MTH-2102-3
GEOMETRIC
REPRESENTATIONS AND TRANSFORMATIONS
GEOM ET R IC
R E P R E S E N TA T I O N S A N D T R A N S F O R M AT I O N S
MTH-2102-3
Learning Guide
Common Core Basic Education Curriculum (Secondary Cycle One)
Mathematics, Science and Technology
Program of study: Mathematics
Secondary I
MTH-1101-3 Finance and Arithmetic
MTH-1102-3 Statistics and Probability
Secondary II
MTH-2101-3 Algebraic Modelling
MTH-2102-3 Geometric Representations and Transformations
Geometric Representations and Transformations
This learning guide was produced by Société de formation à distance des commissions scolaires du
Québec (SOFAD).
Project Coordinator:
Ronald Côté (SOFAD)
Project Coordinator (initial version):
Jean-Simon Labrecque (SOFAD)
Author:
Jean-Claude Hamel
Content Revisors:
Steeve Lemay
Judith Sévigny
Translator:
Claudia de Fulviis
Desktop Publishing:
Daniel Rémy (I.D.Graphique inc.)
Serge Mercier
Text Revision:
André Dumas
Illustrations:
Marc Tellier
Cover Page:
Marc Teliier
Jitze Couperus (Photograph of nautilus)
Despite the following statement, SOFAD authorizes all adult education centres that use this learning
guide to reproduce the scored activities available at http://cours1.sofad.qc.ca/ressources.
© SOFAD
All rights for translation and adaptation, in whole or in part, reserved for all countries. Any reproduction by mechanical or electronic means, including microreproduction, is forbidden without the written permission of a duly authorized representative of the Société de formation à distance des commissions scolaires du Québec.
Legal Deposit – 2012
Bibliothèque et Archives nationales du Québec
Library and Archives Canada
ISBN: 978-2-89493-434-0
September 2012
TAB L E OF C ONT E NT S
Table of Contents
Table of Contents
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Organization and Use of the Guide . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Additional Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Work Pace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Instructional Support . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Evaluation for Certification Purposes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Essential Knowledge Covered in Each Part of the Course . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
9
12
12
12
13
13
Part 1 – Representations of the Physical Environment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
Situation 1 – TWO DIFFER EN T SYSTE MS OF ME ASURE ME NT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Exploration Activity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Activity 1.1 – Body-Based Units of Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Activity 1.2 – One Container, Two Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Activity 1.3 – One Weight, Two Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Activity 1.4 – One Way or the Other . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Activity 1.5 – Two Measurements, One Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Integration Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Review Activity – Planning a Recipe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17
17
20
22
28
33
39
45
51
53
Situation 12 – TH E IMPORTAN CE OF ME ASURE ME NT IN B UIL DING P ROJE C T S . . . . . . . . . . . . . . . . . . . . . .
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Exploration Activity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Activity 2.1 – The Language of Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Activity 2.2 – Cutting a Piece of Wood . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Activity 2.3 – Building a Fence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Activity 2.4 – A Concrete Slab . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Integration Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Review Activity – Building a Flower Box . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Instructions for completing Scored Activity 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
57
57
60
63
71
77
85
94
96
98
Situation 13 – USIN G PLAN S TO DESCRIB E OB JE C T S AND P LAC E S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Exploration Activity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Activity 3.1 – Green City Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Activity 3.2 – Inside the House . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Activity 3.3 – Adrian's Living Room . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Integration Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Review Activity – Landscaping a Backyard . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
99
99
102
104
111
119
128
130
Situation 14 – CR EATIN G BEAUTIFUL OB JE C T S T HAT ARE US E F UL T OO . . . . . . . . . . . . . . . . . . . . . . . . . . .
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Exploration Activity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Activity 4.1 – A Simple Cardboard Box . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Activity 4.2 – A Nice-Looking Container . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Activity 4.3 – Nice, But Not So Simple! . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Integration Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Review Activity – Describing a Lamp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Instructions for completing Scored Activity 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
135
135
138
141
149
155
164
166
169
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Part 2 – Transformations in the Environment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
Situation 15 – M ODELLIN G MOTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Inroduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Exploration Activity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Activity 5.1 – It Moves! . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Activity 5.2 – And It Turns! . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Activity 5.3 – Negotiating a Turn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Activity 5.4 – The Grenville's Are Moving . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Integration Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Review Activity – Describing Parking Maneuvers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
173
173
176
177
185
192
199
206
208
Situation 16 – A PPR ECIATIN G WOR K S OF ART F ROM DIF F E RE NT C ULT URE S . . . . . . . . . . . . . . . . . . . . . . . 211
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211
Exploration Activity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214
Activity 6.1 – Geometric Patterns in a Carpet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217
Activity 6.2 – Meditation on Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223
Activity 6.3 – Different Types of Frieze Patterns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230
Integration Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240
Review Activity – Analyzing and Constructing a Rosette . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242
Instructions for completing Scored Activity 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244
Situation 17 – D R AWIN G AN D MATH EMATICAL RE AS ONING . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Exploration Activity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Activity 7.1 – A Question of Distance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Activity 7.2 – Matthew's Drawing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Activity 7.3 – Scanned Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Integration Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Review Activity – Designing a Stage Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
245
245
248
249
258
265
274
276
Situation 18 – I N DIR ECT MEASUR EMEN T . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Exploration Activity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Activity 8.1 – Aerial Photographs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Activity 8.2 – Measuring the Width of a River . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Integration Exercices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Review Activity – Finding the Height of a Utility Pole . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Instructions for completing Scored Activity 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
279
279
282
283
291
300
302
304
Self-Evaluation Activity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321
Answer Key . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1 – Two Different Systems of Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
1 – The Importance of Measurement in Building Projects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3 – Using Plans to Describe Objects and Places . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
4 – Creating Beautiful Objects That Are Useful Too . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
5 – Modelling Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
6 – Appreciating Works of Art from Different Cultures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
7 – Drawing and Mathematical Reasoning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
8 – Indirect Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
Self-Evaluation Activity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
323
324
329
339
351
360
371
381
389
398
Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 409
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ANSWER
KEY
INT RODUC T I ON
Introduction
N
o doubt you have often had to measure objects, areas or quantities, whether it was to sew
a garment, repair a fence or simply make a recipe. You may have also had to consult scale
drawings; or perhaps you have made scale drawings yourself to carry out renovation projects
or to describe a place to a friend. When you observe the objects around you, you see various shapes
that sometimes involve complex designs. Perhaps you'd like to create similar objects. Motion and
change are everywhere around you, and you may have tried to describe these to someone else. If so,
were you able to find the right words? Would you be able to analyze decorative elements such as frieze
patterns or tessellations so that you could then reproduce them? All of these real-life situations require
mathematical knowledge and skills related to geometry in particular, but also to arithmetic.
Welcome to the Geometric Representations and Transformations course, which is designed to provide you
with the tools you will need to solve different problems related to such situations. These problems deal with
measurements and representations, with transformations in the physical environment and with our perception of the world around us.
The structure of the course is simple as it consists of two parts that deal with specific themes.
The first part (learning situations 1 to 4), entitled Representations of the Physical Environment,
deals primarily with measurement: converting measurements from the imperial system of units to the
International System of Units, and vice versa, and using plans, nets or the decomposition of solids to
represent places or objects. In this first part, you will calculate quantities in order to solve problems
that involve proportions. You will also apply and build on what you already know about fractions, their
representations and the operations involving them. You will calculate areas and volumes. In this part of
the course, you will apply your prior knowledge and will therefore strengthen and build on what you have
learned in previous courses of the common core basic education program.
The second part (learning situations 5 to 8), entitled Transformations in the Environment, deals with
a number of new concepts related to geometrical transformations. The motion of objects in space, symmetry in works of art, perspective in drawings and enlargements and reductions of figures are among the
topics you will study. In this second part, you will use your geometry set (i.e., set squares, a compass and
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TABLE OF
CONTENTS
ANSWER
KEY
a protractor) to construct geometric figures. You will also apply basic knowledge relating to the properties of geometric figures. Lastly, you will again be required to use proportional reasoning to determine
measurements related to similar figures.
In this course, you will be able to develop two operational competencies: to communicate and to reason
logically.
In order to develop your ability to communicate, you will be required, among other things, to describe
shapes and transformations, to interpret plans, and to clearly explain the procedure you use to solve a
problem. Each time, you will be expected to use the appropriate mathematical symbols, notations and
terms.
In order to develop your ability to reason logically, you will learn about the properties of figures through
a number of new situations. You may be required to back up your answers and problem-solving approach
with logical arguments and to deduce results from certain geometric properties (e.g., calculating the
unknown measurements of angles or segments).
These competencies will come in handy, especially in evaluation activities. You must send your tutor four
assignments, namely after completing situations 2, 4, 6 and 8. These assignments take the form of scored
activities and are presented in separate booklets.
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Organization and Use of the Guide
This learning guide is organized according to the main characteristics of individualized learning and the
principles of learning through concrete situations.
It is therefore designed to:
• make you as active a participant as possible,
• make you responsible for your own learning,
• accommodate your personal work pace, and
• allow you to make the most of your experience and knowledge.
As you make your way through the course, you will be able to recognize your successes or failures, determine the reasons for them and identify what you can do to continue learning.
You can consult your tutor at any time during the course. If you find a particular topic especially difficult,
don't hesitate to ask him or her for help. Your tutor will be glad to provide you with the advice, guidance,
constructive criticism and feedback you need.
Learning situations
This guide contains eight learning situations designed to help you learn new concepts and apply them
competently. All the learning situations are organized in the same way. Each one begins with an introduction that describes the assignment you will be required carry out at the end of the learning situation. This
is followed by the exploration activity, which will allow you to review certain previously learned concepts
and help you carry out the given assignment.
Each learning situation is divided into learning activities. In each activity, you will be presented with a
problem and questions. As the questions deal with new concepts, you may not be able to provide satisfactory answers to all of them; however, you are encouraged to do your best. The correct answers and
additional explanations are given after each set of questions. It is important that you try to understand
all of the new concepts that are explained to you. Following the explanations is a summary of the new
concepts as well as exercises. The exercises will allow you to determine your understanding of the newly
learned concepts. The answers to these exercises are found at the end of the guide.
Once you have completed the integration exercises, you can do the review activity, which is a continuation of the assignment described in the introduction. Afterwards, you will be required to assess your
ability to communicate and to reason logically, and you will be given pertinent suggestions for developing
and using these competencies. Each learning situation ends with a "List of New Knowledge" section. You
can refer to this list as needed to make sure you have understood all of the key concepts.
Throughout this guide, different icons will guide you in your work.
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Key p
oints
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The paper clip icon indicates key points to keep in mind.
The foldback clip icon indicates the last few pages of each
List of New Knowledge
learning situation. These pages summarize the essential
knowledge covered in the learning situation.
Tip
A light bulb icon appears in boxes containing tips to make
your work easier.
Reminder
Boxes with a bulletin board tack icon contain reminders of
concepts covered in previous courses.
Words with dotted underlining are defined in the glossary
Underlining
Did you
know
at the end of this guide.
?
The magnifying glass indicates additional information. This
information is not, strictly speaking, part of the course material, and none of the questions on the final examination
will deal with the information found in these sections.
The last section of the guide, the conclusion, summarizes what you have covered in the course. It also
contains a self-evaluation activity to help you determine whether you have a firm grasp of what you have
learned and whether you are ready to sit for the final examination. The conclusion includes the answer
key for the exercises in the guide and the self-evaluation activity, and the glossary.
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Scored activities
This learning guide is accompanied by four distinct scored activities. You are required to do these activities
after learning situations 2, 4, 6 and 8 and to send them to your tutor for correction. In addition to evaluating
your knowledge, scored activities 2 and 4 each include a complex problem designed to assess your ability
to deal with such problems.
Evaluation Situations
Scored Activity 1
Topics Covered
Measurement
(Learning situations 1 and 2)
Scored Activity 2
Representation
(Learning situations 3 and 4)
Scored Activity 3
Isometries
(Learning situations 5 and 6)
Scored Activity 4
Similarities
(Learning situations 7 and 8)
Self-evaluation activity
Completing the self-evaluation activity is a step in preparing for the final examination. Before you tackle
it, you can complete your study of the material by reviewing the table of essential knowledge found in this
introduction. Then, do the self-evaluation activity without referring to the learning guide or the answer
key. Compare your answers with those in the answer key and go back and review certain sections of the
guide if necessary. Like the final examination, the self-evaluation activity consists of two parts so that you
can better prepare for the exam.
The self-evaluation grid that accompanies the self-evaluation activity will help you identify the concepts
you have mastered and those you should review before sitting for the certification exam. Instructions
regarding the concepts to be reviewed are also given in the grid.
Answer key
The answer key for the exercises in the guide is found after the self-evaluation activity. Refer to it after
each set of exercises to make sure you have fully understood all of the concepts, before continuing the
activity or going on to the next learning situation. This section also includes the answer key for the selfevaluation activity.
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Glossary
The glossary found at the end of this guide gives the definitions of the words with dotted underlining.
These terms are listed in alphabetical order. Don't hesitate to refer to the glossary to help you better
understand some of the terms you come across.
Additional Materials
Have all the materials you need handy.
• Your learning guide and a notebook in which to summarize all of the key concepts found in the list of
essential knowledge given in this introduction.
• A dictionary, a calculator, a pencil to write your answers and notes in your guide, a ballpoint pen to
correct your answers, a highlighter (or a light-coloured felt pen) to underline key concepts, an eraser,
etc. For certain exercises, you will need a geometry set (a ruler graduated in inches and in centimetres, a protractor, a set square and a compass).
Work Pace
Here are some tips on how to organize your study time. This course involves approximately 75 hours of
work.
• Draw up a study schedule, taking into account your availability and needs, as well as your family, work
and other obligations.
• Try to devote a few hours a week to your studies, preferably setting aside two hours at a time.
• Stick to your schedule as much as possible.
Instructional Support
Your tutor will help you throughout this course: he or she will be available to answer any questions you
may have, and will correct your scored activities.
This is the resource person you must call if you need any kind of help. If his or her availability and contact
information have not been provided with this guide, you will receive them shortly. Do not hesitate to
consult your tutor if you are having any difficulties with the explanations or the exercises, or if you need
encouragement to continue with your work. Make a note of any questions in writing and contact your
instructor at the appropriate time and, if necessary, write to him or her.
Your tutor will guide you throughout the learning process and provide you with advice, constructive criticism and feedback that will help you succeed in your studies.
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Evaluation for Certification Purposes
In order to earn the three credits for this course, you must obtain a mark of at least 60% on the final examination that will be held in an adult education centre. To be able to write this examination, you should have
an average of at least 60% on the scored activities that test your understanding of the material in this guide.
The final examination for the Geometric Representations and Transformations course consists of two
sections. These sections are included in the same booklet and are to be completed during the same exam
session.
The first section of the examination is intended to evaluate your knowledge of the material, and consists of
short-answer questions and questions requiring more elaborate answers.
The second section is designed to evaluate competencies. This section consists of problems presented in
one or more realistic situations.
Authorized materials for both sections of the examination are:
• a regular or scientific calculator;
• a geometry set;
• a ruler in imperial units;
• a checklist inserted in the examination.
The examination lasts 2 hours 30 minutes.
Essential Knowledge Covered in Each Part of the Course
Learning Situations
Essential Knowledge
1.Two Different Systems of
Measurement
Units of length, capacity, mass and temperature in the
imperial system
Converting measurements of length, capacity, mass and
temperature from one system of units to the other
2.
The
Importance
of
Locating rational numbers on the number line
Measurement in Building
Review of operations on fractions
Projects
Units of area and volume in the imperial system
Converting measurements of area and volume from one
system of units to the other
3.Using Plans to Describe
Objects and Places
Ways of representing a plan's scale
Reading and constructing scale plans
Calculating an actual measurement from a length
represented in a plan
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Learning Situations
4.Creating Beautiful Objects
That Are Useful Too
ANSWER
KEY
Essential Knowledge
Possible nets for a solid (cube, right prisms and right
cylinders)
Review of the calculation of the area of solids
Decomposing a complex solid into simple solids
Estimating the volume of a solid
5. Modelling Motion
Geometric transformations (translations and rotations)
Properties of congruent figures
Constructing the image of a figure under a translation or a
rotation
6.Appreciating Works of Art
from Different Cultures
Geometric transformations (reflections, isometries)
Constructing the image of a figure by a reflection
Symmetry and invariance
7.Drawing and Mathematical
Reasoning
Properties of similar figures
Calculating the similarity ratio of two similar figures
Geometric transformations (dilatations)
Constructing the image of a figure under a dilatation
Calculating the scale factor of a figure and its image
Finding the length of a segment from the similarity ratio or
the scale factor
8. Indirect Measurement
Similar figures that do not result from a dilatation (similarity
transformation)
Finding the measure of an angle or the length of a segment
in a figure by using the measurements of a similar figure or
a congruent figure
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Part 1
Representations
of the Physical Environment
The four learning situations in this first part of the guide deal with real-life contexts that involve
measurement and representation.
1
Two Different Systems of Measurement
2
The Importance of Measurement in Building
Projects
3
Using Plans to Describe Objects and Places
4
Creating Beautiful Objects That Are Useful Too
After you have completed these four learning situations, you will be able to describe your environment
by means of different types of representations, use one of the two systems studied to take
measurements, convert these measurements from one system to the other and calculate quantities
(e.g., area and volume).
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1
ANSWER
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Two Different Systems
of Measurement
Introduction
T
he process of measuring things has always been an essential activity of humans,
whether it be to feed themselves, build things, share resources, trade goods, and so
on. The systems of measurement used by different civilizations throughout history
have been many; however, we will focus here on the two main systems currently in use.
In Québec today, two systems of measurement exist side by side, which can sometimes
complicate things. These two systems are the International System of Units (abbreviated
SI from the French: Système international d'unités), which is the official system currently
used in Canada, and the old imperial system of measurement, which still plays an important
role. Being able to convert from one system to the other in order to compare quantities and
prices is a skill every informed consumer should have.
In this first learning situation, you will
discover or review the main imperial
units for measuring length, capacity and mass. Through observation
and logical reasoning, you will establish the conversion rates between
imperial units and SI units. This will
enable you to solve real-life problems
in which measures of length, capacity
and mass must be converted from
one system to the other.
© Maksim Shmeljov/Shutterstuck.com
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If you were asked to state your weight in a medical questionnaire, how would you indicate it? In kilograms
or in pounds? If you were asked to state your height, would you give it in metres or in feet? You could ask
these questions of people you know who are of different ages. What units do you think they would use?
Figure 1.1 I’m 5 feet 2 inches
tall and I weigh
110 pounds.
I’m 1 metre
35 centimetres tall
and I weigh 26 kilos.
Although the SI system, which is based on the metre, was adopted in Canada in 1971, many people still
use the imperial system of units (e.g., feet, pounds, gallons).
This is especially true when it comes to food.
In the supermarket, for example, both systems of measurement exist side by side to indicate the weight
and volume of certain products. As well, customers often ask for meat or fish in pounds rather than
grams.
Figure 1.2 One pound of
sole, please.
18
Hmm… is
450 grams OK?
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SIT UAT ION 1 – TWO DIF F E RE NT SY STE MS OF M E ASURE M E NT
Another example: In cooking recipes, ingredient quantities are often indicated in imperial units.
Figure 1.3 Ingredients
•
•
•
•
•
Brownies
¼ lb unsalted butter
5 oz unsweetened chocolate
1 ¼ cups sugar
2 eggs
1 tsp vanilla extract
•
•
•
•
1 tbs instant coffee
½ tsp salt
1 cup all-purpose flour
2 cups miniature
marshmallows
Preparation
Grease a square 9 x 9 inch pan and line it with parchment
paper.
Melt the butter and chocolate together. Stir in the sugar and
eggs and whip the mixture until smooth. Then gradually mix in
the flour and salt. Pour the mixture into the pan.
Bake in a 350 °F oven for about 25 min.
Remove the pan from the oven and garnish with the marshmallows. Place the pan under the broiler for 5 min or until the
marshmallows are a golden colour.
Cut into sixteen 2 ¼ in squares .
© Used with the authorization of Zone 3
Adaptation of a recipe by Josée di Stasio.
Your
assignment
Planning a Recipe
Imagine that you want to make this recipe.
The first problem you may encounter is that the ingredients are generally
sold in packages or containers with the quantities given in millilitres or grams.
Your task is to express the measurements in this recipe in SI units so that you
will buy the right quantities.
You will also have to make sure that your baking pan is the right size for the
recipe. If not, you will have to adjust the recipe accordingly.
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Exploration Activity
The assignment's level of difficulty very much depends on what you already know about the two systems
of measurement.
Let's begin with the imperial system. How much do you know about it? If you are familiar with it, the
assignment will seem easy. If not, don't worry, you will be learning about imperial units in the next three
activities.
1.1
Various imperial units of measure are listed in the margin below.
Place each unit in the correct column of the following table depending on whether it is used to measure
length, capacity or weight. In each column, write the units in increasing order, from the smallest to the
largest. If necessary, consult a dictionary or the Internet.
Table 1.1 – Selected imperial units of measure
Gallon
Measures of length
Measures of capacity
Measures of weight
Pound
Mile
Ounce
Fluid ounce
Foot
Quart
Inch
Cup
Ton
Yard
To be able to convert measurements from one system to the other, you should have a good grasp of the
International System of Units, which you studied in previous courses.
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SIT UAT ION 1 – TWO DIF F E RE NT SY STE MS OF M E ASURE M E NT
Check your knowledge by completing the following equalities.
a)450 cm =
m
c)2 kg =
g
e)0.6 km =
m
b)50 cL =
mL
d)250 mL =
L
f) 500 g =
kg
1 dm
There is a relationship between measures of capacity and measures of volume. For example,
a cubic container with edges measuring 1 dm has a capacity of 1 litre (if we disregard the
1 dm
1 dm
thickness of its sides).
1.3
What is the capacity of a cubic container with edges measuring 1 cm?
To determine the capacity of a pan, you must calculate the volume of the solid that models the inside of
the pan. Here, for instance, is the diagram of a pan in the shape of a prism with a rectangular base.
1.4
Look at the pan shown on the right.
a)What is its capacity in litres?
b)If we pour exactly one litre of liquid into this pan,
4 cm
how high will the liquid come up the sides of the pan?
30 cm
15 cm
Lastly, in this learning situation, you will also be required to convert measures of temperature. The imperial
system uses degrees Fahrenheit (°F) and the SI system uses degrees Celsius (°C).
1.5
Find out what you know about these two units of temperature by answering the following questions.
a)At what temperature does water freeze
1) in °C ?
2) in °F ?
b)Which is colder, 0°C or 0°F?
c)At what temperature does water start to boil
1) in °C ?
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Activity 1.1 – Body-Based Units of Measurement
Goal
• To identify the units of length commonly used in the imperial system
• To compare imperial units of length with SI units of length
• To convert imperial units of length to SI units of length
The oldest units of length were based on the human body.
Figure 1.4 Thumb width
Hand span
Cubit
Foot
Step
Study these measurements on your own body and then estimate
• the number of thumb widths in a hand span • the number of hand spans in a foot • the number of hand spans in a cubit • the number of cubits in a step Using your own body as a reference, how many thumb widths are in a foot?
How many feet are in a step?
The advantage of these ancient units of measure is that there is no need to carry around measuring instruments given that lengths are based on the human body. However, there is one obvious disadvantage in
that they vary from one person to the next, so if you need accuracy, they're not the best choice.
For instance, here are the answers an adult learner gave to the previous questions.
According to him, there are about 8 thumb widths in a hand span and 1 12 hand spans
in a foot. A cubit is equal to 2 hand spans and a step measures about 1 12 cubits. Based
on these measurements, he deduced that there are 12 thumb widths in a foot
(8 × 1 12 = 12) and about 3 feet in a step (2 × 1 12 = 3).
Did you obtain the same values? Your answers are probably different.
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To be able to understand one another, especially in business, it is necessary to establish some common
standards.
Did you
know
?
The imperial system of units was first defined in the British Weights and Measures Act of 1824,
which set out the standard of length, the yard, and established the equivalences between this standard of measure and the other units of length, including the inch and the foot.
In the imperial system, a yard, which is approximately the length of a large step, is equal to 3 feet and 1
foot is equal to 12 inches.
Look at the broken ruler illustrated below, which is divided in both inches and centimetres.
Figure 1.5 ← Centimetres (cm)
← Inches (in)
Note that a foot is equal to a little more than 30 cm, but we could be more precise.
How many centimetres are in one inch? Refer to the above illustration to find the best possible approximation. Give your answer to the nearest hundredth of a centimetre.
Using the previous answer, give a more precise equivalence between feet and centimetres.
(The ≈ symbol means approximately.)
1 foot ≈
centimetre (cm)
Determine the relationship between the yard and the metre.
1 yard ≈
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To estimate the length of an inch in centimetres, we will draw a vertical line through the 1-inch mark.
Figure 1.6 Note that an inch is between 2.5 cm and 2.6 cm long. But what if we wanted a more precise estimation,
say to the nearest hundredth of a centimetre?
You will no doubt agree that the expression of a length in inches is proportional to its expression
in centimetres. If we double (or triple) a length expressed in inches, clearly we must also double (or
triple) the same length expressed in centimetres. This is true regardless of the multiplication or division
factor involved.
Now, if we draw a vertical line through the 10-inch mark on the longer piece of ruler, we see that it passes
though the 25.4-cm mark.
Figure 1.7 Let's estimate the answer by using proportional reasoning.
÷ 10
10 in = 25.4 cm
1 in =
÷ 10
25.4 ÷ 10 = 2.54
? cm
One inch (1 in) is therefore approximately equal to 2.54 cm.
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NOTE: In 1959, a new definition of the inch was adopted internationally in order to make it easier to
convert units from one system to the other. According to this new definition, an inch is exactly equal to
2.54 cm. Therefore: 1 in = 2.54 cm.
This relationship allows us to establish other equivalences.
One foot (ft) is 12 times longer than one inch,
One yard (yd) is 3 times longer than one foot,
and therefore 12 times longer than 2.54 cm.
and therefore 3 times longer than 30.48 cm.
2.54 × 12 = 30.48
30.48 × 3 = 91.44
1 ft = 30.48 cm ≈ 30.5 cm
1 yd = 91.44 cm = 0.914 4 m ≈ 0.9 m
There are other units of length in the imperial system. The most common one is the mile which, like the
kilometre in the SI system, is used to measure large distances.
One mile is exactly equal to 1760 yards.
Did you
know
?
Many people wonder why one mile is equal to 1760 yards. Isn't it odd? Why this number and not
another?
The mile is derived from an ancient measure used in the Roman Empire, which was equal to 1000
paces of a Roman legion (in this case, one pace is considered to be two steps). Furthermore, in
England, there was another unit of measure called the furlong, which was used in agriculture and
which was equal to approximately 220 yards. With the adoption of the imperial system, the length
of one furlong was set at exactly 220 yards and the mile was defined as being equal to 8 furlongs.
Thus, one mile is equal to 8 times 220 yards, or exactly 1760 yards.
Express a distance of one mile in kilometres. Round off your answer to the nearest tenth.
Compare your answer to the equivalence given at the bottom of the next page.
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oints
Key p
Imperial units of length
It is a general characteristic of the imperial system to use several different units to measure quantities.
This is especially true of length.
Table 1.2 – Comparison of the units of length in the two systems
Imperial system
Symbol
•The base unit is the yard.
yd
International system
•The base unit is the metre.
•This unit originates from a set of
•This unit is related to a terrestrial
body-based measurements. In
meridian. Originally, the metre was
practice, one yard corresponds to
defined as the length of one ten-mil-
approximately the length of a large
lionth of the earth's meridian along
step.
a quadrant.
•Commonly used derived units
include:
- the foot
- the inch
- the mile
Symbol
m
•By adding prefixes to the base unit,
dm
ft
we obtain smaller units (decimetre,
cm
in
centimetre, millimetre) or larger
mm
mi
units (decametre, hectometre,
dam
kilometre).
hm
•There are 3 feet in one yard.
•There are 12 inches in one foot.
•These units of measurement are
•One mile is equal to 1760 yards.
all based on multiples of 10. For
•One mile is equal to 5280 feet.
example, one kilometre is equal to
km
1000 metres (i.e., 10 x 10 x 10).
Equivalences between the units of length in the two systems
By convention, 1 inch is equal to exactly 2.54 cm.
In many everyday situations, it may be sufficient to use the following approximations to convert measurements from one system to the other.
1 in ≈ 2.5 cm 1 ft ≈ 30 cm
1 yd ≈ 0.9 m
1 mi ≈ 1.6 km
It should be remembered, however, that all of these approximations underestimate the actual values. For
example, one foot is closer to 30.5 cm than to 30 cm. A 10-foot table therefore measures a little more than
3 m, or 3.05 m to be exact (since 10 ft = 10 × 1 ft ≈ 10 × 30.5 cm ≈ 305 cm ). The level of precision required
often depends on the situation.
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SIT UAT ION 1 – TWO DIF F E RE NT SY STE MS OF M E ASURE M E NT
Exercises for Activity 1.1
1.6
According to the imperial system,
a)how many inches are in one yard?
b)how many feet are in one mile? 1.7
Using a ruler graduated in inches, Caroline measured the length of one of her normal steps and found it
to be roughly 25 inches long. How many steps must she take in order to walk a mile?
1.8
1.9
Express the following lengths in the specified SI units.
a)5 inches in centimetres: b) 12 feet in metres: c)10 yards in metres: d) 5 miles in kilometres: e) 1 12 feet in centimetres: f) 14 inch in millimetres: A tourist brochure for New York City indicates that the full height of the Statue of Liberty
from the ground to the tip of the torch is 305 feet 1 inch. It also states that the statue, from
the base to the torch, stands 151 feet 1 inch. Express these measurements in metres.
1.10
Gregory claims that 3 feet 3 83 inches is equal to 1 metre. Is he right?
Justify your answer.
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Activity 1.2 –One Container, Two Measurements
Goal
• To identify the units of capacity commonly used in the imperial system
• To compare imperial units of capacity with SI units of capacity
• To convert imperial units of capacity to SI units of capacity
You may have noticed that on the label of certain food cans the quantity each one contains is indicated
in two ways: in millilitres (mL) and in fluid ounces (fl oz). The fluid ounce is an imperial unit of capacity.
Take a look at some of the products in your pantry or pay close attention to the labels the next time you
go to the grocery store. You will surely see food cans that look like the ones illustrated below.
Figure 1.8 Did you
know
?
The Consumer Packaging and Labelling Act, which was adopted in 1971 by the Canadian
Parliament, requires that metric units be used on consumer products. However, as shown in the
illustration above, it is customary to indicate the equivalent imperial units.
According to the information given on these cans, how many millilitres are in one fluid ounce? Round off
your answer to the nearest tenth.
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SIT UAT ION 1 – TWO DIF F E RE NT SY STE MS OF M E ASURE M E NT
What quantity in millilitres should be indicated on a 25-fluid ounce container?
The gallon is another imperial unit of capacity that was commonly used when this system was in force
in Canada. In fact, in the imperial system, the gallon is the base unit of capacity used to define the other,
derived units. For example, one fluid ounce is exactly equal to
1
160
of a gallon.
Figure 1.9 You can still find one-gallon containers like the oil shown here beside the onelitre container. Use your answer on the previous page to determine the amount
in litres that should be indicated on this oil can?
Round off your answer to the nearest tenth. E e
R l
E e
U b
Peta
R l
U b
g
P ta
e
V
I LVeg L
I
e
O
G
e
tr
n
lo
al
1
Did you
know
G
1
Li
5
6
?
that of the imperial gallon.
Figure 1.10 Another unit of capacity, the cup, is commonly used in cooking. According to the standard that used to apply in Québec, one imperial cup is equal to 8 fluid ounces. As this
measurement has proven to be very practical, another standard was established when
the metric system was adopted. One metric cup is exactly equal to 250 millilitres (mL).
As shown in the illustration on the right, most measuring cups include both types of
units.
Compare the capacity in millilitres held by a metric cup with that held by an imperial
cup.
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e
tr
1
The same name for a unit of capacity could have different meanings in different countries. For
example, the capacity of the American gallon is only
e
p
ra e
G ic
Ju
O
n
lo
al
1
e
p
ra e
G ic
u
J
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No doubt you have understood that the expression of a capacity in millilitres is proportional to
its expression in fluid ounces. To answer the first question in the activity, we simply use proportional
reasoning based on the information given on one of the cans.
For instance, for the can of coconut milk, we get the following:
÷ 14
14 fl oz = 398 mL
÷ 14
398 ÷ 14 = 28.4
1 fl oz = ? mL
1 fluid ounce is therefore approximately equal to 28.4 mL.
Tip
You can find a more precise approximation on the Internet by typing the expression "fluid ounce in
millilitres" into a search engine. If you carry out this search, you will find the following equivalence:
1 fl oz ≈ 28.4131 mL.
Given that one fluid ounce is approximately equal to 28.4 mL, we can answer the other questions on the previous page. For example, the 25-fluid ounce container should indicate 710 mL, since 25 fl oz = 25 × 1 fl oz ≈ 25 × 28.4 mL ≈ 710 mL.
The capacity of the one-gallon can is 160 fl oz. If we use the same reasoning as above, we get:
1 gallon = 160 fl oz = 160 × 1 fl oz ≈ 160 × 28.4 mL ≈ 4500 mL ≈ 4.5 L
This can should indicate 4.5 litres.
NOTE: For a more precise equivalence, we could use the approximation of 28.4131 mL for the fluid
ounce in the above calculation, thus obtaining the following equivalence: 1 gallon ≈ 4.546 L.
Lastly, the capacity of an imperial cup may be expressed in millilitres.
8 fl oz = 8 × 1 fl oz ≈ 8 × 28.4 mL ≈ 227 mL
This is a little smaller than the metric cup, which holds 250 mL; however, for practical purposes, we can
say that it is roughly the same quantity.
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Key p
Units of capacity in the imperial system
Table 1.3 – Comparison of the units of capacity in the two systems
Imperial system
Symbol
• The base unit is the gallon.
gal
International system
• The base unit is the litre.
• To give you an idea: a container in
• One litre is the capacity of a cubic
the shape of prism with a square
container with edges measuring
base measuring 5 in by 5 in and
1 dm.
Symbol
L
with a height of 11 in has a capacity
of about 1 gallon.
• Other units, called derived units,
• Derived units (millilitre, centilitre,
mL
are used in everyday situations.
decilitre, decalitre, hectolitre, kilo-
cL
For example, the fluid ounce and
litre) are obtained by adding pre-
dL
the cup are used in cooking to
fixes to the base unit.
daL
measure small quantities.
• One fluid ounce is equal to
1
160
a gallon.
of
fl oz
• These units of measure are all based
hL
on multiples of 10. For example,
kL
1 litre = 10 dL = 100 cL = 1000 mL.
• One cup is equal to 8 fl oz.
Equivalences between the units of capacity in the two systems
One fluid ounce is slightly more than 28.4 mL, one gallon is equal to 4.546 L and one cup is approximately
equal to 227 mL. However, in everyday situations, the following approximations are generally sufficient for
converting measures of capacity from one system of units to the other.
1 fl oz ≈ 30 mL
1 c ≈ 250 mL
1 gal ≈ 4.5 L
The level of precision needed always depends on the situation.
For example, in a cooking recipe, a quantity of 1 12 cups is usually converted to 375 mL
( 1 12 × 250 = 1.5 × 250 = 375 ) even if, in reality, this quantity is roughly equal to 340 mL
( 1 12 × 227 = 1.5 × 227 ≈ 340 ).
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Exercises for Activity 1.2
1.11
According to the imperial system,
a)how many cups are in one gallon? b)one fluid ounce is equal to what fraction of a cup? 1.12
1.13
Express the following capacities in the SI unit indicated.
a)12 fl oz in millilitres: b)4 gallons in litres: c) 14 cup in millilitres: d)50 fl oz in centilitres: In addition to the cup, tablespoons and teaspoons are commonly used in cooking as units of measure.
When the imperial system was still in force in Québec, one tablespoon was equal to
1
2
fluid ounce. There
are three teaspoons in one tablespoon.
Given that one fluid ounce is equal to 28.4 mL, express 1 tbs and 1 tsp in millilitres.
Did you
know
?
According to the new metric standard in cooking, one tablespoon is equal to 15 mL and one teaspoon is equal to 5 mL.
1.14
At one time in Québec, milk was sold in quarts. Each morning, the milkman would deliver bottles of milk
to his customers' homes. Today, some people still use the expression "a quart of milk" when referring to
a litre of milk.
Given that the quart in Québec was equal to 40 fluid ounces, compare the capacity of one
quart with the capacity of one litre.
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SIT UAT ION 1 – TWO DIF F E RE NT SY STE MS OF M E ASURE M E NT
Activity 1.3 – One Weight, Two Measurements
Goal
• To identify the units of mass commonly used in the imperial system
• To compare imperial units of mass with SI units of mass
• To convert imperial units of mass to SI units of mass
You have surely noticed that, in Québec grocery stores, the pound is commonly used to indicate weight,
particularly with respect to product cost.
Figure 1.11 How many kilos of
tomatoes would
you like, sir?
Uh… I prefer to buy
by the pound... it seems
less expensive!
What do you think of the customer's answer in the above illustration? Could buying by the pound (lb) be
cheaper than buying by the kilogram (kg)?
Obviously, the important thing is the quantity of tomatoes purchased. Whether this quantity is measured
in pounds or in kilograms should make no difference. The sign indicating the price of the tomatoes is very
useful for comparing pounds and kilograms.
Suppose, for example, that the customer buys $3.40 worth of tomatoes. Find the exact quantity of tomatoes he bought, to the nearest thousandth
in pounds:
in kilograms:
Given your answers above, indicate how many kilograms are in one pound.
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To find the quantity of tomatoes in pounds, simply divide the total cost ($3.40) by the price per pound
($1.22/lb).
3.40 ÷ 1.22 ≈ 2.787
The customer bought 2.787 lb of tomatoes.
The quantity of tomatoes in kilograms is calculated in the same way, but this time by dividing the total
cost by the price per kilogram ($2.69/kg).
3.40 ÷ 2.69 ≈ 1.264
The customer bought 1.264 kg of tomatoes.
From this, we can conclude that 2.787 lb is equal to 1.264 kg.
To find how many kilograms are in a pound, we simply use proportional reasoning.
2.787 lb = 1.264 kg
÷ 2.787
÷ 2.787
1.264 ÷ 2.787 = 0.454
1 lb = ? kg
One pound is equal to about 0.454 kg, or 454 g.
Tip
If you type "pounds to kilograms conversion" into a search engine, you will get a more precise
approximation: 1 pound = 0.45359 kg.
You will notice that one pound is equal to approximately half a kilogram. It is not surprising, therefore,
that the price per pound displayed in supermarkets is generally about half of the price per kilogram!
Tip
The previously determined equivalence could have been found in another way.
We could have said that the customer bought $1.22 worth of tomatoes, or exactly
one pound. In this case, we could have calculated the number of kilograms he
bought by dividing the total cost ($1.22) by the price per kilogram ($2.67/kg). Since
1.264 ÷ 2.787 ≈ 0.454, we can conclude that a pound is exactly equal to 0.454 kg.
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SIT UAT ION 1 – TWO DIF F E RE NT SY STE MS OF M E ASURE M E NT
There are other units of mass in the imperial system. For example, one pound contains 16 ounces (oz).
One ounce is equal to how may grams?
NOTE: The ounce (oz) is a unit of mass, whereas the fluid ounce (fl oz) is a unit of capacity. There is,
however, a close connection between these two units: one fluid ounce weighs exactly one ounce.
The ounce is used to measure small masses or to obtain a more precise measurement in pounds. For
example, it is very common to use pounds and ounces to describe the weight of a newborn, even though
officially this weight is registered in kilograms.
Give the weight of an 8 lb 5 oz newborn in kilograms.
We use the ton to measure the mass of much heavier objects. It should be noted, however, that there are
several types of tons. Originally, in the imperial system, the ton (also referred to as the long ton) was
defined as being equal to 2240 pounds. However, the United States today (as did Canada in the past),
uses the short ton, which is equal to 2000 pounds. There is also the metric ton, which is equal to 1000 kg.
Compare the long ton to the metric ton. Which is heavier?
How many kilograms are in a short ton?
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Now let's go back to the previous questions.
By definition, one ounce is equal to
1
16
pound. We also know that one pound is equal to 454 g. Therefore:
1 oz = 454 g ÷ 16 ≈ 28.4 g.
Reminder
Does the number 28.4 remind you of something?
In the previous activity, we established that one fluid ounce is equal to 28.4 mL. It is not a coincidence
that we get the same number here.
As we saw earlier, the imperial units have been defined in such a way that one fluid ounce weighs
exactly one ounce. In the SI system, units of mass are defined such that one litre of water weighs 1 kg
or, if we divide these two measurements by 1000, one millilitre of water weighs 1 g.
1 oz = the mass of 1 fl oz of water = the mass of 28.4 mL of water = 28.4 g
We can find the newborn's weight in different ways. For example, we can express pounds in grams and
ounces in grams and then add the results together.
8 lb 5 oz = 8 × 1 lb + 5 × 1 oz = 8 × 454 g + 5 × 28.4 g ≈ 3 774 g ≈ 3.8 kg
The newborn weighs 3.8 kg.
Let's conclude by comparing the different types of tons.
1 long ton = 2240 lb ≈ 2240 × 0.454 kg ≈ 1017 kg
This is heavier than a metric ton, which is equal to 1000 kg.
1 short ton = 2000 lb = 2000 × 0.454 kg ≈ 908 kg
The short ton used in the United States is equal to about 0.9 metric tons.
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SIT UAT ION 1 – TWO DIF F E RE NT SY STE MS OF M E ASURE M E NT
oints
Key p
Units of mass in the imperial system
Table 1.4 – Comparison of the units of mass in the two systems
Imperial system
Symbol
• The base unit is the pound.
• Since
one
gallon
of
lb
International system
• The base unit is the kilogram
water
think.
responds to the mass of 2 cups,
• One kilogram is the mass of one
or 16 fl oz, of water.
units,
kg
and not the gram as one might
weighs 10 lb, one pound cor-
• Other
Symbol
litre of water.
called
derived
oz
units, are used in everyday si-
• In general, the gram and the metric
g
ton are the only derived units in
t
t
tuations. These are the ounce
the SI system that are used in eve-
and the ton.
ryday situations to measure mass.
• There are 16 ounces in one pound.
• There are 1000 grams in one
• According to the standard used in
kilogram.
the United States, a short ton is
• One metric ton is equal to 1000 kg.
exactly equal to 2000 pounds.
Equivalences between the units of mass in the two systems
Since one pound is equal to 454 g, one ounce is equal to 28.4 g. However, in everyday situations, the following approximations are generally sufficient to convert measures of mass from one system to the other.
1 oz ≈ 30 g
Example:
1 lb ≈ 454 g
Someone wants to prepare 15 hamburgers, each containing a
1
4
lb of ground beef. How
much beef (in kilograms) should he buy?
15 ×
1
4
lb = 3 43 lb ≈ 3 43 × 454 g ≈ 1700 g
He will have to buy about 1.7 kg of beef.
To convert the masses of heavy objects, we can use the following equivalence:
1 short ton ≈ 0.9 metric ton
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Exercises for Activity 1.3
1.15
According to the imperial system,
a)how many ounces are in half a pound?
b)500 lb is equal to what fraction of a short ton? 1.16
Express the masses below in the SI unit given.
a)25 lb in kilograms: b) 12 lb in grams: c)3 lb 4 oz in kilograms: d) 10 12 oz in grams: 1.17
According to the 2010 data available on the Internet, hockey player Sidney Crosby, a star of the Pittsburgh
Penguins, weighs 200 lb. On another site, we see that Fernando Torres, star player for the Spanish soccer
team, weighs 70 kg. Compare the weights of these two athletes.
1.18
An American magazine claims that the largest mammal on earth is the African elephant, which can weigh
up to 7 tons. How would you convey this information to a friend from France?
1.19
The pound is a unit of measure with a long history. This is why the pound refers to slightly different units
of measure in different fields. For example, to weigh gemstones and precious metals, we use the "troy
pound," which is equal to 12 troy ounces. One troy ounce is approximately equal to 31.1 g. Saïd says that
one pound of gold weighs less than a pound of butter. Is he right? Explain.
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SIT UAT ION 1 – TWO DIF F E RE NT SY STE MS OF M E ASURE M E NT
Activity 1.4 – One Way or the Other
Goal
• To convert SI units to imperial units
Let's go back to the two people we met in the Introduction who described their height and weight in
different units.
Figure 1.12 I’m 5 feet 2 inches
tall and I weigh
110 pounds.
I’m 1 metre
35 centimetres tall
and I weigh 26 kilos.
The woman wonders what the difference is between her height and weight and those of the girl. As she
is more familiar with imperial units than SI units, she would like the answer in inches and pounds. Help
her find the answer to her question.
First, find the height difference in inches. Round off your answer to the nearest unit.
Then, find the weight difference in pounds. Again, round off your answer to the nearest unit.
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To be able to compare the woman's height and weight with the girl's, we must express the measurements
using units from the same system.
In Activity 1.1, you learned to convert imperial units of length to SI units of length by using the equivalence of 1 in = 2.54 cm. Since the woman wants the answer in inches and not in centimetres, we must
convert the girl's height of 135 cm into inches in order to compare it with the woman's. But how can we
convert these units? As you will see, there are several ways to go about it.
First method: using proportional reasoning
We know that the expression of a length in inches is proportional to its expression in centimetres. We can
therefore use the following reasoning:
÷ 2.54
1 in = 2.54 cm
÷ 2.54
1 ÷ 2.54 ≈ 0.394
× 135
0.394 × 135 ≈ 53
? in = 1 cm
× 135
? in = 135 cm
First, we express one centimetre in inches by dividing each term in the original equivalence by 2.54. This
gives us an equivalence of 0.394 in ≈ 1 cm. To convert 135 cm into inches, we simply multiply 0.394 by
135. We find that the girl's height is 53 inches. Since the woman is 62 inches (or 5 × 12 + 2) tall, the difference between her height and the girl's height is 9 inches (or 62 – 53).
We can calculate their difference in weight in the same way, given that 1 lb ≈ 0.454 kg and that the girl
weighs 26 kg. If you complete this line of reasoning, you should obtain a weight difference of 53 lb.
1 lb ≈ 0.454 kg
÷ 0.454
lb ≈ 1 kg
lb ≈ 26 kg
Weight difference: 110 –
=
Second method: using a formula
The diagram on the right illustrates the equivalence relationship
× 2.54
between inches and centimetres:
1 in = 2.54 cm
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SIT UAT ION 1 – TWO DIF F E RE NT SY STE MS OF M E ASURE M E NT
This relationship can also be described using the formula below.
l(­cm) = 2.54 l(in)
where l(­cm) is the length in centimetres;
and
l(in) is the length in inches.
This formula simply means the following: to convert a length in inches to a length in centimetres,
multiply the number of inches by 2.54.
However, we want to convert from centimetres to inches. To do so, we can transform the formula by isolating l(in).
l(cm)
l
After dividing by 2.54, we obtain the expression = l , which can also be written: l = (cm) .
( in)
2.54 ( in)
2.54
This new formula can be interpreted as follows: to convert a length in centimetres to a length in
inches, divide the number of centimetres by 2.54.
1 in = 2.54 cm
The diagram on the right also illustrates this rule.
Let's apply this rule to find the girl's height in inches.
In this case, l(cm) = 135 . If we divide by 2.54, we obtain: l( in)
135
=
≈ 53 .
2.54
÷ 2.54
The girl's height is therefore 53 inches.
The same procedure could be used to convert the girl's weight.
The equivalence relationship between pounds and kilograms is illustrated by the diagram
on the right.
× 0.454
1 lb
0.454 kg
It is also be described by the formula below:
m(­kg) ≈ 0.454 m(lb)
where m(­kg) is the mass in kilograms;
and
m(lb) is the mass in pounds.
Use the formula to convert 26 kg into pounds.
You may have initially used a different method to find the answer to the woman's question and obtained
the same answers. The important thing is to use logical reasoning.
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oints
Key p
Different representations of an equivalence relationship
between two units
Whether it be length, capacity or mass, the expression of a measure in imperial units is always proportional
to its expression in SI units. This allows us to establish different representations of an equivalence relationship between two units.
Example:
Below are representations derived from the equivalence 1 gallon ≈ 4.5 litres.
Unit-rate method
Conversion factor
Formula
××
4.5
4.5
1 gallon
1 gallon 4.5
4.5
litres
litres
÷÷
4.5
4.5
÷÷
4.5
4.5
0.22
0.22
gallon
gallon
1 1gallon
gallon 4.5
4.5litres
litres
C( litre) ≈ 4.5C( gallon)
1 litre
1 litre
÷÷
4.5
4.5
Converting measures from one system to the other
All the measurements in one system can be converted to measurements in the other system by using equivalences that give the value of imperial units in SI units, or vice versa. This can be done simply by using one
of the above representations, as needed.
Exemple 1: Given that one gallon equals 4.5 litres, find the capacity of a 500-gallon tank in kilolitres.
(1 kL = 1000 L)
We convert from gallons to litres (and then to kilolitres). You can find the solution simply
by using proportional reasoning.
Since 500 gallons is 500 times greater than one gallon,
500 gallons ≈ 500 × 4.5 L = 2250 L ÷ 1000 = 2.25 kL. The tank has a capacity of 2.25 kL.
Example 2: Given that one fluid ounce is equal to 28.4 mL, what is the capacity in fluid ounces of a
2-litre bottle?
In this case, we convert from litres to fluid ounces. We can use one of the above
representations to complete the reasoning. For example, by using the unit-rate method:
÷ 28.4
1 fl oz
28.4 mL
? fl oz
1 mL
÷ 28.4
1 ÷ 28.4 ≈ 0.035
One millilitre is therefore equal to 0.035 fl oz.
2 litres = 2000 mL ≈ 2000 × 0.035 fl oz ≈ 70 fl oz
The bottle holds 70 fl oz.
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SIT UAT ION 1 – TWO DIF F E RE NT SY STE MS OF M E ASURE M E NT
Exercises for Activity 1.4
If necessary, refer to the table at the end of this learning situation.
1.20
How many inches are in one centimetre? Round off your answer to the nearest tenth.
1.21
a)How many cups are in one litre?
b)How many pounds are in one kilogram?
1.22
Complete the table below. For the formulas, use the letters l and C as subscripts to indicate measures of
length and capacity respectively.
Equivalence
Formula
1 yard ≈ 0.9 metre
1 pound ≈
100 m in yards:
m(g) = 454 m(lb)
1 fl oz ≈ 28.4 mL
1.23
Conversion
1
2
pound in grams:
750 mL in fl oz:
a)One kilometre is equal to what fraction of a mile?
b)Convert 40 km into miles. 1.24
In the United States, speed limits on roads are indicated in miles per hour (mph). This is why near the
U.S. border, we see the following sign which alerts American tourists that speed limits in Québec are
indicated in different units.
In your opinion, is the information on this sign correct? Explain.
METRIC
SIGNAGE
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1.25
TABLE OF
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ANSWER
KEY
Anaïs wants to buy a new computer with a larger screen than the one she has now. In stores, screen sizes
are always indicated in whole numbers of inches. This length is the diagonal screen size.
a)Anaïs uses a ruler graduated in centimetres to find the diagonal of her computer screen. The screen
measures 43 cm. What is the size of her screen in inches?
b)She ends up buying a 22-inch screen. Express this quantity in centimetres.
1.26
a)How many gallons of water are in a 12-litre container?
b)How many litres are in a 4.2-gallon container?
1.27
For each pair of measurements given below, circle the larger one.
a)10 feet or 3 metres
b)4 ounces or 100 grams
c)10 cups or 2 litres
d)30 short tons or 25 metric tons
e)200 miles or 300 kilometres
1.28
You are selling potatoes in a vegetable market. Your sign indicates the price per kilogram. What would you
tell a customer who asks you what the price is per pound?
Potat
oes
$1.80/
kg
1.29
Veronica will soon give birth to her baby. She has read that the newborn's ideal weight should be between
3 kg and 3.5 kg. Give this information in pounds and ounces by using whole numbers.
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Activity 1.5 – Two Measurements, One Temperature
Goal
• To compare imperial and SI units of temperature
• To convert temperature measurements from one system of units to the other
The weather is the most common topic of conversation in our society; and since we are great travellers, especially to the United States, it is not surprising that as many people are familiar with degrees
Fahrenheit as they are with degrees Celsius. As well, all thermometers on the market display both scales.
Examine the thermometer shown on the right. You are already familiar with rulers gradua-
Figure 1.13 ted in centimetres and inches, cups graduated in millilitres and fluid ounces, or scales graduated in kilograms and pounds. What is different about this thermometer compared with
50
all these measuring instruments? Name two differences.
110
40
120
30
100
90
80
As you can see on the thermometer, 0°C = 32°F. If the temperature increases by 10°C, by
how many degrees Fahrenheit does it increase? How about if it increases by 20°C or 30°C?
20
70
60
10
50
40
The relationship between these two scales is modelled by the formula
t(°C) = 59 × (t(°F) − 32), where t(°C) is the temperature in degrees Celsius and t(°F) is the temperature in degrees Fahrenheit. Use this formula to convert the following temperatures to
0
20
-10
degrees Celsius:
50 °F: 80 °F: 0 °F: -40 °F: Using the previous formula, find another formula that would allow you to do the reverse,
30
-20
10
0
-10
-30
°C
-20
°F
that is, convert degrees Celsius to degrees Fahrenheit? Check your formula by using a few
temperature values displayed on the thermometer.
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Temperature measurements have two basic characteristics that distinguish them from other types of
measurements we have seen so far. First, note that there are negative temperature values, which is
not the case for the other types of measures. This is due to the meaning given to the number 0. For
measures of length, capacity or mass, a measure of 0 means that there is no length, capacity or mass. A
glass containing 0 mL of water is empty. A segment 0 cm long does not exist. In the case of temperature,
however, 0 has another meaning. It is a point of reference. It is not a minimum, because we can have
temperatures of less than 0°C or 0°F.
Since the 0 on the temperature scale is only a point of reference, the position of the 0 can be different on
different scales. This is true here since 0°C is not equal to 0°F. This is yet another basic characteristic
that differentiates measures of temperature from other types of measures (since we can have 0 cm = 0
in, 0 kg = 0 lb, etc.).
Did you
know
?
Scientists have shown that there is an absolute temperature of 0 that has the same meaning as the
zeros in other units. This temperature, which is approximately equal to -273°C, is the absolute minimum temperature; it is impossible to have a lower temperature. For scientific applications, another temperature scale, the Kelvin (K) scale, was created to take this phenomenon into account. A
temperature of absolute zero is equal to 0 K.
We can therefore state that a temperature in degrees Celsius IS NOT proportional to its expression
in degrees Fahrenheit. Thus, all that applies to converting measures from one system to the other
(unit-rate method, conversion factor) does not apply to temperature values. We must proceed differently
to convert temperature measurements from one system to the other.
In studying thermometers, you have surely noticed that an increase of 10°C corresponds to an increase
of 18°F, whereas an increase of 20°C corresponds to an increase of 36°F, and an increase of 30°C corresponds to an increase of 54°F. You may have also noticed that the ratio between the two increases
is always 5:9. In fact, the factions
are all equal to the irreducible fraction of
. This
allows us to deduce the following formula:
t(°C) = 59 × (t(°F) − 32)
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SIT UAT ION 1 – TWO DIF F E RE NT SY STE MS OF M E ASURE M E NT
To convert degrees Fahrenheit to degrees Celsius, simply apply this formula. For instance:
for 50°F, we get t(°C) = 59 × (50 − 32) = 59 × 18 = 10 . Thus 50°F = 10°C.
for 80°F, we get t(°C) = 59 × (80 − 32) = 59 × 48 ≈ 27 . Thus 80°F ≈ 27°C.
Similarly, you will find that 0°F ≈ -18°C and that -40°F = -40°C.
To convert degrees Celsius to degrees Fahrenheit, we transform this formula by isolating t(°F).
Transformation of the formula
t(°C) = 59 × (t(°F) − 32)
9
5
9
5
× t(°C) = t(°F) − 32
× t(°C) + 32 = t(°F)
Result: t(°F) = 95 × t(°C) + 32
You
can
check
that
this
last
formula
is
correct.
For
example,
for
10°C,
9
5
we get t(°F) = × 10 + 32 = 18 + 32 = 50 . Thus, 10°C = 50°F.
Figure 1.14 © FreeSoulProduction/Shutterstock.com
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Key p
Units of temperature
In the imperial system, temperature is measured in degrees Fahrenheit (°F). In the SI, it is measured in
degrees Celsius (°C).
The 0 has a different meaning on each scale. In 1742, the Swedish physicist Anders Celsius constructed
the Celsius thermometer, with 100° for the freezing point of water and 0° for the boiling point. However, in
1745, after Celsius' death, the Swedish botanist Carl Linnaeus inverted the scale. In 1724, Daniel Gabriel
Fahrenheit proposed a temperature scale in which the freezing point of water is set at 32°F and the boiling
point of water is set at 212°F.
Comparisons of temperature measurements
In Celsius
In Fahrenheit
Freezing point of water
0°C
32°F
Ideal indoor temperature
20°C
68°F
Temperature of the human body
37°C
98.6°F
Boiling point of water
100°C
212°F
Converting temperature measurements
In order to convert temperature measurements from one system to the other, we can use the following
formulas in which t(°C) represents the temperature in degrees Celsius and t(°F) , the temperature in degrees
Fahrenheit.
To convert degrees Fahrenheit to degrees Celsius: t(°C) = 59 × (t(°F) − 32)
To convert degrees Celsius to degrees Fahrenheit: t(°F) = 95 × t(°C) + 32
Example:
On a summer day, it was 86°F in New York City and 30°C in Montréal. Which city had the
higher temperature?
To compare these temperatures, we must express them in the same unit. For example, we
can express the temperature in New York City in degrees Celsius.
We get: t(°C) = 59 × (86 − 32) = 59 × 54 = 30 . Like Montréal, New York City had a temperature
of 30°C. The temperature was the same in both cities.
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SIT UAT ION 1 – TWO DIF F E RE NT SY STE MS OF M E ASURE M E NT
Exercises for Activity 1.5
1.30
The highest temperature measured on Earth since the beginning of the 20th century is 56.7°C, on July
10, 1913 in Death Valley, United States. The coldest temperature was measured on July 21, 1983, at
Vostok station, in Antarctica. The thermometer in this case indicated – 89.2°C. Give these two measurements in Fahrenheit. Round off your answer to the nearest tenth.
1.31
On Martin's old stove, the control knob for adjusting the oven's temperature is graduated in degrees
Fahrenheit, from 150°F to 500°F. Determine to what temperature in degrees Celsius each of these measurements corresponds. In each case, round off your answer to the nearest multiple of 5.
Off
Broil
150
200
500
250
450
300
400
1.32
350
According to a Web site, there is a risk that bacteria will grow in a water heater if the temperature is kept
below 120°F. Hydro-Québec recommends that water heaters be set at a temperature of 60°C. Compare
these two measurements. Which is higher? By how many degrees?
1.33
Naomi calls Info-Santé, because her baby is running a fever. "This morning, her temperature was normal,
but since then her temperature has risen three and a half degrees," she said to the nurse. The nurse
asked: "Did you measure her temperature in Celsius or Fahrenheit?" Which temperature, degrees Celsius
or degrees Fahrenheit, would indicate a more serious situation? Explain.
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Taking It Further
A European who comes to Québec is probably astonished to see the sizes of containers on our supermarket shelves. Why a 1.89-litre bottle of juice? Why not 2 litres? And if he wants to buy 500 g of butter, he'll
have to make do with 454 g!
Seeing all of these measurements indicated on containers, whether it be on a 737-g box of cookies or
a 341-mL bottle of cooler, one might wonder whether the SI system has truly taken root in Canada.
Wouldn't it make more sense to have a 750-g box of cookies? Why 737 g? And why a 341-mL bottle of
cooler? Is the manufacturer trying to save 9 mL?
Elsewhere in the world, and especially in Europe, products are sold in containers that are generally multiples of 50 g or 50 mL. Most bottles of wine, for instance, have a capacity of 750 mL.
1.34
Look in your refrigerator or your pantry. Gather a sample of about 20 different products. Notice their
sizes and the measures of capacity or mass indicated on them. In your opinion, what percentage of the
sample consists of containers that are truly metric in size?
What is the reason for this situation? Here is one explanation.
Some products are sold in Canada and in the United States in the same type of container. It is very likely
then that the container's capacity corresponds to an exact measurement in the American system of units.
Take, for instance, the 1.89-litre bottle of juice. Where does this strange number come from?
Below are some facts about American units of capacity.
• The American fluid ounce is slightly larger (about 4% larger) than the imperial fluid ounce.
• 1 American gallon contains 128 American fluid ounces.
1.35
Using this information, explain why stores sell 1.89-litre bottles of juice.
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SIT UAT ION 1 – TWO DIF F E RE NT SY STE MS OF M E ASURE M E NT
Integration Exercises
1.36
The body mass index (BMI) allows us to assess whether a person's weight is
appropriate for his or her height. The BMI can be calculated using the following
m
formula: BMI = 2 , where m is the person's mass in kilograms and H, his or her
H
height in metres. Here is a similar formula with imperial units.
BMI = 703 ×
m( lb)
H( in)
2
where m(­lb) is the mass in pounds;
and Interpretation of BMI
BMI
Classification
< 18.5
Underweight
18.5 to 25
Normal
> 25.0
Overweight
≥ 30
Obese
H(in) is the height in inches.
a)Give your weight and height in imperial units and in SI units.
b)Calculate your BMI using both formulas. Do you get the same results?
c)Anne, who is 1.80 m tall, realized that she is overweight since she has a BMI of 29. How many pounds
should she lose in order to achieve an appropriate weight for her height?
1.37
In 2010, the Moroccan Hicham El Guerrouj held the men's record for a one-mile race with a time of
3 min 43.13 s. The Olympics no longer holds this event. The race that comes closest to it is the 1500-m race.
In 2010, the same runner held the World record for the 1500-m race with a time of 3 min 26 s.
In which of the two races did El Guerrouj run faster?
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Tip
It's always a good idea to check that your answer is plausible. In your opinion, over what distance
can someone run faster: a longer distance or a shorter one?
1.38
You work in a pet store where you are in charge of the fish department. A customer would like to add
3 fishes to his aquarium, which already has 7. He asks you if his aquarium is large enough to hold all of
these fishes. He does not know the exact capacity of his tank, but he knows that it measures about 2 feet
by 1 foot and contains 9 inches of water. Given that the norm is one fish per 5 litres of water, what would
you tell him?
1.39
Sandra wants to buy a van, but she is worried about gas consumption. The specifications for the model
she is interested in indicate 10.5 litres/100 km. Sandra's father, who is with her, nostalgically remembers
the first car he bought in 1967, a Volkswagen beetle, the most popular car in the world. "I got at least
25 miles to the gallon with my beetle," he said. According to this information, which car is more fuel
efficient?
1.40
A certain amount of energy raises the temperature of one gallon of water by 9°F. According to the physical properties of water, we can say that the same amount of energy could raise the temperature of
lon of water by 18°F or of
1
4
1
2
gal-
gallon of water by 36°F. By how many degrees Celsius will the temperature
of 1 litre of water be increased if we use the same amount of energy?
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SIT UAT ION 1 – TWO DIF F E RE NT SY STE MS OF M E ASURE M E NT
Review Activity – Planning a Recipe
You should now be able to carry out the assignment described at the beginning of this learning situation.
1.41
Brownie recipe
Here again are the ingredients for the brownie recipe given in at the beginning of this learning situation. The
amounts, expressed in imperial units, are in parentheses.
Convert each measurement to SI units.
Ingredients
( 14 lb) unsalted butter
(5 oz) unsweetened chocolate
(1 14 cups) sugar
2 eggs
(1 tsp) vanilla extract
(1 tbs) instant coffee
( 12 tsp) salt
(1 c) all-purpose flour
(2 c) miniature marshmallows
In the preparation, other measurements are given in imperial units. Also convert these measurements.
Preparation
Butter a
(9 x 9 inch) pan and line it with parchment paper.
Bake for about 25 min in an oven preheated to Cut into sixteen
© S O FAD
(350°F ).
(2 14 in) squares.
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Adjusting the recipe for your pan
As indicated in the recipe, your brownie pan is square, but you notice that each side measures 28 cm.
a)If you make this recipe without adjusting it, what will happen to the height of the mixture in your
pan? Will it be greater than, equal to or less than the height of the mixture in the pan called for in the
recipe? Justify your answer.
b)You are concerned that the brownies won't bake properly if the height of the mixture in your pan is not
the same as that indicated in the recipe. What adjustments would you have to make to the quantities
of each ingredient to ensure that the height of the mixture in your pan is correct?
c)Your brownies are a success! They came out perfectly and all you have to do now is cut them into
squares. You cut them into 25 identical squares. Compare the dimensions of your brownies with those
indicated in the recipe.
© Used with autorization from Zone 3
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geometric Representations and transformations
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ISBN : 978-2-89493-434-0
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