Ministry of Education. The Ontario Curriculum Exemplars Grades 1 8. Mathematics. Samples of Student Work: A Resource for Teachers

Transcription

1 Ministry of Education The Ontario Curriculum Exemplars Grades 1 8 Mathematics Samples of Student Work: A Resource for Teachers Interim Edition 2001

2 Contents Introduction Purpose Features Mathematics Tasks Mathematical Reasoning How the Rubrics Were Developed and Applied Other Methods of Assessment in Mathematics How the Samples Were Assessed and Selected Using the Mathematics Samples Teachers and Administrators Parents Students Student Samples Grade 1: Geometry and Spatial Sense Grade 2: Patterning and Algebra Grade 3: Geometry and Spatial Sense Grade 4: Patterning and Algebra Grade 5: Measurement Grade 6: Measurement Grade 7: Number Sense and Numeration Grade 8: Data Management and Probability Appendix Glossary Trois publications sont disponibles en français sous les titres suivants : Le curriculum de l Ontario : copies types de la 1 re à la 3 e année Mathématiques, 2001; Le curriculum de l Ontario : copies types de la 4 e à la 6 e année Mathématiques, 2001; et Le curriculum de l Ontario copies types de la 7 e et 8 e année Mathématiques, This publication is available on the Ministry of Education s website at The ministry grants permission to reproduce material in this publication for non-commercial purposes.

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4 Introduction In 1997, the Ministry of Education and Training published a new curriculum for Ontario elementary students. The new curriculum is more specific than the previous curriculum regarding both the knowledge and the skills that students are expected to acquire in each grade. In the mathematics document, The Ontario Curriculum, Grades 1 8: Mathematics, 1997, teachers are provided with the curriculum expectations for mathematics in each of five strands number sense and numeration, measurement, geometry and spatial sense, patterning and algebra, and data management and probability and with brief descriptions of four levels of student achievement on which to base their assessments of students work (see page 9). This resource booklet of examplars is an interim document. It focuses on the curriculum expectations for mathematics and provides teachers, parents, and students with a selection of representative end-of-year samples of students responses to assessment tasks in each of the five strands. Other booklets of exemplars, similar to this one, are or will be available for reading, for writing, for science and technology, and for social studies and history and geography. A comprehensive document to represent the four levels of mathematics achievement in each of the five strands for Grades 1 8 will also be developed. The samples are not intended to be used as standards for the province. For the mathematics exemplars project, Ontario school boards were invited to provide samples of student responses in mathematics. Teachers and administrators from all regions were invaluable in developing the material for this booklet. They designed the tasks and the rubrics (scoring scales) related to each of the tasks, field-tested them in classrooms, suggested changes, administered the final tasks, and marked the student work holistically. Specific samples of those provided by the school boards were chosen for use in this resource booklet to represent the four levels of mathematics achievement in a selected number of strands for Grades 1 8. The choice of the samples used reflects the professional judgement of teachers participating in the project. Those included have been reproduced as is, with no editing of the students work. No students, teachers, or schools have been identified. Purpose This booklet has been developed to: show the characteristics of student work at each level of achievement in one strand for each grade; promote greater consistency in the assessment of student work from grade to grade and across the province; 3

5 provide an approach to improving student learning by demonstrating the use of clear criteria applied to student work written in response to clearly defined tasks in mathematics and by including examples of possible feedback to students in the form of teachers notes; show the connections between what students are expected to learn (the expectations) and how their work can be assessed on the basis of levels of achievement. Teachers, parents, and students are encouraged to examine these student samples, to think about the characteristics and descriptions of work at each level of achievement in Grades 1 8, and to develop an understanding of how one level of achievement differs from another. Teachers might also wish to discuss the strategies they could use to enhance student learning and to promote student achievement in mathematics. Features This booklet contains: the mathematics tasks for a selected number of strands; for each of the selected strands, a task-specific assessment rubric based on the categories and descriptions from the achievement-levels chart; samples of student work in mathematics that reflect the four levels of achievement in the strand; teachers notes and comments that explain why a particular achievement level was assigned to each piece of student work; an appendix, Assessment in Mathematics ; a glossary of assessment terms. Mathematics Tasks Mathematical Reasoning Each task begins with a full-class component in which the teacher introduces the scenario for the task that follows. The teacher often reviews the background knowledge required and poses a question that students discuss as a large group. Next, the class breaks up into small groups that explore an extension to the original problem, using concrete materials, active learning, and group interaction. Having gone through an extensive investigation of the mathematics surrounding the problem, students are prepared to begin the On My Own section, in which they are presented with a problem similar to the one in the previous investigation, and within a specific context. Each student must independently internalize the problem and choose a strategy to address it. During the tasks, students are often asked to justify their choices of strategies and the relevance of their solutions, given the context. Often, an additional piece of information is introduced, one that may require rethinking on the part of students. Once again, students are required to explain their thinking and to justify their strategies. 4 The Ontario Curriculum Exemplars, Grades 1 8: Mathematics

6 How the Rubrics Were Developed and Applied In this booklet, the term rubric means a scale. In this case, the scale describes levels of achievement for a particular task and then guides the scoring of the task according to relevant criteria. To assess student achievement, the teacher chooses from different descriptions of work that are specific to each level of achievement. In this project, a rubric was used for each mathematics task to provide an effective means of assessing the particular level of student performance, to allow for consistent scoring of student performance, and to provide information to students on how to improve their work. A rubric was developed for the tasks in each strand on the basis of the achievementlevels chart on page 9 of The Ontario Curriculum, Grades 1 8: Mathematics, The achievement levels for mathematics focus on four categories of knowledge and skills: problem solving, understanding of concepts, application of mathematical procedures, and communication. The brief descriptions in the achievement-levels chart apply in a general way to all mathematics assessment. Each rubric contains the following components: the categories and the achievement levels (i.e., the framework) from page 9 the relevant criteria (descriptions of student learning) from page 9 the expectations for the grade level (level 3 on the achievement-levels chart is the provincial standard) the required components of each mathematics task (e.g., describing and naming two-dimensional shapes) The rubrics for the mathematics tasks in the exemplars project are similar to the mathematics scales used by the Education Quality and Accountability Office (EQAO) for the Grade 3 and Grade 6 provincial assessments in that both the rubrics and the EQAO scales are based on the curriculum expectations and the achievement levels for mathematics in Ontario. The rubrics differ from the EQAO scales in that they were developed to be used in the context of classroom instruction and assessment for each strand. Each student participating in the mathematics project prepared responses to one assessment task. Other Methods of Assessment in Mathematics Although rubrics were used effectively in this project to assess written responses in mathematics, they are only one way to assess student achievement in mathematics. Other forms of classroom assessment include anecdotal records, checklists, tests, quizzes, teacher observation, journals, and teacher-student conferencing. Teachers select and use many assessment tools to assess and evaluate student achievement. It is important to remember that some students who are competent in mathematics have difficulty expressing themselves in writing. A variety of assessment forms should be used to accommodate such learners. Teachers should make informed decisions on the most appropriate strategies for their students. Introduction 5

7 How the Samples Were Assessed and Selected After the elaborated descriptions in the rubrics had been reviewed and revised by all the teachers participating in the mathematics exemplars project, they were used to assess student work in mathematics at both the district school board level and the provincial level. The teachers used a process that is sometimes called consensus marking or teacher moderation : The teachers first reviewed all their students work samples and assigned a holistic score (from level 1 to level 4) to each sample on the basis of the task-specific rubric. The teachers then reviewed the samples a second time, looking at all four categories in the rubric to provide an analytic score for each category (e.g., level 1 in problem solving, level 2 in communication). Following these steps, the teachers used the rubric to assign each sample an overall level based on both the holistic score and the analytic score, with reference to specific criteria requirements that were to be met in the student sample. At the district school board level, groups of three or four other teachers for each grade level then reviewed the student work until they reached consensus on the assigned level. This was done to ensure that the work being selected clearly illustrated that level of performance. All the samples were then submitted to a provincial selection team of teachers, who chose the samples for each level in the selected strand. The comments of the selection team are included in this booklet so that teachers, parents, and students will be able to see how a rubric for a particular mathematics task has been applied to the samples of student work in mathematics. However, these comments are presented as if they were written by a single teacher, to provide an example of how a teacher might summarize a student s work. The following should be noted: One sample of student work in mathematics is included for each of the four achievement levels in each of the selected strands. The tasks in this project assess students ability to solve problems, to demonstrate their understanding of concepts and the application of procedures, and to communicate mathematical ideas. Although the samples of student work in this booklet were selected to show a level of achievement that was largely consistent across the four categories, individual student achievement will frequently vary across the categories. Although the examples in this booklet show responses to most questions, students working at level 1 and level 2 will often omit answers or will provide incomplete responses to some questions. Students effort was not assessed, since this is evaluated separately by teachers as part of the learning skills component in the Provincial Report Card, Grades 1 8. This booklet does not include any student samples that were assessed by means of the rubrics and judged to be below grade level. Teachers are expected to assist students who score below level 1, and to work with their parents, so that the students can improve their performance. 6 The Ontario Curriculum Exemplars, Grades 1 8: Mathematics

8 Using the Mathematics Samples Teachers and Administrators The samples of student work included in this booklet will help teachers and administrators by: providing student samples and criteria for assessment that will enable them to help students improve their achievement in mathematics; providing a basis for conversations among colleagues, parents, and students about the assessment and evaluation of student achievement in mathematics; facilitating communication with parents regarding the learning expectations and levels of achievement in the selected strands; promoting fair, consistent, and objective assessment of mathematics within and across grade levels. Teachers may choose to: use the teaching/learning activities outlined in the mathematics tasks; adapt the mathematics tasks and rubrics to design comparable mathematics tasks and rubrics, using the questions in this document as models for performance-based assessment; use the examples of student work for each level of achievement as reference points when assessing student work; develop other assessment rubrics with colleagues and students; share student work with colleagues for consensus marking; partner with other schools to design tasks and rubrics, and to select samples for other mathematics tasks and other subject areas; have conversations with students about their work to help clarify the students performance and the assignment of a level of achievement; have students demonstrate their learning in ways other than writing (e.g., making oral presentations). Administrators may choose to: encourage and facilitate teacher collaboration regarding assessment; facilitate sessions for parents and school councils on this booklet; participate in future exemplars projects within their district school boards or on behalf of the Ministry of Education. Parents The mathematics tasks in this booklet are samples of learning activities that represent tasks related to the mathematics curriculum for Grades 1 8. In addition, this booklet invites the involvement and support of parents as they work with their children to improve their achievement in mathematics. Parents may use the samples of student work in mathematics and the rubrics as: models to help monitor their children s progress from level to level and from grade to grade; Introduction 7

9 a basis for communication with teachers about their children s achievement; a source of information to help their children improve their achievement; models to illustrate the application of the levels of achievement; a resource to help them understand their children s report cards. Students Students use mathematics every day, and mathematics is essential to learning in all curriculum areas. When students are in a mathematics program where they are given clear expectations for learning, clear criteria for assessment, and immediate and helpful feedback, their performance improves. Their performance also improves as they are encouraged to take responsibility for their own achievement and to reflect on their own progress and next steps. It is anticipated that the contents of this booklet whether presented to students by their teachers or used directly by them will help them in all or some of the following ways: Students will be introduced to a type of task that will be used to assess their learning. Students will better understand how rubrics can be used as an assessment tool. The mathematics tasks and the samples will help clarify the curriculum expectations for learning. The rubrics and teacher comments will help clarify the assessment criteria. With an increased awareness of the tasks and rubrics, students will be more likely to communicate effectively about their achievement with their teachers and parents, and to ask relevant questions about their own progress. Students can use the criteria and the range of student samples to help them assess their own written responses to mathematics tasks. 8 The Ontario Curriculum Exemplars, Grades 1 8: Mathematics

10 Grade 1 Geometry and Spatial Sense

11 10 The Ontario Curriculum Exemplars, Grades 1 8: Mathematics Geometry and Spatial Sense The Tasks Students made a picture of a structure by pasting at least four different shapes, chosen from shapes provided by their teachers, within a rectangular frame. They labelled each shape, sorted the shapes into two groups, and then explained how they sorted them, using words, pictures, and numbers. The following are the curriculum s overall expectations that relate to these tasks: By the end of Grade 1, students will: describe and classify two-dimensional shapes using concrete materials and drawings. During these tasks, students worked on the following selected expectations in specific areas from the Grade 1 curriculum: Students will: explore and identify two-dimensional shapes using concrete materials and drawings (e.g., circle, rectangle, triangle); identify attributes of two-dimensional shapes; use two-dimensional shapes to construct a picture of objects in the environment (e.g., stickers, stamps); compare and sort two-dimensional shapes according to attributes they choose; describe and name two-dimensional shapes (e.g., circle, square, rectangle, triangle). Previous Learning Experiences It was suggested that before attempting these tasks, students should have had experience with the following: identifying, sorting, classifying, tracing, and drawing twodimensional shapes, using mathematics manipulatives and a variety of objects found in the classroom environment communicating, sorting, and classifying processes orally and in writing using the term structure (e.g., building, bridge) The Process Used Introductory activities. Teachers reviewed two-dimensional shapes, using concrete materials (e.g., pattern blocks, stamps, stickers). Students then brainstormed a list of different types of structures from their environment that included these shapes (e.g., buildings, furniture). Following this, students went outside to identify and discuss two-dimensional shapes that they found in the exterior of the school and the playground. Teachers divided their classes into small groups and gave each group a variety of circles, squares, rectangles, and triangles in different sizes and colours (e.g., pattern blocks, attribute blocks, stamps, stickers). Each group then used all four two-dimensional shapes to make a picture of a school.

12 Students transferred their designs to paper by tracing the shapes or by using stickers. They then labelled the shapes used in their designs. For this task, the pattern blocks and paper shapes were viewed as two-dimensional. On my own. Teachers briefly reviewed the introductory activities and then explained the tasks outlined on the previous page under The Tasks. Students were expected to complete the tasks independently. To help students evaluate their work before submitting it, teachers explained the rubric reproduced on the next page, rephrasing the information so that students could understand it. Students were also provided with a checklist of assessment criteria on the On My Own page. Students were encouraged to do their best work. Finally, teachers handed out the student work sheets and the concrete materials required for the first activity. Evaluation. After students completed their response sheets independently, teachers evaluated the work holistically, using the following rubric and noting strengths and weaknesses in any of the four individual categories (problem solving, understanding of concepts, application of mathematical procedures, communication). Answers: For these activities, depending on the types of shapes that students are given, there will be a variety of solutions. Students will sort their shapes, using such criteria as edges and vertices. 11 Grade 1 Geometry and Spatial Sense

13 12 The Ontario Curriculum Exemplars, Grades 1 8: Mathematics Rubric for Geometry and Spatial Sense Categories Level 1 Level 2 Level 3 Level 4 Problem solving The student: uses a limited range of problem-solving strategies and solves a few simple problems uses appropriate problemsolving strategies and solves some problems selects and uses appropriate problemsolving strategies to solve problems selects and uses appropriate problemsolving strategies, modifies strategies, or creates new strategies to solve problems Understanding of concepts The student: describes and names a limited number of twodimensional shapes describes and names some two-dimensional shapes describes and names a variety of two-dimensional shapes describes, names, and compares a variety of twodimensional shapes Application of mathematical procedures The student: makes some major errors when sorting shapes according to one chosen attribute makes several minor errors when sorting shapes according to one chosen attribute accurately sorts shapes according to one chosen attribute accurately sorts shapes according to more than one chosen attribute Communication of required knowledge related to concepts, procedures, and problem solving The student: unclearly and imprecisely describes a simple sorting process, using some simple mathematical terminology describes the sorting process with some clarity and some precision, sometimes using appropriate mathematical terminology describes the sorting process clearly and precisely, using appropriate mathematical terminology describes the sorting process clearly, precisely, and sequentially, using appropriate mathematical terminology Note: This rubric does not include criteria for assessing student performance that falls below level 1.

14 Grade 1 Geometry and Spatial Sense: Level 1 A B Geometry and Spatial Sense Student Work Sheets - I 2 Geometry and Spatial Sense Student Work Sheets - I 3 13 Grade 1 Geometry and Spatial Sense

15 14 The Ontario Curriculum Exemplars, Grades 1 8: Mathematics C Teacher s Notes Problem solving The student solves a few simple problems, using a limited range of strategies. Understanding of concepts The student describes and names a few two-dimensional shapes. The student may confuse some shapes (e.g., a square with a circle, a square with a rectangle). Application of mathematical procedures The student makes some major errors in sorting. The student may include some of the same shapes in both groups. Communication The student describes the sorting process unclearly and imprecisely. The student uses some simple mathematical terminology. Comments The student shows little understanding of the concepts, makes some major errors in applying procedures, and uses some simple mathematical terminology. Geometry and Spatial Sense Student Work Sheets - I 4

16 Grade 1 Geometry and Spatial Sense: Level 2 A B Geometry and Spatial Sense Student Work Sheets - I 2 Geometry and Spatial Sense Student Work Sheets - I 3 15 Grade 1 Geometry and Spatial Sense

17 16 The Ontario Curriculum Exemplars, Grades 1 8: Mathematics C Teacher s Notes Problem solving The student uses appropriate strategies to solve some problems. The student does not use all the shapes in sorting. Understanding of concepts The student describes and names some two-dimensional shapes. Application of mathematical procedures The student makes some minor errors when sorting on the basis of one chosen attribute. The student may use an inappropriate attribute. Communication The student describes the sorting process with some clarity and some precision, sometimes using appropriate mathematical terminology. Comments The student s work shows some understanding of the concepts, and the student uses some of the concepts appropriately, making some minor errors. Some of the tasks are unfinished. The student s use of mathematical terminology is not always appropriate. Geometry and Spatial Sense Student Work Sheets - I 4

18 Grade 1 Geometry and Spatial Sense: Level 3 A B Geometry and Spatial Sense Student Work Sheets - I 2 Geometry and Spatial Sense Student Work Sheets - I 3 17 Grade 1 Geometry and Spatial Sense

19 18 The Ontario Curriculum Exemplars, Grades 1 8: Mathematics C Teacher s Notes Problem solving The student selects and uses an appropriate strategy to solve problems. Understanding of concepts The student accurately describes and names four different two-dimensional shapes. The student demonstrates a clear understanding of the term structure. Application of mathematical procedures The student accurately sorts shapes on the basis of one chosen attribute (e.g., with or without four corners). Communication The student clearly and precisely describes the sorting process. The student uses appropriate mathematical terminology (e.g., corners/no corners, square/circle/rectangle). Comments The student demonstrates the ability to sort shapes according to one attribute. The student communicates clearly and precisely and uses correct mathematical terminology. Geometry and Spatial Sense Student Work Sheets - I 4

20 Grade 1 Geometry and Spatial Sense: Level 4 A B Geometry and Spatial Sense Student Work Sheets - I 2 Geometry and Spatial Sense Student Work Sheets - I 3 19 Grade 1 Geometry and Spatial Sense

21 20 The Ontario Curriculum Exemplars, Grades 1 8: Mathematics C Teacher s Notes Problem solving The student selects and uses appropriate strategies and modifies or creates new strategies as necessary to solve problems. Understanding of concepts The student describes, names, and compares two-dimensional shapes. Application of mathematical procedures The student accurately sorts on the basis of more than one chosen attribute (e.g., shape, corners, or size). Communication The student clearly, precisely, and logically describes the sorting process. The student uses appropriate mathematical terminology. Comments The student s problem-solving techniques are logical and well-organized and the concepts are explained clearly and precisely. The student uses language effectively to communicate, explain, and justify. Geometry and Spatial Sense Student Work Sheets - I 4

22 Student Learning Tasks The work sheets completed by students the learning tasks for each strand are reproduced on the following pages. These are only examples of many imaginative learning tasks that can be used. Teachers are encouraged to use these as a resource to help them select other tasks, prepare appropriate rubrics, and assess some or all of the students in their classes. Teachers can also use the Appendix, Assessment in Mathematics, to develop and extend their own assessment plans. Minor revisions have been made to the original work sheets to reflect suggestions made by teachers during the field test. These revisions include improved wording and page layout or additional directions. Some selected answers for the activities for each of the five mathematics strands described in this booklet are provided in the Evaluation subsection (under Answers ) under the heading The Process Used. 21 Grade 1 Student Learning Tasks

23 22 The Ontario Curriculum Exemplars, Grades 1 8: Mathematics Geometry and Spatial Sense Student Work Sheets - I 2

24 Geometry and Spatial Sense Student Work Sheets - I 3 Geometry and Spatial Sense Student Work Sheets - I 4 23 Grade 1 Student Learning Tasks

25 24 The Ontario Curriculum Exemplars, Grades 1 8: Mathematics Geometry and Spatial Sense Student Work Sheets - I 5

26 Grade 2 Patterning and Algebra

27 26 The Ontario Curriculum Exemplars, Grades 1 8: Mathematics Patterning and Algebra The Tasks 1. Students created a pattern, using at least two different fish, and described their patterns, using pictures, numbers, and words. 2. Students extended a shell pattern and described it, using pictures, numbers, and words. 3. Students identified and extended number patterns and explained them, using pictures, numbers, and words. 4. Students created their own number patterns and described them, using pictures, numbers, and words. The following are the curriculum s overall expectations that relate to these tasks: By the end of Grade 2, students will: identify, extend, and create number, geometric, and measurement patterns, and patterns in their environment; explore patterns and pattern rules; identify relationships between and among patterns. During these tasks, students worked on the following selected expectations in specific areas from the Grade 2 curriculum: Students will: recognize that patterning results from repeating an operation (e.g., addition); describe and make models of patterns encountered in any context (e.g., wallpaper borders, calendars), and read charts that display the patterns; identify patterns (e.g., in shapes, sounds); combine two attributes in creating a pattern (e.g., size and position); identify patterns in addition and subtraction sentences; relate growing and shrinking patterns to addition and subtraction; explain a pattern rule; transfer patterns from one medium to another (e.g., actions, words, symbols, pictures, objects, calculator). Previous Learning Experiences It was suggested that before attempting the tasks, students should have had experience with the following: identifying, explaining, and creating patterns, using concrete objects transferring patterns from one medium to another (e.g., actions, symbols, letters) recognizing growing and shrinking patterns in addition and subtraction

28 The Process Used Introductory activities. Teachers handed out the student work sheets and read the poem together with the class. They wrote the poem on chart paper and illustrated the pattern by drawing coloured fish on the chalkboard or chart paper. Students then coloured in the first two lines of boxes on page 1 of their work sheets. As a group, they identified the patterns in lines 1 and 2 of the poem. They represented the pattern together, using letters (e.g., AA, BB, AA). Teachers divided the class into groups of two and had each group colour the third line of boxes on the work sheet. When they had finished, the groups presented their work to the whole class. On my own. Teachers reviewed the introductory activities and assigned the On My Own work sheets. Students were expected to complete the tasks independently. To help students evaluate their work before submitting it, teachers explained the rubric reproduced on the next page, rephrasing the information so that students could understand it. Students were also provided with a checklist of assessment criteria on the On My Own page. Students were encouraged to do their best work. Evaluation. After students completed their work sheets independently, teachers evaluated students work holistically, using the following rubric and noting strengths and weaknesses in any of the four individual categories (problem solving, understanding of concepts, application of mathematical procedures, communication). Answers: There are a variety of answers for question 1. In 2a, a possible answer is that there are eight and ten shells respectively. In 3a, the next numbers are 20, 24, 28 and 28, 32, 36, 40. In 3c, the numbers are 30, 27, 24, 21 and 42, 39, 36, 33, 30. In 3e, the numbers are 41, 46, 51, 56, 61, 66, 71. In 3g, the numbers are 69, 66, 63, 60, 57, 54, 51. In question 4, a variety of answers are possible; however, teachers should ensure that students patterns and descriptions are accurate and match each other. 27 Grade 2 Patterning and Algebra

29 28 The Ontario Curriculum Exemplars, Grades 1 8: Mathematics Rubric for Patterning and Algebra Categories Level 1 Level 2 Level 3 Level 4 Problem solving The student: uses a limited range of problem-solving strategies to solve a few simple problems uses appropriate problemsolving strategies to solve some problems selects and uses appropriate problemsolving strategies to solve problems selects and uses appropriate problemsolving strategies, modifies strategies, or creates new strategies to solve problems Understanding of concepts The student: identifies and describes a few simple patterns identifies and describes several simple patterns identifies and describes a variety of patterns identifies and describes a complex pattern Application of mathematical procedures The student: makes some major errors when creating simple patterns that combine two attributes makes some major errors when attempting to follow pattern rules makes several minor errors when creating and extending patterns that combine two attributes makes several minor errors when following pattern rules creates and extends patterns that combine two attributes makes only a few minor errors when following pattern rules creates and extends a complex pattern makes practically no errors when following pattern rules Communication of required knowledge related to concepts, procedures, and problem solving The student: unclearly and imprecisely describes some simple patterns, using limited mathematical terminology describes patterns with some clarity and some precision, sometimes using appropriate mathematical terminology and symbols describes patterns clearly and precisely, using appropriate mathematical terminology and symbols describes a complex pattern clearly and precisely, using appropriate mathematical terminology and symbols Note: This rubric does not include criteria for assessing student performance that falls below level 1.

30 Grade 2 Patterning and Algebra: Level 1 A B Patterning and Algebra Student Work Sheets - II 4 Patterning and Algebra Student Work Sheets - II 6 29 Grade 2 Patterning and Algebra

31 30 The Ontario Curriculum Exemplars, Grades 1 8: Mathematics C D Patterning and Algebra Student Work Sheets - II 7 Patterning and Algebra Student Work Sheets - II 8

32 E F Patterning and Algebra Student Work Sheets - II 9 Patterning and Algebra Student Work Sheets - II Grade 2 Patterning and Algebra

33 32 The Ontario Curriculum Exemplars, Grades 1 8: Mathematics G H 4. Patterning and Algebra Student Work Sheets - II 11 Patterning and Algebra Student Work Sheets - II 12

34 Teacher s Notes Problem solving The student uses limited strategies to solve a few simple problems. Understanding of concepts The student identifies and describes some simple patterns. Application of mathematical procedures The student makes some major errors when creating simple patterns that combine two attributes. The student also makes some major errors when attempting to follow pattern rules. Communication The student describes simple patterns unclearly and imprecisely, using limited mathematical terminology. Comments The student uses limited strategies to solve a few simple patterns, makes some major errors when recognizing and describing some simple patterns, and describes patterns unclearly and imprecisely, using limited mathematical terminology. 33 Grade 2 Patterning and Algebra

35 34 The Ontario Curriculum Exemplars, Grades 1 8: Mathematics Grade 2 Patterning and Algebra: Level 2 A B Patterning and Algebra Student Work Sheets - II 4 Patterning and Algebra Student Work Sheets - II 6

36 C D Patterning and Algebra Student Work Sheets - II 7 Patterning and Algebra Student Work Sheets - II 8 35 Grade 2 Patterning and Algebra

37 36 The Ontario Curriculum Exemplars, Grades 1 8: Mathematics E F Patterning and Algebra Student Work Sheets - II 9 Patterning and Algebra Student Work Sheets - II 10

38 G H 4. Patterning and Algebra Student Work Sheets - II 11 Patterning and Algebra Student Work Sheets - II Grade 2 Patterning and Algebra

39 38 The Ontario Curriculum Exemplars, Grades 1 8: Mathematics Teacher s Notes Problem solving The student uses appropriate strategies to solve some problems. Understanding of concepts The student identifies and describes simple patterns. Application of mathematical procedures The student makes several minor errors when creating and extending patterns that combine two attributes. The student also makes several minor errors when following pattern rules. Communication The student describes patterns with some clarity and precision, sometimes using appropriate mathematical terminology. Comments The student identifies and describes simple patterns, makes several minor errors when following rules to create and extend patterns, and solves some problems and describes the process with some clarity and precision, sometimes using appropriate mathematical terminology.

40 Grade 2 Patterning and Algebra: Level 3 A B Patterning and Algebra Student Work Sheets - II 4 Patterning and Algebra Student Work Sheets - II 6 39 Grade 2 Patterning and Algebra

41 40 The Ontario Curriculum Exemplars, Grades 1 8: Mathematics C D Patterning and Algebra Student Work Sheets - II 7 Patterning and Algebra Student Work Sheets - II 8

42 E F Patterning and Algebra Student Work Sheets - II 9 Patterning and Algebra Student Work Sheets - II Grade 2 Patterning and Algebra

43 42 The Ontario Curriculum Exemplars, Grades 1 8: Mathematics G H 4. Patterning and Algebra Student Work Sheets - II 11 Patterning and Algebra Student Work Sheets - II 12

44 Teacher s Notes Problem solving The student selects and uses appropriate strategies to solve problems. Understanding of concepts The student identifies and describes a variety of patterns. Application of mathematical procedures The student creates and extends patterns that combine two attributes. The student makes few minor errors when following pattern rules. Communication The student clearly and precisely describes patterns, using appropriate mathematical terminology and symbols. Comments The student uses pattern rules to solve problems and to create and extend a variety of patterns based on two attributes. The student clearly and precisely uses appropriate mathematical terminology and symbols to describe the patterns. 43 Grade 2 Patterning and Algebra

45 44 The Ontario Curriculum Exemplars, Grades 1 8: Mathematics Grade 2 Patterning and Algebra: Level 4 A B Patterning and Algebra Student Work Sheets - II 4 Patterning and Algebra Student Work Sheets - II 6

46 C D Patterning and Algebra Student Work Sheets - II 7 Patterning and Algebra Student Work Sheets - II 8 45 Grade 2 Patterning and Algebra

47 46 The Ontario Curriculum Exemplars, Grades 1 8: Mathematics E F Patterning and Algebra Student Work Sheets - II 9 Patterning and Algebra Student Work Sheets - II 10

48 G H 4. Patterning and Algebra Student Work Sheets - II 11 Patterning and Algebra Student Work Sheets - II Grade 2 Patterning and Algebra

49 48 The Ontario Curriculum Exemplars, Grades 1 8: Mathematics Teacher s Notes Problem solving The student selects and uses appropriate problem-solving strategies and modifies strategies or creates new strategies to solve problems. Understanding of concepts The student identifies and describes a complex pattern. Application of mathematical procedures The student creates and extends a complex pattern. The student makes practically no errors when following pattern rules. Communication The student clearly and precisely describes a complex pattern, using appropriate mathematical terminology and symbols. Comments The student uses appropriate strategies to solve problems involving a complex pattern and describes the pattern clearly and precisely, using appropriate mathematical terminology.

50 Student Learning Tasks The work sheets completed by students the learning tasks for each strand are reproduced on the following pages. These are only examples of many imaginative learning tasks that can be used. Teachers are encouraged to use these as a resource to help them select other tasks, prepare appropriate rubrics, and assess some or all of the students in their classes. Teachers can also use the Appendix, Assessment in Mathematics, to develop and extend their own assessment plans. Minor revisions have been made to the original work sheets to reflect suggestions made by teachers during the field test. These revisions include improved wording and page layout or additional directions. Some selected answers for the activities for each of the five mathematics strands described in this booklet are provided in the Evaluation subsection (under Answers ) under the heading The Process Used. 49 Grade 2 Student Learning Tasks

51 50 The Ontario Curriculum Exemplars, Grades 1 8: Mathematics Patterning and Algebra Student Work Sheets - II 2

52 Patterning and Algebra Student Work Sheets - II 3 Patterning and Algebra Student Work Sheets - II 4 51 Grade 2 Student Learning Tasks

53 52 The Ontario Curriculum Exemplars, Grades 1 8: Mathematics Patterning and Algebra Student Work Sheets - II 5 Patterning and Algebra Student Work Sheets - II 6

54 Patterning and Algebra Student Work Sheets - II 7 Patterning and Algebra Student Work Sheets - II 8 53 Grade 2 Student Learning Tasks

55 54 The Ontario Curriculum Exemplars, Grades 1 8: Mathematics Patterning and Algebra Student Work Sheets - II 9 Patterning and Algebra Student Work Sheets - II 10

56 Patterning and Algebra Student Work Sheets - II 11 Patterning and Algebra Student Work Sheets - II Grade 2 Student Learning Tasks

57 56 The Ontario Curriculum Exemplars, Grades 1 8: Mathematics Patterning and Algebra Student Work Sheets - II 13 Patterning and Algebra Student Work Sheets - II 14

58 Grade 3 Geometry and Spatial Sense

59 58 The Ontario Curriculum Exemplars, Grades 1 8: Mathematics Geometry and Spatial Sense The Tasks 1. Students each built a three-dimensional structure, using a number of solids. They then chose four three-dimensional solids used in their structures and completed charts that classified the number of faces, edges, and vertices for each shape. Students sketched pictures of their structures and explained how they made the structures. 2. Students used some pattern blocks to make a picture containing lines of symmetry. They then drew as many lines of symmetry as possible. Finally, they selected one line of symmetry and explained why it was a line of symmetry. The following are the curriculum s overall expectations that relate to these tasks: By the end of Grade 3, students will: investigate the attributes of three-dimensional figures and two-dimensional shapes using concrete materials and drawings; draw and build three-dimensional objects and models. During these tasks, students worked on the following selected expectations in specific areas from the Grade 3 curriculum: Students will: sketch a picture of a structure or model created from three-dimensional figures; compare and sort two-dimensional shapes according to two or more attributes; explain the process followed in making a structure or a picture from three-dimensional figures or two-dimensional shapes; explore the concept of lines of symmetry in two-dimensional shapes. Previous Learning Experiences It was suggested that before attempting the tasks, students should have had experience with the following: solving two-dimensional puzzles identifying two-dimensional shapes and three-dimensional solids in their environment building and sketching two-dimensional and three-dimensional structures writing instructions for others to follow reflecting on the accuracy of their work identifying symmetry in, and drawing lines of symmetry on, two-dimensional shapes

60 The Process Used Introductory activities. Students brought a variety of recyclable materials to class (e.g., boxes, cartons, cylinders) for building structures, although three-dimensional blocks and pattern blocks were used in some classrooms. Teachers reviewed the names and attributes of various three-dimensional solids, using a chart like the following to record the information: Name of Names of Number of Number of Number of Recyclable 3-D Solids Faces Edges Vertices Object Teachers explained to students that they would be building structures, using the materials collected in the classroom (or the three-dimensional blocks and pattern blocks), and that they would be asked to sketch and describe one of their structures. On my own. Teachers briefly reviewed the introductory activities and then distributed the student sheets and the concrete materials (i.e., recyclable materials or three-dimensional blocks and pattern blocks) required for the tasks outlined on the previous page under The Tasks. Students were expected to complete the tasks independently. To help students evaluate their work before submitting it, teachers explained the rubric reproduced on the next page, rephrasing the information so that students could understand it. Students were also provided with a checklist of assessment criteria on the On My Own page. Students were encouraged to do their best work. Evaluation. After students completed their work sheets independently, teachers evaluated students work holistically, using the following rubric and noting strengths and weaknesses in any of the four individual categories (problem solving, understanding of concepts, application of mathematical procedures, communication). Answers: In question 1, students may have some problems completing the chart if they choose non-rectangular shapes such as a sphere, cone, and cylinder. However, students should consider their shapes and make reasonable decisions when completing the chart. The following are examples of questions they will have to answer: Does a cone have one or two faces? Does a sphere have one face? Does a cylinder have three faces? Does a cylinder have two edges? Mathematicians are not in agreement regarding such questions; faces, edges, and vertices are used to describe polyhedra. There will be a variety of answers for the rest of the tasks. 59 Grade 3 Geometry and Spatial Sense

61 60 The Ontario Curriculum Exemplars, Grades 1 8: Mathematics Rubric for Geometry and Spatial Sense Categories Level 1 Level 2 Level 3 Level 4 Problem solving The student: uses a limited range of problem-solving strategies to solve a few simple problems uses appropriate problemsolving strategies to solve some problems selects and uses appropriate problemsolving strategies to solve problems selects and uses appropriate problemsolving strategies, modifies strategies, or creates new strategies to solve problems Understanding of concepts The student: provides some incomplete and some inappropriate descriptions and names of three-dimensional solids in terms of faces, edges, and vertices provides appropriate but sometimes incomplete descriptions and names of three-dimensional solids in terms of faces, edges, and vertices provides appropriate and complete descriptions, names, and comparisons of three-dimensional solids in terms of faces, edges, and vertices provides appropriate and complete descriptions, names, comparisons, and explanations of threedimensional solids in terms of faces, edges, and vertices Application of mathematical procedures The student: makes some major errors or omissions when drawing lines of symmetry makes some major errors or omissions when attempting to sketch a picture of a threedimensional structure makes several minor errors or omissions when drawing lines of symmetry makes several minor errors or omissions when sketching a picture of a threedimensional structure makes only a few minor errors or omissions when drawing lines of symmetry makes only a few minor errors or omissions when sketching a picture of a three-dimensional structure makes practically no errors or omissions when drawing lines of symmetry makes practically no errors or omissions when sketching a picture of a three-dimensional structure Communication of required knowledge related to concepts, procedures, and problem solving The student: describes a process unclearly and imprecisely, using some simple mathematical terminology describes a process with some clarity and some precision, sometimes using appropriate mathematical terminology describes a process clearly and precisely, using appropriate mathematical terminology describes a process clearly and precisely, extensively using appropriate mathematical terminology Note: This rubric does not include criteria for assessing student performance that falls below level 1.

62 Grade 3 Geometry and Spatial Sense: Level 1 A B Geometry and Spatial Sense Student Work Sheets - III 2 Geometry and Spatial Sense Student Work Sheets - III 3 61 Grade 3 Geometry and Spatial Sense

63 62 The Ontario Curriculum Exemplars, Grades 1 8: Mathematics C D Geometry and Spatial Sense Student Work Sheets - III 4 Geometry and Spatial Sense Student Work Sheets - III 5

64 E Teacher s Notes Problem solving The student uses a limited range of strategies to solve a few simple problems. Understanding of concepts The student provides some incomplete and some inappropriate answers when describing and naming three-dimensional solids. Application of mathematical procedures The student makes some major errors when attempting to sketch a threedimensional structure. The student also makes some major errors when drawing lines of symmetry. Communication The student either omits explanations or imprecisely and unclearly describes the tasks, using limited mathematical terminology. Comments The student shows a limited grasp of mathematical concepts involving three-dimensional structures. The student s explanations are sometimes incomplete and inaccurate and contain a limited range of mathematical terminology. Geometry and Spatial Sense Student Work Sheets - III 6 63 Grade 3 Geometry and Spatial Sense

65 64 The Ontario Curriculum Exemplars, Grades 1 8: Mathematics Grade 3 Geometry and Spatial Sense: Level 2 A B Geometry and Spatial Sense Student Work Sheets - III 2 Geometry and Spatial Sense Student Work Sheets - III 3

66 C D Geometry and Spatial Sense Student Work Sheets - III 4 Geometry and Spatial Sense Student Work Sheets - III 5 65 Grade 3 Geometry and Spatial Sense

67 66 The Ontario Curriculum Exemplars, Grades 1 8: Mathematics E Teacher s Notes Problem solving The student uses appropriate strategies to solve some problems. Understanding of concepts The student describes and names some three-dimensional solids, providing appropriate but sometimes incomplete responses. Application of mathematical procedures The student makes several minor errors when sketching a three-dimensional structure. The student also makes several minor errors when drawing lines of symmetry. Communication The student describes the process with some clarity and precision, sometimes using appropriate mathematical terminology. Comments The student shows evidence of understanding the concepts; explains the processes used with some clarity, sometimes using appropriate mathematical terminology; and makes several minor errors when naming three-dimensional solids and sketching three-dimensional structures. Geometry and Spatial Sense Student Work Sheets - III 6

68 Grade 3 Geometry and Spatial Sense: Level 3 A B Geometry and Spatial Sense Student Work Sheets - III 2 Geometry and Spatial Sense Student Work Sheets - III 3 67 Grade 3 Geometry and Spatial Sense

69 68 The Ontario Curriculum Exemplars, Grades 1 8: Mathematics C D Geometry and Spatial Sense Student Work Sheets - III 4 Geometry and Spatial Sense Student Work Sheets - III 5

70 E Teacher s Notes Problem solving The student solves problems, using appropriate strategies. Understanding of concepts The student describes and names three-dimensional solids and provides both complete and appropriate answers. The student compares three-dimensional solids in terms of faces, edges, and vertices. Application of mathematical procedures The student makes only a few minor errors when sketching a three-dimensional structure. The student also makes only a few minor errors when drawing more than one line of symmetry. Communication The student clearly and precisely describes the three-dimensional structures and the locations of lines of symmetry, using appropriate mathematical terminology. Comments The student solves most problems; clearly describes three-dimensional solids in terms of faces, edges, and vertices, making only a few minor errors; uses appropriate mathematical terminology to communicate the processes and results; and makes only a few minor errors when locating lines of symmetry. Geometry and Spatial Sense Student Work Sheets - III 6 69 Grade 3 Geometry and Spatial Sense

71 70 The Ontario Curriculum Exemplars, Grades 1 8: Mathematics Grade 3 Geometry and Spatial Sense: Level 4 A B Geometry and Spatial Sense Student Work Sheets - III 2 Geometry and Spatial Sense Student Work Sheets - III 3

72 C D Geometry and Spatial Sense Student Work Sheets - III 4 Geometry and Spatial Sense Student Work Sheets - III 5 71 Grade 3 Geometry and Spatial Sense

73 72 The Ontario Curriculum Exemplars, Grades 1 8: Mathematics E Teacher s Notes Problem solving The student chooses appropriate strategies and modifies and extends them as necessary to solve problems (i.e., produce a picture). Understanding of concepts The student describes and names three-dimensional solids. The student compares three-dimensional structures in terms of faces, edges, and vertices. Application of mathematical procedures The student accurately locates lines of symmetry in a complex design. The student makes practically no errors when sketching three-dimensional structures. Communication The student clearly and precisely describes the process used, extensively employing appropriate mathematical terminology. Comments The student provides clear, precise descriptions of problem-solving strategies, makes practically no errors or omissions, and describes and names three-dimensional solids and locates lines of symmetry. Geometry and Spatial Sense Student Work Sheets - III 6

74 Student Learning Tasks The work sheets completed by students the learning tasks for each strand are reproduced on the following pages. These are only examples of many imaginative learning tasks that can be used. Teachers are encouraged to use these as a resource to help them select other tasks, prepare appropriate rubrics, and assess some or all of the students in their classes. Teachers can also use the Appendix, Assessment in Mathematics, to develop and extend their own assessment plans. Minor revisions have been made to the original work sheets to reflect suggestions made by teachers during the field test. These revisions include improved wording and page layout or additional directions. Some selected answers for the activities for each of the five mathematics strands described in this booklet are provided in the Evaluation subsection (under Answers ) under the heading The Process Used. 73 Grade 3 Student Learning Tasks

75 74 The Ontario Curriculum Exemplars, Grades 1 8: Mathematics Geometry and Spatial Sense Student Work Sheets - III 2

76 Geometry and Spatial Sense Student Work Sheets - III 3 Geometry and Spatial Sense Student Work Sheets - III 4 75 Grade 3 Student Learning Tasks

77 76 The Ontario Curriculum Exemplars, Grades 1 8: Mathematics Geometry and Spatial Sense Student Work Sheets - III 5 Geometry and Spatial Sense Student Work Sheets - III 6

78 Geometry and Spatial Sense Student Work Sheets - III 7 77 Grade 3 Student Learning Tasks

79

80 Grade 4 Patterning and Algebra

81 80 The Ontario Curriculum Exemplars, Grades 1 8: Mathematics Patterning and Algebra The Tasks 1. Students were provided with descriptions of two alternative sales offers of free fish with the purchase of an aquarium. They then determined the offer that would give them the most fish at the end of a week, using pictures, numbers, and words to explain their choice of offer. 2. Students examined a pattern that gave the amount of water required for different numbers of fish and used the pattern to determine the amount of water required to hold first nine and then eighty fish. They explained their answers, using pictures, numbers, and words. 3. Students extended a given pattern and explained the pattern and its underlying rule, using pictures, numbers, and words. 4. Students created a pattern problem about a pet store and then solved it. The following are the curriculum s overall expectations that relate to these tasks: By the end of Grade 4, students will: demonstrate an understanding of mathematical relationships in patterns using concrete materials, drawings, and symbols; identify, extend, and create linear and non-linear geometric patterns, number and measurement patterns, and patterns in their environment; recognize and discuss patterning rules; apply patterning strategies to problem-solving situations. During these tasks, students worked on the following selected expectations in specific areas from the Grade 4 curriculum: Students will: describe patterns encountered in any context; identify and extend patterns to solve problems in meaningful contexts (e.g., ploughed fields, haystacks, architecture, paintings); pose and solve problems by applying a patterning strategy; analyse number patterns and state the rule for any relationships; given a rule expressed in informal language, extend a pattern. Previous Learning Experiences It was suggested that before attempting the tasks, students should have had experience with the following: analysing number patterns displaying number patterns in a chart

82 The Process Used Introductory activities. Working as a whole class, students built a web around the topic Pets. They then developed a pet parade pattern, using pictures, numbers, or words (e.g., 1 cat, 2 dogs, 3 fish, 4 rats; fur, feather, fur, feather). Students then worked in small groups to develop other patterns related to the subject of pets taken from the web. On my own. Teachers reviewed the introductory activities just completed and handed out the student work sheets. Students were expected to complete the tasks independently. To help students evaluate their work before submitting it, teachers explained the rubric reproduced on the next page, rephrasing the information so that students could understand it. Students were also provided with a checklist of assessment criteria on the On My Own page. Students were encouraged to do their best work. Evaluation. After students completed their work sheets independently, teachers evaluated students work holistically, using the following rubric and noting strengths and weaknesses in any of the four individual categories (problem solving, understanding of concepts, application of mathematical procedures, communication). Answers: In task 1, there are two possible patterns: 1, 2, 4, 8, 16, 32, 64 or 1, 2, 4, 7, 11, 16, 22. However, in both cases offer 2 is better than offer 1, since you would receive either 127 or 63 fish. In task 2, the aquarium would be represented by 18 squares to hold 9 fish. Each fish requires 2 squares; so 40 fish require 80 squares. In task 3, there would be 35 jellyfish, 45 piranhas, and 56 sunfish. In task 4, answers will vary. 81 Grade 4 Patterning and Algebra

83 82 The Ontario Curriculum Exemplars, Grades 1 8: Mathematics Rubric for Patterning and Algebra Categories Level 1 Level 2 Level 3 Level 4 Problem solving The student: uses a limited range of problem-solving strategies to solve a few simple problems uses appropriate problemsolving strategies to solve some problems selects and uses appropriate problemsolving strategies to solve most problems selects and uses appropriate problem-solving strategies, modifies strategies, or creates new strategies to solve almost all problems Understanding of concepts The student: provides some incomplete and some inappropriate explanations when attempting to identify and describe simple patterns provides appropriate but incomplete explanations when identifying and describing simple patterns provides both complete and appropriate explanations when identifying and describing a variety of patterns provides both complete and appropriate explanations when identifying and describing complex patterns, and demonstrates that he or she can apply the concepts in a variety of contexts Application of mathematical procedures The student: makes some major errors or omissions when attempting to create simple patterns makes several minor errors or omissions when creating and extending simple patterns makes only a few minor errors or omissions when creating and extending patterns makes almost no errors when creating and extending complex patterns Communication of required knowledge related to concepts, procedures, and problem solving The student: unclearly and imprecisely describes simple patterns, using limited mathematical terminology describes patterns with some clarity and some precision, sometimes using appropriate mathematical terminology and symbols describes patterns clearly and precisely, using appropriate mathematical terminology and symbols describes patterns clearly and precisely, using appropriate mathematical terminology and symbols extensively Note: This rubric does not include criteria for assessing student performance that falls below level 1.

84 Grade 4 Patterning and Algebra: Level 1 A B Patterning and Algebra Student Work Sheets - IV 2 Patterning and Algebra Student Work Sheets - IV 3 83 Grade 4 Patterning and Algebra

85 84 The Ontario Curriculum Exemplars, Grades 1 8: Mathematics C D Patterning and Algebra Student Work Sheets - IV 4 Patterning and Algebra Student Work Sheets - IV 5

86 E F Patterning and Algebra Student Work Sheets - IV 6 Patterning and Algebra Student Work Sheets - IV 7 85 Grade 4 Patterning and Algebra

87 86 The Ontario Curriculum Exemplars, Grades 1 8: Mathematics G Teacher s Notes Problem solving The student uses a limited range of strategies to solve a few simple problems. Understanding of concepts The student provides some incomplete and inappropriate explanations when identifying and describing simple patterns. Application of mathematical procedures The student makes some major errors or omissions when creating and extending simple patterns. Communication The student describes simple patterns, using limited mathematical terminology. Comments The student uses a limited range of strategies to solve some simple problems; makes some major errors when identifying and describing simple patterns and when creating and extending simple patterns; and describes simple patterns, using limited mathematical terminology. Patterning and Algebra Student Work Sheets - IV 8

88 Grade 4 Patterning and Algebra: Level 2 A B Patterning and Algebra Student Work Sheets - IV 2 Patterning and Algebra Student Work Sheets - IV 3 87 Grade 4 Patterning and Algebra

89 88 The Ontario Curriculum Exemplars, Grades 1 8: Mathematics C D Patterning and Algebra Student Work Sheets - IV 4 Patterning and Algebra Student Work Sheets - IV 5

90 E F Patterning and Algebra Student Work Sheets - IV 6 Patterning and Algebra Student Work Sheets - IV 7 89 Grade 4 Patterning and Algebra

91 90 The Ontario Curriculum Exemplars, Grades 1 8: Mathematics G Teacher s Notes Problem solving The student uses some appropriate strategies to solve some problems. Understanding of concepts The student identifies and describes simple patterns, and provides appropriate but incomplete solutions. Application of mathematical procedures The student makes several minor errors or omissions when creating and extending patterns. Communication The student describes patterns with some clarity and some precision, sometimes using appropriate mathematical terminology and symbols. Comments The student uses a number of appropriate problem-solving strategies to solve some problems; makes several minor errors and omissions when identifying, describing, and extending some simple patterns; and communicates with some clarity and some precision, sometimes using appropriate mathematical terminology and symbols. Patterning and Algebra Student Work Sheets - IV 8

92 Grade 4 Patterning and Algebra: Level 3 A B Patterning and Algebra Student Work Sheets - IV 2 Patterning and Algebra Student Work Sheets - IV 3 91 Grade 4 Patterning and Algebra

93 92 The Ontario Curriculum Exemplars, Grades 1 8: Mathematics C D Patterning and Algebra Student Work Sheets - IV 4 Patterning and Algebra Student Work Sheets - IV 5

94 E F Patterning and Algebra Student Work Sheets - IV 6 Patterning and Algebra Student Work Sheets - IV 7 93 Grade 4 Patterning and Algebra

95 94 The Ontario Curriculum Exemplars, Grades 1 8: Mathematics G Teacher s Notes Problem solving The student selects and uses appropriate strategies to solve problems. Understanding of concepts The student identifies and describes a variety of patterns and provides both appropriate and complete explanations. Application of mathematical procedures The student makes only a few minor errors when creating and extending patterns. Communication The student clearly and precisely describes patterns, using appropriate mathematical terminology and symbols. Comments The student identifies, describes, creates, and extends patterns in a variety of situations with a few minor errors, and is able to communicate clearly and precisely, using appropriate mathematical terminology and symbols. Patterning and Algebra Student Work Sheets - IV 8

96 Grade 4 Patterning and Algebra: Level 4 A B Patterning and Algebra Student Work Sheets - IV 2 Patterning and Algebra Student Work Sheets - IV 3 95 Grade 4 Patterning and Algebra

97 96 The Ontario Curriculum Exemplars, Grades 1 8: Mathematics C D Patterning and Algebra Student Work Sheets - IV 4

98 E F Patterning and Algebra Student Work Sheets - IV 6 Patterning and Algebra Student Work Sheets - IV 7 97 Grade 4 Patterning and Algebra

99 98 The Ontario Curriculum Exemplars, Grades 1 8: Mathematics G Teacher s Notes Problem solving The student selects and uses appropriate strategies or creates new ones to solve problems. Understanding of concepts The student identifies and describes complex patterns in a variety of contexts, and provides complete explanations. Application of mathematical procedures The student makes practically no errors when creating and extending complex patterns. Communication The student clearly and precisely describes complex patterns and pattern rules, using appropriate mathematical terminology and symbols extensively. Comments The student identifies, describes, creates, and extends complex patterns in a variety of problem-solving situations; makes practically no errors when making calculations; and communicates clearly and precisely, using appropriate mathematical terminology and symbols extensively. Patterning and Algebra

100 Student Learning Tasks The work sheets completed by students the learning tasks for each strand are reproduced on the following pages. These are only examples of many imaginative learning tasks that can be used. Teachers are encouraged to use these as a resource to help them select other tasks, prepare appropriate rubrics, and assess some or all of the students in their classes. Teachers can also use the Appendix, Assessment in Mathematics, to develop and extend their own assessment plans. Minor revisions have been made to the original work sheets to reflect suggestions made by teachers during the field test. These revisions include improved wording and page layout or additional directions. Some selected answers for the activities for each of the five mathematics strands described in this booklet are provided in the Evaluation subsection (under Answers ) under the heading The Process Used. 99 Grade 4 Student Learning Tasks

101 100 The Ontario Curriculum Exemplars, Grades 1 8: Mathematics Patterning and Algebra Student Work Sheets - IV 2

102 Patterning and Algebra Student Work Sheets - IV 3 Patterning and Algebra Student Work Sheets - IV Grade 4 Student Learning Tasks

103 102 The Ontario Curriculum Exemplars, Grades 1 8: Mathematics Patterning and Algebra Student Work Sheets - IV 5 Patterning and Algebra Student Work Sheets - IV 6

104 Patterning and Algebra Student Work Sheets - IV 7 Patterning and Algebra Student Work Sheets - IV Grade 4 Student Learning Tasks

105 104 The Ontario Curriculum Exemplars, Grades 1 8: Mathematics Patterning and Algebra Student Work Sheets - IV 9 Patterning and Algebra Student Work Sheets - IV 10

106 Grade 5 Measurement

107 106 The Ontario Curriculum Exemplars, Grades 1 8: Mathematics Measurement The Tasks 1. Students designed plans for a school-yard playground, given perimeters and areas of different activity centres to be included (e.g., a rugby field, a baseball field, a climbing area). On the grids provided, they first produced drafts and then final designs. 2. Students calculated the amount of the free space remaining on their playgrounds. 3. Students explained the difference between perimeter and area, using pictures, numbers, and words. 4. Students described additional things that they learned about perimeter and area while doing the task. The following are the curriculum s overall expectations that relate to these tasks: By the end Grade 5, students will: solve problems related to the calculation of the perimeter and the area of regular and irregular two-dimensional shapes. During these tasks, students worked on the following selected expectations in specific areas from the Grade 5 curriculum: Students will: estimate and calculate the perimeter and area of rectangles and squares; explain the rules used in calculating the perimeter and area of rectangles and squares; develop methods of using grid paper to track and measure the perimeter and area of polygons and irregular two-dimensional shapes. Previous Learning Experiences It was suggested that before attempting the tasks, students should have had experience with estimating and calculating area and perimeter, using concrete materials. The Process Used Introductory activities. Teachers presented the scenario outlined on the first page of the activity sheets and then displayed an overhead transparency containing geometric shapes on graph paper representing different areas of a school yard. They discussed area and perimeter with students and possible ways to calculate these measurements. Teachers then distributed the first two pages of the student work sheets. Working in small groups, students completed the small-group activity. On completion, one or two of the groups presented their answers and strategies for finding area and perimeter to the class.

108 On my own. Teachers handed out the student work sheets and explained that the students would be expected to complete several measurement activities independently. To help students evaluate their work before submitting it, teachers explained the rubric reproduced on the next page, rephrasing the information if necessary so that students could understand it. Students were also provided with a checklist of assessment criteria on the On My Own page. Students were encouraged to do their best work. Answers: There will be a variety of answers for task 1. In task 2a, one solution could be that the total area taken up with playground items is 248 cm 2 and the amount of space remaining is 41 cm 2. Other solutions are possible. In questions 2b and 2c, there will be a variety of answers. Evaluation. After students completed their work sheets independently, teachers evaluated students work holistically, using the following rubric and noting strengths and weaknesses in any of the four individual categories (problem solving, understanding of concepts, application of mathematical procedures, communication). 107 Grade 5 Measurement

109 108 The Ontario Curriculum Exemplars, Grades 1 8: Mathematics Rubric for Measurement Categories Level 1 Level 2 Level 3 Level 4 Problem solving The student: uses a limited range of problem-solving strategies to solve a few simple problems uses appropriate problemsolving strategies to solve some problems selects and uses appropriate problemsolving strategies to solve problems selects and uses appropriate problemsolving strategies or modifies strategies to solve problems Understanding of concepts The student: provides some inappropriate and some incomplete explanations of area and perimeter provides appropriate but sometimes incomplete explanations of area and perimeter provides accurate and complete explanations of area and perimeter provides accurate and complete explanations of area and perimeter in a variety of contexts Application of mathematical procedures The student: makes some major errors when measuring and calculating area and perimeter makes several minor errors when measuring and calculating area and perimeter makes only a few minor errors when measuring and calculating area and perimeter makes practically no errors when measuring and calculating area and perimeter Communication of required knowledge related to concepts, procedures, and problem solving The student: describes processes unclearly and imprecisely, using limited mathematical terminology describes processes with some clarity and some precision, sometimes using appropriate mathematical terminology describes processes clearly and precisely, using appropriate mathematical terminology describes processes clearly and precisely, and justifies the strategies used, extensively using appropriate mathematical terminology Note: This rubric does not include criteria for assessing student performance that falls below level 1.

110 Grade 5 Measurement: Level 1 A B Measurement Student Work Sheets - V 2 Measurement Student Work Sheets - V Grade 5 Measurement

111 110 The Ontario Curriculum Exemplars, Grades 1 8: Mathematics C D Measurement Student Work Sheets - V 4 Measurement Student Work Sheets - V 5

112 E F Measurement Student Work Sheets - V 6 Measurement Student Work Sheets - V Grade 5 Measurement

113 112 The Ontario Curriculum Exemplars, Grades 1 8: Mathematics G Teacher s Notes Problem solving The student uses a limited range of problem-solving strategies to solve a few problems. The student makes some major errors in designing the playground with the required perimeter and area. Understanding of concepts The student gives some incomplete and inappropriate explanations of perimeter and area. Application of mathematical procedures The student makes some major errors when measuring and calculating perimeter and area. Communication The student describes tasks unclearly and imprecisely, using limited mathematical terminology. Comments The student uses a limited range of problem-solving strategies, shows a lack of understanding of perimeter and area, makes some major errors when measuring and calculating perimeter and area, and describes tasks unclearly and imprecisely, using limited mathematical terminology. Measurement Student Work Sheets - V 8

114 Grade 5 Measurement: Level 2 A B Measurement Student Work Sheets - V 2 Measurement Student Work Sheets - V Grade 5 Measurement

115 114 The Ontario Curriculum Exemplars, Grades 1 8: Mathematics C D Measurement Student Work Sheets - V 4 Measurement Student Work Sheets - V 5

116 E F Measurement Student Work Sheets - V 6 Measurement Student Work Sheets - V Grade 5 Measurement

117 116 The Ontario Curriculum Exemplars, Grades 1 8: Mathematics G Teacher s Notes Problem solving The student uses some appropriate problem-solving strategies to solve problems. The student includes all the playground areas in the plan, but the shapes provided are sometimes inappropriate and the labelling is sometimes inappropriate or inaccurate. Understanding of concepts The student provides appropriate but sometimes incomplete explanations of perimeter and area. Application of mathematical procedures The student makes several minor errors when measuring and calculating perimeter and area. Communication The student describes the processes used with some clarity and some precision, sometimes using appropriate mathematical terminology. Comments The student applies appropriate problem-solving strategies, provides appropriate but incomplete solutions to perimeter and area problems, makes several minor errors when measuring and calculating, and describes processes with some clarity and some precision, sometimes using appropriate mathematical terminology. Measurement Student Work Sheets - V 8

118 Grade 5 Measurement: Level 3 A B Measurement Student Work Sheets - V 2 Measurement Student Work Sheets - V Grade 5 Measurement

119 118 The Ontario Curriculum Exemplars, Grades 1 8: Mathematics C D Measurement Student Work Sheets - V 4 Measurement Student Work Sheets - V 5

120 E F Measurement Student Work Sheets - V 6 Measurement Student Work Sheets - V Grade 5 Measurement

121 120 The Ontario Curriculum Exemplars, Grades 1 8: Mathematics G Teacher s Notes Problem solving The student selects and uses appropriate problem-solving strategies to solve problems. The student creates a legend for the playground areas. Understanding of concepts The student gives accurate explanations of perimeter and area. Application of mathematical procedures The student makes only a few minor errors when measuring and calculating perimeter and area. Communication The student describes the processes used clearly and precisely, using appropriate mathematical terminology. Comments The student selects and uses appropriate strategies; gives accurate explanations of perimeter and area, using appropriate mathematical terminology; and makes only a few minor errors when measuring and calculating perimeter and area. Measurement Student Work Sheets - V 8

122 Grade 5 Measurement: Level 4 A B Measurement Student Work Sheets - V 2 Measurement Student Work Sheets - V Grade 5 Measurement

123 122 The Ontario Curriculum Exemplars, Grades 1 8: Mathematics C D Measurement Student Work Sheets - V 4 Measurement Student Work Sheets - V 5

124 E F Measurement Student Work Sheets - V 6 Measurement Student Work Sheets - V Grade 5 Measurement

125 124 The Ontario Curriculum Exemplars, Grades 1 8: Mathematics G H Measurement Student Work Sheets - V 8

126 Teacher s Notes Problem solving The student selects, uses, and modifies appropriate problem-solving strategies to solve problems. The student calculates the area of the sandbox from the perimeter. The student calculates perimeter, using multiplication instead of addition. Understanding of concepts The student provides accurate and complete explanations of perimeter and area in a variety of contexts. Application of mathematical procedures The student makes almost no errors when measuring and calculating perimeter and area. Communication The student makes extensive use of appropriate mathematical terminology (e.g., area expressed in square centimetres and perimeter in centimetres). The student describes and justifies processes clearly, precisely, and sequentially. Comments The student gives accurate and complete explanations of perimeter and area, makes few or no errors when calculating, selects and modifies appropriate strategies, and describes and justifies processes clearly, precisely, and sequentially, using appropriate mathematical terminology. 125 Grade 5 Measurement

127 126 The Ontario Curriculum Exemplars, Grades 1 8: Mathematics Student Learning Tasks The work sheets completed by students the learning tasks for each strand are reproduced on the following pages. These are only examples of many imaginative learning tasks that can be used. Teachers are encouraged to use these as a resource to help them select other tasks, prepare appropriate rubrics, and assess some or all of the students in their classes. Teachers can also use the Appendix, Assessment in Mathematics, to develop and extend their own assessment plans. Minor revisions have been made to the original work sheets to reflect suggestions made by teachers during the field test. These revisions include improved wording and page layout or additional directions. Some selected answers for the activities for each of the five mathematics strands described in this booklet are provided in the Evaluation subsection (under Answers ) under the heading The Process Used.

128 Measurement Student Work Sheets - V Grade 5 Student Learning Tasks

129 128 The Ontario Curriculum Exemplars, Grades 1 8: Mathematics Measurement Student Work Sheets - V 3 Measurement Student Work Sheets - V 4

130 Measurement Student Work Sheets - V 5 Measurement Student Work Sheets - V Grade 5 Student Learning Tasks

131 130 The Ontario Curriculum Exemplars, Grades 1 8: Mathematics Measurement Student Work Sheets - V 7 Measurement Student Work Sheets - V 8

132 Measurement Student Work Sheets - V Grade 5 Student Learning Tasks

133

134 Grade 6 Measurement

135 134 The Ontario Curriculum Exemplars, Grades 1 8: Mathematics Measurement The Tasks 1. Students were asked to design a game board, which was to have an area of 300 cm 2, and to include a number of geometric shapes with given areas. After making a rough sketch, each student made a good copy on geopaper. 2. Using their game boards, students were asked to explain (a) how a parallelogram and a rectangle can have the same area and (b) how they drew triangles for the board game. 3. Students completed a chart involving the perimeter and area of each shape they used. 4. Students designed a game box to hold their game boards and pieces. They calculated the volume of their game boxes and explained their answers, using pictures, words, and numbers. The following are the curriculum s overall expectations that relate to these tasks: By the end of Grade 6, students will: identify relationships between and among measurement concepts (linear, square, cubic, monetary); solve problems related to the calculation and comparison of the perimeter and the area of regular polygons. During these tasks, students worked on the following selected expectations in specific areas from the Grade 6 curriculum: Students will: determine the relationship between linear, square, and cubic units (e.g., compare cubic centimetres and cubic metres by constructing a cubic metre with rolled newspaper); understand the relationship between the area of a parallelogram and the area of a rectangle, between the area of a triangle and the area of a rectangle, and between the area of a triangle and the area of a parallelogram; estimate and calculate the area of a parallelogram and the area of a triangle, using a formula; understand the relationship between area and lengths of sides and between perimeter and lengths of sides for squares, rectangles, triangles, and parallelograms; sketch a rectangle, square, triangle, or parallelogram given its area and/or perimeter.

136 Previous Learning Experiences It was suggested that before attempting the tasks, students should have had experience with the following: estimating and calculating the area of two-dimensional shapes developing rules for calculating area and volume The Process Used Introductory activities. Teachers used the activities on the first page of the student work sheets to develop students understanding of the relationships between triangles, rectangles, and parallelograms. On my own. Teachers briefly reviewed the introductory activities just completed. They then handed out the student work sheets and explained that the students would be expected to complete several measurement activities independently. To help students evaluate their work before submitting it, teachers explained the rubric reproduced on the next page, rephrasing the information if necessary so that students could understand it. Students were also provided with a checklist of assessment criteria on the On My Own page. Students were encouraged to do their best work. Evaluation. After students completed their work sheets independently, teachers evaluated students work holistically, using the following rubric and noting strengths and weaknesses in any of the four individual categories (problem solving, understanding of concepts, application of mathematical procedures, communication). Answers: Students answers will vary, depending on the size of the board chosen and the location of the polygons. It is important to ensure that students answers in the chart are correct, that they reflect the polygons on the game board, and that the game board fits inside the rectangular prism. Students should be encouraged to use geoboards before using geopaper. 135 Grade 6 Measurement

137 136 The Ontario Curriculum Exemplars, Grades 1 8: Mathematics Rubric for Measurement Categories Level 1 Level 2 Level 3 Level 4 Problem solving The student: uses a limited range of problem-solving strategies to solve a few simple problems uses appropriate problemsolving strategies to solve some problems selects and uses appropriate problemsolving strategies to solve problems selects and uses appropriate problemsolving strategies or modifies strategies to solve problems Understanding of concepts The student: provides some incomplete and some inappropriate explanations of area, perimeter, and volume provides appropriate but sometimes incomplete explanations of area, perimeter, and volume provides accurate explanations of area, perimeter, and volume provides accurate and complete explanations of area, perimeter, and volume, and demonstrates that he or she can apply the concepts in a variety of contexts Application of mathematical procedures The student: makes some major errors when measuring and calculating area, perimeter, and volume makes some major errors when constructing polygons with given areas makes several minor errors when measuring and calculating area, perimeter, and volume makes several minor errors when constructing polygons with given areas makes only a few minor errors when measuring and calculating area, perimeter, and volume makes only a few minor errors when constructing polygons with given areas makes almost no errors when measuring and calculating area, perimeter, and volume makes almost no errors when constructing polygons with given areas Communication of required knowledge related to concepts, procedures, and problem solving The student: describes processes unclearly and imprecisely, using a limited range of mathematical terminology describes processes with some clarity and some precision, sometimes using appropriate mathematical terminology describes processes and strategies clearly and precisely, using appropriate mathematical terminology describes processes and strategies clearly and precisely, extensively using appropriate mathematical terminology Note: This rubric does not include criteria for assessing student performance that falls below level 1.

138 Grade 6 Measurement: Level 1 A B Measurement Student Work Sheets - VI 2 Measurement Student Work Sheets - VI Grade 6 Measurement

139 138 The Ontario Curriculum Exemplars, Grades 1 8: Mathematics C D Measurement Student Work Sheets - VI 4 Measurement Student Work Sheets - VI 5

140 E F Measurement Student Work Sheets - VI 6 Measurement Student Work Sheets - VI Grade 6 Measurement

141 140 The Ontario Curriculum Exemplars, Grades 1 8: Mathematics Teacher s Notes Problem solving The student uses a limited range of strategies to attempt to solve some simple problems. Understanding of concepts The student gives some incomplete and inappropriate explanations related to perimeter, area, and volume. Application of mathematical procedures The student makes some major errors when calculating area, perimeter, and volume. The student also makes some major errors in the construction of polygons with given areas. Communication The student s descriptions are unclear and imprecise and include limited mathematical terminology. Comments The student uses a limited range of strategies to solve some simple problems; demonstrates a limited understanding of perimeter, area, and volume; and makes some major errors when calculating area, perimeter, and volume, and when constructing polygons.

142 Grade 6 Measurement: Level 2 A B Measurement Student Work Sheets - VI 2 Measurement Student Work Sheets - VI Grade 6 Measurement

143 142 The Ontario Curriculum Exemplars, Grades 1 8: Mathematics C D Measurement Student Work Sheets - VI 4 Measurement Student Work Sheets - VI 5

144 E F Measurement Student Work Sheets - VI 6 Measurement Student Work Sheets - VI Grade 6 Measurement

145 144 The Ontario Curriculum Exemplars, Grades 1 8: Mathematics Teacher s Notes Problem solving The student selects and uses appropriate strategies to solve some problems. Understanding of concepts The student demonstrates an appropriate but sometimes incomplete understanding of area, perimeter, and volume. Application of mathematical procedures The student makes several minor errors when calculating area, perimeter, and volume. The student also makes several minor errors when constructing polygons with given areas. Communication The student describes the processes used with some clarity and some precision, sometimes using appropriate mathematical terminology. Comments The student selects and uses appropriate strategies to solve some problems; demonstrates an incomplete understanding of area, perimeter, and volume; makes several minor errors in calculations and when constructing polygons with given areas; and describes the processes used with some clarity and some precision, sometimes using appropriate mathematical terminology.

146 Grade 6 Measurement: Level 3 A B Measurement Student Work Sheets - VI 2 Measurement Student Work Sheets - VI Grade 6 Measurement

147 146 The Ontario Curriculum Exemplars, Grades 1 8: Mathematics C D Measurement Student Work Sheets - VI 4 Measurement Student Work Sheets - VI 5

148 E F Measurement Student Work Sheets - VI 6 Measurement Student Work Sheets - VI Grade 6 Measurement

149 148 The Ontario Curriculum Exemplars, Grades 1 8: Mathematics Teacher s Notes Problem solving The student selects and uses appropriate strategies to solve problems. Understanding of concepts The student demonstrates complete understanding of perimeter, area, and volume. Application of mathematical procedures The student makes only a few minor errors when measuring and calculating area. The student accurately constructs polygons with given areas. Communication The student clearly and precisely describes the processes employed, using appropriate mathematical terminology. Comments The student selects and uses appropriate strategies to solve problems; clearly demonstrates understanding of perimeter, area, and volume; makes only a few minor errors when measuring and calculating area, perimeter, and volume; accurately constructs polygons with given areas; and describes the processes employed, using appropriate mathematical terminology.

150 Grade 6 Measurement: Level 4 A B Measurement Student Work Sheets - VI 2 Measurement Student Work Sheets - VI Grade 6 Measurement

151 150 The Ontario Curriculum Exemplars, Grades 1 8: Mathematics C D Measurement Student Work Sheets - VI 4 Measurement Student Work Sheets - VI 5

152 E F Measurement Student Work Sheets - VI 6 Measurement Student Work Sheets - VI Grade 6 Measurement

153 152 The Ontario Curriculum Exemplars, Grades 1 8: Mathematics Teacher s Notes Problem solving The student selects and uses appropriate strategies, modifies strategies, or creates new ones as necessary to solve problems. Understanding of concepts The student explains the concepts of perimeter, area, and volume accurately and completely, and demonstrates that he or she can apply the concepts in a variety of contexts. Application of mathematical procedures The student makes practically no errors when measuring and calculating perimeter, area, and volume. The student makes almost no errors when constructing polygons. Communication The student explains the processes used clearly and precisely, and justifies answers, extensively using appropriate mathematical terminology. Comments The student selects and uses strategies, modifies strategies, or creates new ones to solve problems; explains the concepts of perimeter, area, and volume accurately and completely, and demonstrates that he or she can apply the concepts in a variety of contexts; makes practically no errors when calculating or when constructing polygons; and clearly and precisely describes the processes used and justifies solutions, extensively using appropriate mathematical terminology.

154 Student Learning Tasks The work sheets completed by students the learning tasks for each strand are reproduced on the following pages. These are only examples of the imaginative learning tasks that can be used. Teachers are encouraged to use these as a resource to help them select other tasks, prepare appropriate rubrics, and assess some or all of the students in their classes. Teachers can also use the Appendix, Assessment in Mathematics, to develop and extend their own assessment plans. Minor revisions have been made to the original work sheets to reflect suggestions made by teachers during the field test. These revisions include improved wording and page layout or additional directions. Some selected answers for the activities for each of the five mathematics strands described in this booklet are provided in the Evaluation subsection (under Answers ) under the heading The Process Used. 153 Grade 6 Student Learning Tasks

155 154 The Ontario Curriculum Exemplars, Grades 1 8: Mathematics Measurement Student Work Sheets - VI 2

156 Measurement Student Work Sheets - VI 3 Measurement Student Work Sheets - VI Grade 6 Student Learning Tasks

157 156 The Ontario Curriculum Exemplars, Grades 1 8: Mathematics Measurement Student Work Sheets - VI 5 Measurement Student Work Sheets - VI 6

158 Measurement Student Work Sheets - VI 7 Measurement Student Work Sheets - VI Grade 6 Student Learning Tasks

159 158 The Ontario Curriculum Exemplars, Grades 1 8: Mathematics Measurement Student Work Sheets - VI 9

160 Grade 7 Number Sense and Numeration

161 160 The Ontario Curriculum Exemplars, Grades 1 8: Mathematics Number Sense and Numeration The Tasks 1. Students were given the regular price and the percent of discount for each of two types of bicycle that were on sale. They then calculated the amount that would be saved and the sale price for each, and chose one bicycle, explaining their choice. 2. To buy their bicycles, students needed part-time jobs to pay for them. They were given a choice of three jobs that paid different amounts and were asked to determine the weekly salary for each job. 3. Students were told their current bank balance and asked to figure out different ways of paying off the bike they chose in task one, with reference to the job options. Students presented their findings on budget sheets and chose one method of payment, explaining why they liked that option. The following are the curriculum s overall expectations that relate to these tasks: By the end of Grade 7, students will: solve and explain multi-step problems involving simple fractions, decimals, and percents; explain, in writing, the process of problem solving using appropriate mathematical language. During these tasks, students worked on the following selected expectations in specific areas from the Grade 7 curriculum: Students will: perform three-step problem solving that involves whole numbers and decimals related to real-life experiences, using calculators; explain the process used and any conclusions reached in problem solving and investigations; solve problems involving fractions and decimals using the appropriate strategies and calculation methods. Previous Learning Experiences It was suggested that before attempting the tasks, students should have had experience with the following: using a calculator explaining and reflecting on mathematical reasoning dealing with budgets (e.g., using vocabulary such as income, expenses, salary, and earnings) calculating percent organizing data

162 The Process Used Introductory activities. Teachers familiarized students with vocabulary related to completing a budget spreadsheet and then handed out the work sheets. Teachers used a chart paper copy of table 2 on the first page of the work sheets and worked with the class to complete the table. They then asked students to fill it out on their own work sheets. On my own. Teachers reviewed the introductory activities and reviewed the instructions for the remaining tasks on the work sheets, which students were expected to complete independently. Teachers also reviewed the assessment criteria on page 3 of the student work sheets. To help students evaluate their work before submitting it, teachers also explained the rubric reproduced on the next page, rephrasing the information so that students could understand it. Students were also provided with a checklist of assessment criteria on the On My Own page. Students were encouraged to do their best work. Evaluation. After students completed their work sheets independently, teachers evaluated students work holistically, using the following rubric and noting strengths and weaknesses in any of the four individual categories (problem solving, understanding of concepts, application of mathematical procedures, communication). Answers: In task 1, the amount saved for the Superbike is $60 and its sale price is $180. The amount saved for the Mountain Bike is $96 and its sale price is $224. In task 2, the weekly salaries are $37.50 for cutting the grass, $72.80 for delivering the flyers, and $27 for walking the dogs. In task 3, teachers should check to see that each student selected an appropriate way to pay for the bike chosen and used this way when completing the budget page. 161 Grade 7 Number Sense and Numeration

163 162 The Ontario Curriculum Exemplars, Grades 1 8: Mathematics Rubric for Number Sense and Numeration Categories Level 1 Level 2 Level 3 Level 4 Problem solving The student: uses a limited range of problem-solving strategies to solve a few simple problems uses a few appropriate problem-solving strategies to solve some problems selects and uses appropriate problemsolving strategies to solve most problems selects and uses appropriate problemsolving strategies, modifies strategies, or creates new strategies to solve almost all problems Understanding of concepts The student: attempts to compare data to reach simple conclusions compares data to reach some accurate conclusions compares data to reach accurate conclusions compares data to reach accurate conclusions, and justifies choices Application of mathematical procedures The student: makes some major errors when performing operations with decimals and percent makes several minor errors when performing operations with decimals and percent makes only a few minor errors when performing operations with decimals and percent makes practically no errors when performing operations with decimals and percent Communication of required knowledge related to concepts, procedures, and problem solving The student: describes tasks unclearly and imprecisely, using a limited range of mathematical terminology describes tasks with some clarity and some precision, sometimes using appropriate mathematical terminology describes tasks clearly and precisely, using appropriate mathematical terminology describes tasks clearly, precisely, and sequentially, using appropriate mathematical terminology extensively Note: This rubric does not include criteria for assessing student performance that falls below level 1.

164 Grade 7 Number Sense and Numeration: Level 1 A B Number Sense and Numeration Student Work Sheets - VII 3 Number Sense and Numeration Student Work Sheets - VII Grade 7 Number Sense and Numeration

165 164 The Ontario Curriculum Exemplars, Grades 1 8: Mathematics C D Number Sense and Numeration Student Work Sheets - VII 5 Number Sense and Numeration Student Work Sheets - VII 6

166 E Teacher s Notes Problem solving The student uses a limited range of strategies to solve a few simple problems. Understanding of concepts The student reaches some inappropriate and incomplete conclusions from the data provided. The student demonstrates difficulty in identifying the meaning of sale price and savings. Application of mathematical procedures The student makes some major errors when calculating with decimals and percents. The student solves some one-step problems with some accuracy. Communication The student provides explanations that are unclear and imprecise, using limited mathematical terminology. Comments The student uses a limited range of strategies to solve problems involving decimals and percent, making some major errors in the process; gives some solutions that are inappropriate and incomplete; and provides unclear and imprecise explanations, using limited mathematical terminology. Number Sense and Numeration Student Work Sheets - VII Grade 7 Number Sense and Numeration

167 166 The Ontario Curriculum Exemplars, Grades 1 8: Mathematics Grade 7 Number Sense and Numeration: Level 2 A B Number Sense and Numeration Student Work Sheets - VII 3 Number Sense and Numeration Student Work Sheets - VII 4

168 C D Number Sense and Numeration Student Work Sheets - VII 5 Number Sense and Numeration Student Work Sheets - VII Grade 7 Number Sense and Numeration

169 168 The Ontario Curriculum Exemplars, Grades 1 8: Mathematics E Teacher s Notes Problem solving The student uses appropriate problem-solving strategies to solve some problems. Understanding of concepts The student provides appropriate but sometimes incomplete solutions to problems. The solutions show an unclear understanding of the real-life context of the problems. Application of mathematical procedures The student makes a few errors when performing operations with decimals and percent. Given the options, the student does not realize the least number of weeks it would take to pay for the bicycle. The student compares data accurately to make some simple decisions. Communication The student explains solutions with some clarity and some precision, sometimes using appropriate mathematical terminology. Comments The student uses appropriate strategies to solve some problems involving decimals and percent; shows a limited understanding of the real-life context of savings and sale price ; gives solutions that are appropriate but incomplete; and explains the solutions with some clarity and some precision, sometimes using appropriate mathematical terminology. Number Sense and Numeration Student Work Sheets - VII 7

170 Grade 7 Number Sense and Numeration: Level 3 A B Number Sense and Numeration Student Work Sheets - VII 3 Number Sense and Numeration Student Work Sheets - VII Grade 7 Number Sense and Numeration

171 170 The Ontario Curriculum Exemplars, Grades 1 8: Mathematics C D Number Sense and Numeration Student Work Sheets - VII 5 Number Sense and Numeration Student Work Sheets - VII 6

172 E Teacher s Notes Problem solving The student selects and uses appropriate problem-solving strategies to solve problems. The student often uses a combination of methods to solve problems. Understanding of concepts The student tries to give some real-life context to the solutions by mentioning bicycle efficiency and accessories. The student s solutions are reasonable, appropriate, and complete. The student compares data to make appropriate decisions. Application of mathematical procedures The student makes only a few errors when performing operations with decimals and percent, indicating some organization. Communication The student describes processes clearly and precisely and gives solutions, using appropriate mathematical terminology. Comments The student uses appropriate strategies to solve problems involving decimals and percent; gives solutions that are reasonable, appropriate, and complete, and that reflect an understanding of the real-life context of the problem; and gives explanations that are clear and precise and that use appropriate mathematical terminology. Number Sense and Numeration Student Work Sheets - VII Grade 7 Number Sense and Numeration

173 172 The Ontario Curriculum Exemplars, Grades 1 8: Mathematics Grade 7 Number Sense and Numeration: Level 4 A B Number Sense and Numeration Student Work Sheets - VII 3 Number Sense and Numeration Student Work Sheets - VII 4

174 C D Number Sense and Numeration Student Work Sheets - VII 5 Number Sense and Numeration Student Work Sheets - VII Grade 7 Number Sense and Numeration

175 174 The Ontario Curriculum Exemplars, Grades 1 8: Mathematics E Teacher s Notes Problem solving The student selects and uses appropriate problem-solving strategies, modifies strategies, or creates new ones as necessary to solve problems. The student uses a combination of methods to solve problems and includes integers when recording personal expenses. Understanding of concepts The student gives appropriate and complete solutions, providing real-life, emotional justification for the choices made. Application of mathematical procedures The student makes no errors when performing operations with decimals and percent, and gives complete, well-organized explanations. Communication The student gives clear and precise solutions, and justifies the answers provided, extensively using appropriate mathematical terminology. Comments The student selects, modifies, and uses problem-solving strategies, or creates new ones as necessary, to solve real-life problems; gives solutions that are organized, appropriate, and complete and explanations that are clear and precise, using appropriate mathematical terminology extensively; and justifies the solutions provided. Number Sense and Numeration Student Work Sheets - VII 7

176 Student Learning Tasks The work sheets completed by students the learning tasks for each strand are reproduced on the following pages. These are only examples of many imaginative learning tasks that can be used. Teachers are encouraged to use these as a resource to help them select other tasks, prepare appropriate rubrics, and assess some or all of the students in their classes. Teachers can also use the Appendix, Assessment in Mathematics, to develop and extend their own assessment plans. Minor revisions have been made to the original work sheets to reflect suggestions made by teachers during the field test. These revisions include improved wording and page layout or additional directions. Some selected answers for the activities for each of the five mathematics strands described in this booklet are provided in the Evaluation subsection (under Answers ) under the heading The Process Used. 175 Grade 7 Student Learning Tasks

177 176 The Ontario Curriculum Exemplars, Grades 1 8: Mathematics Number Sense and Numeration Student Work Sheets - VII 2

178 Number Sense and Numeration Student Work Sheets - VII 3 Number Sense and Numeration Student Work Sheets - VII Grade 7 Student Learning Tasks

179 178 The Ontario Curriculum Exemplars, Grades 1 8: Mathematics Number Sense and Numeration Student Work Sheets - VII 5 Number Sense and Numeration Student Work Sheets - VII 6

180 Number Sense and Numeration Student Work Sheets - VII 7 Number Sense and Numeration Student Work Sheets - VII Grade 7 Student Learning Tasks

181

182 Grade 8 Data Management and Probability

183 182 The Ontario Curriculum Exemplars, Grades 1 8: Mathematics Data Management and Probability The Tasks 1. Students drew a circle graph to represent the ages of new members at the Prime Time Fitness Institute over a threemonth period and discussed why this information might be important for a fitness club. 2. Using a graph showing the percent of time that a club member, Maria, spent on different activities, students determined how long she spent on two activities each day, given the total number of hours and days she worked out. 3. Students chose a type of graph to show the number of fitness club renewals over a sixteen-month period and explained their choice of graph. They then used the graph to predict future trends. The following are the curriculum s overall expectations for data management that relate to these tasks: By the end of Grade 8, students will: systematically organize and analyse data; interpret displays of data and present the information using mathematical terms; evaluate data and draw conclusions from the analysis of data. During these tasks, students worked on the following selected expectations in specific areas from the Grade 8 curriculum: Students will: organize data on stem-and-leaf plots; read and report information about data presented on line graphs, comparative bar graphs, and circle graphs, and use the information to solve problems; determine the effect on a measure of central tendency of adding or removing a value; construct line graphs and circle graphs and use the information to solve problems; make inferences and convincing arguments that are based on data analysis; determine trends and patterns by making inferences from graphs.

184 Previous Learning Experiences It was suggested that before attempting the tasks, students should have had experience with the following: organizing data on a stem-and-leaf plot constructing line or circle graphs and using the information to solve problems The Process Used Introductory activities. Teachers reviewed stem-and-leaf plot and circle graphs. They handed out the student work sheets and had the class complete the whole class activity outlined on page 2, which asked students to determine the range, mean, and median of the data. Evaluation. After students completed their work sheets independently, teachers evaluated students work holistically, using the following rubric and noting strengths and weaknesses in any of the four individual categories (problem solving, understanding of concepts, application of mathematical procedures, communication). Answers: In task 1, more than half of the members are between the ages of 20 and 40. The company could use this information to help plan additional programs and to schedule the use of machines. In task 2, Maria averages 6 min a day in warm-up and cool-down activities and 24 min a day swimming. In task 3, answers will vary. On my own. Teachers reviewed the introductory activities. Students were expected to complete the tasks on the work sheets independently. Teachers reviewed the assessment criteria on page 4 of the work sheets with students. To help students evaluate their work before submitting it, teachers also explained the rubric reproduced on the next page, rephrasing the information so that students could understand it. Students were also provided with a checklist of assessment criteria on the On My Own page. Students were encouraged to do their best work. 183 Grade 8 Data Management and Probability

185 184 The Ontario Curriculum Exemplars, Grades 1 8: Mathematics Rubric for Data Management and Probability Categories Level 1 Level 2 Level 3 Level 4 Problem solving The student: uses a limited range of problem-solving strategies to solve a few simple problems uses appropriate problemsolving strategies to solve some problems selects and uses appropriate problemsolving strategies to solve problems selects and uses appropriate problem-solving strategies, modifies strategies, or creates new strategies to solve problems Understanding of concepts The student: demonstrates limited ability to interpret data demonstrates some attempt to interpret data accurately uses data to make accurate interpretations uses data to make accurate interpretations and justifies the interpretations Application of mathematical procedures The student: makes some major errors or omissions when attempting to record, organize, and graph data makes some major errors or omissions when constructing a circle graph makes several minor errors or omissions when recording, organizing, and graphing data makes several minor errors or omissions when constructing a circle graph makes only a few minor errors or omissions when recording, organizing, and graphing data makes only a few minor errors or omissions when constructing a circle graph makes almost no errors or omissions when recording, organizing, and graphing data and uses the most efficient and appropriate method makes almost no errors or omissions when constructing a circle graph Communication of required knowledge related to concepts, procedures, and problem solving The student: explains data and processes unclearly and imprecisely, using a limited range of mathematical terminology explains data and processes with some clarity and some precision, sometimes using appropriate mathematical terminology explains data and processes clearly and precisely, using appropriate mathematical terminology explains data and processes clearly, precisely, and sequentially, and justifies the methods and solutions used, extensively using appropriate mathematical terminology Note: This rubric does not include criteria for assessing student performance that falls below level 1.

186 Grade 8 Data Management and Probability: Level 1 A B Data Management and Probability Student Work Sheets - VIII 2 Data Management and Probability Student Work Sheets - VIII Grade 8 Data Management and Probability

187 186 The Ontario Curriculum Exemplars, Grades 1 8: Mathematics C D Data Management and Probability Student Work Sheets - VIII 4 Data Management and Probability Student Work Sheets - VIII 5

188 E F Data Management and Probability Student Work Sheets - VIII 6 Data Management and Probability Student Work Sheets - VIII Grade 8 Data Management and Probability

189 188 The Ontario Curriculum Exemplars, Grades 1 8: Mathematics G H Data Management and Probability Student Work Sheets - VIII 8 Data Management and Probability Student Work Sheets - VIII 9

190 Teacher s Notes Problem solving The student uses a limited range of strategies to solve a few simple problems. Understanding of concepts The student demonstrates some ability to draw conclusions from data in circle graphs. The student makes unreasonable or incomplete predictions based on data from a graph. Application of mathematical procedures The student makes minor errors when constructing a circle graph. The student makes some errors or omissions when organizing information, using a graph. The student attempts to use data from a circle graph. The student makes some errors when manipulating numerical data to solve problems. Communication The student explains data and processes unclearly and imprecisely, using a limited range of mathematical terminology. Comments The student applies a limited range of strategies to attempt to organize, record, and interpret data; makes minor errors or omissions when constructing graphs; and explains data and processes unclearly and imprecisely, using a limited range of mathematical terminology. 189 Grade 8 Data Management and Probability

191 190 The Ontario Curriculum Exemplars, Grades 1 8: Mathematics Grade 8 Data Management and Probability: Level 2 A B Data Management and Probability Student Work Sheets - VIII 2 Data Management and Probability Student Work Sheets - VIII 3

192 C D Data Management and Probability Student Work Sheets - VIII 4 Data Management and Probability Student Work Sheets - VIII Grade 8 Data Management and Probability

193 192 The Ontario Curriculum Exemplars, Grades 1 8: Mathematics E F Data Management and Probability Student Work Sheets - VIII 6 Data Management and Probability Student Work Sheets - VIII 7

194 G H Data Management and Probability Student Work Sheets - VIII 8 Data Management and Probability Student Work Sheets - VIII Grade 8 Data Management and Probability

195 194 The Ontario Curriculum Exemplars, Grades 1 8: Mathematics Teacher s Notes Problem solving The student uses appropriate strategies to solve some problems. Understanding of concepts The student draws some conclusions, using data from circle graphs. The student makes some reasonable predictions based on data from a graph. Application of mathematical procedures The student makes several minor errors when drawing a circle graph. The student makes some minor errors or omissions when organizing information, using an appropriate graph. The student attempts to use data from a circle graph. The student makes some errors when manipulating numerical data to solve problems. Communication The student explains data and processes with some clarity and some precision, sometimes using appropriate mathematical terminology. Comments The student uses appropriate strategies to organize, record, and interpret some data accurately; makes some minor errors or omissions when constructing graphs; and explains data and processes with some clarity and some precision, sometimes using appropriate mathematical terminology.

196 Grade 8 Data Management and Probability: Level 3 A B Data Management and Probability Student Work Sheets - VIII 2 Data Management and Probability Student Work Sheets - VIII Grade 8 Data Management and Probability

197 196 The Ontario Curriculum Exemplars, Grades 1 8: Mathematics C D Data Management and Probability Student Work Sheets - VIII 4 Data Management and Probability Student Work Sheets - VIII 5

198 E F Data Management and Probability Student Work Sheets - VIII 6 Data Management and Probability Student Work Sheets - VIII Grade 8 Data Management and Probability

199 198 The Ontario Curriculum Exemplars, Grades 1 8: Mathematics G H Data Management and Probability Student Work Sheets - VIII 8 Data Management and Probability Student Work Sheets - VIII 9

200 Teacher s Notes Problem solving The student selects and uses appropriate strategies to solve problems. Understanding of concepts The student uses data from a circle graph to make accurate interpretations. The student makes reasonable predictions, using data from a graph. Application of mathematical procedures The student makes only a few minor errors (e.g., when using a protractor) when constructing a circle graph. The student is able to read data presented in a circle graph and to manipulate them accurately to solve problems. The student organizes information, using an appropriate graph, with a few minor errors. Communication The student clearly and precisely explains data and processes, using appropriate mathematical terminology. Comments The student uses appropriate strategies to accurately record, organize, and interpret data; makes only a few minor errors or omissions when constructing graphs; and clearly and precisely explains data and processes, using appropriate mathematical terminology. 199 Grade 8 Data Management and Probability

201 200 The Ontario Curriculum Exemplars, Grades 1 8: Mathematics Grade 8 Data Management and Probability: Level 4 A B Data Management and Probability Student Work Sheets - VIII 2 Data Management and Probability Student Work Sheets - VIII 3

202 C D Data Management and Probability Student Work Sheets - VIII 4 Data Management and Probability Student Work Sheets - VIII Grade 8 Data Management and Probability

203 202 The Ontario Curriculum Exemplars, Grades 1 8: Mathematics E F Data Management and Probability Student Work Sheets - VIII 6 Data Management and Probability Student Work Sheets - VIII 7

204 G H Data Management and Probability Student Work Sheets - VIII 8 Data Management and Probability Student Work Sheets - VIII Grade 8 Data Management and Probability

205 204 The Ontario Curriculum Exemplars, Grades 1 8: Mathematics Teacher s Notes Problem solving The student selects, uses, and modifies appropriate strategies to solve problems. Understanding of concepts The student uses data from the circle graph to draw accurate conclusions in task 1b. The student makes a reasonable prediction, using data from a graph, and justifies the prediction. Application of mathematical procedures The student makes almost no errors when constructing a circle graph. The student uses the most efficient and appropriate method for calculating the number of degrees in the circle graph. The student makes almost no errors or omissions when organizing information, using an appropriate graph. Communication The student clearly, precisely, and sequentially explains data and processes, extensively using appropriate mathematical terminology. The student justifies the methods used and solutions provided. Comments The student uses appropriate strategies to record, organize, and interpret data; makes almost no errors or omissions when constructing graphs; and explains data and processes clearly, precisely, and sequentially, extensively using appropriate mathematical terminology.

206 Student Learning Tasks The work sheets completed by students the learning tasks for each strand are reproduced on the following pages. These are only examples of many imaginative learning tasks that can be used. Teachers are encouraged to use these as a resource to help them select other tasks, prepare appropriate rubrics, and assess some or all of the students in their classes. Teachers can also use the Appendix, Assessment in Mathematics, to develop and extend their own assessment plans. Minor revisions have been made to the original work sheets to reflect suggestions made by teachers during the field test. These revisions include improved wording and page layout or additional directions. Some selected answers for the activities for each of the five mathematics strands described in this booklet are provided in the Evaluation subsection (under Answers ) under the heading The Process Used. 205 Grade 8 Student Learning Tasks

207 206 The Ontario Curriculum Exemplars, Grades 1 8: Mathematics Data Management and Probability Student Work Sheets - VIII 2

208 Data Management and Probability Student Work Sheets - VIII 3 Data Management and Probability Student Work Sheets - VIII Grade 8 Student Learning Tasks

209 208 The Ontario Curriculum Exemplars, Grades 1 8: Mathematics Data Management and Probability Student Work Sheets - VIII 5 Data Management and Probability Student Work Sheets - VIII 6