Mathematics 10C Introduction

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Table of Contents

Welcome to Mathematics 10C

Mathematics 10C Textbook and Website Support

Mathematics 10C Partners

Mathematics 10C Co-Developers

Learning in an Online Environment

LearnAlberta.ca

Alternative Learning Environments and Distributed Learning

Instructional Design

Visual Cues (Icons)

Glossary

Toolkit

Assessment

Using the Mathematics 10C Folder

Welcome to Mathematics 10C

In Mathematics 10C you will be encouraged to develop positive attitudes and to gain knowledge and skills through your own exploration of mathematical ideas—often with the help of study partners. As you progress through this course, you will also be encouraged to make connections to what you already know from your personal experiences. Building on your own experiences will give you a solid base for your understanding of mathematics.

There are seven mathematical processes that are critical aspects of learning, doing, and understanding mathematics. The Alberta Program of Studies incorporates the following interrelated mathematical processes. You will undergo these processes on a regular basis to help you achieve the goals of this course and future mathematics courses.

Process	Rationale	Application
Communication	Students must be able to communicate mathematical ideas in a variety of ways and contexts.	You will write, read, and discuss mathematical ideas with your peers and teacher.
Connections	Through connections, students begin to view mathematics as useful and relevant.	You will connect the math that you learn to meaningful contexts.
Mental Mathematics and Estimation	Mental mathematics and estimation are fundamental components of number sense.	You will make predictions about the outcome of events. You will also determine whether mathematical results are reasonable.
Problem Solving	Learning through problem solving should be the focus of mathematics at all grade levels.	You will learn multiple strategies for approaching problem solving.
Reasoning	Mathematical reasoning helps students think logically and make sense of mathematics.	You will analyze patterns, make generalizations, and test them.
Technology	Technology enables students to explore and create patterns, examine relationships, test conjectures or hypotheses, and solve problems.	You will use interactive multimedia, calculators, or computers to explore mathematical concepts.
Visualization	The use of visualization in the study of mathematics provides students with opportunities to understand and make connections among concepts.	You will use concrete materials, technology, and a variety of visual representations.

The Alberta Program of Studies outlines what students are expected to learn in mathematics courses. These expectations are written in statements called general outcomes.

There are three general outcomes in Mathematics 10C. They are:

• Develop spatial sense and proportional reasoning.
• Develop algebraic reasoning and number sense.
• Develop algebraic and graphical reasoning through the study of relations.

The general outcomes are further divided into specific learning outcomes related to the topics you will be studying in this course. Specific learning outcomes are subdivided into achievement indicators. These achievement indicators form the basis for the outcomes for each lesson.

Mathematics 10C is composed of four units. These units are:

• Unit 1: Measurement
• Unit 2: Algebra and Number
• Unit 3: Relations and Functions
• Unit 4: Linear Equations and Systems

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Mathematics 10C Textbook and Website Support

There are two approved textbooks for this course. They are Math 10 (McGraw-Hill Ryerson) and Foundations and Pre-calculus Mathematics 10 (Pearson). You will be using one of these textbooks throughout this course. You will find additional support at each textbook’s online website—http://www.mhrmath10.ca for McGraw-Hill Ryerson and http://www.pearsoncanada.ca/wncpmath10 for Pearson. You can find tips for success in mathematics, master sheets, general web links, a digital version of the textbook, web interactive, and other useful learning tools. By choosing a chapter from the pull-down menu, you can access interactive quizzes and web resources for individual chapters.

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Mathematics 10C Partners

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Mathematics 10C Co-Developers

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Learning in an Online Environment

This course is delivered to you in an online environment. You can look forward to using resources, such as interactive multimedia and the Internet, for various activities. You will also have access to computer simulations, computer multimedia, computer graphics, and electronic information to support your learning.

Remember that exploring the Internet can be educational and entertaining. However, be aware that these computer networks may not be censored. You may unintentionally come across offensive or inappropriate articles on the Internet. With that in mind, be aware that perspectives presented on the Internet are there for you to analyze critically and to accept or reject based on that analysis. Since information sources are not always cited, you should always confirm facts with a second reliable source.

Some school jurisdictions may limit access to social networking sites. In such circumstances, you should consult with your teacher, as your teacher may need to adapt lessons to accommodate co-operative learning.

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LearnAlberta.ca

LearnAlberta.ca is a protected digital learning environment for Albertans. This Alberta Education portal, found at http://www.learnalberta.ca/, is a place where you can access resources for projects, homework, help, review, or study.

For example, LearnAlberta.ca contains a large Online Reference Centre that includes multimedia encyclopedias, journals, newspapers, transcripts, images, maps, and more. The National Geographic site contains many current video clips that have been indexed for Alberta Programs of Study. The content is organized by grade level, subject, and curriculum objective. Use the search engine to quickly find key concepts. Check this site often, as new interactive multimedia segments are being added all the time.

LearnAlberta.ca now contains all of the available distributed learning online materials in the “T4T Courses” tab. If you are experiencing technical difficulties with the materials for this course, you can find the materials on LearnAlberta.ca.

If you find a password is required, contact your teacher or school to get one. No fee is required.

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Alternative Learning Environments and Distributed Learning

Distributed Learning is a model through which learning is distributed in a variety of delivery formats and mediums—print, digital (online), and traditional delivery methods—allowing teachers, students, and content to be located in different, non-centralized locations. Mathematics 10C students will be completing this course in a variety of learning environments, including traditional classrooms, online/virtual schools, home education, outreach programs, and alternative programs.

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Instructional Design

Explanation

The learning model used in Mathematics 10C is designed to be engaging and to have you participate in inquiry and problem solving. You will actively interpret and critically reflect on your learning process. Learning begins within a community setting at the centre of a larger process of teaching and learning. You will be encouraged to share your knowledge and experiences by interaction, feedback, debate, and negotiation.

Components

This course uses the following structure and instructional design to connect you to the relevant curriculum and scientific concepts in Mathematics 10C. These components are used consistently throughout the course and will help you in seeing the context and overall content of the program. The components of the course are described below in the order that you would see them in a typical lesson.

Component	Description
Focus	In the Focus section, the lesson theme is introduced. A real-world context and link to the unit or module theme is established, objectives are identified, and lesson questions are posed.
Assessment	The Assessment section provides a list of activities you are expected to submit as a record of achievement. These items may include a posting to a discussion board, assigned questions from the textbook, a portion of the unit project, or some other work assigned by your teacher. A Lesson Assignment document will help you to know what is to be submitted as part of your assessment.
Launch	This area helps students to prepare for the lesson ahead. Included in this section are the Are You Ready?, Refresher, and Materials headings.
Are You Ready?	This section provides a short pre-test to help you assess your mastery of prior skills and knowledge. If you are successful with these questions, you can move on to the Discover or Explore sections. If you encounter difficulty with these questions, you can move on to the Refresher to relearn prior skills.
Refresher	The Refresher addresses skills and knowledge gained previously, which will help you tackle the concepts of the lesson. This section may include an overview of a formula (e.g., Pythagorean theorem) or procedure (e.g., how to factor) or a short set of Self-Check questions for practice. This section may also include links or references to previous lessons or multimedia elements that will allow you to review prior skills.
Discover	Discover establishes the inquiry for the lesson. Activities in this section expose you to relationships and concepts to be addressed in the Explore section. Activities within the section will lead you to identify and analyze patterns or trends. Discussion with peers or with teachers will occur to further support inquiry-based learning.
Watch and Listen	Watch and Listen includes both passive and interactive multimedia content (e.g., podcasts, videos, interactive Flash activities). This section also includes a description of what you are supposed to focus on while using the multimedia in order to be active learners.
Try This	Try This includes opportunities to practise and to apply learned concepts outside of a lab environment. These can be simulation activities, questions, webquests, or other activities that provide you with a space to explore different ways of applying new concepts. You will find Try This questions placed in the Lesson Assignment document.
Math Lab	Math Lab is an activity where you complete an investigation that allows for data collection and analysis. The Math Lab often involves a hands-on component. Math Lab activities are also found in the Lesson Assignment document.
Share	Share allows you to use the discussion board to gather information from your peers or your teacher and to compare such information to your own results. This component provides opportunities for you to reflect, communicate, and build consensus about work completed during the Discover activities.
Explore	Explore supports the lesson inquiry initiated in the Discover section by formally introducing and developing concepts. You will be introduced to theorems, formulas, and concepts that will enable you to build on prior knowledge.
Read	Read is used to introduce sections of the textbook used for content or skill development. The relevance of the passage to context and lesson inquiry is defined.
Self-Check	Self-Check provides opportunities for you to check your understanding of new concepts learned in the lessons and to make connections to prior learning. These will be in auto-marked form. You can judge by your results in these sections whether you need to seek further clarification from your teacher on certain concepts.
Did You Know?	This section includes information that enriches your learning. Here, you could find historical information or math trivia that may be of interest to you.
Tips	This section includes alternate strategies, algorithms, or shortcuts for calculating values or implementing procedures.
Caution	This component alerts you to common misconceptions or procedural errors that would lead to incorrect work.
Connect	This heading comprises all of those activities which invite you to reflect on the knowledge and skills gained in the lesson and to connect that to the Big Picture. You will also have the opportunity to extend and enrich your learning with the Going Beyond section.
Project Connection	Aspects of the lesson related to the unit project are identified in this section. An activity related to completion of the unit project is described.
Reflect and Connect	Opportunity for you to consider what knowledge and skills have been gained or expanded during a lesson. You are asked to use a variety of reflective techniques (e.g., concept maps, summaries, answering questions). This may involve reflection on specific lesson elements or connecting lesson topic to the unit theme.
Going Beyond	Going Beyond entails the investigation of a sub-topic or examples related to the lesson theme. These sub-topics typically extend beyond the curriculum but may be of interest to you.
Lesson Summary	The Lesson Summary provides information about what has been accomplished in the lesson. The Lesson Summary also addresses the lesson questions posed at the beginning of the lesson and answers those questions based on the material covered in the lesson. All lesson summaries build toward the unit and course summaries and make connections to the Big Picture introduced at the module level.

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Visual Cues (Icons)

You will see icons throughout the course. These icons are clues regarding the type of activity you are about to begin.

Read	Are You Ready?	Assessment	Caution	Did You Know?
Going Beyond	Math Lab	Reflect and Connect	Project Connection	Refresher
Self-Check	Share	Tips	Try This	Watch and Listen

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Glossary

You will create your own glossary in this course. In Module 1: Lesson 1 you will discover the Glossary Terms document. You will add new terms to this handout and save it in a secure location, such as a course folder. (The course folder may be a document folder on your computer, or it may be a physical location such as a binder for storing print outs and pages.) As you encounter new terms, you can add them to your Glossary Terms document. You can update the Glossary Terms document each time you encounter new terms.

You will find the definitions to these terms in the lessons themselves, as well as in your textbook. You can further enrich your understanding of these terms by doing further research on the Internet or by sharing ideas with other students. The glossary is intended to be personal. You should define terms in a way that makes sense to you. You can add examples of how those terms are used. These examples can be in the form of diagrams, illustrations, or worked problems.

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Toolkit

The Toolkit is a collection of resources that provide you with further explanations or guidance for completing activities and assignments. The Toolkit includes grid paper templates, Math Lab templates, virtual manipulatives, computer video, and other objects to assist you as you work through each lesson.

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Assessment

Your work will be assessed in a number of different ways. Assessment items can be either formative or summative. In a typical lesson, you may be asked to share or discuss the concepts learned with a peer or with your teacher. The results of these discussion items should be recorded for future reference or for your teacher to examine. Other assessment items can include selected textbook questions, Project Connection pieces, and personal reflections. You may also have the opportunity to select your best work for grading.

As a way to help you to recognize the assessment items, you will have a Lesson Assignment document for each lesson. This document contains all of the assessment items for each particular lesson. Save this document to your folder at the beginning of every lesson and add to it as you proceed through the lesson. At the end of the lesson, you can submit the Lesson Assignment document to your teacher. Some of these items will be contribute to your mark, and other items may only serve to help your teacher know how to support your learning.

In this course there will also be opportunities for self-assessment. When you come to the end of a learning section, you can test your knowledge and skills by answering questions related to the section. The solutions are provided as a way for you to compare your own work.

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Using the Mathematics 10C Folder

The Mathematics 10C folder serves as the organized collection of samples of your work in Mathematics 10C. It gives you an ongoing record of your efforts, achievements, self-reflection, and progress throughout the course. When you want to show your friends or family what you’ve been learning, your work is all there for them.

In addition to being able to show others what you have done, the course folder lets you see your progress. It lets you see how your knowledge and skills are growing. It also lets you review and annotate work you have already completed. You may find your course folder useful in preparing for tests, quizzes, and your unit project.

Throughout the course, you will be asked to place your work in the Mathematics 10C folder. This folder may be an electronic folder on a server or a physical folder such as a binder. If you are unsure of the process, your teacher will help you.

Periodically, you will be asked to share items from your course folder with your teacher. This is not always for grading, as often your teacher may use these items to learn more about you and your interests or as a way of tailoring other work assigned to you.

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Mathematics 10C Learn EveryWare © 2009 Alberta Education

Mathematics 10C Copyright and Terms of Use

Copyright

Mathematics 10C Learn EveryWare

ISBN: 978-0-7741-3303-6

Release Date: 2010

Copyright © 2010, Alberta Education. This resource is owned by the Crown in Right of Alberta, as represented by the Minister of Education, Alberta Education, 10155 – 102 Street, Edmonton, Alberta, Canada T5J 4L5. All rights reserved.

This courseware was developed by or for Alberta Education. Third-party content has been identified by a © symbol and/or a credit to the source and must be used as is in the context in which it has been placed. Every effort has been made to acknowledge the original source and to comply with Canadian copyright law. If cases are identified where this effort has been unsuccessful, please notify Alberta Education so corrective action can be taken.

Users of this Resource are subject to the following Terms of Use Agreement.

THIS COURSEWARE IS NOT SUBJECT TO THE TERMS OF A LICENCE FROM A COLLECTIVE OR LICENSING BODY, SUCH AS ACCESS COPYRIGHT.

Terms of Use

NOTICE TO USER: This Terms of Use Agreement is a complete and legal Agreement between You and Her Majesty the Queen in Right of Alberta as represented by the Minister of Education (the “Minister”) regarding the use of Mathematics 10C Learn EveryWare (this “Resource”). By using this Resource, You accept all the conditions of this Agreement. If You do not agree with the following terms and conditions, do NOT use this Resource and remove all associated files from your system.

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This Resource is licensed, not sold, to You by the Minister for use on single workstations and/or on a computer network in an Alberta school jurisdiction.

A “computer network” is any combination of two or more terminals electronically linked and capable of sharing the use of a single electronic resource.

To “use” the Resource is to load included files into either temporary (e.g., RAM) or permanent (e.g., hard disk, CD-ROM) memory of a computer.

2. Permitted Uses

You may

• reproduce this Resource as is for unlimited, non-profit use within your school jurisdiction provided You include the accompanying Copyright notice, this Agreement, and all other proprietary notices contained in the original Resource

• include portions of this Resource, as is or modified, in your own electronic or printed lesson materials. Only material owned by the Minister may be altered. Third-party content, which is acknowledged by a credit to the source, must be used as is in the context in which it has been placed. It may not be altered or repurposed for use in another resource without permission from the original rights holder.

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• do anything with this Resource that is not expressly permitted in this Agreement
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External Internet sites listed in this Resource are provided as an added source of information on an as is basis and without warranty of any kind. Alberta Education is not responsible for the content on external sites nor do URL listings in this Resource constitute or imply an endorsement of the site’s content. The Crown, its agents, employees, or contractors will not be liable to You for any direct or indirect damages arising from the use of any of these sites.

Exploring the Internet can be educational and entertaining. However, be aware that these computer networks may not be censored. Students may unintentionally or purposely find offensive or inappropriate articles on the Internet. Since information sources are not always cited, students should be encouraged to confirm facts with a second source.

Rev. May 26, 2009

Mathematics 10C © 2010 Alberta Education

Mathematics 10C Introduction

Technical Requirements

Minimum System Requirements:

PC Compatible

• Intel(R) Pentium(R) III or AMD-K6(R)-2 processor-based computer
• 450 MHz CPU
• Microsoft(R) Windows(R) 2000/XP
• 512 MB RAM
• monitor capable of 1024 x 768 screen resolution and 16-bit colour
• 16-bit sound card and speakers
• 1x DVD-ROM drive (print course)
• A printer is recommended.

Macintosh(R)

• Power Macintosh(R) G3
• 500 MHz CPU
• Mac OS(R) X
• 512 MB RAM
• monitor capable of 1024 x 768 screen resolution and thousands of colours
• 16-bit sound card
• 1x DVD-ROM drive (print course)
• A printer is recommended.

Minimum Software Requirements*:

For All Platforms

• Adobe(R) Reader(R) 6.0 (Download from http://www.adobe.com/downloads/)
• Adobe(R) Shockwave(R) Player 8.5 (Download from http://www.adobe.com/downloads/)
• Adobe(R) Flash(TM) Player 9 (Download from http://www.adobe.com/downloads/)
• Microsoft(R) Office Excel(R) 2003 (optional)
• Microsoft(R) Office Word 2003
• QuickTime(R) Player 7.0 (Download from http://www.apple.com/quicktime/download)
• Euclid fonts for clear viewing of equations in Microsoft(R) Office Word 2003 (Download from http://www.dessci.com/en/dl/fonts/getfont.asp)

For Windows(R) 2000/XP

• Microsoft(R) Internet Explorer 6.0 for Windows(R) (Download from http://www.microsoft.com/windows/ie/ie6/downloads/default.mspx) OR
• Mozilla(R) Firefox(R) 2 (Download from http://www.mozilla.org/download.html)
• Java(TM) 2 Platform Standard Edition (J2SE 1.4.1) (Download from http://java.sun.com/javase/downloads/index.jsp)

For Mac OS(R) X

• Safari(TM) 1.3 OR
• Mozilla(R) Firefox(R) 2 (Download from http://www.mozilla.org/download.html)
• Java(TM) 2 Platform, Standard Edition, version 1.4.1 for Mac OS(R) X (Download from http://apple.com/java)

* Please note that vendors may have a more current version of their players and/or plug-ins available, which you can download from their sites, than the minimum software requirements listed above.

Mathematics 10C Learn EveryWare © 2010 Alberta Education

Unit 1 Introduction

 

 

The Art Gallery of Alberta, located in downtown Edmonton, opened its doors to the public in 2010. Designed by Randall Stout Architects, Inc., the building houses both national and international exhibitions.

 

With its many curves and jutting shapes suggesting prisms and cones, the architecture of the building truly is capable of capturing the observer’s attention and imagination. The design of the building complements the design of other buildings in the vicinity including Edmonton’s City Hall and the Winspear Centre.

 

What are the shapes that you have observed in your community? Are there buildings with interesting designs? Are there monuments or works of art that feature prisms, cones, or spheres?

 

While there is no doubt that the design of the Art Gallery of Alberta and other such buildings are intended to appeal to the observer, the design of most other objects follow function rather than form.

 

Nowadays, you can do “one-stop shopping” at the local home improvement centre. A few pieces of lumber to finish building your deck, a bookcase for your room, or a plant for your kitchen are some of the many things you can buy at these supercentres.

 

The next time you stop at one of these stores, pay attention to the different objects you can find there and how the shape of an object serves its function. For example, books are placed in rectangle-shaped spaces. The drinking glasses in your kitchen cupboard are cylindrical. Why do these objects have these particular shapes?

 

Can you imagine placing your favourite books on a triangular shelf or drinking from a glass in the shape of a sphere? All of these objects are designed to fulfill their functions. Even the medicine pills sold in pharmacies are designed to be more easily swallowed, chewed, and packaged.

 

There was a time long ago when the furniture you bought at a store was handmade. No two pieces of furniture were made in the same way. The bookcases and TV stands that you buy from today’s stores are all factory-made or prefabricated. In fact, the object is often packaged in pieces, ready for assembly by the buyer. Since the pieces are prefabricated, you can replace the pieces.

 

When you bring that entertainment unit or bookcase home from the store, you often have to assemble it yourself. It is rare to get a piece that is too wide or a screw that is too short. Each piece in a do-it-yourself kit is made to exact measurements. Each piece is made to exact specifications so that if you needed to order another part, you can get one that is identical in shape and size to the original.

 

When you plan where to place a new bookcase, you may need to know the dimensions of the bookcase in imperial units. Knowing how SI units convert into imperial units will help you to find the best fit possible.

 

In this unit you will investigate the following questions:

 

Lesson	Title	Lesson Questions
1	Basic Measurement Systems and Referents	How can referents be used to estimate measurements? Why are there two systems of measurement?
2	Using Measurement Instruments	How do you choose the appropriate techniques, tools, and formulae to describe the dimensions of an object? How can you measure the dimensions of objects of irregular shape or size?
3	Measurement Systems and Conversions	How do the strategies for converting units in the SI compare with those used in the imperial system? When can proportions be used to solve problems?
4	Surface Area of 3-D Objects	How is the concept of surface area applied to understanding the design of structures? How do you determine the surface area of a 3-D shape?
5	The Volume of 3-D Objects	How is the concept of volume applied to understanding the design of structures? How are the formulas for the volumes of solids related to each other?
6	Surface Area and Volume Problem Solving	Why is visualization important to the study of the surface area and volume of 3-D objects? How does changing the dimensions of an object affect its surface area and volume?
7	Introduction to Trigonometry	In what situations can the concepts of trigonometry be used to solve problems? How are the sine, cosine, and tangent ratios used to determine information about a right triangle?
8	Solving Right Triangle Problems	How do you approach problems whose solutions are based on trigonometry and its principles? How is trigonometry used to determine heights and distances that cannot be directly measured?

 

In this unit you will be working on a project as you learn new concepts in each lesson. This project will be based on a place that is special to you, whether it is a place in your home, in the neighborhood, or in your imagination.

 

You will start by examining measurement systems by using interactive multimedia and Math Labs. By using referents, which approximate SI and imperial units, you will be able to obtain good measurement estimates. You will learn how to convert between SI and imperial units and determine which units are appropriate to use for a given measurement task and a given measurement instrument. These skills will help you as you develop your project.

 

You will also investigate the surface area and volume of solids and learn how the properties of 3-D objects are used in design. In this part of the unit, you will conduct hands-on math labs using objects you can find around the house to help you discover the properties of spheres, cones, and pyramids. This knowledge will be transferred to your project where you will describe or design the objects that are found in your special place.

 

The last two lessons of the unit will focus on the concepts of trigonometry and how these can be used to solve problems where direct measurements are difficult to obtain. You will use these concepts in the analysis of the objects in your special place.

Mathematics 10C Learn EveryWare © 2010 Alberta Education

Module 1: Measurement and Its Applications

 

Lesson 1: Systems of Measurement and Personal Referents

 

Focus

 

Most people have a favourite place. That spot might be your bedroom at home, a cabin at the lake, or a beach in a tropical location. Perhaps your favourite place only exists in your imagination; a place where you can go to create, meditate, or take refuge. How would you describe your favourite place to someone who has never been there? One way you would likely describe your place is to explain how big it is—in other words, you could describe its dimensions.

 

In Canada, two measurement systems are commonly used—the SI (International System of Units) or metric system, and the imperial system. The SI is the measurement system officially adopted by Canada, but the imperial system is used frequently in the trades and in day-to-day conversations. For example, many people only know their height and weight in imperial units of measure—feet and inches and pounds, for example. In this lesson you will take a look at both systems of measurement.

 

Since most people don’t usually carry tape measures around with them, you will also relate these measures to common objects, which will allow you to quickly estimate a measurement. Such objects are called referents.

 

Outcomes

 

At the end of this lesson, you will be able to

• provide referents for linear measure including millimetre, centimetre, metre, kilometre, inch, foot, yard, and mile and explain the choices
• compare SI and imperial units, using referents
• estimate a linear measure using a referent, and explain the process used

Lesson Questions

• How can referents be used to estimate measurements?
• Why are there two systems of measurement?

Assessment

• Glossary Terms
• Project Connection
• Lesson 1 Assignment

In this lesson you will complete the Lesson 1 Assignment. Save a copy of the Lesson 1 Assignment to your course folder. You will receive more information about how to complete the assignment later in this lesson.

 

In this course you may come across Self-Check questions, Try This questions, and other activities that are not marked.

 

Remember that these questions provide you with the practice and feedback that you need to successfully complete this course. You should respond to all the questions and place those answers in your course folder. Your teacher may wish to view the work that you have stored in your course folder to check on your progress and to see if you require assistance in some way.

Mathematics 10C Learn EveryWare © 2010 Alberta Education

Module 1: Measurement and Its Applications

 

Launch

 

This section checks to see if you have the prerequisite knowledge and skills to be able to complete this lesson successfully. Go to Lesson 1: Are You Ready? and answer the questions in this section. If you are experiencing difficulty, you may want to use the information and the multimedia in the Refresher section to clarify concepts before completing these exercises.

 

Once you have completed these exercises to the best of your ability, submit your responses to your teacher. Your teacher will give you feedback and provide directions as to how to continue with this lesson.

 

Refresher

 

The formal system of measurement used in Canada is the SI, but imperial units are still used in ordinary conversation. Imperial unit usage is found, for example, in recipes, construction, house renovation, and gardening. The imperial system of measurement actually has a longer history than the SI. Imperial units of length were initially based on human dimensions. In this lesson you will create a body ruler to help you understand these units.

 

As you start making measurements in the imperial system, you will need to be familiar with fractions. Use the multimedia piece titled “Exploring Fractions” to practise basic fraction operations. (Make sure you maximize the screen by clicking on the button in the top-right corner of the video.)

 

Please note that clicking on the link takes you to the LearnAlberta website. This page includes both a video and an interactive component. Go to the left side of the web page, and choose “Exploring Fractions (Object Interactive).”

 

Materials

• rulers and measuring tapes marked in both SI and imperial units

Mathematics 10C Learn EveryWare © 2010 Alberta Education

Module 1: Measurement and Its Applications

Discover

Go through the Referents Table, which introduces the idea of using referents. For example, you may discover that the thickness of 10 sheets of paper is a good approximation of 1 mm.

REFERENTS TABLE

Referent	Description	Example
Millimetre	The millimetre is 1/1000th of a metre. The millimetre is roughly the thickness of a dime. You can also think of 1 mm as the thickness of 10 sheets of paper.
Centimetre	The centimetre is 1/100th of a metre. The centimetre is approximately the thickness of 10 dimes. The centimetre can also be thought of as the width of the fingernail on your smallest finger.
Metre	One metre is the same as the height of a small child. One metre could also be the length of a running stride.	PhotoObjects.net/Thinkstock
Kilometre	One kilometre is equal to 1000 m. You can think of 1 km as the length of 7 football fields. Another way to think about 1 km is the distance you can walk in 10 minutes.
Inch	One inch is an imperial unit. It can be approximated by the width of your thumbnail.	Mark Poprocki/iStockphoto/Thinkstock
Foot	A foot is equal to 12 inches. There’s no better referent for a foot than your own foot! The foot can also be approximated by the length from your wrist to your elbow. Another convenient referent for 1 ft is the length of a 30-cm ruler. This is one of the most accurate estimations of 1 ft.
Yard	One yard is equal to 3 ft or 36 in. One referent for 1 yd is the width of a doorway. You can also think of 1 yd as the length of a normal walking stride.	© Zdenek Krchak /shutterstock
Mile	One mile is equal to 1760 yards or 5280 feet. One mile is the distance of 15 to 20 city blocks. Another way to think of 1 mile is the distance the average person can jog in 10 minutes.	© Laurens Parsons Photography/shutterstock

Every person’s body is different. The length of your foot is most likely different from that of a classmate. In this lab you will determine the measurements of your body parts described in a chart in order to use the body parts as referents for measurement.

The multimedia piece “Body Parts as Referents” demonstrates the way body parts can be used as referents.

Math Lab: Body Referents

Go to the Lesson 1 Assignment that you saved to your course folder. Then complete Math Lab: Body Referents.

Share

You now have an opportunity to share, with other students, the answers to Math Lab: Body Referents questions you have just completed.

To make the most of this sharing opportunity you need to do the following:

• Ensure you have completed all the questions to the best of your ability and place them in a form that is convenient for sharing.
• Post your answers to your class discussion area, or share them using another method as instructed by your teacher.
• Review the results you recorded for steps 1 to 3. Are the results of other students similar to yours? Can you explain reasons for any differences?
• Review the answers provided to questions 1 and 2. If possible, discuss the answers to these questions with the students who posted them. Your discussion might focus on clarifying meaning, or developing a clearer understanding of other students’ strategies and ideas.
• Finally, if necessary, revise your answers for questions 3 and 4 to incorporate what you have learned from the sharing you have done. Save a copy of your revised work in your course folder, along with a record of your discussion.

Mathematics 10C Learn EveryWare © 2010 Alberta Education

Module 1: Measurement and Its Applications

Explore

Glossary Terms

In this course you will often come across math-related words that may be unfamiliar to you. These words will likely be used over and over again, so it is important that you understand the meaning of these words. You will also need to record the words and their meanings so that you can refer to them when necessary.

In this course you will create your own glossary. Use the document titled Glossary Terms to keep a record of the math terms that you come across in Mathematics 10C.

In this lesson the suggested glossary terms you should add to Glossary Terms include the following:

• imperial measurement
• referent
• SI measurement

When you have finished adding definitions to Glossary Terms, you should save the Word document in your course folder. You will refer to it again in other lessons to add new terms.

Read

Read the textbook that you are using for this course.

A non-standard measurement unit is a unit that you would not normally use to report a measurement. Non-standard units are not found on measuring devices. For example, at one time the height of a horse was measured in hands. Personal referents, such as the ones you developed in the Math Lab, are examples of non-standard measurement units. Referents help you to estimate lengths in standard units. For example, you may know that the length of your foot is 25 cm. If you determine that the width of the hallway is as long as eight of your feet, then you can estimate that the hallway is 8 × 25 cm = 200 cm, or 2-m wide.

Try This

Go to the Lesson 1 Assignment that you saved to your course folder, and complete TT 1, TT 2, and TT 3. Once you have completed these questions, make sure to save your updated Lesson 1 Assignment to your course folder.

Share

Post your results to TT 3 on the discussion board, and consider the responses that others have posted there. Use examples from those posts to support or revise your answers to the questions in this Share section.

Self-Check

So far, you have learned about referents and the two systems of measurement. Test yourself now to see how much you remember. Go to Lesson 1 Self-Check.

Mathematics 10C Learn EveryWare © 2010 Alberta Education

Module 1: Measurement and Its Applications

Connect

Project Connection

Think some more about the special place that will be the basis of your Unit 1 Project. Do you see places where you will use metric measures and other places where you will use imperial measures? How will you decide?

Go to the navigation tree and view the Unit 1 Project to review the initial requirements of the project. Please make a few notes and store them in your course folder.

In Lesson 2 you will be ready to represent your place visually using simple shapes such as cylinders, cones, rectangular solids, and spheres. You will also calculate the volumes and surface areas of these solids using both imperial and metric units.

Reflect and Connect

Return to the Lesson 1 Assignment that you saved to your course folder. Then complete RC 1, RC 2, RC 3, and RC 4. Make sure to save your updated Lesson 1 Assignment to your course folder when you have completed the questions. Then submit the Lesson 1 Assignment to your teacher for marks.

Going Beyond

There are other units of measure based on referents. Such units have their origins in agriculture and navigation, for example. Use your favourite Internet search engine to extend your learning by researching the origins of such units of measure as bolt, furlong, league, milestone, and chain. In your search, identify the imperial and metric equivalents of these measures, as well as the referents that are associated with these measures, and explain why these particular referents were chosen.

Mathematics 10C Learn EveryWare © 2010 Alberta Education

Module 1: Measurement and Its Applications

Lesson Summary

In Lesson 1 you investigated the following questions:

• How can referents be used to estimate measurements?
• Why are there two systems of measurement?

In this lesson you used referents to approximate both SI units and imperial units. You examined referents for linear measure including millimetre, centimetre, metre, kilometre, inch, foot, yard, and mile. You used referents to estimate linear measurements, and then you compared those estimates to the actual measurements. You also learned about the origins of the SI and imperial systems of measurement. In your discussions with your peers and with tradespeople in your community, you learned that some trades have adopted the SI, whereas others continue to use the imperial system.

In the next lesson you will use your knowledge of referents to choose appropriate units for measuring, and you will also learn strategies for solving measurement problems.

Mathematics 10C Learn EveryWare © 2010 Alberta Education

Module 1: Measurement and Its Applications

Lesson 2: Using Measurement Instruments

Focus

In Lesson 1 you learned how to estimate SI (metric) and imperial measurements using referents. Estimation is a very important skill that helps you to plan ahead and also to check that your results are reasonable. You will continue to build on this skill in Lesson 2.

Perhaps as part of your unit project, you need to create a cylindrical shape, a very small opening, or a quarter pipe. How can you figure out the required measurements? In what units will you measure your object, and with what instrument?

In this lesson you will expand on your ability to measure while you think about how you can measure large, small, or curved objects.

Outcomes

At the end of this lesson, you will be able to

• justify the choice of units used for determining a measurement in a problem-solving context

• solve problems that involve linear measure, using instruments such as rulers, calipers, or tape measures

• describe and explain a personal strategy used to determine a linear measurement; e.g., the circumference of a bottle, the length of a curve, or the perimeter of the base of an irregular 3-D object

Lesson Questions

• How do you choose the appropriate techniques, instruments, and formulae to describe the dimensions of an object?
• How can you measure the dimensions of objects of irregular shape or size?

Assessment

Your assessment for this lesson includes the following:

• Glossary Terms

• Project Connection

• Lesson 2 Assignment

In this lesson you will complete the Lesson 2 Assignment. Save a copy of the Lesson 2 Assignment to your course folder. You will receive more information about how to complete the assignment later in this lesson.

In this course you may come across Self-Check questions, Try This questions, and other activities that are not marked.

Remember that these questions provide you with the practice and feedback that you need to successfully complete this course. You should respond to all the questions and place the answers in your course folder. Your teacher may wish to view the work that you have stored in your course folder to check on your progress and to see if you require assistance in some way.

Mathematics 10C Learn EveryWare © 2010 Alberta Education

Module 1: Measurement and Its Applications

Launch

This section checks to see if you have the prerequisite knowledge and skills to be able to complete this lesson successfully. Go to Lesson 2: Are You Ready? and answer the questions in this section. If you are experiencing difficulty, you may want to use the information and the multimedia in the Refresher section to clarify concepts before completing these exercises.

Once you have completed these exercises to the best of your ability, submit your responses to your teacher. Your teacher will give you feedback and provide directions as to how to continue with this lesson.

Refresher

Below you will find links to some video clips that may help refresh your memory. These interactive pieces will help you to answer the following questions:

• How do I determine the circumference of a circle?

• How do I calculate the perimeter of a polygon or an enclosed shape with straight sides?

The multimedia lesson titled “Parts of a Circle and Circumference” reviews the parts of a circle and explores the relationships between the diameter, radius, and circumference of a circle. The value of pi is discussed, and the lesson includes a game and practical math problems that require using the formula for the circumference of a circle.

The LearnAlberta resource from the Mathematics Glossary defines the term perimeter. Go to “Perimeter”. It contains an animation to illustrate the definition. Try “Example” at the bottom of the web page.

Materials

• measuring tapes and rulers

• 30-cm long string

• 3-D modelling program (e.g., Google SketchUp)

Mathematics 10C Learn EveryWare © 2010 Alberta Education

Module 1: Measurement and Its Applications

Discover

When you are measuring shapes, sometimes you have to get creative. For example, if you want to measure the circumference of a circle but you only have a tape measure and a piece of string, what can you do?

Try This

Go to the Lesson 2 Assignment that you saved to your course folder. Then complete TT 1. Save your result to your course folder so that you can use your answer for a later comparison.

Watch and Listen

Now that you have tried to measure the diameter of a circle that you have drawn, watch the video clip titled “Measuring a Non-Linear Path.”

Try This

Return to the Lesson 2 Assignment that you saved to your course folder. Then complete TT 2, TT 3, TT 4, TT 5, and TT 6. Return your Lesson 2 Assignment to your course folder when you have completed these questions.

Mathematics 10C Learn EveryWare © 2010 Alberta Education

Module 1: Measurement and Its Applications

Explore

trundle wheel: © Bart Coenders/iStockphoto; Vernier caliper: © Adem Demir/shutterstock

Glossary Terms

In this course you will often come across math-related words that may be unfamiliar to you. In Lesson 1 you started a list of glossary terms and saved the document to your course folder.

Add the following terms to your “Glossary Terms” document:

• circumference

• diameter

• dimensions

• irregular shape

• trundle wheel

• vernier caliper

Then save the updated document in your course folder.

You have the SI (metric) units and the imperial units estimated as body referents, your ruler, an online caliper, and a tape measure, so where do you start? Well, it depends on what you are trying to measure and how accurate a measurement you require.

In construction, sometimes you need a sledge hammer and sometimes you need a ball-peen hammer. The same idea applies in measurement: different instruments provide different scales. You will investigate measurement instruments and their usefulness.

Watch and Listen

Have you ever had to measure a distance that is too long for a tape measure or a metre-stick? Maybe you want to know how far it is to the end of your street. Instead of using a stretched-out tape measure over and over, you can use a trundle wheel. The video titled “Trundle Wheel” will show you how.

A plumber may have to measure the diameter of a PVC (polyvinyl chloride) pipe in order to know if the pipe will be suitable for a repair job. One way to obtain the proper measurement is by using a vernier caliper. Use the multimedia piece titled “Vernier Calipers” to practise using a vernier caliper.

Try This

Return to the Lesson 2 Assignment that you saved to your course folder. Then complete TT 7, TT 8, TT 9, and TT 10. Return your Lesson 2 Assignment to your course folder when you have completed these questions.

Whenever you have been required to measure something in this lesson, you have also been told which measurement instrument to use. However, as you work on projects in your home, you will need to decide for yourself what are the most appropriate instruments and units to use.

When deciding which measurement instrument to use, you will need to consider the following.

Question	Considerations
What measurement instruments do I have available?	You may have limited options. You may have to borrow or purchase the right instrument.
Am I measuring something that is short or long?	You likely won’t want to measure the length of a hallway with a ruler or the thickness of a dime with a trundle wheel.
Is the scale on the measurement instrument big enough?	You can’t measure the diameter of a beach ball using a vernier caliper!
Are the divisions on the scale small enough to give a precise measurement?	How precise you want the measurement to be may determine which instrument you need to use.

You will also need to decide what unit of measurement to use when measuring an object. Some questions you will need to answer include these ones.

Question	Considerations
Do I need the measurement to be in SI units or imperial units?	You may need to report measurements in SI units if previous measurements were also in SI units.
Am I measuring something that is short or long?	If something is long, like the length of a building, you would likely avoid using inches or millimetres as the units of measure.
What measurement instrument am I using?	The instrument that you use will have a certain scale; e.g., inch or cm or m.
Which units will give me a reasonable answer?	The length of a swimming pool could be reported as 50 000 mm, 5000 cm, 50 m, or 0.05 km. The most appropriate example is 50 m.

Self-Check

SC 1. Match the following scenarios with the correct measurement instruments.

measuring the height of your wall	trundle wheel
measuring the diameter of a table-tennis ball	caliper
measuring the width of a sheet of paper	ruler
measuring the distance across the grocery store parking lot	tape measure

SC 2. You have used your caliper to determine that a Canadian dime has a thickness of 1.22 mm and a diameter of 18.03 mm.

1. Determine the circumference of this dime.
2. How many dimes can fit into a container that has the following dimensions?
   
   

Compare your answers.

Mathematics 10C Learn EveryWare © 2010 Alberta Education

Module 1: Measurement and Its Applications

Connect

Project Connection

Think about the special place that is the basis for your Unit 1 Project, and complete the Lesson 2 portion.

Reflect and Connect

Go to the Lesson 2 Assignment that you saved to your course folder. Complete the Reflect and Connect activity.

Going Beyond

Because some things are so small, they are impossible to detect with the naked eye. Take a look at the multimedia piece titled “Cell Size and Scale1” to get a feeling for how small some objects and organisms are.

How would you measure the dimensions of something so small that you cannot see it with the naked eye? For example, how can you measure the width of a plant or animal cell? In this case, you would have to measure the item under a microscope. Use an Internet search engine to discover the steps for measuring the dimensions of a microscopic object or organism.

1Genetic Science Learning Center, University of Utah, http://learn.genetics.utah.edu.

Mathematics 10C Learn EveryWare © 2010 Alberta Education

Module 1: Measurement and Its Applications

Lesson 2 Summary

In Lesson 2 you investigated the following questions:

• How do I choose the appropriate techniques, instruments, and formulae to describe the dimensions of an object?

• How can you measure the dimensions of objects of irregular shape or size?

In this lesson you examined various measuring instruments. You learned that a vernier caliper can measure the dimensions of small objects, such as the diameter of a dime. You learned that a trundle wheel can be used to measure the dimensions of large objects, such as the distance around a room. In order to accurately describe the dimensions of an object, you need to choose an appropriate measuring instrument and also the proper units. Knowing if an object is long or short, if the measurement should be in SI or imperial units, and also what instruments you have available will help you to choose an appropriate tool for measuring an object.

In this lesson you also found ways to measure the dimensions of irregular shapes. One way is to first use a piece of string to completely trace around the shape. Then stretch the string into a straight line and measure with a ruler. Another way is to input a related value, such as the diameter of a circle, into a mathematical formula.

In Lesson 3 you will learn how to convert measurements to different units within the same measurement system. You will also learn how to convert measurements between the SI system and the imperial system.

Mathematics 10C Learn EveryWare © 2010 Alberta Education

Module 1: Measurement and Its Applications

Lesson 2 Summary

In Lesson 2 you investigated the following questions:

• How do I choose the appropriate techniques, instruments, and formulae to describe the dimensions of an object?

• How can you measure the dimensions of objects of irregular shape or size?

In this lesson you examined various measuring instruments. You learned that a vernier caliper can measure the dimensions of small objects, such as the diameter of a dime. You learned that a trundle wheel can be used to measure the dimensions of large objects, such as the distance around a room. In order to accurately describe the dimensions of an object, you need to choose an appropriate measuring instrument and also the proper units. Knowing if an object is long or short, if the measurement should be in SI or imperial units, and also what instruments you have available will help you to choose an appropriate tool for measuring an object.

In this lesson you also found ways to measure the dimensions of irregular shapes. One way is to first use a piece of string to completely trace around the shape. Then stretch the string into a straight line and measure with a ruler. Another way is to input a related value, such as the diameter of a circle, into a mathematical formula.

In Lesson 3 you will learn how to convert measurements to different units within the same measurement system. You will also learn how to convert measurements between the SI system and the imperial system.

Mathematics 10C Learn EveryWare © 2010 Alberta Education

Module 1: Measurement and Its Applications

Launch

This section checks to see if you have the prerequisite knowledge and skills to be able to complete this lesson successfully. Go to Lesson 3: Are You Ready? and answer the questions in this section. If you are experiencing difficulty, you may want to use the information and the multimedia in the Refresher section to clarify concepts before completing these exercises.

Once you have completed these exercises to the best of your ability, submit your responses to your teacher. Your teacher will give you feedback and provide directions as to how to continue with this lesson.

Refresher

Review how to multiply fractions by completing the multimedia piece titled “Multiplying Fractions.”

Try the interactive applet “Equivalent Fractions” to brush up on equivalent fractions. To navigate to the correct lesson, do the following:

• Choose “Fractions and Equivalent Fractions (Home)” in the left-hand column.

• Then choose “Topics” in the upper left-hand corner.

• Select “4. Equivalent Fractions.”

Blueprints are the basic planning documents for building houses. On a blueprint, all of the dimensions of a home are drawn to scale. This means that a specific length in the blueprint equals a specific distance in real life. The ratio is the same for all of the lengths. If you know what the ratio or scale is for a blueprint, then you can convert the measures on the diagram into actual measures.

You can convert measurements within the SI or the imperial system. You can even convert measurements between the two systems. To do so, you need to understand the concepts of ratio and proportion.

Watch and Listen

Go to “Exploring Rate, Ratio and Proportion.” (Make sure you maximize the screen by clicking on the button in the top-right corner of the video.) The website features both a video and an interactive piece. First, view the video. On the left-hand side of the page, select “Exploring Rate, Ratio and Proportion (Video Interactive).” Then try the interactive component to refresh your understanding of ratio and proportion. Click on the “Interactive” button on the right-hand side of the website.

Materials

• results from Math Lab: Body Referents in Lesson 1

• calculator

Mathematics 10C Learn EveryWare © 2010 Alberta Education

Module 1: Measurement and Its Applications

Launch

This section checks to see if you have the prerequisite knowledge and skills to be able to complete this lesson successfully. Go to Lesson 3: Are You Ready? and answer the questions in this section. If you are experiencing difficulty, you may want to use the information and the multimedia in the Refresher section to clarify concepts before completing these exercises.

Once you have completed these exercises to the best of your ability, submit your responses to your teacher. Your teacher will give you feedback and provide directions as to how to continue with this lesson.

Refresher

Review how to multiply fractions by completing the multimedia piece titled “Multiplying Fractions.”

Try the interactive applet “Equivalent Fractions” to brush up on equivalent fractions. To navigate to the correct lesson, do the following:

• Choose “Fractions and Equivalent Fractions (Home)” in the left-hand column.

• Then choose “Topics” in the upper left-hand corner.

• Select “4. Equivalent Fractions.”

Blueprints are the basic planning documents for building houses. On a blueprint, all of the dimensions of a home are drawn to scale. This means that a specific length in the blueprint equals a specific distance in real life. The ratio is the same for all of the lengths. If you know what the ratio or scale is for a blueprint, then you can convert the measures on the diagram into actual measures.

You can convert measurements within the SI or the imperial system. You can even convert measurements between the two systems. To do so, you need to understand the concepts of ratio and proportion.

Watch and Listen

Go to “Exploring Rate, Ratio and Proportion.” (Make sure you maximize the screen by clicking on the button in the top-right corner of the video.) The website features both a video and an interactive piece. First, view the video. On the left-hand side of the page, select “Exploring Rate, Ratio and Proportion (Video Interactive).” Then try the interactive component to refresh your understanding of ratio and proportion. Click on the “Interactive” button on the right-hand side of the website.

Materials

• results from Math Lab: Body Referents in Lesson 1

• calculator

Mathematics 10C Learn EveryWare © 2010 Alberta Education

Module 1: Measurement and Its Applications

Discover

When you convert measurements, you need to know how units of measurement relate to each other. For example, you should know from your previous math studies that 1 cm = 10 mm or 1 ft = 12 in (see questions 5.b. and 5.d. in Are You Ready?). But do you know how many millimetres there are in a kilometre or how many kilometres there are in a mile?

Open up the document SI and Imperial Conversions Sheet. Use the Internet and your textbook to find the correct relationships between the SI units and the imperial units stated on the sheet. Complete the sheet by writing the correct numbers in the blanks.

Save your completed sheet to your folder so that your teacher can check it for accuracy.

Explore

Glossary Terms

Find the “Glossary Terms” document that you saved to your course folder. Add the following words to the document:

• unit analysis

• unit conversion

You may also choose to add other terms to help you understand the math you are studying.

You learned in a previous lesson that both the SI and the imperial systems are used in the trades. In order for two tradespeople who use different measurement systems to understand each other, it may be necessary to use measurement conversions.

You can begin your study of measurement systems by examining conversions within the SI.

Use the multimedia piece titled “International System of Units” to explore unit conversions within the SI. Pay close attention to the following:

• basic units

• prefixes

• conversions

Here are some examples of problems you might encounter when it comes to unit conversions.

Example: Converting Between SI Units

Problem

Convert 450 cm to millimetres and kilometres.

Solution

One way that you can solve a conversion problem is to set up a proportion, and then use cross-multiplication to find the answer.

Since 1 cm = 10 mm,

The variable x will be in units of centimetres.

Then,

There are 4500 mm in 450 cm.

Another way that you can solve a conversion problem is to use the technique of unit analysis. Unit analysis helps you to keep track of the measurement units to ensure that your result will be expressed in the correct units.

Recall that 1 m = 100 cm and 1 km = 1000 m. If you want the final result to be expressed km, you can show your work in the following way:

There is 0.0045 km in 450 cm.

Example: Converting Between Imperial Units

Problem

On a particular Canadian Football League team, the average height of the players is 6 ft 3 in.

First, determine how tall the average football player is in inches only on this team.

Second, figure out, on average, how many of these football players, lined up head to toe, it would take to stretch across a regulation football field with length 110 yards.

Solution

You already have part of the height in inches, so you just need to convert 6 ft into inches, before adding the extra 3 in.

Since 1 ft = 12 in,

The variable x will be in inches.

Then,

There are 72 inches in 6 ft.

Therefore, the average height is 72 + 3 = 75 in.

Next, you can use the unit analysis method to convert 110 yd to inches. Then you can determine how many times 75 in goes into the result.

Recall that 1 ft = 12 in and 3 ft = 1 yd.

So,

The football field is 3960 in.

Now, 3960 in ÷ 75 in = 52.8.

Therefore, it would take about 53 football players, lying head-to-toe, to line the length of a CFL football field.

Mathematics 10C Learn EveryWare © 2010 Alberta Education

Module 1: Measurement and Its Applications

Try This

Practice converting units within a system by completing TT 1 in the Lesson 3 Assignment that you saved to your course folder.

The next step is to learn how to convert measurements between the SI and the imperial system. In the Math Lab: Body Referents in Lesson 1, you established referents for both measurement systems. You can use these referents to make sure each of your calculations is reasonable.

To do so, you would estimate the answer using an appropriate referent; then compare your estimate with your calculation. If the numbers are close, then your calculation is reasonable. If the numbers are different, stop to think about why the numbers are different and where you might have gone wrong in your calculations. Keep this in mind as you read the next section. You will have an opportunity to use referents to estimate in a subsequent Self-Check section.

Read

How do you convert between SI and imperial units? The strategies to do this are the same as those used in the previous conversions. The first thing you have to do is find out the relationship between the units. Once you know this, you can either set up a ratio or prepare to convert using unit analysis.

Read the textbook that you are using for this course to see how these strategies are put into place.

When you are done, you can test yourself in the Self-Check section.

Self-Check

For each of the following, choose the correct answer.

SC 1. A measure of 2 cm is (larger than, smaller than) an inch.

SC 2. A measure of a mile is (larger than, smaller than) a kilometre.

SC 3. A measure of a yard is (larger than, smaller than) a metre.

SC 4. A measure of 25 cm is (larger than, smaller than) a foot.

SC 5. Convert 90 in to yards, demonstrating unit analysis. For this question, please show all your steps to the solution.

Compare your answers.

Try This

Go to the Lesson 3 Assignment that you saved to your course folder. Complete TT 2.

Share

You now have learned several ways of converting measurements from one unit to another. You can convert measurements by setting up a proportion and using cross-multiplication. Alternatively, you could use unit analysis. You also have an idea of how metric units compare to imperial units.

Can you describe why working with proportions is a good strategy for doing unit conversions? Have you developed other strategies of your own? Post your ideas to the discussion board.

Mathematics 10C Learn EveryWare © 2010 Alberta Education

Module 1: Measurement and Its Applications

Connect

Project Connection

Think about the special place that is the basis for your Unit 1 Project, and complete the Lesson 2 portion.

Reflect and Connect

Go to the Lesson 2 Assignment that you saved to your course folder. Complete the Reflect and Connect activity.

Going Beyond

Because some things are so small, they are impossible to detect with the naked eye. Take a look at the multimedia piece titled “Cell Size and Scale1” to get a feeling for how small some objects and organisms are.

How would you measure the dimensions of something so small that you cannot see it with the naked eye? For example, how can you measure the width of a plant or animal cell? In this case, you would have to measure the item under a microscope. Use an Internet search engine to discover the steps for measuring the dimensions of a microscopic object or organism.

1Genetic Science Learning Center, University of Utah, http://learn.genetics.utah.edu.

Mathematics 10C Learn EveryWare © 2010 Alberta Education

Module 1: Measurement and Its Applications

Lesson 3 Summary

In this lesson you investigated the following questions:

• How do the strategies for converting units in the SI compare with those used in the imperial system?

• When can proportions be used to solve problems?

In this lesson you learned how proportional reasoning can be used to convert a measurement within or between the SI and the imperial system.

The SI is based on powers of 10. As a result, conversions in the SI system involve multiplication or division by 10, 100, 1000, and so on.

The imperial system, on the other hand, is not based on powers of any specific value. When using the imperial system, it is more advantageous to set up a proportion. Proportions are best suited to solving problems when an equivalent relationship can be established. Once the proportion is established, you can convert first by cross-multiplying and then by dividing.

You also learned to verify using unit analysis. Unit analysis helps you to convert measurements by cancelling unwanted units. In addition, you solved problems that involve the conversion of units and justified, using mental mathematics, the reasonableness of the solution.

Now continue on to Lesson 4 of this module.

Mathematics 10C Learn EveryWare © 2010 Alberta Education

Module 1: Measurement and Its Applications

 

Lesson 4: Surface Area of 3-D Objects

 

Focus

 

Now that you are familiar with estimating and converting between SI (metric) units and imperial units, you will expand your knowledge to include 3-D objects that include curved surfaces. These objects include cylinders, spheres, cones, prisms, and pyramids. You will use the skills gained in Lesson 2 where you explored measuring curved surfaces.

 

Our surroundings are full of various 3-D objects, many of which can be broken into smaller, basic objects whose surface area and volume can be calculated. You will investigate surface area in this lesson, and by the end of this lesson you will be able to apply and use the surface area calculations needed in your project.

 

Outcomes

 

At the end of this lesson, you will be able to solve, using SI and imperial units, problems that involve the surface area of objects, including:

• right cones
• right cylinders
• prisms
• pyramids
• spheres

Lesson Questions

 

By the end of this lesson you should feel comfortable solving the following questions:

• How is the concept of surface area applied to understanding the design of structures?
• How do you determine the surface area of a 3-D object?

Assessment

• Glossary Terms
• Share: Surface Area Formulas
• Project Connection
• Lesson 4 Assignment

In this lesson you will complete the Lesson 4 Assignment. Save a copy of the Lesson 4 Assignment to your course folder. You will receive more information about how to complete the assignment later in this lesson.

 

In this course you may come across Self-Check questions, Try This questions, and other activities that are not marked.

 

Remember that these questions provide you with the practice and feedback that you need to successfully complete this course. You should respond to all the questions and place those answers in your course folder. Your teacher may wish to view the work that you have stored in your course folder to check on your progress and to see if you require assistance in some way.

Mathematics 10C Learn EveryWare © 2010 Alberta Education

Module 1: Measurement and Its Applications

 

Launch

 

This section checks to see if you have the prerequisite knowledge and skills to be able to complete this lesson successfully. Go to Lesson 4: Are You Ready? and answer the questions in this section. If you are experiencing difficulty, you may want to use the information and the multimedia in the Refresher section to clarify concepts before completing these exercises.

 

Once you have completed these exercises to the best of your ability, submit your responses to your teacher. Your teacher will give you feedback and provide directions as to how to continue with this lesson.

 

Refresher

 

How did you do? Did you remember the difference between a prism and a pyramid? Did you remember how to find the area of basic shapes? Great work, if you remembered! If you did not remember, please carefully read this Refresher section.

 

Let’s take a look at what area means and how to find areas of basic shapes. This information will not only be helpful as you prepare to find the surface area of objects in this lesson but also when you explore the volume of those objects in the next lesson.

 

The Mathematics Glossary defines the term area. Go to “Area” to learn more. It contains an interactive Java applet and Flash animations to illustrate the definition.

 

The mathematics lesson “Finding Area with Unit Squares” explores the measurement of square units. The lesson presents the formulas for the area of a rectangle, parallelogram, and triangle, and it includes math problems that involve the practical application of these formulas.

 

The mathematics lesson “Estimating Area Using a Grid” uses inscribed polygons, circumscribed polygons, and the concept of limits to explain how the area of a circle can be measured. The lesson includes a game and a math problem that demonstrate the practical application of the formula for the area of a circle.

 

Materials

• paper
• prism net and pyramid net
• soup can with label
• scissors
• scotch tape
• a rectangular prism, such as a wooden block, a shoe box, or a cereal box

You will also require these materials to complete Math Lab: The Surface Area of an Orange:

• orange
• string
• ruler

Mathematics 10C Learn EveryWare © 2010 Alberta Education

Module 1: Measurement and Its Applications

 

Discover

 

Math Lab: The Surface Area of an Orange

 

Go to the Lesson 4 Assignment that you saved to your course folder. Complete Math Lab: The Surface Area of an Orange. In order to complete the Math Lab, you will need to go to 1 cm × 1 cm Grid Paper and print a copy.

Mathematics 10C Learn EveryWare © 2010 Alberta Education

Module 1: Measurement and Its Applications

 

Explore

 

Glossary Terms

 

In Lesson 1 you were asked to open a file called Glossary Terms and compile and save a list of math terms that you come across in this course. Go to your course folder and get the Glossary Terms document.

 

In this lesson the suggested glossary terms are the following:

• 3-D object
• apex
• lateral area
• net
• prism
• pyramid
• regular polygon
• right cone
• right cylinder
• sector
• sphere
• surface area

Once you are finished, save Glossary Terms and return the updated document to your course folder.

 

Imagine taking a 3-D object and submerging it in a tub of water. The area of the object that is in contact with the water is called the surface area. You can also think of surface area as the measure of how much exposed area that a solid object has.

 

How would you determine the surface area of an object?

 

One way to do determine the surface area of the object is to peel off the outer layer of the object and then calculate the area of the peel. The peel is called the net. In Math Lab: The Surface Area of an Orange, you obtained a net of the orange (i.e., the orange peel sections) and added the areas of each part of the net (i.e., each peel section) to obtain the surface area of the orange.

 

As you move through this lesson, you will examine the nets of other 3-D objects. By adding together the sections of a net, you can determine the surface area of those objects.

 

Watch and Listen

 

Use “Surface Area of Prisms” to find out how to use the net of a rectangular prism to determine its surface area.

 

Try This: Prisms, Pyramids, and Cylinders

 

Work with a partner to examine the nets of prisms, pyramids, and cylinders. Use the following document titled Surface Area and Volume Investigation to summarize the information you will collect during this investigation. You may want to use the materials outlined in the Launch section as part of the investigation.

 

For each 3-D object, create a net that can be easily folded into the 3-D object. To get started, you can use the following descriptions. However, you are not limited by the descriptions. You are free to use other ways of creating a net for the object.

 

Remember to add the information you discover to your “Surface Area and Volume Investigation” document. Then save a copy of your completed handout in your course folder.

 

Use the cereal box to create the net of a rectangular prism.

• Carefully unfold the cereal box and flatten it.
• Cut off any flaps that are not part of the surface area. (Some flaps will remain because they are a part of the exterior of the box.)

Use an empty Toblerone box to create the net of a triangular prism.

• Carefully unfold the box and flatten it.
• Cut off any flaps that are not part of the surface area.

To create a net of a square pyramid, go to Square Pyramid Net and print a copy.

• Cut out the net.
• Fold along the lines so it forms a three-dimensional pyramid.

Use the soup can with a label to create a net of a cylinder. For an idea of how you can do this, follow this procedure.

• Trace each circular end onto a sheet of paper.
• Cut out each of the ends.
• Remove the label from the soup can.
• Tape the circular ends to the label so that the net can be rolled into a 3-D cylinder.

Try This

 

Go to the Lesson 4 Assignment that you saved to your course folder. Then complete TT 1 to TT 3.

 

 

Share

 

You and your partner have come up with some possible formulas for each of the four 3-D objects and added them to your Surface Area and Volume Investigation Sheet document. You may be certain about some of the formulas, but you may uncertain of other ones. You may be able to use the discussion board to get some ideas from other pairs.

 

Go to Surface Area Exploration and answer the questions.

 

Read

 

Read the textbook that you are using for this course.

 

 

 

Self-Check

 

You have seen in the examples how a formula is applied in finding the surface area of a pyramid. Now use the formulas that you created to find the solutions to the following problems.

 

Note: Make sure the formulas that you created have been checked by your teacher. You should have submitted the formulas to your teacher after you completed Share: Surface Area Formulas.

 

SC 1. Determine the surface area of the following pyramid, to the nearest in2.

 

 

SC 2. Determine the surface area of the cylinder to the nearest cm2.

 

 

SC 3. Determine the surface area of the following prism, rounded to the nearest cm2.

 

 

SC 4. Consider the net shown. What does this net represent?

 

1. A cube with a side 10 cm.
2. A box with a length and width 10 cm and a height 5 cm.
3. A box with a length and width 5 cm and a height 10 cm
4. This net does not represent a box.

Compare your answers.

 

Mathematics 10C Learn EveryWare © 2010 Alberta Education

Module 1 Appendix

 

Suggested Answers

 

Lesson 4

 

SC 1.

 

surface area = area of base + area of side 1 + area of side 2 + area of side 3 + area of side 4

 

surface area = 6 in × 6 in + 0.5(6 in × 4 in) + 0.5(6 in × 4 in) + 0.5(6 in × 4 in) + 0.5(6 in × 4 in)

 

surface area = 36 in2 + 12 in2 + 12 in2 + 12 in2 +12 in2

 

surface area = 84 in2

 

SC 2.This shows an illustration of the net of the cylinder.

 

 

surface area = area of the two circles (top and bottom) + area of the rectangular middle

surface area = 2 × area of circle + rectangular area

 

SC 3.

 

surface area = 2 × area of triangular base + area of side 1 + area of side 2 + area of side 3

surface area = 2 × 0.5(8 cm × 6 cm) + (6 cm × 12 cm) + (8 cm × 12 cm) + (10 cm × 12 cm)

 

 

SC 4. D (It is not a box at all. Make a copy of the net, cut it out, and try to make a box.)

Mathematics 10C Learn EveryWare © 2010 Alberta Education

Module 1: Measurement and Its Applications

 

Try This

 

Retrieve your Lesson 4 Assignment from your course folder, and complete TT 4.

 

Now, see what happens if you take the ratio and rearrange it.

 

 

You can take the formula further by substituting d = 2r.

 

This is the formula for the surface area of a sphere, in terms of the radius.

 

Add this formula to your list of formulas. You should save your list of formulas to your course folder.

 

Read

 

Read the textbook that you are using for this course.

 

 

 

Self-Check

 

SC 5. Find the surface area of the following sphere to the nearest square metre.

 

 

SC 6. Determine the radius of a sphere with a surface area of 64cm2. Report your answer to the nearest centimetre.

 

Compare your answers.

 

Try This

 

From the examples, you have learned that you can determine the surface area of a sphere using the formula A = 4r2. You have also seen how the formula can be used to determine the radius of a sphere, if you know the surface area. Demonstrate what you have learned by going to the Lesson 4 Assignment that you saved to your course folder and completing TT 5.

 

There are other ways of determining the surface area of 3-D objects besides analyzing their nets. Often in mathematics, you can discover properties of unfamiliar objects by examining the properties of familiar ones.

 

For example, the cone is a 3-D object that is shaped much like a pyramid. Like a pyramid, a cone has only one base and the lateral faces of the cone meet at a point called the apex.

 

 

The illustration above shows that as you increase the number of sides on the base, the number of faces also increases. The area of each face also becomes smaller.

 

Eventually, the polygon base approaches the shape of a circle and the lateral area of the pyramid approaches the lateral area of the cone.

 

You can figure out the formula for the surface area of a cone with this idea in mind.

 

 

Consider the formula for the surface area of a rectangular pyramid, as shown in the illustration. The height of the triangular faces, or slant height, is labelled s. The sides of the base are labelled a, b, c, and d.

 

 

In the case of a cone, the perimeter of the base is really the circumference of a circle, so its surface area formula would be

 

 

Read

 

Read the textbook that you are using for this course.

 

 

 

Self-Check

 

Now that you have watched some videos and had a chance to talk with your classmates, it is your turn to try some Self-Check questions to see if you have figured out surface area.

 

 

SC 7. When assembled, the net in the preceding illustration will create a

1. cube
2. cylinder
3. cone
4. prism

 

SC 8. Determine the surface area of the following cone to the nearest square foot.

 

 

Compare your answers.

 

Try This

 

Go to the Lesson 4 Assignment that you saved to your course folder. Complete TT 6. Be sure to pay attention to the information that is given. You may have to use a given value to find another value before you can apply the formula.

 

Share

 

You’ve had a chance to try the Self-Check questions to see if you can calculate the surface area of the various 3-D objects. Post your thoughts about which surface area was the most difficult to calculate and why you think it was the most difficult for you. Then read and respond to postings from two other students by suggesting tips or strategies that you tried when you were determining the surface area of different shaped objects.

Mathematics 10C Learn EveryWare © 2010 Alberta Education

Module 1 Appendix

 

Suggested Answers

 

SC 5.

 

 

SC 6.

 

 

The radius is 4 cm.

Mathematics 10C Learn EveryWare © 2010 Alberta Education

Module 1 Appendix

 

Suggested Answers

 

SC 7. B

 

SC 8. This shows the net of a cone with a radius of 18 ft and a slant length of 35 ft.

 

 

Mathematics 10C Learn EveryWare © 2010 Alberta Education

Module 1: Measurement and Its Applications

 

Connect

 

Project Connection

 

Think about the place that is the focus of your project. What 3-D objects are found in your place? Are there prisms, cones, cylinders, or spheres? If your place is fairly empty of objects, think of the 3-D objects that could occupy your place.

 

Now go to the Unit 1 Project and complete the Lesson 4 portion of your project.

 

Reflect and Connect

 

Go to the Lesson 4 Assignment to complete RC 1, RC 2, and RC 3. Then save your updated Lesson 4 Assignment to your course folder. Then you will submit the Lesson 4 Assignment to your teacher for marks.

 

Going Beyond

 

Did yu know that taller trees generally have more leaves? The water in a tree needs to get from the roots to where photosynthesis happens, which is in the leaves and green parts. If you have a really tall tree, the plant has to force the water against gravity up to its leaves.

 

How is this possible? Consider the study of surface area. If a plant has large leaves, or numerous smaller leaves, then there is more surface area for evaporation to take place. In a plant, this is called transpiration. When transpiration occurs, the water leaving the plant is replaced by water coming up from the ground—and this water has dissolved nutrients in it.

 

For more information, initiate an Internet search using the keyword “transpiration” to see what else you can learn about a plant and the surface area of its leaves.

Mathematics 10C Learn EveryWare © 2010 Alberta Education

Module 1: Measurement and Its Applications

 

Lesson 4 Summary

 

In Lesson 4 you investigated the following questions:

• How is the concept of surface area applied to understanding the design of structures?
• How do you determine the surface area of a 3-D object?

In this lesson you examined the nets of various 3-D objects. These objects included the prism, pyramid, cylinder, sphere, and cone. From the nets, you were able to determine the surface area formulas for each object. These formulas can be used to determine the surface area of any of the 3-D objects investigated, as long as the required information is given.

 

Whether you are figuring out how much paint you need to buy or how much wood is needed to build a shed, solving problems involving area comes in very handy.

 

In the next lesson you will investigate volume, another important concept in design. In Lesson 6 you will apply what you have learned about surface area and volume to solve problems.

Mathematics 10C Learn EveryWare © 2010 Alberta Education

Module 1: Measurement and Its Applications

 

Lesson 5: The Volume of 3-D Objects

 

Focus

 

Have you ever moved from one residence to another? A move can take a great deal of planning, co-ordination, and time.

 

You may need to rent a large truck to contain all of your belongings, or you may have to temporarily store your belongings in a storage facility like the one pictured. The more furniture, clothing, and other possessions that you need to transport, the larger the truck or storage unit that you need for the move.

 

The amount of space becomes an important consideration. This is known as volume. In this lesson you will investigate the concept of volume and learn how to determine the volume of 3-D shapes.

 

Outcomes

 

At the end of this lesson, you will be able to determine the volume of a right cone, a right cylinder, a right prism, a right pyramid, or a sphere using an object or its labelled diagram.

 

Lesson Questions

• How is the concept of volume applied to understanding the design of structures?

• How are the formulas for the volumes of solids related to each other?

Assessment

 

Your assessment for this lesson includes the following:

• Glossary Terms

• Project Connection

• Lesson 5 Assignment

In this lesson you will complete the Lesson 5 Assignment. Save a copy of the Lesson 5 Assignment to your course folder. You will receive more information about how to complete the assignment later in this lesson.

 

In this course you may come across Self-Check questions, Try This questions, and other activities that are not marked.

 

Remember that these questions provide you with the practice and feedback that you need to successfully complete this course. You should respond to all the questions and place those answers in your course folder. Your teacher may wish to view the work that you have stored in your course folder to check on your progress and to see if you require assistance in some way.

Mathematics 10C Learn EveryWare © 2010 Alberta Education

Module 1: Measurement and Its Applications

 

Launch

 

This section checks to see if you have the prerequisite knowledge and skills to be able to complete this lesson successfully. Go to Lesson 5: Are You Ready? and answer the questions in this section. If you are experiencing difficulty, you may want to use the information and the multimedia in the Refresher section to clarify concepts before completing these exercises.

 

Once you have completed these exercises to the best of your ability, submit your responses to your teacher. Your teacher will give you feedback and provide directions as to how to continue with this lesson.

 

Refresher

 

This resource from the Mathematics Glossary defines the term volume. Go to “Volume.” You will find an animation to illustrate the definition.

 

In the interactive mathematics lesson titled “Volume and Displacement,” you can calculate the volume of rectangular prisms. You will also learn that the volume of an irregular object can be found by measuring the amount of water the object displaces.

 

Discover

 

Math Lab: Comparing the Volume of a Cylinder and a Sphere

 

Go to the Lesson 5 Assignment that you saved to your course folder. Then complete Math Lab: Comparing the Volume of a Cylinder and a Sphere.

 

Explore

 

© skvoor/shutterstock

 

Glossary Terms

 

Retrieve your personal glossary handout titled “Glossary Terms” from your course folder, and update it with the following terms:

• area

• base

• volume

Return your updated “Glossary Terms” to the course folder.

 

 

Recall that area is the amount of square units occupying an enclosed shape or two-dimensional space. To find the area of a shape, you need to multiply two dimensions of the shape together. For example,

 

1 cm × 1 cm = 1 cm2

 

On the other hand, volume measures the amount of cubic units occupying a three-dimensional space. To find the volume of an object, you need to multiply three dimensions of the object together. For example,

 

1 cm × 1 cm × 1 cm = 1 cm3

 

Prisms

 

Watch and Listen

 

Watch the multimedia presentation titled “Volume of a Prism.” See if you can remember the three steps that are needed to find the volume of a prism.

 

Generally speaking, the formula for the volume of a prism is the following:

 

volume = area of base × height

 

Self-Check

 

Use the formula for the volume of prisms to solve the following problems.

 

SC 1. Find the volume of the triangular prism shown here.

 

 

SC 2. Find the volume of a rectangular prism that has a base measuring 6 in by 4 in and a height of 8 in.

 

 

Compare your answers.

Mathematics 10C Learn EveryWare © 2010 Alberta Education

Module 1 Appendix

 

Suggested Answers

 

Lesson 5

 

SC 1. The area for the base triangle is the following:

 

 

The next calculation uses the base and the height to give the volume of the prism.

 

 

SC 2. First, find the area of the rectangle representing the base of this rectangular prism.

 

 

Multiply the base by the height in order to find the volume.

 

Mathematics 10C Learn EveryWare © 2010 Alberta Education

Module 1: Measurement and Its Applications

 

Cylinders

 

A cylinder is a prism with a circular base. Although you might not call a cylinder a circular prism, that’s exactly what a cylinder is.

 

Using the formula for the volume of a prism, what could be the formula for the volume of a cylinder?

 

Save your answer to your course folder; then check in your textbook to see if your answer is correct. If it isn’t correct, what parts of your formula were correct? What parts of the formula need to be changed?

 

Review Example 1 and Example 2.

 

Example 1

 

Determine the volume of a soup can with a diameter of 3.5 in and a height of 4.5 in. Show your solution to the nearest tenth of a square inch.

 

Solution

 

 

Example 2

 

The volume of a cylinder is 200 cm3. Determine the radius of the cylinder to the nearest hundredth of a centimetre if the height of the cylinder is 8 cm.

 

Solution

 

 

Try This

 

Return to the Lesson 5 Assignment that you saved to your course folder. Now complete TT 1.

 

Cones

 

Math Lab: Volume of a Cone

 

Go to the Lesson 5 Assignment and complete Math Lab: Volume of a Cone.

 

Read

 

Go to your textbook to find the formulas for a cylinder and a cone and note how they are similar and how they are different. What fraction of the volume of a cylinder is the volume of a cone with the same height and radius?

 

Compare your experimental result in the cone investigation with the formulas you have just read in your textbook. Is your experimental result confirmed? Do your results support the finding that the volume of a cone is the volume of a cylinder with the same height and radius?

 

If your results do not support the ratio, give some reasons why you think this might be the case. Incorporate these comments into your Math Lab. Your teacher will not penalize you if you did not obtain the theoretical ratio. However, he or she will be looking at how you intrepret and explain the data you did obtain.

 

Read

 

Read the textbook that you are using for this course.

 

Rectangular Pyramids

 

You have learned that the volume of a cone is the volume of a cylinder with the same radius and height. This ratio is the same for pyramids. In other words, the volume of a pyramid is the volume of the prism with the same height and base area.

Mathematics 10C Learn EveryWare © 2010 Alberta Education

Module 1: Measurement and Its Applications

 

Read

 

Take a look at the following example, which compares the relationship between the volume of a right rectangular pyramid and the volume of a right rectangular prism. Read the textbook that you are using for this course.

 

Self-Check

 

SC 3. Find the volume of the square pyramid shown in the diagram.

 

 

Compare your answers.

 

Spheres

 

Retrieve your analysis from Math Lab: Comparing the Volume of a Cylinder and a Sphere that you saved to your course folder.

 

The height of the can is equal to twice the radius of the ball.

 

To determine the formula for the volume of a sphere, you can do what you did for the cone. First, what was the ratio of the volume of the tennis ball compared to the volume of the juice container? Did you observe that the volume of the tennis ball was about that of the container?

 

This means that the volume of the sphere would be .

 

This formula is correct, but there’s a way to simplify the formula by finding another way to express the can’s height.

 

Think about the fact that the can and the ball have the same radius and the same height. Does it make sense to you that the height of the can would be equal to twice the radius of the ball?

 

So you could write the volume of a sphere as or, more simply, .

 

Read

 

Read the textbook that you are using for this course.

 

Try This

 

Go to the Lesson 5 Assignment that you saved to your course folder. Complete TT 2 to practise applying the formula for the volume of the sphere. You may have to do something more than apply the formula for the context-based questions.

 

The following table summarizes the different types of 3-D objects you have examined in this lesson. Included are the formulas that have been developed for these objects.

 

Cylinder	Cone	Rectangular Pyramid	Sphere

 

Self-Check

 

SC 4. Sheila is excavating the basement for her house on a small lot in town. The dimensions of the basement excavation need to be 40-ft long by 30-ft wide and 9-ft deep. The excavated soil is placed in a circular area beside the excavation. The radius of this area is 20 ft. As more soil is added, the soil pile forms the shape of a cone. The highest the excavator can lift the soil is 24 ft.

 

Backhoe © ownway/shutterstock

 

Will the excavator be able to put all the soil from this excavation into this one cone-shaped pile? (Show your calculations to the nearest whole number.)

 

Compare your answers.

Mathematics 10C Learn EveryWare © 2010 Alberta Education

Module 1 Appendix

 

Suggested Answers

 

Lesson 5

 

SC 3. First, find the area of the base of this square pyramid. That is the area of a square.

 

 

Now you need to use the base and the height to find the volume. Do you remember that the volume of a pyramid is the volume of a rectangular solid?

 

Mathematics 10C Learn EveryWare © 2010 Alberta Education

Module 1 Appendix

 

Suggested Answers

 

Lesson 5

 

SC 4. This is a two-step problem.

 

Step 1

 

First, you have to figure out the volume of soil that needs to be removed by excavation. That will be the volume of a rectangular solid.

 

First, you find the area of the base. The house is to be 40 ft by 30 ft—.

 

 

The height of the basement excavation needs to be 9 ft.

 

So, the volume of soil will be as follows:

 

 

Step 2

 

From Step 1, you calculated the volume of soil to be removed. What you need to do now is to find the height of the cone that is produced as the backhoe removes the soil.

 

You need to calculate the area of the circular base to the cone.

 

 

To find the height of the cone-shaped soil pile, you need to remember that the volume of a cone is the volume of a cylinder. You can input the known volume into the formula:

 

 

Since the backhoe’s upper limit to lift the soil is 24 ft, the backhoe will not be able to place all of the excavated soil on the cone-shaped pile.

Mathematics 10C Learn EveryWare © 2010 Alberta Education

Module 1: Measurement and Its Applications

 

Connect

 

Project Connection

 

In this lesson you are ready to determine the volume of some of the shapes that make up your place.

 

You should go to the Unit 1 Project and complete the Lesson 5 portion of the project.

 

Reflect and Connect

 

Go to the Lesson 5 Assignment that you saved to your course folder and complete RC 1, RC 2, RC 3, and RC 4. Save your responses to your course folder.

 

Going Beyond

 

Thanks to advances in technology, the world is truly changing. You can have a different perspective on the concept of place by using Google Earth.

 

Initiate an Internet search using the keywords “real world math” and “volume of solids.” Using these search terms, you should find a website titled Real World Math. There’s an exercise titled “Volume of Solids” that shows you how to determine the surface area and volume of some of the world’s more famous places.

Mathematics 10C Learn EveryWare © 2010 Alberta Education

Module 1: Measurement and Its Applications

 

Lesson 5 Summary

 

In this lesson you examined the following questions:

• How is the concept of volume applied to understanding the design of structures?

• How are the formulas for the volumes of solids related to each other?

In this lesson you looked in your surroundings for various three-dimensional shapes including right cones, right cylinders, prisms, pyramids, and spheres. You discovered the relationships between the volumes of related 3-D objects through hands-on labs and by using interactive activities.

 

You developed strategies for determining the volume of a right cone, a right cylinder, a right prism, a right pyramid, or a sphere using an object or its labelled diagram.

 

In the next lesson you will use what you have learned about surface area and volume to solve problems in real-world situations.

Mathematics 10C Learn EveryWare © 2010 Alberta Education

Module 1: Measurement and Its Applications

 

Lesson 6: Surface Area and Volume Problem Solving

 

Focus

 

For the Unit 1 Project, you have been describing your favourite place. That place might be your bedroom at home, a cabin at the lake, or an ancient castle from your imagination.

 

How would you describe your favourite place to someone who has never been there? You could definitely use photos and drawings; but you also need a way to describe the size or dimensions of your place.

 

In the unit project you will need to represent your place with basic three-dimensional objects. You will then calculate the volume of those 3-D objects.

 

As you have been working on your project and spending time around your home and school, maybe you have noticed some basic geometric shapes in your environment. A single structure or space, such as the pictured castle, often includes a variety of geometrical shapes—prisms, pyramids, cones, cylinders, and spheres, for example.

 

If you have a structure that is made of more than one shape, how could you calculate its surface area and volume? What strategies can you develop to investigate the surface area and volume of complex shapes?

 

Outcomes

 

At the end of this lesson, you will be able to do the following:

• Solve problems that involve surface area and the volume of 3-D objects.
  
• Determine an unknown dimension of a right cone, cylinder, prism, pyramid, or sphere, given the object’s surface area or volume and the remaining dimensions.
  
• Solve problems that involve surface area or volume, given a diagram of the composite 3-D object.
  
• Describe the relationship between the volumes of right cones and cylinders with the same height and base and right pyramids and prisms with the same height and base.

Lesson Questions

• Why is visualization important to the study of the surface area and volume of 3-D objects?
• How does changing the dimensions of an object affect its surface area and volume?

Assessment

 

Your assessment for this lesson includes the following:

• Glossary Terms
• Share (Your contribution to the discussion board.)
• Project Connection: The Woodshed
• Lesson 6 Assignment

In this lesson you will complete the Lesson 6 Assignment. Save a copy of the Lesson 6 Assignment to your course folder. You will receive more information about how to complete the assignment later in this lesson.

 

In this course you may come across Self-Check questions, Try This questions, and other activities that are not marked.

 

Remember that these questions provide you with the practice and feedback that you need to successfully complete this course. You should respond to all the questions and place those answers in your course folder. Your teacher may wish to view the work that you have stored in your course folder to check on your progress and to see if you require assistance in some way.

Mathematics 10C Learn EveryWare © 2010 Alberta Education

Module 1: Measurement and Its Applications

 

Launch

 

This section checks to see if you have the prerequisite knowledge and skills to be able to complete this lesson successfully. Go to Lesson 6: Are You Ready? and answer the questions in this section. If you are experiencing difficulty, you may want to use the information and multimedia in the Refresher section to clarify concepts before completing these exercises.

 

Once you have completed these exercises to the best of your ability, submit your responses to your teacher. Your teacher will give you feedback and provide directions as to how to continue with this lesson.

 

Refresher

 

The interactive applet titled “Exploring Surface Area, Volume, and Nets—Use It” can help you review many aspects of surface area and volume. You will find the interactive piece on the right-hand side of the website.

 

Select the “Use It” tab at the bottom of the activity to review the nets of all of the 3-D objects that were previously explored. The activity tests your ability to visualize the net of a given 3-D object.

 

Select the “Explore It” tab to review the surface area and volume formulas for all of the 3-D objects previously introduced. This applet also enables you to do a side-by-side comparison of two formulas. For example, you can use this feature to compare

• the surface area formula of a sphere and the volume formula for a sphere
• the volume formulas of a cylinder and a cone

You can also access a video that demonstrates how math is used in designing large inflatable shapes. On the left-hand side of the website, choose “Exploring Surface Area and Volume (Video Interactive)” to view the video.

Mathematics 10C Learn EveryWare © 2010 Alberta Education

Module 1: Measurement and Its Applications

 

Discover

 

How does adjusting the dimensions of a 3-D object affect its surface area and volume? In the following Math Lab, you will investigate this question.

 

Math Lab: Surface Area and Volume Analysis

 

Go to the Lesson 6 Assignment that you saved to your course folder. Complete Math Lab: Surface Area and Volume Analysis.

Mathematics 10C Learn EveryWare © 2010 Alberta Education

Module 1: Measurement and Its Applications

 

Explore

 

As you look around your surroundings, you may find objects that resemble the 3-D objects studied in the previous two lessons. A soup can, a box, and a ball are examples of cylinders, prisms, and spheres, respectively. A pylon used to alert motorists of traffic obstructions resembles a cone, and some games use pyramid-shaped dice.

 

While there are many examples of prisms, pyramids, cylinders, spheres, and cones, you may notice that many other objects are actually composites of two or more of these basic 3-D objects.

 

The image of the industrial propane tank is an example of a composite figure. Can you tell what 3-D objects are used in the tank’s design?

 

Glossary Terms

 

Throughout Module 1, you have been adding and saving math terms to “Glossary Terms” in your course folder.

 

In this lesson the suggested terms for your glossary are

• composite figure
• hemispherical
• surface area to volume ratio

Return your updated “Glossary Terms” to your course folder.

 

The steps below will help you to solve a surface area or volume problem:

 

Step 1: Decide which 3-D object or objects can be used to model the problem.

 

Step 2: Draw a rough sketch of this 3-D object, and label its dimensions.

 

Step 3: Decide which formulas you will use. You may need to select more than one formula in the case of composite figures. You might also find that you will need to modify the formula to fit the problem.

 

Step 4: Substitute given values into your formulas to solve the problem.

 

Read

 

Work through the following textbook examples that show how problems involving composite figures are solved. In the solutions, pay attention to

• the importance of an accurate drawing
• any preliminary calculations that need to be made
• the sequence of steps

Read the textbook that you are using for this course.

 

 

 

Self-Check

 

SC 1. A grain storage bin has a diameter of 4.8 m. The height of the straight side wall is 10 m. The cone top has an additional height of 1.5 m.

1. What is the total volume of this bin?

2. Suppose you want to paint the outside of one of the grain storage bins. If the slant height of the conical roof is 2.83 m, then what total surface area needs to be painted?
   

SC 2. Mr. Vanilla charges 0.5 cents/cm3 for his ice cream. How much would you pay for one spherical-shaped scoop of ice cream if a scoop of ice cream has a radius of 3.5 cm?

1. $8.99
2. $1.50
3. 26 cents
4. 90 cents

SC 3. The right cylinder and right cone shown have the same radius and volume. The cylinder has a height of 12 in. What is h, the height of the cone?

 

1. 18 in
2. 24 in
3. 36 in
4. 42 in

SC 4. A glass containing water is in the shape of a right circular cylinder with a radius of 3 cm. The height of the water in the glass is 10 cm.

1. What is the volume of the water in the glass? Be sure to include units of measure in your answer. Show or explain how you obtained your answer.
   
2. Five spherical marbles of equal size are dropped into the glass. The water in the glass rises to a height of 11 cm. What is the increase in the volume of the glass contents? Be sure to include units of measure in your answer. Show or explain how you obtained your answer.
   
3. What is the volume of one marble? Be sure to include units of measure in your answer. Show or explain how you obtained your answer.
   
4. What is the radius of one marble? Be sure to include units of measure in your answer. Show or explain how you obtained your answer.

Compare your answers.

 

Try This

 

How did you do on the Self-Check questions? If you did well, go to the Lesson 6 Assignment and answer TT 1 to practise the steps in solving surface area and volume problems. If you need some help with the Self-Check questions, take the time to review the solutions or ask your teacher for help.

 

Share

 

Take a look at the results from Math Lab: Surface Area and Volume Analysis that you saved to your course folder.

 

It seems reasonable to assume that as the dimensions of an object are doubled, the surface area and volume are also doubled. Yet the results of the investigation proved otherwise. In fact, you may have found that as the dimensions are doubled, the surface area is quadrupled, or increases by a factor of four. At the same time, the volume undergoes an increase by a factor of eight! Where do these numbers come from?

 

Review your lab results and see if you can see any patterns in the ratios.

 

You may want to extend the investigation by quadrupling each dimension; then recalculate the surface area or volume.

 

Develop an explanation for how to predict the increase in surface area or volume.

 

Use your explanation to predict how the surface area and volume will change when the dimensions of an object are increased by a factor of 7.

 

Post your explanation on the discussion board.

 

Respond to postings from two other students whose explanations are different from yours.

• Test their explanations by checking if they get the same surface area and volume as you did for an increase by a factor of 7.
  
• Offer suggestions for improvement, or provide alternative strategies.

Once you have received feedback on your own explanation, make any necessary revisions.

 

Save your revised explanation in your folder for your teacher to check.

 

 

Share

 

Use the Internet to investigate these myths, and find a counterexample of each myth. A counterexample is an example that proves that a statement is false.

 

Share your counterexamples on the discussion board. Post a counterexample for each of the myths in the Caution. Be sure to include the reason or reasons why a counter example shows that the myth is false. Copy four other counterexamples from the postings on the discussion board. Save your findings to your course folder.

Mathematics 10C Learn EveryWare © 2010 Alberta Education

Module 1 Appendix

 

Suggested Answers

 

Lesson 6

 

SC 1.

1. This is a two-step problem. You need to calculate the volume of the lower cylinder and then the upper cone part of the grain bin.
   
   For the cylinder, you will calculate the area of the base first. That will be the area of a circle.
   
   

    

   
   
   For the volume of the cylinder part, you multiply the base by the height.
   
   
   
   
   Now you need to find the volume of the cone part. Do you remember that the volume of a cone is the volume of a cylinder? You need the base of the cone, and it is the same circle area as the base of the cylinder. So you are ready to find the volume.
   
   
   
   
   Before you write the final statement, does the answer make sense? Look at the picture. Is the volume of the cone that you calculated definitely less than the volume of the cylinder?
   
   So the final step is to add the two volumes:
   
   181.0 m3 + 9.05 m3 = 190.0 m3
   
2. You need to use the surface area formulas for a cylinder and a cone.
   
   
   
   You will not need to include the areas of the bases of these objects, because they will not be painted. The bottom of the grain bin is sitting on the ground, so it will not need to be painted. The top part of the grain bin, where the base of the cone meets the top face of the cylinder, will also not need to be painted. So the modified formula will be:
   
   
   
   Therefore, the surface area is as follows:
   
   

SC 2. D

 

SC 3. C

 

SC 4.

1. 
   
2. New volume:
   
   
   
   
   Old volume:
   
   
   
   
   The difference is 310.9 cm3 – 282.6 cm3 = 28.3 cm3.
   
3. The volume change was all due to the marbles.
   
   Since there are five marbles and they are the same size, you divide the volume by the number 5.
   
   So, the volume of one marble is 5.7 cm3.
   
4. Since you know the volume of the marble and you know it is a sphere, you can use the formula for the volume of a sphere.
   
   
   

Mathematics 10C Learn EveryWare © 2010 Alberta Education

Module 1: Measurement and Its Applications

 

Connect

 

Project Connection: The Woodshed

 

Up to this point in the project, you have all been working with your own place. For this lesson, you will do a common problem. You will consider the dimensions of a woodshed.

 

Of course, it is still a collaborative approach—you can still work together with a partner.

 

Go to the Unit 1 Project and complete the Lesson 6 portion of the project.

 

Reflect and Connect

 

Go to the Lesson 6 Assignment and complete RC 1, RC 2, and RC 3.

 

Going Beyond

 

The ratio between an organism’s surface area and volume has an enormous impact on its biology. For example, a cell that is shaped like a sphere has a low surface area to volume (SA : V) ratio.

 

 

Cells that are extended (e.g., cylinder) have much more membrane per unit of cytoplasm, which means these cells have more surface area for each bit of goo inside of them. Extending the outer surface of a cell into fingers, like an amoeba, or indentations, like the red blood cells shown above, can greatly increase the total surface area.

 

Interestingly, scientists are identifying the ratio between the surface area and volume as a crucial factor in work with nanotechnology. Take a look at the nanobot in the second visual of red blood cells. What do you suppose its function is?

 

To find out more, perform an Internet search using the following keywords: “nanotechnology,” “surface area,” “volume,” “nanobot,” and “red blood cells.”

Mathematics 10C Learn EveryWare © 2010 Alberta Education

Module 1: Measurement and Its Applications

 

Lesson 6 Summary

 

In this lesson you examined the following questions:

• Why is visualization important to the study of the surface area and volume of 3-D objects?
• How does changing the dimensions of an object affect its surface area and volume?

In this lesson you solved problems involving the surface area and volume of 3-D objects. You learned how to approach problems involving composite figures. You learned that it is important to visualize each problem before putting pencil to paper.

 

You also learned that doubling the dimensions of a 3-D object, such as a prism or a cone, actually increases its surface area by a factor of 4 and its volume by a factor of 8.

 

You also looked at common myths regarding surface area and volume. Through research, you were able to find explanations that dispelled these myths.

 

In Lesson 7 you will continue your study of measurement by looking at the measures of right triangles. This branch of mathematics is known as trigonometry.

Mathematics 10C Learn EveryWare © 2010 Alberta Education

Module 1: Measurement and Its Applications

 

Lesson 7: Introduction to Trigonometry

 

Focus

 

treehouse: © Ewan Chesser/shutterstock; surveyor: © Joe Gough/shutterstock

 

As you have worked through the previous lessons, you have been adding shapes to create your own personal place. You have been asked to provide measurements and calculations for your shapes. Using a program like Google SketchUp makes it pretty easy to show measurements, since you can simply click on the tape measure and then drag the measure alongside your shapes. If only life were that simple!

 

Imagine building a tree house. There would be some measurements that are difficult to take, like the height of the tree—it’s not too easy (or safe) to climb a tree with a tape measure in your hands!

Luckily, there is a great mathematical concept called trigonometry. In the construction industry, surveying allows people to use trigonometry to calculate unknown lengths, so direct measurements are not required.

 

In this lesson you will explore trigonometry and answer questions such as, what does trigonometry mean and how does it work?

 

When you have completed this lesson, you will move on to Lesson 8 where you will apply trigonometric concepts to real-world problems.

 

Outcomes

 

At the end of this lesson, you will be able to use the primary trigonometry ratios of sine, cosine, and tangent to solve right triangle problems.

 

Lesson Questions

 

By the end of this lesson, you should feel comfortable solving the following questions:

• In what situations can the concepts of trigonometry be used to solve problems?

• How are the sine, cosine, and tangent ratios used to determine information about a right triangle?

Assessment

 

Your assessment for this lesson includes the following:

• Glossary Terms

• Share (your contribution to the discussion board)

• Project Connection

• Lesson 7 Assignment

In this lesson you will complete the Lesson 7 Assignment. Save a copy of the Lesson 7 Assignment to your course folder. You will receive more information about how to complete the assignment later in this lesson.

 

In this course you may come across Self-Check questions, Try This questions, and other activities that are not marked.

 

Remember that these questions provide you with the practice and feedback that you need to successfully complete this course. You should respond to all the questions and place those answers in your course folder. Your teacher may wish to view the work that you have stored in your course folder to check on your progress and to see if you require assistance in some way.

Mathematics 10C Learn EveryWare © 2010 Alberta Education

Module 1: Measurement and Its Applications

 

Lesson 7: Introduction to Trigonometry

 

Focus

 

treehouse: © Ewan Chesser/shutterstock; surveyor: © Joe Gough/shutterstock

 

As you have worked through the previous lessons, you have been adding shapes to create your own personal place. You have been asked to provide measurements and calculations for your shapes. Using a program like Google SketchUp makes it pretty easy to show measurements, since you can simply click on the tape measure and then drag the measure alongside your shapes. If only life were that simple!

 

Imagine building a tree house. There would be some measurements that are difficult to take, like the height of the tree—it’s not too easy (or safe) to climb a tree with a tape measure in your hands!

Luckily, there is a great mathematical concept called trigonometry. In the construction industry, surveying allows people to use trigonometry to calculate unknown lengths, so direct measurements are not required.

 

In this lesson you will explore trigonometry and answer questions such as, what does trigonometry mean and how does it work?

 

When you have completed this lesson, you will move on to Lesson 8 where you will apply trigonometric concepts to real-world problems.

 

Outcomes

 

At the end of this lesson, you will be able to use the primary trigonometry ratios of sine, cosine, and tangent to solve right triangle problems.

 

Lesson Questions

 

By the end of this lesson, you should feel comfortable solving the following questions:

• In what situations can the concepts of trigonometry be used to solve problems?

• How are the sine, cosine, and tangent ratios used to determine information about a right triangle?

Assessment

 

Your assessment for this lesson includes the following:

• Glossary Terms

• Share (your contribution to the discussion board)

• Project Connection

• Lesson 7 Assignment

In this lesson you will complete the Lesson 7 Assignment. Save a copy of the Lesson 7 Assignment to your course folder. You will receive more information about how to complete the assignment later in this lesson.

 

In this course you may come across Self-Check questions, Try This questions, and other activities that are not marked.

 

Remember that these questions provide you with the practice and feedback that you need to successfully complete this course. You should respond to all the questions and place those answers in your course folder. Your teacher may wish to view the work that you have stored in your course folder to check on your progress and to see if you require assistance in some way.

Mathematics 10C Learn EveryWare © 2010 Alberta Education

Module 1: Measurement and Its Applications

 

Launch

 

This section checks to see if you have the prerequisite knowledge and skills to be able to complete this lesson successfully. Go to Lesson 7: Are You Ready? and answer the questions in this section. If you are experiencing difficulty, you may want to use the information and the multimedia in the Refresher section to clarify concepts before completing these exercises.

 

Once you have completed these exercises to the best of your ability, submit your responses to your teacher. Your teacher will give you feedback and provide directions as to how to continue with this lesson.

 

Refresher

 

Are you able to tell if a triangle is a right triangle? Test your ability to do so by going to the multimedia piece titled “Right Triangle.” On the bottom of the website is an interactive definition of a right triangle.

 

Use the multimedia item titled “Pythagorean Theorem” to see another visual definition.

 

The multimedia piece titled “Exploring the Pythagorean Theorem” allows you to change the side lengths of a right triangle to see the effect on the length of the hypotenuse. At the website, choose the “Interactive” button near the middle of the page. You will be presented with a visual explanation of the Pythagorean theorem.

Materials

• ruler

• protractor

• calculator

• graph paper

In addition, specific materials are required for the Going Beyond section where you have the opportunity to build a simple sundial. The materials you will require to build the sundial will depend on the design that you choose. You will find out what materials you need by doing an Internet search.

Mathematics 10C Learn EveryWare © 2010 Alberta Education

Module 1: Measurement and Its Applications

 

Discover

 

Watch and Listen

 

Have you ever wondered how pilots calculate a safe angle of descent when they are landing 747 planes? Go to “Exploring Trigonometry” and view the video, which talks about how trigonometry is used at airports. You will find the video on the right-hand side of the website.

 

Try This

 

Go to the Lesson 7 Assignment that you saved to your Math 10C course folder and complete TT 1 to TT 4.

 

Share

 

Post the results of TT 1 to TT 4 to the discussion board. Specifically, include the

• lengths of your triangle

• values of your three ratios

Examine the results that were posted by two or more other students.

• Compare the size of your triangle (the lengths of a, b, and c) with the other ones that have been posted on the discussion board. What do you notice?

• Compare the ratios , , and with others in your class. What do you notice?

• Can you make a generalization from your results?

Try This

 

Go to the Lesson 7 Assignment that you saved to your course folder. Complete TT 5 and TT 6.

Mathematics 10C Learn EveryWare © 2010 Alberta Education

Module 1: Measurement and Its Applications

 

Explore

 

Glossary Terms

 

Retrieve your handout titled “Glossary Terms” from your course folder. The glossary terms for this lesson that you should add to your handout are

• adjacent side

• cosine ratio

• hypotenuse

• opposite side

• proportional

• Pythagorean theorem

• reference angle

• right triangle

• sine ratio

• solving a triangle

• tangent ratio

The three sides of a right triangle are known by three different names. You need to be able to identify the opposite side, the adjacent side, and the hypotenuse.

 

The term hypotenuse may already be familiar to you. Have a look at the diagrams where the orange arrows each point to the hypotenuse. Can you see a pattern? If you were given a right triangle, how could you identify the hypotenuse?

 

 

The great thing about the hypotenuse is that it never changes locations—it is always directly across from the right angle. Did you discover that pattern when you viewed the diagrams?

 

The two other sides that you need to identify are the opposite side and the adjacent side. The location of these other two sides will change, depending on which angle is being considered in the question.

 

 

 

 

For example, if you had the triangle to the right, you can see that the angle we need to find is labelled x. So, in this case, x is the reference angle. Can you tell which side is opposite from angle x?

 

 

 

If you said the bottom, you are correct!

 

See how the bottom side is opposite (or directly across) from angle x? In this case, we would label the bottom as the opposite side.

 

 

 

 

What about in the case to the left? Can you identify the opposite side if you are using angle y as the reference angle?

 

In this case, the left side is opposite from the reference angle.

 

Now that you can label the hypotenuse and the opposite side of a right triangle, all you need to do is label the remaining side as the adjacent side. You can also think of the adjacent side as the side between the reference angle and the 90° angle. Please review the examples below.

 

 

 

In the Discover section, you calculated three ratios which compare the lengths of the sides of a right triangle. Now that you know the names of the sides, you can apply them to the following triangle.

 

The ratios that you calculated in the Discover section were , , and .

 

These ratios have names.

• The sine ratio is the ratio of the length of the side opposite the reference angle to the length of the hypotenuse.

• The cosine ratio is the ratio of the length of the side adjacent to the reference angle to the length of the hypotenuse.

• The tangent ratio is the ratio of the length of the side opposite the reference angle to the length of the side adjacent to the reference angle.

The definitions can be summarized by the following:

 

 

Notice that the ratios are based on the location of the reference angle.

 

A great way to remember the trigonometric ratios is by using the following:

 

SOH CAH TOA

 

If you chant SOH CAH TOA many times, the chant will stay in your head. (It sounds like “soak-a-toe-ah!”) Here’s what it means:

 

 

Learn and Listen

 

The interactive piece titled “Shape and Space” allows you to practise identifying the sides of a triangle by dragging and dropping the opposite, adjacent, and hypotenuse sides into their correct locations. You can find the interactive piece on the right-hand side of the website.

 

Now that you know how to label the sides of a right triangle and know what the different ratios are, you are ready to see how these ratios can help you solve basic trigonometry problems.

 

Mathematics 10C Learn EveryWare © 2010 Alberta Education

Module 1: Measurement and Its Applications

 

Read

 

Here is a chance for you to see how the sine, cosine, and tangent ratios are evaluated and used to determine angles. Pay careful attention to the difference between a trigonometric ratio and the angle that it can be used to find.

 

Read the textbook that you are using for this course.

 

Example

 

© City of Edmonton. Used under Creative Common - Attribution-Noncommercial-Share Alike 2.5 Canada Licence

 

The remainder of this lesson is broken into two main parts:

• How to find an angle using the sine, cosine, or tangent ratio.

• How to find a length using the sine, cosine, or tangent ratio.

Let’s look at some detailed examples of the six possible question types. Go to Finding Angles and Lengths Using Sine, Cosine, and Tangent Ratios. Under the Finding Angles box, choose “Sine,” “Cosine,” and “Tangent” to see examples of each. Then under the Finding Lengths box, choose “Sine,” “Cosine,” and “Tangent” to see examples of each.

 

Self-Check

 

Now that you have some experience with trigonometry, see how well you can answer the following questions.

 

SC 1. In the following triangle, what side is adjacent to angle MLN?

 

 

SC 2. Calculate the cosine of angle MLN.

 

SC 3. Calculate the angle measurement of angle MLN, to the nearest degree.

 

SC 4. Find the length of x in the following diagram.

 

 

Compare your answers.

 

You have learned how to determine the measure of an unknown length or unknown angle in a triangle. You can use sine, cosine, or tangent ratios to set up an equation to solve for the unknown measure. With these techniques, you can determine the measures of all of the unknown lengths and unknown angles in a triangle. This is known as solving a triangle.

 

Of course, you need to know some information before you can solve a triangle. The minimum information you need to know is one of the following:

• The measure of one length and one acute angle.

• The measure of two lengths.

Read

 

Find out how to solve a triangle when you are given different measures of a triangle. Read the textbook that you are using for this course.

 

Pay attention to the numbers that are used in each calculation. Every time a new measure is calculated, one more number can be used to determine the next unknown measure.

 

What reasons are there for using the original given measures? What reasons are there for using calculated measures?

 

Try This

 

Use what you have learned in this lesson to solve some triangles. Remember to use a calculated value to solve for another value only if you are sure of the correctness of the calculated value. If the calculated value is incorrect, then the subsequent values will also be incorrect.

 

Go to the Lesson 7 Assignment that you saved to your course folder. Then complete TT 7.

Mathematics 10C Learn EveryWare © 2010 Alberta Education

Module 1 Appendix

 

Suggested Answers

 

Lesson 7

 

SC 1. Side ML is adjacent to angle MLN.

 

SC 2.

 

 

SC 3.

 

 

SC 4.

 

Mathematics 10C Learn EveryWare © 2010 Alberta Education

Module 1: Measurement and Its Applications

 

Connect

 

Project Connection

 

At this time, you should work on your Unit 1 Project. Go to the Unit 1 Project, and complete the Lesson 7 portion of the project.

 

Reflect and Connect

 

Go to the Lesson 7 Assignment that you saved to your course folder. Now complete RC 1 and RC 2.

 

Going Beyond

 

There are many uses of trigonometry. A sundial is a device that measures time based on the position of the Sun. A sundial is designed in such a way that the Sun casts a shadow from a sharp, straight edge onto a flat surface marked with lines that indicate the hours of the day.

 

In theory, a stick stuck in the ground could form the basis of a sundial. In reality, it’s not that simple. Earth’s axis is tilted, which means that the apparent movement of the Sun through the sky changes every day. If this isn’t accounted for, a sundial that tells perfect time today will be slightly wrong next week and very wrong next month.

 

See if you can find out how to build a sundial by doing a search on the Internet. Use the search terms “how to build a sundial.” Then look through a few of the websites to find a simple set of instructions.

 

(After you construct a real sundial, perhaps you will be interested in adding a virtual sundial into your unit project!)

 

In order to construct a sundial accurately, you will need to find the direction “due north” and you will also need to know the latitude of the town or city where you reside.

 

Note: Grande Prairie has a latitude of approximately 55°, Calgary is 51°, and Edmonton is 53°. You can go online to look up the latitude of your community. The multimedia piece Latitude Table shows a table that includes the latitude of many communities in Alberta.

 

Latitude & Longitude Table for Alberta

Place	Latitude Deg (N)	Latitude Min	Longitude Deg (W)	Longitude Min
Acadia Valley	51	12	110	04
Alix	52	27	113	13
Alsands	57	30	111	25
Barrhead	54	08	114	24

Mathematics 10C Learn EveryWare © 2010 Alberta Education

Module 1: Measurement and Its Applications

 

Lesson 7 Summary

 

In this lesson you examined the following key questions:

• In what situations can the concepts of trigonometry be used to solve problems?

• How are the sine, cosine, and tangent ratios used to determine information about a right triangle?

You learned in this lesson that trigonometry can be used to solve right triangles. A triangle is solved when the given information about the triangle is used to determine the unknown lengths and angles. A triangle is solvable when the following minimum information is known:

• the measures of two sides

• the measure of one side and the measure of an acute angle

In all other situations where the minimum information is not known, a triangle cannot be solved.

 

Trigonometry is used to solve problems in design, policing, and air traffic control, to name a few instances. In Lesson 8 you will be applying trigonometric techniques you have learned to problems arising from everyday contexts.

Mathematics 10C Learn EveryWare © 2010 Alberta Education

Module 1: Measurement and Its Applications

 

Lesson 8: Solving Right Triangle Problems

 

Focus

 

You may enjoy listening to music. The music may have been recorded digitally—a process that uses trigonometry. Perhaps the music is in MP3 format using data compression, which uses an understanding of the human ear’s ability to distinguish between sounds, and this format also requires trigonometry.

 

You may travel over a bridge today. That bridge was built using an understanding of forces acting at different angles. You will notice that bridges involve many triangles—trigonometry was used when designing the lengths and strengths of those triangles.

 

You will often see a surveyor at work in your community. Trigonometry helps the surveyor determine sides of a triangle that are difficult to access. An angle that cannot be reached may be measured from places that can be reached.

 

You saw an example of this principle in Lesson 7 when you read about building a tree house. The heights of trees and tall buildings can be determined by knowing the distance to the base of the tree or building. Similarly, distances across rivers or busy roads can be determined by using angle and length measurements from one side of the river or the road.

 

In the last lesson you used the trigonometric ratios and the Pythagorean theorem to find sides and angles in right triangles. In this lesson you will apply those skills to solve real world problems.

 

Outcomes

 

At the end of this lesson, you will be able to

• solve a problem that involves right triangles
  
• solve a problem that involves one or more right triangles by applying the primary trigonometric ratios or the Pythagorean theorem
  
• solve a problem that involves indirect measurement using the trigonometric ratios, the Pythagorean theorem, and measurement instruments such as a clinometer or a metre-stick

Lesson Questions

• How do you approach problems whose solutions are based on trigonometry and its principles?

• How is trigonometry used to determine heights and distances that cannot be directly measured?

Assessment

 

Your assessment for this lesson includes the following:

• Glossary Terms
• Project Connection: The Staircase
• Lesson 8 Assignment

In this lesson you will complete the Lesson 1 Assignment. Save a copy of the Lesson 8 Assignment to your course folder. You will receive more information about how to complete the assignment later in this lesson.

 

In this course you may come across Self-Check questions, Try This questions, and other activities that are not marked.

 

Remember that these questions provide you with the practice and feedback that you need to successfully complete this course. You should respond to all the questions and place those answers in your course folder. Your teacher may wish to view the work that you have stored in your course folder to check on your progress and to see if you require assistance in some way.

Mathematics 10C Learn EveryWare © 2010 Alberta Education

Module 1: Measurement and Its Applications

 

Launch

 

This section checks to see if you have the prerequisite knowledge and skills to be able to complete this lesson successfully. Go to Lesson 8: Are You Ready? and answer the questions in this section. If you are experiencing difficulty, you may want to use the information and the multimedia in the Refresher section to clarify concepts before completing these exercises.

 

Once you have completed these exercises to the best of your ability, submit your responses to your teacher. Your teacher will give you feedback and provide directions as to how to continue with this lesson.

 

Refresher

 

In this lesson you will be using the trigonometric ratios and the Pythagorean theorem to solve problems you encounter in your everyday life. These problems will all involve right angle triangles. Do you remember what makes a triangle a right angle triangle?

 

“Space and Shape: Trigonometry” provides a good review before you begin problem solving. On the right-hand side of the website, click on “Interactive.” When you are finished, choose “Video.”

 

The video at “Space and Shape: Similarity and Congruence” shows the difference between similar and congruent shapes. For example, are stop signs congruent or similar to other stop signs? View the video to find out. On the right-hand side of the website, choose “Video.”

 

Once you have viewed the video, choose “Interactive.” Viewing the interactive information is a fantastic way to figure out if you are truly comfortable with congruent and similar shapes. The site allows you to move and reshape triangles to determine if they are congruent, similar, or neither when compared to each other.

 

Please review your work from Lesson 7 for the definition of the trigonometric ratios. You will also need to solve triangles in which two pieces of information are given and you have to find either a length or an angle. Remember the six types of solving triangles from Lesson 7.

 

Materials

• tape measure
• clear plastic ruler
• clear plastic protractor
• clear tape
• cotton string
• small weight (e.g., a metal washer)

You will also need the following items to complete Math Lab: Clinometer.

• 1 plastic protractor
• 1 soda straw
• 1 paper clip
• 1 toothpick or another paper clip
• 1 6–8 in (15–20 cm) length of thread
• 1 roll of fishing line or relatively inflexible string (The length depends upon use.)
• 2 pieces of tape (e.g., transparent, masking, duct)

Discover

 

Math Lab: Creating a Clinometer

 

Go to the Lesson 8 Assignment that you saved to your course folder. Then complete Math Lab: Creating a Clinometer. It is best to complete this Math Lab with a partner.

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Module 1: Measurement and Its Applications

 

Explore

 

Glossary Terms

 

Find the “Glossary Terms” handout that you have saved to your course folder. Add these terms to your glossary:

• angle of depression
• angle of elevation
• clinometer
• congruent triangles
• similar triangles

Return your updated “Glossary Terms” to your course folder.

 

Sometimes you will see either the expression angle of elevation or angle of depression when you are solving a word problem involving angles.

 

The angle of elevation is the angle you measured with your clinometer. It is the angle from the horizontal to the line of sight as shown in the diagram.

 

The angle of elevation is useful to know in problems where the observer is looking upwards at something.

 

The angle of depression, on the other hand, refers to those instances when the observer is looking downwards at something. Like the angle of elevation, the angle of depression is the angle between the horizontal and the line of sight. The difference is that the line of sight, in this case, is directed downwards.

 

Try This

 

Go to the Lesson 8 Assignment that you saved to your course folder. Complete TT 1 to TT 5.

Mathematics 10C Learn EveryWare © 2010 Alberta Education

Module 1: Measurement and Its Applications

 

Share

 

Post your conclusions on the discussion board. Check the responses posted by other students to see if other students agree with you. If not, discuss the differences with the other students and try to reach a resolution.

 

Explain how you reached a resolution. Save your work to your course folder.

 

Trigonometry problems vary in complexity. Some problems involve only one right triangle and one or two steps. Other problems may involve two triangles and may require several calculations. You can approach these problems by following these guidelines:

• Sketch the scenario: Set up the problem with a drawing.

• Find the right triangle(s): You will need to identify the triangle or triangles in your sketch. Label the given lengths and/or angles. Also, label the length or the angle that you need to find.

• Write a trigonometric equation: Use SOH CAH TOA and the information given in the problem to select an equation—sine, cosine, or tangent—to solve.

• Solve the equation: Rearrange the equation, and solve for the unknown length or angle.

Example

 

Retrieve your data from Math Lab: Creating a Clinometer that you saved to your course folder. This example will take you through the steps that were just outlined to help you determine the height of the structure you measured.

 

One of the required measurements from the Math Lab is the angle of elevation from the horizontal to the top of the structure that you measured. Read the Caution bubble to see how to make sure you read the information correctly.

 

 

For this example, assume that the measurements taken were the following:

• The distance between you and the base of the tree is 10 m.
• The string on the clinometer passes through 140°.
• The distance from the ground to the eye level of the person taking the measurement is 1.5 m.

Follow the steps to solve for the height of the tree.

 

Sketch the Scenario

 

The angle of elevation is 140° − 90° or 50°.

 

A sketch might look like the following.

 

Find the Right Triangle(s)

 

The right triangle is coloured red. The distance between you and the tree is 10 m. The angle of elevation is 50°.

 

The length that needs to be found is y.

 

Write a Trigonometric Equation

 

Using 50° as the reference angle, the known side is the adjacent side, and the required side is the opposite side.

 

The ratio that contains both of these sides is tangent (TOA). So the equation is

 

 

Solve the Equation

 

Multiplying both sides of the equation by 10 gives the following:

 

 

Don’t forget to add the height to eye-level!

 

The height of the tree is 11.9 m + 1.5 m = 13.4 m.

 

Try This

 

Go to the Lesson 8 Assignment that you saved to your course folder. Complete TT 6.

 

Read

 

You now have an idea of the approach you can take with trigonometry problems. Read the following examples to see how you can apply this same approach given different circumstances. Read the textbook that you are using for this course.

 

 

Mathematics 10C Learn EveryWare © 2010 Alberta Education

Module 1: Measurement and Its Applications

 

Try This

 

Go to the Lesson 8 Assignment that you saved to your course folder. Complete TT 7.

 

Self-Check

 

Choose the correct answer.

 

SC 1. In a triangle, the sum of all angles equals ________.

1. 180°
2. 90°
3. 360°
4. There is not enough information given to answer this question.

SC 2. A kite has a string 150-m long. If the string makes an angle of 41° with the ground, find the height of the kite above the ground.

1. 98 m
2. 113 m
3. 150 m
4. 168 m

 

 

SC 3. A 10.0-m ladder leans against a vertical wall. The angle between the ladder and the ground is 73°. Find the distance from the foot of the ladder to the wall.

1. 2.9 m
2. 7.6 m
3. 9.6 m
4. 15.9 m

SC 4. A person in a hot air balloon is 150 m above the ground. An object is 285 m away from the balloon on a line directly beneath the balloon. What is the angle of depression of the person’s line of sight to the object on the ground rounded to the nearest degree?

1. 28°
2. 32°
3. 62°
4. 58°

Compare your answers.

 

Some trigonometry problems need to be modelled with two triangles. For this type of problem, you will need to determine the measure of either a length or an angle from each triangle. You may need to calculate the measure of a length or angle on one triangle before you can determine the length on another triangle.

 

Example

 

In the diagram, the given distances are as follows:

 

AB = 8 ft

BC = 11 ft

CD = 9 ft

 

What is the distance from point A to point D?

 

Solution

 

The length AD is labelled y in triangle ACD. There is not enough information to solve triangle ACD. You would need to know the length of another side in triangle ACD in order to solve the triangle. However, there is enough information to solve triangle ABC.

 

Notice that triangles ACD and ABC have length AC in common. You can solve triangle ABC for length AC. Once you know the length of AC, you will have enough information to solve for all unknown measures of triangle ACD, including length AD.

 

In triangle ABC,

 

 

In triangle ACD, you now know the length of the side opposite 62°, and you are solving for the hypotenuse y.

 

 

Read

 

Read the textbook that you are using for this course. Read through the following additional example to see how a trigonometry problem can be modelled by two right triangles. Pay attention to the extra step at the end. Why is this step necessary?

 

 

 

Try This

 

Remember as you model and solve problems involving two triangles that you can only solve a triangle if you have enough information. If you don’t have enough information in one triangle, then you need to solve the other triangle to get the values you need for the first triangle.

 

Go to the Lesson 8 Assignment and complete TT 8.

Mathematics 10C Learn EveryWare © 2010 Alberta Education

Module 1 Appendix

 

Suggested Answers

 

Lesson 8

 

SC 1. A

 

SC 2. A

 

The following is the detailed solution:

 

 

SC 3. A

 

The following is the detailed solution:

 

 

SC 4. A

 

The following is the detailed solution:

 

 

The angle of depression is 90° − 62° = 28°.

Mathematics 10C Learn EveryWare © 2010 Alberta Education

Module 1: Measurement and Its Applications

 

Connect

 

You have seen that trigonometry can be used to model real-world problems. In fact, the concepts of trigonometry are applicable in the design of new homes, in the reconstruction of traffic accidents, in the use of navigation, and in other areas.

 

In this section of Lesson 8 you will have an opportunity to connect what you have learned to a number of everyday contexts.

 

Project Connection: The Staircase

 

It is time for you to do another common problem. Of course, it is still a collaborative approach—you can still work together with a partner. Go to the Unit 1 Project and complete the Lesson 8 portion of the project.

 

Reflect and Connect

 

Go to the Lesson 8 Assignment and complete the Reflect and Connect activity.

 

Going Beyond

 

handheld GPS: © Olaru Radian-Alexandru/shutterstock; GPS illustration: © Scott Maxwell/LuMaxArt/shutterstock

 

Did you know that trigonometry is the main math tool behind the technology of GPS (global positioning system)? Your car or your phone may have built-in GPS, so you never need to be lost! So how does GPS work?

 

GPS measures the satellite signals closest to your location by receiving signals from a minimum of three satellites, which is referred to as triangulation.

 

GPS locks onto a position and uses trigonometry to calculate its position. This position is measured in latitude and longitude. From that point, as long as the satellite stays locked onto your location, then GPS can provide the speed, the distance, and, most important of all, a map to your destination. Initiate an Internet search for GPS for a much more detailed explanation.

 

The trigonometry used in GPS systems does go beyond what you have learned in this course. An Internet search for GPS and spherical trigonometry can help you learn more.

Mathematics 10C Learn EveryWare © 2010 Alberta Education

Module 1: Measurement and Its Applications

 

Lesson 8 Summary

 

In this lesson you investigated the following questions:

• How do you approach problems whose solutions are based on trigonometry and its principles?

• How is trigonometry used to determine heights and distances that cannot be directly measured?

At the beginning of this lesson, you learned the steps for approaching problems based on trigonometry. You learned that the key first step is to sketch a diagram of the problem.

 

Then it is so important to correctly identify the right triangle in the problem and label the opposite, adjacent, and hypotenuse sides. You can then set up a sine, cosine, or tangent equation. This equation can then be solved to yield a solution.

 

In this lesson you also built a simple clinometer which helps you to measure heights which are otherwise difficult to measure directly. The height of a tree or a flagpole or your house can be measured from the safety of the ground. There is no need to climb the structure and drop down a measuring tape!

 

You have now completed the final lesson of Module 1. If you have not already done so, you should go to the Unit 1 Project and make sure that all of your activities have been completed and submitted to your teacher for marks.

Mathematics 10C Learn EveryWare © 2010 Alberta Education

Unit 1 Conclusion

 

 

As you look closely at your home and your community, you will see many objects of interesting shape and size. In some instances, an object’s form is designed to be visually inspiring. For example, the design of buildings in modern architecture often emphasizes creativity and beauty.

 

On the other hand, even in architecture, the buildings need to be designed to fulfill their functions. A library will be designed in a different way than a hardware store. Likewise, a bank will incorporate security into its design while a shopping mall will emphasize accessibility.

 

On a smaller scale, a drinking glass is designed in the shape of a cylinder for many practical reasons. One reason is that a cylindrical glass is easier to hold. Such a glass is also easily stored with other glasses of the same shape.

 

Similarly, a bookcase has rectangular openings so that more books can be stored and retrieved than if they were placed on a bookcase with circular openings.

 

In this unit you examined the 3-D objects around you. You began by reviewing the SI and imperial measurement systems. You used referents to obtain approximate measurements of various objects. You also learned how to convert measurements between the SI and imperial systems.

 

In your project work, you used a 3-D rendering program to create your own special place. This place contained objects whose dimensions you measured.

 

In this unit you built upon your knowledge of volume and surface area by extending those concepts to cylinders, cones, pyramids, and spheres. You investigated these properties by conducting math labs and using interactive multimedia.

 

Later, you calculated the volume and surface area of the 3-D objects found in your special place that you described in the Unit 1 Project.

 

At the end of this unit, you learned about trigonometry and how its concepts are used to determine measurements that are not easily obtained directly. For example, you learned to use a clinometer to determine the angle of elevation to the top of a tall structure. You then used the tangent ratio to find the height of the structure without actually measuring it.

 

At the end of the unit, you had the opportunity to apply trigonometry concepts to word problems. You learned that sketching the scenario and choosing the right ratio were important steps in finding the solutions to these problems.

 

The following table summarizes the learning outcomes and corresponding learning activities in this unit.

 

Specific Outcome	Major Learning Activities That Address Specific Outcome
Solve problems that involve linear measurement using SI and imperial units of measure estimation strategies measurement strategies	Lesson 1 Math Lab: Body Referents Lesson 2 Video: Measuring a Non-Linear Path
Apply proportional reasoning to problems that involve conversions between SI and imperial units of measure.	Lesson 3 SI and Imperial Conversions Sheet
Solve problems, using SI and imperial units, that involve the surface area and volume of 3-D objects, including: right cones right cylinders right prisms right pyramids spheres	Lesson 4 Math Lab: The Surface Area of an Orange Interactive Math Lesson: Surface Area of Prisms Try This: Surface Area and Volume Investigation Project Connection Lesson 5 Math Lab: Comparing the Volume of a Cylinder and Sphere Watch and Listen: Volume of a Prism Math Lab: Volume of a Cone Project Connection Lesson 6 Math Lab: Surface Area and Volume Analysis Project Connection: The Woodshed
Develop and apply the primary trigonometric ratios (sine, cosine, tangent) to solve problems that involve right triangles.	Lesson 7 Try This: TT 1, TT 2, TT 3, TT 4 Try This: TT 5 and TT 6 Project Connection Lesson 8 Math Lab: Creating a Clinometer Try This: TT 6 Project Connection: The Staircase

Mathematics 10C Learn EveryWare © 2010 Alberta Education

Unit 1 Project: Place

 

Do you have a special place—somewhere you enjoy and feel really good about when you are there? What kind of a place is it? How would you describe the place to someone else? Is it a place created by people or a place in nature? “Place” has different meanings across different cultures.

 

 

Watch and Listen

 

Watch the video titled Interview with an Elder, Part 1 that describes how the concept of measurement is perceived in another culture. Pay careful attention to the description of how measurements were taken by the Nehiyawak (Cree) people.

 

The video titled Interview with an Elder, Part 2 is one example of a meaningful place—a Nehiyawak (Cree) teepee used to create a home in nature that is friendly to Mother Earth. Listen to Elder Bill Sewepagaham describe the critical steps in constructing a teepee.

 

Complete each part of the Unit 1 Project. You will find instructions in each lesson under the Project Connection heading about when you should work on your project.

 

Save your responses and your work to the course folder. Have a discussion with your teacher about how your project will be submitted. Your teacher may want you to submit each component as it is completed, or your teacher may want to wait until the entire project is completed prior to submission.

 

You should also contact your teacher about scoring criteria for the project.

 

Introduction

 

Working with a group or on your own, your task is to create an interpretation of what place means to you. Place may be in the form of a geographical place, a home, a sculpture, something as fanciful as a castle, or anything else that comes to mind.

 

The whole project will be more fun if this place you describe has real meaning to you. Feel free to use a drawing, a model, a sculpture, or a photo to visually represent your place. This is the first step in your project.

 

Lesson 1

 

Think some more about the special place that will be the basis of your Unit 1 Project. Do you see places where you will use metric measures and other places where you will use imperial measures? How will you decide?

 

Lesson 2

 

What shapes are found in your special place? Use a 3-D modelling program to add a shape to your project creation, and make sure you record its dimensions. Save your 3-D model to the project folder in your course folder.

 

Lesson 3

 

There is no Project Connection for Lesson 3.

 

Lesson 4

 

Search the Internet to locate a drawing application. You might want to try an application called Google SketchUp. You may wish to view the tutorials that explain how to use this application.

 

Use the application to draw at least one of each of the following:

• right cone
• right cylinder
• rectangular prism
• pyramid
• sphere

Save your 3-D objects.

 

Select at least three of these objects that occupy or could occupy your place. Add these to your Unit 1 Project. For each of these objects, include the following:

• a drawing
• the measurements, such as length, radius, slant height, and so on
• detailed calculations of the surface area

Lesson 5

1. Choose two or three of the basic shapes—i.e., prism, pyramid, sphere, cylinder, or cone—that you have studied from your Google SketchUp or model.
2. Give reasons why you chose the shapes you did for your place.
3. Decide what dimensions make up the base and the height.
4. Take the measurements you need using Google SketchUp or physical measurements from your model.
5. Show the calculation of the volume along with an explanation of your strategies. (This information will look like the solutions to the Self-Check questions that you have completed in Lesson 5.)

Lesson 6: The Woodshed

 

Let’s apply what you have been learning to a very practical situation. Ian wants to build a shed to store the wood he will need to fuel his wood-burning stove over the winter.

 

He has room for a woodshed 8-ft deep and 16-ft long. He wants to stack the wood 6-ft high inside the shed. Ian has figured out that this wood will last him all winter.

1. He wants the shed to be covered on three sides and the roof with steel cladding that costs $3.99 per square foot. The front opening is to be 8-ft high and the back wall 6-ft high, so the shed will have a roof slanted to help snow slide off. The roof will have a 2-ft overhang in both the front and the back and no overhang on the sides. Calculate how much cladding steel he will need and its cost.

2. A cord of wood is 4 ft × 4 ft × 8 ft. How many cords does his woodshed hold?

3. On average, one piece of firewood is 2-ft long, with a diameter of 8 in. How many pieces of wood will Ian be able to fit into his new shed?

Save your work in your folder.

 

Lesson 7

 

You have been introduced to six different question types in this lesson:

• finding a missing side length using each of the three trigonometry ratios
• finding a missing angle using each of the three trigonometry ratios

 

For this part of your project, please create three questions using three of the question types. These questions should relate to your place.

 

As well as submitting your questions and detailed solutions to your teacher, you will also post the three questions to your class discussion board. Note: Do not post your solutions; only post the questions.

 

Then you will find solutions for the three questions that another student has posted. First, you will place your solutions to the other student’s questions in your course folder. Next, you will provide your solutions to the other student.

 

When a classmate answers your three questions, let the student know if he or she correctly answered your questions. If the student made a mistake, let this person know where the error occurred and provide the correct detailed answer.

 

You will be assessed on the following:

• the three questions you post, along with the diagram(s) that go with your questions
• your answers to a classmate’s questions
• the feedback you provided to a classmate who has answered your questions (includes accuracy and a detailed solution, as well as politeness)

For example, your project in Google SketchUp may resemble the image below. You could then submit a question and solution, such as the following:

 

Question 1: If the teepee is 14-ft tall and the teepee makes an angle of 60° with the ground, what is the radius of the teepee?

 

 

Solution to question 1: The opposite side is 14 ft. The variable x is the adjacent side.

 

So when you think of SOH CAH TOA, you can see that you will use TOA or tan.

 

 

So, the teepee has a radius of 8 ft.

 

Lesson 8: The Staircase

 

Anya has just purchased her first house. It is beautifully finished on the outside.

 

The inside does need some work, the house is not very large, and there are some space challenges.

 

Anya has found one problem. The staircase to her upper bedroom is very steep—almost like a slanted ladder. She wants a staircase that is easy to climb. Unfortunately, Anya is unsure of how to design this staircase in the space she has. Can you help?

 

 

Your first task is to figure out a new location for the staircase and to sketch a diagram. Since Anya wants a staircase that is easy to climb, you need to look up standard rise and run. You could research using the Internet by entering the keywords “staircase standard rise run.”

 

You will not likely have a diagram detailed enough to answer Anya’s questions about your design. Your new knowledge of trigonometry and the Pythagorean theorem will help you answer these questions.

1. What is the length of the new staircase?
2. What is the height and width of each new stair?
3. What is the difference between the angle made by the new staircase and the old staircase?

Mathematics 10C Learn EveryWare © 2010 Alberta Education