Invitations to Mathematics Investigations in Number Sense and Estimation “All ’Round Numbers” at the SuggestedGrade for students 6 level x 4 Multipl ying a 5 by less 24 number Start 4 x 3 than 1 decreases End the va lue of the x 6 nu mber 3rd Edition An activity of The CENTRE for EDUCATION in MATHEMATICS and COMPUTING Faculty of Mathematics, University of Waterloo Waterloo, Ontario, Canada N2L 3G1 © 2010 The Centre for Education in Mathematics and Computing Copyright © 2008, 2010 The Centre for Education in Mathematics and Computing Faculty of Mathematics University of Waterloo Waterloo, Ontario Canada N2L 3G1 Limited reproduction permission: 1. The Centre for Education in Mathematics and Computing grants permission to indi- vidual teachers to reproduce the Black Line Masters as needed for use with their own students. 2. The Centre for Education in Mathematics and Computing grants permission to an educator providing a professional development workshop to make up to 35 copies of the Black Line Masters for any individual activity for use once with one group. Reproduction of text pages for an entire school or school district or for commercial use is prohibited. Investigations in Number Sense and Estimation Grade 6: All ’Round Numbers Preface The Centre for Education in Mathematics and Computing at the University of Waterloo is dedicated to the development of materials and workshops that promote effective learning and teaching of mathematics. This unit is part of a project designed to assist teachers of Grades 4, 5, and 6 in stimulating interest, competence, and pleasure in mathematics among their students. While the activities are appropriate for either individual or group work, the latter is a particular focus of this effort. Students will be engaged in collaborative activities which will allow them to construct their own meanings and understanding. This emphasis, plus the “Extensions” and related activities included with individual activities/projects, provide ample scope for all students’ interests and ability levels. Related “Family Activities” can be used to involve the students’ parents/care givers. Each unit consists of a sequence of activities intended to occupy about one week of daily classes; however, teachers may choose to take extra time to explore the activities and extensions in more depth. The units have been designed for specific grades, but need not be so restricted. Activities are related to the Ontario Curriculum but are easily adaptable to other locales. “Investigations in Number Sense and Estimation” is comprised of activities which explore the properties, estimation, and uses of whole numbers and fractions in mathematical and everyday settings. A reasonably level of numeracy is essential to navigating the complexities of the highly technical world in which we live. The activities in this unit develop many facets of number sense and apply them to a wide variety of practical situations. Preface i Grade 6: All ’Round Numbers Investigations in Number Sense and Estimation Acknowledgements Contributing Teachers Nancy Dykstra (Waterloo County Board of Education) Jo-Anne Judge (Waterloo County Board of Education) Ron Sauer (Waterloo County Board of Education - retired) Authors/Co-editors Bev Marshman (University of Waterloo) Lorna Morrow (Mathematics Consultant) We wish to acknowledge the support of the Centre for Education in Mathematics and Computing, and in particular, of Peter Crippin and Ian VanderBurgh. A special thank you goes to Linda Schmidt for prompt, accurate type-setting and creative diagrams, and for her patience in converting our efforts, both past and present, to new software. ii Acknowledgements Investigations in Number Sense and Estimation Grade 6: All ’Round Numbers Table of Contents Preface ..........................................................................................................................................................................i Acknowledgements .....................................................................................................................................................ii Table of Contents ....................................................................................................................................................iii Overview ....................................................................................................................................................................... 1 COMMON BELIEFS......................................................................................................................................................... 1 ESSENTIAL CONTENT ................................................................................................................................................... 1 CURRICULUM EXPECTATIONS .........................................................................................................................................2 PREREQUISITES ............................................................................................................................................................3 LOGOS ..........................................................................................................................................................................3 MARGINAL PROBLEMS DESCRIPTION .............................................................................................................................3 MATERIALS ..................................................................................................................................................................4 LETTER TO PARENTS .....................................................................................................................................................5 Activity 1: Using Numbers .......................................................................................................................................7 Activity 2: Comparing & Ordering .........................................................................................................................10 Activity 3: Number Properties ..............................................................................................................................18 Activity 4: Fractions ............................................................................................................................................... 23 Activity 5: Estimation ............................................................................................................................................. 28 BLM 1: Placing Numbers ......................................................................................................................................... 33 BLM 2: Numbers in Place ........................................................................................................................................ 34 BLM 3: Being Reasonable ........................................................................................................................................ 35 BLM 4: Scaling Large Objects .............................................................................................................................. 36 BLM 5: Scaling Small Objects .............................................................................................................................. 37 BLM 6: Errors, Errors, Errors! ............................................................................................................................. 38 BLM 7: The Range Game ......................................................................................................................................... 39 BLM 8: Range Points ................................................................................................................................................ 40 BLM 9: Fraction Tiles ...............................................................................................................................................41 BLM 10: Truths and Half–truths ......................................................................................................................... 42 BLM 11: Relative Sizes ........................................................................................................................................... 43 BLM 12: Greater or Less? ..................................................................................................................................... 44 BLM 13: Name That Number ................................................................................................................................ 45 BLM 14: Should Remainders Remain? ................................................................................................................. 46 BLM 15: ‘Four’ Patterns ......................................................................................................................................... 47 BLM 16: Division Patterns ..................................................................................................................................... 48 BLM 17: Find the Path ............................................................................................................................................ 49 BLM 18: Reach A Number ...................................................................................................................................... 50 BLM 19: The Problem with Fractions ...................................................................................................................51