mathematics: the ontario curriculum exemplars, grade 8 · pdf filethe ontario curriculum...

The Ontario Curriculum – ExemplarsGrade 8

Mathematics

Ministry of Education

2002

Samples of Student Work:A Resource for Teachers

Contents

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3Purpose 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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

Teachers and Administrators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8Parents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9Students . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

Geometry and Spatial Sense . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11Exploring the Pythagorean Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

The Task . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12Expectations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12Prior Knowledge and Skills . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12Task Rubric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13Student Samples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15Teacher Package . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

Patterning and Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63Smothered in Chocolate! . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64


Data Management and Probability / Number Sense and Numeration . . . . . . . . . . 103Rolling in Sales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104


This publication is available on the Ministry of Education’s website athttp://www.edu.gov.on.ca.

3

In 1997, the Ministry of Education and Training published a new mathematics cur-riculum policy document for Ontario elementary students entitled The OntarioCurriculum, Grades 1–8: Mathematics, 1997. The new curriculum is more specificthan previous curricula with respect to both the knowledge and the skills that studentsare expected to develop and demonstrate in each grade. The document contains thecurriculum expectations for each grade and an achievement chart that describes fourlevels of student achievement to be used in assessing and evaluating student work.

The present document is part of a set of eight documents – one for each grade – thatcontain samples (“exemplars”) of student work in mathematics at each of the four levelsof achievement described in the achievement chart. The exemplar documents areintended to provide assistance to teachers in their assessment of student achievementof the curriculum expectations. The samples represent work produced at the end ofthe school year in each grade.

Ontario school boards were invited by the Ministry of Education to participate in thedevelopment of the exemplars. Teams of teachers and administrators from across theprovince were involved in developing the assessment materials. They designed theperformance tasks and scoring scales (“rubrics”) on the basis of selected Ontario cur-riculum expectations, field-tested them in classrooms, suggested changes, adminis-tered the final tasks, marked the student work, and selected the exemplars used in thisdocument. During each stage of the process, external validation teams and Ministryof Education staff reviewed the tasks and rubrics to ensure that they reflected theexpectations in the curriculum policy documents and that they were appropriate forall students. External validation teams and ministry staff also reviewed the samples ofstudent work.

The selection of student samples that appears in this document reflects the professionaljudgement of teachers who participated in the project. No students, teachers, or schoolshave 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.

Introduction

4 The Ontar io Curr iculum – Exemplars , Grade 8: Mathematics

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.

Use of the exemplar materials will be supported initially through provincial in-servicetraining.

Purpose of This Document

This document was developed to:

• show the characteristics of student work at each of the four levels of achievementfor Grade 8;

• 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 documentand consider them along with the information in the Teacher’s Notes and Comments/Next Steps sections. They are encouraged to examine the samples in order to developan understanding of the characteristics of work at each level of achievement and theways in which the levels of achievement reflect progression in the quality of knowl-edge 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 over a school year.

Features of This Document

This document contains the following:

• a description of each of three performance tasks (each task focuses on a particularstrand or combination of strands), as well as a listing of the curriculum expectationsrelated to the task

• a task-specific assessment chart (“rubric”) for each task

• two samples of student work for each of the four levels of achievement for each task

• Teacher’s Notes, which provide some details on the level of achievement for eachsample

1. In this document, parent(s) refers to parent(s) and guardian(s).

5Introduct ion

• Comments/Next Steps, which offer suggestions for improving achievement

• the Teacher Package that was used by teachers in administering each task

It should be noted that each sample for a specific level of achievement represents thecharacteristics of work at that level of achievement.

The Tasks

The performance tasks were based directly on curriculum expectations selected fromThe Ontario Curriculum, Grades 1–8: Mathematics, 1997. The tasks encompassed thefour categories of knowledge and skills (i.e., problem solving; understanding of con-cepts; application of mathematical procedures; communication of required knowledgerelated to concepts, procedures, and problem solving), requiring students to integratetheir knowledge and skills in meaningful learning experiences. The tasks gave studentsan opportunity to demonstrate how well they could use their knowledge and skills ina specific context.

Teachers were required to explain the scoring criteria and descriptions of the levels ofachievement (i.e., the information in the task rubric) to the students before theybegan the assignment.

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 dif-ferent levels of achievement given in the rubric to assess student achievement on aparticular task.

The rubric for a specific performance task is intended to provide teachers and studentswith an overview of the expected product with regard to the knowledge and skillsbeing assessed as a whole.

The achievement chart in the curriculum policy document for mathematics provides a standard 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 andgeneral in nature, it provides a reference point for all assessment practice and a frame-work within which to assess and evaluate student achievement. The descriptionsassociated with each level of achievement serve as a guide for gathering and trackingassessment information, enabling teachers to make consistent judgements about thequality of student work while providing clear and specific feedback to students andparents.

For the purposes of the exemplar project, a single rubric was developed for each per-formance task. This task-specific rubric was developed in relation to the achievementchart 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.

• 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 improvestudents’ learning. In order to improve their work, students need to be provided withuseful feedback. Students find that feedback on the strengths of their achievement andon areas in need of improvement is more helpful when the specific category of knowl-edge or skills is identified and specific suggestions are provided than when theyreceive only an overall mark or general comments. Student achievement should beconsidered in relation to the criteria for assessment stated in the rubric for each cate-gory, and feedback should be provided for each category. Through the use of a rubric,students’ strengths and weaknesses are identified and this information can then beused as a basis for planning the next steps for learning. In this document, the Teacher’sNotes indicate the reasons for assessing a student’s performance at a specific level ofachievement, and the Comments/Next Steps give suggestions for improvement.

In the exemplar project, a single rubric encompassing the four categories of knowledgeand skills was used to provide an effective means of assessing the particular level ofstudent performance in each performance task, to allow for consistent scoring of stu-dent 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 of additionalrubrics if they need to assess student achievement on a specific task in greater detailfor one or more of the four categories. For example, it may be desirable in evaluating awritten report on an investigation to use separate rubrics for assessing understandingof concepts, problem-solving skills, ability to apply mathematical procedures, andcommunication skills.

7Introduct ion

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 in thecontext of classroom instruction to assess achievement in a particular assignment.

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,interviews, 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 educatorsin the following way:

• The teams selected a cluster of curriculum expectations that focused on the knowl-edge and skills that are considered to be of central importance in the subject area.Teams were encouraged to select a manageable number of expectations. The partic-ular selection of expectations ensured that all students would have the opportunityto demonstrate their knowledge and skills in each category of the achievementchart in the curriculum policy document for the subject.

• The teams drafted three tasks for each grade that would encompass all of theselected expectations and that could be used to assess the work of all students.

• The teams established clear, appropriate, and concrete criteria for assessment, andwrote 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 participat-ing in the assessment project.

• The tasks were field-tested in classrooms across the province by teachers who hadvolunteered to participate in the field test. Student work was scored by teams ofeducators. 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 finalize the tasks, which were then administeredin the spring of 2001.

In developing the tasks, the teams ensured that the resources needed for completingthe tasks – that is, all the worksheets and support materials – were available.

Prior to both the field tests and the final administration of the tasks, a team ofvalidators – 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 at the districtschool board level by teachers of the subject who had been provided with training inthe scoring. These teachers evaluated and discussed the student work until they wereable to reach a consensus regarding the level to be assigned for achievement in eachcategory. This evaluation was done to ensure that the student work being selectedclearly illustrated that level of performance. All of the student samples were then for-warded to the ministry. A team of teachers from across the province, who had beentrained by the ministry to assess achievement on the tasks, rescored the student sam-ples. They chose samples of work that demonstrated the same level of achievement inall four categories and then, through consensus, selected the samples that best repre-sented the characteristics of work at each level of achievement. The rubrics were theprimary tools used to evaluate student work at both the school board level and theprovincial level.

The following points should be noted:

• Two samples of student work are included for each of the four achievement levels.The use of two samples is intended to show that the characteristics of an achievementlevel can be exemplified in different ways.

• Although the samples of student work in this document were selected to show alevel of achievement that was largely consistent in the four categories (i.e., problemsolving; understanding of concepts; application of mathematical procedures; com-munication of required knowledge), teachers using rubrics to assess student workwill notice that students’ achievement frequently varies across the categories (e.g., astudent may be achieving at level 3 in understanding of concepts but at level 4 incommunication of required knowledge).

• Although the student samples show responses to most questions, students achievingat level 1 and level 2 will often omit answers or will provide incomplete responsesor incomplete demonstrations.

• Students’ effort was not evaluated. Effort is evaluated separately by teachers as partof the “learning skills” component of the Provincial Report Card.

• The document does not provide any student samples that were assessed using therubrics and judged to be below level 1. Teachers are expected to work with studentswhose achievement is below level 1, as well as with their parents, to help the stu-dents improve their performance.

Use of the Student Samples

Teachers and Administrators

The samples of student work included in the exemplar documents will help teachersand 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;

9Introduct ion

• facilitating communication with parents regarding the curriculum expectations andlevels of achievement for each subject;

• promoting fair and consistent assessment within and across grade levels.

Teachers may choose to:

• use the teaching/learning activities outlined in the performance tasks;

• use the performance tasks and rubrics in the 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 next stepsfor learning;

• share student work with colleagues for consensus marking;

• partner with another school to design tasks and rubrics, and to select samples forother performance tasks.

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.

Parents

The performance tasks in this document exemplify a range of meaningful and relevantlearning activities related to the curriculum expectations. In addition, this documentinvites the involvement and support of parents as they work with their children toimprove their achievement. Parents may use the samples of student work and therubrics 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’ performanceimproves as they are encouraged to take responsibility for their own achievement andto 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 expectationsfor learning.

• The rubrics and the information given in the Teacher’s Notes section will help clarifythe 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 thedifferences in the levels of achievement. By analysing and discussing these differences,students will gain an understanding of ways in which they can assess their ownresponses and performances in related assignments and identify the qualitiesneeded to improve their achievement.

Geometry and Spatial Sense


Exploring the Pythagorean Theorem

The Task

This task required each student to draw or construct as manynon-congruent squares as possible on either five by five geopaperor a five by five geoboard, and to determine the area of each ofthe squares. Each student was then asked to construct a right-angled triangle, to draw a square on each side of the triangle,and to determine the area of each of the squares formed on thesides of the triangle. Students then compared the sizes of thesquares drawn on the sides of the right-angled triangles andstated the relationship among the squares – the Pythagorean theorem. Finally, they were asked to construct semicircles on thesides of a right-angled triangle and to determine whether thearea of the semicircle on the hypotenuse is equal to the com-bined areas of the semicircles on the other two sides.

The Geometer’s Sketchpad software, which is referenced for use inthis task, is licensed to the Ontario Ministry of Education (1999).

Expectations

This task gave students the opportunity to demonstrate theirachievement of all or part of each of the following selectedoverall and specific expectations from the strand Geometry andSpatial Sense. Note that the codes that follow the expectationsare from the Ministry of Education’s Curriculum Unit Planner(CD-ROM).

Students will:

1. investigate geometric mathematical theories to solve problems(8m59);

2. use mathematical language effectively to describe geometricconcepts, reasoning, and investigations (8m60)

3. investigate the Pythagorean relationship using area modelsand diagrams (8m65);

4. apply the Pythagorean relationship to numerical problemsinvolving area and right triangles (8m70);

5. explain the Pythagorean relationship (8m73).

Prior Knowledge and Skills

To complete this task, students were expected to have someknowledge or skills related to the following:

• the concepts of area, perimeter, square root, and perfectsquares

• the properties of right-angled, obtuse, and acute triangles

• the difference between drawing a figure and constructing one

For information on the process used to prepare students for the taskand on the materials, resources, and equipment required, see theTeacher Package reproduced on pages 56–61 of this document.

13 Geometry and Spatial Sense

Task Rubric – Exploring the Pythagorean Theorem

Expectations*

1, 3, 4

1, 3

4

Level 1

– selects and applies a problem-

solving strategy to determine the

areas of few of the different-

sized squares that can be drawn

on a 5 x 5 geoboard or on 5 x 5

geopaper, arriving at an incom-

plete or inaccurate solution


solving strategy to solve a prob-

lem related to the Pythagorean

theorem, arriving at an incom-

plete or inaccurate solution

– demonstrates a limited under-

standing of the Pythagorean theo-

rem when analysing the data

and looking for relationships

– applies mathematical proce-

dures with many errors and/or

omissions when investigating the

Pythagorean theorem

Level 2

– selects and applies an appropri-

ate problem-solving strategy to

determine the areas of some of

the different-sized squares that

can be drawn on a 5 x 5

geoboard or on 5 x 5 geopaper,

arriving at a partially complete

and/or partially accurate solution

– selects and uses an appropriate

problem-solving strategy to solve

a problem related to the

Pythagorean theorem, arriving at

a partially complete and/or par-

tially accurate solution

– demonstrates some understand-

ing of the Pythagorean theorem

when analysing the data and

looking for relationships


dures with some errors and/or


Pythagorean theorem

Level 3



determine the areas of many of

the different-sized squares that

can be drawn on a 5 x 5


arriving at a generally complete

and accurate solution





a generally complete and accu-

rate solution

– demonstrates a general under-





dures with few errors and/or


Pythagorean theorem

Level 4



determine the areas of most or

all of the different-sized squares

that can be drawn on a 5 x 5


arriving at a thorough and accu-

rate solution





a thorough and accurate solution

– demonstrates a thorough under-





dures with few, if any, minor

errors and/or omissions when

investigating the Pythagorean

theorem

Problem solving

The student:

Understanding of concepts

The student:

Application of mathematical procedures

The student:


Expectations*

2, 5

Level 1

– uses mathematical language and

notation with limited clarity to

analyse and describe geometric

relationships and concepts

Level 2


notation with some clarity to



Level 3


notation clearly to analyse and

describe geometric relationships

and concepts

Level 4


notation clearly and precisely to



Communication of required knowledge

The student:

*The expectations that correspond to the numbers given in this chart are listed on page 12.

Note: This rubric does not include criteria for assessing student performance that falls below level 1.


Exploring the Pythagorean Theorem Level 1, Sample 1

A B


C D


E F


G Teacher’s Notes

Problem Solving

– The student selects and applies a problem-solving strategy to determine the

areas of few of the different-sized squares that can be drawn on a 5 x 5

geoboard or on 5 x 5 geopaper, arriving at an incomplete or inaccurate

solution (e.g., in question 1a, draws five distinct squares and states, “Those

dots represent 1 cm. So I would take each side and mutiply them to get the

area” – an ambiguous statement).

– The student selects and applies a problem-solving strategy to solve a problem

related to the Pythagorean theorem, arriving at an incomplete or inaccurate

solution (e.g., in question 3c, gives a simple, qualitative relationship; in ques-

tion 4, adds his or her own system of linear measurement to the rectangle

and comes up with a close but inaccurate solution; in question 5, labels the

diagram and states some comparisons without proving them).

Understanding of Concepts

– The student demonstrates a limited understanding of the Pythagorean

theorem when analysing the data and looking for relationships (e.g., in

question 3d: “In all of them except the ones where the numbers are all the

same like 3, 3, 3 or 6, 6, 6”).

Application of Mathematical Procedures

– The student applies mathematical procedures with many errors and/or

omissions when investigating the Pythagorean theorem (e.g., in question 2,

neither of the areas of the square on the hypotenuse has been found

accurately).

Communication of Required Knowledge

– The student uses mathematical language and notation with limited clarity to

analyse and describe geometric relationships and concepts (e.g., in ques-

tion 1b, explains: “. . . each square in the 5 x 5 geopaper is 1 cm so 5 x 5 = 25.

Those dots represent 1 cm”).


Comments/Next Steps

– The student needs to work with concrete materials, such as pattern blocks

and geoboards, for his or her investigations.

– The student should develop more problem-solving strategies, such as

recording information in lists and charts when conducting investigations.

– The student should make use of mathematical software, such as The

Geometer’s Sketchpad, to explore and develop an understanding of

geometric relationships.



A B


C D


E F


G H


I Teacher’s Notes

Problem Solving

– The student selects and applies a problem-solving strategy to determine the

areas of few of the different-sized squares that can be drawn on a 5 x 5

geoboard or on 5 x 5 geopaper, arriving at an incomplete or inaccurate

solution (e.g., in question 1, draws three of the squares, and uses the stan-

dard algorithm to determine the area of a square).

– The student selects and applies a problem-solving strategy to solve a problem

related to the Pythagorean theorem, arriving at an incomplete or inaccurate

solution (e.g., in question 2a, draws a sketch of a right-angled triangle on

geopaper with rectangles instead of squares on the sides).


– The student demonstrates a limited understanding of the Pythagorean


question 3d, displays an understanding in his or her answer that the great-

est number rolled must be the hypotenuse, but does not apply the

Pythagorean theorem; in question 5, states, “Yes because the HYPOtenuses

area is still greater then the right angles area”).




a correct answer is obtained using the lengths of the triangle, but no

attempt to explain the reasoning is evident; in question 5 the formula for the

area of a circle is not applied).


– The student uses mathematical language and notation with limited clarity to

analyse and describe geometric relationships and concepts (e.g., attempts

to use diagrams to explore and explain relationships, but provides explana-

tions that are simplistic and difficult to interpret; uses terminology, such as

hypotenuse, incorrectly; in question 1b, counts dots instead of spaces when

using dot paper to calculate the area of squares).


Comments/Next Steps

– The student needs to use concrete materials to investigate the properties of

shapes.

– The student should develop more problem-solving strategies, such as creat-

ing systematic lists and charts.

– The student needs to draw diagrams accurately, using the geopaper to

guide the drawing process and the subsequent problem-solving task.

– The student should make use of mathematical software, such as The

Geometer’s Sketchpad, to explore and develop an understanding of geometric

relationships.

– The student should learn to use and interpret measurements on dot paper

correctly.



A B


C D


E F

*

*

* Page D in this sample


G Teacher’s Notes

Problem Solving

– The student selects and applies an appropriate problem-solving strategy to

determine the areas of some of the different-sized squares that can be

drawn on a 5 x 5 geoboard or on 5 x 5 geopaper, arriving at a partially com-

plete and/or partially accurate solution (e.g., in question 1, draws various

squares, showing their dimensions and subdividing the last one drawn into

1 cm squares to show 16 cm2).


solve a problem related to the Pythagorean theorem, arriving at a partially

complete and/or partially accurate solution (e.g., in questions 2 and 3,

draws or constructs only isosceles right-angled triangles and then con-

cludes that the area of the square on the hypotenuse is “double the area of

the other two squares” [“double the area of the squares on the legs”]

rather than the sum of the areas of the other two squares).


– The student demonstrates some understanding of the Pythagorean theorem

when analysing the data and looking for relationships (e.g., in question 2c,

realizes that the square on the hypotenuse is larger than the other two

squares, but identifies it as “double the area of the other two squares”).


– The student applies mathematical procedures with some errors and/or


applies the Pythagorean theorem correctly to find the length of the

hypotenuse of the right-angled triangle, but the area of the rectangle is inac-

curate; in question 5, identifies the formula for the area of a circle, but

applies the formula to one circle rather than to all these semicircles).


– The student uses mathematical language and notation with some clarity to


tion 5, makes the following statement, but does not fully explain it: “The rela-

tionship is not the same because the radius2 of the circle will not be the

same as the length of a square”).


Comments/Next Steps

– The student should continue using mathematical software, such as The

Geometer’s Sketchpad, when solving problems and looking for relationships.

– The student needs to review his or her completed work, working backward

to check solutions for accuracy.

– The student should record discovered relationships as equations in an

effort to explain the relationships that are found.

– The student should make a conclusion based on a variety of scale drawings,

not leap to a conclusion based on too special an example.



A B


C D


E F


G Teacher’s Notes

Problem Solving


determine the areas of some of the different-sized squares that can be

drawn on a 5 x 5 geoboard or on 5 x 5 geopaper, arriving at a partially com-

plete and/or partially accurate solution (e.g., in question 1, draws different

sizes of squares, uses a strategy to determine the areas of the squares, but

experiences problems in determining the lengths of the diagonals).


solve a problem related to the Pythagorean theorem, arriving at a partially

complete and/or partially accurate solution (e.g., in question 3d, lists many

of the possible outcomes when rolling three dice; some of these will not

form a triangle and only one – 3-4-5 – will form a right-angled triangle).


– The student demonstrates some understanding of the Pythagorean theorem

when analysing the data and looking for relationships (e.g., in question 2c,

identifies the Pythagorean theorem, but presents the area of the hypotenuse

in the diagram of the right-angled triangle incorrectly as 38 cm2 instead

of 32 cm2).




uses the Pythagorean formula correctly to find the length of the rectangle,

but does not use it correctly to find the rectangle’s area).


– The student uses mathematical language and notation with some clarity to


tion 3c, uses some proper terminology in the explanation of the Pythagorean

theorem: “the sum of the length squared on the right angle is equal to the

length of the hypotenuse squared”).


Comments/Next Steps

– The student should create charts to organize data and to identify patterns

or relationships that become apparent.

– The student should use the correct conventions when mentioning polygons.

– The student needs to accurately apply the Pythagorean theorem when

solving problems.

– The student should provide written explanations when asked to explain how

an answer was derived.

– The student should consider using mathematical software, such as The

Geometer’s Sketchpad, when solving problems and looking for relationships.



A B


C D


E F


G H


Comments/Next Steps

– The student should use mathematical language that relates precisely to the

Pythagorean theorem.

– The student should continue to use written explanations to support his or

her reasoning.

– The student should use precise measurements when completing calculations

that will be used in determining patterns.

– The student should use mathematical software, such as The Geometer’s

Sketchpad, when solving problems and looking for relationships.

Teacher’s Notes

Problem Solving


determine the areas of many of the different-sized squares that can be

drawn on a 5 x 5 geoboard or on 5 x 5 geopaper, arriving at a generally

complete and accurate solution (e.g., in question 1, draws six of the eight

squares and correctly determines their areas).


solve a problem related to the Pythagorean theorem, arriving at a generally

complete and accurate solution (e.g., in question 3c, uses a systematic

method to check combinations, arriving at correct solutions).


– The student demonstrates a general understanding of the Pythagorean


question 2a, the data from the sketch of the right-angled triangle are inter-

preted correctly in the equation a2 + b2 = c2).


– The student applies mathematical procedures with few errors and/or omis-

sions when investigating the Pythagorean theorem (e.g., in question 4, uses

the Pythagorean theorem to find the length of the hypotenuse of the right-

angled triangle, and applies the formula for a parallelogram to calculate the

area of the rectangle; in question 5, applies the formula for the area of a cir-

cle correctly, but assigns the value of 3 to pi, instead of 3.14, and comes up

with the wrong conclusion because of a mechanical error, 3 x 6.25 = 18.75,

not 12.5).


– The student uses mathematical language and notation clearly to analyse

and describe geometric relationships and concepts (e.g., in question 3d,

provides a written explanation of process; in question 4, includes a sketch

to support the explanation and the calculations).



A B


C D


E F


G H


Comments/Next Steps

– The student could use mathematical software, such as The Geometer’s

Sketchpad, when solving problems.

– The student should use mathematical language that relates precisely to the

Pythagorean theorem.

– The student should continue to provide written explanations and sketches to

fully explain what was discovered during the problem-solving process.

Teacher’s Notes

Problem Solving


determine the areas of many of the different-sized squares that can be

drawn on a 5 x 5 geoboard or on 5 x 5 geopaper, arriving at a generally

complete and accurate solution (e.g., in question 1, draws six of the eight

squares and calculates their areas correctly).


solve a problem related to the Pythagorean theorem, arriving at a generally

complete and accurate solution (e.g., in question 3d, uses trial and error and

methodically lists many possible cases in order to determine the correct

solution).


– The student demonstrates a general understanding of the Pythagorean


question 2c, gives an explanation that reveals a general understanding; in

question 3a, demonstrates understanding by constructing two different

right-angled triangles with squares on each of the sides, but does not show

or explain the lengths of the hypotenuses).


– The student applies mathematical procedures with few errors and/or


applies the formula for the area of a circle, but makes an error in calcula-

tions and assigns a value of 3 to π).


– The student uses mathematical language and notation clearly to analyse

and describe geometric relationships and concepts (e.g., in question 3d,

provides a clear written explanation and a sketch to justify the final answer).



A B


C D


E F

*

*

* Page D in this sample


G Teacher’s Notes

Problem Solving


determine the areas of most or all of the different-sized squares that can be

drawn on a 5 x 5 geoboard or on 5 x 5 geopaper, arriving at a thorough and

accurate solution (e.g., in question 1, draws all eight squares and correctly

calculates their areas).


solve a problem related to the Pythagorean theorem, arriving at a thorough

and accurate solution (e.g., in question 3c, uses The Geometer’s Sketchpad

to explore and represent the relationship between the squares on the sides

of right-angled triangles).


– The student demonstrates a thorough understanding of the Pythagorean


question 3a, “In other words, the area of the squares opposite the hypotenuse

added together will always equal the area of the square on the side of the

hypotenuse”).


– The student applies mathematical procedures with few, if any, minor errors

and/or omissions when investigating the Pythagorean theorem (e.g., in

question 5, applies the formula for the area of a circle accurately and pre-

cisely to prove that the Pythagorean theorem also works for semicircles).


– The student uses mathematical language and notation clearly and precisely

to analyse and describe geometric relationships and concepts (e.g., in ques-

tion 3d, uses “a pythagoreon triple” to describe the measurements of a

right-angled triangle; in question 4, presents the solution in a concise and

precise manner).


Comments/Next Steps

– The student could show calculations that demonstrate the areas of the

squares shown on the drawings.

– The student should continue to use mathematical software, such as The

Geometer’s Sketchpad, when solving problems.

– The student should provide concrete examples to enhance his or her

explanations.



A B


C D


E F


G H


Comments/Next Steps

– The student should use mathematical software, such as The Geometer’s

Sketchpad, for problem-solving investigations.

– The student needs to review all measurements and calculations to try to

eliminate computational errors.

– The student should continue to provide detailed written explanations to

support his or her calculations.

– The student should explain the errors due to approximation in question 3c.

Teacher’s Notes

Problem Solving


determine the areas of most or all of the different-sized squares that can be

drawn on a 5 x 5 geoboard or on 5 x 5 geopaper, arriving at a thorough and

accurate solution (e.g., in question 1, draws most of the eight possible

squares and correctly determines their areas; creates new strategies to

calculate the areas of squares 4 and 5).

– The student selects and uses an appropriate problem-solving strategy to

solve a problem related to the Pythagorean theorem, arriving at a thorough

and accurate solution (e.g., in question 4, works backwards to find unknown

lengths by making an insightful connection between the Pythagorean theo-

rem and the area of the rectangle).


– The student demonstrates a thorough understanding of the Pythagorean


question 2c, identifies the relationship between the sum of the two squares

adjacent to the right angle and the square on the hypotenuse).



and/or omissions when investigating the Pythagorean theorem (e.g., in

question 4, applies the formula for the area of a triangle to calculate the

area of the rectangle; in question 5, applies the formula for the area of a

circle to prove that the Pythagorean theorem also works for semicircles,

although a minor error in the radius of one side results in an incorrect

conclusion).


– The student uses mathematical language and notation clearly and precisely

to analyse and describe geometric relationships and concepts (e.g., in ques-

tion 3a, provides clear and precise labelling of the diagrams; in question 4,

uses the square root symbol and a persuasive written explanation to support

his or her calculations).


1

Title: Exploring the Pythagorean Theorem

Time requirement: 175 minutes (total)• 40 minutes for pre-task 1• 45 minutes for pre-task 2• two periods of 45 minutes each for the exemplar task

Description of the Task

This task requires each student to draw or construct as many non-congruent squares as possibleon either five by five geopaper or a five by five geoboard, and to determine the area of each ofthe squares. Each student is then asked to construct a right-angled triangle, to draw a square oneach side of the triangle, and to determine the area of each of the squares formed on the sides ofthe triangle. Students will then compare the sizes of the squares drawn on the sides of the right-angled triangles and will state the relationship among the squares – the Pythagorean theorem.Finally, they will be asked to construct semicircles on the sides of a right-angled triangle and todetermine whether the area of the semicircle on the hypotenuse is equal to the combined areas ofthe semicircles on the other two sides.

Expectations Addressed in the Exemplar Task

Note that the codes that follow the expectations are from the Ministry of Education’s CurriculumUnit Planner (CD-ROM).

Students will:

1. investigate geometric mathematical theories to solve problems (8m59);

2. use mathematical language effectively to describe geometric concepts, reasoning, andinvestigations (8m60);

3. investigate the Pythagorean relationship using area models and diagrams (8m65);

4. apply the Pythagorean relationship to numerical problems involving area and righttriangles (8m70);

5. explain the Pythagorean relationship (8m73).

Mathematics Exemplar TaskGrade 8 – Geometry and Spatial Sense

Teacher Package

2

Teacher Instructions

Prior Knowledge and Skills Required

To complete this task, students should have some knowledge or skills related to the following:• the concepts of area, perimeter, square root, and perfect squares• the properties of right-angled, obtuse, and acute triangles• the difference between drawing a figure and constructing one

The Rubric*

The rubric provided with this exemplar task is to be used to assess students’ work. The rubric isbased on the achievement chart given on page 9 of The Ontario Curriculum, Grades 1–8:Mathematics, 1997.

Before asking students to do the task outlined in this package, review with them the concept of arubric.

Accommodations

Accommodations that are normally provided in the regular classroom for students with specialneeds should be provided in the administration of the exemplar task.

Materials and Resources Required

Before students attempt a particular task, provide them with the appropriate materials fromamong the following:• a copy of the Student Package (see Appendix 1) for each student• 100 tiles per pair of students (squares of construction paper could be substituted)• numbered cubes – dice (a minimum 3 per student)• rulers• paper• protractors• calculators• compasses• geostrips (if available)• optional: centimetre grid paper, toothpicks, geoboards and elastics, The Geometer’s Sketchpad*

* The Geometer’s Sketchpad software is licensed to the Ontario Ministry of Education (1999).

Teacher Package

*The rubric is reproduced on pages 13–14 of this document.


3

Task Instructions

Introductory Activities

The pre-tasks are designed to review and reinforce the skills and concepts that students will beusing in the exemplar task and to model strategies useful in completing the task.

Pre-task 1: Constructing Squares

1. Provide each pair of students with 100 tiles.

2. Ask students to identify a square with an area of one square unit. Then ask them to use thetiles they have been given to construct different-sized squares.

Ask students questions like the following:• How are the dimensions of the square you constructed related to the number of tiles you

used?• How is the number of tiles per side related to the area of the square?• Do you notice anything of interest when you calculate the square root of the area of any

square?• If you double the length of the sides of a square, what happens to the area?

Note that The Geometer’s Sketchpad can be used effectively with this activity.

Pre-task 2: The Geometer’s Sketchpad

This task can be done as a whole-class demonstration or students can work in pairs.Manipulatives (e.g., toothpicks, geoboards, geostrips) should be made available for students’ use.Students may also draw by hand or investigate using The Geometer’s Sketchpad.

Have students roll three numbered cubes (dice) at the same time. Then ask: “Is it always possibleto create a triangle using the numbers on the dice to represent the lengths of the sides? Forexample, if the numbers 2, 2, and 3 were rolled, would you be able to form a triangle with sides of2 units, 2 units, and 3 units? Experiment to find out.”

Ask students to construct triangles from the numbers they rolled and investigate the types oftriangles constructed.

Ask students questions like the following:• Will it always be possible to make a triangle using the numbers rolled as the dimensions of the

sides? Give reasons for your answer.• Assuming that you roll the same number on all three dice, can you make more than one kind

and size of triangle? Why or why not?• What numbers is it possible to roll that cannot be used to create a triangle? Explain why.• What is the probability of forming a triangle from three rolls of the dice?• What is the probability of forming an equilateral triangle from one roll?

Students can continue to investigate the probability of creating a triangle given the numbers onregular dice. For variety, try using different types of dice (e.g, tetrahedron, octahedron,dodecahedron).

4

Exemplar Task

1. Distribute a copy of the Student Package to each student.

2. Students should have access to tiles and The Geometer’s Sketchpad.

3. Tell students that they will be working individually and independently to complete theassigned task.

4. Remind students about the rubric and make sure that each student has a copy of it.

5. Students can use the space provided in the Student Package or additional sheets of paper fortheir responses. Computer printouts are encouraged.

6. The problem that the students will solve independently is provided in the worksheets inAppendix 1.


5

Appendix 1: Student Worksheets

Exploring the Pythagorean Theorem

1. a) How many different sizes of squares can you draw on a 5 by 5 geopaper or make on a 5 by 5 geoboard? [There are more than 5.]

b) Show how you would determine the area of each of the squares.

6

2. a) On a geoboard or geopaper, make or draw a right-angled triangle.

b) On each side of the triangle, make or draw a square.


7

c) How do the squares compare? Record any relationship you observe between or amongstthe areas of the squares. Think of a way of recording your data so that someone looking at it will be able to tell what you are thinking.

8

3. a) Using either geoboards, geopaper, Geometer’s Sketchpad, or geostrips, construct twodifferent right-angled triangles. Show your work below.

b) Construct, as you did in question 1, squares on each of the sides of the triangles you havejust drawn.


9

c) Examine the relationships between and amongst the areas of these new squares you havejust constructed. Summarize what you think is true about squares constructed on the sidesof right-angled triangles.

d) If you were to throw three numbered cubes – with each of the digits 1 to 6 on each cube –and use the three numbers facing up to construct a triangle, in how many of the caseswould you be able to form a right-angled triangle?

10

4. ABCD is a rectangle. Find the area of ABCD using information from triangle AEB.

Explain how you found the area.

A

D

B

CE

6 units 8 units


11

5. This is a diagram of a right-angled triangle with semi-circles constructed on each of the threesides.

Is the same relationship among the squares true for the semi-circles?Investigate.

12

Appendix 2

Geopaper

Patterning and Algebra


Smothered in Chocolate!

The Task

This task required each student to use interlocking cubes toform larger cubes and to investigate the patterns found whenlarge cubes are made from unit cubes. Students then consideredwhat would happen if they were to submerge the cubes they hadmade into a container of chocolate. The specific task was todetermine how many of the unit cubes would have one face, twofaces, or three faces covered with chocolate. Students were alsogiven the opportunity to determine the relationships among thevertices, edges, and faces of cubes. They were requested toexpress their generalizations algebraically and to sketch cubeson isometric paper.

Expectations

This task gave students the opportunity to demonstrate theirachievement of all or part of each of the following selected overalland specific expectations from the strand Patterning and Algebra.Note that the codes that follow the expectations are from theMinistry of Education’s Curriculum Unit Planner (CD-ROM).

Students will:

1. identify, create, and discuss patterns in algebraic terms (8m75);

2. apply and defend patterning strategies in problem-solvingsituations (8m78);

3. describe and justify a rule in a pattern (8m79);

4. write an algebraic expression for the nth term of a numericsequence (8m80);

5. find patterns and describe them using words and algebraicexpressions (8m81).



• the concepts of surface area and volume

• identifying the relationship between surface area and volume

• discussing patterns in algebraic terms

• identifying various types of prisms

• understanding and identifying the following features of a cube:edge, face, and vertex


65 Patterning and Algebra

Task Rubric – Smothered in Chocolate!

Expectations*

3

1

2, 4

5

Level 1


solving strategy that leads to an

incomplete or inaccurate solution


standing of algebraic patterns

– applies mathematical procedures

with many errors and/or omis-

sions when analysing the data

– states generalizations and/or

algebraic expressions for the nth

terms that include many errors

and/or omissions

– uses mathematical language

and/or algebraic notation with

limited clarity to explain and jus-

tify generalizations based on the

model

Level 2


ate problem-solving strategy that

leads to a partially complete

and/or partially accurate solution


ing of algebraic patterns


with some errors and/or omis-

sions when analysing the data



terms that include some errors

and/or omissions


and/or algebraic notation with

some clarity to explain and jus-

tify generalizations based on the

model

Level 3



leads to a generally complete

and accurate solution




with few errors and/or omissions

when analysing the data



terms that include few errors

and/or omissions


and/or algebraic notation clearly

to explain and justify generaliza-

tions based on the model

Level 4



leads to a thorough and accu-

rate solution




with few, if any, minor errors

and/or omissions when

analysing the data



terms that include few, if any,

minor errors and/or omissions


algebraic notation clearly and

precisely to explain and justify

generalizations based on the

model

Problem solving

The student:


The student:


The student:


The student:




Smothered in Chocolate! Level 1, Sample 1

A B


C D


E F


Comments/Next Steps

– The student needs to use manipulatives when problem solving to consolidate

his or her understanding.

– The student should organize his or her data by using charts or systematic

lists to help identify patterns.

– The student needs additional practice in identifying and recording patterns.

Teacher’s Notes

Problem Solving

– The student selects and applies a problem-solving strategy that leads to an

incomplete or inaccurate solution (e.g., in question 2, creates a chart that

categorizes the pattern of the cubes rather than the number of unit cubes

with 3, 2, 1, and 0 faces covered in chocolate).


– The student demonstrates a limited understanding of algebraic patterns

(e.g., in question 2b, describes simple patterns related to the general loca-

tion of the unit cubes with 3, 2, 1, and 0 face(s) covered).



omissions when analysing the data (e.g., in question 3, uses rudimentary

procedures and makes many errors, such as identifying a 5 x 5 dimension

rather than a 5 x 5 x 5 dimension; identifies the correct number of unit

cubes with 3 faces covered in paint but not those with 1 or 2 covered faces).

– The student states generalizations and/or algebraic expressions for the

nth terms that include many errors and/or omissions (e.g., in question 4,

describes the number of cubes with 3 faces covered with a simple mathe-

matical statement: “4 x 2 = 8 cubes”).


– The student uses mathematical language and/or algebraic notation with

limited clarity to explain and justify generalizations based on the model

(e.g., in questions 2b and 3, uses notations 3 x 3 and 5 x 5, rather than

3 x 3 x 3 and 5 x 5 x 5, to describe volume; in question 4, uses rudimentary

language without evidence of mathematical terminology to describe patterns

and procedures: “The reasons why I have chosen these answers are because

I added all the cubes it took to answer how many cubes were covered”).



A B


C D


E F


Comments/Next Steps

– The student needs to use manipulatives when problem solving to consolidate

his or her understanding.

– The student should organize data by using charts and/or systematic lists to

help identify patterns.

– The student needs to begin using algebraic expressions and symbols.

Teacher’s Notes

Problem Solving

– The student selects and applies a problem-solving strategy that leads to an

incomplete or inaccurate solution (e.g., in question 3, appears to have

guessed at a 4 x 4 x 4 cube, and has not used the numbers of covered faces

to support his or her answer).


– The student demonstrates a limited understanding of algebraic patterns

(e.g., in question 2b, lists only one pattern, and vaguely explains the other

patterns).



omissions when analysing the data (e.g., in question 3, does not attempt to

use the number of uncovered cubes to arrive at the size of cube needed).


nth terms that include many errors and/or omissions (e.g., in question 4,

uses a specific example with errors and attempts to explain the generaliza-

tion, but does not see how to apply it to the nth term).



limited clarity to explain and justify generalizations based on the model

(e.g., in question 2b, “I noticed that the you always get 8 corners with 3 faces

covered, that you would get different numbers with 2, 1 and 0 faces”).



A B


C D


E Teacher’s Notes

Problem Solving

– The student selects and applies an appropriate problem-solving strategy

that leads to a partially complete and/or partially accurate solution (e.g., in

question 2, provides three lists to record data, but they contain two

errors; in question 3, draws a diagram but does not show or explain the

problem-solving method used).


– The student demonstrates some understanding of algebraic patterns (e.g., in

question 2b, identifies the correct pattern for the number of unit cubes with

3 faces covered in chocolate, but does not identify patterns for 0, 1, and 2

covered faces).



omissions when analysing the data (e.g., in question 3, gives the correct

number of cubes with 1 and 3 painted faces but not 2 painted faces; draws

the 5 x 5 x 5 cube, but there are no mathematical procedures evident to sup-

port the answer of 5 x 5 x 5).


nth terms that include some errors and/or omissions (e.g., in question 2b,

accurately describes the patterns for 3 faces covered in chocolate but does

not identify patterns for 0, 1, and 2 chocolate-covered faces; in question 4,

uses a chart that contains errors when explaining generalizations).



some clarity to explain and justify generalizations based on the model

(e.g., in question 2b, notices that, however large the cube made from unit

cubes is, the eight corner areas will always be covered: “all cubes (made

out of cubes) have 8 three faces and no matter how big or small it is, it will

be the same”; in question 4, uses simple charts and diagrams for explanations;

provides only simple written explanations).


Comments/Next Steps

– The student needs to use isometric paper to sketch cubes.

– The student should use charts to list data to help him or her identify patterns.

All charts should be accurately labelled to ensure easy interpretation of

the data.

– The student needs additional practice in identifying and recording patterns.

– The student should retrace his or her steps if he or she is unable to find a

pattern.



A B


C D


E F


Comments/Next Steps

– The student should use charts and/or systematic lists to help him or her

identify patterns.

– The student should use variables to write or describe algebraic expressions.

– The student needs to use mathematical procedures (describing, generalizing,

and translating into algebraic expressions to the nth terms) to justify his or

her assertions.

– The student should retrace his or her steps if he or she is unable to find a

pattern.

Teacher’s Notes

Problem Solving


that leads to a partially complete and/or partially accurate solution (e.g., in

question 1, correctly draws three cubes; in question 2, describes many of

the correct numbers of unit cubes with 0, 1, 2, and 3 faces covered in choco-

late for two of the cubes, and the correct number of cubes with

3 chocolate-covered faces for the 2 x 2 x 2 cube; in question 3, states that

he or she built a model “case around it” to find the original cube).


– The student demonstrates some understanding of algebraic patterns (e.g., in

question 2b, identifies and describes only the correct pattern for determining

the number of unit cubes with 3 faces covered in chocolate).



omissions when analysing the data (e.g., in question 3, finds the correct

number of paint-covered faces on the unit cubes in a 5 x 5 x 5 cube without

finding the total number of unit cubes used to create the larger cubes; pro-

vides some correct answers but does not show any calculations).


nth terms that include some errors and/or omissions (e.g., in question 4,

attempts to explain all his or her conclusions, but provides simplistic and

incomplete statements).



some clarity to explain and justify generalizations based on the model

(e.g., in question 2b, describes only one pattern: “I think that the only

recagnisable patern is that no mater how big or small the cube is there will

always be 8 little cubes with 3 sides showing and these cubes are alway’s

the carner cubes”).



A B


C D


E Teacher’s Notes

Problem Solving


that leads to a generally complete and accurate solution (e.g., in question 4,

applies the congruency of the faces to develop a rule).


– The student demonstrates a general understanding of algebraic patterns

(e.g., in question 2, identifies the correct numbers of total cubes and cov-

ered and uncovered faces for the three cubes; in question 3, identifies the

correct patterns for the number of unit cubes with paint on 3 faces, 2 faces,

and 1 face).



sions when analysing the data (e.g., in question 3, applies the patterns dis-

covered in question 2b to find the size of the original cube – 5 x 5 x 5 – and

the number of covered faces but does not explain his or her reasoning).


nth terms that include few errors and/or omissions (e.g., in question 4,

explains algebraic expressions accurately to describe the patterns for 1, 2,

and 3 covered faces with a partial explanation for 0 faces and no algebraic

representations).


– The student uses mathematical language and/or algebraic notation clearly

to explain and justify generalizations based on the model (e.g., describes

most generalizations in statements without the use of algebraic expres-

sions, as in question 4: “3 faces – the number of corners in a cube and that

would always no matter how big or small will always be 8”).

Comments/Next Steps

– The student could use charts to list data to help him or her compare data.

– The student should use variables to write or describe algebraic expressions

and abstract applications.

– The student needs to use math terminology consistently when communicating

discoveries or algebraic expressions.



A B


C D


E F


Comments/Next Steps

– The student should use isometric paper correctly to sketch cubes.

– The student needs to use variables to write or describe algebraic expressions

and abstract applications.

– The student should use correct mathematical terminology consistently and

should complete all explanations of procedures.

Teacher’s Notes

Problem Solving


that leads to a generally complete and accurate solution (e.g., in question 3,

explains how the answer of 5 x 5 x 5 is determined).


– The student demonstrates a general understanding of algebraic patterns

(e.g., in question 2b, identifies the correct pattern for the number of unit

cubes with 3 faces covered in chocolate and provides partial explanations

of patterns for unit cubes with 1 and 2 chocolate-covered faces).


– The student applies mathematical procedures with few errors and/or

omissions when analysing the data (e.g., in question 3, applies the patterns

discovered in question 2b to find the size of the original cube – 5 x 5 x 5 –

and explains how the answer was obtained).


nth terms that include few errors and/or omissions (e.g., in question 4,

provides explanations for the number of cubes with 1 and 2 chocolate-

covered faces).


– The student uses mathematical language and/or algebraic notation clearly

to explain and justify generalizations based on the model (e.g., describes

generalizations in statements without the use of algebraic expressions, as

in question 4: “In a cube, there is always 8 corners, so no matter what the

dimensions are, 8 cubes will always have 3 faces showing”).



A B


C D


E F


G Teacher’s Notes

Problem Solving


that leads to a thorough and accurate solution (e.g., in question 1, labels

and accurately draws cubes and gives dimensions; in question 2, lists all

data in a chart with labelled columns and rows that allows the data to be

easily compared).


– The student demonstrates a thorough understanding of algebraic patterns

(e.g., in question 2b, identifies all patterns correctly for solving the number

of unit cubes with 0, 1, 2, and 3 chocolate-covered faces, using clear lan-

guage and a diagram).



and/or omissions when analysing the data (e.g., in question 3, correctly

applies the patterns discovered in question 2b to find the size of the original

cube – 5 x 5 x 5).


nth terms that include few, if any, minor errors and/or omissions (e.g., in

answering question 4, refers to the correct generalizations made in ques-

tion 2b that describe the patterns for the number of unit cubes with 0, 1, 2,

and 3 faces covered in chocolate).


– The student uses mathematical language and algebraic notation clearly and

precisely to explain and justify generalizations based on the model (e.g., in

question 2b, states: “For three faces to be covered in chocolate, you must

count all of the corners of your three cubes, and all of your figures should

come out to 8, except for a single cube (1 x 1 x 1) for it will have all six faces

covered”; in question 2b, uses exponents and the variable n).


Comments/Next Steps

– The student needs to consolidate his or her understanding of terminology

and variables by using them in algebraic expressions to represent patterns

consistently.

– The student should continue to provide extensive explanations of procedures.

– The student needs to use n to represent the same thing in the algebraic

expressions that he or she uses.



A B


C D


E F


Comments/Next Steps

– The student needs to use algebraic expressions with variables to further

describe patterns.

– The student should continue to use organizers and sketches as strategies

for problem solving.

Teacher’s Notes

Problem Solving


that leads to a thorough and accurate solution (e.g., in question 2, clearly

and accurately lists all the data in a chart, allowing the data to be

compared easily; in question 3, uses a detailed, labelled diagram/model to

support the problem-solving process).


– The student demonstrates a thorough understanding of algebraic patterns

(e.g., in question 2b, identifies all patterns correctly for solving the number

of unit cubes with 0, 1, 2, and 3 faces covered in chocolate).



and/or omissions when analysing the data (e.g., in question 2b, has discov-

ered clear and straightforward ways of describing the patterns for the num-

bers of cubes with 1 and 3 faces covered, and gives an accurate though

convoluted description of the pattern for the number of cubes with 2 faces

covered).


nth terms that include few, if any, minor errors and/or omissions (e.g., in

question 2b, describes algebraic expressions accurately to identify the pat-

terns for the number of unit cubes with 0, 1, 2, and 3 chocolate-covered

faces; describes generalizations in words).


– The student uses mathematical language clearly and precisely to explain

and justify generalizations based on the model (e.g., in question 2, “For the

first cube (2 x 2 x 2), I discovered that there were no individual cubes which

had no chocolate on them, as well as discovering that each cube which was

covered in chocolate had 3 faces covered, so one does not count the cubes

for both 1 face and 2 faces”; uses written statements supported with dia-

grams to describe the found patterns).


1

Title: Smothered in Chocolate!



This task requires each student to use interlocking cubes to form larger cubes and to investigatethe patterns found when large cubes are made from unit cubes. Students then consider whatwould happen if they were to submerge the cubes they have made into a container of chocolate.The specific task is to determine how many of the unit cubes would have one face, two faces, orthree faces covered with chocolate. Students are also given the opportunity to determine therelationships among the vertices, edges, and faces of cubes. They are requested to express theirgeneralizations algebraically and to sketch cubes on isometric paper.



Students will:

1. identify, create, and discuss patterns in algebraic terms (8m75);

2. apply and defend patterning strategies in problem-solving situations (8m78);

3. describe and justify a rule in a pattern (8m79);

4. write an algebraic expression for the nth term of a numeric sequence (8m80);

5. find patterns and describe them using words and algebraic expressions (8m81).

Mathematics Exemplar TaskGrade 8 – Patterning and Algebra

Teacher Package

2



Before attempting the task, students should have some knowledge or skills related to the following:• understanding the concepts of surface area and volume• identifying the relationship between surface area and volume• discussing patterns in algebraic terms• identifying various types of prisms• understanding and identifying the following features of a cube: edge, face, and vertex

The Rubric*



Accommodations



Before students attempt a particular task, provide them with the appropriate materials fromamong the following:• a copy of the Student Package (see Appendix 1) for each student• interlocking cubes (25 per student)• graph paper• calculators

Task Instructions



Pre-task 1

Have students work in groups of four. Give each group ten interlocking cubes. Begin by askingthe following questions:• How many faces are visible on one cube? On two cubes joined together? On four cubes

joined together?• How does the configuration of the connected cubes affect the number of visible faces?

(Answers will vary depending on how the cubes are connected.)

Teacher Package

*The rubric is reproduced on page 65 of this document.


3

Pre-task 2

Give each group of four students fifty interlocking cubes. Ask students to identify the number offaces, edges, and vertices for one cube, eight unit cubes (2 x 2 x 2) connected to make one cube,and twenty-seven unit cubes (3 x 3 x 3) connected to make a bigger cube. Have students recordthe data. Then have them represent their data using a table such as the one below. Reproduce thistable on the chalkboard or on an overhead.

Cube containing ... Vertices Edges Faces

1 cube

8 small cubes

27 small cubes

After students complete the task, ask them the following questions:• What data were the same for the one-unit cube? The eight-unit cube? The twenty-seven-unit

cube?• If you were to submerge the largest cube in a container of chocolate, how many of the unit

cubes would have chocolate on three sides?• If you were to build an even larger cube – say, a 5 x 5 x 5 cube – what are some of the patterns

you would observe?• Does Euler’s rule hold for each of the cubes you have built?

(Euler’s Rule: Faces + Vertices = Edges + 2 or F + V = E + 2)

Exemplar Task




4. The problem that the students must solve independently is provided in the worksheets inAppendix 1.

4


Smothered in Chocolate!

Imagine that each individual cube is a piece of toffee.

Pieces of toffee are joined together to form cubes.The cubes will then be completely dipped in chocolate.

1. Build three cubes of different sizes using interlocking cubes. Sketch the three cubes on theisometric paper provided at the end of this package.

1 piece of toffee


5

2. a) For each of your three cubes you have just built, your task will be to determine the total number of cubes required for each structure you have built, and that have chocolateon exactly:i) 3 facesii) 2 facesiii) 1 faceiv) 0 faces

Present your data so that someone looking at the way you organize your work will be able to see how you solved the problem.

6

b) Describe all of the patterns you observe by looking at the data.


7

3. A number of unit cubes were put together to form a larger cube, which was then painted. Thelarger cube is subsequently taken apart and it is discovered that 27 of the smaller cubes haveno paint on them. How big was the original cube and how many of the smaller cubes hadpaint on only:

a) one face?b) two faces?c) three faces?

Show how you arrived at your answer.

8

4. If you had a large cube made from unit cubes such that the dimensions of the large cube aren x n x n, how many pieces would be covered in chocolate on exactly three faces, exactly two faces, exactly one face, and no faces if the entire cube was covered with chocolate?

Justify your answer.


9

Data Managementand Probability / Number Sense andNumeration


Rolling in Sales

The Task

This task required each student to determine the lowest possiblepercentage discount and the highest possible percentage discountresulting from their rolling two numbered cubes (dice) andadding the numbers shown on the numbered cubes. Studentshad to state which percentage discount was most likely to occurand why; the probability of getting a percentage discount greaterthan 10 percent; and which they would choose, and why, betweena 10 percent discount and rolling the numbered cubes. Finally,students were asked to invent a probability game that uses percent.

Expectations

This task gave students the opportunity to demonstrate theirachievement of all or part of each of the following selected over-all and specific expectations from the strands Data Managementand Probability, and Number Sense and Numeration. Note thatthe codes that follow the expectations are from the Ministry ofEducation’s Curriculum Unit Planner (CD-ROM).

Students will:

1. solve and explain multi-step problems involving fractions,decimals, integers, percents, and rational numbers (8m8);

2. use mathematical language to explain the process used andthe conclusions reached in problem solving (8m9);

3. ask “What if” questions; pose problems involving fractions,decimals, integers, percents, and rational numbers andinvestigate solutions (8m30);

4. explain the process used and any conclusions reached inproblem solving and investigations (8m31);

5. apply percents in solving problems involving discounts, salestax, commission, and simple interest (8m34);

6. identify probability situations and apply knowledge of probability (8m95);

7. list the possible outcomes of simple experiments by usingtree diagrams, modelling, and lists (8m118);

8. identify the favourable outcomes among the total number of possible outcomes and state the associated probability(e.g., of getting closer in a random draw) (8m119).



• the addition and multiplication of integers

• solving multi-step problems involving simple fractions, decimals,and percents

• intuitive concepts of probability and how probability canrelate to chance

• listing the possible outcomes of simple experiments by usingtree diagrams or matrices


105 Data Management and Probabi l i ty / Number Sense and Numeration

Task Rubric – Rolling in Sales

Expectations*

1, 3, 7

6, 7, 8

3, 5, 6, 7

Level 1


solving strategy to solve a multi-

step problem involving percent

and probability, arriving at an

incomplete or inaccurate

solution


standing of percent, fractions,

and probability

– applies a percent algorithm

with many errors and/or omis-

sions to solve problems involving

discounts


dures with many errors and/or

omissions when investigating

situations involving probability

Level 2




and probability, arriving at a par-

tially complete and/or partially

accurate solution


ing of percent, fractions, and

probability


with some errors and/or omis-


discounts


dures with some errors and/or



Level 3




and probability, arriving at a

generally complete and accurate

solution



and probability


with few errors and/or omis-


discounts


dures with few errors and/or



Level 4




and probability, arriving at a

thorough and accurate solution



and probability


with few, if any, minor errors

and/or omissions to solve prob-

lems involving discounts


dures with few, if any, minor

errors and/or omissions when

investigating situations involving

probability

Problem solving

The student:


The student:


The student:


Expectations*

2, 4

Level 1


notation to describe percent and

fractions with limited clarity


notation to explain situations

involving probability with limited

clarity

Level 2



fractions with some clarity



involving probability with some

clarity

Level 3



fractions clearly



involving probability clearly

Level 4



fractions clearly and precisely



involving probability clearly and

precisely


The student:




Rolling in Sales Level 1, Sample 1

A B


C D


E F


Comments/Next Steps

– The student should create an organizational system that will help him or her

find all the possible outcomes when completing problem-solving activities.

– The student needs to ensure that all steps are included in multi-step prob-

lems to arrive at more thorough and accurate answers.

– The student should provide more detail in written explanations, including

specific instructions and rules for invented games.

– The student needs to check all calculations, such as working backwards to

check for accuracy.

– The student should use the appropriate dollar notation (e.g., in question 1b).

Teacher’s Notes

Problem Solving

– The student selects and applies a problem-solving strategy to solve a multi-

step problem involving percent and probability, arriving at an incomplete or

inaccurate solution (e.g., in question 2, starts systematic lists, but fails to

identify all possible outcomes).


– The student demonstrates a limited understanding of percent, fractions,

and probability (e.g., in question 2, does not realize that there would be 36

possible outcomes for rolling the dice, rather than the 21 recorded).


– The student applies a percent algorithm with many errors and/or omissions

to solve problems involving discounts (e.g., in question 1b, states and calcu-

lates the discounts, but makes no attempt to calculate the final discounted

costs; in question 5, makes a computation error [13 + 4 = 18 rather than

17]).


omissions when investigating situations involving probability (e.g., in ques-

tion 2, uses an incomplete list of combinations to determine an inaccurate

probability; in question 6, fails to recognize the equal likelihood of discounts

of 11% and 15% from question 5).


– The student uses mathematical language and notation to describe percent

and fractions with limited clarity (e.g., in question 4, gives reasons why 10%

would be better: “I would chose 10% because the odds are against you 17:21

of the time you wolud get 1-9 and you would get 10-12 4:21 of the time so it’s

easier to chose 10%”).

– The student uses mathematical language and notation to explain situations

involving probability with limited clarity (e.g., in question 5, gives no expla-

nations of how to play the games and includes diagrams that fail to add any

other information about how to play).



A B


C D


E F


Comments/Next Steps

– The student needs to correctly construct and interpret tables and charts for

recording data.

– The student should utilize one organizing method effectively before starting

to use alternative methods.

– To arrive at more thorough and accurate answers, the student needs to

ensure that all steps are included in multi-step problems.

– The student should provide more detail in his or her written explanations,

including specific instructions and rules for invented games.

– The student needs to check all calculations in some way, such as working

backwards to check for accuracy.

Teacher’s Notes

Problem Solving


step problem involving percent and probability, arriving at an incomplete

or inaccurate solution (e.g., in question 2, uses a chart to identify possible

outcomes, but includes both of the headers of the matrix in the calculations;

in question 4, provides a tree diagram, which is incorrect in representing

this problem).


– The student demonstrates a limited understanding of percent, fractions,

and probability (e.g., in questions 1b and 2, shows inaccuracies in the use of

the fractions and decimals, as in 2⁄ 12 = 0.2 = 2%).


– The student applies a percent algorithm with many errors and/or omissions

to solve problems involving discounts (e.g., in question 1b, is able to deter-

mine the percentage discounts but not to apply an appropriate computation

to calculate the discounted amounts).



tion 2, lists all possible outcomes, but does not identify 7 as the most likely

to occur; in question 6, constructs an inaccurate chart based on question 5;

in question 3, selects 12 instead of 36 for the denominator in the probability

calculations; in question 5, is unable to apply probability concepts to this

scenario).



and fractions with limited clarity (e.g., in question 1b, in the calculation 2⁄ 12 = 0.2 = 2%, none of the indicated notations are equivalent).


involving probability with limited clarity (e.g., in question 5, gives no expla-

nation of how to play the game and includes diagrams that fail to add any

other information about how to win a 15% discount).



A B

Note: The student’s colour coding in the chart is done accurately.


C D


E F


Comments/Next Steps

– The student needs to provide more evidence of how solutions are developed

by showing the process used to support the answers given.

– The student should provide more complex responses, providing explana-

tions, supporting evidence, and details that would enhance the answer.

– The student should review his or her completed work to ensure that data

have been accurately used and that the calculations are correct.

Teacher’s Notes

Problem Solving

– The student selects and applies a problem-solving strategy to solve a

multi-step problem involving percent and probability, arriving at a partially

complete and/or partially accurate solution (e.g., in question 2, develops a

colour-coded matrix of the possible outcomes; uses lists throughout the

task that are correct but include only simple data; shows little evidence of

managing complex data throughout the task).


– The student demonstrates some understanding of percent, fractions, and

probability (e.g., in question 3, lists the probability incorrectly, and shows

calculations that are correct but are based on a misunderstanding of what

is required in the question).


– The student applies a percent algorithm with some errors and/or omissions

to solve problems involving discounts (e.g., in questions 1a and 3, attempts

to find solutions by inventing his or her own algorithm to solve the problem

[e.g., dividing by the cost of the clothing outfit] rather than by following the

stated directions).



tion 4, identifies the probability, which is accurate given the student’s ear-

lier misconception, but is not accurate for this question).



and fractions with some clarity (e.g., in question 4, “10% discount. I would

choose this because you chances of getting 10% or greater are 1 in 12. You

have likely better savings with the 10%”).


involving probability with some clarity (e.g., in question 5, provides a simple

explanation: “The probability of a customer getting a 15% discount is 1⁄ 6”).



A B


C D


E F


Comments/Next Steps

– The student should use more diagrams to illustrate the likelihood of an

event occurring.

– The student needs to provide more evidence of how solutions are developed

in a multi-step problem by showing the process used to support his or her

answers.

– The student should provide more complex responses, providing explana-

tions, supporting evidence, and details that would enhance his or her

answers.

Teacher’s Notes

Problem Solving


step problem involving percent and probability, arriving at a partially com-

plete and/or partially accurate solution (e.g., in question 2, develops a

matrix of the possible outcomes; in question 5, draws for the game a simple

diagram that does not provide an opportunity to demonstrate an under-

standing of probability).


– The student demonstrates some understanding of percent, fractions, and

probability (e.g., in question 5, indicates that the probability of getting 15%

is 1⁄ 3, but in question 6, indicates that 5% is most likely to occur “because the

space is bigger then 10% and 15%”).


– The student applies a percent algorithm with some errors and/or omissions

to solve problems involving discounts (e.g., in question 1b, provides the cor-

rect amount for the least that could be paid but does not provide calculations).



tion 3, does not convert the fraction to its lowest terms or provide a per-

centage equivalent).



and fractions with some clarity (e.g., in question 4, “I would choice the 10%

because there would be a very slim chance of getting higher then 10% so I

would choose 10%”).


involving probability with some clarity (e.g., in question 2: “The discount that

shows up most is seven”).



A B


C D


E F


Comments/Next Steps

– The student needs to check his or her completed work for computational

errors.

– The student could develop more complex probability games for problem-

solving tasks.

Teacher’s Notes

Problem Solving


step problem involving percent and probability, arriving at a generally com-

plete and accurate solution (e.g., in questions 2 and 3, creates matrices and

identifies the correct probability; in question 5, however, makes an error in

his or her game by creating a matrix that allows a discount of 16%).


– The student demonstrates a general understanding of percent, fractions,

and probability (e.g., in question 1, uses a percent discount to find a total

monetary discount; in question 3, shows the probability as fractions in their

lowest terms).


– The student applies a percent algorithm with few errors and/or omissions

to solve problems involving discounts (e.g., in question 1, completes conver-

sions and calculations correctly).


sions when investigating situations involving probability (e.g., in question 3,

correctly finds the probability using the matrix, while in question 4, using a

similar approach, makes a small adding error that results in an incorrect

solution).



and fractions clearly (e.g., in question 3, presents the statement “The proba-

bility of getting a percentage discount greater than 10% is 1⁄ 12”, and supports

this answer with a chart and a fraction expressed in the lowest terms).


involving probability clearly (e.g., in question 5, describes the invented game

using a chart and words).



A B


C D


E F


Comments/Next Steps

– The student needs to express probability as fractions in their lowest terms

and should ensure that probability is displayed as fractions, decimals, and

percents where required.

– The student needs to check his or her completed work for computational

errors that result in incorrect solutions.

– The student could develop more complex probability games for problem-

solving tasks.

– The student should review his or her completed questions to ensure that

correct solutions are applied to related subsequent questions.

Teacher’s Notes

Problem Solving


step problem involving percent and probability, arriving at a generally com-

plete and accurate solution (e.g., in question 2, creates a matrix to identify

the total number of outcomes to answer questions 2 and 3).


– The student demonstrates a general understanding of percent, fractions,

and probability (e.g., in question 1, correctly converts a percent to a fraction

and a decimal; in question 6, identifies 6, 7, 8, and 9 as the numbers most

likely to occur, but does not include 10 and 11 in the solution).


– The student applies a percent algorithm with few errors and/or omissions

to solve problems involving discounts (e.g., in questions 1a and 1b, determines

the highest and lowest percentage discounts possible, but calculates only

the most you could pay, not the least).


sions when investigating situations involving probability (e.g., in question 3,

a 3⁄ 36 chance of rolling more than 10% should have been expressed as 1⁄ 12).



and fractions clearly (e.g., uses charts and diagrams to support his or her

written statements).


involving probability clearly (e.g., in question 5, explains the invented games

clearly).



A B


C D


E F


Comments/Next Steps

– The student should continue to provide detailed diagrams and written

explanations.

– The student should continue to solve problems that require written

explanations.

– The student could invent a more complex game in order to demonstrate a

greater understanding of probability concepts

Teacher’s Notes

Problem Solving


step problem involving percent and probability, arriving at a thorough

and accurate solution (e.g., in question 2, uses a chart to list all possible

outcomes).


– The student demonstrates a thorough understanding of percent, fractions,

and probability (e.g., in question 3, correctly calculates the probability of

getting a percentage discount higher than 10%; in question 4, demonstrates

a clear understanding that there is a small probability of rolling a discount

higher than 10% by showing the high probability of receiving a discount

below 10%).


– The student applies a percent algorithm with few, if any, minor errors and/

or omissions to solve problems involving discounts (e.g., in question 1b,

calculates new prices correctly by determining the percent discount and

applying it correctly to find the total dollar discount).


and/or omissions when investigating situations involving probability (e.g., in

questions 3 and 4, uses fractions, decimals, and percents correctly to indi-

cate probability).



and fractions clearly and precisely (e.g., in question 2, shows the answer

clearly using a matrix, a written statement, and a diagram).


involving probability clearly and precisely, (e.g., in question 4, provides an

accurate written explanation, along with clear numerical answers; provides

concise, precise explanations throughout the assignment).



A B


C D


E F


Comments/Next Steps

– The student should use proper notation for repeating decimals.

– The student should continue working on investigations involving multiple

solutions.

– The student should answers all questions completely.

Teacher’s Notes

Problem Solving


step problem involving percent and probability, arriving at a thorough and

accurate solution (e.g., in question 2, systematically lists all combinations;

in question 3, uses a chart with a list of all ordered pairs).


– The student demonstrates a thorough understanding of percent, fractions,

and probability (e.g., in question 4, calculates the probability both as a frac-

tion and a percent).


– The student applies a percent algorithm with few, if any, minor errors

and/or omissions to solve problems involving discounts (e.g., in question 1b,

determines and applies the percent discount correctly to find the total dollar

discount).


and/or omissions when investigating situations involving probability (e.g., in

question 5, accurately calculates which outcome is the most likely in the

invented game).



and fractions clearly and precisely (e.g., in question 3, to support the written

explanation, lists all possible outcomes of rolling two dice).


involving probability clearly and precisely (e.g., in question 5, accurately

describes the game, using diagrams, words, and mathematical expressions).


1

Title: Rolling in Sales



This task requires each student to determine the lowest possible percentage discount and thehighest possible percentage discount resulting from their rolling two numbered cubes (dice) andadding the numbers shown on the numbered cubes. Students must state which percentagediscount is most likely to occur and why; the probability of getting a percentage discount greaterthan 10 percent; and which they would choose, and why, between a 10 percent discount and rollingthe numbered cubes. Finally, students will be asked to invent a probability game that uses percent.



Students will:

1. solve and explain multi-step problems involving fractions, decimals, integers, percents,and rational numbers (8m8);

2. use mathematical language to explain the process used and the conclusions reached inproblem solving (8m9);

3. ask “What if” questions; pose problems involving fractions, decimals, integers,percents, and rational numbers and investigate solutions (8m30);

4. explain the process used and any conclusions reached in problem solving andinvestigations (8m31);

Mathematics Exemplar TaskGrade 8 – Data Management and Probability, and

Number Sense and Numeration

Teacher Package

2

5. apply percents in solving problems involving discounts, sales tax, commission, andsimple interest (8m34);

6. identify probability situations and apply knowledge of probability (8m95);

7. list the possible outcomes of simple experiments by using tree diagrams, modelling, andlists (8m118);

8. identify the favourable outcomes among the total number of possible outcomes andstate the associated probability (e.g., of getting closer in a random draw) (8m119).



Before attempting the task, students should have some knowledge or skills related to thefollowing:• addition and multiplication of integers• solving multi-step problems involving simple fractions, decimals, and percents• intuitive concepts of probability and how probability can relate to chance• listing the possible outcomes of simple experiments by using tree diagrams or matrices

The Rubric*



Accommodations



Before students attempt a particular task, provide them with the appropriate materials fromamong the following:• a copy of the Student Package (see Appendices 1 and 2) for each student• 3 cups• 1 “prize” that will fit under a turned-over cup• 2 numbered cubes per student• grid paper for the matrix• calculators• math dictionaries (optional)

Teacher Package

*The rubric is reproduced on pages 105–106 of this document.


3

Task Instructions



Pre-task 1: The Integer Spinner Game

Have students, working in pairs, use a spinner to determine two integers. (See Appendix 1, whichincludes one example of a spinner that you can make.) Ask one student in each pair to find thesum of the two integers and the other to find their product. The player with the greater numberwins one point. Have students play several rounds and then analyse the game to determine if it isfair. If students determine that the game is not fair, ask them how could they make it fair.

Pre-task 2: The Three Cups Game

Place three cups upside down with a “prize” underneath one of them. Ask students to predict theprobability of selecting the cup concealing the prize. (The answer is 33% or 0.33 or 1⁄ 3.)

Turn over one of the cups that does not conceal the prize. Have students debate whether this newinformation changes their prediction.

Exemplar Task




4. The problem that the students will solve independently is provided in the worksheets inAppendix 2.

You may find the following analyses of the assigned tasks useful.

Question 1

– The cubes are rolled and the numbers shown on each cube are added together. The sum ofthe numbers is your percentage discount.

– The current price of one outfit is $110.

– What is the least you could pay?[6+6 (maximum numbered cube sum) = 12% x 110 = $13.20 reduction thus $96.80]

– What is the most you could pay?[1+1 (minimum numbered cube sum) = 2% x 110 = $2.20 reduction thus $107.80]

4

Students may find it useful to create a matrix that shows the possible outcomes:

+ 1 2 3 4 5 6

1 2 3 4 5 6 7

2 3 4 5 6 7 8

3 4 5 6 7 8 9

4 5 6 7 8 9 10

5 6 7 8 9 10 11

6 7 8 9 10 11 12

Question 2

– What percentage discount is most likely to occur? How do you know?[7% because it has a probability of 6 out of 36 (or 1 out of 6 or 16.6%)]

Question 3

– What is the probability of getting a percentage discount greater than 10%?[The probability is 3⁄ 36 or 1⁄ 12 or 8.3%.]

Question 4

– If you were offered a choice between a 10% discount and rolling the numbered cubes, whichwould you choose? Justify your choice.[Answers may vary. The probability of getting a roll of 10 is 3 out of 36 (1 out of 12), 11 is 2 out of 36(1 out of 18), and 12 is 1 out of 36. So there are 3 chances in 36 or 1 in 12 of getting higher than a10% discount and a 30 out of 36 chance (5 out of 6) of getting less than a 10% discount. This meansyou have a better chance for a higher discount if you choose the 10% discount than if you rolled thenumbered cubes.]

Question 5

– As the store manager, you are prepared to offer a discount of 15%. Invent another game thatwill produce this result. Determine the probability of a customer getting a 15% discount.[Answers may vary as students may use other types of numbered cubes in their game.]


5

Appendix 1

The Integer Spinner Game

Work with a partner, alternating turns as the spinner.

For each turn, the player spins twice, noting the two numbers arrived at (one for each spin). Player A adds the two numbers. Player B multiplies thetwo numbers. The player with the greater result wins a point.

Play several rounds of the game.

6

Analyse the game and state whether it is fair or unfair. Discuss your answer.If the game is unfair, how could you make it fair?

Suppose the game were changed so that player A subtracts the smallernumber from the larger and Player B divides the larger number by thesmaller. How would this affect the results of the game?


7


Rolling In Sales

A local clothing store decides to offer a unique sales promotion based on thesum of two standard numbered cubes.

The cubes are rolled and the numbers shown on each cube are added together.The sum of the numbers is your percentage discount.

The current price of one outfit is $110.00.

1. a) Explain how you would determine the lowest percentage discount andthe highest percentage discount.

b) Calculate the least you could pay and the most you could pay.

8

2. What percentage discount is most likely to occur? How do you know?


9

3. What is the probability of getting a percentage discount greater than 10%?Show your work.

10

4. If you were offered a choice between a 10% discount and rolling thenumbered cubes, which would you choose? Justify your choice.


11

5. As the store manager, you are prepared to offer a discount of 15%. Invent another game that will produce this result. Determine theprobability of a customer getting a 15% discount.

12

6. What discount is most likely to occur? How do you know?

The Ministry of Education wishes to acknowledgethe contribution of the many individuals, groups,and organizations that participated in the develop-ment and refinement of this resource document.

mathematics: the ontario curriculum exemplars, grade 8 · pdf filethe ontario curriculum...

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