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Mathematics 11 Teacher and Student Support Resource December 2013 DRAFT Mathematics 11 Table of Contents Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Teaching and Learning Guidelines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Resources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Websites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 11 Planning. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sample Lesson: Numeracy Games. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sample Lesson: Weather and the Environment . . . . . . . . . . . . . . . . . . . . Sample Lesson: Wheel of Fortune . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sample Lesson: Puzzles, Games, and Measurement. . . . . . . . . . . . . . . . Sample Lesson: Which Phone Would You Buy?. . . . . . . . . . . . . . . . . . . . Sample Lesson: Healthy Eating. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 13 15 17 18 20 22 Theme Overviews and Suggestions for Teaching and Learning . . . . . . . . . . . . . Concept Map of Themes and Outcomes . . . . . . . . . . . . . . . . . . . . . . . . . Outcomes: Arithmetic Operations and Proportional Reasoning . . . . . . . . Theme: Earning and Spending Money . . . . . . . . . . . . . . . . . . . . . . . . . . . Theme: Home . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Theme: Recreation and Wellness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Theme: Travel and Transportation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 26 27 33 37 43 46 Appendices. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix A: Arithmetic Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix B: Proportional Reasoning. . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix C: Earning and Spending Money . . . . . . . . . . . . . . . . . . . . . . . Appendix D: Home. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix E: Recreation and Wellness. . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix F: Travel and Transportation . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 50 55 57 70 101 107 Mathematics 11 These materials were created by writing partnerships of school boards and the provincial government. This document reflects the views of the developers and not necessarily those of the Ministry of Education. Permission is given to reproduce these materials for any purpose except profit. Teachers are also encouraged to amend, revise, edit, cut, paste, and otherwise adapt this material for educational purposes. Any references in this document to particular commercial resources, learning materials, equipment, or technology reflect only the opinions of the developers of this Mathematics 11 course overview, and do not reflect any official endorsement by the Ministry of Education or by the partnership of school boards that supported the production of the document. Acknowledgments Michelle Dament Prairie Spirit School Division Dalmeny, Saskatchewan Samantha Olenick Greater Saskatoon Catholic School Division Saskatoon, Saskatchewan Heather Granger Prairie South School Division Avonlea, Saskatchewan Wanda Pihowich Saskatoon Public School Division Saskatoon, Saskatchewan Shelda Hanlan Stroh Greater Saskatoon Catholic School Division Saskatoon, Saskatchewan Kelly Russell Lloydminster Catholic School Division Lloydminster, Saskatchewan Mathematics 11 Introduction Recommended Prerequisite: Mathematics 9 This course is designed for theme-based instruction, which should enable students to broaden their understanding of mathematics as it is applied in important areas of day-to-day living. There is a need for learning to be meaningful in order to be transferable. Learning mathematics should provide students an opportunity to explore mathematics in their lives. Earning and Spending Money My Life In this course, emphasis is placed on Travel and making informed decisions about Transportation finances, home design and maintenance, recreation and personal wellness, and travel and transportation. All mathematics relate to the themes: Earning and Spending Money, Home, Recreation and Wellness, and Travel and Transportation. Students can draw on their own or others experiences in the workforce to develop and extend their Recreation and knowledge about earning and spending money. They will Wellness also apply mathematics for the purpose of designing, building, and maintaining a home and yard. Students will apply reasoning and problem solving skills to make predictions and decisions in recreational and wellness activities. As well, they will investigate and solve problems related to planning a trip. Page 1 Mathematics 11 Home Teaching and Learning Guidelines The teacher of a Mathematics 11 course should: Choose themes and topics from the curriculum appropriate to student background, interests, and motivation. Identify the appropriate teaching/learning and assessment/evaluation strategies to help students achieve the outcomes. Use resources that best suit students’ competencies and interests, and include both print and web-based resources. Plan the delivery of the themes, using the support materials as a guideline, to provide students with a variety of learning experiences that focus on active learning, understanding, and engagement. Students in a modified course typically benefit from instruction that: Provides students with a clear overview of the course, each unit of study, and expectations. Provides students with activities that involve developing critical thinking and decision-making skills. Helps students organize new knowledge, understand the relationships among the new knowledge, and connect it to knowledge already learned. Helps students understand where they have been, where they are now, and where they are going in the learning process (Lenz, 2000). Diagnoses the students’ current understanding and skill level. Identifies and builds on student’s prior knowledge. Differentiates what students will learn in order to achieve the outcomes and teaches the prerequisite skills if they are missing. Differentiates the instructional approach and instructional groups (alone, pairs, small group, total group). Structures individual lessons in a systematic and organized manner, and presents course content in a structured manner. Integrates technology and uses a variety of resources. Uses current and local information to promote relevance. Models and uses scaffolded instructional strategies. Teaches students strategies that are specific to particular learning tasks. Provides enough guidance and practice so that students can master the strategies. Teaches students self-management, self-reflection, and self-regulation strategies to assist students in accomplishing tasks. Provides timely and constructive feedback to students. Provides assessment criteria for tasks to students. Bases students’ assessment and evaluation on the knowledge, skills, and strategies that help students achieve the outcomes. Page 2 Mathematics 11 Uses the information obtained from assessment and evaluation to individualize and inform upcoming instruction. Shares assessment and evaluation information (e.g., rubrics, checklists, etc.) with students before those items are used, to help students track personal growth and set learning goals. Strategies Teachers use multiple teaching, learning, and assessment strategies to ensure that students have had the opportunity to learn the curriculum content and improve skills prior to evaluation. When deciding which strategy to use, consider the following questions: Can all learners use this strategy to show thinking and learning? Will this strategy inform my instruction and provide a way to give feedback to students? Will patterns of understanding or confusion emerge as a result of using this strategy? Is this strategy convenient to design, use, and administer? (Cris Tovani, 2011, So What Do They Really Know?, p. 74) The following is a partial list of strategies that could be used in the Mathematics 11 course to help students achieve the outcomes. Strategy Description 12 word summary In 12 words or less, have students summarize important aspects of a particular chunk of instruction. 3-2-1 Students jot down 3 ideas, concepts, or issues presented. Students jot down 2 examples or uses of the idea or concept. Students write down 1 unresolved question or a possible misunderstanding. 60 second think Use in your classroom at any time as no equipment is required. Ask students to stop, and have a 60-second think about how their learning is going right then. Accurately “time” the 60-seconds to allow quiet thinking time. Circular check In groups, students are each given a different problem with a definite answer. The first student completes the first step without contribution from others in the group and passes it to the next student. The second student corrects any mistakes in the first step and completes the next step without input from the group. The problem is passed to the next student and the process continues until the group has the correct answer. Page 3 Mathematics 11 Class vote Present several possible answers or solutions to a question or problem and have students vote on what they think is best. Concept circle Ask students to quickly sketch a concept circle like this image (noting that any number of spikes can be drawn). Students then do an “individual brainstorm”, trying to recall the key concepts that are related to the work they are doing now. Students then highlight or draw a box around, any concepts that they are having trouble understanding. These concepts are then recorded by the student in their learning logs for further examination or they can be discussed with the teacher next time there is an opportunity to do so. Enter/exit slips Ask students a specific question about the lesson (or refer to Phrases and Prompts for ideas to respond to). Students respond on the slip and give it to the teacher, either on their way out or on their way in the next day. Teacher can then evaluate the need to re-teach or questions that need to be answered. Feedback sandwich Good news “I did really well on … ” Bad news “I think these parts need to be changed … because …” Good News “Some ways I can improve it are …” Flash cards After 10 minutes into a lecture or concept presentation, have students create a flash card that contains the key concept or idea. Toward the end of the class, have students work in pairs to exchange ideas and review the material. Four corners Page 4 Teacher posts questions, concepts, or vocabulary words in each of the corners of the room. Each student is assigned a corner. Once in the corner, the students discuss the focus of the lesson in relation to the question, concept, or words. Students may report out or move to another corner and repeat. Mathematics 11 Frayer model Write a term in the middle (e.g. rational number). Complete the other four boxes in regards to the term. Definition Facts Term: Examples Give one/Get one Non-Examples Students are given papers and asked to list 3-5 ideas about the learning. Students draw a line after their last idea to separate his/her ideas from their classmate’s lists. Students get up and interact with one classmate at a time. Exchange papers, read your partner’s list, and then ask questions about new or confusing ideas. Graphic organizers Page 5 Mathematics 11 ICE tactic Students ask themselves: What are the: Ideas (basics, details, facts, terminology)?, Connections (relationships, synthesis, patterns)? Extensions (transfer, hypotheticals, creative adaptations, going beyond the obvious)? This is a simple way to keep students focused on the big picture even while they are on the run, learning, during any lesson (Young and Wilson, 2000). Idea wave Each student lists 3-5 ideas about the assigned topic. One volunteer begins the “idea wave” by sharing his idea. The student to the right of the volunteer shares one idea; the next student to rights shares one idea. Teacher directs the idea wave until several different ideas have been shared. At the end of the formal idea wave, a few volunteers who were not included may contribute. Jigsaw Students first meet in their “expert group”, where each student has the identical assignment. The students become a team of specialists, gathering and synthesizing information, becoming experts on their topic, and rehearsing their presentations. Then the students change groups to their jigsaw groups. Each student in each group educates the whole group about her or his specialty. Learning cell Students develop questions and answers on their own (possibly using the QMatrix). Working in pairs, the first student asks a question and the partner answers and vice versa. Each student can correct the other until a satisfactory answer is reached. Learning logs Use learning logs or learning journals for students to reflect on their recent work (perhaps at the end of their work each week). Refer to Phrases and prompts for ideas. Muddiest point Students are asked to write down the muddiest point (what was unclear) in the lesson. Page 6 Mathematics 11 Old school Ipads Give students whiteboards, paper plates, index cards, or large sheets of paper when they enter. When asking a question have ALL students write the answer and at your signal, have ALL students hold up the Ipad so that you can see who/ how many got the answer. Discussion to elaborate can follow. Phrases and prompts What have I learnt? What am I most pleased with about my work? What did I find difficult? How can I try to improve? What did I learn today? What did I do well? What am I confused about? What do I need help with? What do I want to know more about? What am I going to work on next? (Weeden et al., 2002) The part I liked best was… The part I found confusing was… Two things I learnt were… One question I have is… I was surprised that… I already knew that… One thing I know that wasn’t mentioned is… I would like to know more about… I would like to spend more time on… Some questions I know how to do… One thing I want to get better at is … One word web card… (Davies, 2012) This week I have learned… For next week I am focusing on… I will know I am getting better when… I feel confident when … My strength today was … I’m proud of this because… I feel frustrated when … I need to find out more about … I need help with … My highest priority learning goal is .. Next time I do this I will … When I wasn’t sure, I asked [my friend’s name] about … When I wasn’t sure, I asked [my teacher’s name] about … One thing I am still not sure about is … I will work on this by … (Office of Learning and Teaching, DE&T Page 7 Mathematics 11 http://www.sofweb.vic.edu.au/blueprint/fs1/assessment.asp). What is the most important point you learnt today? What point remains least clear to you? How is ___________ similar to/different from ____________ ? What are the characteristics/parts of ____________ ? In what other ways might we show/illustrate ____________ ? How does ____________ relate to ____________ ? Give an example of ____________ . What approach/strategy could you use to ____________ ? Provide three examples of ____________ and one non-example. Explain to a student in grade X (or who was absent today) what you learned about ____________ today. Write about the work we did today. What was easy? What was hard? What do you still have questions about? If you got stuck today in solving a problem, where did you get stuck? Why do you think you had trouble there? If you did not get stuck, what idea helped you solve the problem? The hardest part of this chapter so far is …. I need help with ____________, because … To me, ____________ (e.g. geometry) means … ____________ (e.g. measuring angles) can be useful for …. ____________ (e.g. fractions) are challenging when … Place mat Each group member writes ideas in a space around the centre of a large piece of paper. Afterwards, the group compares what each member has written, and common items are compiled in the centre of the paper. Portfolio In the process of selection and explanation as to why students have chosen specific pieces for their portfolios there is already a self-assessment process in place. However, this can be taken further by more specifically asking students to respond to the following process and questions: 1. Arrange all your work from most to least effective 2. Reflecting on your two best works, and on a separate sheet(s) of paper for each work, answering the following questions. What makes this your best (second best) work? How did you go about it? Page 8 Mathematics 11 What problems did you encounter? How did you solve them? What goals did you set for yourself? How did you go about accomplishing them? 3. Answering these two questions on a single sheet(s) of paper at the front of your portfolio. What makes your most effective work different from your least effective work? What are your goals for your future work?” (http://www.ncrel.org/sdrs/areas/issues/students/learning/lr2port.htm) Quick write Students write for 2-3 minutes about what they learned or heard from the explanation. Also it could be an open ended question from teacher (refer to Phrases and prompts for ideas). Studentgenerated lists Top 10 things I need to find out … Questions I have about my work … Strategies I can use to improve my work … Think, Pair, Share Think about your answers and write them down, Pair with a partner to discuss and add comments to your answers, Share your answers with the class. Thumbs up - thumbs down To check for understanding, have students hold up their thumb; thumb up means “I got it”, thumb horizontal means “I’m not sure, maybe”, and thumb down means “I’m lost. I have questions”. Traffic Lights The traffic lights can be used in a range of different ways. To check for understanding, during individual or group work, provide students with a set of green, yellow, and red stacking cups. All students start with the green cup displayed, stacked over the other two cups. As students work, they can change the cup that is displayed to indicate to the teacher that their progress is green (good understanding and do not need assistance), yellow (partial understanding, getting answers, but with difficulty, minor errors, or have a basic question), or red (no understanding, stalled, need an explanation before moving forward). For self-assessing their own work, students label their work green, yellow or red according to whether they have good (“I got it”), partial (“I’m not sure, maybe”), or little (“I’m lost, I have questions”) understanding. (Black et al, 2003). Examine your work and highlight where you feel • Stopped • Cautious Page 9 Mathematics 11 • Going straight ahead. Use a red marker or a pink highlighter to mark in the margins where you feel “stopped” because you don’t understand. Write a learning goal about this. Use an orange or yellow marker or highlighter to mark in the margins where you feel “cautious” because you are unsure or don’t understand it very well. Use a green marker or highlighter to mark in the margin where you feel you are “going straight ahead” because you understand it well. For assessing a peer’s oral presentation: Green: better than I could have done/I learnt something from this Yellow: about the same as I could have done/no major omissions or mistakes Red: not as good as I could have done/some serious omission or mistakes” Students could then go on and give their peers feedback on specific strengths and weaknesses. (Black et al., 2003) Transfer and apply Students list what they have learned and how they might apply it to their lives. Students list interesting ideas, strategies, concepts learned in class. They write some possible ways to apply this learning in their lives, another class, or in their community. Wall posters Regular prompt questions can be made into wall posters. Refer to Phrases and prompts for ideas. Which face? 3 boxes are labelled with: put their work into. and students choose which box to (Unless otherwise referenced, the above strategies are from Office of Learning and Teaching, DE&T http://www.sofweb.vic.edu.au/blueprint/fs1/assessment.asp). Page 10 Mathematics 11 Resources Each theme makes reference to the use of specific websites. Teachers need to consult their board policies regarding use of any copyrighted materials. Before reproducing materials for student use from printed publications, teachers need to ensure that their board has a Can copy licence and that this licence covers the resources they wish to use. Before screening videos/films with their students, teachers need to ensure that their board/school has obtained the appropriate public performance licence. Teachers are reminded that much of the material on the Internet is protected by copyright. The copyright is usually owned by the person or organization that created the work. Reproduction of any work or substantial part of any work on the Internet is not allowed without the permission of the owner. Websites The URLs for the websites were verified by the developers prior to publication. Given the frequency with which these designations change, teachers should always verify the websites prior to assigning them for student use. Page 11 Centre for Innovation in Mathematics Teaching http://www.cimt.plymouth.ac.uk/ Coolmath 4 Kids http://www.coolmath4kids.com/ Figure This! Math Challenges http://www.figurethis.org/index.html Fun Math Lessons http://math.rice.edu/~lanius/Lessons/ Index of EARAT Manuals: The Apprenticeship Network http://www.theapprenticeshipnetwork.com/earat/manuals/ Interactive Mathematics http://www.cut-the-knot.org/content.shtml Intermath Online Mathematics Dictionary http://intermath.coe.uga.edu/dictnary/homepg.asp Math Central http://mathcentral.uregina.ca/ Math in Daily Life http://www.learner.org/interactives/dailymath/ Math is Fun http://www.mathisfun.com/ Math TV http://www.mathtv.com/ Math Worksheets http://www.math-aids.com/ Mudd Math Fun Facts http://www.math.hmc.edu/funfacts/ National Library of Virtual Manipulatives http://nlvm.usu.edu/en/nav/vlibrary.html The Math Forum @ Drexel University http://mathforum.org/ Trades Math Workbook http://www.hrsdc.gc.ca/eng/jobs/les/tools/support/trades_math_workbook.shtml Virtual Math http://www.virtualmaths.org Your Financial Toolkit http://www.fcac-acfc.gc.ca/ft-of/home-accueil-eng.html Mathematics 11 Planning Traditionally, teachers start unit planning with interesting activities and textbooks in mind, rather than starting with the big ideas or concepts they want the students to master. If learning is to be effective for the students, the teacher must begin with the final destination in mind. Teachers should be clear about what learning outcome(s) and goal(s) will be set for the students and what assessments will be used to provide evidence that the students have mastered the learning outcome(s) and goal(s) (Wiggins, G. and McTighe, J. (1998). Understanding by Design). A concern with teaching any mathematics course is the time it takes to cover the content. Employing a conceptual approach allows the teacher to become a facilitator or guide to coach learners in building on what they already know. This constructivist approach allows learners to: build on their prior knowledge place less emphasis on memorization and rote learning see mathematical skills as useful tools and processes build a depth of knowledge develop an understanding of the connections in mathematics build self-confidence and a positive disposition towards mathematics. (ABE Level Three: Mathematics Curriculum Guide, pp. 128) When a teacher uses a conceptual approach, instruction framed around context focuses on concepts rather than content. According to the National Council of Teachers of Mathematics (2000): In planning individual lessons, instructors should strive to organize the mathematics so that fundamental ideas form an integrated whole. Big ideas encountered in a variety of contexts should be established carefully, with important elements such as terminology, definitions, notation, concepts, and skills emerging in the process. (p.15) As teachers design and plan their course, lessons should reinforce basic skills, include a variety of instructional strategies and activities, and connect to the larger mathematical concepts. Sample lessons have been included as examples that incorporate overlapping outcomes, indicators, and themes and use a variety of strategies, resources, and activities. Page 12 Mathematics 11 Sample Lesson: Numeracy Games Outcome(s): M11.1 Extend understanding of arithmetic operations to rational numbers to solve problems within the home, money, recreation, and travel themes. M11.2 Demonstrate understanding of reasoning by analyzing puzzles and games. Suggested Theme(s): Recreation and Wellness Resource(s): Math Link. MathLinks 9 (2009). pp. 54, 62, 71 Material(s): Deck of playing cards Two dice Coin Game 1: Play the following game with a partner or in a small group. You will need one deck of playing cards. Remove the jokers, aces, and face cards from the deck. Red cards represent positive integers. Black cards represent negative integers. In each round, the dealer shuffles the cards and deals two cards to each player. Use your cards to make a fraction that is as close as possible to zero. In each round, the player with the fraction closest to zero wins two points. If there is a tie, each tied player wins a point. The winner is the first player with ten points. If two or more players reach ten points in the same round, keep playing until one player is in the lead by at least two points. Game 2: Play this game with a partner or in a small group. You will need two dice and one coin. For each turn, roll the two dice and toss the coin. Then, repeat. Create numbers of the form . from the result of rolling the two dice. Tossing heads means the rational numbers are positive. Tossing tails means the rational numbers are negative. Record the two pairs of numbers. Choose one number from each pair so that the sum of the chosen numbers is as close as possible to zero. Record the sum of the chosen numbers. In each round, the player with the sum closest to zero wins two points. If there is a tie, each tied player wins one point. The winner is the first player with ten points. If two or Page 13 Mathematics 11 more players reach ten points in the same round, keep playing until one player is in the lead by at least two points. Game 3: Play this game with a partner or in a small group. You will need a deck of playing cards. Remove the jokers, face cards, and 10s from the deck. Red cards represent positive integers. Black cards represent negative integers. Aces represent 1 and -1. In each round, the dealer shuffles the cards and deals four cards to each player. Use your four cards to make two fractions with a product that is as far from zero as possible. In each round, the player with the product that is furthest from zero wins two points. If there is a tie, each tied player wins a point. The winner is the first player with ten points. If two or more players reach ten points in the same round, keep playing until one player is in the lead by a least two points. Game 4: In a magic square, the sum of each row, column, and diagonal is the same. Try Magic Squares (http://illuminations.nctm.org/LessonDetail.aspx?id=L263). Page 14 Mathematics 11 Sample Lesson: Weather and The Environment Outcome(s): M11.1 Extend understanding of arithmetic operations to rational numbers to solve problems within the home, money, recreation, and travel themes. M11.3 Demonstrate understanding of data collection and analysis within the home, recreation, and travel themes. M11.4 Demonstrate understanding of measurement in the Système International (metric) and Imperial System within the home and travel themes. Suggested Theme(s): Home Recreation and Wellness Travel and Transportation Resource(s): Data Analysis and Probability http://mdk12.org/instructio n/clg/public_release/algebr a_data_analysis/G3_E2_I 2.html Skill Building: The most extreme change in temperature in Canada took place in January 1962 in Pincher Creek, AB. A warm, dry wind, known as a Chinook, raised the temperature from -19°C to 22°C in one hour. How many degrees did the temperature rise in Celsius? In Fahrenheit? (MathLinks 9, p. 345). Introduction: Wind makes the air feel colder than the actual temperature. This is called wind chill. The graph below shows the wind chill temperatures for various wind speeds when the actual air temperature is 8° Celsius. A curve of best fit has been drawn. MathLinks 9 (2009) Material(s): Page 15 Mathematics 11 Questions: For what wind speed is the wind-chill temperature 0° Celsius? Use mathematics to explain how you determined your answer. Use words, symbols, or both in your explanation. Use the graph to explain how the wind-chill temperature changes in comparison to the wind speed. Include an estimate of when the effect of the wind levels off. Activity It’s Raining, It’s Pouring http://www.glencoe.com/sec/math/t_resources/lab_m anual/pdfs/mac1_04/scimath_lab21.pdf Project Global Warming MathLinks 9 (2009). pp. 448 Page 16 Mathematics 11 Sample Lesson: Wheel of Fortune Outcome(s): M11.2 Demonstrate understanding of reasoning by analyzing puzzles and games. M11.3 Demonstrate understanding of data collection and analysis within the home, recreation, and travel themes. Suggested Theme(s): Recreation and Wellness Resource(s): Authentic Activities for Connecting Mathematics to the Real World: Wheel of Fortune http://www.wfu.edu/~mcco y/mprojects.pdf Material(s): Wheel of Fortune episode Page 17 Watch: an episode of Wheel of Fortune Activity: Complete an analysis of the letters used in the show. Make a list of all the letters of the alphabet and chart the letters used on Wheel of Fortune using tally marks. Do not count letters guessed but not found in the phrase. Find the total for each letter and the grand total of all letters in the entire show. For each letter, divide the total for that letter by the grand total to find the percent of each letter used. Create a circle graph displaying the letter analysis. Questions: Which are the top ten letters used (in order)? Which are the bottom five letters least used? How many vowels are in the top ten? Which consonants would be the most useful? Which vowel might be the least useful? What percentage of all the letters surveyed were vowels? If you watched three different days of Wheel of Fortune, do you think your results will be similar? Test and see. Mathematics 11 Sample Lesson: Puzzles, Games, and Measurement Outcome(s): M11.2 Demonstrate understanding of reasoning by analyzing puzzles and games. M11.4 Demonstrate understanding of measurement in the Système International (metric) and Imperial System within the home and travel themes. M11.6 Demonstrate understanding of the Pythagorean Theorem to solve problems within the home theme. Suggested Theme(s): Recreation and Wellness Introduction: Millions of Canadians enjoy the challenge and fun of playing chess. Early versions of this game existed in India over 1400 years ago. The modern version of chess emerged from southern Europe over 500 years ago. Many Canadians also enjoy the challenge and fun of completing Sudoku puzzles. Sudoku is a Japanese logic puzzle completed on a 9-by-9 square grid. The grid includes nine 3-by-3 sections. Resource(s): Problem Solving With Games. MathLinks 9 (2009). pp. 45, 81 Material(s): Chessboard and pieces Ruler Sudoku puzzle Watch: How to Play Chess in 10 minutes (http://www.youtube.com/watch?v=t-uwGvx4V_A) Game: Play the game of chess. Describe strategies of how you win the game. Award prizes for the student(s) that win the most games. Activity: Math is used to play, strategize, and win chess. However, math is also used in the design of the game board. If each of the small squares on a chessboard has a side length of 3 cm, what is the total area of the dark squares? Measure the chessboard you are using. Do the small squares have side lengths of 3 cm? If not, what are the side lengths? What is the total area of the dark squares? If the total area of a chessboard is 1024 cm2. What is the side length of each of the smallest squares? What is the length of the diagonal of the board? Compare your solutions with your classmates’ solutions. Watch: How to Do a Sudoku (http://www.youtube.com/watch?v=z6mGHf9bq3I) Page 18 Mathematics 11 Puzzle: Complete a Sudoku puzzle. Describe strategies used to complete the puzzle. Activity: If the smallest squares on the grid have a side length of 1.1 cm, what is the area of the whole grid? Measure the Sudoku puzzle you completed. Do the small squares have side lengths of 1.1 cm? If not, what are the side lengths? What is the area of the whole grid? If the whole grid has an area of 182.25 cm2, what are the dimensions of each 3 by 3 section? Page 19 Mathematics 11 Sample Lesson: Which Phone Would You Buy? Outcome(s): M11.3 Demonstrate understanding of data collection and analysis within the home, recreation, and travel themes. M11.4 Demonstrate understanding of measurement in the Système International (metric) and Imperial System within the home and travel themes. M11.7 Demonstrate understanding of proportional reasoning within the home, money, recreation, and travel themes. M11.9 Demonstrate understanding of responsible spending habits. Suggested Theme(s): Earning and Spending Money Home Recreation and Wellness Resource(s): Material(s): Introduction: The market for mobile phones is ever changing and ever expanding. Tech-savvy consumers are continuously replacing their smart phones with a newer, faster, and better model that has the latest technologies. How does a consumer choose which phone they should buy? When shopping for a mobile phone, there are many features to consider such as size, appearance, ergonomics, robustness, screen size, display, and price. Activity: One feature to consider when purchasing a mobile phone is the size. Some consumers like a larger phone which results in larger screen for displaying images and videos. Others prefer a smaller phone that is more Page 20 Mathematics 11 compact for fitting into pockets and purses. What is your preference? Have students choose four smart phones to compare and contrast. They can compare and contrast current models or phones from previous years with current models. Have students research the dimensions to calculate the perimeter and area of the phone and research dimensions of the screen to calculate perimeter and area of the screen. They can organize the information in a table. Based on size, have students rank the phones in order of their first through fourth choices. Discuss: Use the following questions to generate discussion: What affect does changing the dimensions have on the size of the icons? How many icons fit on the screen? What are the trends of phones in regards to size? Have phones increased or decreased in size? Has screen size changed? How does adding a case affect the size of the phone? Activity: Another consideration when purchasing a mobile phone is the initial cost of the phone. Have students research the cost of each of the phones. They can add two more columns to their table and add the initial cost and the length of the plan for each of the phones previously researched. Based on the initial cost, have students rank the phones in order of their first through fourth choices. Discuss: Have a class discussion about cell phone packages and what students deem as necessary. Group the students according to their similar needs. Activity: Have students research the cost of cell phone plans for the four phones from at least three providers. Based on the research and class discussion, have students find the total cost to purchase each of the phones. Ask students “Which phone would you purchase and why?” Ask students to be prepared to share and justify their answers with the class. Page 21 Mathematics 11 Sample Lesson: Healthy Eating Outcome(s): M11.3 Demonstrate understanding of data collection and analysis within the home, recreation, and travel themes. M11.4 Demonstrate understanding of measurement in the Système International (metric) and Imperial System within the home and travel themes. M11.7 Demonstrate understanding of proportional reasoning within the home, money, recreation, and travel themes. M11.9 Demonstrate understanding of responsible spending habits. Suggested Theme(s): Recreation and Wellness Resource(s): Real-Life Math, Tables, Charts and Graphs, Second edition, Tom Campbell. Food Labels. http://classroom.kidshealth. org/9to12/personal/nutrition/ food_labels.pdf Material(s): Food labels 20-oz. soft drink Page 22 Introduction: When you drop your money into a vending machine or onto a fast food counter, are you thinking about the Nutrition Facts food labels of what you are about to buy? Food companies spend a lot of time and money marketing to teens, so it’s important that you learn to think critically about what you eat. A balanced diet is one of the keys to good health, physical and mental development, and an active lifestyle. Making healthy food choices requires knowledge of your nutritional needs and of the nutrients found in foods. This project is meant to analyze what one eats, what it costs, and use mathematical applications of percent and reading and interpreting data to better understand one’s own health and nutrition. Discussion: Use the following questions to generate discussion: How often do you read Nutrition Facts food labels? What information do you look for? Why is serving size such a crucial piece of information on the food label? Do you pay attention to serving size when you’re drinking a can of soda or eating a bag of chips? Unless you grow all of your own food, you probably eat food that’s been processed. What does “processed” mean? How can you tell if a food is fresh, minimally processed, or highly processed? Can you think of an example of each? If nutrition information were available on restaurant menus, would it affect what people order? Would it make a difference to you? While food labels are helpful to everyone, why are they necessary for people with food allergies or Mathematics 11 certain health problems, like diabetes or heart disease? Activity: 1. Bring three nutrition labels (not a full bag of chips) from three of your favorite foods or condiments (ketchup, sour cream, granola bars, chips, Doritos, etc.). 2. Read Deciphering Food Labels (http://kidshealth.org/parent/nutrition_center/healthy_e ating/food_labels.html) and Food Labels (http://kidshealth.org/teen/food_fitness/nutrition/food_l abels.html) 3. Look at the labels you brought and analyze what is in that food. The guideline of 30% of calories from fat is used for healthy living. Calculate the percentage of fat in the foods using the three labels. 4. Notice the recommended serving size on your food label. Is that the amount you would typically eat or drink at one sitting? For most people it’s not — we eat much larger portions, and therefore more calories, sugar, and fat. Keep track of your portion sizes for 1 day. Are they over, under, or about even with the recommended serving sizes? Investigate: Bring in a 20-oz. soft drink. Note the grams of sugar per serving (for example, some colas have as much as 65 grams). If 4 grams of sugar is equal to 1 teaspoon, measure out the amount in the bottle into a clear cup (16¼ teaspoons). What if you drank two of those every day (32½ teaspoons)? Or three (48¾ teaspoons)? Discuss how empty calories really add up. Research: Research the difference in fat content when foods are labeled no fat, low fat, and fat free? What are the differences when we are talking about calories from fat? Read Figuring Out Fat and Calories (http://kidshealth.org/teen/food_fitness/nutrition/fat_calori es.html). Activity: 1. In the table, you will record what you eat in one day. Be as specific as possible with brands and record information from the food labels in the table. Use the applications My Fitness Pal, Lose It or other apps to assist in collecting data. Page 23 Mathematics 11 2. Do you take vitamins or minerals? Which ones? Keep track of the vitamins and minerals you take. Read Vitamins and Minerals (http://kidshealth.org/teen/food_fitness/nutrition/vitami ns_minerals.html) 3. How would you classify your diet? Check one. Super health freak! Healthy. Not too bad. Needs work. Cholesterol is hardening as we speak. 4. What modifications could you make to improve your diet and reduce the calories from fat? Be very specific in your plan. Activity: 1. Using grocery store flyers and fast food or restaurant menus, create a meal plan for one week of your favourite or typical food choices. Determine the cost of your food for the week. 2. Read The Food Guide Pyramid Becomes a Plate (http://kidshealth.org/teen/food_fitness/dieting/myplate .html). 3. Using Canada’s Food Guide (http://hc-sc.gc.ca/fnan/food-guide-aliment/index-eng.php), create a healthy meal plan for one full week for you. Include breakfast, lunch, supper and snacks. Ensure you are getting the right number of servings every day in each food group. 4. Read Supermarket Shopping (http://kidshealth.org/teen/food_fitness/nutrition/grocer y_shopping.html). 5. Using the store flyers again, make a grocery list and price out the healthy meal plan items for you for that week. 6. Compare your results from steps 1. and 3. Which is more expensive? Why? 7. Compare your results from steps 1. and 3. with a classmate. What similarities or differences do you notice? 8. Adapt your meal plan and grocery list for your family. How much would you have to budget for food/groceries every month if you were responsible for feeding your family? Page 24 Mathematics 11 Food Page 25 Total Calories Grams of Fat Calories from Fat % of calories from fat Classification Mathematics 11 Theme Overviews and Suggestions for Teaching and Learning This resource was created as a teaching, learning, and assessment support to give teachers an idea of how modified Mathematics 11 could be approached. Support materials have been developed as a guideline and do not need to be followed precisely or in a particular order. Concept Map of Themes and Outcomes The following concept map frames the themes and outcomes in Math 11. Page 26 Mathematics 11 A Theme Overview chart for each of the themes offers a recommended clustering of expectations and provides a starting point from which teachers can plan the course. Following each theme overview are suggested teaching and learning experiences, which may be used as a guideline for the teacher and may include: Resources Materials Introduction Pre-Assessment Activities Investigate Assessment Extension Skill Building Instruction Practice Questions Interactive Project Problem Connections Game Watch Research Terminology Brainstorm Discuss Adaptations Conclusion The Arithmetic Operations (M11.1) and Proportional Reasoning (M11.7) outcomes overlap in all four themes and the intent is that these outcomes may be taught in one or more of the themes. However, if an outcome has been covered, it is not necessary to revisit it in all four themes. Arithmetic Operations Outcome Proportional Reasoning Outcome M11.1 Extend understanding of arithmetic operations to rational numbers to solve problems within the home, money, recreation, and travel themes. At a Glance M11.7 Demonstrate understanding of proportional reasoning within the home, money, recreation, and travel themes. At a Glance Positive and negative numbers Operations on whole numbers Operations on integers Operations on fractions Operations on decimals Operations on percents Compare and convert fractions, decimals and percents Place value Ratios Rates Unit rates Unit costs Currency conversions Guiding Questions Guiding Questions What characteristics, patterns, and properties of numerical values and operations did your recognize? Can you apply the meanings and relationships of arithmetic operations to Page 27 Where are proportions, ratios, and rates used? How are ratios and rates similar and different? Can you apply ratios, rates, and proportions as a way to make Mathematics 11 compute and make reasonable estimations? Can you compare and order positive and negative numbers? How can you identify opposite numbers? How can you identify equivalent numbers? What is your understanding of arithmetic operations and can you apply your understanding to the context? Do you understand that multiplication makes numbers bigger, which is not always the case with fractions? Can the sum of two rational numbers be less than both of the rational numbers? Do you understand the relationship between fractions, decimals, and percents? Do you understand place value? Which successful and efficient strategies did you use to answer questions with rational numbers? comparisons? Do you recognize percent as a ratio comparing a value to 100? Can you identify when two ratios are equivalent? How can you identify if something is proportional? When you are shopping, how do you determine which item is the better deal? Why are unit costs necessary? How can you calculate unit rate and unit cost? How do you convert from one unit to another? Sample Applications or Problems Sample Applications or Problems Work in the school canteen at noon or during a school event. Order numbers according to value by listing or placing numbers on a number line. Bring in a set of measuring cups and look to see which is larger: ¼, ⅓, or ¾. (ABE Level Three: Mathematics Curriculum Guide (2006). pp. 83). Use examples of bank accounts in overdraft, two under par versus three under par, or temperature to compare values. (ABE Level Three: Mathematics Curriculum Guide (2006). pp. 80). The temperature at 7:00am was reported as -5° C. If the temperature rose 10° C by noon, what was the Page 28 If you make $11/hour, how long will you have to work to make a purchase worth $400? Show that 3:6 represents the same ratio as 2:4 by showing that a ramp with a height of 3 m and a base of 6 m and a ramp with a height of 2 m and a base of 4 m are equally steep. You take 100 mL of a liquid vitamin supplement every morning. You can buy a 100 mL size for $6.50 or a 500 mL size for $25.00. If the supplement keeps in the refrigerator for only 72 h, investigate which size is the better buy. Explain your reasoning. Compare the cost of one bottle of juice from a case of 12 versus the cost of one Mathematics 11 temperature? (ABE Level Three: Mathematics Curriculum Guide (2006). pp. 82). Find the difference of the average boiling point of 98.5°C, found in the experiment, and the accepted boiling point of 100°C. (ABE Level Three: Mathematics Curriculum Guide (2006). pp. 82). Consider the numbering of houses. Can you follow a pattern to determine the order of the house numbers? Can you predict the next house number? Add 2⅛ + 3¼. Validate results using a 12-inch ruler. Use skill testing questions from contests to illustrate the order of operations. Which is the better deal, ¼ off or 30% off? Use wrenches to compare and convert fractions. Determine a 15% discount on a purchase already discounted 50%. Estimate the change from a $20 bill if a purchase of $12.87 is made. Estimate the amount it will cost for the groceries in your shopping cart. bottle bought from a vending machine. Age, gender, body mass, body chemistry, and habits such as smoking are some factors that can influence the effectiveness of a medication. For which of these factors might doctors use proportional reasoning to adjust the dosage of medication? What are some possible consequences of making the adjustments incorrectly? Bring the label from a bag of chips to class. Use the information on the label to calculate how many calories and how much fat you would consume if you ate the whole bag. Then search out information on a form of exercise you could choose for burning all those calories. For what length of time would you need to exercise? Use currency manipulatives to explain why someone might offer $15.02, rather than $15.00, to pay a charge of $13.87. Use a collection of foreign currency to convert to Canadian currency. What are the various costs included in the final total for purchasing a digital audio player online from an American source? Using an online calculator, calculate the final cost, and describe how it compares with the cost of the purchase from a major retailer in Saskatchewan. Activities and Resources Activities and Resources Understanding Arithmetic Operations Appendix A.1. How Many Students in the Class? Appendix B.1. Arithmetic Operations Choice Board Appendix A.2. What’s the Cost of Those Bananas? Appendix B.2. Basic Skills Choice Board Appendix A.3. Proportional Reasoning Choice Board Appendix B.3. Cut It Appendix A.4. Rate and Ratio http://staff.argyll.epsb.ca/jreed/math7/stran d1/1208.htm Page 29 Mathematics 11 Using Integers http://www.helpingwithmath.com/printable Equivalent Ratios s/worksheets/numbers/int0601negative_ http://www.skwirk.com.au/p-c_s-12_u01.htm 208_t-573_c-2129/VIC/8/Equivalentratios/Ratios/Ratios-and-rates/Maths/ Comparing and Ordering Integers http://www.helpingwithmath.com/by_subj Unit Rate ect/integers/int_comparing.htm http://www.icoachmath.com/math_dictionar y/unit_rate.html Comparing and Ordering Integers http://www.helpingwithmath.com/printable s/worksheets/numbers/int0601integers_0 1.htm Comparing and Ordering Rational Numbers MathLinks 9 (2009). pp. 46 - 54 Problem Solving With Rational Numbers in Decimal Form MathLinks 9 (2009). pp. 55 - 62 What’s a Fraction? http://www.tv411.org/math/fractionsdecimals-percentages/whats-fraction Fraction Division via Rectangles http://fawnnguyen.com/2013/05/21/20130 518.aspx Problem Solving With Rational Numbers in Fraction Form MathLinks 9 (2009). pp. 63 - 71 What’s a Percent? http://www.tv411.org/math/fractionsdecimals-percentages/whats-percent Using Percents http://www.tv411.org/math/fractionsdecimals-percentages/using-percents Multiple Percents http://www.tv411.org/math/fractionsdecimals-percentages/multiple-percents Counting and Making Change Page 30 Mathematics 11 http://www.worksheetworks.com/math/mo ney/change.html Canadian Money Work Sheets http://www.homeschoolmath.net/workshe ets/canadian-money.php Making Change http://www.mathscore.com/math/practice/ Making%20Change/ Combining Percents http://www.mcgrawhill.ca/school/learning centres/file.php/9780070973381/olc2/dl/5 05863/4_4_Combining_Percents.pdf Multiple Percents http://www.tv411.org/math/fractionsdecimals-percentages/multiple-percents Real World Applications Real World Applications Redeeming coupons Spending money Paying bills Taking inventory Estimating amount spent when ordering off a menu Reconciling bank statements Calculating amount of change due back Calculating total of purchasing items Calculating gross pay Calculating net pay Calculating pay deductions Altering recipes Evaluating the appreciation of property and depreciation of a vehicle Reading an odometer or a gauge Recognizing values used in wrench and socket measurements Calculating the days left in a month or minutes left in an hour Comparing time in different time zones Comparing temperatures Keeping scores or running tallies in Page 31 Rates of pay (hourly, weekly, or annually) Price comparison through the purchase of bulk items Measurement conversions (100cm/km) Currency conversions Density in science (e.g. 1.00g/L) Understanding recommended dosages for medication (e.g. 1 tablet every 3 hours) Adjusting recipes Nutrition (labelling, serving size, calories burned during certain activities, calories from fat, recommended daily values of nutrients) Fitness (e.g. the number of repetitions per set in weight training) Calculating total time to travel Speed (e.g. 100 m/ 8.2 s, 500 km/ 4.5 h) Speed limits (e.g. 100 km/h) Gasoline prices (e.g. $1.229/L) (ABE Level Three: Mathematics Curriculum Guide (2006). pp. 97 – 98, 100) Mathematics 11 games (Yahtzee, Kaiser, Whist, Rummy) Recognizing the values used in sports such as bowling, hockey, football, baseball, soccer, and golf (ABE Level Three: Mathematics Curriculum Guide (2006). p. 86) Page 32 Mathematics 11 Theme Overview: Earning and Spending Money Theme Introduction Some students may have already entered the workforce and will have some knowledge about earning and spending money. The intent of this theme is to have students develop a deeper understanding of different ways people earn money and spend responsibly by exploring gross income, net income, and spending habits. Outcomes that overlap in all four themes M11.1 Extend understanding of arithmetic operations to rational numbers to solve problems within the home, money, recreation, and travel themes. M11.7 Demonstrate understanding of proportional reasoning within the home, money, recreation, and travel themes. Outcomes M11.8 Demonstrate understanding of income. M11.9 Demonstrate understanding of responsible spending habits. At a Glance Number sense Estimation Proportional reasoning Time Wages Money Currency Guiding Questions Page 33 What are the different ways that employees can be paid? How often are you paid? What is your understanding of net pay? Gross pay? What deductions would you expect on your earnings? How old do you have to be for the employer to begin to make deductions? If you work part time, do you have the same deductions as someone who works full time? What are union dues? What occupations have to pay union dues? What other factors, besides amount of pay, are important to you when choosing a job? How much money do you spend in a week? What are the advantages of tracking your spending? What are tax exemptions and when do they apply? Mathematics 11 What factors should you consider before making a purchase? How does the value of your purchase change over time? Career Connections Cashier Bank teller Payroll clerk TOPIC: Server Salesperson Accountant OUTCOME: SUGGESTED TEACHING AND LEARNING: EARNING MONEY Methods of Earning Income M11.8 Activity Does Money Grow On Trees? Appendix C.1. Resources Wages and Salaries MathWorks 10 (2010). pp. 54 – 78 Reading Timesheets and Pay Stubs M11.8 Activity Reading Timesheets Appendix C.2. Activity Reading Pay Stubs Appendix C.3. Resources Wages and Salaries, Additional Earnings MathWorks 10 (2010). pp. 56, 61, 73 Timesheet Information: False Timesheets http://www.brockport.edu/hr/payroll/timesheets.html How to Check a Timesheet http://www.ehow.com/how_2093546_checktimesheet.htmling How to Calculate Total Hours from a Timesheet http://www.ehow.com/how_5853052_calculate-totalhours-timesheet.html Gross and Net Income Page 34 M11.8 Activity Gross and Net Income Appendix C.4 Mathematics 11 Calculating Gross Monthly Income Appendix C.5. Discussion What other factors, besides amount of pay, are important to you when choosing a job? (wages, hours, type of work, location, distance from work, promotion, responsibility, job security, unemployment, experience) Resources Wages and Salaries MathWorks 10 (2010). pp. 54 – 63 Alternative Ways to Earn Money MathWorks 10 (2010). pp. 64 – 71 Additional Earnings MathWorks 10 (2010). pp. 72 - 91 Math at Work 10 (2011). pp. 147 – 148 Canadian Income Tax Rates for Individuals http://www.cra-arc.gc.ca/tx/ndvdls/fq/txrts-eng.html Calculating Deductions http://www.craarc.gc.ca/tx/bsnss/tpcs/pyrll/clcltng/menu-eng.html Payroll Deductions Tables http://www.cra-arc.gc.ca/tx/bsnss/tpcs/pyrll/t4032/menueng.html TOPIC: OUTCOME: SUGGESTED TEACHING AND LEARNING: RESPONSIBLE SPENDING HABITS Spending M11.9 Brainstorm Brainstorm items and services that students spend their money on. After brainstorming, organize the items into categories (e.g. phone, transportation, dining out, snacks, coffee, entertainment, clothing, etc.). Activity Spending Log Appendix C.6. Activity Needs Versus Wants Appendix C.7 Page 35 Mathematics 11 Activity Online Need Versus Want Activity http://web.extension.illinois.edu/money/needsVsWants.c fm (has ideas for Math 21 credit and loans as well). Purchasing and Buying M11.9 Activity PST and GST Appendix C.8 Activity What are You Buying? Appendix C.9 Resource Canadian tax rates http://www.watchbuys.net/kb/questions/159/What+are+t he+GST%7B47%7DHST%7B47%7DPST+rates+for+Ca nada%3F Purchasing Power M11.9 Project Research and report on the estimated costs involved in a large expense from one of the three themes: HOME: Plan a home renovation/improvement. Include the cost of contractors, equipment, supplies, etc. RECREATION and WELLNESS: Choose a leisure activity or sport to participate in. Include the cost of equipment, fees, uniform, travel, fundraising, etc. Page 36 TRAVEL and TRANSPORTATION: Plan a trip. Include the cost of gasoline, accommodations, food, entertainment, car rental, etc. Mathematics 11 Theme Overview: Home Theme Introduction The intent of this theme is to develop a deeper understanding of the applications of data collection and analysis, measurement, and geometry for the purpose of designing, building, and maintaining a home and yard. Outcomes that overlap in all four themes M11.1 Extend understanding of arithmetic operations to rational numbers to solve problems within the home, money, recreation, and travel themes. M11.7 Demonstrate understanding of proportional reasoning within the home, money, recreation, and travel themes. Outcomes M11.3 Demonstrate understanding of data collection and analysis within the home, recreation, and travel themes. M11.4 Demonstrate understanding of measurement in the Système International (metric) and Imperial System within the home and travel themes. M11.5 Demonstrate understanding of angles to solve problems within the home theme. M11.6 Demonstrate understanding of the Pythagorean Theorem to solve problems within the home theme. At a Glance Number sense Estimation Proportional reasoning Conversions Measurement Referents Dimensions Perimeter Area Angles Pythagorean theorem Guiding Questions Page 37 What is the purpose of collecting and analyzing data? Why display data? Can the data be presented in such a way to give a different message? Which is the best way to display the data? Can you make predictions from data? Can you make decisions from data? Can you identify trends? What is the purpose of measurement? What is the difference between counting and measuring? What do you measure? Mathematics 11 What devices do you use to measure? What kinds of measurements can a ruler give you? If you didn’t have a standard measurement tool, what could you use instead? What is metric measurement? What is imperial measurement? How are metric and imperial measurements the same? How are metric and imperial measurements different? How are length, height, and width different? How are length, height, and width the same? What is perimeter? Does every shape have a perimeter? How can you estimate the perimeter? How can you calculate perimeter? What is area? Does every shape have an area? How can you estimate the area? How can you calculate area? Why are the units for area always square units or units squared? What is an angle? Can you demonstrate replication of an angle? When do angles need to be precisely measured and when are estimations adequate? How is the Pythagorean Theorem used? How can you calculate the length of a diagonal? What is a 3:4:5 ratio? Career Connections Framer Surveyor Landscaper Lumber yard employee Counter top installer Window installer Finishing carpenter Kitchen designer Appliance salesman Electrician Roofer Mudder Farmer Page 38 Cabinet maker Flooring buyer Plumber Painter Buyer for lumber Designer Estimator Business owner Draftsperson Dry Waller Flooring installer Rancher Mathematics 11 TOPIC: OUTCOME: SUGGESTED TEACHING AND LEARNING: DATA COLLECTION AND ANALYSIS Read and Interpret Graphs M11.3 Pre-Assessment (replicate in Recreation and Wellness and Travel and Transportation) Reading and Interpreting Graphs Appendix D.1. Collect and Analyze Data M11.3 Pre-Assessment (replicate in Recreation and Wellness and Travel and Transportation) Collect, Organize, and Analyze Data Appendix D.2. Activity: Explore the cost of a home renovation by collecting data through reading newspapers, catalogues, and online sources and create an organized list. Activity: Graph the average house cost in your community for the last 20 years. Practice: Data Analysis MathLinks 9 (2009). pp. 410 – 429. TOPIC: OUTCOME: SUGGESTIONS FOR TEACHING AND LEARNING: M11.4 Activity Throughout time, cultures have invented their own systems of measurement – using the cycles of the moon, knots in a string, the length of a hand or a foot, the observation of the night sky, or other clues in nature. Have students research units of measure and creatively present the information. MEASUREMENT History Resource The Math Teacher's Book of Lists: Grades 5-12, 2nd Edition [Paperback] Judith A. Muschla Robert Muschla Systems of Measurement Page 39 M11.4 B001IQZOA6 B001H6RTNM (Author), Gary (Author) Activity Have students devise their own system of measurement and Mathematics 11 explain it to others. Students can use an item with no calibrations marked, for example the length of their arm or a piece of string. This will reinforce the need for standardized measures. Activity Examine and discuss what is measured and with what tools. For example, time is measured with a clock and temperature is measured with a thermometer. Activity Based on an estimate of the length of your foot, estimate the dimensions of your bedroom, and compare your estimate with the measurement you get using a tape measure. Activity Exploring the Relationship Between Metric and Imperial Measures for Length Appendix D.3. Resource Systems of Measurement MathWorks 10 (2010). pp. 94 - 103 Conversions M11.4 Activity Use a ruler to draw and label lines for the following measurements: 10 inches, 5 inches, 3 centimeters, 15 centimeters, 1 foot, 1 inch, 3 inches, and 10 centimeters. Activity Measure an object in your home and convert between SI and Imperial units. What maximum size fridge/stove will fit in your kitchen? Resource Converting Measurements MathWorks 10 (2010). pp. 106 – 113 Perimeter M11.1 M11.4 Activity What is Perimeter? Appendix D.4. Application Perimeter Measurement and Conversions Appendix D.5. Page 40 Mathematics 11 Area M11.1 M11.4 Connections Exponents and The Product and Quotient Laws Appendix D.6. Activity What is Area? Appendix D.7. TOPIC: ANGLES Angles OUTCOME: M11.5 SUGGESTIONS FOR TEACHING AND LEARNING: Activity What is an Angle? Appendix D.8. Activity Using Pattern and Fraction Blocks to Understand Angles Appendix D.9. Resource Measuring, Drawing, and Estimating Angles MathWorks 10 (2010). pp. 174 – 186 Angle Construction and Bisection M11.5 Instruction Construct and Bisect Angles Appendix D.10. Resource Angle Bisectors and Perpendicular Lines MathWorks 10 (2010). pp. 187 – 195 Complementary, Supplementary, and Vertically Opposite Angles M11.1 M11.5 Investigate Complementary, Supplementary and Vertically Opposite Angles Appendix D.11. Game Go Fish or Memory Card Game Appendix D.12. Angles in Construction Page 41 M11.5 Project Consider the design of a shed, dog house, ice fishing shack, etc. Describe the cutting angles for framing a window with Mathematics 11 casing or a room with baseboards where mitre cuts are required. Resources Draw a Floor Plan MathWorks 10 (2010). p. 105 Basic Carpentry Framing Tips http://www.carpentry-pro-framer.com/index.html J & H Builders Window Measure Information Form: http://www.jhbuilders.com/common/pdf/window_measure_f orm042910.pdf J & H Builders Door Measure Information Form: http://www.jhbuilders.com/common/pdf/door_measure_for m042910.pdf TOPIC: OUTCOME: SUGGESTIONS FOR TEACHING AND LEARNING: M11.6 Activity Right Triangles And The Pythagorean Theorem Appendix D.13. PYTHAGOREAN THEOREM The Right Triangle And Pythagorean Theorem Activity Pythagorean Theorem Choice Board Appendix D.14. Application What is a 3:4:5 ratio? Find at least 5 sets of Pythagorean triples that are not multiples of each other. Use the 3:4:5 ratio to determine if objects in the classroom or your house are square. Resource The Pythagorean Theorem MathWorks 10 (2010). pp. 272 – 282 Irrational Numbers M11.6 Activity Square Roots and Irrational Numbers Appendix D.15. Resource MathLinks 9 (2009). pp. 72 – 81 Math Makes Sense 9 (2009), pp. 4 – 21 Page 42 Mathematics 11 Theme Overview: Recreation and Wellness Theme Introduction Recreational activities such as playing games, solving puzzles, and participating in sporting events as well as activities connected to personal wellness will be used to teach problem solving strategies and reasoning skills. Students will use data collection and analysis to make predictions and inform decisions in order to effect changes in their own lives in terms of recreation and personal wellness. Outcomes that overlap in all four themes M11.1 Extend understanding of arithmetic operations to rational numbers to solve problems within the home, money, recreation, and travel themes. M11.7 Demonstrate understanding of proportional reasoning within the home, recreation, and travel themes. Outcomes M11.2 Demonstrate understanding of reasoning by analyzing puzzles and games. M11.3 Demonstrate understanding of data collection and analysis within the home, money, recreation, and travel themes. At a Glance Number sense Estimation Proportional reasoning Strategizing Solving puzzles Reasoning Data collection Data analysis Guiding Questions Page 43 When you think of your favorite game, what comes to mind? What strategies do you use to win a game? What is an effective strategy? Can games/puzzles be solved more than one way? What is your favorite approach to solving a game/puzzle? What is the purpose of collecting and analyzing data? Why display data? Can the data be presented in such a way to give a different message? Which is the best way to display the data? Can you make predictions from data? Can you make decisions from data? Mathematics 11 Can you identify trends? Career Connections Weather forecaster Lifestyle analyst Gamer Nutritionist Athlete Data Analyst Education policy analysis and researchers Business owners TOPIC: PUZZLES AND GAMES OUTCOME: Dietician Personal trainer Sports analyst Statistician Health Care Analyst Manager Scientific researcher Financial Analyst SUGGESTED TEACHING AND LEARNING: Analyze and Strategize M11.2 Activity Puzzles and Games Appendix E.1. Create a Game M11.2 Task What’s the Name of the Game? Appendix E.2. TOPIC: OUTCOME: SUGGESTED TEACHING AND LEARNING: DATA COLLECTION AND ANALYSIS Read and Interpret Graphs M11.3 Pre-Assessment (replicate in Home and Travel and Transportation) Reading and Interpreting Graphs Appendix D.1. Collect and Analyze Data M11.3 Pre-Assessment (replicate in Home and Travel and Transportation) Collect, Organize, and Analyze Data Appendix D.2. Activity Personal Wellness Appendix E.3. Page 44 Mathematics 11 Practice: Data Analysis MathLinks 9 (2009). pp. 410 – 429. TOPIC: PROPORTIONAL REASONING OUTCOME: SUGGESTED TEACHING AND LEARNING: Unit Rates and Unit Pricing M11.7 Skill Building You are training for an upcoming cross-country meet. You run13 km, three times a week. Your goal is to increase you average speed by 1.5 km/h, so that you can complete each run in 1 ¼ h. How long does it take you to complete each run now, to the nearest tenth of a minute? (Answer: 1 h 27.6 min) (MathLinks 9, p. 321). Activity Which is Your Cookie of Choice? Appendix E.4. Activity (replicate in Travel and Transportation) What Are We Going To Do For Entertainment? Appendix E.5. Activity Exercise and Fitness Appendix E.6. Activity Measuring Heartbeat http://www.glencoe.com/sec/math/t_resources/lab_manu al/pdfs/mac1_04/scimath_lab02.pdf Target Heart Rate: Calculating the Math http://www.kcautv.com/Global/story.asp?s=275590 The Beat Goes On Activity – Heartbeat Math Worksheet http://www.teachengineering.org/collection/cub_/activities/ cub_human/cub_human_lesson05_activity2_mathworksh eet.pdf Resources Proportional Reasoning MathWorks 10 (2010). pp. 12 - 22 Unit Price Page 45 Mathematics 11 MathWorks 10 (2010). pp. 23 – 27 Page 46 Mathematics 11 Theme Overview: Travel and Transportation Theme Introduction In the Travel and Transportation theme, students will use data collection and analysis to inform decisions in regards to planning a trip. Students will explore the parts of travel such as time, time zones, temperature, measurement conversions, price comparison, and currency exchange. Outcomes that overlap in all four themes M11.1 Extend understanding of arithmetic operations to rational numbers to solve problems within the home, money, recreation, and travel themes. M11.7 Demonstrate understanding of proportional reasoning within the home, money, recreation, and travel themes. Outcomes M11.3 Demonstrate understanding of data collection and analysis within the home, recreation, and travel themes. M11.4 Demonstrate understanding of measurement in the Système International (metric) and Imperial System within the home and travel themes. At Glance Number sense Estimation Proportional reasoning Data collection Data analysis Time Time zones Temperature Measurement Conversions Unit pricing Price comparisons Currency Guiding Questions Page 47 What is the purpose of collecting and analyzing data? Why display data? Can the data be presented in such a way to give a different message? Which is the best way to display the data? Can you make predictions from data? Can you make decisions from data? Can you identify trends? How does the concept of time factor into travel plans? How is time a measurement? What words are used to refer to time? What units are used for time? How does the concept of temperature factor into travel plans? Mathematics 11 How is temperature a measurement? What words are used to refer to temperature? What units are used for temperature? How does the concept of measurement conversions factor into travel plans? How do Canadian prices compare to foreign prices for the same item? Career Connections Travel agent Bank teller TOPIC: Travel show host Flight attendant OUTCOME: SUGGESTED TEACHING AND LEARNING: DATA COLLECTION AND ANALYSIS Read and Interpret Graphs M11.3 Pre-Assessment (replicate in Home and Recreation and Wellness) Reading and Interpreting Graphs Appendix D.1. Collect and Analyze Data M11.3 Pre-Assessment (replicate in Home and Recreation and Wellness) Collect, Organize, and Analyze Data Appendix D.2. Activity Survey the class on the mode of transportation they take to school each day. Represent the data graphically. Activity Where Should We Stay? Appendix F.1. Practice: Data Analysis MathLinks 9 (2009). pp. 410 – 429. Page 48 Mathematics 11 TOPIC: MEASUREMENT CONVERSIONS OUTCOME: Time M11.4 SUGGESTED TEACHING AND LEARNING: Activity How Old Am I? Appendix F.2. Activity Happy New Year! Appendix F.3. Temperature M11.3 M11.4 Activity What’s The Temperature? Appendix F.4. Linear and Weight Measurements M11.4 Activity Carry-On Safety Requirements Appendix F.5. M11.3 M11.4 Activity Come Fly With Me Appendix F.6. TOPIC: PROPORTIONAL REASONING OUTCOME: Unit Rates and Unit Pricing M11.7 SUGGESTED TEACHING AND LEARNING: Activity (replicate in Recreation and Wellness) What Are We Going To Do For Entertainment? Appendix E.5. Resources Proportional Reasoning MathWorks 10 (2010). pp. 12 - 22 Unit Price MathWorks 10 (2010). pp. 23 – 27 Currency Page 49 M11.4 M11.7 Activity Discuss items that students have recently bought. Have students recall the exact price. Research the item in another country, find the price and convert to Canadian dollars. How do the prices compare? Students can use online exchange rate charts and calculators to compare costs of items. Mathematics 11 Resources Currency Exchange Rates MathWorks 10 (2010). pp. 41 - 48 Converting Canadian and U.S. Dollars Math Essentials 10 (2005). p. 238 Currency Converter http://www.canadianforex.ca/currency-converter M11.3 M11.4 M11.7 Page 50 Project Let’s Travel! Appendix F.7. Mathematics 11 Appendices Appendix A: Arithmetic Operations Appendix A.1 Understanding Arithmetic Operations Whole Numbers and Integers Jillian was on page 56 of her book. She read 15 more pages. On what page did she end up? How far is it from 109 to 143? How many strategies can you use to multiply 34 and 78? Chris can put 8 pictures on one page of his photo album. If he has 123 pictures, how many pages will he need? Why might it be easier to calculate 297 ÷ 3 mentally by thinking of 297 as 300 - 3? Fractions 1 Jack and Jill ordered two medium pizzas, Gemma had 4 4 of licorice twists. After her 5 one cheese and one pepperoni. Jack ate 6 brother asked her for some, Gemma broke 1 1 of 2 of one licorice twist and gave it to him. of a pizza and Jill ate 2 of a pizza. How How much licorice does Gemma have left? much pizza did they eat together? Elementary and Middle School Mathematics Teaching Developmentally 2nd Ed. Van de Walle and Folk (p. 332) 1 First Steps in Mathematics Operation Sense (p. 38) 12 Lori bought 6 4 kilograms of potatoes. She Why might it be easier to calculate 15 ÷ used 1 8 of a kilogram for mashed potatoes. How much did she have left over? 3 instead of 5 ÷ 3 using mental math? 7 4 Big Ideas from Dr. Small Creating a Comfort Zone for Teaching Mathematics Grades 4-8 Small (p. 59) Decimals Mark jumps 2.38 m. If he jumps another 0.87 m, he will have jumped the same distance as Sonya. How far did Sonya jump? First Steps in Mathematics Operation Sense (p. 39) Yesterday the minimum temperature was 15.2°C in the morning. By the afternoon, the temperature rose to a maximum of 38.8°C. How much did the temperature rise? First Steps in Mathematics Operation Sense (p. 39) A painter needs to calculate the area of a How many students will get 0.75 litres of wall to know how much paint to order. The juice from a 21-litre container? First Steps in Mathematics Operation Sense (p. 66) wall is 7.1 m long and 3.5 m high. What is the area? First Steps in Mathematics Operation Sense (p. 55) Page 51 Mathematics 11 Appendix A.2 Arithmetic Operations Choice Board Complete three activities in a row (vertical, horizontal, or diagonal). On a $10 purchase, Tom was offered 3 successive discounts of 20%, 10%, and 5% in any order he wished. He selected the discounts in the order 5%, 10% and 20%. Which of the following order of discounts would have been better for him? A) 20, 10, 5 B) 20, 5, 10 C) 5, 20, 10 D) 10, 20, 5 Find a sale item and decide whether or not it is a good sale. Provide justification for your response. How does adding one to the digits of any fraction affect its size? What is 30% of 40% of 50? Precisely one of the numbers: 234 2345 23 456 234 567 Represent your answer with 2 345 678 a picture. 23 456 789 is a prime number. Which one must it be? Write a justification for your answer. In January, fares went up by 20%. In August, they went down by 20%. Sue claims that "The fares are now back to what they were before the January increase." Show a number between 0.6 and 0.7. Is the effect always the same? Explain your answer using a whole number fare of $100. Use a number line to support your answers. Write a justification for your answer. What if you double the numerator of a fraction and halve the denominator, what happens to the value of the fraction? Go to the provided website and solve the indicated problem. http://figurethis.org/challeng es/c30/challenge.htm If an item is reduced by 10% and then 10% again, then what single percent reduction is equivalent? 3 4 Compare 10 with 10. 3 4 Compare 10 with 11. Experiment with some numbers and draw a conclusion based on your results. Page 52 Create your own question that is similar to the one on the website. Now between 0.6 and 0.61. How many numbers are between 1 and 2? Represent your answer concretely, pictorially, or symbolically. Mathematics 11 Appendix A.3 Basic Skills Choice Board Complete three activities in a row (vertical, horizontal, or diagonal). Do stores order a percentage of each size? Why is my size always gone when things go on sale? What do you see in a photo? take a photo describe fractions found within the picture What does your stride length say about you? Figure out the puzzle. http://www.youtube.com/wat ch?v=Dkvu4TE9cLg Enlarge a photo on a photocopier to 200%. Which one is “more square”; the original photo or the enlargement? Do the fractions, ratios, and percents become 200% larger? How have sizes changed over the years? plate sizes and silverware drink cups clothing sizes Perfect 10. http://www.yummymath.co m/2012/perfect-10/ What proportions have gotten smaller over the years? How much smaller? Which location is the sunniest? What does your stride length say about you? Can the length of your leg and foot size predict the length of your stride? Take measurements and investigate the relationship between the length of your leg and the length of your stride. How have sizes changed over the years? Read Vanity sizing http://en.wikipedia.org/wiki/Vanity_sizing Which location is the sunniest? Explain how you know that one location will be the sunniest using fractions. Page 53 Mathematics 11 Page 54 Mathematics 11 Appendix A.4 Cut It Interactive: Try the Slice It application on Google or Android. Practice: Task: Develop your own Cut It game on paper. Activity: Print or trace the Cut It pieces on graph paper. Count the square units of each piece and calculate the percentage that you cut the original shape into. Page 55 Mathematics 11 Appendix B: Proportional Reasoning Appendix B.1 How Many Students in the Class? Introduction: This activity can be used to get an understanding of the prior knowledge of proportional reasoning that your students have. Problem: 1. Suppose there are two groups in an after school program, one with 20 students and one with 25. If the first class has 10 boys, and the second class has 12, which class has more boys? (Possible student response: the program with 12 girls has more). 2. If class C has 42 students, how many boys would there be if it is in proportion to the class with 25 students? 3. If there are 5 boys in class D, how many boys would there be if it is in proportion to the class with 20 students? Questions: How did you find your answers? What strategies did you use? Did you compare ratios? What is a ratio? What is a proportion? How are ratios and proportions similar? Appendix B.2 What`s The Cost Of Those Bananas? Activity: Have students go to the local supermarket and select one item from the produce department that is paid for by weight. Have them calculate the cost of the object using the hanging pan scale present in the department. Record their data. At the checkout counter, have the students record the weight given on the electronic balance used by the cashier. Have students record the cost of the item. How do the two measurements and costs compare? Have students explain the significance of the number of digits (precision) of the scales. This application can be done in the classroom if there is access to a pan scale and an electronic balance. If done in the classroom, provide items for students to measure— bunch of bananas, two or three potatoes. Page 56 Mathematics 11 Appendix B.3 Proportional Reasoning Choice Board Complete a row, column or diagonal line of activities. How much smaller is Mini What’ s your favorite percent Me in Austin Powers? of chocolate milk? http://robertkaplinsky.co m/work/mini-me/ http://www.yummymath.com/ 2010/chocolate-milk-andmixture-problems/ Would Usain Bolt still be the fastest if they organized the race according to body proportions? http://www.mathalicious.co m/lesson/on-your-mark/ Write out 8 proportionality statements you believe are true and 2 you believe are false. How far apart are the freeway exits? How fast are you going? http://robertkaplinsky.com/wo rk/freeway-exits/ http://mrpiccmath.weebly.co m/1/post/2011/08/roadlines.html What would Barbie’s proportions be if she was human sized? How much money is that? How many Pepsi points would it take to earn a harrier? http://180days201213.fawnnguyen.com/201 3/06/03/day-166---barbieproportions.aspx Page 57 http://robertkaplinsky.com/wo rk/drug-money/ http://mrpiccmath.weebly.co m/1/post/2012/07/3-actspepsi-points.html Mathematics 11 Appendix C: Earning and Spending Money Appendix C.1 Does Money Grow on Trees? Brainstorm: Pose the question “If you have a job, how do you get paid?” and brainstorm with students the things they know. Discuss: Use the following questions to generate discussion: Are there other ways of getting paid? What are the different ways that employees can be paid? How often are you paid? Research: Have students search job ads in the newspaper or online to find examples of other ways of getting paid. Instruction: Gross pay is the amount earned before deductions of income tax, employment insurance, Canada pension plan, and other secondary deductions. Net pay is the amount of money an employee earns after deductions. The following is a list of ways employees can be paid depending on where they work. The calculations in each of these examples will be gross pay. 1. A wage is the amount an employee earns for each regular hour of work. A regular work week usually consists of 40 hours. According to Stats Canada, part time work consists of less than 30 hours of work per week. Minimum wage in Saskatchewan is _______. Any hours over 40, employers must pay overtime or give equivalent time off to the employee. Overtime rates vary but usually are time and a half (1.5 × wage). If you work on a statutory holiday you will be paid double time. There are additional earning possibilities (e.g. tips) for people who receive a wage. Example: Alice works for minimum wage at a restaurant. She works 43 hours this week and earns $250 in tips. Twenty-five percent of her tips must be shared with the other employees. Calculate the wage she earns this week. 2. A salary is a fixed payment made by an employer at regular intervals, usually monthly, bimonthly or occasionally weekly. Example: An advertisement seeking people to teach English in Japan states the applicant will receive $42 000/yr. Calculate the weekly gross salary and monthly gross salary. 3. Straight commission is when an employee receives a percentage of the total sales made, regardless of how much he/she sold. Page 58 Mathematics 11 Example: The Falcon Real Estate agency pays a commission rate of 2%. If you sold $425 000 worth of property in a month, what would you receive? 4. Graduated commission is a commission which offers incentive to employees to sell more, because they receive a higher rate of commission as the volume of sales increases. Example: Russell is a computer retailer. He receives 4% commission on sales up to $8000, for sales between $8000 and $15 000 the rate is 6%, for sales over $15 000 he receives 8% on the amount over the $15 000. If Dec Russ sold $33 500 worth of goods, calculate his earnings: a) On the first $8000 – 4% b) On the next $7000 ($8000 – $15 0000) – 6% c) On anything over $15 000 - 8% 5. Salary or wage plus commission is a fixed amount of money plus commission on top of your regular income. 6. Piece work is when you are paid for the amount of work you have done and not the hours it has taken you. Examples include tree planters and servers (tips can be considered piece work because it depends on the number of tables you have served). Practice: 1. Jane works at the recreation center for 8 hours Saturday and 6 hours on Sunday. She is paid at a rate of $10.70/hr. During the week she referees 4 games of basketball and is paid $25 per game. What is her total gross wage in a week? 2. Mark is a word processor operator. He makes $11.50 /hr. Determine his gross earnings for a week if he worked 52 hours. 3. Mary earns a gross salary of $28 000 a year. She also gets a commission of 7%. She sold $2673.19 in merchandise in her first two weeks of work. Determine her gross wage for these two weeks. 4. As a travel agent, Sheila who has built a large clientele who travel frequently. Her company pays her minimum wage plus 5% commission on total sales. February is a great month for sales to warm vacations. She works 40 hours per week for each week in February, and her total sales are $40 000. Determine her gross wage. 5. Duke is a server at an upscale restaurant. He earns minimum wage and has worked 30 hours this week. The customers have been very generous with tips and he collected $643, however he must give 35% of this to the dishwashers. Determine this week’s gross wage. Page 59 Mathematics 11 6. Robin sells houses for a major real estate company in Saskatoon and works on graduated commission. For total sales up to $300 000, she receives a commission of 1.3% and anything over $300 000 the commission rises to 2.5%. Last month Robin sold 3 houses at the following prices: $153 000, $165 000, $292 000 which totals to ____________. Determine the gross wage this month on her total sales. 7. Sam has been offered a job that pays $497.35 for a 35 hour work week. A second company offers Sam a job earning $16.75/hour, but will only guarantee 30 hours /week. Which job would you take if you were Sam? 8. Two sales positions are available in retail stores. One pays an hourly rate of $11.45 for 40 hours/week. The other pays a weekly salary of $405 for the same number of hours, plus a commission of 5% of sales. If average sales for the position are $750/week, then in which position would you earn the most? Activity: Have students consider two careers that they are interested in and research how they would get paid in those careers. Have students identify advantages and disadvantages of the method of each payment. Career: __________________ Starting wage/salary: _________ Method of Payment: ________________ Advantages Disadvantages Extension: Have students research the history of minimum wages in Saskatchewan. Page 60 Mathematics 11 Appendix C.2 Reading Timesheets Activity: Display timesheets that have been acquired from a local business, the internet or a textbook (MathWorks 10 (2010). pp. 61, 73). Questions: Do you use timesheets at your job? Have you ever completed a timesheet? What is the purpose of a timesheet? How many hours and minutes did each of the employees work each day? What are the similarities between the timesheets? What are the differences between the timesheets? Appendix C.3 Reading Pay Stubs Activity: Collect a variety of pay stubs and have students bring in their own examples of pay stubs. Use the pay stubs as a starting point for discussion. A discussion should lead into deductions and net pay. Read: Reading a Pay Stub http://www.tv411.org/finance/earning-spending/reading-paystub Page 61 Mathematics 11 Appendix C.4 Gross and Net Income Discuss: Use the following questions to generate discussion: What is your understanding of gross pay? What is your understanding of net pay? What deductions would you expect on your earnings? How old do you have to be for the employer to begin to make deductions? If you work part time, do you have the same deductions as someone who works full time? What are union dues? What occupations have to pay union dues? Practice: Calculating Gross Monthly Income Appendix C.5 Activity: Using the two careers that your researched earlier, find the wage. Calculate the yearly, monthly and weekly gross wages available for each job. Instruction: (Note: Values will change annually, so blanks have been inserted for the most current numbers. As well, information on Employment Insurance may need to be updated with the most current practices). Gross pay is subject to two types of deductions: Basic and Secondary Basic Deductions include Income Tax, Canada Pension Plan and Employment Insurance Premiums. 1. Income Tax is known as a progressive tax because the tax rate increases as your gross wage increases. Income taxes are used to finance government services. Federal tax rates 20___ 15% on the first $___________ 22% for more than $________________ (on the next ___________) 26.0% for more than _________________ (on the next ____________) 29.0% on any remainder (over $_______________) There are also Provincial taxes which are approximately 50% of the Federal tax. 2. Canada Pension Plan (CPP) contributions provide you and your dependents with some financial protection throughout your life (retirement pension as early as age 60, disability benefits, or survivor benefits). Employees, employers and the self employed contribute to the CPP. You pay CPP if you are 18 years of age or older and make more than $______________. CPP pensionable earnings ceiling $______________. Contributors who earn more than the $______________ ceiling on pensionable earnings in the year 20____ are not required or allowed to contribute more to the CPP. Page 62 Mathematics 11 Employee and employer contribution rates for the year 20_____ are _________%. The maximum employee and employer contribution to the plan will be $______________. 3. Employment insurance (EI) has no age restriction. Every dollar is insurable. You can receive regular EI benefits if you lose your job through no fault of your own (for example, due to shortage of work, seasonal or mass lay-offs) and can't find work, providing you meet these requirements: you must apply; you have paid into the EI account; you have been without work and without pay for at least seven consecutive days; you have worked for the required number of insurable hours based on where you live and the unemployment rate in your economic region at the time of filing your claim for benefits. In some instances, you may need more hours of insurable work to qualify. For example : if you are in the work force for the first time you will need a minimum of 910 hours of insurable work in the last 52 weeks to qualify; if you are re-entering the work force after an absence of two years, you are a re-entrant and will, in most instances, need a minimum of 910 hours of insurable work in the last 52 weeks to qualify; Employment Insurance premium rate for the year 20_____ is $_______ or ______%. As provided in legislation, employers will pay 1.4 times the employee rate, or $_______ per $100 of employee earnings. Maximum Insurable Earnings will remain at $____________ for 20_____. ($________/year) Secondary Deductions vary greatly depending on your work and your contract with your employer. Deductions may include group life insurance, union dues, disability insurance, etc. Page 63 Mathematics 11 Net Pay (take home pay) = Gross Pay – Deductions Example: If you made a salary of $2800 a month, and had secondary deductions of $70/month in union dues, determine your annual and monthly net wage. Solution: It is easier to calculate deductions on a yearly basis. Yearly Gross = _________________ a) Federal Income Tax b) CPP c) EI Total Fed ______________________ Provincial (Total Fed x 0.5) ____________________ Total Income Tax ____________________ Yearly Gross = _________________ Yearly Net wage: Federal Income Tax Provincial Income Tax CPP EI Secondary Deductions __________ __________ __________ __________ __________ Total Deductions ______________ Gross - Total Deductions _________ - _________ Monthly Net_____________ Determine the percent of total earnings deducted to the gross income: Total deductions × 100% = Gross Wage Practice: 1. Shannon earns a monthly salary of $3625.00. Determine her net yearly and monthly pay, and the percentage of deductions. 2. Terry earns an annual salary of $75 500. She pays into two secondary deductions: union dues of $652/year and union pension of $6264/year. Determine her net yearly and monthly pay and the percentage of deductions. 3. Curtis earns $54 300/year. He belongs to a union and pays yearly dues of $369.72. Calculate his net yearly and monthly pay and the percentage of deductions. Page 64 Mathematics 11 Appendix C.5 Calculating Gross Monthly Income If you are paid hourly $ _________ × _________ × 52 weeks (pay before (# of hours you deductions) work in 1 week) 12 months = $ ____________ (gross monthly income) If you are paid weekly $ _________ × 52 weeks (pay before deductions) 12 months = $ ____________ (gross monthly income) If you are paid bi-weekly $ _________ x 26 (pay before deductions) 12 months = $ ____________ (gross monthly income) If you are paid twice a month $ _________ x 24 (pay before deductions) 12 months = $ ____________ (gross monthly income) If you are paid monthly Page 65 $ ____________ (gross monthly income) Mathematics 11 Appendix C.6 Spending Log Activity: Log your spending over a two week period. Keep your receipts in an envelope (for a later activity). Date Item Purchased Category and Cost Category Total Total Spent: ____________________ Questions: What are two of your main recurring expenses? Were you surprised by the total amount spent in any category? Where you surprised by the total amount? Did you have any unexpected expenses? Did you have to add another category? How much did you spend in a week? How much did you save in a week? Have you saved any money in case you have an opportunity to do something unexpected? For example, go to a concert? What are the advantages of tracking your spending? Did you need to borrow money to make a purchase? Page 66 Mathematics 11 Appendix C.7 Needs Versus Wants Activity: Refer to the Spending Log Appendix C.6 created in the previous activity. Fill in the following table, determining if each item purchased was a need or a want. Be prepared to justify your choices. Needs Item Total Wants Cost Item Cost Total Extension: Cut pictures of desired purchases from magazines or print off the Internet and arrange them under the appropriate category. Discuss how one person’s want could be another person’s need. A variation is to use the SMART board and display pictures of potential purchases. Questions: Are there any items in both columns? Was it challenging to classify each item? In which category did you spend more money? Are there items in the “Want” category that could be eliminated? What would you eliminate in order to save for a special event? Did you need to borrow money in order to make a purchase? Page 67 Mathematics 11 Appendix C.8 PST and GST Use the receipts you collected. Identify items that had GST and/or PST added. Make a list below. PST and GST on my purchases Item Cost Before Taxes No Tax PST GST Cost After Taxes Total Questions: Which items had NO tax added? Which items had GST only? Did any items have only PST? What percent of the original cost of an item is the PST? The GST? What is the purpose of each tax? What are tax exemptions and when do they apply? Activity: You have $35 to spend in a store. You know you can’t buy anything that costs $35 because tax will be added. From the following list, identify 5 items you think you could afford. DVD............................. $31.20 Hoodie ........................ $32.80 Video Game ................ $27.00 Jeans .......................... $33.99 Phone Card ................. $30.00 Jacket ......................... $29.99 Headphones ................ $31.99 Shoes .......................... $31.82 Page 68 Mathematics 11 Record each item you circled. Estimate the amount of PST, GST, and total cost for each item. Item Price Estimated PST Estimated GST Estimated Total Cost Record each item you circled. Calculate the exact PST, GST and total cost for each item. Item Page 69 Price PST GST Total Cost Under/Over $35 Under Over Under Over Under Over Under Over Under Over Mathematics 11 Appendix C.9 What are You Buying? Brainstorm and Discuss: Pose the question “What do teenagers buy?” and brainstorm with students what they buy. Extend the conversation to discuss where they shop (online, in stores, malls, etc.) and determine if incentives increase their likelihood of buying an item. Have a class discussion of past purchases that had incentives (e.g., SPC cards, Group on, Best Buy price zone points, Shoppers Optimum, Safeway Club Card, BOGO, percent discounts, pre-sale gift with purchase). Research: Search online to find out what teenagers are purchasing. Activity: If you had $500 to spend on any items you would like, what would they be? Find/take photos of the item(s), find the exact price and where you would purchase it from. Why did you purchase this? Instead of spending the $500 on yourself, spend the money on friends or family. What would you buy, who would you buy it for, and what is the exact cost? Activity: You are putting together a travel kit for a trip. Estimate the cost of each individual toiletry, estimate the taxes and determine the total cost. Research on-line or use flyers to determine the actual cost. Page 70 Mathematics 11 Appendix D: Home Appendix D.1 Reading and Interpreting Graphs Activity: Search online for information that is related to the theme (home renovations, sports or recreational activities, travel) that are displayed graphically. Questions: What information can I interpret from the graph? Could a different message be interpreted? Can I identify any trends? Can I make any predictions from the data? Page 71 Mathematics 11 Appendix D.2 Collect, Organize, and Analyze Data Introduction: Research involves asking questions, collecting data, organizing the information, and analyzing the data to draw conclusions. Pre-Assessment: To review what students know about data collection and analysis, give the class a survey with a question such as: How many extracurricular activities (in and out of school) are you involved in? List all of the activities and indicate the number of hours for each activity per week. Collect the surveys and use to introduce the topic of collecting and analyzing data. Discussion: Discuss that data can be numerical and non-numerical. Brainstorm ideas of numeric and non-numeric and record. Numeric Age # of siblings Non-numeric mode of transportation favorite food or color Activity: Use the data from the surveys to: Explain the differences between population and sample. Determine how the information gathered from the surveys can be organized and displayed on the board. Represent graphically. Show visuals of graphs (bar graph, broken-line graph, histogram) and discuss when best to use. Make conclusions about the data. Example: The following are the marks from a previous assessment: 74 82 65 51 88 76 68 84 59 73 81 92 73 Organize Data in a Stem and Leaf Plot: Page 72 Mathematics 11 Graphically Display Data: Circle Graph Page 73 Mathematics 11 Activity: 1. As a class, develop a question to determine the opinions of students at your school about a topic of choice. For a topic, you might choose favourite foods, sports, actors, or musicians. 2. As a class, write and edit the survey question. 3. Survey everyone in the class. 4. Organize the results. Based on the results of the class survey, predict the entire school’s response to the question. Questions: Does your prediction accurately reflect the opinions of all students in the school? Explain. Is the class a population or a sample? Explain. What are other examples within your school that explain the terms sample and population? How else might you choose people for your survey to reflect the opinions of all students in your school? Discuss: Use the following questions to generate discussion about data collection and analysis: What is the purpose of collecting data? What is the purpose of analyzing data? Why display data? What are the different ways you have seen data displayed? Can the data be presented in such a way to give a different message? How do you compile and interpret data? Which is the best way to display the data? Can you make predictions from data? Can you make decisions from data? Can you identify trends? Page 74 Mathematics 11 Appendix D.3 Exploring the Relationship Between Metric and Imperial Measures for Length Instructions: Measure how many centimetres make up an inch. This can be done using a paper ruler or sewing tape measure. Create a table of values and graph the values. Use paper and pencil or the following website for graphing: Meta Calculator http://www.meta-calculator.com/online/ Sample of student work using http://www.meta-calculator.com/online/ Page 75 Mathematics 11 Appendix D.4 What is Perimeter? Brainstorm: Pose the question “What do you know about perimeter?” and brainstorm with students the things they know or recall. Discuss: Use the following questions to generate discussion: What is perimeter? What do you need to know to measure perimeter? How is perimeter measured? Does every shape have a perimeter? How can you estimate the perimeter? How can you calculate perimeter? Investigate: Provide students with diagrams and ask them to find the perimeter of the shapes. Find the perimeter of different shapes in the classroom. Appendix D.5 Perimeter Measurement and Conversions Prior Knowledge: Review adding and subtracting like terms. For example, a movie is 2 hours and 30 minutes. How long is the movie? If you have 3 dimes and 2 nickels, then how much money do you have? These responses lead to the idea that conversion may be necessary before you can add or subtract lengths. Introduction: Measurements cannot be added if they have different units of measure. Conversion is required. Decisions need to be made about which unit is being converted. There is choice depending on preference or context of the question. Activity: Which room in your house requires the most, and which requires the least, amount of baseboard? What is the difference in the two amounts? Activity: Find the perimeter of baseboards, fencing, border around flower bed, or retaining wall. Dimensions that are not in the same units and must first be converted prior to adding. How many linear feet of fencing is required to go around your yard? How many linear feet of garden edging are required to build a garden? How many linear feet of baseboard/siding/window trim are needed to complete a renovation at your home/living room? Page 76 Mathematics 11 Appendix D.6 Exponents and the Product and Quotient Laws How are these expressions the same? How are they different? 2+2+2 2×2×2 How are these expressions the same? How are they different? feet + feet feet × feet How are exponents used when finding the units for perimeter and area? Perimeter cm + cm = cm (more centimetres directly measured and then counted) Area cm × cm = cm2 (derived measure in units squared) How does changing the dimensions change the perimeter and the area? 1’ 3 feet 1’ 1’ 1’ 3 feet 1’ 1’ 6 inches 2 feet 1’ Page 77 Mathematics 11 1’ Appendix D.7 What is Area? Brainstorm: Pose the question “What do you know about area?” and brainstorm with students the things they know or recall. Discuss: Use the following questions to generate discussion: What is area? What is the difference between perimeter and area? What do you need to know to measure area? How is area measured? Does every shape have an area? How can you estimate the area? How can you calculate area? Why are the units for area always square units or units squared? Investigate: Provide students with diagrams and ask them to find the area of the 2D shapes. Find objects of different shapes in the classroom to find the area of. Watch: Area of a Circle, How to Get the Formula http://www.youtube.com/watch?v=YokKp3pwVFc Proof Without Words: The Circle http://www.youtube.com/watch?v=whYqhpc6S6g Activity: Develop the formulas for the area of a square, triangle, and circle. Activity: Which room in your house requires the most, and which requires the least, amount of carpet? What is the difference in the two amounts? Activity: Calculate the area to paint or wall paper a wall, to tile a floor, to lay sod or patio slabs. Activity: You are asked to plant a garden with an area of 30 ft2. Using the seed packs provided, decide how you will plant your crops to allow enough growing space. Applications: How much paint is needed to paint the walls of the classroom? How many 2” × 4” boards are needed to build a fence in your backyard and stain it? How much sod would be required to redo your front lawn? Additional Questions: The environmental club has permission to use a rectangular plot of land in the school yard for composting and recycling storage. If they know the dimensions of Page 78 Mathematics 11 the plot of land, how can you they determine the area? If they know the area and the length of the land, how can they determine the width? (MathLinks 9, p. 99) How can we determine the area of irregular shapes by dividing it into regular shapes? How can we determine the area of composite figures? Project: Draw plans for a landscape design. Include one of the following design elements, which will be in the shape of a rectangle: swimming pool, concrete patio, hockey rink, or beach volleyball pit. Also include a design element that is in the shape of a triangle and one that is in the shape of a circle. (MathLinks 9, p. 271) Page 79 Mathematics 11 Appendix D.8 What is an Angle? Materials: protractor Brainstorm: Pose the question “What is an angle?” and brainstorm with students the things they know or recall. Discuss: Use the following questions to generate discussion. If an angle is a measure of rotation, how many degrees is one rotation? How can you estimate the measure of an angle? How are angles measured? Can you measure a reflex angle? Skill Building: Create referent angles of 30°, 45°, 60°, 90°, 180° using a clock face, folding paper, etc. Sketch an angle of given measure (e.g. 38°) using the referents. Activity: Use a circle to create and label acute, obtuse, right, straight and reflex angles. Activity: Using objects in the classroom, identify and name types of angles (e.g. acute, obtuse, right, straight and reflex). Practice: Angle Worksheets www.math-aids.com/Geometry/Angles/ Page 80 Mathematics 11 Appendix D.9 Using Pattern and Fraction Blocks to Understand Angles Resource: Grade 6-9 Math Workshop: Symmetry, Lines, and Angles, pp. 10 - 11 Materials: Pattern and fraction blocks, protractor Page 81 Mathematics 11 (Understanding Fractions http://mathcentral.uregina.ca/RR/database/RR.09.95/hanson4.html) Activity: Have students: 1. Identify all of the different angles in the pattern block shapes as acute, right, or obtuse. 2. Using the square as the benchmark, determine the size of the different angles (without a protractor). For each angle, trace the pattern block, identify on the diagram the angle being referred to and give a written explanation of how they determined the angle. 3. Share the different ways that they have determined the size of the angle. 4. Find the reflex angle on a pattern block. What is its measure? 5. By combining two or more pattern blocks, create a new angle. Have a classmate find the measure of the angle using benchmarks. What type of angle is this? 6. Using pattern blocks, create a straight angle. How many different ways can you do this? 7. Use the blocks to measure other angles in the environment. Page 82 Mathematics 11 Appendix D.10 Construct and Bisect Angles Resource: Grade 6-9 Math Workshop: Symmetry, Lines, and Angles, pp. 17 - 18 Materials: Mira, compass, protractor, straight edge Instruction: Constructions Angle Bisectors Paper Folding (Informal Construction) Fold paper at the vertex, making sure that one ray is directly matched over the other. MIRA (Informal Construction) Place MIRA on the vertex, through the middle of the angle. Reflect one ray onto the other. Draw in the dotted line. Compass and Straight Edge (Formal Construction) Note: It is easier to draw the intersecting arcs when the radius of the compass is greater than half of the length of the line segment shown. No quantitative measuring is involved in constructions! Place compass on vertex. Draw an arc of the same length on each ray. Place compass on points where arcs intersect the rays and draw a new arc from each. Draw a line connecting vertex and the new intersection point. Skill Building: Practice constructing and bisecting angles of different measures using the angle construction and bisection methods. Page 83 Mathematics 11 Appendix D.11 Complementary, Supplementary and Vertically Opposite Angles Resource: Grade 6-9 Math Workshop: Symmetry, Lines, and Angles, p. 11 Materials: Brads Straight angle (2) and right angle templates (made of cardstock) Paper strip Protractor Introduction: Complementary and supplementary angles come in pairs and as the measure of one angle of the pair increases, the other decreases. The purpose of this activity is to have students physically manipulate angles to develop a better understanding of complementary and supplementary angles. Students will also develop a better understanding of vertically opposite angles. Investigate: Have the students: 1. Using the right angle template, attach the ray template to the vertex of the angle using a brad, and then move the ray up and down, measuring the resulting angles with a protractor. Record the measures of angles 1 and 2 in the table below and determine the relationship. ∠1 ∠2 2. Using the straight angle template, attach the ray template to the center of the line using a brad, and then move the ray up and down, measuring the resulting angles with a protractor. Record the measures of angles 1 and 2 in the table below and determine the relationship. ∠1 ∠2 3. Using two straight angle templates, attach them in the center of each line using a brad, creating an X. Measure the angles with a protractor. Record the measures of angles 1 and 2 in the table below and determine the relationship. Page 84 Mathematics 11 ∠1 ∠2 Page 85 Mathematics 11 Appendix D.12 Go Fish or Memory Card Games Resource: Grade 6-9 Math Workshop: Symmetry, Lines, and Angles, pp.11 - 12 Materials: Cards numbered 0° to 180° plus any extras. Instructions: Each group is provided with a set of cards that are to be shuffled. The game can be played in a variety of ways: 1. Each player is dealt 6 cards. Taking turns, each player asks someone else if he/she has a particular card. The card that is requested must be one that is a supplement to a card in the player’s hand. If the card is a supplement, he/she must give it to the requester. If not, the player requesting draws one card. If the card is a supplement to a card in the player’s hand, he/she may lay the two cards down. If not, the player keeps the card in his/her hand and play moves to the next player. The game is completed when one player has laid down all of his/her cards as supplementary angles. 2. Each player is dealt 6 cards. The goal of this version is also to collect pairs of angle cards that are supplements, but in this version a player does not request cards from other players. Rather he/she draws a single card at the start of each turn or takes the last previously discarded card. The player ends his/her turn by discarding one card. When a player has all of the angle cards in his/her hand matched as supplements, the game is completed. 3. Spread the cards out face down. Players take turns flipping over two cards. If the cards are supplementary angles, the player removes the two cards and draws again. If they are not supplementary, the cards are flipped back over and the next player starts his/her turn. The game ends after a set period of time or once there are no remaining pairs. The player who has collected the most pairs is the winner. Adaptations: Have students write the rules for their favorite version of the game. Play these games using complementary angles. Remove all cards which have obtuse and straight angle measures. The “go fish” version may also be played where rather than asking for a particular value (40) the person asks for the complement to 50 degrees. Increase the difficulty level by playing both supplementary and complementary at the same time. Once students are familiar with integers, negative angles could be introduced. Cards including angles with decimals may also be added to the game (e.g., 37.5, 78.6). Extension: In any of the versions of the game, students with remaining angle cards could be asked to either construct the angle and its complement and/or supplement or to sketch them. Page 86 Mathematics 11 Page 87 0° 5° 10° 15° 20° 25° 30° 35° 40° Mathematics 11 Page 88 45° 50° 55° 60° 65° 70° 75° 80° 85° Mathematics 11 Page 89 90° 95° 100° 105° 110° 115° 120° 125° 130° Mathematics 11 Page 90 135° 140° 145° 150° 155° 160° 165° 170° 175° Mathematics 11 Page 91 180° FREE FREE 15° 20° 25° 30° 35° 40° Mathematics 11 Page 92 45° 50° 55° 60° 65° 70° 75° 80° 85° Mathematics 11 Page 93 90° 95° 100° 105° 110° 115° 120° 125° 130° Mathematics 11 Page 94 135° 140° 145° 150° 155° 160° 165° 170° 175° Mathematics 11 Page 95 180° FREE FREE 0° 5° 10° 0° 5° 10° Mathematics 11 Page 96 15° 20° 25° 180° FREE FREE 30° 35° 40° Mathematics 11 45° 50° 55° DRAW 2 DRAW 2 DRAW 2 60° Page 97 65° 70° Mathematics 11 Page 98 75° 80° 85° 90° 95° 100° 105° 110° 115° Mathematics 11 Page 99 120° 125° 130° 135° 140° 145° 150° 155° 160° Mathematics 11 165° Page 100 170° 175° Mathematics 11 Page 101 Mathematics 11 Appendix D.13 Right Triangles and Pythagorean Theorem Materials: shoes, measuring tape Interactive: To learn about right triangles, try this site and select right triangles: Triangles http://www.mathsisfun.com/geometry/triangles-interactive.html Interactive: To learn about Pythagoras` Theorem, try this site: Pythagoras http://www.mathsisfun.com/pythagoras.html Instruction: In a right-angled triangle, we use certain words to describe its parts. Pythagoras, a Greek, noticed a very special relationship among the sides of a rightangled triangle. He noticed that for any right-angled triangle: a2 + b2 = c2 (leg)2 + (leg)2 = (hypotenuse)2 Any letters can be used to represent the angles and sides of the right-angled triangle. If you will see the letters A, B, C used for angles and a, b, c used for the sides: Page 102 Mathematics 11 a is opposite angle A (A) b is opposite B c is opposite C Activity: Got a Lot of Shoes? http://www.mathsisfun.com/activity/pythagoras-theoremshoes.html Activity: A Walk in the Desert http://www.mathsisfun.com/activity/walk-in-desert.html Page 103 Mathematics 11 Appendix D.14 Pythagorean Theorem Choice Board Complete a row, column or diagonal line of activities. Draw a right triangle and label the right angle, legs, and hypotenuse. State the relationship of the sides of a triangle. Write a song that contains one page of lyrics that explain or describe Pythagoras’ Theorem or a parody of an existing song or an original work. Determine a set of 8 Pythagorean triples. Prove them with equations. Name a career in which one would have to use the Pythagorean Theorem. Give an example of when, where, and how it would be used. Draw a comic strip that demonstrates Pythagoras’ Theorem. The comic strip must contain at least eight panels, clearly drawn characters, an explanation of a mathematical technique, concept, or rule, and element(s) of humor, irony, drama. Write a descriptive essay about Pythagoras: his life, accomplishments, and failures. Design a teaching tool with a diagram of a proof of the Pythagorean Theorem. Create four contextual problems that use the Pythagorean Theorem. Show all the solutions. Find another mathematical theorem. State it, diagram its proof, and write a paragraph about why, how, and where it works. (Choice Menus. http://curry.virginia.edu/uploads/resourceLibrary/nagc_choice_menus.pdf) Page 104 Mathematics 11 Appendix D.15 Square Roots and Irrational Numbers Investigate: To determine your students’ understanding of square roots, perfect squares, non-perfect squares and irrational numbers, have students try the Investigate (Math Makes Sense 9, pp. 6 – 7). Use perfect squares as benchmark to explain how to estimate non-perfect squares (p. 15). Example: To estimate√12, 12 is almost halfway between 9 and 16, yet closer to 9. A reasonable estimate may be: √12 ≈ 3.4 Check with a calculator. Skill Building: Ask students for some of their favorite numbers (not too large!). For example, hockey shirt, house address, birth date, birth year, number of Facebook friends, etc. Use these numbers to create irrational numbers. Without the use of calculators, have students order the irrational numbers on a number line. Page 105 Mathematics 11 Appendix E: Recreation and Wellness Appendix E.1 Puzzles and Games Introduction: Many people have an interest in puzzles and games. Sometimes, it is difficult to determine how puzzles work. However, with a few simple strategies, you can usually figure them out. Problem: On a shelf, there are 10 books with 100 pages each. If a bookworm starts at the first page of the first book and eats through the last page of the last book, how many pages does the bookworm eat through (excluding covers)? (Answer: 802) Use a strategy such at Think, Pair, Share as students work on the solution. During Think, individual students can work through the answer to the solution. During Pair, they can share their solution with a partner as well as discuss the strategies they used to solve the puzzle. During Share, the pairing can share the strategies they used to solve the puzzle. The students or the teacher can record them on the board. Discuss: the strategies that were used to solve the puzzle. Examples of strategies that may have been used are: guess and check look for a pattern make a systematic list draw or model eliminate possibilities solve a simpler problem work backwards develop alternative approaches The strategies that were not used by the students can be modelled by the teacher. Problem Solving Strategies http://pred.boun.edu.tr/ps/ps3.html Activity: Try the Handcuffs Puzzle http://britton.disted.camosun.bc.ca/jbhandcuff.htm Task: In groups of 3 or 4, provide students with a variety of puzzles that require the different strategies. Have each group work on one puzzle that is different from each of the other groups and ask that they find their solution using two different strategies. When each group member is confident with finding the solution and with the two strategies used, have the students form a jigsaw. One member from each group joins a new group. In the newly formed groups, each member will present their puzzle, provide time for the other members of the group to solve the puzzle, assisting when necessary and then providing the solution along with strategies as well. Activity: Provide students with puzzles that have incorrect solutions, so they can analyze them for errors. Page 106 Mathematics 11 Games: Create stations with different games such as Tetris, Rubik’s cube, Blokus, chess, checkers, Backgammon, Mastermind, Tic-Tac-Toe, Connect Four or Five, Battleship, Cathedral World, and Mancala. Provide students with YouTube videos if they are uncertain of how to play. Ask them to play and describe strategies of how they win each game. Award prizes for the student(s) that win the most games. Resources: Problem Solving Strategy Guess and Check http://library.thinkquest.org/25459/learning/problem/psguess.html Math Playground http://www.mathplayground.com/games.html Page 107 Mathematics 11 Appendix E.2 What’s The Name of the Game? Introduction: When you think of your favorite game, what comes to mind? If may be a computer game or video game. You may also enjoy playing games that have been around a lot longer. These may include the use of a game board and may involve cards, dice, or specially designed playing pieces. Examples of these games include chess, checkers, dominoes, euchre, bridge, Monopoly, and Scrabble (MathLinks 9, p. 42). Discuss: You may want to use the following probing questions to generate discussion. What kind of games do you play? (e.g. computer, video, board, card) What are your favorite games? Why? What makes a great game? Activity: You are a game designer. You are developing an online computer game for you and your friends to play. Describe the game and explain why it is going to be a great game. Task: Individually or in pairs, have students create their own game. They must provide a materials list, provide the materials, and make the game. Have students describe the rules of the game, including how the winner is decided, provide written instructions, and provide a demonstration to their classmates. Extension: Pair students with a younger grade. Have your students teach the younger ones how to play. Your students will be required to observe the younger ones playing, noting any strategies used in a math journal. Discuss as a class. Page 108 Mathematics 11 Appendix E.3 Personal Wellness Task: For a two week period, record the following information each day: Number of hours of sleep Number of hours of screen time (television, phone, computer, video games, etc.) Number of hours of vigorous activity Number of hours of leisurely activity Number of hours of sitting (at desk during school, homework, reading, traveling, etc.) Number of hours of at work Number of well balanced meals Number of unhealthy snacks, meals, etc. Assessment: Try one or more of the following: How Healthy is Your Lifestyle? http://definitionofwellness.com/wellnesshandouts/How_healthy_lifestyle.pdf Wellness Assessment http://definitionofwellness.com/wellness-assessment.html Personal Wellness Quiz http://www.definitionofwellness.com/wellnessassessments/personal-wellness-quiz.pdf Personal Health Assessment http://www.definitionofwellness.com/wellnessassessments/personal-wellness-quiz.pdf Fit Together: Personal Health Assessment http://www.fittogethernc.org/HRA.aspx Activity: As a class, determine how each of the pieces of data will be displayed (e.g. bar graph, line graph, broken-line graph, histogram, circle graph). Create eight different graphs for each of the points to be displayed around the classroom so that students can add their own information to the graphs. Discuss: Can you make predictions from data? Can you make decisions from data? Can you identify trends? In comparison to the rest of the class, are you an outlier or average? After viewing the data, are there some changes you want to make to your lifestyle? Note: Ask students to keep this data for Mathematics 21. They will be asked to complete the same data collection task again and they may want to compare and contrast or use the same information they have already gathered. Page 109 Mathematics 11 Appendix E.4 Which is Your Cookie of Choice? Materials: boxes of cookies Instructions: Display pictures of cookie boxes without explanation. Have students come up with some possible questions that could go along with these pictures. Students should try and figure out which is the best buy. This will lead into how to find the best buy and unit pricing. Appendix E.5 What Are We Going to Do for Entertainment? Brainstorm and Discuss: Pose the question “What are we going to do for entertainment on our trip?” As a class, brainstorm things to do (e.g. museum, amusement park, sporting event, concert, zoo, sightseeing, drama production). Activity: Individually, students pick three ideas and provide a short description for each attraction. Research the cost per person and the group cost. Which one is a better deal? Problems: 1. You bought five concert tickets and paid $4.50 handling fee for each ticket. The total cost, before tax, was $210.00. What was the cost of each ticket, excluding the handling fee? (Answer: $37.50) (MathLinks 9, p. 320). 2. A local golf course offers two plans for paying for buckets of balls at the driving range. The standard plan is $6 per bucket and the member’s plan is a $98 monthly fee plus $1.50 per bucket. If you use 25 buckets of balls per month, what is the price per bucket with the member’s plan? What is the total cost with the member’s plan? What is the total cost with the standard plan? Which is the better deal? (MathLinks 9, p. 366). 3. You are going to rent a climbing wall for a school fun night. The rental charge for the wall is $145/h. You have $800 to spend. For how many hours can you rent the wall and stay within your spending limit? (Answer: 5.5 h) (MathLinks 9, p. 451). Activity: Some amusement parks offer single-ride tickets, where you pay each time you ride, and all-day passes, where you pay once for unlimited rides. The prices for both types of tickets need to be high enough for the amusement park to earn a profit but low enough that people decide to come. Search for information about ticket prices at amusement parks. If you plan on going on seven rides at the amusement park, which is the better option? (MathLinks 9, p. 359). Page 110 Mathematics 11 Appendix E.6 Exercise and Fitness Resource: Kids Health: Fitness. http://classroom.kidshealth.org/classroom/9to12/personal/fitness/fitness.pdf Introduction: To improve personal wellness, we need to make healthy eating choices and increase our physical activity. Teens should be getting at least 60 minutes of physical activity each day, but computers, TVs, and video games can make this more difficult. Discuss: Use the following questions to generate discussion: What is fitness? What does it mean to be physically f it? How can people get fit? Why is exercise important? How does it benefit both the body and mind? Why is it sometimes difficult for people to stay fit? Make a list of 10 simple things people can do to maintain fitness. Activity: Exercise has many benefits — it can make you feel good, look good, and even ward off some diseases. It’s recommended that teens exercise at least 60 minutes a day — but if you don’t like sports (or you just aren’t very active) this can seem like a daunting task. 1. Outline a 5-day program, Get off the Couch http://classroom.kidshealth.org/classroom/9to12/personal/fitness/fitness_handout 1.pdf, which includes simple and fun exercises, to help you become more fit. The ideal exercise program combines strength training, aerobic exercise, and stretching. Consider different types of exercise and how you can break down the 60 minutes so it isn’t overwhelming. Be sure to incorporate warming up and cooling down, too. 2. Keep track of your exercise each day. Take note of the type of exercise (strength training, aerobic exercise, and stretching) you are doing and how many minutes. 3. At the end of the 5-day program, determine the proportion of strength training, aerobic exercise, and stretching you did each day. Then determine the amount of each you did on average over the 5 days. Are these the proportions of each type of exercise that you want to be doing? If so, restart your 5-day program. If not, reexamine and make changes to your 5-day program. Page 111 Mathematics 11 Appendix F: Travel and Transportation Appendix F.1 Where Should We Stay? Resource: Math Essentials 10 (2005). pp. 228 – 231. Introduction: When you travel out of town, one option for accommodations is a hotel. Often, a hotel’s prices depend on its location and features. A location close to restaurants, shopping, theatres, and other attractions is more desirable than a location in the suburbs. Some hotels have features such as a pool, games room, fitness club, free wireless, kitchenettes, and complimentary breakfast. Practice: Math Essentials 10 (2005). pp. 228 – 231. Activity: As a class, design a questionnaire about where you would like to go on a trip. Narrow down the choices then use the top five locations to survey the class, grade, family members, etc. (decide on the population and the sample). Based on the results of the survey, use the top location as the destination and collect data for a select date on hotel options and costs. The following table is an example of how the data could be collected: Hotel Name Average Room Rate per Night Stars Ratings Amenities In groups of 4, each group member takes one of the columns (e.g. average room rate per night) and graphically displays the information. Use the place mat activity for students to individually, based on their own information and graph, to decide and justify the best hotel choice and then as a group to synthesize the information and come to a consensus on which hotel is the best choice. Conclude with a class discussion about which hotel is the best choice. Page 112 Mathematics 11 Appendix F.2 How Old Am I? Introduction: Introduce the concept of time by writing the question on the board “How Old Are You?” Engage students in a discussion about age and how it relates to time. Practice: There are: _________ weeks in a year _________ months in a year _________ hours in a day _________ days in a year _________ days in a week _________ weeks in a month (Explain why this is difficult to answer accurately) Practice: Planning Trip Dates. Warm Up. Math Essentials 10 (2011). p. 194 Activity: To review what students know about time, pose the following questions: When is your birthday? (e.g. February 20, 1977) Can you write your birthday in numerical form? (e.g. 20/02/77) (For Explanation and Practice, refer to Math Essentials 10 (2011). pp. 195-197) How old are you in years? How old are you in months (rounded to one decimal place)? How old are you in days? How old are you in minutes? How old are you in seconds? If your birthday were February 29, 2004, how old are you? If your birthday were February 29, 2004, how many times would you have celebrated your birthday on your actual birth date? Brainstorm and Discuss: Pose the following questions to be discussed in class: How does the concept of time factor into travel plans? How is time a measurement? What words are used to refer to time? What units are used for time? Problem: A nurse wrote that I was born at 22:05. What time was I born at? Instruction and Practice: Time – AM/PM vs. 24 Hour Clock http://www.mathsisfun.com/time.html Practice: 24 Hour Clock Sort http://www.collaborativelearning.org/12and24hourclocksort.pdf Extension: Ask your parent(s)/guardian(s) or find the card given at the hospital that has the time you were born. Write this time as both the 12 and 24 hour times. Find your exact age in years, days, minutes and seconds. Page 113 Mathematics 11 Appendix F.3 Happy New Year! Problem: Pose the problem: Your friend claims that when she graduates, she will spend New Year’s Eve in Times Square, New York City (show an image or a video). Another friend claims he will celebrate New Year’s with her in Times Square, and then fly home to Saskatchewan to ring in the New Year with you. Is this possible? Practice: Time Zones. Math Essentials 10 (2011). pp. 198 – 201. Crossing Time Zones. Math Essentials 10 (2011). pp. 202 – 203. Problem: When a football game starts in Toronto at 3 p.m. ET, what time will the game start in Saskatchewan in August? In November? Resources: World Clock Chart http://www.timeanddate.com/worldclock/ Daylight Savings Time? http://www.timeanddate.com/time/dst/ Practice: Getting There by Airplane. Math Essentials 10 (2011). pp. 204. – 207. Arrival Times. Math Essentials 10 (2011). pp. 208 – 209. Travel Times (12-h Clock). Math Essentials 10 (2011). pp. 210 – 211. Schedules and the 24-h Clock. Math Essentials 10 (2011). pp. 212 – 214. Travel Times (24-h Clock). Math Essentials 10 (2011). pp. 215. Activity: Have students use the West Jet, Air Canada, Delta or United websites to look up flights to a destination requiring an overlay (e.g., London, England). Have the student look at the departure/arrival times versus the actual flying time (differences due to time changes. Brainstorm and Discuss: Revisit the Happy New Years! problem. Is it possible? If so, how? If not, could changes be made to make it possible? Where is the first place in the world to ring in the New Year? Where is the last place? Where and how many places could you ring in the new year taking into consideration flights, airport line ups/security, travel time during flights, to and from airport, etc. Where should you start? Where should you end? If you had unlimited funds (e.g. private jet, private car), how would this change your previous answer? Could you now visit more locations? Which ones? Page 114 Mathematics 11 Appendix F.4 What’s the Temperature? Problem: Pose the problem: Can it be 32° in January? Brainstorm: What should we consider before answering this question? What are the units of measure: °C and °F? What is the location? What is the time of day? If the location is Saskatchewan, can it be 32° in January? Activity: Using your community as the location, pick a time frame (5 or 10 years) to collect and organize the average monthly temperatures. Display each month’s average temperature (in both Celsius and Fahrenheit) graphically. Resources: What’s the Temperature? Math Essentials 10 (2011). pp. 216 – 219. Temperature Conversions MathWorks 10 (2010). pp. 138 - 145 Page 115 Mathematics 11 Appendix F.5 Carry-On Safety Requirements Materials: rulers, tape measures, weigh scale, clear plastic bags in a variety of sizes, a variety of toiletries (toothpaste, toothbrush, shampoo, conditioner, soap, deodorant, contact solution, medicine, etc.), a variety of items to pack in carry-on (clothing, books, electronics, shoes), carry-on suitcases, computer bags, bags, or purses in a variety of sizes. Introduction: Consider how the length + width + height and weight measurement is used in air travel. Research: Search online for carry-on safety requirements for airport security. Take note of toiletry limitations and requirements. Search airline requirements for carry-on luggage requirements and restrictions. Discuss: Discuss how an understanding of linear and weight measurement is necessary to pack toiletries and carry-on luggage. Practice: Convert the following measurements: 1. 18 m to ft. 2. 36 mm to inches 3. 5 feet and 8 inches to cm 4. 120 km to miles 5. 42 inches to feet 6. Inches to feet 7. 96 inches to yards 8. 5 miles to yards If more practice with measurement and conversions is needed, refer to Home or the following: Linear Measurement: Metric. Math Essentials 10 (2005). pp. 57 – 71. Linear Measurement: Imperial. Math Essentials 10 (2005). pp. 79 – 95. How Much Should I Bring? Math Essentials 10 (2005). Pp. 232 – 235. Systems of Measurement. MathWorks 10 (2010). pp. 94 – 103 Activity: Using the researched information and the toiletry items available, decide which system (imperial or metric) is more consistently used and change all measurements to that system. Task: Using the researched information, first have students pack their toiletry bag, meeting the safety requirements. Second, have students pick a carry-on, pack it and check the dimensions and weight to ensure that they could travel with it. Extension: Create a mock security line up. Have students act as security guards and others act as travellers. Travellers present their imitation passports, boarding pass to security and are asked to remove items such as shoes, hats, jackets, belts, items from pockets, etc. Security guards search carry-on and check toiletries for restricted items. Page 116 Mathematics 11 Appendix F.6 Come Fly With Me Materials: boarding pass Introduction: The purpose of this project is for students to plan a trip taking into consideration location, time, and temperature. Brainstorm and Discuss: Where are we going on our trip? The location of the trip and accommodations were previously determined in the activity Where Should We Stay? Appendix F.1. If this activity was not completed, then have the class brainstorm, discuss and choose a travel location (that requires air travel). Research: Have students determine when the best time of year to travel to their location would be. When the best month of the year has been agreed on, find the monthly average low and high temperatures in both Celsius and Fahrenheit. Research and Discuss: Use the internet to search for flights in that month. Determine which dates are the best to travel based on price and flight times, the total number of days on the trip, the number of travel days, the number of days that accommodations will be needed, etc. When a flight has been decided, record the departure and return information including location, times, and travel time. Have a discussion about the number of hours of air travel, the number of hours of lay over, and total traveling time. If a boarding pass is available, display and have students work individually or in pairs to gather the important information to determine the number of hours of air travel, the number of hours of lay over, and total traveling time. Activity: Now that flights and airline(s) have been decided, have students go home and measure their luggage to determine if the pieces they would choose as carry-on and checked would meet the airline requirements for size and weight. Page 117 Mathematics 11 Appendix F.7 Let’s Travel! Introduction: Your task is to create a travel itinerary and trip journal to a destination of your choice. You will need to plan your entire trip, including: accommodations, transportation, entertainment, and meals. Make sure that you take into account currency exchange if applicable. Task: 1. Where and when are you going? Make a tentative plan of where you will be going, possible dates and timeline for travel. 2. Accommodations: Where will you stay when you get there? Present three options, advantages and disadvantages, about where you might stay while on your vacation (e.g. hotel, resort, hostel, bed and breakfast, with family/friends). The presentation must include prices and amenity considerations. In a summary, identify your final lodging decisions and how you arrived at those decisions. Include information about amenities as well as meals or other perks. Make sure you also include your research and any other information you used to arrive at your decision. 3. Transportation: Where are you going and how are you getting there? Present three options, advantages and disadvantages, about how you might travel to and from your destination. The presentation must include prices and time considerations. Tell how you will get around while on the vacation (e.g. car, bus, taxi, walking, rental car). You need to include prices. In a summary, identify your final travel decisions and how you arrived at those decisions. Make sure you also include your research and any other information you used to arrive at your decisions. 4. Entertainment: What will you do when you get there? Present 15 options for activities for you to do while on your vacation. The presentation must include prices, travel and time considerations. Make sure all activities are school appropriate. In a summary, identify your final activity decisions and how you arrived at those decisions. Include the cost of the activities. The time frame for each event should also be in your summary. Make sure you also include your research and any other information you used to arrive at your decision. 5. Meals: What will you eat when you get there? Present 15 options for meals while on your vacation. The presentation must include prices and transportation considerations. You need to eat at least two really nice meals while on vacation. A really nice meal is eaten at a restaurant where you must sit down and be waited on. In a summary, identify your final meal decisions and how you arrived at those decisions. Include the estimated cost of each meal as well as a description of the type of cuisine. Give a brief description of the each restaurant and an approximate Page 118 Mathematics 11 location. Make sure you also include your research and any other information you used to arrive at your decision. 6. Packing: Prior to packing, you need to research the weather. What are the average temperatures during the time of your travel (convert to the SI measurement system where appropriate)? How will this affect the activities you chose? How will this affect what you pack in your luggage? How will you pack? What size/weight of luggage are you allotted? Include the size and weight of luggage in both SI and Imperial systems where appropriate. How much spending money will you take? Are you going to a different country? If so, convert your spending money from Canadian currency to the local currency. How much money do you have? 7. Present a day by day trip journal of transportation, lodging, entertainment and meals. Add all of your expenses for a grand total cost of your trip. Organize the information graphically and in a written format. Be prepared to present your travel itinerary. Page 119 Mathematics 11