Samples of Student Work:A Resource for Teachers
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Ministry of Education
The Ontario Curriculum
ExemplarsGrade 9
Mathematics
Contents
IInnttrroodduuccttiioonn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
Purpose of This Document . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4Features of This Document . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4The Tasks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5The Rubrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5Development of the Tasks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7Assessment and Selection of the Samples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8Use of the Student Samples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
Teachers and Administrators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9Parents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10Students . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
PPrriinncciipplleess ooff MMaatthheemmaattiiccss,, AAccaaddeemmiicc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
Designing a Perfume Bottle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12The Task . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12Expectations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12Prior Knowledge and Skills . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13Task Rubric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14Student Samples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15Teacher Package . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
FFoouunnddaattiioonnss ooff MMaatthheemmaattiiccss,, AApppplliieedd . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
Footprints on the Road . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44The Task . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44Expectations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44Prior Knowledge and Skills . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45Task Rubric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46Student Instructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47Student Samples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48Teacher Package . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
Une publication équivalente est disponible en français sous le titresuivant: Le curriculum de l’Ontario : Copies types de 9e année – Mathématiques, 2000.
This publication is available on the Ministry of Education’s website at http://www.edu.gov.on.ca.
3
In 1999, the Ministry of Education published a new curriculum for Ontario secondaryschool students in Grades 9 and 10. The new curriculum is more specific than previouscurricula with respect to both the knowledge and the skills that students are expectedto develop and demonstrate in each grade. In the curriculum policy document foreach discipline, teachers are provided with the curriculum expectations for eachcourse within the discipline and an achievement chart that describes four levels ofstudent achievement to be used in assessing and evaluating student work.
The document entitled The Ontario Curriculum, Grades 9–12: Program Planning andAssessment, 2000 states that “assessment and evaluation will be based on the provincialcurriculum expectations and the achievement levels outlined in this document and in the curriculum policy document for each discipline” (p. 13). The document alsostates that the ministry is providing a variety of materials to assist teachers in improv-ing their assessment methods and strategies and, hence, their assessment of studentachievement. The present document is one of the resources intended to provide assistance to teachers in their assessment of student achievement. It contains samples(“exemplars”) of student work at each level of achievement.
Ontario school boards were invited by the ministry to participate in the developmentof exemplars. Forty-seven district school boards responded to this invitation. Teams of subject specialists from across the province were involved in developing the assess-ment materials. They designed the performance tasks and scoring scales (“rubrics”)based on selected Ontario curriculum expectations, field-tested them in classrooms,suggested changes, administered the final tasks, marked the student work, andselected the exemplars used in this document. During each stage of the process, external validation teams reviewed the subject material to ensure that it reflected theexpectations in the curriculum and that it was accessible to and appropriate for allstudents. Ministry staff who had been involved in the development of the curriculumpolicy documents also reviewed the tasks, rubrics, and exemplars.
The selection of student samples that appears in this document reflects the profes-sional judgement of teachers who participated in the project. No students, teachers, or schools have been identified.
The procedures followed during the development and implementation of this projectwill serve as a model for boards, schools, and teachers in designing assessment taskswithin the context of regular classroom work, developing rubrics, assessing theachievement of their own students, and planning for the improvement of students’learning.
The samples in this document will provide parents1 with examples of student work tohelp them monitor their children’s progress. They also provide a basis for communica-tion with teachers.
Introduction
1 In this document, parent(s) refers to parent(s) and guardian(s).
4 The Ontar io Curr iculum – Exemplars , Grade 9: Mathematics
Use of the exemplar materials will be supported initially through provincial in-servicetraining. A variety of additional opportunities (e.g., discipline- or subject-specificworkshops and summer institutes) will be available to secondary school teachers tosupport the use of the exemplars.
Purpose of This Document
This publication was developed to:
– show the characteristics of student work at each of the four levels of achievementfor Grade 9;
– promote greater consistency in the assessment of student work across the province;
– provide an approach to improving student learning by demonstrating the use ofclear criteria applied to student work in response to clearly defined assessmenttasks;
– show the connections between what students are expected to learn (the curriculumexpectations) and how their work can be assessed using the levels of achievementdescribed in the curriculum policy document for the subject.
Teachers, parents, and students should examine the student samples in this document and consider them along with the information in the Teacher’s Notes andComments/Next Steps sections. They are encouraged to examine the samples in orderto develop an understanding of the characteristics of work at each level of achieve-ment in Grade 9 and the ways in which the levels of achievement reflect a progressionin the quality of knowledge and skills demonstrated by the student.
The samples in this document represent examples of student achievement obtainedusing only one method of assessment, called performance assessment. Teachers willalso make use of a variety of other assessment methods and strategies in evaluatingstudent achievement in a course over a term or school year.
Features of This Document
This document contains the following:
– a description of each performance task, as well as the curriculum expectationsrelated to the task
– the task-specific assessment chart, or rubric
– two samples of student work for each of the four levels of achievement
– Teacher’s Notes, which provide some details on the level of achievement for eachsample
– Comments/Next Steps, which offer suggestions for improving achievement
– the Teacher Package that was used by teachers in administering the task
It should be noted that each sample for a specific level of achievement represents thecharacteristics of work at that level of achievement.
5Introduct ion
The Tasks
The performance tasks for mathematics were based directly on curriculum expecta-tions selected from the Grade 9 courses in mathematics. The tasks encompassed thefour categories of knowledge and skills in mathematics (i.e., Knowledge/Understanding,Thinking/Inquiry/Problem Solving, Communication, and Application), requiring students to integrate their knowledge and skills in meaningful learning experiences.The tasks gave students an opportunity to demonstrate not only how well they hadlearned to use the required knowledge and skills in one context, but how well theycould use their knowledge and skills in another context.
Teachers were required to explain the scoring criteria and descriptions of the levels of achievement (i.e., the information in the task rubrics) to the students before theybegan the assignment (for the rubrics, see pages 14 and 46).
The Rubrics
In this document, the term rubric refers to a scoring scale that consists of a set ofachievement criteria and descriptions of the levels of achievement for a particulartask. The scale is used to assess students’ work; this assessment is intended to helpstudents improve their performance level. The rubric identifies key criteria by whichstudents’ work is to be assessed, and it provides descriptions that indicate the degreeto which the key criteria have been met. The teacher uses the descriptions of the different levels of achievement given in the rubric to assess student achievement on a particular task.
The rubric for a specific performance task is intended to provide teachers and stu-dents with an overview of the expected final product with regard to the knowledgeand skills being assessed as a whole.
The achievement chart in the curriculum policy document for mathematics provides astandard province-wide tool for teachers to use in assessing and evaluating their students’achievement over a period of time. While the chart is broad in scope and general in nature,it provides a reference point for all assessment practice and a framework within which toassess and evaluate student achievement. The descriptions associated with each level ofachievement serve as a guide for gathering and tracking assessment information, enablingteachers to make consistent judgements about the quality of student work while providingclear and specific feedback to students and parents.
For the purposes of the exemplar project, a single rubric was developed for a perfor-mance task in each course. This task-specific rubric was developed in relation to theachievement chart in the curriculum policy document.
The differences between the achievement chart and the task-specific rubric may besummarized as follows:
– The achievement chart contains broad descriptions of achievement. Teachers use itto assess student achievement over time, making a summative evaluation that isbased on the total body of evidence gathered through using a variety of assessmentmethods and strategies.
6 The Ontar io Curr iculum – Exemplars , Grade 9: Mathematics
– The rubric contains criteria and descriptions of achievement that relate to a specifictask. The rubric uses some terms that are similar to those in the achievement chartbut focuses on aspects of the specific task. Teachers use the rubric to assess studentachievement on a single task.
The rubric contains the following components:
– an identification (by number) of the expectations on which student achievement inthe task was assessed
– the four categories of knowledge and skills
– the relevant criteria for evaluating performance of the task
– descriptions of student performance at the four levels of achievement (level 3 onthe achievement chart is considered to be the provincial standard)
As stated earlier, the focus of performance assessment using a rubric is to improve students’ learning. In order to improve their work, students need to be provided withuseful feedback. Students find that feedback on the strengths of their achievement and on areas in need of improvement is more helpful when the specific category ofknowledge or skills is identified and specific suggestions are provided than when they receive only an overall mark or general comments. Student achievement shouldbe considered in relation to the criteria for assessment stated in the rubric for eachcategory, and feedback should be provided for each category. Through the use of arubric, students’ strengths and weaknesses are identified and this information canthen be used as a basis for planning the next steps for learning. In this document, the Teacher’s Notes section indicates the reasons for assessing a student’s performanceat a specific level of achievement, and the Comments/Next Steps section indicatessuggestions for improvement.
In the exemplar project, a single rubric encompassing the four categories of knowl-edge and skills was used to provide an effective means of assessing the particular levelof student performance in the performance task, to allow for consistent scoring of student performance, and to provide information to students on how to improve their work. However, in the classroom, teachers may find it helpful to make use ofadditional rubrics if they need to assess student achievement on a specific task ingreater detail for one or more of the four categories. For example, it may be desirable in evaluating an oral report to use one rubric for assessing the content(Knowledge/Understanding), one for the research (Thinking/Inquiry/Problem Solving), one for the writing (Communication), and one for the delivery of the oralpresentation itself (Application).
The rubrics for the tasks in the exemplar project are similar to the scales used by theEducation Quality and Accountability Office (EQAO) for the Grade 3, Grade 6, andGrade 9 provincial assessments in that both the rubrics and the EQAO scales arebased on the Ontario curriculum expectations and the achievement charts. Therubrics differ from the EQAO scales in that they were developed to be used only inthe context of classroom instruction to assess achievement in a particular assignmentin a course.
7Introduct ion
Although rubrics were used effectively in this exemplar project to assess responsesrelated to the performance tasks, they are only one way of assessing student achieve-ment. Other means of assessing achievement include observational checklists, tests,marking schemes, or portfolios. Teachers may make use of rubrics to assess students’achievement on, for example, essays, reports, exhibitions, debates, conferences, inter-views, oral presentations, recitals, two- and three-dimensional representations, journals or logs, and research projects.
Development of the Tasks
The performance tasks for the exemplar project were developed by teams of subjectspecialists in the following way:
– The teams selected a cluster of curriculum expectations that focused on the knowl-edge and skills in the course that are considered to be of central importance in thesubject. Teams were encouraged to select a manageable number of expectations toenable teachers to focus their feedback to students. The particular selection ofexpectations ensured that all students in the course would have the opportunity to demonstrate their knowledge and skills in each category of the achievementchart in the curriculum policy document for the subject. Different tasks were developed for the academic courses and applied courses.
– For each course, the teams drafted two tasks that would encompass all of theselected expectations and that could be used to assess the work of all students inthe course. (Only one of these tasks would eventually be used for the final adminis-tration of the task.)
– The teams established clear, appropriate, and concrete criteria for assessment, and wrote the descriptions for each level of achievement in the task-specific rubric,using the achievement chart for the subject as a guide.
– The teams prepared detailed instructions for both teachers and students participating inthe assessment project.
– The two tasks were field-tested in classrooms across the province – one in the fall of 1999, the other in the winter of 2000 – by teachers who had volunteered toparticipate in the field test. Student work was scored by teams of teachers of thesubject. In addition, classroom teachers, students, and board contacts providedfeedback on the task itself and on the instructions that accompanied the task. Suggestions for improvement were taken into consideration in the revision of thetasks, and the feedback helped to determine which of the two tasks would actuallybe used for the final administration of the tasks in May 2000.
In developing the tasks, the teams ensured that the resources needed for completingthe task – that is, all worksheets and support materials – were provided. It was alsosuggested that students could consult the teacher-librarian at the school about addi-tional print and electronic materials.
Prior to both the field tests and the final administration of the tasks, a team of validators – including research specialists, gender and equity specialists, and subjectexperts – reviewed the instructions in the teacher and student packages, making further suggestions for improvement.
Assessment and Selection of the Samples
After the final administration of the tasks, student work was scored by trained boardpersonnel. The student samples were then forwarded to the ministry, where a team of teachers from across the province, who had been trained by the ministry to assessachievement on the tasks, scored and selected the student samples that would serve as the exemplars for each level of achievement.
The rubrics were the primary tool used to evaluate student work at both the districtschool board level and the provincial level. The samples that appear in this documentwere selected in the following way:
– At the district school board level, after some training was provided, teachers of thesubject evaluated and discussed the student work until they were able to reach aconsensus regarding the level to be assigned for achievement in each category. This evaluation was done to ensure that the student work being selected clearlyillustrated that level of performance.
– Student work was then sorted into two groups: (1) work that demonstrated thesame level of achievement in all four categories; and (2) work that demonstratedachievement at more than one level over the four categories.
– All the samples were submitted to a provincial selection team of teachers, who re-scored and validated the samples of work that demonstrated the same level ofachievement in all four categories, and chose, through consensus, two samples thatbest represented the characteristics of work at that level.
The following points should be noted:
– Two samples of student work are included for each of the four achievement levelsin each subject for which there is written work. The use of two samples is intendedto show that the characteristics of an achievement level can be exemplified in different ways.
– Although the samples of student work in this booklet were selected to show a level of achievement that was largely consistent in the four categories of Knowl-edge/Understanding, Thinking/Inquiry/Problem Solving, Communication, andApplication, teachers using rubrics to assess student work will notice that students’achievement frequently varies across the categories (e.g., a student may be achiev-ing at level 3 in Knowledge/Understanding but at level 4 in Communication).
– Although the student samples show responses to most questions, students achiev-ing at level 1 and level 2 will often omit answers or will provide incompleteresponses or incomplete demonstrations.
– Students’ effort was not evaluated. Effort is evaluated separately by teachers as partof the “learning skills” component of the Provincial Report Card.
– This document does not include any student samples that were assessed using therubrics and judged to be below level 1. (Work judged to be below level 1 is workon which a student achieves a mark of less than 50%. A student whose overallachievement at the end of a course is below 50% will not obtain a credit for thecourse.) Teachers are expected to work with students whose achievement is below level 1, as well as with their parents, to help the students improve their performance.
8 The Ontar io Curr iculum – Exemplars , Grade 9: Mathematics
9Introduct ion
Use of the Student Samples
Teachers and Administrators
The samples of student work included in this document will help teachers and administrators by:
– providing student samples and criteria for assessment that will enable them to helpstudents improve their achievement;
– providing a basis for conversations among teachers, parents, and students about thecriteria used for assessment and evaluation of student achievement;
– facilitating communication with parents regarding the curriculum expectations andlevels of achievement for each subject or course;
– promoting fair and consistent assessment within subjects and courses.
Teachers may choose to:
– use the teaching/learning activities outlined in the performance tasks;
– use the performance tasks and rubrics in this document in designing comparableperformance tasks;
– use the samples of student work at each level as reference points when assessingstudent work;
– use the rubrics to clarify what is expected of the students and to discuss the criteriaand standards for high-quality performance;
– review the samples of work with students and discuss how the performances reflectthe levels of achievement;
– adapt the language of the rubrics to make it more “student friendly”;
– develop other assessment rubrics with colleagues and students;
– help students describe their own strengths and weaknesses and plan their nextsteps for learning;
– share student work with colleagues for consensus marking;
– partner with other schools to design tasks and rubrics, and to select samples forother performance tasks and other subject areas.
Administrators may choose to:
– encourage and facilitate teacher collaboration regarding standards and assessment;
– provide training to ensure that teachers understand the role of the exemplars inassessment, evaluation, and reporting;
– establish an external reference point for schools in planning student programs andfor school improvement;
– facilitate sessions for parents and school councils using this document as a basis fordiscussion of curriculum expectations, levels of achievement, and standards;
– participate in future exemplar projects within their district school boards or withthe Ministry of Education.
10 The Ontar io Curr iculum – Exemplars , Grade 9: Mathematics
Parents
The performance tasks in this document exemplify a range of meaningful and relevantlearning activities related to the curriculum expectations for Grade 9 mathematicscourses. In addition, this document invites the involvement and support of parents as they work with their children to improve their achievement. Parents may use thesamples of student work and the rubrics as:
– resources to help them understand the levels of achievement;
– models to help monitor their children’s progress from level to level;
– a basis for communication with teachers about their children’s achievement;
– a source of information to help their children monitor achievement and improvetheir performance;
– models to illustrate the application of the levels of achievement.
Students
Students are asked to participate in performance assessments in all curriculum areas.When students are given clear expectations for learning, clear criteria for assessment,and immediate and helpful feedback, their performance improves. Students’ perfor-mance improves as they are encouraged to take responsibility for their own achieve-ment and to reflect on their own progress and “next steps”.
It is anticipated that the contents of this document will help students in the followingways:
– Students will be introduced to a model of one type of task that will be used toassess their learning, and will discover how rubrics can be used to improve theirproduct or performance on an assessment task.
– The performance tasks and the exemplars will help clarify the curriculum expecta-tions for learning.
– The rubrics and the information given in the Teacher’s Notes section will help clarify the assessment criteria.
– The information given under Comments/Next Steps will support the improvementof achievement by focusing attention on two or three suggestions for improvement.
– With an increased awareness of the performance tasks and rubrics, students will bemore likely to communicate effectively about their achievement with their teachersand parents, and to ask relevant questions about their own progress.
– Students can use the criteria and the range of student samples to help them see the differences in the levels of achievement. By analysing and discussing these differences, students will gain an understanding of ways in which they can assesstheir own responses and performances in related assignments and identify the qualities needed to improve their achievement.
Principles ofMathematicsAcademic
12 The Ontar io Curr iculum – Exemplars , Grade 9: Mathematics
Designing a Perfume Bottle
The Task
This task required students to:
• determine the maximum volume possible for a cylindricalbottle having a total surface area of less than 150 cm2;
• determine the volume of the bottle to the nearest 10 mL;
• report the dimensions of the bottle and the correspondingsurface area and volume.
For the specific problem as it was posed to students, see page 37of this document.
Students recorded the results of their investigation in a reportthat included an explanation of the process; a rationale for theselection of values (e.g., radius, height); a demonstrated knowl-edge of how to use and manipulate the formulas for surface areaand volume; and a conclusion that stated the maximum volumeto the nearest 10 mL, the dimensions rounded to one decimalplace, and the total surface area rounded to one decimal place.
Expectations
This task gave students the opportunity to demonstrate achieve-ment of the following selected expectations from two strands –Number Sense and Algebra, and Measurement and Geometry.
Number Sense and Algebra
Students will:
1. substitute into and evaluate algebraic expressions involvingexponents, to support other topics of the course;
2. communicate solutions to problems in appropriate mathemat-ical forms and justify the reasoning used in solving the prob-lems.
Measurement and Geometry
Students will:
3. determine the optimal values of various measurementsthrough investigations facilitated, where appropriate, by the use of concrete materials, diagrams, and calculators or computer software;
4. solve problems involving the surface area and the volume of three-dimensional objects;
5. identify, through investigation, the effect of varying the dimen-sions of a rectangular prism or cylinder on the volume or sur-face area of the object;
6. identify, through investigation, the relationships between the volume and surface area of a given rectangular prism orcylinder;
7. explain the significance of optimal surface area or volume invarious applications;
13 Principles of Mathematics , Academic
8. solve simple problems, using the formulas for the surface areaand the volume of prisms, pyramids, cylinders, cones, andspheres;
9. solve multi-step problems involving the volume and the sur-face area of prisms, cylinders, pyramids, cones, and spheres.
Prior Knowledge and Skills
To complete this task, students were expected to have someknowledge or skills relating to the following:
• the meaning of the concepts of surface area and volume andthe units in which surface area and volume are measured
• the formulas for the surface area and the volume of a cylinder
• the ability to use the formulas, including the ability to makecorrect substitutions
• the effective use of a scientific calculator (e.g., the use of thevalue for π instead of 3.14)
• rounding to the nearest 10 mL and to one decimal place
For information on the process used to prepare students for the taskand on the materials and equipment required, see the TeacherPackage reproduced on pages 37–41 of this document.
14 The Ontar io Curr iculum – Exemplars , Grade 9: Mathematics
Task Rubric – Designing a Perfume Bottle
Expectations*
5
2, 3, 5, 6,
2
1, 3, 4, 9
Criteria
– demonstrates an under-
standing of the effect on
surface area and volume
of varying dimensions
– selects values in the
problem-solving process
systematically
– demonstrates reasoning in
the selection of values
– forms accurate conclusions
– clearly identifies calcula-
tions and presents them in
a logical, organized way
– uses mathematical
language, symbols, and
units accurately
– demonstrates skill in the
application of concepts and
procedures (the formula)
when determining the optimal
size of the perfume bottle
Level 1
– demonstrates a limited
understanding of the effect
on surface area and volume
of varying dimensions
– selects values in a systematic
manner to a limited degree
– demonstrates limited
reasoning in the selection
of values
– forms conclusions with
limited accuracy
– identifies and presents cal-
culations with limited clarity
– uses language, symbols, and
units with limited accuracy
– demonstrates limited skill in
the application of the formula
Level 2
– demonstrates some under-
standing of the effect on sur-
face area and volume of
varying dimensions
– selects values in a some-
what systematic manner
– demonstrates some reason-
ing in the selection of values
– forms conclusions with
some accuracy
– identifies and presents cal-
culations with some clarity
– uses language, symbols, and
units with some accuracy
– demonstrates some skill in
the application of the formula
Level 3
– demonstrates considerable
understanding of the effect
on surface area and volume
of varying dimensions
– selects values in a generally
systematic manner
– demonstrates considerable
reasoning in the selection of
values
– forms conclusions with
considerable accuracy
– identifies and presents cal-
culations with considerable
clarity
– uses language, symbols,
and units with considerable
accuracy
– demonstrates considerable
skill in the application of the
formula
Level 4
– demonstrates a thorough
understanding of the effect
on surface area and volume
of varying dimensions
– selects values in a highly
systematic manner
– demonstrates a high degree
of reasoning in the selection
of values
– forms conclusions with a
high degree of accuracy
– identifies and presents cal-
culations with a high degree
of clarity
– uses language, symbols, and
units with a high degree of
accuracy
– demonstrates a high degree
of skill in the application of
the formula
Knowledge/Understanding
The student:
Thinking/Inquiry/Problem Solving
The student:
Communication
The student:
Application
The student:
* The expectations that correspond to the numbers given in this chart are listed on page 12. Note that, although all of the expectations listed there were addressed through instruction relating to
the task, student achievement of expectations 7 and 8 was not assessed in the final product.
Note: A student whose overall achievement at the end of a course is below level 1 (that is, below 50%) will not obtain a credit for the course.
Principles of Mathematics , Academic15
Designing a Perfume Bottle Level 1, Sample 1
A B
Comments/Next Steps– The student’s use of decimal values for radius and height shows insight,
but radius and height should be investigated in a more systematic manner,
specifically by fixing volume or surface area.
– The student could check for accuracy when selecting values to substitute
into formulas.
– The student could demonstrate more reasoning in the selection of values.
Teacher’s Notes
Knowledge/Understanding– The student demonstrates a limited understanding of the effect on surface
area and volume of varying radius and height.
Thinking/Inquiry/Problem Solving– The student selects values in a systematic manner to a limited degree,
selecting only a small number of decimal values for radius and height.
– The student demonstrates limited reasoning in the selection of values.
– The student forms conclusions with limited accuracy (e.g., states the
height but not the radius; describes volume inaccurately).
Communication– The student uses units and symbols with limited accuracy (e.g., has units in the
chart but does not always use them appropriately in discussion or in the sample
calculation).
Application– The student demonstrates limited skill in the application of the formula
(e.g., the surface area calculation contains the value 0.8, which was likely
meant to be 8.0, rounded from 7.9).
16 The Ontar io Curr iculum – Exemplars , Grade 9: Mathematics
Principles of Mathematics , Academic17
Designing a Perfume Bottle Level 1, Sample 2
Teacher’s Notes
Knowledge/Understanding– The student demonstrates a limited understanding of the effect on surface
area and volume of varying dimensions (e.g., by fixing the radius at 2.5 cm
and then using a limited selection of heights).
Thinking/Inquiry/Problem Solving– The student selects values in a systematic manner only to a limited degree
(e.g., the radius is fixed but the height selections are random).
– The student forms conclusions with limited accuracy.
– The student demonstrates limited reasoning in the selection of values
(e.g., focuses on surface area without regard to maximizing the value of
the volume).
Communication– The student presents calculations with limited clarity (e.g., shows the
calculation steps but does not identify the formulas in the sample calculation).
– The student’s use of language and symbols has limited accuracy.
– In the student’s explanation, the radius appears to have been set at 2.5 cm;
however, in the chart, calculations based on that radius do not work out.
For example, for radius 2.5 cm and height 9 cm:
V = πr 2h SA = 2πr 2 + 2πrh
= 176.7 cm3 = 180.6 cm2
= 129.5 cm3 (student solution) = 149.8 cm2 (student solution)
Application– The student demonstrates limited skill in the application of the formula
(e.g., does not provide the general form of the formula but makes the correct
substitutions in the samples).
Comments/Next Steps– The student could improve the overall solution by using a more systematic
approach involving both surface area and volume with a more in-depth
investigation.
– The student could clearly identify concluding statements with a title or add
the word “conclusion” in the statements.
● ●
Designing a Perfume Bottle Level 2, Sample 1
18 The Ontar io Curr iculum – Exemplars , Grade 9: Mathematics
A B
C
Principles of Mathematics , Academic19
D
E F
20 The Ontar io Curr iculum – Exemplars , Grade 9: Mathematics
Comments/Next Steps– The student needs to be more precise in making calculations, and should
show the calculations clearly in the written report.
– The student could use mathematical language with more clarity
(e.g., “I then proceded to minus 1 (one column at a time) until I had tried
5 cylinders”).
Teacher’s Notes
Knowledge/Understanding– The student demonstrates some understanding of the effect on surface area
and volume of some random varying of radius and height, as shown in the
chart.
Thinking/Inquiry/Problem Solving– The student forms conclusions with some accuracy on the basis of
calculations made.
– The student demonstrates some reasoning in a somewhat systematic selection
of values through trial and error and then a narrowing down of choices for the
radius and the height.
Communication– The student presents calculations with some clarity, as shown in the rough
work.
– The student uses units with some accuracy by including them in the chart
but leaves some units out in the written explanation.
Application– The student demonstrates some skill in the application of the formula but
omits some steps (e.g., does not show the value for the radius; does rounding
early in the substitution of values).
Principles of Mathematics , Academic21
22 The Ontar io Curr iculum – Exemplars , Grade 9: Mathematics
A B
Designing a Perfume Bottle Level 2, Sample 2
C
Principles of Mathematics , Academic23
D
Comments/Next Steps– The student might consider varying the radius to decimal values for a
volume of 140 mL.
– The student should notice that the differences between the computer
solutions and the student’s sample calculations result from using the
π button in the first instance and 3.14 for π in the second.
Teacher’s Notes
Knowledge/Understanding– The student demonstrates some understanding of the effect on surface area
and volume of some varying of dimensions (e.g., provides the variations of
the radius in integer values only; considers a small number of volumes to
determine variations in height).
Thinking/Inquiry/Problem Solving– The student selects values (e.g., for the radius and the height) in a some-
what systematic manner.
– The student demonstrates some reasoning in the selection of values
(e.g., satisfies the conditions of the problem by settling on values for
the radius and the volume).
Communication– The student uses units and mathematical language with some accuracy in
the discussion.
– The student uses symbols with some accuracy, except in the charts.
Application– The student demonstrates some skill in the application of the formula in the
sample calculations (e.g., uses brackets).
24 The Ontar io Curr iculum – Exemplars , Grade 9: Mathematics
Principles of Mathematics , Academic25
A B
Designing a Perfume Bottle Level 3, Sample 1
C
Teacher’s Notes
Knowledge/Understanding– The student demonstrates considerable understanding of the effect on
surface area, volume, and height of varying the radius.
Thinking/Inquiry/Problem Solving– The student shows a generally systematic selection of volumes from 100 mL
to 140 mL, in increments of 10 mL.
– The student demonstrates considerable reasoning in the selection of values
(e.g., “The two bottle designs still can be produced but design #2 looks like
it might be reaching its limit”).
– The student forms conclusions with considerable accuracy.
Communication– The student uses a combination of charts and words with considerable clarity.
– The student uses language, symbols, and units with considerable accuracy
(e.g., the unit symbols are appropriately included in the headings of charts,
at the end of the calculations, and in the conclusion).
Application– The student demonstrates considerable skill in the application of the
formula (e.g., shows a good use of brackets and of the order of operations).
Comments/Next Steps– The student could improve the accuracy of rounding (e.g., in the sample
calculation, the result of π ✕ 4 ✕ 7 should be rounded to 88.0. not 87.9).
– The student might select a sample volume of 150 mL to confirm that all pos-
sible solutions have been considered.
– In the selection of radius values, the student should consider using decimal
values.
– In the conclusion, the student could state the dimensions rounded to one
decimal place, as outlined in the instructions.
– The student could use brackets for substitution. For example:
V = πr 2h could be V = πr 2h
= π ✕ 32✕ 6 = π(3)2(6)
– The student could state the values of the variables used in the formulas.
For example: r = 3
h = 6
V = ...
26 The Ontar io Curr iculum – Exemplars , Grade 9: Mathematics
Principles of Mathematics , Academic27
Designing a Perfume Bottle Level 3, Sample 2
A B
C
Teacher’s Notes
Knowledge/Understanding– The student demonstrates considerable understanding of the effect on
surface area and volume of varying dimensions (e.g., uses radii from 1 cm
to 10 cm).
Thinking/Inquiry/Problem Solving– The student selects values in a generally systematic manner (e.g., tries
volumes from 100 mL to 150 mL).
– The student demonstrates considerable reasoning in the selection of values.
– The results show that the student forms conclusions with considerable
accuracy.
Communication– The student identifies and presents calculations with considerable clarity
(e.g., uses brackets appropriately).
– The student uses language, symbols, and units with considerable accuracy
(e.g., uses correct units in the chart headings, states the given information
before substituting into formulas, uses symbols consistently, calculates
samples correctly for the most part).
Application– The student demonstrates considerable skill in the application of the formula
(i.e., manipulates formulas correctly; substitutes all numbers correctly).
Comments/Next Steps– The student would improve the accuracy of the solution by making further
investigation of radii of approximately 3 cm (e.g., 2.5 cm, 2.6 cm).
– The student’s concluding statement should include all the required information,
with the requested degree of accuracy.
– The student could proofread for spelling.
– The surface area in the student’s sample calculation should be 149.86, not
149.88.
– The student could improve the consistency of rounding (e.g., could round
consistently either to one decimal place or to two decimal places).
28 The Ontar io Curr iculum – Exemplars , Grade 9: Mathematics
Principles of Mathematics , Academic29
A B
Designing a Perfume Bottle Level 4, Sample 1
C D
30 The Ontar io Curr iculum – Exemplars , Grade 9: Mathematics
E F
Principles of Mathematics , Academic31
G H
32 The Ontar io Curr iculum – Exemplars , Grade 9: Mathematics
Comments/Next Steps– The student could include the height in the written conclusion.
– The student provides a net diagram that includes all the appropriate
measurements.
Teacher’s Notes
Knowledge/Understanding– The student demonstrates, by a thorough investigation, a complete under-
standing of the effect on surface area and volume of varying radii and their
corresponding heights (e.g., chooses radii ranging from 1.0 cm up to 4.5 cm,
with the appropriate heights).
Thinking/Inquiry/Problem Solving– The student selects values in a highly systematic manner to arrive at a final
answer.
– The student demonstrates a high degree of reasoning in the selection of
values (e.g., “I had decided that if these variables did not come up with total
surface areas of less than 150 cm2 , I would change my methods”).
– The student forms highly accurate conclusions (e.g., realizes that the height
and the diameter would be equal for an optimum container).
Communication– The student uses technology to present calculations with a high degree of
clarity.
– The student uses language, symbols, and units with a high degree of accuracy
(e.g., the symbols in the net diagram).
Application– The student demonstrates a high degree of skill in the application of the formula.
Principles of Mathematics , Academic33
Designing a Perfume Bottle Level 4, Sample 2
34 The Ontar io Curr iculum – Exemplars , Grade 9: Mathematics
A B
C
Principles of Mathematics , Academic35
D
Teacher’s Notes
Knowledge/Understanding– The student demonstrates a thorough understanding of the effect on volume
and surface area of varying dimensions through his or her insightful recognition
that increasing the radius beyond 3 cm results in a decreasing volume.
Thinking/Inquiry/Problem Solving– The student uses a highly systematic approach in selecting radii.
– The student demonstrates a high degree of reasoning in selecting radii
between 2 cm and 3 cm.
Communication– The student identifies and presents the sample calculations with a high
degree of clarity.
– The student consistently uses language, symbols, and units with a high
degree of accuracy.
Application– The student demonstrates a high degree of skill in the application of the formula.
– The student uses π and not 3.14 in the calculations.
Comments/Next Steps– The student could improve spelling through proofreading by self or a peer.
– The student could improve consistency in rounding calculations.
– The student should use a consistent number of significant digits.
– The student should use brackets instead of multiplication signs in
substitutions.
– The student should round only at the end of a calculation.
36 The Ontar io Curr iculum – Exemplars , Grade 9: Mathematics
E
Principles of Mathematics , Academic37
Mathematics Exemplar TaskGrade 9 Principles of Mathematics, Academic
Teacher Package
Title: Designing a Perfume Bottle
Time requirement: Up to 320 minutes
Preparation for the task: Up to 110 minutes
The student task: 210 minutes (70 minutes ✕ 3)
Description of the TaskThe Goodsmell perfume producing company has a new line of perfume and
is designing a new bottle for it. Because of the expense of the glass required
to make the bottle, the surface area must be less than 150 cm2. The company
also wants the bottle to hold at least 100 mL of perfume. The design under
consideration is in the shape of a cylinder.
Determine the maximum volume possible for a cylindrical bottle that has a
total surface area of less than 150 cm2. Determine the volume to the nearest
10 mL. Report the dimensions of the bottle and the corresponding surface
area and volume.
Final ProductPrepare a written report that includes:
• a clear and complete explanation of the process that you used to solve
the problem
• the calculations that you made, presented in an organized fashion
• a rationale for your selection of values (e.g., radius, height)
• if you used technology, sample calculations of surface area and volume to demonstrate that you know how to use and manipulate the
formulas (e.g., show substitution into formulas as if technology were
not used)
• a conclusion that states the maximum volume to the nearest 10 mL,
the dimensions rounded to one decimal place, and the total surface
area rounded to one decimal place
Note: Students do not need to indicate calculator or computer
instructions as part of the process of preparing their final reports.
Assessment and Evaluation• You will need to use the task-specific rubric* to assess and evaluate
the student work.
• In order to score the application row of the rubric if the student used
technology, evaluate the sample calculations for surface area and
volume that the student was asked to provide in the written report.
Expectations Addressed in the Exemplar Task
Teacher Package
Number Sense and AlgebraStudents will:
1. substitute into and evaluate algebraic expressions involving
exponents, to support other topics of the course;
2. communicate solutions to problems in appropriate mathematical
forms and justify the reasoning used in solving the problems.
(continued)
* The rubric is reproduced on page 14 of this document.
38 The Ontar io Curr iculum – Exemplars , Grade 9: Mathematics
Note that, although all of the expectations listed will be addressed
through instruction relating to the task, student achievement of expecta-
tions 7 and 8 will not be assessed in the final product.
Teacher Instructions
Prior Knowledge and Skills RequiredTo complete this task, students are expected to have some knowledge or skills
relating to the following:
• the meaning of the concepts of surface area and volume and the units
in which surface area and volume are measured;
• the formulas for the surface area and the volume of a cylinder;
• the ability to use the formulas, including the ability to make correct
substitutions;
• the effective use of a scientific calculator (e.g., the use of the value for
π instead of 3.14);
• rounding to the nearest 10 mL and to one decimal place.
AccommodationsAccommodations that are normally provided in regular classrooms for
students with special needs should be provided in the administration of
this performance task.
You may wish to review the relevant course profile for specific suggestions
for accommodations appropriate for students in special education programs.
Materials and Resources Required• Student instructions
• Use of a computer lab, a set of graphing calculators, or a set of
scientific calculators
Measurement and GeometryStudents will:
3. determine the optimal values of various measurements through
investigations facilitated, where appropriate, by the use of con-
crete materials, diagrams, and calculators or computer software;
4. solve problems involving the surface area and the volume of
three-dimensional objects;
5. identify, through investigation, the effect of varying the dimen-
sions of a rectangular prism or cylinder on the volume or
surface area of the object;
6. identify, through investigation, the relationships between the
volume and surface area of a given rectangular prism or cylinder;
7. explain the significance of optimal surface area or volume in
various applications;
8. solve simple problems, using the formulas for the surface area
and the volume of prisms, pyramids, cylinders, cones, and
spheres;
9. solve multi-step problems involving the volume and the sur-
face area of prisms, cylinders, pyramids, cones, and spheres.
Principles of Mathematics , Academic39
PreparationThe following are suggestions and instructions about the students’ use of
technology in completing the task:
• The exemplar task lends itself to the use of a spreadsheet, and spread-
sheet use should be encouraged whenever possible.
• If the use of a spreadsheet is not feasible, then students should use
scientific calculators, and it is recommended that students work in
pairs to share the calculation load.
• If students work in pairs, only one solution per pair should be handed in.
• Regardless of the calculation method used, students must report their
calculations.
• If students use technology in completing work that is to be assessed
extermally, it can be difficult to ascertain whether they understand how
to use and manipulate formulas. For this reason, students are asked to
provide sample calculations of surface area and volume so that the
teacher can see whether students are able to substitute appropriately.
The following are comments about a sample solution of the exemplar
task, which has been included in the Teacher Package for the use of theteacher only:• A sample solution for the performance task is provided as the last
section of this Teacher Package.
• The sample solution does not represent a model for student answers. It
is only meant to show how a student mightapproach the problem.
• The sample solution was completed by means of a spreadsheet.
• The sample solution represents one strategy for finding a solution;
students may use different or more sophisticated strategies.
• To familiarize yourself with the rubric for the exemplar task, you may
wish to use the rubric to assess the model solution.
Please do not share the sample solution with students.
Note: Students may provide answers within an acceptable range, depend-
ing on the investigation, line of reasoning, or strategy they pursue, or the
value of π they use.
RubricIntroduce the task-specific rubric to the students at least one day before
the administration of the task. Review the rubric with the students,allow-
ing ample class time to ensure that each student understands the criteria
and the descriptions for achievement at each level.
Some students may perform below level 1. It will be important to note the
characteristics of their work in relation to the criteria in the assessment
rubric and to provide feedback to help them improve.
Task Instructions
Pre-task Activities: Up to 110 minutesIf you have not yet covered surface area and volume of a cylinder:
• review the meaning of the concepts of surface area and volume and
the units in which surface area and volume are measured;
• introduce the information for the surface area and volume of a cylinder;
• demonstrate the use of formulas, including correct substitution, and the
effective use of a scientific calculator.
If you have not yet covered optimal value of surface area and volume:
• try a sample problem (e.g., determine the minimum surface area for a
cylinder having a volume of 300 mL);
• let students work in pairs, attempting to solve the problem;
40 The Ontar io Curr iculum – Exemplars , Grade 9: Mathematics
• discuss the methods students used to solve the problem (encourage the
discussion of many different methods) and the conclusions that they
reached;
• discuss the importance of a systematic and efficient method for
examining values;
• hand out the assessment rubric that will be used with the actual
exemplar task;
• discuss the assessment rubric in conjunction with the students’
solutions to the sample problem just completed.
The Task: 210 minutes• Hand out the student package.
• Read through with students the section “Description of the Exemplar
Task”.
• Have the students read the task silently.
• Answer any questions they may have about the activity.
• Remind students about the rubric, and ensure that each student has a
copy of it.
• Set students to work on the task, allowing 210 minutes for completion.
Assessment Instructions• Review the rubric with students at the beginning of each day.
• Remind students who have used technology to include in their report
the sample calculations for surface area and volume.
SAMPLE SOLUTION
FOR TEACHER USE ONLY!!
The Search for Maximum Volume With Total Surface Area LessThan 150 cm2
r (cm) h (cm) TSA V (mL)(cm2)
1.00 31.83 206.28 100.002.00 7.96 125.133.00 3.54 123.22 ***4.00 1.99 150.535.00 1.27 197.086.00 0.88 259.537.00 0.65 336.458.00 0.50 427.129.00 0.39 531.16
10.00 0.32 648.32
r (cm) h (cm) TSA V (mL)(cm2)
1.00 63.66 406.28 200.002.00 15.92 225.133.00 7.07 189.884.00 3.98 200.535.00 2.55 237.086.00 1.77 292.867.00 1.30 365.028.00 0.99 452.129.00 0.79 553.38
10.00 0.64 668.32
r (cm) h (cm) TSA V (mL)(cm2)
1.00 47.75 306.28 150.002.00 11.94 175.133.00 5.31 156.554.00 2.98 175.535.00 1.91 217.086.00 1.33 276.197.00 0.97 350.738.00 0.75 439.629.00 0.59 542.27
10.00 0.48 658.32
r (cm) h (cm) TSA V (mL)(cm2)
1.00 41.38 266.28 130.002.00 10.35 155.133.00 4.60 143.22 *******4.00 2.59 165.535.00 1.66 209.086.00 1.15 269.537.00 0.84 345.028.00 0.65 434.629.00 0.51 537.83
10.00 0.41 654.32
Principles of Mathematics , Academic41
My problem solving process:I set the volume at 100 mL because the question said that this was the
least it could be. Then I chose r values from 1 to 10. I rearranged the
volume formula so that I could calculate h for each value of r. Then I
substituted each r and h into the total surface area formula. I looked to
make sure that a surface area of less than 150 cm2 was possible.
Then I repeated this for volume = 200 mL. This time, a surface area less
than 150 cm2 was not possible. So then I repeated the calculation for
volume = 150 mL. Still not possible.
After that, I tried 130 mL, which worked. And then I tried 140 mL. It
also worked. I was supposed to find the maximum volume to the nearest
10 mL. So it must be 140 mL.
Sample calculations:
for r = 3 cm, h = 4.95 cm
Surface Area = 2πr2 + 2πrh Volume = πr2h
= 2π(3)2 + 2π(3)(4.95) = π(3)2(4.95)
= 56.5 + 93.3 = 139.96 – rounded to 140 mL
= 149.8 cm2
My conclusion is:A maximum volume of 140 mL occurs when r = 3.0 cm and h = 5.0 cm
(rounded from 4.95 cm). The surface area is 149.8 cm2.
r (cm) h (cm) TSA V (mL)(cm2)
1.00 44.56 286.28 140.002.00 11.14 165.133.00 4.95 149.88This one
works!4.00 2.79 170.535.00 1.78 213.086.00 1.24 272.867.00 0.91 347.888.00 0.70 437.129.00 0.55 540.05
10.00 0.45 656.32
Foundations ofMathematicsApplied
44 The Ontar io Curr iculum – Exemplars , Grade 9: Mathematics
Footprints on the Road
The Task
This task required students to:
• engage in small group and whole class discussions to generateproblems involving the investigation of relationships betweenthings like foot size, stride, leg length, and/or height;
• engage in whole class activity to identify data needed for theinvestigations, to collect the data from the whole class, and to prepare a class summary sheet of data;
• work individually to select a problem, analyse the data, andwrite a report.
Each student analysed a set of data and wrote a report. The reportincluded a clear statement of the problem, a hypothesis of thesolution, an explanation of factors that might have affected thevalidity and accuracy of the data gathered, a table of values and a scatter plot, a description of the dispersion of the data in thescatter plot, a conclusion that made direct reference to the analysisof the data, and reference to another situation involving a relation-ship between two variables.
Expectations
This task gave students the opportunity to demonstrate achieve-ment of the following selected expectations from the Relationshipsstrand.
Relationships
Students will:
1. determine relationships between two variables by collectingand analysing data;
2. compare the graphs of linear and non-linear relations;
3. describe the connections between various representations ofrelations;
4. pose problems, identify variables, and formulate hypothesesassociated with relationships;
5. demonstrate an understanding of some principles of samplingand surveying and apply the principles in designing and carry-ing out experiments to investigate the relationships betweenvariables;
6. collect data, using appropriate equipment and/or technology;
7. organize and analyse data, using appropriate techniques andtechnology;
8. describe trends and relationships observed in data, makeinferences from data, compare the inferences with hypothesesabout the data, and explain the differences between the inferencesand the hypotheses;
9. communicate the findings of an experiment clearly and con-cisely, using appropriate mathematical forms;
10. demonstrate an understanding that straight lines representlinear relations and curves represent non-linear relations.
45 Foundations of Mathematics , Appl ied
Prior Knowledge and Skills
To complete this task, students were expected to have someknowledge or skills relating to the following:
• the ability to describe relationships
• a knowledge of variables that can affect relationships
• the ability to construct graphs by hand
• a knowledge of scatter plots and their interpretation
• the ability to write reports
For information on the process used to prepare students for the taskand on the materials and resources required, see the Teacher Packagereproduced on pages 67–71 of this document.
46 The Ontar io Curr iculum – Exemplars , Grade 9: Mathematics
Task Rubric – Footprints on the Road
Expectations*
5, 10
1, 3, 7, 8
7, 9
6
Criteria
– demonstrates understand-
ing of a mathematical rela-
tion by stating a working
hypothesis
– identifies and explains factors
that might affect the validity
and accuracy of the data
collected
– describes the dispersion of
the data with reference to
trends and relationships
– draws a valid conclusion
and makes an argument for
the relationship stated
– prepares an organized and
clear report, using appro-
priate mathematical forms
– communicates graphically,
using proper form
– identifies and describes
a realistic application
Level 1
– demonstrates a limited under-
standing of a relational
hypothesis
– identifies factors and explains
them with a limited rationale
– describes the dispersion of
the data with limited accuracy
and detail
– makes an argument for the
relationship in the conclusion
in limited detail
– demonstrates limited organ-
ization and clarity
– communicates graphically,
demonstrating limited skill
in the use of proper form
– identifies and describes
a realistic application in
limited detail
Level 2
– demonstrates some under-
standing of a relational
hypothesis
– identifies factors and
explains them with a some-
what effective rationale
– describes the dispersion of
the data with some accuracy
and detail
– makes an argument for the
relationship in the conclu-
sion in some detail
– demonstrates some organi-
zation and clarity
– communicates graphically,
demonstrating some skill
in the use of proper form
– identifies and describes
a realistic application in
some detail
Level 3
– demonstrates considerable
understanding of a relational
hypothesis
– identifies factors and explains
them with a considerably
effective rationale
– describes the dispersion of
the data with considerable
accuracy and detail
– makes an argument for the
relationship in the conclusion
in considerable detail
– demonstrates considerable
organization and clarity
– communicates graphically,
demonstrating considerable
skill in the use of proper form
– identifies and describes
a realistic application in
considerable detail
Level 4
– demonstrates a thorough
understanding of a relational
hypotheses
– identifies factors and explains
them with a highly effective
rationale
– describes the dispersion of
the data with a high degree
of accuracy and detail
– makes an argument for the
relationship in the conclusion
in a high degree of detail
– demonstrates a high degree
of organization and clarity
– communicates graphically,
demonstrating a high degree of
skill in the use of proper form
– identifies and describes
a realistic application in
thorough detail
Knowledge/Understanding
The student:
Thinking/Inquiry/Problem Solving
The student:
Communication
The student:
Application
The student:
* The expectations that correspond to the numbers given in this chart are listed on page 44. Note that, although all of the expectations listed there were addressed through instruction relating to
the task, student achievement of expectations 2 and 4 was not assessed in the final product.
Note: A student whose overall achievement at the end of a course is below level 1 (that is, below 50%) will not obtain a credit for the course.
47 Foundations of Mathematics , Appl ied
Student Instructions
Part I: Read the following text
• In your group of three or four students, brainstorm at least three ques-
tions that could be answered through further investigation, based on
the conversation between Henri and Kim.
• Share these questions with the whole class.
• As part of a whole class discussion, identify problems for research
arising from the discussions.
• Participate in gathering data relevant to the problems for the class
summary data sheet.
Two friends, Henri and Kim, were walking along a dirt roadafter a heavy rain. They noticed that they were leavingwell-defined sets of footprints and began talking aboutthe differences in their tracks.
“Our footprints are about the same size,” said Henri, “butmy stride is longer than yours.”
“That’s true,” replied Kim, “but notice that I am keepingup with you while we walk.”
The two friends continued walking and talking, discussingthe information that might be gained about someone, justby examining their footprints in the dirt.
Part II: Final Report
Hand in a written report containing the information required in each
step below. Do all work in good form and write all explanations in
complete sentences.
1. a) Select one of the approved research problems identified from the
discussions in Part 1. Make a clear statement of the problem.
b) Hypothesize a solution to the problem.
c) Include an explanation of factors that might affect the validity
and accuracy of the data gathered.
2. a) Construct a table of values and a scatter plot.
b) Describe the dispersion of the data in the scatter plot (i.e., do the
pieces of data cluster into a recognizable trend or are they random-
ly distributed?). If you see a trend, identify it. Explain the presence
of any pieces of data that do not fit the trend.
3. a) Write a conclusion to the problem, making direct reference to
your analysis of the data.
b) If you believe a relationship exists, describe it as completely as
possible, so that someone reading your work could make a pre-
diction about himself/herself.
4. You have examined a relationship between the two variables.
Describe another situation in which it would be important to identify
a relationship between two variables, and explain the importance.
48 The Ontar io Curr iculum – Exemplars , Grade 9: Mathematics
Footprints on the Road Level 1, Sample 1
A B
Foundations of Mathematics , Appl ied
C
49
Teacher’s Notes
Knowledge/Understanding– The student provides a hypothesis that demonstrates a limited understanding
of the spectrum of the relationship (e.g., hypothesizes that “if you have big
feet you will have a big stride length”, with no mention of the range “bigger”
or its opposite “smaller”).
– The student states factors with limited explanation (e.g., uses “size of shoes”
to imply the size of the feet and “male or female” to imply the difference
between genders in overall size that results in different stride lengths).
Thinking/Inquiry/Problem Solving– The student describes the dispersion of the data with limited accuracy
(e.g., uses “close together” as a description of the trend).
– The student provides limited detail in the argument for the pieces of data that
do not fit the trend (e.g., “some people are taller than others” addresses one
type of outlier but does not describe the effect of outliers in the relationship).
– The student’s explanation of the relationship in the conclusion restates the
hypothesis with limited detail, and omits any reference to the data (i.e., the
table of values or the graph), any mention of a spectrum, and any provision
for predictability.
Communication– The student demonstrates limited organization and clarity in reasoning (e.g., the
lack of logical sequence and layout contributes to a lack of flow; the graph
is not with the table of values; the steps are not numbered in sequence).
– The student demonstrates limited skill in the use of proper form when com-
municating graphically (e.g., shows limited accuracy in the plotting of points;
constructs an inaccurate y-axis; omits the scale and the labels or units).
Application– The student identifies a realistic relationship but provides limited detail in
explaining the importance of studying the relationship (e.g., “Tall people
are gonna have big feet”).
Comments/Next Steps– The student should move from simply stating the factors and the presence
of outliers to explaining them.
– The student must complete all components of the task (e.g., allow for pre-
dictability in the conclusion [question 3] and explain the importance of the
new situation [question 4]).
– The graphs should include appropriate and correct scales, labels, and units.
– The student would improve the flow and clarity of the information provided
by reporting in an organized way the reasoning used.
50 The Ontar io Curr iculum – Exemplars , Grade 9: Mathematics
Foundations of Mathematics , Appl ied51
Footprints on the Road Level 1, Sample 2
A B
C
Teacher’s Notes
Knowledge/Understanding– The student makes a hypothesis about the investigation but demonstrates a
limited understanding of a relationship (e.g., gives no indication of exceptions
to the rule “If your foot size is big then your height will be high”).
– The student refers to factors that might affect the validity of the data (e.g., “be
consistent”, “be precise”) but provides limited explanation of their importance.
Thinking/Inquiry/Problem Solving– The student describes, with limited accuracy and detail, the dispersion of
the data collected.
– The student states the trend and the conclusion, providing only limited
detail in his or her argument.
Communication– The student demonstrates limited reasoning in reporting (e.g., presents the
graph first and the data table at the end).
– The student communicates graphically, demonstrating limited skill in the
use of proper form (e.g., does not present one of the data points, because
of the range chosen).
Application– The student states a realistic relationship with limited and unclear detail as
to its importance.
Comments/Next Steps– The student should include labels, titles, and units when constructing
graphs.
– The student could reorder various parts of the report to reflect improved
reasoning.
– Throughout the report, the student must provide more explanation and
detail.
52 The Ontar io Curr iculum – Exemplars , Grade 9: Mathematics
Foundations of Mathematics , Appl ied53
Footprints on the Road Level 2, Sample 1
A B
C
Teacher’s Notes
Knowledge/Understanding– The student demonstrates some understanding of a relational hypothesis by
looking for a “connection between” the variables.
– The student shows knowledge of a continuum (e.g., “as the hieght increases . . .”)
– The student’s rationale in identifying and explaining factors that affect the
validity of the data collected is somewhat effective.
Thinking/Inquiry/Problem Solving– The student demonstrates some skill in accurately recognizing the random
dispersion of the data (e.g., “There is no recognizable trend”).
– The student draws a conclusion from the data and addresses the relational
contradiction between the hypothesis and the conclusion with some detail
(e.g., argues for “more information from a larger age range”).
Communication– The student chooses a scale that shows the data with some organization
and clarity.
– The student communicates graphically, demonstrating some skill in the use
of proper form, but uses breaks in the axes inappropriately.
Application– The student identifies a realistic application with some detail (e.g., “the
older you are the more worn out your heart gets”).
Comments/Next Steps– The student should include an appropriate title for the graph and should be
careful to identify units in the table of values and the graph.
– The student needs to give more detail in the explanations and the reasoning.
54 The Ontar io Curr iculum – Exemplars , Grade 9: Mathematics
Foundations of Mathematics , Appl ied55
Footprints on the Road Level 2, Sample 2
A B
C
56 The Ontar io Curr iculum – Exemplars , Grade 9: Mathematics
Teacher’s Notes
Knowledge/Understanding– The student states a hypothesis that includes only one end of a spectrum,
demonstrating some understanding of a relational hypothesis.
– The student identifies some factors that affect the validity and accuracy of
the data collected and gives only some explanation of those factors.
Thinking/Inquiry/Problem Solving– The student uses a line of best fit to indicate that a trend exists and describes
the dispersion with some accuracy and detail (e.g., “the data . . . are ramdonly
distributed”).
Communication– The student communicates graphically, showing some skill in graph construction
(e.g., uses appropriate axes and scales but omits a title and some necessary
labels and units).
– The student demonstrates some organization and clarity of reasoning (e.g.,
“Anothe situation in which it would be inportant to identify a relation ship
between two variables are height and leg length”).
Application– The student identifies a realistic application.
– The student states some detail in the explanation about the application
(e.g., “the taller you are the longer the legs”).
Comments/Next Steps– The student should include appropriate titles, labels, and units when graphing.
– The student could provide more detail in explanations to better convey the
reasoning.
Foundations of Mathematics , Appl ied57
Footprints on the Road Level 3, Sample 1
A B
58 The Ontar io Curr iculum – Exemplars , Grade 9: Mathematics
C D
Foundations of Mathematics , Appl ied59
Teacher’s Notes
Knowledge/Understanding– The student demonstrates considerable understanding of the relational
hypothesis (e.g., “the longer your legs are, the longer your stride will be”).
– The student identifies factors that may affect the validity of the data col-
lected and explains with considerable effectiveness how they may do so.
Thinking/Inquiry/Problem Solving– The student describes the dispersion of the data with considerable accu-
racy through the text (e.g., addresses one ordered pair that does not fit
the trend and provides a possible reason for its extremity).
– The student provides, with considerable detail, a conclusion that allows
for predictability (e.g., “If you have longer legs your stride would be bigger
than some one with shorter legs”).
Communication– The student demonstrates considerable organization and clarity of reasoning.
– The student communicates graphically, demonstrating considerable skill
in graph construction (e.g., plots the ordered pairs accurately).
Application– The student describes a realistic application, using considerable detail
(e.g., “how much you practice deturmins on how well you play”).
Comments/Next Steps– The student could improve the accuracy of the work done, include more
details, and provide a better explanation for exceptions.
60 The Ontar io Curr iculum – Exemplars , Grade 9: Mathematics
Footprints on the Road Level 3, Sample 2
A B
Foundations of Mathematics , Appl ied61
C
Teacher’s Notes
Knowledge/Understanding– The student demonstrates considerable understanding of a relational
hypothesis by formulating a working hypothesis of the relationships
between leg length and stride length.
– The student identifies three factors that might affect the validity of the
data collected, and gives considerably effective explanations for them.
Thinking/Inquiry/Problem Solving– The student describes the dispersion of the data with considerable accuracy
(e.g.,“bunched up in the top, right hand corner”).
– The student considers data that do not fit the trend and gives reasons
(e.g.,“bad measuring or any of the other explanations that I mentioned in
my hypothesis”).
– The student draws a valid conclusion and makes an argument for the rela-
tionship stated in the conclusion with considerable detail (e.g., allows for
predictability in considering a situation of walking behind or “ahead of the
crowd”).
Communication– The student demonstrates considerable organization and clarity of mathe-
matical reasoning.
– The student communicates graphically, demonstrating considerable skill in graph
construction by including uniform and varied scales (e.g.,“1 square = 5 units”,
“Two squares = 5 units”) and circling the point that represents the double
ordered pair.
Application– The application is realistic (e.g., “the distance someone is walking and how
fast they are moving”), and the student supports it with considerable detail.
Comments/Next Steps– The student could improve labelling by using the task numbering system
within the report.
– The graph and the data could be better integrated with the text (e.g., by
using the order text, table of values, graph, text).
62 The Ontar io Curr iculum – Exemplars , Grade 9: Mathematics
Footprints on the Road Level 4, Sample 1
A B
Foundations of Mathematics , Appl ied63
C
Another relationship between two variables would be the older youare, the taller you are (in most cases). That’s how doctors can tell if you will grow when you get older, by seeing how tall you are whenyou are little, and see if you will be tall or average or even short bymeasuring your height throughout your childhood. And to see thedoctor if you are healthy and growing properly by measuring yourheight when you are young. And if you are not growing properly,they can find out whats wrong.
Teacher’s Notes
Knowledge/Understanding– The student makes a statement to be investigated that includes both ends
of a continuum, demonstrating a thorough understanding of a relational
hypothesis.
– The student identifies and explains many factors that could affect the
investigation (e.g., “if people exagerate their stride length”).
Thinking/Inquiry/Problem Solving– The student not only describes dispersion in words but also refers to an
alternative method of recognizing the trend, demonstrating a high degree
of accuracy and detail (e.g., “but I have made a line of best fit and as you
can see it is slightly going up”).
– The student states the conclusion clearly, with a high degree of detail, as
an affirmation of the hypothesis, with reference to both aspects of the
relationship, including the exceptions to the rule (e.g., “People with long
legs had longer strides and people with shorter legs had shorter strides.
Some people with short legs had big strides . . .”).
Communication– The report is highly organized, with a high degree of clarity of mathematical
reasoning (e.g., the data and the graph are integrated into the body of the
text).
– The student communicates graphically, demonstrating a high degree of skill
in the use of proper form (e.g., the graph is well planned and properly labelled).
Application– The student identifies a realistic application and provides a highly detailed
and thorough description of it, followed by an explanation of its importance
(e.g., “the older you are, the taller you are . . . That’s how doctors can tell
if you will grow . . . , by seeing how tall you are when you are little . . .”).
Comments/Next Steps– The student could give more thought to an appropriate title for the data
table and the graph.
– The student makes a well-organized and thorough analysis of the investigation.
Footprints on the Road Level 4, Sample 2
A B
64 The Ontar io Curr iculum – Exemplars , Grade 9: Mathematics
Foundations of Mathematics , Appl ied65
C D
Teacher’s Notes
Knowledge/Understanding– The student demonstrates a thorough understanding of a relational
hypothesis by using a physical argument to justify the belief that the
hypothesis will be shown to be true.
– The student identifies and explains with a high degree of effectiveness three
key factors that will affect the validity of the data collected (e.g., “not all
bodies are proportionate”).
Thinking/Inquiry/Problem Solving– The student describes the dispersion of the data with a high degree of
accuracy and detail (e.g., illustrates the trend by using a line of best fit
and addresses the “positive correlation”).
– The student demonstrates a thorough understanding of a linear relationship
(e.g., “The graph showed a positive and strong correlation between a person’s
height and their stride”).
Communication– The report is highly organized and demonstrates a high degree of mathe-
matical reasoning.
– The student communicates graphically, demonstrating a high degree of
skill in the use of proper form (e.g., the graph is clear, is well labelled, and
has an appropriate title).
Application– The student identifies a realistic application and predicts an outcome
(e.g., “I would anticipate a positive correlation”), putting into practice
the skills developed throughout the investigation.
Comments/Next Steps– The student should address the outliers in the analysis of the dispersion of
the data.
– The student should use SI units instead of imperial units.
66 The Ontar io Curr iculum – Exemplars , Grade 9: Mathematics
Foundations of Mathematics , Appl ied67
Mathematics Exemplar Task Grade 9 Foundations of Mathematics, Applied
Teacher Package
Title: Footprints on the Road
Time requirement: 150–225 minutes
The student task: Part I – Up to 75 minutes
Part II – 150 minutes
Description of the TaskThrough small group and whole class discussion, students will generate prob-
lems involving the investigation of relationships. Students will identify the data
needed for the investigations, collect the data from the whole class, and prepare
a class summary data sheet (Part I). Each student will then select one problem,
analyse the data, make a conclusion, and submit a written report (Part II).
Part I: Class Discussion (time: 75 minutes )• In small groups, students generate questions based on the text they read,
a scenario involving two friends, Henri and Kim.*
• As a whole class, students identify problems or research arising from a
discussion of the questions.
• In small groups, students gather the data identified as necessary for
analysing the research problems selected.
• Students prepare a class summary data sheet to facilitate the preparation
of their individual final reports.
Part II: Individual Work (time: 150 minutes)• Each student analyses a set of data and writes a report.
Note: The time required for Part I will vary according to the extent of
students’ familiarity with carrying out investigations and describing
relationships. Time may also be needed to prepare students for Part II.
Allow 150 minutes (75 minutes × 2) for the completion of Part II.
Assign homework as appropriate.
Assessment and Evaluation• You will need to use the task-specific rubric* to assess and evaluate
the student work.
• You will need to approve the relationship that a student chooses to
investigate. Some relationships will not provide students with as much
data as others. See the section “Task Instructions” for ideas of good
relationships to investigate.
Expectations Addressed in the Exemplar Task
Teacher Package
RelationshipsStudents will:
1. determine relationships between two variables by collecting
and analysing data;
2. compare the graphs of linear and non-linear relations;
3. describe the connections between various representations of
relations;
4. pose problems, identify variables, and formulate hypotheses
associated with relationships;(continued)
* The text of the scenario is reproduced on page 47 of this document. * The rubric is reproduced on page 46 of this document.
68 The Ontar io Curr iculum – Exemplars , Grade 9: Mathematics
Note that, although all the expectations listed will be addressed through
instruction relating to the task, student achievement of expectations 2 and 4
will not be assessed in the final product.
Teacher Instructions
Prior Knowledge and Skills Required• Have students practise describing relationships by discussing some obvious
relationships and the language used to characterize them. For example:
– Describe the relationship between age and natural hair colour.
– Describe the relationship between marks and the amount of time
spent doing homework and studying.
• For the purposes of this exemplar, students will construct graphs by
hand. Note also that the choice of scale, or decisions to break the scale
on an axis, can have an influence on the appearance of the dispersion.
Consider the graphs below, which represent a set of measurements
involving foot length and height. The first graph starts both scales at
zero; the second breaks the scale on both axes and uses finer scales;
the third breaks the scale on the horizontal axis only.
5. demonstrate an understanding of some principles of sampling and
surveying and apply the principles in designing and carrying out
experiments to investigate the relationships between variables;
6. collect data, using appropriate equipment and/or technology;
7. organize and analyse data, using appropriate techniques and
technology;
8. describe trends and relationships observed in data, make infer-
ences from data, compare the inferences with hypotheses about
the data, and explain the differences between the inferences
and the hypotheses;
9. communicate the findings of an experiment clearly and concisely,
using appropriate mathematical forms;
10. demonstrate an understanding that straight lines represent linear
relations and curves represent non-linear relations.
69 Foundations of Mathematics , Appl ied
• If students are unfamiliar with interpreting scatter plots, spend some
time on simple examples, such as the following:
• The descriptions of the dispersion might make reference to:
– randomness versus clustering;
– the nature of the clustering: a line or a curve (linear or non-linear);
– some description of the line or curve (e.g., points up to the right or
points down to the right);
– the identification of outliers and some explanation for them.
AccommodationsAccommodations that are normally provided in regular classrooms for
students with special needs should be provided in the administration of
this performance task.
You may wish to review the relevant course profile for specific suggestions
for accommodations appropriate for students in special education programs.
Materials and Resources Required• Student instructions
• Scientific calculators
• Graphing materials: pencils, rulers, erasers, graph paper
RubricIntroduce the task-specific rubric to the students at least one day beforethe administration of the task. Review the rubric with the students,allowing ample class time to ensure that each student understands thecriteria and the descriptions for achievement at each level.
Some students may perform below level 1. It will be important to note thecharacteristics of their work in relation to the criteria in the assessmentrubric and to provide feedback to help them improve.
Task Instructions
Part I : 75 minutes• Organize students into groups of three or four.• Hand out the “Student Instructions” page.*
Have students read the whole page silently. Answer any questionsabout the overall activity.
• Give students 5 or 10 minutes to work on Part I. Indicate that each student should keep a record of the questions.
• Facilitate a whole group discussion of the questions that have beengenerated, including the following:What is meant by foot length?Who is taller, Henri or Kim?Is Kim male or female?Who is walking faster? Who is taking more steps per minute?Is it reasonable that two people who have the same foot length wouldhave different stride lengths?
• As questions are generated, display them on the overhead. (If you areusing the board, you might have one student record them all for you.)
Note: If key questions, such as the second, third and fifth examples, donot come up, suggest them and add them to the list.
* The student instructions are reproduced on page 47 of this document.
70 The Ontar io Curr iculum – Exemplars , Grade 9: Mathematics
• Have students work in their groups for about 5 or 10 minutes to begin
suggesting answers to the questions. Discuss what makes a good
hypothesis.
• Facilitate a whole class discussion for 5 or 10 minutes to share some of
the hypothesized answers. Many of the answers to the posed questions
will depend on mathematical relationships. For example, “Who is taller,
Henri or Kim?” assumes a mathematical relationship, perhaps between
stride length and height.
• Draw an example such as this from the discussion and rephrase it as
a research problem: Is there a relationship between stride length and
height in humans? If so, describe it.
• Ask students to identify other research problems that might arise from
the discussion:
Is there a relationship between foot length and height in humans? If so,
describe it.
Is there a relationship between stride length and foot length in
humans? If so, describe it.
Is there a relationship between stride length and leg length in humans?
If so, describe it.
Is there a relationship between stride length and gender? If so, describe it.
Note: If students have difficulty recognizing the problems for research,
you might lead them to likely problems by focusing on the relationships
that arose during the discussions above.
• Once a set of research problems has been posed, identify two or three
problems that you will allow for use in Part II. Make this decision with
student input, but finalize the choice on the basis of your own judgement
of which problems would lead to appropriate investigations.
• Good investigations to consider are:
– foot length and height
– leg length and height
– leg length and stride length
– height and stride length
Note: Some students may generate questions and research problems
that extend the investigation. For example, students might reflect on
the predictive value of the depth of the footprint. Gathering data would
then be a more extensive and time-consuming activity. As well, depth
of footprint might lead to questions about the relationship to a person’s
weight, which may be a sensitive issue for some students.
Encourage students to ask wide-ranging questions and to generate
interesting research problems, but keep time lines and sensitivity in
mind when selecting problems to approve for Part II.
• For each research problem, facilitate a discussion of the points below.
Instruct students to take careful notes, because they will be required to
explain the discussion as part of the write-up in Part II.
The following are the discussion points for each research problem:
– the variables in the relationship, and the data that would be collected
to examine the relationship;
Discuss independent and dependent variables. Note that the classifi-
cation of variables may not be obvious. For example, is foot length
dependent on height, or is height dependent on foot length? This is
especially interesting with Grade 9 students, whose growth and height
may not have caught up with their foot length. In this discussion,
focus on students’ explaining their reasoning.
71 Foundations of Mathematics , Appl ied
– factors that might affect the validity of the data and that would be
considered in gathering a representative sample of data (e.g., gender,
age, height);
Note that the research problems are phrased in terms of human rela-
tionships. Discuss the concept of representative samples and question
whether, on the basis of the variables identified, the class is a repre-
sentative sample of humanity. Discuss how a representative sample
might be identified.
– factors that might affect accuracy in measuring (e.g., Should shoes
be on or off when measuring height? What is meant by foot length?
Is the measurement of one stride sufficient?).
• Using the class as a sample, have the students work in small groups to
gather the data identified above.
• Prepare a class summary sheet, and have each student record his or her
data (without identification), both in his or her notebook and on the
class summary sheet.
• Photocopy the summary sheet for use in the next class.
Part II: 150 minutes (75 minutes × 2)• Ensure that students are now working as individuals.
• Hand out the student instructions for Part II and have students read all
of Part II and the assessment rubric.
• Answer any questions and highlight areas that you feel need explanation
or emphasis.
• Review and discuss the rubric.
• Hand out the class summary sheet of data.
• Reinforce the importance of clarity and explanation in students’ written
work.
• Discuss the meaning of each step of the report and the type of answer
that would be involved in each step. (See the descriptions from the
section “Prior Knowledge and Skills Required”.)
• Students work to complete their written reports (Part II).
Follow-upAfter students have handed in their written reports, direct a whole class
discussion:
• Ask students to share their results on the various research questions.
• Ask students to share their ideas about other instances in which it
would be important to identify a relationship between variables.
• Return student attention to the original scenario involving Henri and Kim.
Pose some of the original questions generated in Part I and ask students
to answer them now, having examined a relationship in some depth.
The Ministry of Education wishes to acknowledge the contributions of the many individuals, groups, and organizations that participated in the development and refinement of this resource document.
ISBN 0-7794-0248-0
00-210
© Queen’s Printer for Ontario, 2000