- Email: [email protected]

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MathWorks 10 Teacher Resource

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Teacher Resource

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Pacific Educational Press Vancouver, Canada

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MathWorks

Copyright Pacific Educational Press 2010 ISBN 978-1-895766-53-0 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise without the prior written permission of the publisher.

PA re EV CIF pr A IC od L E uc UA DU tio TI C n ON AT of pa CO ION rt P AL or Y— P al N R lo O E f t T SS he FO . co R ALL nt D R en IS IG ts TR H in IB TS an UT R y IO ES fo N E rm R VE is D st ric tly pr oh ib ite

Writers Katharine Borgen, Vancouver School Board and University of British Columbia Catherine Edwards, Pacific Educational Press Sheeva Harrysingh-Klassen, J.H. Bruns Collegiate, Winnipeg Mark Healy, West Vancouver Secondary School, West Vancouver Craig Yuill, Prince of Wales Secondary School, Vancouver

Consultants Katharine Borgen, PhD, Vancouver School Board and University of British Columbia John Willinsky, PhD, Public Knowledge Project Jordie Yow, Mathematics Reviewer Design, Illustration, and Layout Warren Clark Laraine Coates Sharlene Eugenio Five Seventeen

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Cover photo courtesy Colin Pickell Editing Christa Bedwin Theresa Best Diana Breti Laraine Coates Barbara Dominik Catherine Edwards Leah Giesbrecht Deborah Hutton Barbara Kuhne

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Printed and bound in Canada

Developed for the Western and Northern Canadian Protocol Apprenticeship and Workplace Mathematics Program.

Contents Introduction How to Use the Student Resource How to Use the Teacher Resource

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Currency Exchange

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Introduction Curriculum and Chapter Overview The Mathematical Ideas Planning Chapter 1 Chapter Project: The Party Planner Proportional Reasoning Puzzle It Out: Magic Proportions Unit Price Setting a Price On Sale! Currency Exchange Rates Reflect on Your Learning: Unit Pricing and Currency Sample Chapter Test Sample Chapter Test: Solutions Blackline Masters Alternative Chapter Project: Food Planning at a Wilderness Lodge Alternative Chapter Project: Blackline Masters

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Introduction Curriculum and Chapter Overview The Mathematical Ideas Planning Chapter 2 Chapter Project: A Payroll Plan for a Summer Business 2.1 Wages and Salaries

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Contents continued 99 104 105 110 116 119 123

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2.2 Alternative Ways to Earn Money Puzzle It Out: A Weird Will 2.3 Additional Earnings 2.4 Deductions and Net Pay Reflect on Your Learning: Earning an Income Sample Chapter Test Sample Chapter Test: Solutions Blackline Masters Alternative Chapter Project: Outdoor Rock Concert Alternative Chapter Project: Blackline Masters

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3 Length, Area, and

Volume

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3.1 3.2 3.3 3.4

Introduction Curriculum and Chapter Overview The Mathematical Ideas Planning Chapter 3 Chapter Project: Design an Ice-Fishing Shelter Systems of Measurement Converting Measurements Surface Area Volume Puzzle It Out: The Decanting Puzzle Sample Chapter Test Sample Chapter Test: Solutions Blackline Masters Alternative Chapter Project: Design and Build a Play Structure Alternative Chapter Project: Blackline Masters

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Volume

Introduction Curriculum and Chapter Overview

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Contents continued

5 Angles and Parallel Lines Introduction Curriculum and Chapter Overview The Mathematical Ideas Planning Chapter 5 Chapter Project: Create a Perspective Drawing Measuring, Drawing, and Estimating Angles Angle Bisectors and Perpendicular Lines Non-Parallel Lines and Transversals Parallel Lines and Transversals Puzzle It Out: The Impossible Staircase Sample Chapter Test Sample Chapter Test: Solutions Blackline Masters Alternative Chapter Project: Create a Model of a Tower Alternative Chapter Project: Blackline Masters

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5.1 5.2 5.3 5.4

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4.1 4.2 4.3

The Mathematical Ideas Planning Chapter 4 Chapter Project: Culinary Competition Temperature Conversions Mass in the Imperial System Mass in the Système International Puzzle It Out: The Counterfeit Coin Making Conversions Sample Chapter Test Sample Chapter Test: Solutions Blackline Masters Alternative Chapter Project: Measuring Snowload Alternative Chapter Project: Blackline Masters

6 Similarity of Figures Introduction Curriculum and Chapter Overview The Mathematical Ideas

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Trigonometry of

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Planning Chapter 6 Chapter Project: Design a Community Games Room Similar Polygons Determining if Two Polygons Are Similar Drawing Similar Polygons Puzzle It Out: Rationing Chocolate Bars Similar Triangles Sample Chapter Test Sample Chapter Test: Solutions Blackline Masters Alternative Chapter Project: Build a Miniature Town Alternative Chapter Project: Blackline Masters

Introduction Curriculum and Chapter Overview The Mathematical Ideas Planning Chapter 7 Chapter Project: Design a Staircase for a Home 7.1 The Pythagorean Theorem 7.2 The Sine Ratio 7.3 The Cosine Ratio 7.4 The Tangent Ratio 7.5 Finding Angles and Solving Right Triangles Puzzle It Out: 16 Squares Sample Chapter Test Sample Chapter Test: Solutions Blackline Masters Alternative Chapter Project: Draw a Scale Map of the School Grounds Alternative Chapter Project: Blackline Masters

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introduction

Conceptual Framework

In keeping with the philosophy of the Common Curriculum Framework for Grades 10-12 Mathematics, the student textbook and teacher resource incorporate the following aspects of learning mathematics:

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communication connections mental mathematics and estimation problem solving reasoning technology visualization critical thinking cultural considerations adapting instruction for diverse student needs

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Communication

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Students are provided with opportunities to learn by reading, listening, doing, and speaking. Solving realistic workplace problems and engaging in a variety of hands-on activities will enable students to gather information and knowledge in various ways, express their learning, and communicate with others. The numerous opportunities for class or small group discussion of contextual problems encourage students to share their experiences

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Connections The student textbook contains a wealth of real-world examples and problems, especially those related to apprenticeship programs and to employment that students can enter after completing secondary school. Connections between mathematical processes and real-world applications of those processes are made explicit. Concrete examples describe how math is used on the job, and word problems and activities are contextualized to ensure that students can make connections between the mathematical ideas and the workplace. In addition, connections are made across the chapters so that students will be able to apply mathematical ideas in different contexts when they encounter them.

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The Apprenticeship and Workplace Mathematics pathway was designed for students who may want to pursue post-secondary studies in trades, certified occupations, or direct entry into the workforce. Consequently, MathWorks 10 delivers the curriculum outcomes through projects, activities, and problems set in real-world contexts, enabling students to make connections between school mathematics and the workplace.

and prior knowledge, and thereby develop mathematical understanding. Many features of the textbook are flexible, so teachers can decide which communication mode works best in their classroom.

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MathWorks 10 was developed to deliver the curriculum of the Workplace and Apprenticeship Mathematics Grade 10 course.

Mental mathematics and estimation

Mental mathematics and estimation problems appear throughout the student textbook. Realistic problem scenarios show students that mental math and estimation are used in daily life as well as in the workplace.

Problem solving Problem solving is fundamental in this textbook. Students are encouraged to critique given solutions, identify errors in given strategies, and develop their own strategies for approaching problems. They are given many opportunities to develop approaches to problems individually, in pairs, and in small groups. Examples with worked solutions range from simple to multistep processes that build upon prior knowledge

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Reasoning

Technology

Critical thinking is key to problem solving. The textbook includes many opportunities for students to develop analytical and critical thinking skills by strategizing solutions to problems and evaluating the options presented. Cultural considerations To reflect the educational interests of western Canadian students, the images, problems, activities, and projects incorporate realistic western and northern contexts. This text is mindful of the multiethnic composition of Canadian schools. In particular, First Nations, Métis, Inuit, and francophone perspectives are represented.

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Hands-on activities, puzzles, and projects in which there is no one set method and no one set solution challnge students to use analytical skills to find a solution. Group discussion of mathematics problems develops students’ ability to make predictions and conjectures and encourages participation by students who have difficulty with rote algebraic mathematics. It also helps students to connect the abstract math to a familiar, concrete workplace situation.

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and skills. Students are challenged to see familiar mathematics in new scenarios and apply new mathematics to solve the multi-step questions.

Adapting instruction for diverse student needs

A variety of technologies can be used to complete the projects and solve many of the problems in the textbook. However, technologies are not equally available to all students, so there is flexibility and choice. The use of communications technologies, such as the internet, and presentation software, such as PowerPoint, will further expand students’ abilities to collect data and to communicate mathematical ideas to others. Visualization

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The development of visualization skills, spatial sense, and measurement sense are fostered through the use of technology, graphic organizers, manipulatives, and diagrams. The culminating activities of many of the chapter projects are presented in a visual form, encouraging students to make the connection between abstract mathematical concepts and the physical world. In addition, the strong visual components of the textbook, including illustrations, photographs, graphs, and charts, enrich the presentation of the material.

Many students learn best through experiential learning. With a range of hands-on activities and opportunities to adapt teaching strategies, the textbook accommodates these learners. The resources are flexible and adaptable to a variety of learning styles. Hands-on activities, discussion topics, and projects that can be completed by pairs or small groups as well as individually maximize opportunities to customize the course for particular classrooms. Alternative instructional strategies described in the teacher resource support this as well. In some cases, students may not have mastered mathematics from earlier grades. The teacher resource lists essential mathematics students may know from earlier grades and includes review materials, and the teacher can decide whether or not students would benefit from a review.

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MathWorks 10 Teacher Resource

Assessment Teachers use assessment as an investigative tool to find out as much as they can about what their students know and can do and what confusions, preconceptions, or gaps in learning they might

have. Workplace Mathematics 10 supports the Workplace and Apprenticeship Mathematics Grade 10 curriculum by incorporating assessment for learning, assessment as learning, and assessment of learning. Assessment for learning

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student reflection on the information they gather and the decisions they make to complete activities and projects; hands-on activities and projects that allow students to learn through discovery, see patterns, make connections, draw conclusions, and make predictions; hands-on activities and projects that require students to work with mathematics in a non-algebraic format that challenges their preconceived notions of mathematics, helping them to discover a new way of conceptualizing math; puzzles with multiple possible solutions that encourage students to try to find a solution in any manner that suits their needs; detailed worked examples that allow students to see a step-by-step algebraic process to solve a problem; review and practice questions with an answer key, so students can gauge their progress.

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ongoing dialogue that allows the student to reflect on his or her work and the teacher to uncover the student’s mathematics misconceptions; • group discussions of math from prior grades as well as the new concepts, which enable the teacher to gauge a student’s prior knowledge of the topic and decide how much review is necessary; • group discussions of applications of mathematics to real-world examples, which enable students to compare the processes they would use to answer the question and see that there are multiple ways to solve a problem. This sharing allows students to clarify confusions they may have about the mathematics.

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Teachers use assessment for learning to uncover what students believe to be true and to learn more about the connections students are making and their prior knowledge, preconceptions, knowledge gaps, and learning styles. In this textbook, assessment for learning is addressed through •

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Assessment of learning

Assessment of learning includes strategies designed to confirm what students know, demonstrate whether or not they have met curriculum outcomes or the goals of their individualized programs, or certify proficiency and make decisions about students’ future programs or placements. It is designed to provide evidence of achievement to parents, other educators, the students themselves, and sometimes to outside groups (such as employers or other educational institutions). In this textbook, assessment of learning is addressed through

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Assessment as learning is an active process of cognitive restructuring that occurs when individuals interact with new ideas. For students to be actively engaged in creating their own understanding, they must become adept at personally monitoring what they are learning, and they must use what they discover from the monitoring to make adjustments, adaptations, and even major changes in their thinking. In this textbook, assessment as learning is addressed through

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project presentations that give students the opportunity to demonstrate their understanding of the math concepts using visuals, technology, and written or oral reports; • chapter tests that give students the opportunity to demonstrate their mathematical understanding in written form. •

Introduction

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Introduction

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How to Use the Student Resource

Each chapter begins with an introduction to the mathematical concepts addressed in the chapter and their relevance to the workplace, the learning outcomes, and the key mathematical terms students will encounter.

Chapter Project

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Each chapter contains a project in which students apply the mathematical concepts in a realworld scenario. The project provides students with opportunities to reflect on their learning and draw connections between the mathematical ideas and tools they encounter and real-world applications. Students will return intermittently to the project as they work through the chapter and will complete a culminating activity at the end that allows them to synthesize the various mathematical concepts they have learned to use.

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MathWorks 10 Teacher Resource

Math on the Job Each numbered section within the chapter begins with a Math on the Job scenario that briefly describes a job or workplace and specifically mentions the ways in which mathematics is used in that job. The scenario concludes with a problem to be solved as a class, guided by the teacher.

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Explore the Math The lessons are called Explore the Math and contain a brief explanation of the mathematical ideas being considered and realworld contexts in which the math is applied. Definitions

Definitions of mathematical terms relevant to the lesson are provided. Definitions are also included in the end-of-book glossary.

Each lesson includes one to four worked examples that model problem-solving strategies and techniques for students. Where appropriate, the worked examples include alternative solutions.

Hints

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Examples

In some sections, hints are provided to help activate students’ prior knowledge, remind them of concepts addressed in the chapter, or encourage reflection.

Introduction

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Discuss the Ideas

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Once students have some familiarity with the material, they are presented with a contextual problem to consider and solve. Students can work on these in pairs or small groups or the teacher can lead a brief class discussion.

Mental Math and Estimation

Mental math problems are realistic situations in which estimation or mental math is required to arrive at a solution.

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Activities

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Each chapter contains several handson activities that provide opportunities for students to work collaboratively and apply their learning in a realistic context.

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MathWorks 10 Teacher Resource

Build Your Skills

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The practice problems in each chapter enable students to build their skills and gain confidence in their ability to strategize solutions. These problems can be used flexibly: they can be assigned as homework, completed in the classroom, or solved by pairs or small groups of students working collaboratively. Answers are included at the back of the student book, providing an opportunity for selfassessment.

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Extend Your Thinking

Extension questions are in-depth problems students solve once they have completed the Build Your Skills questions.

Puzzle It Out Each chapter contains a puzzle or game that reflects the mathematical ideas in the chapter and offers a light-hearted approach to mathematical strategy.

Introduction

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The Roots of Math

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Students are introduced to the history of mathematics through this short essay on a topic related to the chapter’s focus. Where appropriate, Canadian history is emphasized.

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Each chapter concludes with a summary of the concepts learned in the chapter.

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MathWorks 10 Teacher Resource

Time Allotment MathWorks 10 is structured on the assumption that teachers have 90 instructional hours available. The following chart shows the estimated instructional

time for each chapter, expressed as a percentage of total instructional time.

mathworks 10 Time Allotment Chapter

% Time

Unit Pricing and Currency Exchange

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Earning an Income

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Linear and Area Measurement Mass and Temperature

Angles and Parallel Lines Similarity of Figures Trigonometry

How to use the Teacher Resource

This teacher resource is a comprehensive resource for both new and experienced teachers. It outlines and discusses the repertoire of instructional and assessment strategies that may be used and identifies the features of the student book and their underlying rationale. For each chapter in the student book, the teacher resource contains the following.

is summarized under the heading Why Are These Concepts Important? The section concludes with a list of the prior skills and knowledge that students are expected to bring to the chapter. You may choose to review these concepts with your students, depending on individual classroom needs. Planning for Instruction and Assessment

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The chapter introduction locates the chapter within the Apprenticeship and Workplace Mathematics 10 curriculum and maps it to the general and specific outcomes addressed in the chapter.

The technology icon alerts you to activities in which the students may benefit from using technology such as computers or the internet.

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The Mathematical Ideas

Rubrics have been provided to assist you in allocating class time, preparing materials, and designing your assessment strategy.

In this section, the “big ideas” of the chapter are described, with examples. This provides some mathematical background for teachers, if needed, and explains the chapter’s mathematical focus. The workplace relevance of the mathematical concepts

Chapter Project A detailed description of the chapter project provides information on the project’s goals, outcome, prerequisites, and activities. This overview will assist you to plan class time for

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project work during the course of the chapter. The teaching suggestions will assist you with integrating the mathematical concepts into the project. A project assessment rubric is provided, as well as a student self-assessment rubric.

The hands-on chapter activities allow for a range of teaching and learning strategies to be used to meet the needs of students with varying interests, backgrounds, and aptitudes.

Each chapter includes an alternative chapter project with Blackline Masters and a project assessment rubric, to accommodate different class interests and learning styles and to provide variety from year to year.

Puzzle it out

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Puzzles and games provide ample opportunities for students to demonstrate mathematical reasoning and to apply new skills in an engaging way. In addition to solutions to the puzzles that are in the student book, the teacher resource includes alternative puzzles with solutions. Many more spatial puzzles and games are available online, including on the website of the National Library of Virtual Manipulatives. Use the following key word searches: virtual math games; interactive math games; math puzzles; spatial puzzles; spatial games; spatial math games; and virtual math games.

Chapter Subsections

For each chapter subsection, the teacher resource follows the format of the student book.

Worked solutions have been provided for all questions, including alternative methods of arriving at solutions and, in some cases, extension activities for students ready for more in-depth work.

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Teaching notes include alternative teaching strategies. For example, features such as Math on the Job and Discuss the Ideas can be used as discussion starters in the classroom, and the teacher resource contains numerous suggestions for connecting students’ work and life experiences to the mathematical concepts.

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MathWorks 10 Teacher Resource

Appendix

Each chapter concludes with a sample chapter test and worked solutions, graphic organizers and other Blackline Masters for the chapter project and activities, and the alternative chapter project teacher and student materials with Blackline Masters and an assessment rubric.

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Unit Pricing and Currency Exchange

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introduction This is one of two chapters in the student textbook that deliver the outcomes of the Number strand of Apprenticeship and Workplace Mathematics 10. In this chapter, students will be introduced

to unit pricing and currency exchange. This outcome comprises part of the Number strand and integrates the Algebra strand. The chart below locates this chapter within the curriculum.

Number, Grades 10–12

This chart illustrates the development of the Number strand in the Apprenticeship and Workplace Mathematics pathway through senior secondary school. The highlighted cells contain the outcomes that chapter 1 addresses. Grade 11

Grade 12

General Outcome

General Outcome

General Outcome

Develop number sense and critical thinking skills.

Develop number sense and critical thinking skills.

Develop number sense and critical thinking skills.

Specific Outcome

Specific Outcome

Specific Outcome

It is expected that students will:

It is expected that students will:

It is expected that students will:

Solve problems that involve unit pricing and currency exchange, using proportional reasoning.

Analyze puzzles and games that involve numerical reasoning, using problemsolving strategies.

Analyze puzzles and games that involve logical reasoning, using problem-solving strategies.

Demonstrate an understanding of income, including: wages, salary, contracts, commissions, piecework to calculate gross pay and net pay

Solve problems that involve personal budgets.

Solve problems that involve the acquisition of a vehicle by: buying, leasing, leasing to buy.

Demonstrate an understanding of compound interest.

Critique the viability of small business options by considering: expenses, sales, profit or loss.

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Grade 10

Demonstrate an understanding of financial institution services used to access and manage finances. Demonstrate an understanding of credit options, including: credit cards, loans.

Algebra, Grade 10 General Outcome Develop algebraic reasoning.

Chapter 1 Unit Pricing and Currency Exchange

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Curriculum and chapter overview

General outcome: Algebra Develop algebraic reasoning.

General outcome: Number Develop number sense and critical thinking skills.

Integrated throughout the chapter.

Specific outcome: Solve currency exchange problems using proportional reasoning.

Section 1.1: Proportional Reasoning

Math on the Job Practise Your Prior Skills: Ratio Discuss the Ideas: Adapting a recipe Activity 1.1: Visualize a proportion

Section 1.2: Unit Price

Section 1.3: Setting a Price

Section 1.4: On Sale!

Section 1.5: Currency Exchange Rates

Math on the Job

Math on the Job

Math on the Job

Math on the Job

Explore the Math

Explore the Math

Explore the Math

Explore the Math

Activity 1.3: Which price is right?

Discuss the Ideas: Concert promoter

Mental Math and Estimation

Mental Math and Estimation

Build Your Skills

Discuss the Ideas: Seasons and holidays

Activity 1.4: Taking advantage of sales promotions

Activity 1.5: What’s your ride? survey

Build Your Skills

Activity 1.6: Calculate foreign exchange

Mental Math and Estimation

Activity 1.2: Fruit drink taste tester

Build Your Skills

Practise Your Prior Skills: Rate Discuss the Ideas: Cindy Klassen, speed skater

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Mental Math and Estimation Practise Your New Skills

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Specific outcome: Solve unit pricing problems using proportional reasoning.

Chapter Project: The Party Planner Reflect on Your Learning Practise Your New Skills

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MathWorks 10 Teacher Resource

The Roots of Math: Canadian currency

Build Your Skills

The Mathematical Ideas Proportional Reasoning

The lesson structure built into the student textbook introduces students to each topic by first having them read about or listen to a Math on the Job situation that incorporates the concept to be explored. Students then explore how algebra can be used to solve related problems. Once students have worked through the algebraic examples, they will build upon the algebra by discussing a scenario that uses the concept (Discuss the Ideas), trying a mental math question, and by working on hands-on activities. Project activities allow students to synthesize their skills and knowledge and apply them holistically to a real-life situation. The activities model proportional reasoning in two ways: using tables to see patterns; and solving algebraic expressions.

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In this chapter, students will explore proportional reasoning using tables in the chapter project, and in Activity 1.2: Fruit Drink Taste Tester, Activity 1.3: Which Price Is Right?, Activity 1.4: Taking Advantage of Sales Promotions, and Activity 1.5: What’s Your Ride? Survey. In each activity, students will generate data for their table using patterns, and then use their data to answer questions.

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Example

The scale on a map states that 1 centimetre represents an actual distance of 5 kilometres. The map distance between two towns is 8 centimetres. What is the actual distance? solution

The students will start the table with the given information, that the ratio between map and real distance is 1:5. They would build the table to 8 cm by following a multiplication pattern, as follows. 8 × 5 = 40

The table they would use would be like this one:

Finding Actual Distance Map distance (cm)

1

2

3

4

5

6

7

8

Actual distance (km)

5

10

15

20

25

30

35

40

or

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Each of these is described in more detail below. Using Tables

The underlying concept that students will understand is that the multiplicative relationship is always y = mx, where m is one of the constants of proportionality. In the following example, you will see that the constant m is 5.

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Many aspects of our world operate according to proportional rules. In the workplace, the use of proportions can be seen in a wide range of fields, including nursing, pharmacy, construction and other building trades, baking, graphic arts, photography, land surveying, commodity trading, and many others. In this chapter we will examine proportional reasoning through the lens of the workplace.

Algebraic Expressions Each section in this chapter uses algebra to solve unit cost and currency exchange problems. As detailed below, students have used these algebraic methods in previous grades to solve proportional reasoning questions; however, since the concept is being modelled in new contexts and scenarios, students may have some difficulty setting up the initial proportion.

Chapter 1 Unit Pricing and Currency Exchange

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Why Are These Concepts Important?

The important concept that students will learn is that all rate pairs describing a given proportional situation are equivalent. The same statement is true of the reciprocal of these rate pairs. These two constants of proportionality define the multiplicative relationship. Example

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$21.00 = _____ £x ______ $3.00 £2.00

1. Concepts

a) Ratios, rates, proportional reasoning;

Multiply each side of the equation by the common denominator, 3.00 multiplied by 2.00, or 6.00. 6.00 _____ 21.00 = ____ x 6.00 3.00 2.00

b) Interpolation, extrapolation;

) ( )

c) Percents;

2. Mathematics Skills

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a) Identifying equivalent ratios;

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126.00 ______ = _____ 6.00x 3.00 2.00

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b) Using a fractional equation to solve for an unknown.

42.00 = 3.00x

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Simplify each side of the equation by dividing the numerator by the denominator.

42.00 _____ =x 3.00 14.00 = x

Therefore, you would receive £14.00 for $21.00 CAD.

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Examining the relationships among different representations is important. Different representations highlight different aspects of the situation, each fostering insights and interconnections to the other. This allows students to understand both why and how the strategies work.

Student work in this chapter will build on certain WNCP outcomes from earlier grades. Students will review these mathematical concepts and skills and apply them in a new context to real-life problems involving unit pricing and currency exchange. The following is a list of concepts and mathematics skills to which students have been exposed in grades 8 and 9.

The students would first set up a proportion between the Canadian dollar and the English pound and then solve for the unknown quantity: pounds dollars ______ = _______ dollars pounds or

•

Prior Skills and Knowledge

solution

Understanding proportionality by using several representations enables students to evaluate problem situations critically and to determine whether the context is proportional or nonproportional.

d.

If you travel to a foreign country, you exchange Canadian dollars for the currency used there. In England, you could exchange $3.00 CAD for £2.00. How many pounds could you exchange for $21.00?

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MathWorks 10 Teacher Resource

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3. Technology: presentation software, basic calculator functions, spreadsheets, and internet search skills. (NB: Some students may not have been exposed to spreadsheets, presentation software, or the internet.)

planning Chapter 1 This chapter will take 2–3 weeks of class time to complete. Class period estimates are based on a class length ranging from 60 to 75 minutes. These estimates may vary depending on individual classroom needs. planning for instruction

1.1

Introduce the chapter project “The Party Planner”

Materials

20 minutes for a class discussion on the opening questions about the project

internet, newspapers, flyers, magazines, local stores Blackline Master 1.2

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Estimated Time

d.

Section Lesson Focus

Math on the job: Northern nurse

40 minutes

Practise your prior skills: Ratio

Discuss the ideas: Adapting a recipe Examples 1, 2 1.1

Activity 1.1: Visualize a proportion

15 minutes

Blackline Master 1.1

1.1

Activity 1.2: Fruit drink taste tester

45 minutes

Blackline Master 1.5 or spreadsheet software

1.1

Practise your prior skills: Rate

1 class

Discuss the ideas: Cindy Klassen, speed skater Examples 1, 2

Mental math and estimation Practise your skills

Puzzle it out: Magic proportions 1.2

Math on the job: Organic farmer Explore the math Examples 1, 2

Activity 1.3: Which price is right?

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Math on the job: Construction cost estimator

1 class, if students are using the internet at school

internet or local stores

1 class

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Examples 1, 2

Discuss the ideas: Concert promoter; Seasons and holidays

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Build your skills

1 class

Mental math and estimation Build your skills

Chapter 1 Unit Pricing and Currency Exchange

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planning for instruction Section Lesson Focus 1.4

Math on the job: Stone and tile business owner

Estimated Time

Teacher Notes

1 class

internet, newspapers, flyers, magazines

1 class

Blackline Master 1.3 or spreadsheet software, internet (if giving time to research project)

Explore the math Examples 1, 2 Mental math and estimation Activity 1.4: Taking advantage of sales promotions Build your skills

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Project: Research your ideas

1.5

The roots of math: Canadian currency

15 minutes

internet

1.5

Math on the job: Agricultural exporter

1 class

internet

Explore the math

Mental math and estimation

Activity 1.5: What’s your ride? survey 1.5

Examples 1, 2

1 class

Activity 1.6: Calculate foreign exchange Build your skills

Project: Make a presentation

1 class to finish projects

Presentation of projects to class

1–2 classes

Reflect on your learning

1 class

Practise your new skills

1 class

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Chapter test

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word-processing software, spreadsheet software, internet

Planning for Assessment Assessment for Learning

•

Chapter launch

•

Project discussions (ongoing)

•

Math on the job scenarios

•

Exploration of new concepts

•

Activities

•

Discuss the ideas

•

Mental math and estimation

•

Puzzle it out: Magic proportions

Assessment as Learning

Assessment of Learning

Teacher Notes •

Make a checklist to keep track of how much work the students have done on their project

•

Observe how students participate during discussions

•

Observe how students work through activities in small groups, pairs, or individually

d.

In the Chapter

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Purpose

Reflection and Practice •

Build your skills problems

•

Prompt students’ self-assessment

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Review student work, provide feedback

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Reflect on your learning

Chapter Review •

Chapter project: Planning a party

•

Quizzes

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Chapter test

•

Review assessment records and add unit results to ongoing records

Check daily homework and provide feedback on questions

•

Challenge your students to find relationships without always using a formula

•

Have students present their final project to the class and allow students to give feedback to presenters

•

Give small quizzes as the chapter progresses to give as much feedback as possible

•

Keep a log or journal of observations to aid in reporting

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Learning Skills/ Observe and record throughout the unit how Mathematical Disposition students are working with new language and concepts

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Chapter 1 Unit Pricing and Currency Exchange

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chapter proJect—The Party Planner

Prerequisites: Students need to understand ratios, proportions, and basic calculator functions. If students want to use a spreadsheet for the pricing calculations, then some prior spreadsheet experience would be an asset. They will also need to be familiar with presentation software if they choose to use it for their presentation. If they are familiar with any layout software, they could use it to make party invitations. Familiarity with internet research may also be helpful in this project.

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About this Project: This project is divided into three parts. Initially, students will plan their project and identify areas that they need to research. Partway through the chapter, students will apply what they have learned about unit pricing and cost comparisons to decide on the purchases they will make and work out the costs in several ways. As a final activity, they will develop a presentation for their team or club mates, including a table or a spreadsheet listing all the components of the party and their respective costs. Students will give this presentation to the class. Allow 3–5 minutes for each student.

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An alternative project, “Food Planning at a Wilderness Lodge,” is included on pp. 70–79. This project can be done by small groups or pairs of students as well as individuals.

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Outcome: In this project, students will integrate the concept of unit pricing into a real-world scenario in which they create a party concept, plan within a set budget and given parameters, work with technology, and practise and further develop presentation skills.

activity more successfully. This project could also be completed by small groups of students. A selfassessment rubric, Blackline Master 1.4 (p. 168), should be handed out to students early in the project. It outlines the criteria for evaluation of their project and suggests some ways to reflect on their learning.

d.

Goals: To use the concept of proportional reasoning to find unit prices, to build skills, and to synthesize learning in this chapter.

Introduce the project to your students as you begin this chapter. This initial part of the project allows for group brainstorming as a class. Most students will have attended a party, allowing them to draw upon personal experiences about what activities students enjoy at a party. Suggest that they can make their own choices about the party, but prepare a few suggestions to help them get started. To begin the project, have students decide on the location they will use. Next, ask them to list the things they will need to consider and buy while planning the party. If they need help with this activity, items to consider (a checklist is included on Blackline Master 1.2, p.66) include the following: • • • •

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Students should be given a few class periods to work on this project during the time spent on this chapter. This will allow for questions/feedback from the teacher as well as allowing the teacher to observe the quality of work as it is done, rather than at the end of the chapter. Interim guidance can help students complete the culminating

1. Start to plan

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MathWorks 10 Teacher Resource

• • •

What decorations will you use? What will the invitations look like? What activities or entertainment will you plan for the party? What kind of music will you play? Do you need to organize a sound system? What food do you plan to serve? How will you handle food allergies? Will you need to order plates, cutlery, or glasses? Where might you purchase supplies?

Emphasize that even though they are planning the party, students must keep the total budget in mind.

Suggest sources of information that students can use, such as magazines, party supply stores, catalogues, flyers, newspapers, and websites. There are a number of online sources of party supplies that students may find helpful. A keyword search string that may generate useful options is “party supplies” plus the place name of your community or a larger centre nearby where such supplies might be found, for example, “party supplies Vancouver.”

This segment of the project requires the largest amount of work on the part of students. Here they are practising both their research and their unit costing skills. Students are expected to develop a cost analysis that is within their budget, including all the supplies they would need to purchase and any other costs, such as venue rental charges. All their work should be recorded in a table (an example is shown in the student book and reproduced on Blackline Master 1.3, p. 67) or on a spreadsheet.

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1. Start to plan •

Record your observations. Provide students with numeric information on how they will be assessed using a scheme that meets your reporting needs.

2. Research your ideas •

Have students make a checklist of all items that should be in their project to allow them to reflect on their progress. Blackline Master 1.2 (p. 66) contains a checklist that students may use.

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2. Research your ideas

Assessing the Project

d.

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At the end of this segment of the project, discuss progress with your students to ensure that all requirements have been met. 3. Make a presentation

3. Make a presentation

Use the following rubric as a gauge to accompany a numerical grading rubric you have created. • Ask students to self-assess their project using Blackline Master 1.4 (p. 68). • If time doesn’t allow for presentations, have students set up their projects on their desks and allow one row of the class at a time to walk around the classroom to view and comment on all the projects. • You might want to take photos of students with their projects to put in their school portfolio. •

In this segment of the project, students will synthesize their planning and research activities, and practise their presentation skills. Presentations to clients are often done with handouts and other tools, including presentation software or posters. Encourage students to use such tools to enhance their project presentation, which they will give to the class.

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Chapter 1 Unit Pricing and Currency Exchange

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Project Assessment rubric Not Yet Adequate

Adequate

Proficient

Excellent

Conceptual Understanding •

Explanations show understanding of unit pricing, budgeting parameters, research methods, client needs

shows very limited understanding; explanations are omitted or inappropriate

shows partial understanding; explanations are often incomplete or somewhat confusing

shows understanding; explanations are appropriate

shows thorough understanding; explanations are effective and thorough

limited accuracy; major errors or omissions

partially accurate; some errors or omissions

generally accurate; few errors or omissions

accurate and precise; very few or no errors

For example:

For example:

For example:

Accurately:

writes and evaluates unit costs while adhering to the budget creates tables from the information given using pen/paper or spreadsheets calculates total costs, unit costs, and taxes for all items purchased theme, invitations, decorations, style are appropriate

writes down all sources

presentation includes handouts, posters or electronic display, listed items, and a cost analysis

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For example:

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items are listed and unit costs are calculated correctly

•

items listed and unit costs are calculated correctly

•

may have some needed items missing

•

total cost within budget

•

total cost within budget

•

total cost within budget using erroneous unit cost

•

sources are listed

•

sources are listed

•

activities and entertainment are appropriate for the event and somewhat imaginative

•

activities are appropriate for the event and show unusual creativity

•

•

has all presentation handouts, poster, or electronic presentation

•

very few calculation errors

has all presentation handouts, poster, or electronic presentation

•

no calculation errors

•

adds some extra creativity to the project

unit costs are calculated incorrectly

•

total cost not within the budget

•

activities and entertainment not appropriate for the event

•

•

sources missing

•

•

presentation poster, handouts, or electronic presentation not created

or

project is incomplete

•

activities and entertainment are appropriate for the event

presentation poster, handouts, or electronic presentation are good project could use some more work to ensure calculations are done correctly

•

project is completed but there is nothing beyond what is listed as a minimum

uses few effective strategies; does not solve problems

uses some appropriate strategies, with partial success, to solve problems; may have difficulty explaining the solutions

uses appropriate strategies to successfully solve most problems and explain solutions

uses effective and often innovative strategies to successfully solve problems and explain solutions

does not present work and explanations clearly; uses few appropriate mathematical terms

presents work and explanations with some clarity, using some appropriate mathematical terms

presents work and explanations clearly, using appropriate mathematical terms

presents work and explanations precisely, using a range of appropriate mathematical terms

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Uses appropriate strategies to solve problems successfully and explain the solutions

items are listed, but the unit costs are not calculated correctly

item costs are missing

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•

•

•

Problem-Solving Skills

d.

Procedural Knowledge

Communication •

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Presents work and explanations clearly, using appropriate mathematical terminology

MathWorks 10 Teacher Resource

1.1

Proportional Reasoning

Time required for this section: 3 Classes

Math on the Job

Simplify each side of the equation by dividing by the denominator.

Students have used ratios and proportions in grade 8. Activate their prior knowledge by giving students a few minutes to try to solve the question in this scenario themselves. When presenting the solution, you may want to show students that there is more than one method.

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Method 1: Set up a ratio by aligning the same units. Students may have seen this method in science class, where it is called dimensional analysis. Show the students that the same units (mg) should cancel each other out, leaving the desired units (mL).

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120 mg _____ _______ = 2 mL 300 mg x

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To solve for x, multiply both sides of the equation by the common denominator, 300x.

d.

120x = 600

Divide each side by the coefficient of the variable, 120. 120x = ____ _____ 600 120 120

x = 5 mL

Method 2: Find the unit amount of mg/mL first by dividing the numerator by the denominator. 60 mg 120 mg ______ _______ = 2 mL 1 mL

Solution

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Start your class with a discussion about Sandra Tuccaro’s nursing position. Have a student read the text aloud to the class. Before presenting the mathematical solution, discuss the fact that drugs are packaged in standard doses but the amount of medication that a person needs is individual. Also mention that in some cases the only medical professional available may be a nurse. Thus, the nurse must be capable of adjusting medications to suit a patient’s size/weight to administer the proper dosage.

( ) (

)

120 300x ____ = _____ 2 mL 300x 300 x

36 000x _______ = _____ 600x x 300

Then you can ask students how many mg of drugs they need. Since the nurse wants to give the patient 300 mg of the drug, she can calculate how many 60 mg units she needs. Since each 60 mg of the drug is dissolved in 1 mL of fluid, she will need to give the patient 5 mL of fluid. Method 3: Nurses use this “nursing rule” to figure out the doses they need. Drug prescribed _______________ × Number of measures Dose per “measure”

The drug prescribed is the 300 mg in the scenario. The dose per measure is the diluted solution of 120 mg that has been mixed into the solution. The number of measures is the 2 mL in the solution. This could be stated in another way. What you want _____________ × The amount it comes in What you’ve got

Chapter 1 Unit Pricing and Currency Exchange

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Substitute in the actual numbers. 300 mg × 2 mL = 5 mL _______ 120 mg The dose is 5 mL. You may ask students why it would be beneficial for a nurse to memorize such a rule.

Discuss the Ideas

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d.

practise your prior skills ratio

try the Discuss the Ideas “adapting a recipe” question. Start by asking students if they have ever had to double a recipe. The idea of doubling is represented by the ratio 1:2. In this case, you want to reduce your recipe using the ratio 20:12, which can be simplified to 5:3 or 1.67:1. Consequently, students would have to take _ 35 of each amount in the ingredient list to get the same taste. If students make the mistake of multiplying by _ 5 , you could 3 show that would increase the recipe.

Read through the ratio and proportion lesson with your class. Pair students up and have them

Activity 1.1￼ visualize a proportion

Solutions to Visualize a Proportion

BLackline Master 1.1 graph paper

1. See answer on the next page.

This activity can be done by pairs or small groups of students. It reinforces the idea that a proportion is a multiplicative relationship and not an additive one.

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This activity leads nicely into a discussion about reducing or enlarging pictures. Many students have digital cameras. They may have done some editing or resizing using image editing software and noticed what happens when they “drag” the photo down or across rather than from the corner of the picture. If they do not enlarge or reduce the photo by the same amount on all sides, the photo becomes distorted.

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MathWorks 10 Teacher Resource

2. See answer on the next page.

3. Students should notice that when you multiply or divide the two sides of a triangle by the same amount, the resulting triangle will be proportional to the original. If you add or subtract length, the triangles do not stay proportional.

B 1.

C

A

B B

2. a)

b) B

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A

C

A

C

A

proportional c)

d.

B

C A not proportional

d)

B

B

B

A

C

C A not proportional

B

C

A

C

proportional

A

C

Activity 1.2￼ Fruit Drink Taste Tester

because students could use the “edit—fill down” feature to expand the table to as many batches as they like. The pattern in the table works intuitively since the first batch is the given recipe, 2 batches double each ingredient, 3 batches triple them, and so on.

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In this activity, students could work in small groups to simulate the research teams characteristic of real-life development settings. This activity allows students to visualize proportions in various formats, in this case, algebraically and in tabular form. The activity first builds on patterns, then uses the patterns to solve problems that could be done algebraically or using technology. Students begin the activity by filling in the table. This is an effective activity to use spreadsheets with

Chapter 1 Unit Pricing and Currency Exchange

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solutions to taste tester activity mixing the concentrates Batches

Orange concentrate (cups)

Recipe #2

Water (cups)

Orange concentrate (cups)

Water (cups)

3

7

2

5

2

6

14

4

10

3

9

21

6

15

4

12

28

8

20

5

15

35

10

25

18

42

12

30

21

49

14

35

24

56

16

40

27

63

18

45

30

70

20

50

6 7 8 9 10

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d.

Recipe #1

mixing the concentrates

Recipe #1

Batches

Orange concentrate (cups)

1 2 3 4 5 6 7

Water (cups)

2

5

=$B$3*A4

=$C$3*A4

=$D$3*A4

=$E$3*A4

=$B$3*A5

=$C$3*A5

=$D$3*A5

=$E$3*A5

=$B$3*A6

=$C$3*A6

=$D$3*A6

=$E$3*A6

=$B$3*A7

=$C$3*A7

=$D$3*A7

=$E$3*A7

=$B$3*A8

=$C$3*A8

=$D$3*A8

=$E$3*A8

=$B$3*A9

=$C$3*A9

=$D$3*A9

=$E$3*A9

=$B$3*A10

=$C$3*A10

=$D$3*A10

=$E$3*A10

=$B$3*A11

=$C$3*A11

=$D$3*A11

=$E$3*A11

=$B$3*A12

=$C$3*A12

=$D$3*A12

=$E$3*A12

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Orange concentrate (cups)

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Water (cups)

3

10

8

Recipe #2

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1. You would need 300 cups of orange concentrate. Students can use algebra or fill down on their spreadsheet (3 cups × 100 = 300 cups). 2. Unit amounts: 7 = 2.3 water/concentrate (cups) Recipe #1: __ 3 5 = 2.5 water/concentrate (cups) Recipe #2: __ 2

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Since Recipe #1 has less water per cup of orange concentrate, it would have a stronger orange taste.

3. Proportion: 5 = ___ __ 2.5 2 1 So 2.5 cups of water are needed for 1 cup of concentrate.

4. The original recipe makes 10 cups, so 1 batch equals 10 cups. One way to solve this is first to determine the fraction of a batch that would yield 8 cups. 10 cups _______ = 1 batch

8 cups ________ x batches

Multiply both sides by the common denominator, x.

10x = 8 x = __ 4 5

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10 = __ 8 x x ___ x 1

Note that one of the examples of rates refers to the price of lumber for linear foot. Construction materials are measured and sold in imperial units.

Simplify the fraction to _45 .

To make 8 cups, you need to make _ 45 of a batch. To determine the portion of each water and concentrate needed, multiply the amounts for one batch by _ 45 . 4 × 3 = ___ 12 __ 5 5

Rate of pay = $15.85/h

10

Check: $15.85 × 6 h = $95.10.

To make 8 cups of Recipe #1, you would use 2 _25 cups of concentrate and 5 _ 3 cups of water.

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Discuss the Ideas Cindy Klassen, Speed Skater

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Rate of pay = _____________ amount earned time worked

$95.00 Rate of pay = ______ 6h

= 5 __ 3 5

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For example, Michelle earned $95.00 for working 6 hours at a supermarket checkout. What was her rate of pay?

4 × 7 = ___ __ 28 5 5

Once students have worked out the examples on rate, they could work backwards to check their answers.

solution

= 2 __ 2 5

In this section, you will discuss the concept of rate. Since students are in grade 10, getting their drivers’ licences is a big concern. You could take the opportunity to discuss speed limits and how the posted sign is the maximum rate, that is, 50 km/h, 110 km/h, and so on. The major concept that students need to remember is that a rate compares two different units.

d.

( ) ( )

Practise Your Prior Skills Rate

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5. Add all of the cups up first to get total parts (2 + 3 + 5 = 10 cups). Since you only want 4 4 of each ingredient. cups, you need to take __ 10 pineapple juice: 2 × 0.4 = 0.8 cranberry juice: 3 × 0.4 = 1.2

lemon juice: 5 × 0.4 = 2

Check: 0.8 + 1.2 + 2 = 4 cups

Cindy’s average speed is a rate comparing her speed to the elapsed time. Students could discuss how speed skaters alter their actual speed as the race progresses, usually culminating with a sprint to the finish line. Thus, an average rate may not always be the best indicator of an athlete’s ability. Cindy’s average speed is a rate comparing her speed to the elapsed time. The solution is as follows. 1500 m __________ = 13.01 m/sec 115.27 sec Chapter 1 Unit Pricing and Currency Exchange

33

Mental Math and Estimation This concept of checking leads to the Mental Math, since the mathematics for the pipe straps is the same as the “check.” Have students try the mental math for a set time—about 2 minutes. Then ask the students to share the strategies they used to find the answer.

d.

Show students the actual value using multiplication of the actual values.

Then explain that an estimation technique using rounding would be a better way to aid students that have trouble with mental math. So, have the students round the dollar value to $0.05. Then explain that 5 times 5 equals 25 and 5 times 50 equals 250. Then ask students to look at the decimal value of 0.05 and remind the students that when they multiply, they need to move the decimal over by the place value (hundredths is 2 places). Since the decimal is small, they need to make their answer “250.00” smaller, thus moving the decimal 2 places towards the left to find $2.50.

PA re EV CIF pr A IC od L E uc UA DU tio TI C n ON AT of pa CO ION rt P AL or Y— P al N R lo O E f t T SS he FO . co R ALL nt D R en IS IG ts TR H in IB TS an UT R y IO ES fo N E rm R VE is D st ric tly pr oh ib ite

$0.0497 × 50 = $2.49

Practise your NEW skills: Solutions 1. To simplify, divide the numerator and denominator by 2 to get 4:1.

Ways to write this ratio include the following.

8 to 2

8:2 8 __ 2

55 words = __________ 2000 words 2. ________ x minutes 1 minute 55 = _____ 2000 ___ 1 x

Multiply each side by the common denominator, 1x, or x.

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55 = _____ 2000 x x ___ 1 x

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55x = 2000

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55x ____ = ______ 2000x x 1

2000 55x = _____ ____ 55 55 2000 x = _____ 55

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To rotate the tires on 5 trucks, use the following proportion. 4 tires ______ = _______ 20 tires 15 m xm

3. Each truck has 4 tires, so 5 trucks have 20 tires.

x = 36.36 minutes

It will take the secretary 36 minutes, rounded to the nearest minute.

MathWorks 10 Teacher Resource

4 = ___ ___ 20 15 x

The common denominator is 15x.

( ) ( )

4 = ___ 20 15x 15x ___ x 15

60x = _____ ____ 300x x 15 4x = 300

4x = ____ 300 ___ 4 4 x = ____ 300 4 x = 75 It would take 75 minutes to rotate the tires on 5 trucks. Alternatively, you can multiply 15 minutes (time for one truck) by 5 (the number of trucks) to get 75 minutes.

870s 4350 = ____ _____ 6 145

To rotate 2 tires, divide the time for 4 tires in 2. 15 = 7.5 minutes ___ 2

725 = 6s 725 6s = __ ____ 6 6

It would take 7.5 minutes to rotate 2 tires. 4. First, calculate what the salesperson sold in the first two days.

Next, calculate what he or she sold on the weekend.

6. To calculate the profits on 50 DVDs, use a fractional equation.

PA re EV CIF pr A IC od L E uc UA DU tio TI C n ON AT of pa CO ION rt P AL or Y— P al N R lo O E f t T SS he FO . co R ALL nt D R en IS IG ts TR H in IB TS an UT R y IO ES fo N E rm R VE is D st ric tly pr oh ib ite

To the nearest centimetre, Siu is 121 cm tall.

6 + 4 = 10 cars

d.

121 = s

$2550.00 ________ _________ = x 200 DVDs 50 DVDs

36 – 10 = 26 cars

Since he or she sold the same number of cars on each day, calculate what was sold each day.

2550 ___ _____ = x 200 50

2x = 26

The lowest common denominator is 200. 2500 = ___ x 200 200 _____ 200 50

( ) ( )

26 2x = ___ ___ 2 2

2550 = 4x 4x 2550 = ___ _____ 4 4

x = ___ 26 2

x = 13 cars on each day

Alternatively, since the salesperson sold 26 cars in two days and an equal number of cars were sold on each day, divide 26 by 2 to get 13 cars sold on each day.

$637.50 = x

The total profit on the sale of 50 DVDs is $637.50. Next, calculate the profit on 900 DVDs.

The proportion of cars sold on Saturday is 13:36.

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5. The ratio can be written as _ 5 . 6

2550 ____ _____ = x 200 900

The lowest common denominator is 1800.

( ) ( )

2550 = ____ x 1800 1800 _____ 200 900

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Let s represent Siu’s height.

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Use the following proportion to solve for s. 5 = ____ __ s 6 145

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2x 22 950 = ___ ______ 2 2

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Since this is a fractional equation, multiply both sides by the lowest common denominator, 6 multiplied by 145.

( ) ( ) 870 ( __ 5 )= ( ____ s )870 145 6

6 × 145 __ 5 = ____ s 6 × 145 145 6

22 950 = 2x

$11 475.00 = x

The total profit on the sale of 900 DVDs is $11 475.00.

Chapter 1 Unit Pricing and Currency Exchange

35

2550 = __ _____ x 200 1

( ) ( )

200 _____ 2550 = __ x 200 200 1 2550 = 200x

$12.75 = x

3 Spanish oak:4 red mahogany

$12.75 × 50 = $637.50

$12.75 × 900 = $11 475.00

$15.00 = ______ $75.00 7. ______ 5 kg x kg

So, for Spanish oak, the ratio is 3:7. For red mahogany, it is 4:7. Let s = the amount of Spanish oak needed. 3 = ___ __ s 7 12 The common denominator is 7 multiplied by 12, or 84. 84 __ 3 = ___ s 84 12 7

( ) ( )

252 ____ = ___ 84s 7 12

The numerator, 15, has been multiplied by 5 to get 75. To keep the fractions equivalent, the denominator, 5, must also be multiplied by 5 to equal x.

5 × 5 = 25 x = 25

For $75.00, the restaurant could buy 25 kg of olives. Calculate the cost to buy 20 kg of olives.

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15 = ___ ___ x 5 20

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The lowest common denominator is 20.

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15 = ___ x 20 20 ___ 5 20

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300 ____ 20x = ____ 5 20

60 = x

It would cost $60.00 to buy 20 kg of olives.

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3+4=7

PA re EV CIF pr A IC od L E uc UA DU tio TI C n ON AT of pa CO ION rt P AL or Y— P al N R lo O E f t T SS he FO . co R ALL nt D R en IS IG ts TR H in IB TS an UT R y IO ES fo N E rm R VE is D st ric tly pr oh ib ite

2550 _____ _____ = 200x 200 200

8. First, determine what the proportion is for each stain.

d.

Alternatively, students could find the profit on one DVD and then multiply the number of DVDs sold by this number.

MathWorks 10 Teacher Resource

36 = 7s

7s 36 = __ ___ 7 7

5.14 = s, rounded off

Let r = the amount of red mahogany needed. 4 = ___ r __ 7 12 Again, the common denominator is 84. r 84 84 __ 4 = ___ 7 12

( ) ( )

84r 336 ____ = ___ 7 12

48 = 7r

7r 48 = __ ___ 7 7

6.86 = r, rounded off

For 12 litres, the carpenter needs 6.86 L of red mahogany and 5.14 L of Spanish oak.

Extend your thinking

40 074 = 5x

40 074 5x = ___ ______ 5 5

9. First, determine how long it would take the bullet train to travel the circumference of the Earth. 6 = ______ ___ x 30 40 074

8015 = x

The bullet train could travel the circumference of the earth in 8015 minutes.

6 can be simplified to __ The ratio ___ 1 . 5 30 1 = ______ x __ 5 40 074

Now, convert this to days.

PA re EV CIF pr A IC od L E uc UA DU tio TI C n ON AT of pa CO ION rt P AL or Y— P al N R lo O E f t T SS he FO . co R ALL nt D R en IS IG ts TR H in IB TS an UT R y IO ES fo N E rm R VE is D st ric tly pr oh ib ite

d.

min ________ 8015 = 133.58 hours 60

The common denominator is 5 multiplied by 40 074. 5 × 40 074 __ 1 = ______ x 5 × 40 074 5 40 074

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133.58 hours ___________ = 5.57 days 24

Both Keiko and Yuki underestimated how fast the Shinkasen can go!

Each side of the equation can be simplified to give the following equation.

￼

Puzzle it out

Magic Proportions

There are many solutions to this puzzle. Here is one strategy to help students get started:

1. Start by drawing one large 3 × 3 square on the board.

2. Have students randomly call out numbers from 0–8 to fill in each square with little regard for the ratio, but not repeating a number.

8. This is where the trial and error comes in. Once you have completed one solution as a class, challenge the students to find another solution.

This would work well as a group activity. Some more solutions 2

4

2

1

3

3. Add up each row and write the sum at the outside of the box of each row.

1

3

8

0

5

7

4. Add up each column and write the sum at the bottom of each column.

5

7

6

4

6

8

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0

4

0

1

5

1

5

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7

3

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5. Check to see if the numbers on the outside of the rows and columns form the 1:2:3 ratio. 6. Most likely, they will not. 7. Then ask students to try moving only one number at a time to make the row proportion work. Once the rows work, then check the impact this had on the column sums.

￼

￼

Chapter 1 Unit Pricing and Currency Exchange

37

1.2

Unit Price

Time required for this section: 2 classes

Explore the math

solution

Example

Calculate the unit price by dividing the total price from each wholesaler by the number of plants.

A 200 g bag of chips costs $1.00. A 750 g bag of chips costs $2.70. Which size is the better value?

Company A: $45.95/20 = $2.30/plant

Company B: $48.50/24 = $2.02/plant

solution

Company B’s unit price is $0.28/plant less than Company A’s price.

To find the unit cost per gram, divide the dollar value by the number of grams in the package.

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Other factors to consider include the following:

a) Has Linda bought from this wholesaler before and has she been happy with their products?

b) Are the plants in stock when she needs them?

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The application of proportional reasoning to unit costs is new for the students. Explain that the word unit means 1. Unit price is thus the cost of one item, and a unit rate is a rate with a denominator of 1 (for example, earnings per hour or cost per kilogram). In order to convert bulk prices or rate values to unit prices and unit rate values, the student must see that the denominator must be a 1. Some students, including ESL students, may not realize that the “/” symbol holds the dual purpose of mathematically meaning “divide” and reads also as “per” in a rate question.

PA re EV CIF pr A IC od L E uc UA DU tio TI C n ON AT of pa CO ION rt P AL or Y— P al N R lo O E f t T SS he FO . co R ALL nt D R en IS IG ts TR H in IB TS an UT R y IO ES fo N E rm R VE is D st ric tly pr oh ib ite

Start the class by having a student read aloud the scenario describing Linda Fogarty, a self-employed organic farmer who has a horticultural technology diploma. Discuss the fact that on a farm growing and selling produce, materials are rarely purchased as single items. Therefore, it’s important for the owner to compute unit prices to establish the lowest unit cost. Also discuss some of the other applications of math that Linda mentions in her work. Ask students if they can think of any other ways that math would be used in this context.

d.

Math on the Job

c) Does the wholesaler carry other products that she needs so that she can optimize her buying efficiency?

MathWorks 10 Teacher Resource

For the 200 g bag

$1.00 _____ = $0.005/g 200 g

For the 750 g bag: $2.70 = $0.0036/g _____ 750 g Thus, the best deal is the 750 g bag.

Activity 1.3￼ which price is right? Introduce this activity by having students read the items on the list. Ask the students if they have any of the items in their homes and to think of how the items are packaged. You could prompt the discussion by asking if they buy garbage bags as single items, or as packages of 10, 50, or 100. Then proceed to ask if they (or their family members) buy items based only on price or does brand loyalty, quality, or quantity affect their decisions?

After this discussion, make sure students understand that the mathematical purpose of the activity is to find unit costs. The business purpose is to compare prices in an effort to keep overhead costs to a minimum and make the most profit.

d.

Assign students to work in pairs to discuss questions 1 and 2.

PA re EV CIF pr A IC od L E uc UA DU tio TI C n ON AT of pa CO ION rt P AL or Y— P al N R lo O E f t T SS he FO . co R ALL nt D R en IS IG ts TR H in IB TS an UT R y IO ES fo N E rm R VE is D st ric tly pr oh ib ite

Ask students to complete the extension activity as a take-home assignment. Alternatively, you could ask students to bring in copies of newspaper ads and compare unit prices for selected goods in the next class.

After receiving a few answers, discuss with students the fact that if comparing equally favourable brands, the price does sway customers. But many companies purposely choose to package items in different sizes than their competitors do to make comparison pricing more difficult. Some large stores do put the unit prices (in very small print) next to the price, but that is not necessarily the practice in all regions. Sample Solutions

Extension

Find two ads for the same product in your local newspaper and compare the unit prices.

Comparing Different Brands—Same Size Item

Items per pkg.

Unit price

Brand B

Unit price

Light bulbs

4

$2.29

$0.57

$2.99

$0.75

Paper towels

6

$6.49

$1.08

$9.29

$1.55

Garbage bags

20

$8.79

$0.44

$7.48

$0.37

5

$7.95

$1.59

$7.69

$1.54

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Brand A

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Comparing Different Sizes—Same Brand Item

Smaller size

Price

Unit price

Larger size

Price

Unit price

Light bulbs

3

$2.49

$0.83

6

$4.49

$0.75

Paper towels

3

$3.69

$1.23

6

$6.49

$1.08

Garbage bags

20

$8.79

$0.44

30

$9.99

$0.33

5

$7.95

$1.59

8

$11.99

$1.50

Sponges

Chapter 1 Unit Pricing and Currency Exchange

39

For light bulbs, Brand A offers the lowest unit cost. For paper towels, Brand A offers the lowest unit cost.

For garbage bags, Brand B offers the lowest unit cost.

For sponges, Brand B offers the lowest unit cost.

Part B

For light bulbs, the package of 6 is the better buy.

For paper towels, the package of 6 is the better buy.

For garbage bags, the package of 30 is the better buy.

For sponges, the package of 8 is the better buy.

2. The package containing fewer items might have a lower unit cost if that size package was on a special promotion. In that case, the smaller package would be the better buy. 3. You might not choose to buy the product with the lowest unit price if the number of items in the package doesn’t meet your needs, or you prefer the quality of a different brand. You might have a coupon for a different brand that reduces the unit price, or you might belong to a customer loyalty program that gives points for a particular brand.

PA re EV CIF pr A IC od L E uc UA DU tio TI C n ON AT of pa CO ION rt P AL or Y— P al N R lo O E f t T SS he FO . co R ALL nt D R en IS IG ts TR H in IB TS an UT R y IO ES fo N E rm R VE is D st ric tly pr oh ib ite

d.

1. Part A

build your skills: Solutions $1053.00 = $87.75 1. ________ 12

The unit price of each sink is $87.75. 2. Package A:

$19.99 = $2.86/kg ______ 7

$22.99 ______ = $7.66/shirt 3

b) 2 packages of 3 plus 1 package of 1? 2($22.99) + $9.98 = $55.96

Or 1 package of 3 plus two packages of 2?

Package B:

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$35.95 = $2.57/kg ______ 14

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Package C has the lowest unit cost.

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$120.00 = $30.00/lock 3. _______ 4 $192.00 _______ = $32.00/lock 6 The first supplier has the lower cost per lock. When selecting a lock, you should also consider the quality of the locks, since you want them to be secure. 40

$15.49 4. a) ______ = $7.75/shirt 2

MathWorks 10 Teacher Resource

$22.99 + 2($15.49) = $53.97

The best combination is to buy one package of 3 shirts and two packages of 2.

5. First, convert everything to kilograms so all the denominator units are the same.

500 g = 0.5 kg

Now, calculate unit price. $7.50 ______ = $15.00/kg 0.5 kg $12.50/kg $19.50 = $13.00/kg ______ 1.5 kg So, the second price is the best buy.

Which combination will be the best price for 2.5 kg? Two of the second price plus one of the first? 2($12.50) + $7.50 = $32.50 Or 1 of the second price plus one of the third? $12.50 + $19.50 = $32.00

Kit 3: $70.50 ______ = $0.94 75 Kit 3 has the best unit value. How many does Jason need? First, divide the kits needed by the number of workers to see how many kits are needed. 250 workers ___________ = 3.34 75

The best price for 2.5 kg of meat can be obtained by buying 1 kg at the second price and 1.5 kg at the third.

Then calculate how much these 3 kits would cost.

PA re EV CIF pr A IC od L E uc UA DU tio TI C n ON AT of pa CO ION rt P AL or Y— P al N R lo O E f t T SS he FO . co R ALL nt D R en IS IG ts TR H in IB TS an UT R y IO ES fo N E rm R VE is D st ric tly pr oh ib ite

6. Convert the denominators to kg.

250 g = 0.25 kg

500 g = 0.5 kg

Then calculate unit price. $4.25 _______ = $17.00/kg 0.25 kg $7.95 ______ = $15.90/kg 0.5 kg $29.50 ______ = $14.75/kg 2

Extend your thinking

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The last package of meat has the lowest unit price. Nonetheless, the other store has two unit prices that are lower than this, so it would be better to buy your meat at the other store.

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d.

(You can factor out buying mostly from the first option because it is the most expensive.)

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7. There are a couple of ways to solve this problem. Here is one option. First, you need to figure out which kit has the lowest unit price so you can see which kit is the best value. First, calculate the price per worker each kit can cover. Kit 1: $42.50 ______ = $4.72 9 Kit 2: $58.25 = $1.46 ______ 40

3 × $70.50 = $211.50

How many workers still need kits if Jason buys 3 of kit 3? First, calculate the maximum number of workers these 3 kits will cover.

3 × 75 = 225

Then, calculate how many workers still need to be covered by subtracting 225 workers (already covered by 3 of kit 3) from the total number of workers.

250 − 225 = 25

Kit 1:

25 = 2.78 ___ 9

Calculate the total cost of these 3 kits.

3 × $42.50 = $127.50

Kit 2: One kit 2 will cover all 25 workers at a total cost of $58.25, which is a better buy than three of kit 1. Now, calculate the total cost. $211.50 (3 of kit 3) + $58.25 (1 of kit 2) = $269.75 The least expensive combination is 3 large kits and 1 medium kit at a price of $269.75, before taxes.

Chapter 1 Unit Pricing and Currency Exchange

41

1.3

Setting a Price

Time required for this section: 1 class

Math on the Job

solution

solution

Method 1: 2-step process

d.

Convert 45% to a decimal by dividing by 100 to get 0.45. Then multiply.

PA re EV CIF pr A IC od L E uc UA DU tio TI C n ON AT of pa CO ION rt P AL or Y— P al N R lo O E f t T SS he FO . co R ALL nt D R en IS IG ts TR H in IB TS an UT R y IO ES fo N E rm R VE is D st ric tly pr oh ib ite

Start the class with a student reading aloud the scenario describing Maurice, the cost estimator for a construction company. Discuss the number of factors that must be taken into consideration when estimating a job and the ramifications that can occur when done poorly. For example, estimating too little time can lead to labour shortages (for example, when hiring for different aspects of the job, the architects may be on a different time schedule than the stucco installers), cost overruns, disappointed clients, and so on. Furthermore, the cost estimator must ensure that the final price covers all costs plus makes a profit. Remember that building trades use imperial units of measurement.

$55.00 × 0.45 = $24.75

To find the selling price, add.

$55.00 + $24.75 = $79.75

Method 2: 1-step process

The total that Franka charges is the price she pays plus the markup. Therefore, the selling price is 145% of the price she pays (100% plus 45% markup). Convert 145% to a decimal by dividing by 100 to get 1.45. Then multiply. $55.00 × 1.45 = $79.75

What is the cost per square foot for stuccoing? $30 600.00 _________ = $8.50/sq ft 3600

Example 2

Using the example above, Franka was adding a markup of 145% on the same $55.00 pair of jeans.

Explore the math

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The PST rates given in Figure 1.1 were accurate at the time of publication. Ensure that students use current rates in their calculations.

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The idea of percent is familiar to students. However, many students are used to calculating percents as a two-step process rather than in one step. Help them to incorporate the one-step method into their repertoire of skills.

Method 1: 2-step process

Convert 145% to a decimal by dividing by 100 to get 1.45. Then multiply. $55.00 × 1.45 = $79.75

To get the selling price, add. $55.00 + $79.75 = $134.75 Method 2: 1-step process

Example 1 Franka purchases jeans wholesale for her designer clothing store. She pays $55/pair and charges a markup of 45%. What is the selling price?

42

MathWorks 10 Teacher Resource

Convert 145% to a decimal by dividing by 100 to get 1.45. Add the 145% markup to 100% of the original price paid to find 245%. Convert to a decimal and multiply. $55.00 × 2.45 = $134.75

For some students, you may want to show what a markup of 100% looks like first, then proceed to 145%.

PA re EV CIF pr A IC od L E uc UA DU tio TI C n ON AT of pa CO ION rt P AL or Y— P al N R lo O E f t T SS he FO . co R ALL nt D R en IS IG ts TR H in IB TS an UT R y IO ES fo N E rm R VE is D st ric tly pr oh ib ite

Discuss the ideas Concert Promoter

This discussion allows students to think about how market forces affect pricing decisions. Sometimes a product’s price has an adverse effect on sales. When that happens, the business owner must make decisions to minimize losses. Students can begin by discussing the many factors that can affect concert ticket sales, such as the popularity of the band, the price, the number of events occurring at the same time, and so on. Have students discuss the questions in small groups for about 5 minutes. After that, sample answers could be given to the entire class for a wrap-up. sample answers

1. Raise the prices, see if you can add another show, limit the number of tickets per person.

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2. Lower the prices, increase the promotions/ads, give tickets away as radio prizes.

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the discussion by asking students to think of items that they buy or use in each of the four seasons (for example, snow blowers in the winter, leaf blowers in the fall, lawn mowers in the summer, and rakes in the spring). Then discuss how the prices of those items change as the seasons change. For example, in Winnipeg, local summer produce such as strawberries is much cheaper in July than imported produce in December. Furthermore, holidays often mean that demand for items changes. For example, turkey and ham sales usually increase at Thanksgiving.

d.

Recall that multiplying by 1 adds in the original $55.00 by multiplication rules. Now, changing the 1 to a 2 accounts for the fact that 145% means that you are doubling the original price and adding 45%.

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3. There are not many circumstances, since your goal is to at least break even. However, in some dire circumstances, it may be better to make some money rather than no money. Discuss the ideas Seasons and Holidays This discussion builds upon the Concert Promoter discussion as the change in season and holidays also changes consumer demand. You could begin

Moreover, students may have discussions in social studies about the effect natural disasters have on prices. For example, flooding during the rainy season in India can cause rising prices, hurricanes can ruin beaches in Mexico, affecting the tourist industry, and avalanches in BC and Alberta can affect the ski/snowboarding resorts or equipment businesses. Business owners, therefore, need to prepare for the time of the year and adjust prices to take advantage of consumers’ seasonal demands. sample answers

1. Mother’s Day (May), high school graduation (June), and weddings (summer) tend to be a high volume time. Students may think of other events that may cause a demand for roses (for example, Valentine’s Day). 2. Summer: road trips tend to increase. In light of rising gasoline costs, encourage students to talk about trade and economic fluctuations.

3. The price of toys at Christmas: students will be able to suggest many examples. 4. Certain jewellery pieces, such as blue diamonds or real fresh-water pearls, expensive watches like Patek Philippe or Rolex, rare art works, first edition books, certain foods such as caviar. 5. Answers will vary, but students may notice that prices are often set just below a psychological

Chapter 1 Unit Pricing and Currency Exchange

43

kilogram also creates the impression that items are less expensive than they are. Mental Math and Estimation

d.

Ask students to work in pairs to collaborate on strategies. Students should be able to see that the price difference between the two helmets is $5.00 and multiply that by 25 to arrive at $125.00.

PA re EV CIF pr A IC od L E uc UA DU tio TI C n ON AT of pa CO ION rt P AL or Y— P al N R lo O E f t T SS he FO . co R ALL nt D R en IS IG ts TR H in IB TS an UT R y IO ES fo N E rm R VE is D st ric tly pr oh ib ite

turning point, such as $39.95 instead of $40.00. Other examples in which goods and services are advertised to seem less expensive than they are include plane fares that do not include taxes and fuel surcharges, oneway trips instead of round trips, hotel prices quoted by one night prices when a minimum stay is three nights. Sometimes manufacturers advertise an old price but have reduced the size of the package. Selling foods using the 100gram price rather than the price per pound or

build your skills: Solutions

These solutions were worked out using the tax rates in effect at the time of publication and are included on p. 29 of the student book. 1. As a percentage, the regular price plus the markup is 100% plus 60%, which equals 160%.

1.60 × $22.75 = $36.40 a shirt

2. $49.95 + (2 × $129.95) = $309.85 5 = 0.05 ____ 100

0.05 × $309.85 = $15.49

The total GST paid on the items is $15.49.

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$309.85 + $15.50 = $325.34

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The total cost of the items, including tax, is $325.35.

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3.

4. As a percentage, the regular price plus the markup is 100% plus 25%, or 125%.

The sink:

$89.95 × 1.25 = $112.44

The bathtub:

$639.95 × 1.25 = $799.94

GST is 5%.

You would pay $217.68 for a hard hat and a pair of steel-toed boots in Saskatchewan.

2 × $74.95 = $149.90

$149.90 × 1.25 = $187.38

Add to find the total she charged her customer, excluding tax.

$112.44 + $799.94 + $187.38 = $1099.76

5. a)

50 × $3.50 = $175.00

175 × $3.99 = $698.25

250 × $2.00 = $500.00

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As a percentage, the Fort McMurray price plus the markup is 100% plus 10%, which equals 110%.

2 faucets:

$179.90 × 1.10 = $197.89

As a percentage, the Saskatchewan price with PST and GST is 100% plus 5% GST plus 5% PST, or 110%.

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$197.89 × 1.10 = $217.68

MathWorks 10 Teacher Resource

$175.00 + $698.25 + $500.00 = $1373.25 Her total income is $1373.25. b)

100 × $3.50 = $350.00

100 × $2.00 = $200.00

Extend your thinking 8. First, convert kilograms to grams.

10 kg × 1000 = 10 000 g

Find the unit price per gram. $175.00 = $0.0175/g ________ 10 000 g

6. a) $2.50 × 1.15 = $2.88

Calculate the price for 250 g.

b) First, find the difference in price per person.

250 g × $0.0175 = $4.38

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The new unit price would be $2.88.

$2.88 − $2.50 = $0.38

Then, find the difference for 100 people.

$0.38 × 100 = $38.00

She would make $38.00.

c) Student answers will vary but should show they have considered various options such as trying to source less expensive ingredients and supplies; making her portions per person smaller; or trying to find ways she can be more efficient with her time, including perhaps hiring a delivery person.

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7. Answers will vary but students may suggest discounts of as much as 50%, since that is a fairly common discount for out-of-season items. Students should recognize that Marie’s profits will be lower, but that some revenue is better than no revenue on those items. Marie’s reasoning would be that she has paid for these items and that it’s best if she tries to recover some of the money she spent.

a) Student answers will vary but they should consider overhead costs such as the rent and utilities for the store; equipment costs such as a display fridge for the cheese and knives or other tools; materials for displaying, storing, and packaging the cheese such as plastic wrap; and staff time for stocking, cutting, and serving the cheese. b)

$4.38 × 1.40 = $6.13

It would cost $6.13 for 250 grams.

c) Since the customer is getting a discount of 15%, they are paying 85% of the original price (100% − 15%).

$6.13 × 0.85 = $5.21

The price would be $5.21.

d) Yes, you would still be making a gross profit. $5.21 − $4.38 = $0.83

But you would also need to consider whether this smaller margin would cover your additional costs, as discussed in a).

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d.

She receives $150.00 more income if she sells 100 quarts directly from her farm. Students may suggest that she would sell to a wholesaler because she may receive large orders from wholesalers or she may not be able to sell all her crop directly or at the farmers’ market.

Chapter 1 Unit Pricing and Currency Exchange

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1.4

On Sale!

Time REquired for this section: 2 classes

Math on the Job

Cost Method 1: 2-step process

d.

First find the percent discount on one tile and subtract that amount from the original price. Remember to convert the percent to a decimal.

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Start the class with a student reading aloud the scenario featuring Daniel, the owner of a stone and tile company. Discuss with students how businesses like to keep their stock up-to-date so that customers will want to keep coming back to see what’s new and trendy. Discuss the fact that the price of many building items decreases when larger quantities are purchased (such as the price of limestone per square yard or lumber per linear foot, and so on). In order to turn over their stock, businesses tend to have end-of-season sales and clearances. Have students make a connection to sales that they have attended and when they occurred. Students then could discuss whether a purchase they made was needed or an impulse motivated by the “sale” tag.

$6.99 × 0.15 = $1.0485

$6.99 − $1.05 = $5.94 per tile

$5.94 × 50 = $297.00

Cost Method 2: 1-step process

If the discount per tile is 15%, then you can calculate the discount percentage.

100% − 15% = 85%

$6.99 × 0.85 = $5.94 a tile

Multiply to find the total cost.

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Imperial measurements are standard in the construction trades. Discuss with students the number of tiles needed. Each tile is 12″ × 12″ or 1′ × 1′. Calculate using feet since the price is given per square foot. Since 50 square feet of slate are needed, calculate the cost of 50 tiles.

Explore the math

Discuss with students some current promotions that they have seen on television or in the local store. Ask them if they think the sales promotions work. Are there cases when a store holds too many sales (for example, some electronics stores have weekly sales flyers)? If so, how might that affect consumer behaviour (customers take the sale for granted, thereby hesitating to buy at full price)?

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Note: Some students may not understand the concept of “square feet.” Draw a square on the board and label each side as 4. Then draw in vertical and horizontal lines within the square to total 16 boxes within the space. Each side measures 4 linear feet. The area within the square becomes the square footage since the area of the square (4 × 4) is 16. In this example, 50 square feet could be represented by a 10′ × 5′ rectangle or a 25′ × 2′ rectangle, or some other shape. Point out that the shape does not have to be a square; remind students that it is area that is being discussed.

$5.94 × 50 = $297.00

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MathWorks 10 Teacher Resource

Mental Math and Estimation Students should be able to round the price up to $1000.00 and realize that 20% is $200.00 so they will pay about $800.00 at the sale price before taxes. Ask students to share their personal strategy with a partner.

Activity 1.4￼ taking advantage of sales promotions Sample solutions Sample answers for questions 1 and 2 are shown in the table below.

d.

Ask students to work in pairs for this activity, role-playing roommates in their first apartment. Have them look through flyers or internet sales sites to comparison shop. Remind students that usually first-time homeowners/renters have limited budgets, which is why you need to buy the items on sale.

Item

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Assessing promotions Store name*

Stereo Couch

Product*

Promotional pitch

Regular price

Sale price

Percent discount

Discontinuing item sale

$699.99

$499.99

29%

Pre-holiday sale

$569.99

$484.49

15%

* Students will fill in the exact store and product information from the sources they use.

3. The first store advertised a $200.00 discount. The percent discount is 29%.

The second store advertised a 15% discount. The discount amount is $85.50. The sale price would, therefore, be the original cost minus the discount.

4. The promotions were not misleading but you had to read the fine print on the first store’s sales flyer to see that the reduced price was only in effect for one week. 5. Ask students to compare answers and see who found the best price for each item.

$569.99 – $85.50 = $484.49

build your skills: Solutions

$2.94 ______ = 0.249 $11.78

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1. a) Calculate the price of a package at half price by dividing by 2.

This can be rounded to 25%.

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$5.89 _____ = $2.95, rounded off 2

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This can also be done mentally: one package is 50% off so you are paying 75% of the total price on two packages.

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Calculate the cost of one package at full price and one package at half price.

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$5.89 + $2.95 = $8.84 b) First, calculate the total regular price.

2 × $5.89 = $11.78

Then, calculate the difference between the regular cost and sale costs for the two packages. $11.78 − $8.84 = $2.94 Finally, calculate the percentage.

2. a) At Ross’s store, the total price, including GST, can be found by adding 100% and 5%, converting to a decimal, and multiplying the price by 1.05. $49.95 × 1.05 = $52.45

At Al’s store, subtract the discount percent, 15%, from 100%, and convert to a decimal. Then multiply the price by 0.85 to find the sale price. Chapter 1 Unit Pricing and Currency Exchange

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Find the total savings on labour for 16 hours.

Then, calculate the sale price plus GST. $47.56 × 1.05 = $49.94

b) Al is right—with the sale, the racquet at his store is less expensive.

You would save $28.80 on labour. 5. a) Calculate the total cost of 20 fans at each store. The first wholesaler offers 5% off $157.00.

3. a) Calculate cost of morning highlights. Convert 15% to a decimal and subtract that amount from the original price.

Calculate the cost of one fan with the discount.

$55.00 × 0.15 = $8.25

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$157.00 × 0.95 = $149.15

$55.00 − $8.25 = $46.75

Then, calculate the cost with GST.

Calculate cost of mid-afternoon highlights.

$149.15 × 1.05 = $156.61

$55.00 − $5.00 = $50.00

Find the cost of 20 fans.

Morning appointments will get you the lowest price on highlighting.

b) Answers will vary but students should offer reasons for their answers. For example, the discount will appeal more because it allows you to save more money, or the coupon will appeal more because you know immediately how much you are saving without having to do any calculations. The time of day the person is available will also affect his or her choice. 4. First, calculate the roofer’s savings with a 20% discount. Convert 20% to a decimal.

$156.61 × 20 = $3132.15

The second wholesaler charges $149.00 each for orders of 10 or more. Find the cost of one fan, with GST. $149.00 × 1.05 = $156.45

Find the cost of twenty fans.

$156.45 × 20 = $3129.00

b) The second wholesaler offers a better buy.

6. a) Calculate the percentage of markdown each item has been given. Shirts:

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$27.50 × 0.20 = $5.50

$31.99 – $19.99 = $12.00 discount $12.00 ______ = 0.375 $31.99

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Calculate your cost of materials if he passes on 50% of his discount by dividing by 2.

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$5.50 _____ = $2.75 2

The discount is 38%, rounded to the nearest percent.

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Your discount on a square metre is $2.75.

Shorts:

$27.50 – $2.75 = $24.75

$24.95 − $16.95 = $8.00 discount $8.00 ______ = 0.32 $24.95

Your cost for a square metre would be $24.75. $24.75/m2 × 74 m2 = $1831.50 Calculate his hourly rate with a 5% discount. Convert 5% to a decimal. $36.00 × 0.05 = $1.80 48

$1.80 × 16 = $28.80

d.

$55.95 × 0.85 = $47.56

MathWorks 10 Teacher Resource

The discount is 32%. Jacket: $49.99 − $24.99 = $25.00 discount

$25.00 = 0.5 ______ $49.99 The discount is 50%.

b) Add up all the discounts.

12.00 + $8.00 + $25.00 = $45.00 total $ savings

$26.00/h × 0.95 = $24.70/h

Find the total discounted price before tax.

Or you can calculate the total regular price.

$24.70/h × 55h = $1358.50

$31.99 + $24.95 + $49.99 = $106.93

Find the total price, including tax.

Calculate the total sale price.

$1358.50 × 1.12 = $1521.52

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d.

Calculate the discount price by subtracting 5% from 100% of the total price to find 95%. Convert to a decimal and multiply to find the discounted price.

$19.99 + $16.95 + $24.99 = $61.93

Calculate the savings.

Find the total savings.

$1601.60 − $1521.52 = $80.08

$106.93 – $61.93 = $45.00

Your total savings are $80.08.

b) Similarly, calculate the second person’s savings.

The most money is saved on the jacket. Extend your thinking

7. a) 5% GST + 7% PST = 12% total tax

$26.00 × 60 × 1.12 = $1747.20 regular price

$24.70 × 60 × 1.12 = $1659.84 discount price

Calculate the regular price.

$26.00/h × 55 h = $1430.00

$1747.20 − $1659.84 = $87.36

Add 12% to 100% of the total price to find 112%. Convert to a decimal and multiply.

The total savings are $87.36.

Then, calculate the difference in what you saved.

$1430.00 × 1.12 = $1601.60

The owners of the second house save $7.28 more than you do.

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Solutions

Canadian currency

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the roots of math

$87.36 − $80.08 = $7.28

1. Answers will vary. In BC, the Tsimshian people used Eulachon oil for trade. Other items used for trade by First Nations people included preserved meats, rare stones, tools, and furs. 2. Answers will vary. Possible factors to consider when determining the value of goods or services include the time spent providing a

service, the original monetary value of the item, or the rarity of the item. 3. Answers will vary. Possible answers could include that money is valuable because it can be exchanged for goods or services, or that the value of a country’s currency depends on the strength of its economy. Chapter 1 Unit Pricing and Currency Exchange

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1.5

Currency Exchange Rates

Time REquired for this section: 2 classes

Math on the Job

Explore the math

d.

have learned previously from the media or from another source about exchange rates.

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Start the class with a student reading aloud about Naomi Coates, the office manager for a potato grower. Have students think about items that they use at home that are not usually from Canada, such as electronics, vehicles, or clothes. Since many items come from international sources, the buyer of the items must convert Canadian dollars to the international currency of the country where the product is being purchased. Then have students think of items that Canada exports, for example, wheat, apples, and buses. Again, the seller must convert the amount received from the buyer to Canadian funds. Sellers must also price the product to minimize losses from the fluctuating dollar. This discussion can lead into considering the effect that the rise or fall of the Canadian dollar has on manufacturers, agricultural producers, mills, and other businesses that depend on import and/or export of goods. A linked discussion can address the impact currency exchange rates have on consumer prices. Challenge students to think of examples from their own lives and from what they

After reading the Explore the Math lesson on currency, you may want to try the extension activity on exchange rates. In this extension, students explore the issues that underlie exchange rates and their purpose. Before beginning the activity, you will want to explain a little more about exchange rates and how they are established. Most of the world’s major currencies are flexible in that they rise or fall with changes in the supply and demand for the currency, but nations sometimes intervene to try to manage their currency’s rate of exchange. Changes in exchange rates can affect trade among nations and a nation’s domestic economy. The value of a nation’s currency, and thus its exchange rate compared to other currencies, is influenced by many factors, from the general—a country’s economic and political situation—to the specific—interest and employment rates.

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extension Activity￼ exchange rates around the world

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In this activity, students are divided into two different groups: country A and country B. The residents of each country participate in two auctions. In the first, students can only buy goods produced in their own countries while in the second, they exchange their currency with foreign currency to buy foreign goods. Materials: Prepare 2 bags with 100 paperclips in each and 2 bags with 200 beans in each to be used as currency. For the items to be auctioned,

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there should be 4 of one item (such as pencils) and 2 each of 4 different items (such as 2 postcards, 2 stickers, etc.). Also have 50 to 100 each of 2 different smaller items (such as soft and hard candies). In the following description, specific items are used, but you are free to choose whatever will work best with your class. Now, split these items into two sets so that you have identical sets for auction 1 and 2.

Have students volunteer, or choose leaders for each country. Each leader stands in front of their country’s residents with the items their country has produced in the last year—for country A, a postcard, a pack of gum, a pencil, and the soft candy and for B, a chocolate bar, a bumper sticker, a pencil, and the hard candy. Hold the auctions at the same time. The leader auctions off the three larger goods to the highest bidder within each country. Record the prices they go for. Then students buy the small items with their currency until everyone has something (students may not save their money).

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Collect all the currency after the auction. When the auctions are finished:

Run the auctions for both countries simultaneously as before. Students choose which country’s items they want and thus at which auction they will be. Again, record the prices the larger items go for and, when the auction is finished, let the students buy the candies. When these auctions are complete:

Compare the number of beans and paperclips paid for auctioned items. Again, the number of beans paid was likely more than paperclips. Popular goods may have higher prices in the second auction because there were more potential buyers. • Explain that an exchange rate is defined as the price or value of a nation’s currency in terms of another nation’s currency, and ask students to share their experiences with exchange rates. • Determine the exchange rate in the auction activity. • Ask students to suggest reasons why people in one country would want currency from another. •

Mental Math and Estimation

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Compare the amount of currency paid for the 3 larger items in each country. The country with more currency (beans)probably paid more. • Compare the price of the same item for both countries. Again, those with beans likely paid more. • Ask students to discuss whether the people in the country with more currency (the beans) are richer. Students should come to the conclusion that the amount of currency in circulation does not make the country wealthier. What is important is how much the currency will buy in different countries. •

In order to buy something from another country, you must have the right currency. Before the auction, allow the students to barter with one another to trade currencies. There is no fixed rate and no one has to exchange if they do not want to. When students exchange currency, they must report it on the board, listing how many paperclips were exchanged for how many beans.

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Both countries produce one good that is the same (the pencils) and other goods that are unique (postcards, bumper stickers, etc.). List each country’s goods on the board. For example, country A produces postcards, packs of gum, pencils, and a large supply of soft candy. Country B produces chocolate bars, bumper stickers, pencils, and a large supply of hard candy.

Auction 2: For this auction, trade is allowed between the two countries. Distribute the second set of currency to the students as before and give the leaders the second set of goods.

d.

Auction 1: For this auction, the two countries are not allowed to trade with each other. Country A uses paperclips for money while country B uses beans. Distribute the 100 paperclips among the students in country A and the 200 beans among the students in county B, giving students unequal amounts. This represents the income they earned during the past year.

One strategy for solving this question is suggested here. Students may suggest other methods. Ask students to share their strategy with a partner. First round €95.00 up to €100.00. Then round 1.644 to 1.5. Add 50% of 100 to 100 to get $150.00, the approximate price of the hotel room.

Chapter 1 Unit Pricing and Currency Exchange

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Activity 1.5￼ What’s Your Ride? Survey The Royal Bank website (www.rbcroyalbank.com) lists both buy and sell rates. This is why it is one of the recommended sites. The site www.XE.com is an alternative site that could be consulted. Note, however, that these are external websites and are not endorsed by either the WNCP or Pacific Educational Press.

In doing the currency conversions, students will discover that some websites list separate buy and sell rates, while others list mid-market rates or nominal rates. Mid-market rates are derived from the mid-point between the buy and sell rates of large-value transactions in the global currency market. The Bank of Canada currency converter lists a nominal rate, which shows where the bank estimates the market to have been at noon on that day. Since buy rates and sell rates include overhead and profit margins that are set independently by each foreign currency provider, they will vary depending on the provider and will always be different than the mid-market rates and the nominal rates.

Figure 1.2, on p. 45 of the student book, was compiled from two websites, the Royal Bank of Canada and HSBC.

sample Solutions

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This activity allows students to discover the difference between buy and sell rates as well as to explore different world currencies. To start this activity, have students look at the chart and the range of prices listed in the column Foreign amount. Ask them to guess which vehicle costs the most in Canadian dollars and which costs the least.

Extension

Conduct online research to select two additional vehicles from countries of your choice. Research the models and their prices in the currency of their origin. Using an online currency converter, determine the price in Canadian dollars.

The table below gives sample answers for question 1.

Make and model of vehicle

Name of currency

Exchange rate

Fiat 500

euro

Maruti Gypsy King

rupee

Mini Cooper S

pound

United States

Dodge Ram 3500

US dollar

Japan

Daihatsu Move Latte L

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comparing cars

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Foreign amount

Canadian amount

1.597 95

€10 900.00

$17 427.66

0.024 1358

Rs 537 921.07

$12 983.16

2.029 99

£16 245.00

$32 977.19

1.024 88

US$30 420.00

$31 176.85

0.009 48742

¥1 073 380.00

$10 183.61

Rates as of October 10, 2008

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b) Using the currency convertor at www.rbcroyalbank.com, you determine that converting €10 900.00 to Canadian dollars will yield $17 146.79 CAD.

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3. The following sample answers use the price of the Fiat 500 as listed in the table.

4. a) The bank builds a profit margin and overhead into their buy and sell rates. In other words, they do not convert money for free. The buy rate that the bank offers on the euro on this date is €0.6079 for $1.00 CAD compared to a sell rate of €0.6130.

d.

2. This question allows students to get engaged in the topic by finding photos and prices of vehicles in different countries. Auto show websites, such as for the Dubai motor show, are a good place to find representative makes and models. However, if time is an issue or if the classroom does not have high-speed internet access, simply ask students to choose one vehicle from the list.

a) Using the calculator at the Royal Bank website (www.rbcroyalbank.com), you determine that to buy €10 900.00, you will need $17 781.17 CAD. The rate for $1.00 CAD is €0.6130.

b) The Bank of Canada website lists a nominal rate, which is neither a buy rate nor a sell rate. This question requires students to find the sell rate (the bank is selling the euro to the customer), so the Royal Bank sell rate will likely be higher than the nominal rate. On October 24, 2008, the Bank of Canada rate for €10 900.00 was $17 564.26 CAD while the RBC sell rate was $17 781.17 CAD.

$17 781.17 − $17 146.79 = $634.38

The difference between the buy and sell rates is significant when you are converting money at the bank.

Extension Solution

You may want to alert students that finding prices in the currency where the car originates can be challenging, especially since some websites will not be in English. Also, many websites will list prices in US dollars.

Activity 1.6￼ Calculate Foreign exchange

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This activity allows students to investigate the impacts of buying items from outside Canada, using the currency exchange skills they have developed. Remind students that the fluctuating dollar can have a big impact on whether this is an effective way to buy things, and that there may be extra costs such as duty and shipping to consider.

sample solutions

Answers will vary. For question 4, students may suggest that buying from a foreign source could be less expensive and allow a wider choice than buying locally. They may also suggest that buying from a foreign source could take longer, that it is difficult to assess the quality of an item when buying online, and that duty and shipping may eliminate any price differential.

Chapter 1 Unit Pricing and Currency Exchange

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build your skills: Solutions 1. You would choose the bank selling rate to buy these currencies.

5. Use the bank selling rate because the bank is selling these currencies to you.

a) 1.644 814

$650.00 _________ a) = 395.18 euros 1.644 814 $650.00 _________ = 639.13 francs b) 1.017 007 $650.00 _________ = 3702.48 kronor c) 0.175 558

b) 0.133 451 c) 0.019 360

d) 3702.00 kroner × 0.165 558 = $612.98 CAD

a) 0.009 295 b) 0.950 964 c) 1.004 350

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2. You would choose the bank buying rate to sell these currencies.

3. Use bank buying rates because the bank is buying the currency from you. a) 4500.00 pesos × 0.083 443 = $375.49 CAD

She receives a lower amount back because bank buy and sell rates are different—the banks build in a profit margin for exchanging money.

6. Chris is buying these currencies so he will pay the bank selling rate.

b) $25 000.00 Hong Kong × 0.128 451 = $3211.28 CAD c) 2200.00 euros × 1.580 814 = $3477.79 CAD

d) 8545.00 Scottish pounds × 1.996 146 = $17 057.07 CAD 4. Use the bank sell rate because the bank is selling the currency to you.

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______________ = €729.57 $1200.00 CAD 1.644 814

$5000.00 × 1.038 650 = $5193.25 CAD

St. Andrew’s:

£8500.00 × 2.060 146 = $17 511.24 CAD

Spring City Golf & Lake Resort:

¥26 600.00 × 0.162 600 = $4325.16 CAD SAFRA Resort & Country Club:

S$15 000.00 × 0.762 280 = $11 434.20 CAD Leopoldsdorf:

€4000.00 × 1.644 814 = $6579.26 CAD

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Megan will have €729.57 in the local currency for her expenses in Germany.

Pebble Beach:

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Country

Golf course

United States

Pebble Beach

Scotland

St. Andrew’s

China

Spring City Golf & Lake Resort

Singapore

SAFRA Resort & Country Club

S$15 000.00

$11 434.20

Austria

Leopoldsdorf

€4000.00

$6579.26

MathWorks 10 Teacher Resource

Estimated funds needed

Estimated funds needed in CAD

US$5000.00

$5193.25

£8500.00

$17 511.24

¥26 600.00

$4325.16

Extend your thinking

dollars, which you will then convert to Canadian dollars.

$8.95 CAD = US$8.73 (rate: 1 CAD = 0.975 229 USD)

$8.95 CAD = A$9.39 (rate: 1 CAD = 1.049 05 AUD)

US:

$8.95 + $1.00 = $9.95 $9.95 ÷ 1.004 350 = US$9.91

Australia:

$8.95 + $2.00 = $10.95

$10.95 ÷ 0.950 964 = A$11.51

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b) Use the bank buy rates for this question because you will be selling the vinegar abroad and thus receiving US or Australian

d.

7. a) The answer to this question will depend on when the conversion website was accessed.

You would have to set the US price at $9.91 and the Australian price at $11.51.

Reflect on your Learning Unit pricing and currency

Ask students to review and reflect on the list of new skills and knowledge they have encountered in this chapter.

Practise your new skills: Solutions

1. a) For the first part of this question, you can divide 80 by 2 to find 40 km in half an hour.

10

For the second part of this question, you multiply 80 km by 2 to get 160 km, and then add the 40 km from above to find 200 km in two-and-a-half hours.

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b) To find the amount of Canadian dollars for 10.00 euros, multiply the exchange rate by 10 to get $15.90 CAD.

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2. Using proportional reasoning, the bakery would sell 300 loaves of white bread that day. 30 = 7.5 m/second 3. a) ___ 4 $2.80 b) _____ = $0.23/egg, 12 rounded to the nearest cent

4. For the first part of this question, students should demonstrate that the answer is no. They can demonstrate this by converting both sides to a decimal or finding a common denominator to compare the fractions. 4 ≠ __ __ 5 6 7 5 ≠ ___ 8 __ 7 10 4 ≠ ___ 8 __ 6 10 These are not proportional. If you reduce an 8″ × 10″ photograph, you can make any proportion that is equivalent to 8:10. For example, 4:5 or 2.5:3.125. $1.89 5. a) _____ = $0.38/lb 5 lbs $5.99 ______ = $0.30/lb 20 lbs The 20 lb bag is the better buy.

Chapter 1 Unit Pricing and Currency Exchange

55

$15.00 c) ______ = $0.20/lb 75 lbs

6. Calculate the price at Krazy Krazy. $1299.99 − $300.00 = $999.99

Calculate the price at Too Good To Be True.

$1299.99 × 0.30 = $390.00

$1299.99 − $390.00 = $909.99 $909.99 × 1.05 = $955.49

Too Good To Be True offers the best deal.

7. a) The simplest way to solve this problem is to realize that 1 cup of sugar is double the amount called for in the recipe, so you need to double the flour. 1 cups × 2 = 4 __ 2 cups 2 __ 2 2 2 __ 4 cups = 5 cups of flour 2 b) To make this simpler to solve, students may first want to convert the fractions of cups to a decimal. 1 cups = 2.5 cups 2 __ 2 1 cup = 0.5 cups __ 2 Flour: 2.5 = __ ___ x 12 8

10

20 = 12x

20 = ____ 12x ___ 12 12 1.666 = x

Now, convert 1.666 back to a fraction to get 1 _23 cups of flour. Sugar:

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0.5 = __ ___ x 12 8

( ) ( )

96 ___ 0.5 = __ x 96 12 8

48 = ____ ___ 96x 12 8

4 = 12x

4 = ____ 12x ___ 12 12 0.333 = x

Again, convert back to a fraction to get _ 13 cup of sugar.

8. a) Let x be the number of Canadian dollars. ________ _____ €500.00 = €1.00 x $1.59

(

) (

)

________ _____ $ 1.59x €500.00 = €1.00 $1.59x x $1.59

1.59(500.00) = x

$795.00 = x

It will cost her $795.00 CAD to buy €500.00.

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This is the best buy but you will want to consider whether or not you will use 75 lbs of potatoes.

Each side can be simplified by dividing the numerator by the denominator.

d.

b) You will want to consider the quality of the potatoes and the quantity that you can use. You would also consider the type of potato and perhaps whether it is organic or not. Students may offer a variety of answers.

The common denominator is 12 multiplied by 8, or 96.

( ) ( )

2.5 = __ x 96 96 ___ 12 8

240 = ____ ____ 96x 12 8 56

MathWorks 10 Teacher Resource

b) $795.00 × 1.005 = $798.98 The final cost is $798.98. 9. a) First, calculate the total amount she spent in euros. 15(€28.92) + 40(€9.95) = €831.80

Then, determine the unit rate for 1 euro. $1.00 = 1.5437 _______ €0.6478 Multiply the unit rate by the total amount she spent. 1.5437 × 831.80 = 1284.05

c) Students can plot this using pen and paper or, if using a spreadsheet, they can use the built-in graphing tool and choose the scatter plot feature. Students should extend their original tables first so that they can then extend their graph to answer the following question.

The fabric cost $1284.05 CAD.

d.

10. a) Divide $27.00 by 3 to find $9.00/h.

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You have $1.50 unaccounted for, and $1.50 divided by 3 is $0.50. Add this to $9.00 to get $9.50/hr.

b) Copy the following table or use a spreadsheet to make a table showing the number of hours versus dollars earned. Calculating Earnings Hours

Dollars earned

0

0

1

$9.50

2

$19.00

3

$28.50

4

$38.00

5

$47.50

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To find the dollars earned, multiply the number of hours by the hourly rate.

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On a spreadsheet, students would use the following formulas. They can also use the “edit—fill down” feature to carry the first formula down the table.

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Calculating Earnings Dollars earned

0

=A3*9.50

1

=A4*9.50

2

=A5*9.50

3

=A6*9.50

4

=A7*9.50

5

=A8*9.50

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Hours

d) Student answers for this question will vary since they are reading their own graph, but the values they determine should be close to the following calculated values. 3.5 hours worked:

$9.50(3.5) = $33.25

She will earn $33.25 for working 3.5 hours. 12.5 hours worked:

$9.50(12.5) = $118.75

She will earn $118.75 for working 12.5 hours.

Chapter 1 Unit Pricing and Currency Exchange

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Sample chapter test Name:

Date:

Part A: Multiple Choice Choose the best response to each of the following questions:

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d.

1. Jean and her best friend, Verna, want to buy 3 DVDs that are regularly priced at $20.00 each. Today, three stores, A, B, and C, have the DVDs on sale. At which store will the friends spend the least amount of money if they buy 3 DVDs? DVD Sales

a) Store A

Store

Sale offer

A

Buy 1 and get 2 for half price!

B

All DVDs on sale at 35% off!

C

Buy two and get one free!

b) Store B

c) Store C

d) Stores A and C

2. Lisa is managing a popular new band. Each of the songs the band plays is 3 to 5 minutes long. Lisa needs to let the venue where the band will be performing know approximately how many songs they will play in two hours. Which is the best estimate?

a) 8 songs

b) 12 songs

c) 20 songs

d) 30 songs

3. Whose rate of pay is the highest?

a) Antoine earns $66.00 in 8 hours.

b) Laurie earns $72.00 in 5 hours.

c) Ken earns $51.00 in 5 hours.

d) Sara earns $89.00 in 6 hours.

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a) $25.03

b) $3595.27

c) $359.53

d) $250.30

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4. Janine just came home from a vacation in Cancún, Mexico and has 300.00 pesos left over. If the bank buys pesos at $0.083 443 CAD, how much will Janine get back in Canadian dollars?

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5. Adelina will be going to a construction trade show in Paris this year. Her budget is $1200.00 CAD. If the bank sell rate is one euro for $1.644 814 CAD, how many euros will she have to spend in Paris?

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a) €1973.78

b) €729.57

c) €731.71

d) €1968.00

Part B: Short Answer 1. Find the unit cost of each of the following items. Show your calculations. b) a package of 25 Richelieu screws for $1.45

d.

a) a package of 10 wood floor tiles for $69.07

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2. Write as a unit rate. Show your calculations. a) 25 m of tape for $0.95

b) 120 words typed in 3 minutes

c) driving 240 km in 4 hours

d) $22.80 for 3 hours of work

Part C: Extended Answer

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1. Samir must arrange for catering for a lunch at the office where he works. The caterer tells him that they charge $65.00 for six people and $12.00 for each additional person. There will be between 9 and 15 people at the lunch. Before he can place the order, the accounting office needs Samir to complete a budget form that shows the price for each person.

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a) Find the cost for each person if 10 people attend the lunch. Show your calculations.

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b) Fill in the table and show your calculations below. Number of people

9

10

11

12

13

14

15

Cost

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d.

2. Patrick and Lylah work as lift truck operators in a warehouse. They can unload pallets of goods off transport trucks and shelve them at the same speed. It takes Patrick 1 hour to unload and shelve 17 pallets. How long would it take Lylah to unload and shelve 25 pallets? Show your work.

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3. The Winnipeg Harvest food bank has started its semi-annual food drive. To support the drive, local grocery stores have advertised a sale on canned soup. Two different brands of soup are available in large quantities. Tastes Like Homemade is being sold at $18.89 for 12 cans of 284 mL. Savory Soup is being sold at $30.69 for 24 cans of 284 mL.

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a) Which is the better deal between these two brands? Justify your answer by showing two different ways to solve this question.

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b) If a school raises $500.00 to buy soup for the food bank, how many cans of the lowest-priced soup can the school buy?

4. Stan has a part-time job working 12 hours a week. His gross pay is $110.20 a week. Cecelia has a part-time job working 8 hours a week. Her gross pay is $90.40 a week. a) Find the ratio of the number of hours Stan works to the number of hours Cecelia works during a week.

b) Find Stan’s gross hourly rate of pay.

c) Find Cecelia’s gross hourly rate of pay.

d) Find the ratio of Stan’s gross hourly rate of pay to Cecelia’s gross hourly rate of pay.

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d.

5. You have decided to buy a new car and must choose between a regular model and a hybrid model. The hybrid model uses less fuel since it uses an electric motor to power the car when it is possible. Model

Price including taxes and shipping

Average fuel economy (L/100 km)

Regular model

$24 456.00

12.4

Hybrid model

$25 840.00

10.8

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a) How many litres of fuel will be required to drive each vehicle 24 000 km?

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b) Fuel costs $1.03/litre. How many kilometres would you need to drive to save enough money in fuel to pay for the extra cost to buy the hybrid?

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c) Why might you choose to purchase the hybrid even if you planned to sell the car in 2 years?

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Sample Chapter test: Solutions Part A: Multiple Choice

Store B:

x = 0.083 443(300)

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0.35 × $20.00 = $7.00

$13.00 × 3 = $29.00

x = $25.03

$20.00 − $7.00 = $13.00

The answer is a).

CAD $1.644 814 CAD = $1200.00 ______________ 5. ______________ 1.00 euro x euros

Store C:

$20.00 + $20.00 = $40.00

The answer is b). Store B has the best offer.

2. To estimate how many songs can be played in one hour, use 4 minutes as the average song length and convert 2 hours to minutes.

2 h × 60 = 120 min

4 min ______ = _______ 120xmin 1 song

( ) ( )

x __ 4 = ____ 120 x x 1

4x = 120

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4x = ____ 120 ___ 4 4 x = 30

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The answer is d). They can play about 30 songs in two hours. 3. Students should be able to narrow this down to either b) or d) by looking at the numbers. (Laurie and Sara earn more in less time than either Antoine or Ken.) b)

$72.00 ÷ 5 h = $14.40/h

d)

$89.00 ÷ 6 h = $14.83/h

The answer is d). 62

) ( )

(

1 = ____ 300 0.083 433x 0.083 433x ________ 0.083 443 x

d.

$20.00 + 0.5($20.00) + 0.5($20.00) = $40.00

300.00 pesos 1.00 peso _________ = ____________ x $0.083 443

4.

1. Store A:

MathWorks 10 Teacher Resource

(

) (

)

814 ________ x 1.644 = _______ 1200.00 x 1 x

1.644 814x = 1200

1.644 814x = ________ _________ 1200 1.644 814 1.644 814 x = €729.57

She will have 729.57 euros to spend. The answer is b). Part B: Short Answer

$69.07 1. a) ______ = $6.90/tile 10 tiles

$1.45 b) ________ = $0.06/screw, rounded 25 screws to the nearest cent

$0.95 = $0.04/min, rounded to 2. a) ______ 25 min the nearest cent 120 words = 40 words/min b) _________ 3 min 240 km c) _______ = 60 km/h 4h $22.80 d) ______ = $7.60/h 3h

Part C: Extended Answer 1. a) 10 − 6 = 4 people, in addition to the 6 for $65.00. $65.00 + 4($12.00) = $113.00 total cost $113.00 ÷ 10 people = $11.30/person b) 9 $101.00

$113.00

11 $125.00

12 $137.00

2. First, find how much time it takes Patrick to unload 1 pallet. 60 mins ________ = 3.53 mins/pallet 17 pallets

Then, multiply the number of pallets Lylah will unload and shelve by this unit rate.

13

14

$149.00

$161.00

15 $173.00

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Cost

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d.

Number of people

3.53 × 25 = 88.24

It will take her about 89 minutes to move 25 pallets. 3. a) One way to solve this is to find the unit price of both cans of soup.

(

) ( )

30.96 = ____ 500 12x 12x _____ 12 x

371.52x 6000x _______ = ______ x 12

30.96x = 6000

30.96x = _____ 6000 ______ 30.96 30.96

x = 193.80

Because you cannot buy a portion of a can of soup, round up to get 194 cans.

$18.89/12 cans = $1.57/can

4. a) The ratio of hours worked is 12:8, simplified to 3:2.

$30.69/24 cans = $1.28/can

b) Stan’s gross hourly rate of pay is $110.20 ÷ 12 = $9.18

c) Cecelia’s gross hourly rate of pay is $90.40 ÷ 8 or $11.30.

d) The ratio of Stan’s gross hourly rate of pay to Cecelia’s gross hourly rate of pay is $9.18:$11.30.

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Another way to solve this would be to notice that the price of Tastes Like Homemade can be doubled to get the price of 24 cans so the two brands can be compared.

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Students may suggest other methods. Savory Soup is the better deal. $30.96 b) _______ = _______ $500.00 12 cans x cans

5. a) Regular model: 12.4 L _______ = 0.124 L/km 100 km If the car uses 0.124 L/km and is driven 24 000 km, multiply the L/km by 24 000. 24 000 × 0.124 = 2976 L

Chapter 1 Unit Pricing and Currency Exchange

63

If the car uses 0.108 L/km and is driven 24 000 km, multiply L/km by 24 000. 24 000 × 0.108 = 2592 L The regular model would use 2976 L of fuel, and the hybrid model would use 2592 L if you drove 24 000 km.

$25 840.00 – $24 456.00 = $1384.00

Determine the cost of fuel for 1 km for each car. Regular:

(

) (

)

$0.065 96 = ________ x ________ $1384.00 x 1 x

0.065 96x = 1384.00 1384.00 x = _______ 0.065 96

x = 20 982.41 km

You would need to drive 20 982.41 km to save enough in fuel costs to pay the extra cost of the hybrid model.

c) Your reasons for buying the hybrid could include:

0.124 × $1.03 = $0.127 72

• you wish to save the environment

Hybrid:

• you think the price of fuel will rise

0.108 × $1.03 = $0.111 24

Then consider the difference in fuel costs per km. $0.127 72 – $0.111 24 = $0.065 96

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If you save 6.596 cents per km by driving

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$0.065 96 = ________ $1384.00 ________ 1 x

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b) First find the difference in price between the two models.

the hybrid model, how many km do you need to drive to save $1384.00?

d.

Hybrid model: 10.8 L _______ = 0.108 L/km 100 km

MathWorks 10 Teacher Resource

• you think the hybrid will get a better resale price

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d.

Blackline Master 1.1 graph paper (0.5 cm x 0.5 cm)

Name: Date:

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Blackline Master 1.2 Chapter Project Checklist

Name:

Date:

Party Planning checklist

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Is there a rental fee? If so, how much? Include this expense in your budget.

What decorations will you choose?

What will the invitations look like?

What activities or entertainment will you plan for guests?

What kind of music will you choose?

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What food and drinks will you need? How will you handle food allergies?

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What items such as plates, cutlery, and glasses do you need? How many will you need?

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Where might you purchase supplies?

Other notes?

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d.

Where will the party be held?

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d.

Blackline Master 1.3 Chapter ProjectSupply List

Name:

Purchases Date:

Party Supplies

Name of store Unit cost (if online, include the website address) Number of items needed Taxes (GST and PST) Total cost

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Blackline Master 1.4 Student project Self-Assessment

Date:

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To evaluate how well you did on your project, you will want to consider the following: • the thoroughness of your research • the accuracy of your calculations and budgeting • the effectiveness of your use of technology for researching, organizing, and presenting • the creativity you brought to planning and presenting • your completion of all the assigned tasks on time

d.

Name:

How do you feel you have done overall, given the criteria above?

Were you able to complete all aspects of the project? If not, why? Did you allot your time effectively?

In what areas did you excel?

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Are there areas in which you could improve?

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If you collaborated with a partner or a small group, what strengths did each person bring to the project?

If you had the project to do over again, what would you do differently?

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Blackline Master 1.5 Mixing the Concentrates Table

Name:

Recipe #1

Batches Orange concentrate (cups) Water (cups)

Date:

mixing the concentrates Recipe #2

Orange concentrate (cups) Water (cups)

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Alternative chapter proJect—Food Planning at a wilderness lodge Teacher Materials

PA re EV CIF pr A IC od L E uc UA DU tio TI C n ON AT of pa CO ION rt P AL or Y— P al N R lo O E f t T SS he FO . co R ALL nt D R en IS IG ts TR H in IB TS an UT R y IO ES fo N E rm R VE is D st ric tly pr oh ib ite

OUTCOME: In this project, students will integrate the concept of unit pricing into a real-world scenario in which they role-play the cook at a wilderness ecotourism lodge, planning a menu for a 3-day trip for a family of four. They will work within a set budget, using technology if appropriate, and practise and further develop their presentation skills.

PREREQUISITES: Students need to understand ratios and proportions and basic calculator functions. If students want to use a spreadsheet for the pricing calculations, then some prior spreadsheet experience would be an asset. They will also need to be familiar with presentation software if they choose to use it for their presentations. If they are familiar with any layout software they could use it to create a menu. Familiarity with internet research will also be helpful in this project. For this project, students will use retail prices for the supplies they need.

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ABOUT THIS PROJECT: This project is divided into three parts. Initially, students will create an overall plan for the project using a checklist (Blackline Master 1.2a, p. 76), plan a menu, and identify areas that they need to research. Partway through the chapter, students will apply what they have learned about unit pricing and cost comparisons to decide on the purchases they will make and work out costs in several ways. As a final activity, they will develop a presentation for the manager of their wilderness lodge, including a planned menu for four people for three days, a table or spreadsheet listing all the food items needed for the trip, plus any picnic supplies and their respective costs. Students will give their presentation to the class. Allow 3–5 minutes per student.

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Students should be given a few class periods to work on this project during the time spent on this chapter. This will allow for questions/feedback from the teacher as well as allowing the teacher to observe the quality of work as it is done, rather than at the end of the chapter. Interim guidance can help the students complete the culminating activity more successfully.

d.

GOALS: To use the concept of proportional reasoning to find unit prices, to build skills, and to synthesize learning in this chapter.

MathWorks 10 Teacher Resource

This project could be completed by pairs or small groups of students acting as co-workers at a wilderness lodge, or by individuals. An assessment rubric for this project follows. Blackline Master 1.5a (p. 79), which should be handed out to students early in the project, outlines the criteria for evaluation of their project and suggests some ways in which they can reflect on their learning. 1. Start to plan

Introduce the project to your students as you begin this chapter. This initial part of the project allows for group brainstorming as a class. Students may not be familiar with wilderness lodges and ecotourism trips, so be prepared to share some background information with them (see Blackline Master 1.1a (p. 75) for a brief description that you can hand out to students, or do an internet search for an example in your province or territory). Many students will have gone on a picnic, a camping trip, or out on the land following their people’s traditional way of life, so allow them to connect with their own experiences as a starting point for this project. Explain to students that they will role-play the cook at a wilderness lodge that offers ecotourism trips to families and individuals. In this case, they are to plan the food for a family of four (two adults and two teenagers) for a three-day trip. The family will spend nights at the lodge and will eat breakfast and dinner there. The visitors will spend their days out

Students may find Blackline Master 1.2a (p. 76), which contains a checklist of items to complete in each segment of the project, useful in organizing their project. A graphic organizer, Blackline Master 1.3a (p. 77), is provided so students can record their menu and ensure that each meal is included. 2. Research your ideas

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This segment of the project requires the largest amount of work on the part of students. Here, they are practising both their research and their unit costing skills. Students are expected to develop a cost analysis that is within their budget, including all the food supplies they would need to purchase, and any other costs, such as those needed for the picnic lunches (for example, garbage bags, preferably biodegradable, for packing garbage off the land, “green” cloth napkins, insulated reusable drink containers, and so on). All their work should be recorded in a table or on a spreadsheet. Blackline Master 1.4a (p. 78) will help students record their research. Remind students that a wilderness lodge would try to follow “green” practices in their decision-making.

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delivered to their homes, online sites for grocery stores (most grocery stores chains and some independent stores now have their own websites, including weekly flyers with pricing information). At the end of this segment of the project, discuss progress with your students to ensure that all requirements have been met. 3. Make a presentation

d.

In this segment of the project, students will synthesize their planning and research activities and practise their presentation skills. Presentations to a manager or company owner are often done with handouts and other tools, including presentation software, posters, or folders containing several items (in this case, the menu, the pricing research in a table or spreadsheet, and the spreadsheets or tables reflecting the cost analysis). Provide students with copies of Blackline Master 1.5a (p. 79) to give them an opportunity to reflect on their learning.

PA re EV CIF pr A IC od L E uc UA DU tio TI C n ON AT of pa CO ION rt P AL or Y— P al N R lo O E f t T SS he FO . co R ALL nt D R en IS IG ts TR H in IB TS an UT R y IO ES fo N E rm R VE is D st ric tly pr oh ib ite

on the land and will need a picnic lunch to carry with them. Tell students that they are to develop a healthy menu for the family members. A good starting point for developing a nutritious menu plan is the Canada Food Guide. This is available online at www.hc-sc.gc.ca/fn-an/food-guide-aliment/ index-eng.php or may be available in your school library. Students may need some assistance navigating through the Canada Food Guide site to find the information they need. Using the link for Choose Foods is helpful. Students can click through to a small chart that lists the number of servings for each group for children, teens, and adults by gender that will help them calculate the number of servings. Students may choose to include traditional foods on their menus. The budget for the family’s meals (and any related expenses) is $600.00.

Suggest sources of information that students can use for their research, for example, grocery store flyers in their local newspaper, flyers that are

Extensions

1. Some wilderness lodges have a practice of helping guests turn the results of their berrypicking activities into sweet treats back at the lodge. This practice can be used as the basis of an extension activity in which students use proportional reasoning to find how many pies the cook can bake, given a certain quantity of berries and the amount needed for a pie. This activity could be adapted for muffins, a crumble, or other baking that includes berries as an ingredient. Scenario: The family has spent the day berry-picking. The father has picked a quart of blueberries (if necessary, remind students that a quart contains 4 cups). Each of the two teenagers has only picked two cups. And the mother has picked another quart. If it takes 4 cups of blueberries to make a pie, how many pies will the cook at the lodge be able to make?

Chapter 1 Unit Pricing and Currency Exchange

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SOLUTION

Students will reason that each of the mother, father, and the two teenagers together have picked the 4 cups needed for a pie. So the cook will be able to bake three blueberry pies. Students might also convert the quart to 4 cups, add all the cups together to find 12, and divide by 3.

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PA re EV CIF pr A IC od L E uc UA DU tio TI C n ON AT of pa CO ION rt P AL or Y— P al N R lo O E f t T SS he FO . co R ALL nt D R en IS IG ts TR H in IB TS an UT R y IO ES fo N E rm R VE is D st ric tly pr oh ib ite

2. If you would like to include an activity that requires students to apply their foreign exchange skills, you could develop a related activity in which the wilderness lodge advertises its services in two foreign countries, for example, the United States and England. Students could use a rate of $1000.00 CAD a day/person and convert this rate into US dollars and English pounds. They could then design an advertisement or poster using the converted price.

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Solutions will vary, according to the countries selected and the exchange rates on the day they are researched. Ensure that students use the correct rate. Travellers coming to Canada will be buying Canadian dollars in their local currency and will, therefore, be purchasing at the bank selling rate.

d.

SOLUTION

MathWorks 10 Teacher Resource

Project Assessment rubric: Food planning at a wilderness lodge Not Yet Adequate

Adequate

Proficient

Excellent

Conceptual Understanding Explanations show understanding of unit pricing, budgeting parameters, research methods, client needs

writes and evaluates unit costs while adhering to the budget

shows partial understanding; explanations are often incomplete or somewhat confusing

shows understanding; explanations are appropriate

shows thorough understanding; explanations are effective and thorough

limited accuracy; major errors or omissions

partially accurate; some errors or omissions

generally accurate; few errors or omissions

accurate and precise; very few or no errors

For example:

For example:

For example:

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Procedural Knowledge • Accurately:

shows very limited understanding; explanations are omitted or inappropriate

For example:

creates tables from the information found using pen/paper or spreadsheets calculates total costs, unit costs, and taxes for all items purchased menu is nutritious and appealing

records all sources

presentation includes the menu and a table or spreadsheet with a cost analysis

•

•

items are listed and unit costs are calculated correctly

•

unit costs are calculated incorrectly

items are listed but the unit costs are not calculated correctly

items listed and unit costs are calculated correctly

•

•

total cost within budget

•

total cost within budget

total cost not within the budget

may have some needed items missing

•

•

total cost within budget using erroneous unit cost

sources are listed

•

•

presentation includes a detailed menu and thorough cost analysis

sources are listed

•

presentation includes an appealing and detailed menu and a thorough cost analysis

•

no calculation errors

•

adds some extra creativity to the project

•

item costs are missing

•

•

•

sources missing

•

presentation does not include a menu or cost analysis

•

project is incomplete

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sources included

•

presentation included a basic menu and cost analysis

•

project could use some more work to ensure calculations are done correctly

•

very few calculation errors

•

project is completed but there is nothing beyond what is listed as a minimum

uses some appropriate strategies, with partial success, to solve problems; may have difficulty explaining the solutions

uses appropriate strategies to successfully solve most problems and explain solutions

uses effective and often innovative strategies to successfully solve problems and explain solutions

does not present work and explanations clearly; uses few appropriate mathematical terms

presents work and explanations with some clarity, using some appropriate mathematical terms

presents work and explanations clearly, using appropriate mathematical terms

presents work and explanations precisely, using a range of appropriate mathematical terms

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uses few effective strategies; does not solve problems

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Uses appropriate strategies to solve problems successfully and explain the solutions

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Problem-Solving Skills

d.

•

Communication •

Presents work and explanations clearly, using appropriate mathematical terminology

Chapter 1 Unit Pricing and Currency Exchange

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Alternative chapter proJect—Food Planning at a wilderness lodge Student Materials Start to plan

T

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Your budget for food and any other supplies you may need is $600.00 for the family for three days. Your goal is to plan a healthy menu for the family that you can prepare within this budget. •

First, consult the Canada Food Guide to research the daily serving requirements for different food groups. The Canada Food Guide can be found online at: www.hc-sc.gc.ca/fn-an/ food-guide-aliment/index-eng.php.

•

Using the information you found in the Canada Food Guide, prepare a menu plan for three days for this family. For each day, the family members need:

breakfast at the lodge;

a packed lunch to take out on the land; and

dinner at the lodge.

Remember that you need 4 servings for each family member. •

Next, list the items you will need to research. Are there any non-food items that will be needed for the picnic? Remember that ecotourists need to pack all their garbage out and prefer to use biodegradable or recyclable materials.

Research Your Ideas

Now you will put your menu plan into action.

Using local foodstore flyers, the newspaper, or online grocery store sites, research the cost of the food and other items on your menu. Remember to buy enough for 4 people. Make a table or spreadsheet to record your research. Include the item, the amount you need, the unit price, and the total price.

•

Is the amount within your budget? If not, you may have to revise your menu.

•

Once you have done your research, calculate the cost of each meal.

•

Then, work out the cost of the meals for each person.

•

Once you’ve met your budget, create a printed menu. You can use graphic design or word processing software or calligraphy, if you know it.

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Make a PRESENTATION Your project file will contain the following information:

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d.

In this project, you will imagine that you are the cook at a wilderness lodge in the Northwest Territories that offers ecotourism trips to families. The lodge is located on a lake in the barrenlands. Your project is to plan the menu for a 3-day ecotourism trip for a family consisting of two adults and two teenagers. The family will eat breakfast and dinner in the lodge, but they will need a picnic lunch to take out on the land during the day. You may complete this project on your own or work with a small group to complete the menu plan.

•

the menu for each day of the ecotourism trip;

•

a table or spreadsheet listing all the items you plan to buy, along with their unit price and their total cost; and

•

calculations showing how much each meal costs (for 4 people) and how much each person’s meals over the three days will cost.

MathWorks 10 Teacher Resource

Blackline Master 1.1a Wilderness Lodges and Ecotourism—some background information

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Guests stay at the lodge and spend their time out on the land on a variety of activities, including: • hiking • mountain biking • kayaking and canoeing • fishing (catch and release) • wildlife viewing • nature photography • berry-picking

d.

The model used for this project is a real wilderness lodgeDate: located on a lake Name: in the barrenlands in the Northwest Territories, where visitors go to see great caribou herds, among other things. The practices of this lodge are similar to those found in many other areas.

In the evenings, there are several indoor activities. The cook at the lodge is happy to take the berries that visitors pick and transform them into sweet treats. Guests can learn from local people about the traditions and history of the First Nations and Inuvialuit peoples of the Northwest Territories. On dark nights, guests may be able to view the colourful curtains of light called the aurora borealis, which is Latin for northern lights. Protocols

Guests at a wilderness lodge that specializes in ecotourism should follow safety protocols and show respect for the environment.

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Safety Protocols • respect what your guide tells you because he or she is familiar with the area • don’t go out on the land by yourself • do not approach wildlife closely, particularly potentially dangerous larger animals • let someone at the lodge know where you are going and when to expect you back again • take a map and a GPS, if you have one Respect for the Environment keep a distance from wildlife so as not to disturb them in their natural habitat • pack out all garbage from daytrips • try and use only biodegradable or recyclable items • don’t go out on the land by yourself •

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Blackline Master 1.2a Project Checklist

Name:

Date:

Food Planning checklist

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Have you planned the menu for breakfast, a picnic lunch, and dinner for a family of 4 people? Have you researched food prices?

Did you check your budget and adjust your menu as necessary?

Did you create your menu using software or calligraphy? Have you created a table or spreadsheet listing all your purchases?

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Did you work out the total cost, the cost per meal, and the cost per person?

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Have you assembled the materials you need? (Your menu and your cost analysis.)

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Have you planned your presentation?

Have you created any materials you wish to use to enhance your presentation, such as a poster or presentation software slide show?

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d.

Have you consulted the Canada Food Guide in order to plan a nutritious menu?

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• yoghurt

• coffee

Day 2

d.

• orange juice

• oatmeal with berries and milk

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Blackline Master 1.3a Menu Plan

Name: Date:

A sample breakfast menu is suggested here to get you started.

Day 1

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d.

Blackline Master 1.4a Food Price research

Name:

Source of item Item

grocery store flyer chicken breast Date:

Food price Chart

Amount needed

2 lb.

Unit Price

$8.97/lb.

TOTAL

Total Price

$17.94

Blackline Master 1.5a Student project Self-Assessment

Date:

PA re EV CIF pr A IC od L E uc UA DU tio TI C n ON AT of pa CO ION rt P AL or Y— P al N R lo O E f t T SS he FO . co R ALL nt D R en IS IG ts TR H in IB TS an UT R y IO ES fo N E rm R VE is D st ric tly pr oh ib ite

To evaluate how well you did on your project, you will want to consider the following: • the thoroughness of your research • the accuracy of your calculations and budgeting • the effectiveness of your use of technology for organizing and presenting • the creativity you brought to planning and presenting • your completion of all the assigned tasks on time

d.

Name:

How do you feel you have done, given the criteria above?

Were you able to complete all aspects of the project? If not, why? Did you allot your time effectively?

In what areas did you excel?

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Are there areas in which you could improve?

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If you collaborated with a partner or a small group, what strengths did each person bring to the project?

If you had the project to do over again, what would you do differently?

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