when you buy from an
individual. You must, however take extra precautions when dealing with an individual.
1. Verify that you are dealing with the true owner.
2. Have the vehicle thoroughly inspected by a certified and independent garage.
No matter whom you buy a vehicle from, always conduct a road test and make sure that
you are very comfortable with your choice. It is your money and it will be your
responsibility. Get an opinion from someone who knows about cars. Don't let yourself
be pressured and don't make forced decisions.
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Mathematics 21
Appendix B: Home
Appendix B.1 What Do You Already Know About Measurement?
Brainstorm: To review units of measurement from Math 11, pose the question “What
do you already know about measurement?” and brainstorm with students the things
they know or recall about measuring.
Discuss: You may want to use the following probing questions to generate discussion:
 What’s bigger: a metre or a yard?
 What’s bigger: a centimetre or an inch?
 What’s bigger: a kilometre or a mile?
 How far is Regina from Saskatoon? (People often answer in units of time not
distance. Why do we do this? Is this done in other places, countries?)
 What measuring tool do you use to measure area?
 What item is about 1 foot long? (Leading students to referents)
 How tall is a flagpole or a tall structure? How can you get the measurement?
 How many cm in a foot?
Practice: Measurement Worksheets http://www.math-aids.com/Measurement/
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Mathematics 21
Appendix B.2 Enlargements, Reductions, and Scale Factor
Resource: MathLinks 9 (2009). pp. 129 – 131, 138
Materials: centimetre grid paper, tracing paper, ruler
Introduction: Many occupations require people to design projects using drawings or
models. For example, architects create plans for homes. These plans are called
blueprints. Architects work with ratios and proportions to produce floor plans that
represent accurate dimensions of the various areas of a home. The floor plan helps
people judge if the proposed design is suitable for their lifestyle.
Practice and Questions: Designers. MathLinks 9 (2009). pp. 129
Discuss: Use the following probing questions to generate discussion.
 Why do you think accuracy is important in developing a floor plan?
 Why is it important to maintain the same proportions for the dimensions of an
actual object and its image?
 What are other examples in which ratios are used to compare objects in daily life?
Investigate:
1. Find an illustration (e.g. cartoon character). Brainstorm with a classmate how you
might enlarge the illustration. What different strategies can you develop?
2. Try out two of your strategies and draw an image that is twice as large as the
illustration. What will be the ratio of the lengths of the sides of the enlargement to
the original?
Discuss: Use the following probing questions to generate discussion.
 Which strategy for making an enlargement do you prefer?
 What method might you use to check that the enlarged image is twice as large as
the original?
 How are the enlargement and the original the same?
 How are the enlargement and the original different?
Activity: Create a scale drawing of your classroom. Measure the dimensions of the
classroom and items that can be seen in a top view, including desks, tables, cupboards,
and shelves. Choose a scale factor and draw the scale drawing on grid paper. What
changes would you make to the layout of your classroom? Where would you place
desks or tables? Draw a scale drawing of your new classroom layout.
Practice: Enlargements and Reductions. MathLinks 9 (2009). pp. 136 – 138.
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Mathematics 21
Appendix B.3 Scale Factor, Scale Drawings, and Scale Models
Resource: MathLinks 9 (2009). pp. 141
Materials: Canadian quarter, caliper
Investigate:
 What measurements would help you compare the illustration of the quarter to an
actual quarter? Take the measurements. (Diameter of actual Canadian quarter:
23.88 mm).
 What is the scale factor?
Discuss: Use the following questions to generate discussion:
 How do you determine the actual length of an object using a scale drawing?
 How do you use scale factor to determine the actual length?
Practice: Scale Diagrams. MathLinks 9 (2009). pp. 142 – 145.
Activity: Shadow, Shadow. MathLinks 9 (2009). p. 164
Problem: Your family is moving to a new house with a living room that is 17 ft. by 15 ft.
Cut out and label simple geometric shapes, drawn to scale, to represent every piece of
furniture in your present living room. Place all of your cut-outs on a scale drawing of the
new living room to find out if the furniture will fit appropriately (e.g., allowing adequate
space to move around).
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Mathematics 21
Appendix B.4 Cars, Critters, and Barbie
Resource: Authentic Activities for Connecting Mathematics to the Real World: Project 4
Cars, Critters, and Barbie http://www.wfu.edu/~mccoy/mprojects.pdf
Watch: The Future Channel: Designing Toy Cars
http://thefutureschannel.com/videogallery/designing-toy-cars/
Investigate Part 1: Known Scale Toys
Measure four known-scale toys and record scale and toy measurements. Calculate the
actual measurements using a proportion.
toy scale
toy measure
=
actual scale actual measure
Investigate Part 2: UnKnown Scale Toys
Measure four unknown-scale toys and record toy measurements. Find actual measures
in reference book. Calculate the scale using a proportion.
1
toy measure
=
actual scale actual measure
Investigate Part 3: Barbie
Measure the Barbie doll and determine her real life measures
(select a height).
Barbie height Barbie measure
=
real-life height real-life measure
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Mathematics 21
Doll Measurements
Real-life Measurements
Height
Width at Shoulders
Inseam
Length of Head
Length of Arm
Width of Thigh
Width of Stomach
Length of Foot
Other
Other
Other
Draw your life-sized Barbie on large paper and colour to complete her picture.
Discuss: Generate a discussion about the life-sized Barbie and her proportions.
Discuss her influence on young girls.
Extension: Following the discussion about Barbie have students write about their
mathematical and social findings.
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Mathematics 21
Appendix B.5 Teacher Resource for Cars, Critters, and Barbie
Resource: Authentic Activities for Connecting Mathematics to the Real World: Project 4
Cars, Critters, and Barbie http://www.wfu.edu/~mccoy/mprojects.pdf
Materials:
Known-Scale Toys that work well for this activity are model cars of various sizes. The
scale is usually given on the box (for example, 1:24). A variety of different sizes provide
an interesting context. Students may also bring their own models from home, provided
that they still have the box or information giving the scale. Students measure the cars
and use the given scale to estimate the size of the real car. Once this part of the project
is completed, students may go to websites of car manufacturers to find actual sizes and
check their work.
Unknown-Scale Toys may be any type of animal models. Small zoo or farm animals are
inexpensive and appropriate for this activity. These typically do not include the scale. So
this time students work in reverse. They look up the animal in a reference book and
obtain its average length, width, or height. Then they measure the toy, and calculate the
scale.
Barbie: The model Barbie is sketched on large butcher-paper, and students may use
colored markers to complete her picture. This is a valuable lesson for many reasons,
including the fact that the life-size Barbie is somewhat grotesque. The proportions of the
doll do not translate well to real life size, and this is apparent to students.
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Mathematics 21
Appendix B.6 Gingerbread House
Resource: Stories from the Classroom. Gingerbread math: A sticky investigation
http://www.tc2.ca/pdf/T3_pdfs/GingerbreadMath.pdf
Materials: Gingerbread house, rulers
Key Questions:
 Could we live in our gingerbread houses and ride in the sleigh that came in our
kit?
 How big would the reindeer be if we could build life size replicas in their image?
 What is their scale and how would we have to change their measurements in
order to make our Christmas wish a reality?
Investigate: Give students a gingerbread house kit to assemble. Have them
determine the measurements of height, length and width and any other relevant
dimensions that will assist them in answering the key questions.
Have them redraw their floor plans so that they can use them as blueprints for an actual
house. They should complete a net (a 2D pattern of a 3D figure) of their original house
and a net of their modified house including the four walls, roof, door, windows, and one
other feature of their choice such as a chimney, sleigh, reindeer or Santa.
Additional Questions:
Other examples of questions that arose from students: Stories from the
 How big Santa be based on the gummy figure in the kit?
 How big would the doorknob be based on the size of the house?
Page 87
Mathematics 21
Appendix B.7 Glowing Rectangles
Resources:
Yummy Math: Glowing Rectangles. http://www.yummymath.com/wpcontent/uploads/glowing-rectangles.pdf
Choosing a Television to Suit Your Room. MathLinks 9 (2009). p. 287
Materials: measuring tape, grid paper
Introduction: We spend a lot of time sitting in front of televisions, computers, and other
electronic devices with screens. Have you ever noticed that not all screens are the
same rectangular shape? That is to say, they are not proportional or similar. The
height to width ratio is not the same when you start comparing analog tube televisions,
HD televisions, and movie theatre screens. If you have been to an electronics store
recently, you may have noticed many HD televisions showcased along a wall. HD
screens are all similar to teach other, because the height to width ratio is always the
same. The same can be said for analog tube televisions and for most movie theatre
screens and movie formats.
Instructions: The typical HD screen has a width to height ratio of 16 to 9.
1. Find the missing dimension for each of the HD screens. Assume that all dimensions
are in inches.
24
12
6
48
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2. Give a length/width dimension that is not similar to this screen type.
3. An HDTV screen is 40 inches long. What is the height of the screen?
4. Sketch your own HD screen below. Make sure to label the side lengths. The screen
should be similar to the other HD screens.
5. Here is one type of dimension for a movie theatre screen: Movies and movie theatre
screens are often made in the format 2.35 to 1 (widescreen format, width to height).
Find the missing dimensions. Assume that the dimensions are in feet.
37.6
12
6. Is the screen below similar to a movie theatre screen? Show mathematical evidence
to support your answer.
3
7
7. In the past, there were many drive-in theatres. The giant screen was about 150 feet
wide. How tall was the screen?
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8. Regular, old, analog tube television is a 4 to 3 ratio (width to height). Find the
missing dimensions. Assume all dimensions are in inches.
32
15
9. Give different lengths and widths for screens that are similar to the tube screen. You
might use a ratio table to organize your thinking:
Length
Width
10. At home, measure your own glowing rectangles (e.g. computers, phones, portable
video game systems, digital cameras, I Pod, I Pad, I Phone or anything else with a
screen). Find the dimensions of the screens in millimeters. Use the table below to
record your data:
Product
Length
Width
Simplified Ratio of
L:W
Are any of your screens similar to the widescreen, HD, or analog tube screen? Explain.
11. Sort the following glowing rectangles. Label each as either HD, tube, wide screen,
or none of the above.
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A
B
C
D
E
F
G
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H
Mathematics 21
Project: Choose a television that best suits your needs and considers your room size
and the location for the television. Does a standard or high-definition television (HDTV)
make the most sense for your room? How large of a screen should you get?
The following table gives you the best viewing distance for the screen size for two types
of TVs.
Screen Size (cm)
68.8
81.3
94.0
Viewing Distance (cm)
Standard TV
HDTV
205.7
172.7
243.8
203.2
281.9
233.7
1. Given this information, what size of television would be best? Make a sketch of your
room, including where you plan to place the TV and the best place for a person to view
it from.
2. If the television is 320 cm away from your chair/couch, how large of a standard TV
would be best?
3. How will your answer change if you have a HDTV?
What type and size of TV would be best for your room? Justify your answer.
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Appendix B.8 Geometric Shapes
Materials: Volumetric shapes or Geomodel folding shapes
Brainstorm: Pose the question “What is a geometric shape?” and brainstorm with
students the things they know or recall.
Discuss: You may want to use the following probing questions to generate discussion.
 What is the name of each geometric shape shown?
Activity: Have students group the geometric shapes into categories. Discuss the
categories the students chose and have them justify why they chose those categories.
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Appendix B.9 What is Surface Area?
Brainstorm: Pose the question “What do you know about surface area?” and
brainstorm with students the things they know or recall.
Activity: Provide students with a variety of different sized boxes and wrapping paper.
Ask them how wrapping a gift relates to surface area.
Discuss: You may want to use the following probing questions to generate discussion.
 What is the difference between area and surface area?
 What do you need to know to measure surface area?
 How can you calculate surface area?
 How many surfaces does an object (e.g. cereal box) have?
 How many surfaces are there in a cylinder?
Investigate: Provide students with net diagrams and ask them to find the surface area
of the 3D shape. Have students cut them out and build them or leave them as 2D.
Suggestions: For those students who need more guidance, ask them to recall from
Math 11 how they found the area of an object. Ask them to consider how many
surfaces them now have on the 3D object and how they could find the area of each
object separately and then the 3D object as a whole. They may discover the “shortcut”
by multiplying congruent surfaces.
Additional Questions:
 What is the surface area?
 Do you still have the same number of shapes?
 What are the units?
 Is there a more efficient way to find the surface area?
Practice: MathLinks 9 (2009). pp. 29 – 30.
MathWorks 10 (2010). pp. 121-122.
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Appendix B.10 How Many Sheets of Drywall Are Needed?
Resource: Human Resources and Skills Development Canada. Trades Math
Workbook.
http://www.hrsdc.gc.ca/eng/jobs/les/tools/support/trades_math_workbook.shtml#form
Journeypersons working on a construction site follow specifications from a set of
drawings or prints that show different views of the finished building project.
Journeypersons in all trades scan the drawings for the detailed information they need.
Investigate: Look at the drawings for a residence to estimate the number of drywall
sheets needed for the walls of the ensuite bathroom.
Drywall sheets: 4 ft. × 8 ft.
Width of pocket door: 3 ft.
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Appendix B.11 Heat and Frost Insulators
Resource: Human Resources and Skills Development Canada. Trades Math
Workbook.
http://www.hrsdc.gc.ca/eng/jobs/les/tools/support/trades_math_workbook.shtml#form
Materials: paper towel rolls, toilet paper rolls or Pringles can, paper to imitate
insulation.
Investigate: Heat and frost insulators cover pipes to keep substances hot or cold. How
many square meters of material are needed to insulate a pipe that is 6 m long and has a
diameter of 2 m?
Suggestions: Have students think of the cylinder as being laid out flat so that the
circumference becomes the width measurement.
Use the formula: A   dh
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Appendix B.12 What is Volume?
Materials: 3D objects
Terminology: faces, edges, vertices
Brainstorm: Pose the question “What do you know about volume?” and brainstorm with
students the things they know or recall.
Discuss: You may want to use the following probing questions to generate discussion.
 What is the difference between surface area and volume?
 What do you need to know to measure volume?
 How can you calculate volume?
 If two boxes have the same volume, must they also have the same surface area?
 What is the difference between volume and capacity? (e.g. The capacity of a fuel
tank on a vehicle refers to the volume the tank will hold inside. The volume of the
fuel in the tank refers to the space the fuel takes up).
Investigate: Provide students with 3D objects and ask them the following questions:
 What is the base shape of the 3D object?
 How do you find the area?
 Are there are any unit conversions needed?
 What are the units of area?
 What is the height (or length) of the 3D object?
Suggestions: To bridge surface area to volume relate surface area by height to
volume. Explain that volume is surface area with a height. Use stacks of paper, poker
chips or pennies to show that a nearly flat object can have volume when stacked.
Volume = Surface area of the base × height
Watch: Surface Area and Volume Video
http://www.learnalberta.ca/content/mejhm/index.html?l=0&ID1=AB.MATH.JR.SHAP&ID
2=AB.MATH.JR.SHAP.SURF&lesson=html/video_interactives/areavolume/areaVolume
Small.html
Interactive: Surface Area and Volume Interactive
http://www.learnalberta.ca/content/mejhm/index.html?l=0&ID1=AB.MATH.JR.SHAP&ID
2=AB.MATH.JR.SHAP.SURF&lesson=html/video_interactives/areavolume/areaVolumeI
nteractive.html
Activities:
 Create a cube and calculate the surface area and the volume of the entire
geometric solid.
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
List all the possible ways that 24 one-inch squares of candy might be packed in a
box. Be sure to include the surface area and volume of the box needed for
packing.
 Draw a diagram that explains how to calculate surface area and volume.
 Use a Venn diagram to compare surface area and volume.
 Write 2 word problems that involve surface area and two that involve volume. Be
creative!
 Use manipulatives to explain surface area and volume to your teacher.
 Create a song or rap to explain how to find surface area or volume.
 Write a fairy tale about Queen Area and King Volume.
 Create a game and game board that involves surface area and/or volume.
 Write a poem that illustrates the differences in surface area and volume. It must
have at least eight lines.
(Math Contract – Area and Volume.
http://view.officeapps.live.com/op/view.aspx?src=http%3A%2F%2Fdaretodifferentiate.wi
kispaces.com%2Ffile%2Fview%2FHandout%2B08.doc)
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Appendix B.13 How is Volume Used?
Resource: Human Resources and Skills Development Canada. Trades Math
Workbook.
http://www.hrsdc.gc.ca/eng/jobs/les/tools/support/trades_math_workbook.shtml#cn-tphp
Problems:
1. A construction craft worker needs to know how much material is in the coneshaped pile shown below. Calculate the approximate volume of the pile in cubic
yards. Use this formula to calculate the radius of a pile of material:
r = ¾ × height
27 ft.3 = 1 yd.3
2. A landscape horticulturalist needs to order enough sand to create a border 152
mm deep around a square surface, as shown below. How many cubic meters of
sand are needed?
3. Compare the volumes of concrete needed to build three steps that are 4 ft. wide
and that have the cross-sections shown below. Explain your assumptions and
reasoning.
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4. Dave has a very small yard but needs a rain barrel against his house. What
shape of a rain barrel would maximize volume and minimize the area of the
base? Discuss the different three dimensional shapes and the design of the rain
barrels. For example a cylinder is the area of a circle through a height, a
rectangular prism is the area of a rectangle through a height, triangular prism is
the area of a triangle through a height, a half-cylinder (flat side can go along the
wall) is the area of half a circle through a height.
Extension: Research how to choose a furnace or air conditioner based on the volume
of the house (ABE Level Three: Mathematics Curriculum Guide (2006). p. 106).
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Appendix B.14 Landscape Design
Resource: Landscape Design. MathLinks 9 (2009). p. 253, 281
Introduction: Gardeners and landscapers are often required to calculate areas when
designing a landscape for a backyard, commercial property, or park. When determining
how much soil, gravel, mulch, and seed they need for a project, landscape designers
also calculate volumes. Here is a landscape design created for a property.
Practice and Questions: Landscape Design. MathLinks 9 (2009). p. 253
Project: You have been hired to create a landscape design for a park. The park is
rectangular and covers an area of 500 000 m2. The park includes the following
features:
 A play area covered with bark mulch
 A sand area for playing beach volleyball
 A wading pool
The features in your design include the following shapes:
 A circular area
 A rectangular area
 A parallelogram-shaped area with the base three times the height
Include the following in your design:
 A scale drawing showing the layout of each of the required features
 A list showing the area of each feature and the volume of each material (mulch,
sand, and water) required to complete the park.
Extension: Research the cost of the items in your home community and determine a
budget for your new park.
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Appendix B.15 Angles Formed by Transversals
Resource: Grade 6-9 Math Workshop: Symmetry, Lines, and Angles, pp.13 – 14.
Materials:
 Diagram #1 and #2 handouts
 Angle pairs and counter example cards
 Enlarged alternate diagram
 MIRA
 Compass
 Straight edge
Introduction: When a transversal crosses lines that are not parallel, corresponding
angles are formed. When a transversal crosses lines that are parallel, the
corresponding angles are congruent.
Terminology: transversal, parallel, perpendicular, corresponding angles
Instructions: Have students work in small groups or pairs.
1. Provide each group with a copy of the diagram with parallel lines and one
transversal with numbered angles as well as a beige card with one angle set on it
and a non-example on the back of the card.
2. Have students identify the characteristics defining the pairs of angles that are on
their card. Next, the students are to consider the pair of angles that is given as a
counter-example on the back of the card and to see what characteristic(s) this
particular pair of angles does not have. If they cannot see the contradiction, they
must go back and reconsider their defined characteristics so that there is a
contradiction.
3. Once all of the groups feel they understand the type of angle they have been
given examples of, put up the second diagram that has two non-parallel
transversals. Repeat the process above with the pink cards. Have the class
discuss what lines, if any are parallel, and why; and which lines are transversals
and why. Next, ask the students to identify all pairs of angles in this new diagram
that are the same type as what they have been looking at.
4. Put up the third diagram that is labeled using letters. Have the class discuss what
lines, if any are parallel, and why; and which lines are transversals and why.
How many transversals and how many sets of parallel lines are there? Next, ask
the students to identify all pairs of angles in the new diagram that have the same
characteristics as those they had in the first two diagrams.
5. Following the completion of each group creating their list, select a pair of angles
and ask the students which groups had that type of angle. Have those groups
explain how they know (give the characteristics). It may be beneficial to also post
the original diagram for the students to refer to. As a class, write out the
characteristics of the pairs of angles and give a few minutes for the groups who
had worked with other types of angles to identify all pairs of angles on the new
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diagram that have the same characteristics. Finally, give the students the name
for the type of angles (vertically opposite, adjacent, etc.). Allow the students
some time to discuss how they would remember the name and/or recognize the
angles in other situations, such as non- parallel lines. Repeat the process with
the other types of angles.
Extensions:
 Students could be asked to focus on vertically opposite angles. Can they draw
them in different contexts of the home? How are they related to each other? The
students could be asked to measure, cut out, paper fold, and use the MIRA to
explore this relationship.
 Students could be asked to create a set of parallel lines using paper folding, the
MIRA, or compass and straightedge and then to identify the different pairs of
angles. Using the compass, paper folding, and/or the MIRA, the students could
then be asked to look for relationships. These relationships would then be
discussed within small groups, and each small group could be assigned one type
of angle to report about their findings. An example would be to have the students
create paper dolls/paper chains and describe/highlight different lines that they can
see in their resulting creations – parallel lines, perpendicular lines, transversals.
The paper chain/doll designs are made by repetitively folding paper in half, then
cutting along the edge(s) that have the open side(s) on them. Legal paper works
well.
 Students could be asked to construct non-parallel lines with a transversal. Have
them write a journal entry about what they are able to determine about the
different types of angles in this situation. What is the relationship between the
angles if the lines are not parallel? What is the relationship between the lines if
the angles are not congruent?
 Students could be asked to identify examples of each type of angle from within
their environment. Using a map or pictures, have the students highlight with
different coloured markers, lines which are parallel, perpendicular, or neither.
Also ask the students to highlight lines that are transversals. This activity could
include physical objects, pictures in the media, photos they have taken, or
drawings they have created. Students could use markers to highlight pairs of
angles and to write a description about the type of angles, the relationship
between the different lines, etc.
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Appendix B.16 Diagram 1 and Beige Cards: Parallel Lines
1
5
9
13
Page 104
2
6
10
14
Mathematics 21
Angle Pair Set #2
Angle Pair Set #1
<1 and <6
<2 and <5
<9 and <14
<10 and <13
<1 and <2
<1 and <5
<2 and <6
<5 and <6
<9 and <10
<10 and <14
<9 and <13
<13 and <14
Angle Pair Set #4
Angle Pair Set #3
<5 and <10
<6 and <9
Angle Pair Set #5
<1 and <14
<2 and <13
Page 105
<1 and <9
<2 and <10
<5 and <13
<6 and <14
Angle Pair #6
<5 and <9
<6 and <10
Mathematics 21
Non-example #1
Non-example #2
<1 and <14
<1 and <6
Non-example #3
Non-example #4
<5 and <14
<1 and <10
Non-example #5
<2 and >14
Page 106
Non-example #6
<2 and <13
Mathematics 21
Appendix B.17 Diagram 2 and Pink Cards: Non-Parallel Lines
1
5
9
13
Page 107
2
3
6
7
10
14
11
15
4
8
12
16
Mathematics 21
Angle Pair Set #1
<1 and <6
<2 and <5
<3 and <8
<4 and <7
<9 and <14
<10 and <13
<11 and <16
<12 and <15
Angle Pair Set #2
<1 and <2
<1 and <5
<2 and <6
<5 and <6
<3 and <4
<3 and <7
<4 and <8
<7 and <8
<9 and <10
<9 and <13
<10 and <14
<13 and <14
<11 and <12
<11 and <15
<12 and <16
<15 and <16
Angle Pair Set #4
Angle Pair Set #3
<5 and <10
<6 and <9
<7 and <12
<8 and < 11
Page 108
<1 and <9
<2 and <10
<3 and <11
<4 and <12
<5 and <13
<6 and <14
<7 and <15
<8 and <16
Angle Pair Set #5
Angle Pair #6
<1 and <14
<2 and <13
<3 and <16
<4 and <15
< 5 and <9
<6 and <10
<7 and <11
<8 and <12
Mathematics 21
Page 109
Non-example #1
Non-example #2
<1 and <14
<3 and <8
Non-example #3
Non-example #4
< 6 and <11
<1 and <10
Non-example #5
Non-example #6
<2 and <14
<5 and <11
Mathematics 21
Appendix B.18 Diagram 3 ( l1 // l 2 )
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Appendix B.19 Angles in Construction
Application: In construction of homes and buildings, angles are created when two lines
meet. Pick two of the following items from within your environment:
 Porch
 Veranda
 Railings on stairs
 Trusses
 Fence
Take a photo of the item and highlight with different coloured markers, lines which are
parallel, perpendicular, or neither. Also ask the students to highlight lines that are
transversals.
Additional Questions:
 How do you know if the lines are parallel? Justify your answer.
 How do you know if the lines are perpendicular? Justify your answer.
 What objects represent transversals?
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Appendix B.20 Triangle Properties
Materials: Protractor, straight edge
Brainstorm: To review what students know about triangles, pose the question “What do
you know about the triangle below?”
and brainstorm with students the things they know or recall about triangles.
Students’ knowledge could include: number of sides and angles, the names of the sides
of a right triangle, triangle type (isosceles, scalene, right, and equilateral), measures of
angles inside a triangle, etc.
Investigate:
1. Have students draw triangles using a straight edge and measure and label each
angle in the triangle. What conclusion can they make about the sum of the
measures of the angles in a triangle?
2. Have students draw a right triangle and label the right angle at 90°. Have
students label the hypotenuse.
3. Have students use a protractor and straight edge to draw scalene, isosceles and
equilateral triangles. Show students how to indicate which angles and sides are
congruent.
4. Ask students to draw an equiangular triangle. What conclusion can you make
about this triangle in comparison to the triangles previously drawn?
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Appendix B.21 Building Bridges Teacher Resource
Resource: NSW Department of Education and Training Teaching Trigonometry:
Building Bridges Lesson
http://www.curriculumsupport.education.nsw.gov.au/secondary/mathematics/years7_10/
teaching/trig.htm
Introduction: This activity encourages students to discover the importance of triangles
in real-life constructions. Students build bridges, using a limited supply of resources,
and test the bridges ability to hold weight.
Materials:
 6 drinking straws (preferably the straight plastic variety, not the bendy type)
 2 sticky labels
 2 math textbooks
 a ruler per group
 a pair of scissors per group
 one set of small graduated masses (1g – 200g)
Visualize and Discuss: Using photographs of appropriate bridges in your area or the
PowerPoint Building Bridges
(http://www.curriculumsupport.education.nsw.gov.au/secondary/mathematics/years7_10
/teaching/trig.htm), ask students to think of bridges they have crossed. Perhaps they
have walked across a suspension bridge or have been fortunate enough to cross some
of the great bridges of the world. Ask students to view the bridge photographs then
visualize the bridges they know. Discuss the various features of these bridges and the
following questions.
Questions:
 What do the bridges look like?
 What features do the bridges have in common?
 What two dimensional shapes have occurred repeatedly in these bridge
constructions?
Task: In groups, have students complete the Building Bridges Task (Appendix B.22).
Pose the question: As you have limited resources to build the bridge, what strategies
might you use to ensure you do not waste materials? Bridges are load tested and
results recorded. The group with the best load bearing bridge is presented with merit
certificates.
Discussion:
 What have you learnt while building this bridge?
 Where else do you see triangles?
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Extension: Ask students to bring in as many pictures of triangles as they can over the
next few lessons and make a collage on the classroom wall.
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Appendix B.22 Building Bridges Student Task
Materials: 6 drinking straws (preferably the straight plastic variety, not the bendy type),
2 sticky address labels, 2 maths textbooks, a ruler and a pair of scissors
Task:
Place the two textbooks (supports for the bridge) so that the distance between them is
further than the length of one straw. These books represent the banks of a fast-flowing
river, infested with leeches and people-eating crocodiles.
Your group must build a bridge to carry people from one side to the other. You have 30
minutes to construct this bridge, using any of the given materials except the scissors
and the ruler which may not be part of the bridge.
At the end of this time your bridge will be tested for strength and the results recorded.
The winning group will receive merit certificate.
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Appendix B.23 What is the Problem? Teacher Resource
Resource: NSW Department of Education and Training Teaching Trigonometry: What
is the Problem?
http://www.curriculumsupport.education.nsw.gov.au/secondary/mathematics/years7_10/
teaching/trig.htm
Introduction: This activity assesses a starting point for learning trigonometry. It
encourages divergent thinking and attempts to address the reasons why trigonometry
was developed. Students contribute their own solutions to problems before the
introduction of scale drawings or trigonometric ratios.
Watch: A video or show images of abseiling down a rock face. An example is Abseiling
with Melbourne Adventure Hub http://www.youtube.com/watch?v=29u1ER5Z0Z8.
Discuss and Share: Pose the problem to the class (see What is the problem? Student
Task (Appendix B.24). In groups of 3 or 4, students have 10 minutes to discuss the
problem. Each group shares their ideas with the class. All ideas are recorded.
Questions:
 What information is available?
 How could you measure the height?
 What units of measure could you use?
 How accurate would this method be?
 What problems may arise using the method you have described?
 What is another way of measuring the height?
Task: Each group selects three or four ideas, from the board, to test outside in the
school grounds e.g. on the tallest building, a flag pole, tree or appropriate structure.
Allow 15 minutes maximum and students are not allowed to use a measuring instrument
or tool.
Discuss: Whole class discussion on the accuracy and limitations of the different ways
of calculating the height. You may want to use the following probing questions to
generate discussion.
 What is the accuracy and limitations of the different methods?
 How do you know?
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Appendix B.24 What is the Problem? Student Task
You are in a group which is to abseil down a rock face tomorrow. Your task is to
estimate the height of the face. You have no measuring instruments. You need to
determine the height to know how much rope to take. You cannot take excess rope as
you are at the start of a four day exercise and you must not have extra weight with you.
Tomorrow morning you will walk the track which will take you to the top of the rock face.
Brainstorm as many ways as possible to estimate the height of the rock face. Record all
ideas, even if they appear absurd. Each group will share their ideas with the class.
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Appendix B.25 Same Shape Triangles Teacher Resource
Resource: NSW Department of Education and Training Teaching Trigonometry: Same
Shape Triangles
http://www.curriculumsupport.education.nsw.gov.au/secondary/mathematics/years7_10/
teaching/trig.htm
Introduction: In this activity, students use practical measurement skills and ratio
calculations to find a pattern linking the ratio of sides of a triangle with the angles. This
lesson is designed to develop the concepts of sine, cosine and tangent ratios of angles.
Materials:
 Rulers
 three large charts (enlarge to A3) – one for each ratio (opp/hyp, adj/hyp and
opp/adj) - for a class graph
 one set of triangles per group. Triangle sheets A-G should be photocopied onto
coloured cardboard. Carefully cut out the triangles and place each set (8 triangles
in a set) into plastic bags.
 one calculating ratios for similar triangles worksheet per group
(Triangle Sheets website:
http://www.curriculumsupport.education.nsw.gov.au/secondary/mathematics/year
s7_10/teaching/trig.htm)
Discuss and Practice: Show the students a large right-angled triangle with one angle
marked.
Remind the students about the hypotenuse (from Pythagoras’ theorem) and show them
the opposite and adjacent sides in relation to the marked angle. Discuss the meaning
of the words opposite and adjacent in this context. Practice labelling right-angled
triangles from the board. Explain that the lesson involves investigations of the ratios of
pairs of sides of right-angled triangles with angles of different sizes. Discuss the word
ratio and what it means in this context. Compare opp/hyp for a very large and a very
small angle, as shown in diagram, and have students estimate which one will have the
larger ratio.
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Questions:
 What is the hypotenuse of a right-angled triangle? Where do you find it?
 What is ratio?
 What happens to the opp/hyp ratio when the angle is large?
 What happens to the opp/hyp ratio when the angle is small?
Investigate: Divide class into groups of 3 or 4 students. Hand out one Calculating
ratios for similar triangles worksheet (Appendix C.21) and a set of triangles (Appendix
C.21) to each group. Each student takes two triangles. They measure each side to the
nearest millimetre and complete the worksheet for their triangles writing the ratios as a
fraction and using a calculator to estimate them to 3 decimal places. Each group
completes the worksheet including the mean values for each ratio to 2 decimal places.
Members of the group stack their triangles as neatly as possible on top of each other
and discuss their findings.
Connect and Discuss: When every group worksheet is completed, one member of
each group brings it forward with their group’s stack of triangles and briefly reports their
findings. Each group now plots its mean values on the three class graphs. At this stage,
do not join the plotted points as the next lesson will add more values to the graph. Class
discusses graphs.
Note the fact that triangles which have the same ratios also have the same angles. This
is the basis for scale drawings where although the triangles are different sizes, the
angles are in the same proportion or ratio. Explain to students that these ratios have
special names:
 opp/hyp is sine of the angle (sin)
 adj/hyp is cosine of the angle (cos)
 opp/adj is tangent of the angle (tan)
These ratios are used in a branch of mathematics called trigonometry or trig for short.
You may want to use the following probing questions to generate discussion.
 What did your group find when you stacked the triangles on top of one another?
(They should discover that triangles with the same angles have approximately
equal ratios).
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


What information can you observe from each graph? (They should be able to see
that the ratio increases as the angle increases for the opp/hyp graph; the ratio
decreases as the angle increases for adj/hyp and the ratio increases as the angle
increases for opp/adj).
What occupations use trigonometry in their jobs? (All kinds of engineers,
navigator, surveyor, architect, air traffic controller, cartographer, landscape
architect, meteorologist, electronics designer, oceanographer, roofing contractor,
marine engineer, geologist and sheet metal, heating and air-conditioning
engineers).
Where is the word trigonometry derived? (The word trigonometry is derived from
two Greek words meaning ‘triangle’ and ‘measurement’).
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Appendix B.26 Going the Wrong Way
Resource: Mr. Quenneville’s Website: Unit 2 Trigonometry
https://sites.google.com/a/hdsb.ca/mr-quenneville/home/grade-10-applied-math/unit-2trigonometry
Instructions: There are two problems shown below. For each problem, the answer
provided is incorrect. Partner A will identify the errors in the given solutions. Partner B
will write a correct solution to the problem.
Partner A
Solve for the missing side labelled x.
Partner B
Solve for the missing side labelled x.
hypotenuse
61°
61°
opposite
52 mm
52 mm
x
x
adjacent
cos 61° = 52
x
0.485 = 52
1
x
x=
52
0.485
x = 107.2
Solve for the missing side x.
32
x
2
2
Solve for the missing side x.
2
32
x = 20 + 32
m
m
2
x = 1424
20 mm
x = 1424
x
m
m
20 mm
x = 37.74
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Appendix B.27 Solving Trigonometric Problems
Resources: Math Open Reference: Solving Problems Using Trigonometry
http://www.mathopenref.com/trigprobslantangle.html
Mr. Quenneville’s Website: Unit 2 Trigonometry https://sites.google.com/a/hdsb.ca/mrquenneville/home/grade-10-applied-math/unit-2-trigonometry
Skill Building: Provide students with an overview of how to use a calculator in
trigonometry
 Make sure the calculator is in degree mode
 To find the value of a trigonometric ratio given an angle, in degrees. e.g. sin 30°
=
 To find the angle given a trigonometric ratio. e.g. sin θ = 0.7071
 Rounding to four decimal places. Why are trigonometric ratios rounded to four
decimal places? Why isn’t one decimal place used?
Problem: A ramp has been built to make a stage wheelchair accessible. The building
inspector needs to find the angle of the ramp to see if it meets regulations. He has no
instrument for measuring angles. With a tape measure, he sees the stage is 4ft high
and the distance along the ramp is 28ft.
Solution:
1. Draw a diagram: Include all the information given and label the measure we are
asked to find as x. Draw it as close to scale as you can.
2. Find right triangles: We can assume the side of the stage is vertical and makes a
right angle at the floor (point C). So the ramp itself is a right triangle (ABC).
3. Choose a tool: Right Triangle Toolbox
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Reviewing what we are given and what we need:
 We are asked to find x, the angle at which the ramp goes up to the stage.
 We are given the hypotenuse (AB) and the side opposite the angle
Looking at our toolbox, we are looking for a function that involves an angle, it's opposite
side and the hypotenuse. We see that the sin function meets our needs:
O = the side Opposite the angle, H is the Hypotenuse.
where
4. Solve the equation:
4
 Inserting the values given and the unknown(x): sin (x) = 28



Using a calculator, divide 4 by 28: sin (x) = 0.1429
What angle has 0.1429 as it's sine? For this we use the inverse function arcsine.
It tells us what angle has a given sine: x = sin-1 (0.1429)
Using a calculator again, we find that sin-1 (0.1429) is 8.22°: x = 8.22°
5. Is it reasonable?: We see from our calculation that the ramp angle is somewhere
around 9°. Looking at our diagram we see this looks about right. If you get a very
different answer, the most common error is not setting the calculator to work in degrees.
Practice: Solve the application questions. Draw a diagram where necessary. Find
angles to the nearest degree and distances to the nearest tenth of a unit.
1. A ladder is leaning against a building and makes an angle of 62 with level ground.
If the distance from the foot of the ladder to the building is 4 feet, find, to the nearest
foot, how far up the building the ladder will reach.
?
55
4 ft.
2. The Dodgers Communication Company must run a telephone line between two
poles at opposite ends of a lake as shown below. The length and width of the lake is 75
feet and 30 feet respectively.
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What is the distance between the two poles, to the nearest foot?
3. A ship on the ocean surface detects a sunken ship on the ocean floor at an angle of
depression of 50 . The distance between the ship on the surface and the sunken ship
on the ocean floor is 200 metres. If the ocean floor is level in this area, how far above
the ocean floor, to the nearest metre, is the ship on the surface?
4. Draw and label a diagram of the path of an airplane climbing at an angle of 11 with
the ground. Find, to the nearest foot, the ground distance the airplane has traveled
when it has attained an altitude of 400 feet.
5. If an engineer wants to design a highway to connect New York City directly to Buffalo,
at what angle, x, would she need to build the highway? Find the angle to the nearest
degree.
To the nearest mile, how many miles would be saved by travelling directly from New
York City to Buffalo rather than by travelling first to Albany and then to Buffalo?
6. In order to safely land, the angle that a plane approaches the runway should be no
more than 10. A plane is approaching Pearson airport to land. It is at an altitude of 850
m. It is a horizontal distance of 5 km from the start of the runway. Is it safe for the plane
to land?
7. An 8 m long ramp reaches up a vertical height of 1m. What angle does the ramp
make with the ground?
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8. A tree casts a shadow 42 m long when the sun’s rays are at an angle of 38° to the
ground. How tall is the tree?
Application: We use trigonometry to determine inaccessible distances. Have students
measure an inaccessible distance (e.g. flagpole, tree, height of a tall building) applying
trigonometry and using a clinometer (Constructing a Clinometer Appendix C.24).
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Appendix B.28 Constructing a Clinometer
Resource: Mr. Quenneville’s Website: Unit 2 Trigonometry
https://sites.google.com/a/hdsb.ca/mr-quenneville/home/grade-10-applied-math/unit-2trigonometry
Clinometer
http://www.virtualmaths.org/activities/topic_shapes/theod2/resources/clinometer.pdf
Materials: protractor or protractor template, scissors, cardboard, string, paperclip, straw
Introduction: A clinometer is used to find the angle of elevation of an object.
Instructions: Read all directions carefully before you begin:
1. Cut along the dotted line above, and glue the protractor onto a piece of
cardboard. Carefully cut around the edge of the protractor.
2. Take a 20 cm piece of string, and tie a washer or paperclip to one end. The
other end should be taped to the flat edge of the protractor so that the end
touches the vertical line in the center, and the string can swing freely. This can
best be done by taping the string to the back of the protractor and wrapping it
around the bottom.
3. Glue a straw to the flat edge of the clinometer. The finished product should look
like figure 1 below:
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Figure 1
You can now use your clinometer. To find an angle of elevation, look through the straw
to line up the top of an object. The string hanging down will then be touching the angle
of elevation.
Note: The angle you measure will always be less than 90º when you are reading the
clinometer.
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Appendix B.29 Who Uses Trigonometry? Teacher Resource
Resource: Mr. Quenneville’s Website: Unit 2 Trigonometry 2.3.1 Who Uses
Trigonometry Project https://sites.google.com/a/hdsb.ca/mr-quenneville/home/grade-10applied-math/unit-2-trigonometry
Instructions: Choose a career of interest that uses trigonometry.
Suggestions:
Aerospace
Archaeology
Astronomy
Building
Carpentry
Chemistry
Engineering
Geography
Manufacturing
Navigation
Architecture
Optics
Physics
Sports
Surveying
Process: Decide how you will learn more about the use of trigonometry in your chosen
career.
Suggestions:
Internet research
text research
interview
job shadow
job fair
Product: Select the way you will share what you learn.
Suggestions:
Skit
newspaper story
Brochure
electronic
photo essay
verbal presentation
presentation
poster
report
Personal Selection Chart
Your name:
Due date:
Content
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Process
(you may choose more
than one)
Product
Mathematics 21
Teacher’s comments and
suggestions

Your final submission must include the following:
 the career/activity investigated
 a brief description of your process
 description of the career/activity, including how trigonometry plays a role
 list of sources used

Your final submission can include some of the following:
i) for a career
 type of education/training required
 potential average salary
 employability
 example of job posting (newspaper, Internet, etc.)
ii) for a topic or activity
 historical background
 related issues
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Appendix B.30 Who Uses Trigonometry? Student Task
Resource: Mr. Quenneville’s Website: Unit 2 Trigonometry 2.5.3 Who Uses
Trigonometry Research Assignment https://sites.google.com/a/hdsb.ca/mrquenneville/home/grade-10-applied-math/unit-2-trigonometry
You are to investigate someone who uses trigonometry in their professional lives. You
will be responsible for submitting:
 a report
 a presentation
The Report: The report should describe what the profession is all about. Let us know
what they do and what type of education is needed to enter that profession. The report
should also include a description of how trigonometry is used by the professional in their
work. What types of problems do they need trigonometry for? Include one example of a
problem that could be solved using trigonometry from the field of work you are
researching. A list of resources that you used must be included. These may be articles,
books, websites, magazines, etc…
The Presentation: The presentation should provide a quick snapshot of your research.
Include visuals (pictures, graphics, etc…) related to the profession. The presentation
can be a poster, newspaper article created by you, a brochure that you have created, a
skit, an electronic presentation, etc.
Your presentation should highlight:
 your chosen profession
 education needed (e.g. college / university / workplace) and courses in high
school
 what kind of problems the professional will need to use trigonometry to solve
Where do you get information?
The internet is a great place to start. You can do a search using the title at the top of the
page. This will give you an idea of different professions and then you can investigate the
specific one you pick. If you know someone who actually is in one of those professions,
ask them!! The library is a great place to start and to get help on research.
Types of presentations
If you decide to present a skit it should be 5 minutes and could involve 3 people
maximum. If you select to write a newspaper story it should 350-400 words, one
graphic, proper newspaper format, and includes one interview quote. A presentation
done as a brochure should be 4 or 6 sided and has 2 graphics. If you want to do an epresentation, it should include 12-14 slides and make use of different transitions. A
verbal presentation would be 2-3 minutes and have interaction with the audience. A
visual poster would be bristle board size.
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Appendix B.31 Staircases, Steepness, and Slope
Resource: Finding Ways to Nguyen Students Over: Staircases and Steepness.
http://fawnnguyen.com/2012/05/03/20120503.aspx?ref=rss
Materials: ruler, protractor
Introduction: Pose the question “What do you see?” (Possible responses: going down,
going up, all using their legs, exercise, at an angle).
Activity: Provide students with the Staircases Handout (Appendix C.28). (Answer: D,
A, B, E, C, F or D, A, E, B, C, F because B and E have the same steepness).
Questions:
 What was your ranking?
 What tools did you use to measure? Who measured with a protractor?
 What if we didn’t have a protractor? What if we only had a ruler? What would you
measure instead?
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Mathematics 21

Which of these lengths would you measure? Do you need to measure more than
one? And if you measured more than one, what would you do with the two/three
numbers you have?
Watch: Tutorial – Measure Slope Steepness by Bruce Temper, Director of the Utah
Avalanche Center. http://www.youtube.com/watch?v=hIlFqnvgVlY
Discuss: Use the following questions to generate discussion:
 What is slope?
 How can you measure slope?
 What would you need to change in the staircase to change the slope?
 What would be a reasonable incline to push a wheelchair up if a door step is n
meters from the ground?
 How long would the ramp be?
 How could you determine the length?
Watch: What is the slope of a staircase?
http://www.youtube.com/watch?v=R5wKjst_sMM
Watch: How to calculate, layout and build stairs
Part 1 of 3 http://www.youtube.com/watch?v=531UPCjZTm0
Part 2 of 3 http://www.youtube.com/watch?v=8QrWlMC4qCY
Part 3 of 3 http://www.youtube.com/watch?v=b9etXSmeW1w
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Appendix B.32 Staircases Handout
1. Without measuring
the staircases, put
them in order of
"steepness," starting
with the shape with
the least “steepness.”
2. Explain how you came up with your ranking of “steepness” in #1. Because you were
asked NOT to measure, what “tools” or strategies did you use to make your
decision?
3. Now discuss your ranking in #1 with another classmate. Are you going to change
your ranking? If so, please indicate your new ranking.
 No, I’m sticking with my original ranking.
 I’m changing my ranking to this…
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4. Now discuss your ranking in #3 with a different classmate. Are you going to change
your ranking? If so, please indicate your new ranking.
 No, I’m sticking with my original ranking.
 I’m changing my ranking to this…
5. You may now measure the staircases with whatever tool(s) you need. Use the
space below to keep track of your measurements, calculations, and notes.
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Appendix B.33 Pitch of a Roof
Consider how the pitch of a roof relates to slope:
Questions:
 How do you convert roof pitch from a ratio to degrees? (Tangent ratio)
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Mathematics 21
Appendix C: Recreation and Wellness
Appendix C.1 Inductive and Deductive Reasoning
Instruction:
Inductive Reasoning:
 Inductive reasoning is a conclusion based on several past observations.
Conclusion is probably true, but not necessarily true.
 Inductive reasoning is used when we collect evidence, observe patterns and draw
conclusions from these observed patterns. This evidence does not prove
conclusions, but suggests the conclusion.
Examples:
 After eating mushrooms for the first time, you experienced stomach cramps. You
also developed stomach problems the next three times you ate mushrooms. You
reason inductively that you are allergic to morels.
 On your way to school on Monday, Casey’s dog races out and barks at you. The
dog repeats his performance on Tuesday and again on Wednesday. You reason
inductively that the Casey’s dog doesn’t like you.
Deductive Reasoning:
 Deductive reasoning is a conclusion based on accepted statements such as
definitions, postulates, theorems, given information, and known properties of
mathematics.
 Deductive reasoning uses logic that is based on accepted facts to draw
conclusions.
Examples:
 If a student is on the E.D. Feehan basketball team, then he/she must have at
least a C average. Jody is on the basketball team. If you accept these two
statements as true, you must also be willing to accept the logical conclusion that
Jody must have at least a C average.
Practice: Inductive and Deductive Reasoning. Foundations of Mathematics 11 (2011).
pp. 2 - 64
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Appendix C.2 Puzzles and Games
Problem: On a shelf, there are 10 books with 100 pages each. If a bookworm starts at
the first page of the first book and eats through the last page of the last book, how many
pages does the bookworm eat through (excluding covers)? (Answer: 802)
Use a strategy such at Think, Pair, Share as students work on the solution. During
Think, individual students can work through the answer to the solution. During Pair,
they can share their solution with a partner as well as discuss the strategies they used
to solve the puzzle. During Share, the pairing can share the strategies they used to
solve the puzzle. The students or the teacher can record them on the board.
Task: In groups of 3 or 4, provide students with a variety of puzzles that require the
different strategies. Have each group work on one puzzle that is different from each of
the other groups and ask that they find their solution using two different strategies.
When each group member is confident with finding the solution and with the two
strategies used, have the students form a jigsaw. One member from each group joins a
new group. In the newly formed groups, each member will present their puzzle, provide
time for the other members of the group to solve the puzzle, assisting when necessary
and then providing the solution along with strategies as well.
Activity: Provide students with puzzles that have incorrect solutions, so they can
analyze them for errors.
Games: Create stations with different games such as Cribbage, Magic Square,
Yahtzee, Sudokus, Kakuro, Kaponk, Guesstamations, and Qwirkle. Provide students
with YouTube videos if they are uncertain of how to play. Ask them to play and
describe strategies of how they win each game. Award prizes for the student(s) that win
the most games.
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Appendix C.3 Measures of Central Tendency
Materials: Linking cubes
Introduction: Each year, Canada’s Prairie Provinces produce tens of millions of tonnes
of grains, such as wheat, barley, and canola. The growth of a grain crop partly depends
on the quantity of heat it receives. One indicator of the quantity of heat that a crop
receives in a day is the daily average temperature. This is defined as the average of
the high and low temperatures in a day (MathLinks 9, p. 314).
Brainstorm and Discuss:
 What is an average?
 What are applications/examples of where you have used averages before?
 What are measures of central tendency?
Activity:
1. Ask 5 students to come to the front of the room. Using a number of linking cubes
that is a multiple of the number of people at the front of the room, build trains of
linking cubes of various sizes and give one train to each person. For example, if
5 people are selected, 25 linking cubes could be used and divided into five groups
such as: 1, 2, 4, 6, 6, 6.
2. Ask the group to line up in a logical order. Questions could be raised regarding
whether it matters which person with 6 goes first in the line and why.
3. Ask the group, or the audience, or both to identify the person with the train that is
in the center of the trains (data). Compare this to the median of a roadway being
in the center of the road, and that the number of cars on either side is irrelevant –
it is just the center. Ask the group and/or audience to define what median of a
data set means to them based on this experience.
4. Ask the group what the most common length of train is represented in this set.
Relate this to something being “in the mode” or common or preferred (as in
fashions, etc.). Again, have the group and/or audience create a definition for
mode.
5. Ask the group to determine how they could equally share the linking cubes
amongst themselves – what would the length of the train be? Ask the group for
alternative methods of determining this number – do they use the formula
approach – lump them all together and then divide by 7, do they lump them
together and then deal them out, or do they gradually adjust by people with
greater numbers of cubes giving some to ones with smaller numbers of cubes.
6. Have the group members take back their original number of cubes. Try a number
of different changes to the number of cubes and/or who is holding them. Some
examples are: add two more people and give each 4 cubes – does this change
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Mathematics 21
any of the measures of central tendency; add another person to the group and
give them a train of 25 – does this change any of the measures of central
tendency; give one more cube to one of the original people – does this change
any of the measures of central tendency; remove two cubes from any train, or
from two different trains – does this change any of the measures of central
tendency; and so on. What is important here is that they get an understanding of
the impact that different changes in the data set can result in.
(Grade 6-9 Math Workshop: Subtraction of Fractions, Data Management, and
Probability, pp. 5 – 6)
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Appendix C.4 Personal Wellness
Note: Use the information gathered in Activity Personal Wellness in Mathematics 11 or
have students complete the task again.
Activity: For each statement, e.g. number of hours of sleep, compile individual data
over a two week period and calculate the mean, median, and mode.
Research: Using web-based resources, research what a typical number of hours
should be for each of the categories. Take into consideration gender, age, weight, etc.
What measures of central tendency did the research use? Make an analysis of your
overall individual personal wellness by comparing and contrasting your gathered
information with what you researched.
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Appendix C.5 Recreation and Personal Wellness Budget
Task: Pick an activity, hobby, sport, etc. that you are currently involved in or would like
to be involved in. Gather as many details as possible, for example:
 What is the activity, hobby, or sport? (e.g. gym membership, club volleyball, guitar
lessons, horseback riding, hockey camp, etc.)
 What is the time frame? (e.g. days, weeks, months, years)
 Where is the activity, hobby, or sport?
 How do you get there? (e.g. drive, bike, bus, etc.)
 Is it a group or individual activity?
 Who are the participants? (e.g. coach, friends, family, etc.)
 Is there any out of town travel?
 Is there any special equipment required?
 Is there any special clothing or uniforms required?
 Is there a registration fee?
 Are there any other details that are important?
After gathering as many details as possible, list all of the costs (e.g. registration fees,
uniform, out of town travel, tournament fees, etc.) and create a budget using these
costs.
Questions:
 Was the total cost of the activity lower or higher than you expected?
 How will you finance this activity?
 If you are financing this activity, can you afford it or do you need to reduce some
of your expenses?
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Appendix D: Travel and Transportation
Appendix D.1 Direction, Location, and Distance
Pre-Assessment:
1. Pose the problem:
An exchange student is a new arrival in your math class today. When he is called to the
main office, you give him instructions on how to get there. Write your detailed
instructions considering direction, location, and distance and be thorough.
2. The exchange student only understands a minimal amount of English, so you decide
to illustrate the instructions instead.
Extension: Have students exchange written instructions then maps. In pairs, one
partner will be blindfolded and the other will give read exactly the written instructions
(only divert from the written instructions if the partner is unsafe). The teacher can follow
students as they proceed to the main office to see which students had the best written
instructions and best illustration.
Practice: Chippy’s Journey http://nrich.maths.org/2813
Prior Knowledge: Latitude and longitude is the most common grid system used for
navigation. Each degree of latitude is approximately 111 km apart. Each degree of
longitude varies from 0 to 111 km. A degree of longitude is widest at the equator and
gradually decreases at the poles.
Practice: In partners, pick locations on the map. Practice stating the location, giving
directions and determining the distance.
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Appendix D.2 Map Reading
Materials: Road maps, GPS
Introduction: You are going on a trip with a group of fellow students. Determine the
context:
 What is the purpose of the trip (e.g. class trip, sports team, extracurricular club,
band, drama club, etc.)?
 Who is going? Chaperones? How many people in total?
 Where are you going? (e.g. Edmonton for band trip)
 What is a location of interest or tourist attraction that your group will visit? For
example:
Royal Canadian Mounted Police Academy and Depot Division, Regina, SK
Western Development Museums, Moose Jaw, North Battleford, Saskatoon, and
Yorkton, SK
Fort Walsh National Historic Site, Maple Creek, SK
Heritage Historical Park Village, Calgary, AB
Calgary Tower, Calgary, AB
West Edmonton Mall Water Park, Edmonton, AB
Royal Tyrell Museum of Palaeontology, Drumhellar, AB
Fairmont Hot Springs, Fairmont, BC
Vancouver Aquarium, Vancouver, BC
Butchart Gardens, Victoria, BC
Task: Determine the distance from your home community to your destination. Pick the
most direct route and determine the distance in both kilometres and miles. Then pick a
stretch of highway that is “under construction”. Determine an alternate detour route and
recalculate the distance in both kilometres and miles.
Brainstorm and Discuss: Twice on previous trips, your GPS did not pick up a satellite
signal and wouldn’t work. How could you find your way without the use of technology?
Task: Write detailed directions from your home community departure location to the city
limits of the destination location and illustrate. Follow the same format as a GPS in
detailing the directions.
Extension: Use scale factor and proportional reasoning to estimate and then calculate
fuel economy.
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Appendix D.3 Area of Interest
Activity: Compare and contrast two areas of interest at your destination (determined
previously in the Map Reading Activity or pick a new destination). Use the following
table:
Area of Interest
Hours of operation
Entry costs
Transportation options to
get there
Travel reviews
Safety concerns
Time needed at attraction
Description
Historical significance
Why it is an area of interest
How the area of interest
relates to math
Would you go? Justify.
Other
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Task: Organize the information gathered about your first choice in the form of a
presentation, such as a travel brochure, video or guidebook. If you or someone you
know has been there, use those photos. If not, find photos on the internet. Be prepared
to present to the class and justify why you choose the area of internet.
Research and Journal Entry: Have student research online and create a math journal
entry in regards to the following points based on the chosen area of interest:
 Identify and describe situations, experiences, or locations around the area of
interest that are relevant to self, family, or community.
 Compare social justice issues that are present in the location of choice to those
present in your community or another community.
 Identify and explain cultural activities and/or views of mathematics related to the
location of interest.
 Identify and analyze cultural items related to the mathematics at the location of
interest.
 Identify controversial issues or historical events that are or have occurred at the
location of interest.
 Analyze the influences that historically significant events have had on the current
field of mathematics.
Discuss: Have a class discussion on the findings and the following key questions:
 What is an area of interest?
 What makes an area of interest significant?
 How is math related to an area of interest?
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Appendix D.4 Budgeting for a Trip
Introduction: In the Map Reading and Area of Interest activities, plans have already
been set for going on a trip. The context that was created in these activities can be
expanded on or a new scenario can be created.
Task: Create an itinerary of what you will be doing each day. Indicate what you will be
doing, how you will be getting there, where and when you will be eating, etc.
Activity: To create a budget for your trip, consider the following areas of expense:
1. Determine the cost of accommodations (how many people/room; cost per
person/night; cost per person for the entire trip)
2. Estimate a food allowance (cost per day; cost per trip)
3. Estimate an entertainment allowance (include area of interest; cost per day; cost per
trip)
4. Estimate spending money (consider shopping, gifts, souvenirs)
5. Estimate the cost of transportation (how are you getting there (e.g. chartered bus,
vehicle) and how are you traveling around the location when there (e.g. chartered bus,
vehicle, city bus, taxi cab).
Compile the information in a budget format (***money***), indicate which of the above
are fixed and variable expenses, complete the calculations and indicate the total cost of
the trip.
Task: You have fundraised $______ for your trip, however it is not enough. Analyze
and modify your budget to be able to go and not return from your trip in debt.
Discuss: Have a class discussion on the following key questions:
 What fixed and variable expenses do you consider when creating a travel budget?
 Was the total cost of the trip lower or higher than you expected?
 What expenses were higher than you expected?
 What expenses were lower than you expected?
 If you were planning a trip again, what changes would you make to lower your
expenses?
Resource: Travel Math. http://www.travelmath.com/
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