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Grade Mathematics Support Document for Teachers Gr ade Mathemat ics Support Document for Teachers 2015 M a ni t o b a E d u c a t i o n a n d A d v a n c e d L e a r nin g Manitoba Education and Advanced Learning Cataloguing in Publication Data Grade mathematics [electronic resource] : support document for teachers Includes bibliographical references ISBN: 978-0-7711-5905-3 Mathematics—Study and teaching (Secondary) Mathematics—Study and teaching (Secondary)—Manitoba I Manitoba Manitoba Education and Advanced Learning 372.7044 Copyright © 2015, the Government of Manitoba, represented by the Minister of Education and Advanced Learning Manitoba Education and Advanced Learning School Programs Division Winnipeg, Manitoba, Canada Every effort has been made to acknowledge original sources and to comply with copyright law If cases are identified where this has not been done, please notify Manitoba Education and Advanced Learning Errors or omissions will be corrected in a future edition Sincere thanks to the authors, artists, and publishers who allowed their original material to be used All images found in this document are copyright protected and should not be extracted, accessed, or reproduced for any purpose other than for their intended educational use in this document Any websites referenced in this document are subject to change Educators are advised to preview and evaluate websites and online resources before recommending them for student use Print copies of this resource can be purchased from the Manitoba Text Book Bureau (stock number 80637) Order online at This resource is also available on the Manitoba Education and Advanced Learning website at Available in alternate formats upon request Contents Acknowledgements vii Introduction 1 Overview 2 Conceptual Framework for Kindergarten to Grade Mathematics Assessment 10 Instructional Focus 12 Document Organization and Format 13 Number 1 Number and Shape and Space (Measurement)—8.N.1, 8.N.2, 8.SS.1 Number—8.N.3 25 Number—8.N.4, 8.N.5 41 Number—8.N.6, 8.N.8 57 Number—8.N.7 81 Patterns and Relations Patterns and Relations (Patterns)—8.PR.1 Patterns and Relations (Variables and Equations)—8.PR.2 Shape and Space Shape and Space (Measurement and 3-D Objects and 2-D Shapes)— 8.SS.2, 8.SS.3, 8.SS.4, 8.SS.5 Shape and Space (Transformations)—8.SS.6 Statistics and Probability Statistics and Probability (Data Analysis)—8.SP.1 Statistics and Probability (Chance and Uncertainty)—8.SP.2 13 31 11 Bibliography 1 Contents iii Grade Mathematics Blackline Masters (BLMs) BLM 8.N.1.1: Determining Squares BLM 8.N.1.2: Determining Square Roots BLM 8.N.1.3: I Have , Who Has ? BLM 8.N.1.4: Pythagorean Theorem BLM 8.N.3.1: Percent Pre-Assessment BLM 8.N.3.2: Percent Self-Assessment BLM 8.N.3.3: Percent Grids BLM 8.N.3.4: Percent Scenarios BLM 8.N.3.5: Percent Savings BLM 8.N.3.6: Final Cost BLM 8.N.3.7: Percent Increase and Decrease BLM 8.N.4.1: Ratio Pre-Assessment BLM 8.N.4.2: Meaning of a ? b BLM 8.N.4.3: Problem Solving BLM 8.N.6.1: Mixed Numbers and Improper Fractions BLM 8.N.6.2: Mixed Number War BLM 8.N.6.3: Decimal Addition Wild Card BLM 8.N.6.4: Fraction Multiplication and Division BLM 8.N.6.5: Multiplying and Dividing Proper Fractions, Improper Fractions, and Mixed Numbers BLM 8.N.6.6: Fraction Operations BLM 8.N.7.1: Integer Pre-Assessment BLM 8.N.7.2: Solving Problems with Integers (A) BLM 8.N.7.3: Solving Problems with Integers (B) BLM 8.N.7.4: Solving Problems with Integers (C) BLM 8.N.7.5: Number Line Race BLM 8.PR.1.1: Patterns Pre-Assessment BLM 8.PR.1.2: Determine the Missing Values BLM 8.PR.1.3: Break the Code BLM 8.PR.1.4: Linear Relations BLM 8.PR.1.5: Graphs BLM 8.PR.2.1: Algebra Pre-Assessment BLM 8.PR.2.2: Solving Equations Symbolically BLM 8.PR.2.3: Algebra Match-up BLM 8.PR.2.4: Analyzing Equations BLM 8.PR.2.5: Analyzing Equations Assessment BLM 8.PR.2.6: Solving Problems Using a Linear Equation iv G r a d e M a t h e m a t i c s: S u p p o r t D o c u m e n t f o r Te a c h e r s BLM 8.SS.2.1: Measurement Pre-Assessment BLM 8.SS.2.2: Nets of 3–D Objects BLM 8.SS.2.3: 3–D Objects BLM 8.SS.2.4: Matching BLM 8.SS.3.1: Nets BLM 8.SS.3.2: Surface Area Problems BLM 8.SS.4.1: Volume Problems BLM 8.SS.6.1: Coordinate Image BLM 8.SS.6.2: Tessellating the Plane BLM 8.SS.6.3: Tessellation Slideshow BLM 8.SS.6.4: Tessellation Recording Sheet BLM 8.SS.6.5: Tessellation Transformation BLM 8.SP.1.1: Data Analysis Pre-Assessment BLM 8.SP.1.2: Data BLM 8.SP.1.3: Graph Samples BLM 8.SP.2.1: Probability Pre-Assessment BLM 8.SP.2.2: Tree Diagram BLM 8.SP.2.3: Table BLM 8.SP.2.4: Probability Problems BLM 8.SP.2.5: Probability Problem Practice Grades to Mathematics Blackline Masters BLM 5-8.1: Observation Form BLM 5-8.2: Concept Description Sheet BLM 5-8.3: Concept Description Sheet BLM 5–8.4: How I Worked in My Group BLM 5–8.5: Number Cards BLM 5–8.6: Blank Hundred Squares BLM 5–8.7: Place-Value Chart—Whole Numbers BLM 5–8.8: Mental Math Strategies BLM 5–8.9: Centimetre Grid Paper BLM 5–8.10: Base-Ten Grid Paper BLM 5–8.11: Multiplication Table BLM 5–8.12: Fraction Bars BLM 5–8.13: Clock Face BLM 5–8.14: Spinner BLM 5–8.15: Thousand Grid Contents v BLM 5–8.16: Place-Value Mat—Decimal Numbers BLM 5–8.17: Number Fan BLM 5–8.18: KWL Chart BLM 5–8.19: Double Number Line BLM 5–8.20: Algebra Tiles BLM 5–8.21: Isometric Dot Paper BLM 5–8.22: Dot Paper BLM 5–8.23: Understanding Words Chart BLM 5–8.24: Number Line BLM 5–8.25: My Success with Mathematical Processes BLM 5–8.26: Percent Circle vi G r a d e M a t h e m a t i c s: S u p p o r t D o c u m e n t f o r Te a c h e r s Acknowledgements Manitoba Education and Advanced Learning wishes to acknowledge the contribution of and to thank the members of the Grades to Mathematics Support Document Development Team Their dedication and hard work have made this document possible Writer Grades to Mathematics Support Document Development Team (2006–2008) Manitoba Education and Advanced Learning School Programs Division Staff Anita Fedoruk Louis Riel School Division Holly Forsyth Fort La Bosse School Division Linda Girling Louis Riel School Division Chris Harbeck Winnipeg School Division Heidi Holst Lord Selkirk School Division Steven Hunt Independent School Jan Jebsen Kelsey School Division Betty Johns University of Manitoba Dianna Kicenko Evergreen School Division Kelly Kuzyk Mountain View School Division Judy Maryniuk Lord Selkirk School Division Greg Sawatzky Hanover School Division Darlene Willetts Evergreen School Division Heather Anderson Consultant (until June 2007) Development Unit Instruction, Curriculum and Assessment Branch Carole Bilyk Project Manager Coordinator Development Unit Instruction, Curriculum and Assessment Branch Louise Boissonneault Coordinator Document Production Services Unit Educational Resources Branch Kristin Grapentine Desktop Publisher Document Production Services Unit Educational Resources Branch Heather Knight Wells Project Leader Development Unit Instruction, Curriculum and Assessment Branch Susan Letkemann Publications Editor Document Production Services Unit Educational Resources Branch Acknowledgements vii Suggestions for Instruction QQ QQ Explain how a formatting choice, such as the size of the intervals, the width of bars, or the visual representation, may lead to misinterpretation of the data Identify conclusions that are inconsistent with a data set or graph, and explain the misinterpretation Materials: BLM 8.SP.1.3: Graph Samples, math journals Organization: Small group/individual Procedure: Tell students that they will be analyzing different sets of graphs to determine which graph best represents the given data and explaining why one graph leads to a misinterpretation of the data represented Hand out graph Sample 1, Sample 2, and Sample from BLM 8.SP.1.3: Graph Samples, one at a time For each sample, have students, working in groups, discuss the following: QQ What can be said about these graphs? QQ What scenario the data display? QQ Is there something that can be done to each graph to make it clearer? Explain QQ Do the graphs display the same or different data? Explain QQ What are the advantages and disadvantages of each graph? QQ Which graph is more accurate? Explain QQ Can either of the graphs be misinterpreted? Explain Have students explain, in their math journals, how the format of graphs (how the graphs are made) can lead to a misinterpretation of the data Ask them to use words and diagrams to explain their thoughts Observation Checklist  Observe students’ responses to determine whether they can the following: r Explain how the format of graphs can lead to a misinterpretation of the data 10 G r a d e M a t h e m a t i c s: S u p p o r t D o c u m e n t f o r Te a c h e r s Statistics and Probability (Chance and Uncertainty)—8.SP.2 Enduring Understandings: The principles of probability of a single event also apply to independent events Probability can be expressed as a fraction or decimal between and 1, where indicates an impossible event and indicates a certain event Probabilities can be expressed as ratios, fractions, percents, and decimals General Learning Outcome: Use experimental or theoretical probabilities to represent and solve problems involving uncertainty Specific Learning Outcome(s): Achievement Indicators: 8.SP.2 Solve problems involving the probability of independent events [C, CN, PS, T]  Determine the probability of two independent events and verify the probability using a different strategy  Generalize and apply a rule for determining the probability of independent events  Solve a problem that involves determining the probability of independent events Prior Knowledge Students may have had experience with the following: QQ QQ Describing the likelihood of a single outcome occurring, using words such as QQ impossible QQ possible QQ certain Comparing the likelihood of two possible outcomes occurring, using words such as QQ less likely QQ equally likely QQ more likely Statistics and Probability 11 QQ Demonstrating an understanding of probability by QQ identifying all possible outcomes of a probability experiment QQ differentiating between experimental and theoretical probability QQ determining the theoretical probability of outcomes in a probability experiment QQ QQ QQ QQ QQ determining the experimental probability of outcomes in a probability experiment comparing experimental results with the theoretical probability for an experiment Expressing probabilities as ratios, fractions, and percents Identifying the sample space (where the combined sample space has 36 or fewer elements) for a probability experiment involving two independent events Conducting a probability experiment to compare the theoretical probability (determined using a tree diagram, table, or another graphic organizer) and experimental probability of two independent events Background Information Probability Probability refers to the chance of an event occurring The probability of an event must be greater than or equal to and less than or equal to Note: Students in Grade will be working only with the probabilities of independent events Definitions independent events Events in which the theoretical probability of an event occurring does not depend on the results of another event Example: Rolling a number cube and then selecting a card from a deck 12 What is the probability of rolling a on a number cube and then pulling a from a deck of cards? 1 P(6 cube) = P(6 card) = so P(6 cube and card) = 13 78 G r a d e M a t h e m a t i c s: S u p p o r t D o c u m e n t f o r Te a c h e r s dependent events Events in which the theoretical probability of an event occurring does depend on the results of another event Example: Selecting a red card from a deck and then selecting the Queen of Clubs without putting the first card back 1 P(red) = P(Queen of Clubs) = so P(red, then Queen of Clubs) = 51 102 Organizing Outcomes/Results There are different strategies for organizing favourable outcomes, such as tables and tree diagrams Example: If Joe has six cards numbered to and a regular six-sided number cube, what is the probability of turning a and rolling a at the same time? This scenario can be written as follows: What is P(1,1)? Table Card Number Cube 1,1 2,1 3,1 4,1 5,1 6,1 P(event) = P(1,1) = 1,2 2,2 3,2 4,2 5,2 6,2 1,3 2,3 3,3 4,3 5,3 6,3 1,4 2,4 3,4 4,4 5,4 6,4 1,5 2,5 3,5 4,5 5,5 6,5 1,6 2,6 3,6 4,6 5,6 6,6 favourable outcome total number of outcomes 36 Statistics and Probability 13 Tree Diagram P(1,1) = 14 36 Card Cube Outcome 1 1,1 1,2 1,3 1,4 1,5 1,6 2 2,1 2,2 2,3 2,4 2,5 2,6 3 3,1 3,2 3,3 3,4 3,5 3,6 4 4,1 4,2 4,3 4,4 4,5 4,6 5 5,1 5,2 5,3 5,4 5,5 5,6 6 6,1 6,2 6,3 6,4 6,5 6,6 This can also be expressed as 1:36, ≈3% or ≈ 0.03 G r a d e M a t h e m a t i c s: S u p p o r t D o c u m e n t f o r Te a c h e r s Mathematical Language certain experimental probability impossible independent events less likely likely more likely outcome probability probable simulation theoretical probability Learning Experiences Assessing Prior Knowledge Materials: BLM 8.SP.2.1: Probability Pre-Assessment Organization: Individual Procedure: Tell students that they will be extending their understanding of probability over the next few lessons; however, you first need to find out what they already know about probability Hand out copies of BLM 8.SP.2.1: Probability Pre-Assessment Have students complete the pre-assessment individually Observation Checklist  Observe students’ responses to determine whether they can the following: r Determine all possible outcomes of a specified event r Use an organization method to organize the outcomes r Determine the probability of a specified event r Represent probability as a ratio, fraction, decimal, and percent Statistics and Probability 15 Suggestions for Instruction QQ Determine the probability of two independent events and verify the probabilities using a different strategy Materials: six-sided number cube and coin per group, BLM 5–8.23: Understanding Words Chart, BLM 8.SP.2.2: Tree Diagram Organization: Individual/small group/whole class Procedure: Provide each student with a copy of BLM 5–8.23: Understanding Words Chart, and have students explore their understanding of key mathematical terms Divide the class into small groups, and present students with the following scenario: John rolls a six-sided number cube at the same time that Sue flips a coin Determine all possible outcomes if they complete the task at the exact same time (Provide each group with a number cube and a coin in case they need to manipulate the items to help them determine the outcomes.) Allow groups to organize their work as they see fit When it is time to review their work, write their outcomes on the whiteboard using a tree diagram When all possible outcomes are recorded, explain to students how a tree diagram allows favourable outcomes to be determined in an organized manner Ask students to state P(3, H), P(odd, T), P(1 or 2, H) Have students complete BLM 8.SP.2.2: Tree Diagram Observation Checklist  Observe students’ responses to determine whether they can the following: r Successfully use a tree diagram to determine probabilities 16 G r a d e M a t h e m a t i c s: S u p p o r t D o c u m e n t f o r Te a c h e r s Suggestions for Instruction QQ Determine the probability of two independent events and verify the probabilities using a different strategy Materials: A King, Queen, and Jack of Spades, and a 1, 2, and of Hearts of a deck of cards per group, BLM 8.SP.2.3: Table Organization: Small group/whole class Procedure: Divide the class into small groups, and present students with the following scenario: Rita has the Jack, Queen, and King of Spades from a deck of cards, and Jessica has the 1, 2, and of Hearts from the deck of cards If they both flip a card at the same time, what are all the possible outcomes? (Provide each group with the Jack, Queen, and King of Spades and the 1, 2, and of Hearts from a deck of cards in case they need to manipulate items to help them determine the outcomes.) Tell students that their task is to determine all possible outcomes using a strategy for organizing the outcomes other than a tree diagram Have each group present their method to the class If no one demonstrates how a table could be used, you will need to demonstrate it Ask students to state P(J, 1), P(face card, odd), P(Q, 1) Have students complete BLM 8.SP.2.3: Table Observation Checklist  Observe students’ responses to determine whether they can the following: r Successfully use a table to determine probabilities Statistics and Probability 17 Suggestions for Instruction QQ QQ Generalize and apply a rule for determining the probability of independent events Solve a problem that involves determining the probability of independent events Materials: BLM 8.SP.2.4: Probability Problems, BLM 8.SP.2.5: Probability Problem Practice, chart paper Organization: Small group/whole class Procedure: Divide students into small groups, and provide each group with one problem from BLM 8.SP.2.4: Probability Problems, as well as chart paper Ask groups to solve their respective problems and be prepared to present their solutions As each group presents its problem and solution strategy to the class, allow other groups to ask questions and add to the solution Have students identify the various strategies that the groups used to solve the problems BLM 8.SP.2.5: Probability Problem Practice provides additional problems for practice Discuss rules for finding the probability of independent events (Students should generalize that multiplication can be used to find solutions to the probability questions.) Observation Checklist  Observe students’ responses to determine whether they can the following: r Generalize a rule to determine the number of outcomes for two independent events r Apply a generalized rule and knowledge about probability in order to reason mathematically r Determine the possible outcomes of a probability experiment involving two independent events r Determine the probabilities of favourable outcomes 18 G r a d e M a t h e m a t i c s: S u p p o r t D o c u m e n t f o r Te a c h e r s Gr ade Mathemat ics Bibliography Bibliography Banks, James A., and Cherry Banks Multicultural Education: Issues and Perspectives 2nd ed Boston, MA: Allyn and Bacon, 1993 Print Kilpatrick, J., J Swafford, and B Findell, eds Adding it Up: Helping Children Learn Mathematics Washington, DC: National Academy Press, 2001 Kroll, Virginia Equal Shmequal: A Math Adventure Illus Philomena O’Neill Watertown, MA: Charlesbridge Publishing, Inc., 2005 Print Manitoba Education Grade Mathematics Support Document for Teachers Winnipeg, MB: Manitoba Education, 2009 Available online at ——— Kindergarten to Grade Mathematics: Manitoba Curriculum Framework of Outcomes (2013) Winnipeg, MB: Manitoba Education, 2013 Available online at Manitoba Education and Training Success for All Learners: A Handbook on Differentiating Instruction: A Resource for Kindergarten to Senior Winnipeg, MB: Manitoba Education and Training, 1996 Manitoba Education, Citizenship and Youth Kindergarten to Grade Mathematics Glossary: Support Document for Teachers Winnipeg, MB: Manitoba Education, Citizenship and Youth, 2009 Available online at ——— Rethinking Classroom Assessment with Purpose in Mind: Assessment for Learning, Assessment as Learning, Assessment of Learning Winnipeg, MB: Manitoba Education, Citizenship and Youth, 2006 Available online at OECD (Organisation for Economic Co-operation and Development) PISA 2012 Field Trial Problem Solving Framework September 30, 2012 Ontario Ministry of Education The Ontario Curriculum Grades 1–8: Mathematics Ontario: Ministry of Education, 2005 Available online at The Official M C Escher Website Home Page (7 Oct 2010) Reys, Robert E Helping Children Learn Mathematics Mississauga, ON: J Wiley & Sons Canada, 2009 Print Small, Marian Making Math Meaningful to Canadian Students, K–8 Toronto, ON: Nelson Education, 2008 Print Tessellations.org “Do-It-Yourself.” (7 Oct 2010) Bibliography Van de Walle, J., and Louann H Lovin Teaching Student-Centered Mathematics, Grades 5–8 Boston, MA: Pearson Education, 2006 Print Western and Northern Canadian Protocol (WNCP) The Common Curriculum Framework for K–9 Mathematics Edmonton, AB: Governments of Alberta, British Columbia, Manitoba, Northwest Territories, Nunavut Territory, Saskatchewan, and Yukon Territory, May 2006 Available online at Western Australian Minister of Education and Training First Steps in Mathematics: Number Sense: Whole and Decimal Numbers, and Fractions: Improving the Mathematics Outcomes of Students Don Mills, ON: Pearson Professional Learning, 2006 Print Printed in Canada Imprimé au Canada ... that follow A code is used to identify each SLO by grade and strand, as shown in the following example: 8.N.1 The first number refers to the grade (Grade 8) The letter(s) refer to the strand (Number)... learners in Grade The document is intended to be used by teachers as they work with students in achieving the learning outcomes and achievement indicators identified in Kindergarten to Grade Mathematics:... strands across Kindergarten to Grade Some strands are further subdivided into substrands There is one general learning outcome per substrand across Kindergarten to Grade 9 The strands and substrands,